Production Engineering and Management under Fuzziness (Studies in Fuzziness and Soft Computing, 252) 9783642120510, 3642120512

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Cengiz Kahraman and Mesut Yavuz (Eds.) Production Engineering and Management under Fuzziness

Studies in Fuzziness and Soft Computing, Volume 252 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 235. Kofi Kissi Dompere Fuzzy Rationality, 2009 ISBN 978-3-540-88082-0 Vol. 236. Kofi Kissi Dompere Epistemic Foundations of Fuzziness, 2009 ISBN 978-3-540-88084-4

Vol. 245. Xuzhu Wang, Da Ruan, Etienne E. Kerre Mathematics of Fuzziness – Basic Issues, 2009 ISBN 978-3-540-78310-7

Vol. 237. Kofi Kissi Dompere Fuzziness and Approximate Reasoning, 2009 ISBN 978-3-540-88086-8

Vol. 246. Piedad Brox, Iluminada Castillo, Santiago Sánchez Solano Fuzzy Logic-Based Algorithms for Video De-Interlacing, 2010 ISBN 978-3-642-10694-1

Vol. 238. Atanu Sengupta, Tapan Kumar Pal Fuzzy Preference Ordering of Interval Numbers in Decision Problems, 2009 ISBN 978-3-540-89914-3

Vol. 247. Michael Glykas Fuzzy Cognitive Maps, 2010 ISBN 978-3-642-03219-6

Vol. 239. Baoding Liu Theory and Practice of Uncertain Programming, 2009 ISBN 978-3-540-89483-4

Vol. 248. Bing-Yuan Cao Optimal Models and Methods with Fuzzy Quantities, 2010 ISBN 978-3-642-10710-8

Vol. 240. Asli Celikyilmaz, I. Burhan Türksen Modeling Uncertainty with Fuzzy Logic, 2009 ISBN 978-3-540-89923-5

Vol. 249. Bernadette Bouchon-Meunier, Luis Magdalena, Manuel Ojeda-Aciego, José-Luis Verdegay, Ronald R. Yager (Eds.) Foundations of Reasoning under Uncertainty, 2010 ISBN 978-3-642-10726-9

Vol. 241. Jacek Kluska Analytical Methods in Fuzzy Modeling and Control, 2009 ISBN 978-3-540-89926-6 Vol. 242. Yaochu Jin, Lipo Wang Fuzzy Systems in Bioinformatics and Computational Biology, 2009 ISBN 978-3-540-89967-9 Vol. 243. Rudolf Seising (Ed.) Views on Fuzzy Sets and Systems from Different Perspectives, 2009 ISBN 978-3-540-93801-9 Vol. 244. Xiaodong Liu and Witold Pedrycz Axiomatic Fuzzy Set Theory and Its Applications, 2009 ISBN 978-3-642-00401-8

Vol. 250. Xiaoxia Huang Portfolio Analysis, 2010 ISBN 978-3-642-11213-3 Vol. 251. George A. Anastassiou Fuzzy Mathematics: Approximation Theory, 2010 ISBN 978-3-642-11219-5 Vol. 252. Cengiz Kahraman, Mesut Yavuz (Eds.) Production Engineering and Management under Fuzziness, 2010 ISBN 978-3-642-12051-0

Cengiz Kahraman and Mesut Yavuz (Eds.)

Production Engineering and Management under Fuzziness

ABC

Editors Cengiz Kahraman Istanbul Technical University Management Faculty Department of Industrial Engineering 34367 Macka, Istanbul Turkey E-mail: [email protected] Mesut Yavuz Shenandoah University Harry f. Byrd, Jr. School of Business 1460 University Dr. Winchester, VA 22601 USA E-mail: [email protected]

ISBN 978-3-642-12051-0

e-ISBN 978-3-642-12052-7

DOI 10.1007/978-3-642-12052-7 Studies in Fuzziness and Soft Computing

ISSN 1434-9922

Library of Congress Control Number: 2010927080 c 2010 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Dedicated To my parents, Havva and Ömer Cengiz Kahraman

To Nurhan, my inspiration Mesut Yavuz

Preface

Production engineering and management involve a series of planning and control activities in a production system. A production system can be as small as a shop with only one machine or as big as a global operation including many manufacturing plants, distribution centers, and retail locations in multiple continents. The product of a production system can also vary in complexity based on the material used, technology employed, etc. Every product, whether a pencil or an airplane, is produced in a system which depends on good management to be successful. Production management has been at the center of industrial engineering and management science disciplines since the industrial revolution. The tools and techniques of production management have been so successful that they have been adopted to various service industries, as well. Production management deals with five Ms: men, machines, methods, materials and money. All these elements present great uncertainty in real life. Men vary in their productivity and even one man’s productivity can vary with time. Machines break down; methods yield different results under different circumstances, etc. Such uncertainties and ambiguities make production management all the more complex and difficult. Fuzzy logic was developed by L. A. Zadeh in the 1960s and has been widely used in various fields where one deals with uncertainty and vagueness. The production management literature is rich in the tools and techniques focusing on the five Ms utilizing fuzzy logic. This book presents a collection of production engineering and management techniques under fuzziness. The book is intended to be a valuable resource to undergraduate and graduate students interested in the applications of production management under fuzziness. Chapters represent all areas of production management and are organized to reflect the natural order of production management tasks. In all chapters, special attention is given to applicability and wherever possible, numerical examples are presented. While the reader is expected to have a fairly good understanding of the fuzzy logic, the book provides the necessary notation and preliminary knowledge needed in each chapter. Chapter 1 addresses the typical first step in production management: forecasting. Predicting the events of the future has always been of great importance. Even for the simple-sounding tasks of forecasting the demand or cost of a product, one faces great uncertainty and has to incorporate many qualitative factors into decision

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making. Numerous researchers have studied forecasting under fuzziness and contributed to the literature. The chapter summarizes and classifies forecasting techniques based on crisp logic, fuzzy logic and the grey theory. Inventory, defined as any stored resource that is used to satisfy a current or future need, is one of the most important assets in many production systems. Inventory management has been a key research stream in operations research, starting with the century-old economic order quantity model. Chapter 2 presents the economic order quantity, economic production quantity, single period and periodic review models under fuzziness. Material requirement planning (MRP) is at the very heart of traditional production systems. Advancement of computers popularized the MRP approach and made it applicable in a gamut of industries. Chapter 3 presents MRP under fuzziness, where demand and capacity data are fuzzy, and develops a fuzzy method to solve the emerging linear programming model. Chapter 4 discusses fuzziness in just-in-time (JIT) and lean production systems. JIT manufacturing is widely accepted as the opposite of the traditional push-based (MRP) manufacturing philosophy. A JIT system is typically based on pull signals, i.e., Kanban cards. The chapter presents one- and two-card systems with fuzzy demands and withdrawal and production times. The chapter also addresses one-of-akind production and earliness-tardiness scheduling models with fuzzy due dates. Lead time, defined as the duration between placing an order and receiving the ordered goods, is a key performance indicator in supply chains. From a manufacturer’s point of view lead time, or throughput time, is the time between accepting an order and shipping the finished products. Chapter 5 discusses fuzziness in lead time management through lead time and order fulfillment, due date bargaining and scheduling with lead time objectives. Petri nets are a powerful tool in modeling complex systems. Most modern manufacturing systems are complex due to the number of components in them and the relationships among all those elements. Chapter 6 is concerned with modeling flexible manufacturing systems and presents a two stage modeling approach based on fuzzy sets. Fuzzy logic in general has been instrumental in advanced manufacturing systems. Chapter 7 studies a semiconductor manufacturing facility as a complex system and addresses job remaining cycle time estimation in the system. The chapter proposes a self-organizing map – fuzzy back propagation network to estimate remaining cycle time of each job in the system. Scheduling, the process of determining when each of a number of tasks should be performed, has been one of the central problems in operations research for many decades. Different manufacturing systems have different complexity levels and sources of uncertainty inherent in them. Chapter 8 discusses several production scheduling problems with single and multiple objectives and presents fuzzy logic based tools for their solution. Project evaluation and review technique (PERT) is one of the main tools used in project and production scheduling. Chapter 9 presents interval PERT and its fuzzy extension, which can be used to successfully capture uncertainty in activity durations.

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Today’s complex and competitive business environment often requires companies to rely on others to supply necessary components for production and distribution of finished goods, thereby making the management of supply chains critical. A supply chain can be viewed as a network of companies that work together towards a common goal. Hence, optimization of supply chains requires solving network optimization problems. Chapter 10 presents transportation, assignment and minimum cost flow problems under fuzziness and also develops a fuzzy model of the bullwhip effect. In recent years, incorporation of production and distribution management has received increasing attention from researchers. While the computation power of modern computers allows us to solve more difficult optimization problems in less time, increasing scope of the systems modeled create more uncertainty and leave more room for subjective judgment. Chapter 11 presents a fuzzy simulation model and a bi-criteria optimization method for a production and distribution system. Chapter 12 presents investment planning for production systems under fuzziness. The investment analyses presented in the chapter include fuzzy present worth, fuzzy annual worth, fuzzy rate of return, fuzzy B/C ratio, fuzzy replacement and fuzzy payback period. Budgeting can be defined as financial planning of operations in a system. Budgeting in a production system is related to the sales forecast, production plan, inventory policy, and staffing plan among other things. Chapter 13 presents a fuzzy model for production and operations budgeting and control. Location selection is another key problem in production management. Chapter 14 focuses on fuzzy multi-criteria decision making methods developed for the location selection problem. The chapter presents fuzzy analytical hierarchy process (fuzzy AHP) and fuzzy technique for order preference by similarity to the ideal solution (fuzzy TOPSIS) techniques. The chapter also presents a framework based on the fuzzy information axiom for facility location selection. After selecting the location of a facility, one needs to optimize the layout and material flow within the facility. Chapter 15 presents a multi-criteria model with quantitative and qualitative factors. In the model, fuzzy logic is integrated with the analytical hierarchy process. Chapter 16 is concerned with human reliability analysis in the context of production management. The chapter uses fuzzy cognitive maps, which provide a graphical and mathematical representation of an individual’s system of beliefs. Productivity is a measure relating a quantity or quality of output to the inputs required to produce it. Productivity measurement has an important role in production and service systems. Chapter 17 realizes productivity measurement under vague and incomplete information. Data envelopment analysis is also applied to a productivity optimization problem. Statistical process control charts are widely used to monitor a process and evaluate whether it is in control or not. A crisp process control chart reports the process as either “in control” or “out of control,” whereas the fuzzy control charts do it by using suitable linguistic or fuzzy numbers by offering flexibility for control limits. Chapter 18 presents fuzzy attribute control charts and fuzzy variable control charts.

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Chapter 19 addresses one of the major components of the field of statistical quality control: acceptance sampling. Acceptance sampling is primarily used for the inspection of incoming or outgoing lots, and it refers to the application of specific sampling plans to a designated lot or sequence of lots. In the chapter, two main distributions of acceptance sampling plans (binomial and Poisson) are used with fuzzy parameters and their acceptance probability functions are derived. Then fuzzy acceptance sampling plans are derived based on these distributions. Process capability broadly defined as the ability of a process to meet customer expectations, i.e., specification limits, is one of the key performance indicators of a production system. Chapter 20 focuses on specification limits that are expressed in linguistic terms, and produces fuzzy process capability indices. Chapter 21 focuses on intelligent quality management systems. The chapter provides a review of crisp and fuzzy quality measurement approaches in the literature and also proposes an intelligent quality measurement system using fuzzy association rules. Chapter 22 addresses the strategic investment decision of whether to close a production plant or not. The chapter develops fuzzy real options models for the problem and also presents a case study from forest products industry. As the editors of the book we truly hope that it meets the expectations of the reader, and it is a useful resource for practitioners and researchers in the field. We sincerely thank all our authors for contributing their invaluable chapters to this book. They have been very kind to deliver their chapters in time despite all their other responsibilities. We especially thank Professor Janusz Kacprzyk, the series editor, for making this book possible. We are grateful to the referees whose valuable and highly appreciated works contributed to select and improve the high quality of the chapters included in the book. Last but not least, we sincerely thank Dr. øhsan Kaya for the countless hours he invested in editing this book.

Cengiz Kahraman Istanbul, Turkey Mesut Yavuz Winchester, Virginia, U.S.A.

Contents

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . ˙ Cengiz Kahraman, Mesut Yavuz, Ihsan Kaya

1

Fuzzy Inventory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesut Yavuz

25

Fuzzy Material Requirement Planning . . . . . . . . . . . . . . . . . . . . . . . Josefa Mula and Ra´ ul Poler

39

Fuzziness in JIT and Lean Production Systems . . . . . . . . . . . . . . Mesut Yavuz

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Fuzzy Lead Time Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesut Yavuz

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Manufacturing System Modeling Using Petri Nets . . . . . . . . . . . Cengiz Kahraman, Fatih T¨ uys¨ uz

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Fuzzy Technology in Advanced Manufacturing Systems: A Fuzzy-Neural Approach to Job Remaining Cycle Time Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Toly Chen Fuzzy Project Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Na¨ım Yalaoui, Fr´ed´eric Dugardin, Farouk Yalaoui, Lionel Amodeo, Halim Mahdi Interval PERT and Its Fuzzy Extension . . . . . . . . . . . . . . . . . . . . . . 171 Didier Dubois, J´erˆ ome Fortin, Pawel Zieli´ nski Fuzziness in Supply Chain Management . . . . . . . . . . . . . . . . . . . . . 201 P´eter Majlender

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Fuzzy Simulation and Optimization of Production and Logistic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 L. Dymowa, P. Sevastjanov Fuzzy Investment Planning and Analyses in Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Cengiz Kahraman, A. C ¸ a˘grı Tolga Fuzzy Production and Operations Budgeting and Control . . . 299 Dorota Kuchta Fuzzy Location Selection Techniques . . . . . . . . . . . . . . . . . . . . . . . . 329 Cengiz Kahraman, Selcuk Cebi, Fatih Tuysuz Fuzziness in Materials Flow and Plant Layout . . . . . . . . . . . . . . . 359 Kuldip Singh Sangwan Fuzzy Cognitive Maps for Human Reliability Analysis in Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Massimo Bertolini, Maurizio Bevilacqua Fuzzy Productivity Measurement in Production Systems . . . . 417 Semra Birg¨ un, Cengiz Kahraman, Kemal G¨ uven G¨ ulen Fuzzy Statistical Process Control Techniques in Production Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Cengiz Kahraman, Murat G¨ ulbay, Nihal Erginel, Sevil S ¸ ent¨ urk Fuzzy Acceptance Sampling Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 ˙ Cengiz Kahraman, Ihsan Kaya Fuzzy Process Capability Analysis and Applications . . . . . . . . . 483 ˙ Cengiz Kahraman, Ihsan Kaya Fuzzy Measurement in Quality Management Systems . . . . . . . . 515 George T.S. Ho, Henry C.W. Lau, Nick S.H. Chung, W.H. Ip Fuzzy Real Options Models for Closing/Not Closing a Production Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Christer Carlsson, Markku Heikkil¨ a, Robert Full´er Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

Chapter 1

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems Cengiz Kahraman, Mesut Yavuz, and İhsan Kaya*

Abstract. Forecasting is an important part of decision making as many of our decisions are based on predictions of future unknown events. Forecast is an interesting research topic that has received attention from many researchers in the past several decades. Forecasting has many application areas including but not limited to stock markets, futures markets, enrollments of a school, demand of a product and/or service. Management needs to reduce the risks associated with decisionmaking, which can be done by anticipating the future more clearly. Accurate forecasts are therefore essential for risk reduction. Forecasting provides critical inputs to various manufacturing-related processes, such as production planning, inventory management, capital budgeting, purchasing, work-force scheduling, resource allocation and other important parts of the production system operation. Accurate forecasts are crucial for successful manufacturing and can lead to considerable savings when implemented efficiently. Forecasting literature contains a large variety of techniques from simple regression to complex metaheuristics such as neural networks and genetic algorithms. Fuzzy set theory is also another useful tool to increase forecast efficiency and effectiveness. This chapter summarizes and classifies forecasting techniques based on crisp logic, fuzzy logic and the grey theory. The chapter also presents numerical examples of fuzzy simple linear regression and grey forecasting methodology.

1 Introduction Webster’s dictionary defines forecasting as an activity “to calculate or predict some future event or condition, usually as a result of rational study or analysis of Cengiz Kahraman Department of Industrial Engineering, İstanbul Technical University, 34367 Maçka İstanbul, Turkey Mesut Yavuz Shenandoah University, The Harry F. Byrd, Jr. School of Business, Winchester, Virginia, U.S.A. İhsan Kaya Faculty of Engineering and Architecture, Department of Industrial Engineering, Selçuk University, 42075 Konya, Turkey C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 1–24. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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pertinent data.” Forecasting is the process of estimation in unknown situations. Forecasting is the science and art of predicting the future. Forecasting techniques are used to give a business or organization a clear view of their future situation. Forecasting techniques involve the formulation of models about the world, and the manipulation of the models to form predictions about the future state of the world as it relates to the products or services we are interested in. Forecasting is important in many aspects of our lives. As individuals, we try to predict success in our marriages, occupations, and investments. Organizations invest enormous amounts based on forecasts for new products, factories, retail outlets, and contracts with executives. Government agencies need forecasts of the economy, environmental impacts, new sports stadiums, and effects of proposed social programs. Poor forecasting can lead to disastrous decisions. Decision makers need forecasts only if there is uncertainty about the future. Thus, we have no need to forecast whether the sun will rise tomorrow. There is also no uncertainty when events can be controlled; for example, you do not need to predict the temperature in your home. Many decisions, however, involve uncertainty, and in these cases, formal forecasting procedures (referred to simply as forecasting hereafter) can be useful. There are alternatives to forecasting. A decision maker can buy insurance (leaving the insurers to do the forecasting), hedge (bet on both heads and tails), or use “just-in-time” systems (which minimizes the need to forecast by letting the customers pull the products they need). Another possibility is to be flexible about decisions (Armstrong 2002). Forecasting provides critical inputs to various manufacturing-related processes, such as production planning, inventory management, capital budgeting, purchasing, work-force scheduling, resource allocation, and other important parts of the production system operation. The goal of any manufacturing firm is to produce products that meet and exceed customer based requirements by producing high quality products. Important company decisions covering production planning, workforce scheduling, inventory management, purchasing, capital budgeting, and resource allocations rely on forecasts. Most of the forecasting models proposed in the literature analyze the case of a retailer and a manufacturer who together calculates the demand forecast of the end-consumer. One of the processes which greatly influence decision making within a company is forecasting demand. Nonetheless, demand is one of the greatest sources of uncertainty which companies face, so much so that many companies end up resigning to working with forecasts with numerous errors. Most companies know that their forecasts are inexact, but do not know how to solve this problem. Inaccurate forecasts force companies to find ways to compensate for the uncertainty, and the most often used method to achieve this is by building inventory. Accurate forecasts help to achieve better customer service and lower inventory levels. The rate of successful forecasts depends on the company’s capacity to obtain relevant information and to interpret it. The literature contains numerous forecasting applications in production systems such as a forecasting model for coal production (Mohr and Evans 2009), time series forecasting for machines (Guo et al. 2008), agents and ant colony for emergent forecasting (Valckenaers et al. 2006), grey forecasting model to predict Taiwan's annual semiconductor production (Chang et al. 2005), autoregressive

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integrated moving average with seasonality for stocks (Albertson and Aylen 2003), Litterman Bayesian vector autoregression model for production forecasting of technology industries (Hsu et al. 2002), exponential smoothing for demand forecasting (Gardner Jr. et al. 2001), forecasting techniques for sales and demand (Mady 2000), autoregressive time series for taper turning (Fung et al. 1998), structural time series models for forecasting industrial production (Thury and Witt 1998), forecasting of agricultural production (Allen 1994). Forecasting techniques include uni-variant, multi-variant, and qualitative analysis. Time series used to forecast future trends include exponential smoothing, ARIMA (AutoRegressive Integrated Moving Average) and trend analysis. Multivariant prediction methods include multi regression model, econometrics, and state space. Delphi marketing research, situational analysis, and historical analogue belong to qualitative methodologies. These forecasting methods forecast trends over different time horizons (Box et al. 1994). The term "fuzzy logic" emerged in the development of the theory of fuzzy sets by Lotfi Zadeh (1965). A fuzzy subset A of a (crisp) set X is characterized by assigning to each element x of X the degree of membership of x in A. Many problems in real world deal with uncertain and imprecise data so conventional approaches cannot be effective to find the best solution. To cope with this uncertainty, the fuzzy set theory has been developed as an effective mathematical algebra under vague environment. Although humans have comparatively efficient in qualitative forecasting, they are unsuccessful in making quantitative predictions. Since fuzzy linguistic models permit the translation of verbal expressions into numerical ones, thereby dealing quantitatively with imprecision in the expression of the importance of each criterion, some methods based on fuzzy relations are used. When the system involves human subjectivity, fuzzy algebra provides a mathematical framework for integrating imprecision and vagueness into the models. The fuzzy set theory has some advantages in forecasting. For example one of the most important advantages of fuzzy time series approximations is to be able to work with a very small set of data and not to require the linearity assumption. Fuzzy sets offer a clear insight into the forecasting model and it is especially popular for dealing with non-linear systems and hence it is suitable for forecasting the stock market. Fuzzy forecasting methods are suitable under incomplete data conditions and require fewer observations than other forecasting models. Fuzzy theory was originally developed to deal with problems involving linguistic terms and have been successfully applied to forecasting. The advantages of using fuzzy logic in forecasting applications include the following (Mamlook et al. 2009): • Fuzzy method uses fuzzy sets that enable us to condense large amount of data into smaller set of variable rules. • Fuzzy logic controllers are based on heuristics and therefore able to incorporate human intuition and experience. Over the years there have been successful applications and implementations of the fuzzy set theory in production management. The fuzzy set theory has been recognized as an important problem modeling and solution technique.

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In this chapter, the fuzzy set theory is used to increase the efficiency and effectiveness of forecasting. A literature review on fuzzy forecasting belonging to the year 2009 is given in Section 2. Forecasting methods, grey fuzzy simple linear regression and grey forecasting methodology are detailed in Section 3. Recently developed forecasting techniques based on fuzzy and grey set theories are detailed by numerical examples in the area of in Section 4. Finally Section 5 presents the conclusions and future research directions.

2 A Literature Review on Fuzzy Forecasting Over the last few decades numerous researchers have contributed to the literature of fuzzy forecasting. In this chapter, due to space considerations, we present a brief review of the literature by focusing on the works published in 2009 only. Chu et al. (2009) proposed a dual-factor modified fuzzy time-series model, which takes stock index and trading volume as forecasting factors to predict stock index. They also employed the Taiwan stock exchange capitalization weighted stock index and National Association of Securities Dealers Automated Quotations as experimental datasets and two multiple factor models as comparison models. The experimental results indicated that the proposed model was effective in stock index forecasting. Egrioglu et al. (2009a) proposed a new hybrid method in order to analyze fuzzy seasonal time series. The proposed model contained terms of autoregressive (AR), Seasonal multiplicative autoregressive process (SAR), moving average (MA), and simple moving average (SMA) structures. Xiao et al. (2009) attempted to use forecasting accuracy as the criterion of fuzzy membership function, and proposed a combined forecasting approach based on fuzzy soft sets. They also examined the method with data of international trade from 1993 to 2006 in the Chongqing Municipality of China and compared it with a combined forecasting approach based on rough sets and each individual forecast. They showed that the combined approach proposed in their study improved the forecasting performance of each individual forecast and was free from a rough sets approach's restrictions as well. Aladag et al. (2009) employed feed forward neural networks to define fuzzy relation by trying various architectures for high order fuzzy time series. The proposed approach based on neural networks was applied to wellknown enrollment data for University of Alabama. They compared obtained results with other methods and it was clearly seen that the proposed method had better forecasting accuracy when compared with other methods proposed in the literature. Atsalakis and Valavanis (2009) surveyed more than 100 related published articles that focus on neural and neuro-fuzzy techniques derived and applied to forecast stock markets. Classifications were made in terms of input data, forecasting methodology, performance evaluation, and performance measures. Bekiros (2009) introduced a hybrid neuro-fuzzy system for decision-making and trading under uncertainty. The efficiency of a technical trading strategy based on the neuro-fuzzy model was investigated in order to predict the direction of the market for 10 of the most prominent stock indices of U.S.A, Europe and Southeast Asia. Efendigil et al. (2009) presented a comparative forecasting methodology regarding to uncertain customer demands in a multi-level supply chain (SC) structure via

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neural techniques. They proposed a new forecasting mechanism which was modeled by artificial intelligence approaches including the comparison of both artificial neural networks and adaptive network-based fuzzy inference system techniques to manage the fuzzy demand with incomplete information. Egrioglu et al. (2009b) proposed a new modified method to analyze k-factor and nth order fuzzy time series forecasting model using feed forward artificial neural networks that determines fuzzy logic relations. The proposed method was applied to the total number of annual car road accidents casualties in Belgium and the results obtained from the proposed method were compared with those obtained from other methods. Hassan (2009) presented fuzzy logic in combination with Hidden Markov Model in order to notably improve the prediction accuracy for nonstationary and non-linear stock market datasets. Liu (2009) developed an integrated fuzzy time series forecasting system in which the forecasted value would be a trapezoidal fuzzy number instead of a single-point value. Furthermore, this system could effectively deal with stationary, trend, and seasonal time series and increases the forecasting accuracy. Two numerical data sets were selected to illustrate the proposed method and the forecasting accuracy was compared with four fuzzy time series methods. The results of the comparison showed that the system could produce more precise forecasted values than those of four methods. Leu et al. (2009) proposed a new fuzzy time series model termed as distance-based fuzzy time series to predict the exchange rate. Unlike the existing fuzzy time series models which require exact match of the fuzzy logic relationships, the distancebased fuzzy time series model uses the distance between two fuzzy logic relationships in selecting prediction rules. Chen et al. (2009) presented a new method to forecast enrollments based on automatic clustering techniques and fuzzy logical relationships. Lu and Wang (2009) proposed an enhanced fuzzy linear regression model, in which the spreads of the estimated dependent variables were able to fit the spreads of the observed dependent variables, no matter the spreads of the observed dependent variables are increased, decreased or unchanged as the magnitudes and spreads of the independent variables change. Lai et al. (2009) established a novel financial time series-forecasting model by evolving and clustering fuzzy decision tree for stocks in Taiwan Stock Exchange Corporation. The forecasting model integrated a data clustering technique, a fuzzy decision tree (FDT), and GA to construct a decision-making system based on historical data and technical indexes. Kuo et al. (2009) proposed a new hybrid forecasting model which combined particle swarm optimization with fuzzy time series to improve the forecasted accuracy. The role of the particle swarm optimization was to find proper content of the two factors which were the length of an interval and the content of forecast rules to improve the forecasted accuracy of the proposed model. Kuo et al. (2009b) used the proposed methodology to solve the Taiwan Stock Index Futures forecasting problem. Khashei et al. (2009) used fuzzy logic and artificial neural networks to construct a new hybrid model to overcome some limitations of auto-regressive integrated moving average in an attempt to yield more accurate results. Khemchandani et al. (2009) proposed a novel approach, termed as regularized least squares fuzzy support vector regression, to handle financial time series forecasting. Mamlook et al. (2009) concerned with the short-term load forecasting

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(STLF) in power system operations. In their study, a methodology had been introduced to decrease the forecasted error and the processing time by using fuzzy logic controller on an hourly base. Therefore, it predicted the effect of different conditional parameters (i.e., weather, time, historical data, and random disturbances) on load forecasting in terms of fuzzy sets during the generation process. Ragulskis and Lukoseviciute (2009) established time series forecasting method based on fuzzy networks. They used a concept based on attractor embedding to construct a simple and deterministic rule for the selection of a non-uniform vector of time lags. Hong and White (2009) introduced the dynamic neuro-fuzzy local modeling system that was based on a dynamic Takagi–Sugeno (TS) type fuzzy inference system with on-line and local learning algorithm for complex dynamic hydrological modeling tasks. The model was aimed to implement a fast training speed with the capability of on-line simulation where model adaptation occurred at the arrival of each new item of hydrological data. Yolcu et al. (2009) proposed a new approach which used a single-variable constrained optimization to determine the ratio for the length of intervals which had an important impact on the performance of fuzzy time series forecasting. The proposed approach was applied to the two well-known time series, which were enrollment data at The University of Alabama and inventory demand data. Recently some authors have concentrated on grey forecasting models. Their studies are briefly explained in the remaining part of the chapter. Jia et al. (2009) proposed a prediction method based on grey correlation and adaptive neuro-fuzzy system to forecast synthesis characteristics of hydraulic valve. Grey correlation model was used first to get the main geometric elements affecting the synthesis characteristics of the hydraulic valve and thus simplified the system forecasting model. Lin et al. (2009) proposed a new model named EFGMm(1, 1) by eliminating the error term resulted from the traditional calculation of background value with an integration equation to substitute for such error term. In addition, they integrated Fourier series and exponential smooth technique into the new model to reduce the periodic and stochastic residual errors, respectively. Jia et al. (2009) used grey correlation model firstly to get the main geometric elements affecting the synthesis characteristics of the hydraulic valve and thus simplifies the system forecasting model. Li et al. (2009) used the trend and potency tracking method to analyze sample behavior, extract the concealed information from data, and utilize the trend and potency value to construct an adaptive grey prediction model, AGM (1,1), based on grey theory. The experimental results showed that the proposed model could improve the prediction accuracy for small samples. Xie and Liu (2009) proposed a novel discrete grey forecasting model termed DGM model and a series of optimized models of DGM. They also modified the algorithm of GM(1, 1) model to enhance the tendency catching ability. Hsu and Wang (2009) proposed a new prediction approach using the multivariate grey model combined with grey relational analysis for forecasting of integrated circuit outputs. They also compared the performance of the combined multivariate grey prediction model GM(1, N) with that of univariate and multivariate auto regression models using a time series sample from 1993 to 2004. Hamzaçebi et al. (2009) performed forecasting using direct and iterative methods, and the obtained

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems

7

results of the methods were compared using grey relational analysis to find the method which gave a better result. Hsu (2009) proposed a grey model with factor analysis techniques to deal with the multi-factor forecasting problems. In the grey modeling, the use of genetic algorithm has the ability to search global optimum solution to construct two improved multivariable grey forecasting models that were AGAGM(1,N) and GAGM(1,N). These two models were applied for forecasting Taiwanese integrated circuit output. Zhou et al. (2006) presented a trigonometric grey prediction approach by combining the traditional grey model GM (1, 1) with the trigonometric residual modification technique for forecasting electricity demand. Albert et al. (2003) presented an improved Grey-based prediction algorithm to forecast a very-short-term electric power demand for the demand-control of electricity. They adopted Grey prediction as a forecasting means. Lin and Yang (2003) applied the Grey forecasting model to forecast accurately the output value of Taiwan’s opto-electronics industry from 2000 to 2005. The results show that the average residual error of the Grey forecasting model is lower than 10%. They further show that the Grey forecasting model exhibits high prediction accuracy.

3 Forecasting Techniques Choosing an appropriate forecasting method depends on the situation. For example, for long-range forecasting of the environment or of the market, econometric methods are often appropriate. For short-range forecasting of market share, extrapolation methods are useful. Forecasts of new-product sales could be made judgmentally by experts. Decisions by parties in conflict, such as companies and their competitors, can be predicted by role-playing which is a way of predicting the decisions by people or groups engaged in conflicts (Armstrong 2002). Forecast methods may be broadly classified into qualitative and quantitative techniques. Qualitative or subjective forecast methods are intuitive, largely educated guesses that may or may not depend on past data. Usually these forecasts cannot be reproduced by someone else, since the forecaster does not specify explicitly how the available information is incorporated into the forecast. Even though subjective forecasting is a nonrigorous approach, it may be quite appropriate and the only reasonable method in certain situations. Forecasts that are based on mathematical or statistical models are called quantitative. Once the underlying model or technique has been chosen, the corresponding forecasts are determined automatically; they are fully reproducible by any forecaster. Quantitative methods or models can be further classified as deterministic or probabilistic (also known as stochastic or statistical) (Abraham and Ledolter 1983). Qualitative forecasting methods differ from quantitative methods in that they factor in subjective or personal experiences into their model. This tutorial will not provide an in-depth look at such methods, but we will provide a brief overview. There are several different varieties of qualitative forecasting methods, including: • Delphi Method. This method uses the expertise of senior personnel who examine sales reports and make forecasts based on their previous experience.

C. Kahraman, M. Yavuz, and İ. Kaya

8

• Sales Force Composite. Often used in large organizations, this method combines the forecasts of sales representatives for their particular regions and comes up with a composite forecast for a product. • Consumer Market Survey. Instead of soliciting input from upper management or sales staff, this forecasting method relies on polls of upcoming consumer purchases to predict demand for a product. Among many other forecast criteria, the choice of the forecast model or technique depends on (1) what degree of accuracy is required, (2) what the forecast horizon is, (3) how high a cost for producing the forecasts can be tolerated, (4) what degree of complexity is required, and (5) what data are available (Abraham and Ledolter 1983).

3.1 Forecasting by Regression Regression analysis is concerned with modeling the relationships among variables. It quantifies how a response (or dependent) variable is related to a set of explanatory (independent, predictor) variables. A simple regression model is a formula describing the relationship between one descriptor variable and one response variable. These formulas are easy to explain; however, the analysis is sensitive to any outliers in the data. (Myatt 2006). Multiple regression is a statistical method for studying the relationship between a single dependent variable and one or more independent variables. It is unquestionably the most widely used statistical technique in the social sciences. It is also widely used in the biological and physical sciences (Allison 1999). In the following, crisp and fuzzy forecasting techniques by regression analysis are given. 3.1.1 Estimation in Simple Linear Regression Assume that we have some data ( xi , yi ) , 1 ≤ i ≤ n on two variables x and Y. The values of x are known in advance and Y is a random variable. We assume that there is no uncertainty in the x data. The future value of Y with certainty cannot be predicted so we focus on the mean of Y, E(Y). We have an assumption that E(Y) is a linear function of x. It can be formulated as follows (Buckley 2004): E (Y ) = a + b ( x − x ) .

(1.1)

In this formula, x is the mean value of the x and it is not a fuzzy set. The regression model is as follows shown in Eq. (1.2) (Buckley 2004): Yi = a + b ( xi − x ) + ε i

(1.2)

where ε i are independent and N ( 0,σ 2 ) with σ 2 unknown. The basic regression equation for the mean of Y is y = a + b ( x − x ) and we wish to estimate values a and b. We will need the (1 − β )100% confidence interval for a and b. We should

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems

9

determine the crisp point estimators of a, b, and σ 2 . The crisp estimator of a is aˆ = y , the mean of the yi values. bˆ is B1 B2 . Where, n

n

i =1

i =1

B1 = ∑ yi ( xi − x ), B2 = ∑ ( xi − x )

2

(1.3)

Finally, ⎛1⎞

n

σˆ 2 = ⎜ ⎟ ∑ ⎡⎣ yi − aˆ − bˆ ( xi − x ) ⎤⎦ ⎝ n ⎠ i =1

2

(1.4)

Linear regression equation represented in Eq. (1.2) can be analyzed by using Buckley’s (2004) approach. (1 − β )100% confidence intervals for a and b given in Eq. (1.2) are as follows, respectively: ⎡ ⎢ aˆ − t β 2 ⎣⎢

⎡ ⎢ ⎢bˆ − t β ⎢ 2 ⎢ ⎣

σˆ 2

( n − 2)

nσˆ 2 n

( n − 2 ) ∑ ( xi − x )

2

, aˆ + t β 2

, bˆ + t β 2

i =1

σˆ 2

⎤ ⎥

( n − 2 ) ⎦⎥ ⎤ ⎥ ⎥ n 2 ⎥ ( n − 2 ) ∑ ( xi − x ) ⎥ i =1 ⎦ nσˆ 2

(1.5)

(1.6)

Also the fuzzy estimator for σ 2 can be defined by the following confidence interval: ⎡ 2 2 ⎢ nσˆ , nσˆ 2 2 ⎢χ β χ β ⎣⎢ R , 2 L , 2

where χ R2 , β

2

and χ L2, β

⎤ ⎥ ⎥ ⎦⎥

(1.7)

are the points on the right and left sides of the χ 2 density 2

respectively and where the probability of exceeding (being less than) it is β 2 . This formula is a biased estimate for σ 2 . To obtain an unbiased fuzzy estimator, the following equations are defined: L ( λ ) = [1 − λ ] χ R2 ,0.005 + λ n,

R ( λ ) = [1 − λ ] χ L2,0.005 + λ n

(1.8)

Then the unbiased (1 − β )100% confidence interval for σ 2 should be calculated from Eq. (1.8): ⎡ nσˆ 2 nσˆ 2 ⎤ , ⎥, ⎣⎢ L ( λ ) R ( λ ) ⎦⎥

σ%ˆ 2 = ⎢

0 ≤ λ ≤ 1.

(1.9)

If β is taken into account as an α -cut level, the fuzzy triangular membership function for σ 2 is obtained from Eq. (1.9) as shown in Eq. (1.10). The triangular

C. Kahraman, M. Yavuz, and İ. Kaya

10

fuzzy membership functions can be built by placing these confidence intervals one on top of another.

(σˆ ) 2

α

⎡ ⎤ nσˆ 2 nσˆ 2 , =⎢ ⎥, 2 2 1 1 n n α χ α α χ α − + − + ] R ,0.005 [ ] L,0.005 ⎣⎢ [ ⎦⎥

0 ≤α ≤1

(1.10)

3.1.2 Estimation in Fuzzy Simple Linear Regression Fuzzy theory, originally explored by Zadeh in 1965, describes linguistic fuzzy information using mathematical modeling. Because the existing statistical time series methods could not effectively analyze time series with small amounts of data, fuzzy time series methods were developed. The fuzzy regression equation is as follows: y% ( x ) = a% + b% ( x − x )

(1.11)

where y% ( x ) , a% , and b% are fuzzy numbers and x and x are real numbers. y% ( x ) is our fuzzy number estimator for the mean of Y (E(Y)) given when x. Let a% (α ) = ⎡⎣ a%1 (α ) , a%2 (α ) ⎤⎦ , b% (α ) = ⎡⎣b%1 (α ) , b%2 (α ) ⎤⎦ and y% [ x ] (α ) = ⎡⎣ y ( x )1 (α ) , y ( x ) 2 (α ) ⎤⎦ . Fuzzy calculation will be made by using (α ) -cuts and interval arithmetic as follows: ⎧ y ( x )1 (α ) = a1 (α ) + ( x − x ) b1 (α ) ⎪ ⎪ ⎪ ⎪ y ( x ) 2 (α ) = a2 (α ) + ( x − x ) b2 (α ) ⎪ ⎪ y% [ x ] (α ) = ⎨ ⎪ ⎪ y ( x ) (α ) = a (α ) + ( x − x ) b (α ) 1 2 1 ⎪ ⎪ ⎪ ⎪⎩ y ( x ) 2 (α ) = a2 (α ) + ( x − x ) b1 (α )

if

(x − x) > 0 (1.12)

if

(x − x) < 0

The alpha cuts of a% and b% is calculated by using Eqs. (1.5), and (1.6), respectively.

3.2 Forecasting Time Series A time series is a set of observations xt , each one being recorded at a specific time t . Time-series models are based on a series of discrete and equal time increments. That is, predictions for the next {week, month, quarter, year} are based on, and only on, the past values of the last N periods{weeks, months, years} of the variable we wish to forecast. A time-series has four main components that we wish to use when forecasting. They are its:

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 11

• Trends. The overall direction of the demand over time. • Seasonality. The fluctuations in demand that occur repeatedly and often (i.e. every year). • Cycles. The fluctuations in a variable that occur repeatedly and in the long-term (i.e. every several years). • Random Variations. The element of chance in all data. An important part of the analysis of a time series is the selection of a suitable probability model (or class of models) for the data. Time series forecasting, especially long-term prediction, is a challenge in many fields of science and engineering. Many techniques exist for time series forecasting. In general, the object of these techniques is to build a model of the process and then use this model on the last values of the time series to extrapolate past behavior into future. Forecasting procedures include different techniques and models. Moving averages techniques, random walks and trend models, exponential smoothing, state space modeling, multi-variate methods, vector autoregressive models, cointegrated and causal models, methods based on neural, fuzzy networks or data mining and rule-based techniques are typical methods used in time series forecasting (Ragulskis and Lukoseviciute, 2009). 3.2.1 Moving Averages Moving average is one of the simplest time-series forecasting methods to calculate, and works best when demand is expected to be fairly steady. A n-period moving average is found by adding up the demand for a product or service over the last n periods and dividing that sum by n. After a period elapses, you add that period's demand to the moving average sum and drop the oldest period from the sum. The moving average model for smoothing historical demand proceeds, as the name implies, by averaging a selected number of past periods of data. The average moves because a new average can be calculated whenever a new period’s demand is determined. Whenever a forecast is needed, the most recent past history of demand is used to do the averaging (Vollmann 2004). 3.2.2 Exponential Smoothing A more complex type of moving average estimate is called Exponential Smoothing. Exponential Smoothing forecasts demand by taking its previous forecast, and adding the difference between the previous forecast and the actual demand which is also multiplied by a value called a smoothing constant. The exponential smoothing method is a variation of the weighted moving average. It utilizes an exponential weighting curve to develop the weighting mechanism. Thus, the weights are applied systematically eliminating the need for judgmental views. The exponential smoothing method is probably the most widely used amongst sophisticated logistic managers. (Blumberg 2004).

12

C. Kahraman, M. Yavuz, and İ. Kaya

3.2.3 Seasonal Time Series Seasonality is the periodic and largely repetitive pattern that is observed in time series data over the course of a year. As such, it is largely predictable. A generally agreed definition of seasonality in the context of economics is provided by Hylleberg (1992) as follows: “Seasonality is the systematic, although not necessarily regular, interlayer movement caused by the changes of weather, the calendar, and timing of decisions, directly or indirectly through the production and consumption decisions made by the agents of the economy. These decisions are influenced by endowments, the expectations and preferences of the agents, and the production techniques available in the economy.” This definition implies that seasonality is not necessarily fixed over time, despite the fact that the calendar does not change.

3.3 Fuzzy Time Series The fundamental reason for building a time series model for forecasting is that it provides a way of weighting the data that is determined by the properties of the time series. Structural time series models (STMs) are formulated in terms of unobserved components, such as trends and cycles, that have a direct interpretation. Thus they are designed to focus on the salient features of the series and to project these into the future. They also provide a way of weighting the observations for signal extraction, so providing a description of the series. Fuzzy time series was firstly introduced by Song and Chissom (1993a, 1993b, 1994). Fuzzy time series has been widely studied for recent years for the aim of forecasting. The traditional time series approaches require having the linearity assumption and at least 50 observations. In fuzzy time series approaches, there is not only a limitation for the number of observations but also there is no need for the linearity assumption. The fuzzy time series approaches consist of three steps. The first step is the fuzzification of observations. In the second step, fuzzy relationships are established and the defuzzification is done in the third step (Yolcu et al. 2009).

3.4 Grey Theory Grey theory, originally developed by Deng (1982), focuses on modeling uncertainty and information insufficiency in analyzing and understanding systems via research on conditional analysis, prediction and decision making. In the field of information research, deep or light colors represent information that is clear or ambiguous, respectively. Meanwhile, black indicates that the researchers have absolutely no knowledge of system structure, parameters, and characteristics; while white represents that the information is completely clear. Colors between black and white indicate systems that are not clear, such as social, economic, or weather systems. The grey forecasting model adopts the essential part of the grey system theory and it has been successfully used in finance, integrated circuit industry and the market for air travel. The grey forecasting model uses the operations of accumulated generation to build differential equations. It has the characteristics of requiring less data.

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 13

Grey system theory is a truly multidisciplinary theory dealing with grey systems that are characterized by both partially known and partially unknown information. It has been widely used in several fields such as agriculture, industry and environmental systems studies (Deng 1989, Chang and Tseng 1999). As an essential part of grey system theory, grey forecasting models have gained in popularity in time-series forecasting due to their simplicity and ability to characterize an unknown system by using as few as four data points (Zhou et al. 2006). 3.4.1 Grey Forecasting Model Grey theory, originally developed by Deng (1982), focuses on model uncertainty and information insufficiency in analyzing and understanding systems via research on conditional analysis, prediction and decision making. The grey forecasting model adopts the essential part of the grey system theory and it has been successfully used in finance, integrated circuit industry and the market for air travel (Hsu and Wang 2002, Hsu 2003, Hsu and Wen 1998). The grey forecasting model uses the operations of accumulated generation to build differential equations. Intrinsically speaking, it has the characteristics of requiring less data. The GM(1,1), can be denoted by the function as follows (Hsu 2001): 0 Step 1. Assume an original series to be x( ) ,

(

x( ) = x( 0

0)

(1) , x(0) ( 2 ) , x(0) ( 3) ,..., x(0) ( n ) ) ,

Step 2. A new sequence x (1) is generated by the accumulated generating operation (AGO).

(

)

k

() ( 1 1 1 1 1 1 x ( ) , x ( ) = x( ) (1) , x ( ) ( 2 ) , x ( ) ( 3) ,..., x( ) ( n ) , where x ( k ) = ∑ x 1

i =1

(

0)

( i ).

)

Step 3. Establish a first-order differential equation. dx(1) / dt + az = u where z

(1)

( k ) = α x ( k ) + (1 − α ) x ( k + 1) (1)

(1)

k = 1,2,..., n − 1. α denotes a horizontal adjument coefficient, and 0 p α p 1 . The selecting criterion of α value is to yield the smallest forecasting error rate (Wen et al. 2000). Step 4. From Step 3, we have

u⎞ u ⎛ 0 1 xˆ ( ) ( k + 1) = ⎜ x ( ) (1) − ⎟ e − ak + , a⎠ a ⎝

⎡ ⎛⎜1⎞⎟ ⎢− z⎝ ⎠ ( 2) ⎢ ⎛ ⎞ ⎢ ⎝⎜1⎠⎟ B = ⎢ − z ( 3) ⎢ L ⎢ ⎢ ⎛⎜1⎞⎟ ⎢− z ⎝ ⎠ ( n ) ⎣

where

(1.13)

⎤

1⎥

⎥ ⎥ 1⎥ ⎥ L⎥ 1 ⎥⎥ ⎦

(1.14)

C. Kahraman, M. Yavuz, and İ. Kaya

14 ⎡a ⎤

θˆ = ⎢ ⎥ = ( BT B ) BT Y , u −1

⎣ ⎦

(1.15)

Step 5. Inverse accumulated generation operation (IAGO). Because the grey forecasting model is formulated using the data of AGO rather than original data, IAGO can be used to reverse the forecasting value. Namely ⎡ ⎛⎜1⎞⎟ ⎢− z⎝ ⎠ ( 2) ⎢ ⎛ ⎞ ⎢ ⎜⎝1⎟⎠ B = ⎢ − z ( 3) ⎢ L ⎢ ⎢ ⎛⎜1⎞⎟ ⎢− z ⎝ ⎠ ( n ) ⎣

⎤

1⎥

⎥ ⎥ 1⎥ ⎥ L⎥ 1 ⎥⎥ ⎦

(1.16)

0 1 1 xˆo( ) ( k ) = xˆ ( ) ( k ) − xˆ ( ) ( k − 1) , k = 2,3,..., n.

4 Applications 4.1 Forecasting for Metal Rods Using Grey Model In an automotive company, a machine produces metal rods used in automobile suspension system. A random sample of 13 rods is selected, and the diameter is measured. The resulting data are shown in Table 1.1: Step 1. The original series, x( 0) , of diameters are given in Table 1.1. Table 1.1 Actual Values of rods diameter (mm) Rod Diameter Rod Diameter 1

9.14

8

9.16

2

9.11

9

9.09

3

9.13

10

9.13

4

9.15

11

9.1

5

9.16

12

9.08

6

9.13

13

9.12

7

9.1

Step 2. A new sequence, x(1) , is generated by the accumulated generating operation (AGO). This sequence is given in Table 1.2.

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 15 Table 1.2 The new sequence generated by AGO Rod

x(1)

x(1)

1

9.14 7

63.92

2

18.25 8

73.08

3

27.38 9

82.17

4

36.53 10

91.3

5

45.69 11

100.4

6

54.82 12

109.48

Rod

Step 3. A first-order differential equation is established. α is selected as 0.9. Tables 1.3 and 1.4 are obtained as follows. Table 1.3 B matrix B -9.140911

1

-18.250913

1

-27.380915

1

-36.530916

1

-45.690913

1

-54.82091

1

-63.920916

1

-73.080909

1

-82.170913

1

-91.30091

1

-100.400908

1

Table 1.4 Y matrix Period

1

2

3

4

5

6

7

8

9

10

11

Y

9.11

9.13

9.15

9.16

9.13

9.1

9.16

9.09

9.13

9.1

9.08

Step 4. From Step 3, we find BT as follows: -9.141

1

-18.251

1

-27.381

1

-36.531

1

C. Kahraman, M. Yavuz, and İ. Kaya

16 -45.691

1

-54.821

1

-63.921

1

-73.081

1

-82.171

1

-91.301

1

-100.401 1

and T

-1

T

((B *B) )*B 0.0049795

0.3637

0.0039857

0.3093

0.0029898

0.2547

0.0019917

0.2000

0.0009925

0.1453

-0.0000034

0.0907

-0.0009960

0.0363

-0.0019952

-0.0184

-0.0029868

-0.0727

-0.0039827

-0.1273

-0.0049753

-0.1817

And finally we obtain “a= 0.00043823” and “u=9.14583”. The forecasted values are obtained as in Table 1.5. Table 1.5 The forecasted values by grey model Year GM(1,1) Year GM(1,1) 1

9.1400

8

2

9.1398

9

9.1158 9.1118

3

9.1358

10

9.1078

4

9.1318

11

9.1038

5

9.1278

12

9.0999

6

9.1238

13

9.0959

7

9.1198

4.2 Forecasting by Simple Linear Regression The strength of paper used in the manufacture of cardboard boxes (y) is related to the percentage of hardwood concentration in the original pulp (x). Under

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 17

controlled conditions, a pilot plant manufactures 16 units, each from a different batch of pulp, and measures the tensile strength. The data are shown in Table 1.6 (Montgomery and Runger 2006): Table 1.6 Data for cardboard boxes x

y

x

y

1.0 101.4 2.5 111.0 1.5 117.4 2.5 123.0 1.5 117.1 2.8 125.1 1.5 106.2 2.8 145.2 2.0 131.9 3.0 134.3 2.0 146.9 3.0 144.5 2.2 146.8 3.2 143.7 2.4 133.9 3.3 146.9

B1 and B2 values are determined as 112.15, and 7.17, respectively by using Eq. (1.3). The fuzzy membership function of a can be calculated by using the following equation:

⎡ ⎤ ⎢129.71 − 2.92tα ,129.71 + 2.92tα ⎥ 2 2⎦ ⎣

(1.17)

The membership function of a is obtained as shown in Figure 1.1. Also (α ) -cuts of b can be calculated by using the following equation: ⎡ ⎤ ⎢15.64 − 4.36t β ,15.64 + 4.36t β ⎥ 2 2⎦ ⎣

Fig. 1.1 The Membership function of a

(1.18)

18

C. Kahraman, M. Yavuz, and İ. Kaya

Fig. 1.2 The membership function of b

Fig. 1.3 The membership function of σ 2

The membership function of b is obtained as shown in Figure 1.2 by using Eq. (1.6). Also the σ 2 can be calculated from Eq. (1.10). The membership function of σ 2 is obtained as shown in Figure 1.3.

4.3 Fuzzy Prediction by Simple Linear Regression Consider the example which is detailed above. We will use the same data to predict fuzzy values. The fuzzy prediction for the cardboard boxes will be made.

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 19

Consider the first value that x=1.0. We will predict y% ( x = 1.0 ) by using Eq. (1.12). Since x = 2.33 , ( x − x ) is calculated as ( x − x ) = −1.33 < 0 . By using the second part of Eq. (1.12), the membership function of y% (1.0 ) is obtained as shown in Figure 1.4. We can also predict y% ( x = 2.4 ) by using Eq. (1.12). Since x = 2.33 , ( x − x ) is calculated as ( x − x ) = 0.67 > 0 . The membership function of y% ( 2.4 ) is obtained by using the first part of Eq. (1.12) as shown in Figure 1.5. The other values for fuzzy prediction are presented in Table 1.7.

Fig. 1.4 The membership function of y% (1.0 )

Fig. 1.5 The membership function of y% ( 2.4 )

C. Kahraman, M. Yavuz, and İ. Kaya

20 Table 1.7 Fuzzy predictions for cardboard boxes x

y

y% ( x )

1.0 101.4 (73.57, 108.982, 144.394) 1.5 117.4 (90.268, 116.802, 143.337) 1.5 117.1 (90.268, 116.802, 143.337) 1.5 106.2 (90.268, 116.802, 143.337) 2.0 131.9 (106.966, 124.623, 142.279) 2.0 146.9 (106.966, 124.623, 142.279) 2.2 146.8 (113.646, 127.751, 141.857) 2.4 133.9 (117.662, 130.879, 144.097) 2.5 111.0 (117.45, 132.443, 147.437) 2.5 123.0 (117.45, 132.443, 147.437) 2.8 125.1 (116.816, 137.136, 157.456) 2.8 145.2 (116.816, 137.136, 157.456) 3.0 134.3 (116.393, 140.264, 164.135) 3.0 144.5 (116.393, 140.264, 164.135) 3.2 143.7 (115.97, 143.392, 170.814) 3.3 146.9 (115.759, 144.956, 174.154)

5 Conclusions Forecasting is an important part of decision making, and many of our decisions are based on predictions of future unknown events. Forecasting is important in many aspects of our lives. We try to predict success in our occupations, and investments. Organizations invest enormous amounts based on forecasts for new products, factories, retail outlets, and contracts with executives. Decision makers need forecasts when there is uncertainty about the future. Forecasting serves many needs. It can help people and organizations to plan for the future and to make rational decisions. The ability to obtain good forecasts has been highly valued. Poor forecasting can lead to disastrous decisions. Forecasting practice has improved over time. Knowledge about forecasting has increased rapidly. Many different techniques such as fuzzy set theory, neural networks, and genetic algorithms have been used to develop the efficiency of the forecasting. In the literature, the fuzzy set theory and grey theory have been used to develop quantitative forecasting models such as time series analysis and regression analysis, and in qualitative models such as the Delphi method. In these

Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 21

applications, these theories provide a language by which indefinite and imprecise demand factors can be captured. The structure of these models are often simpler yet more realistic than non-fuzzy and non-grey models which tend to add layers of complexity when attempting to formulate an imprecise underlying demand structure. When demand is definable only in linguistic terms, fuzzy forecasting models must be used. Otherwise, crisp statistical techniques should be used. In this chapter, the fuzzy set theory is used to develop the efficiency, accuracy and sensitiveness of simple linear regression. In future research, neural networks, neuro-fuzzy inference systems, fuzzy rule based systems, genetic algorithms and other metaheuristics can be used together to increase the accuracy of forecasting methods by developing a hybrid model.

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Fuzzy and Grey Forecasting Techniques and Their Applications in Production Systems 23 Jia, Z.Y., Ma, J.W., Wang, F.J., Liu, W.: Characteristics forecasting of hydraulic valve based on grey correlation and ANFIS. Expert Systems with Applications (2009), doi:10.1016/j.eswa.2009.06.003 Khashei, M., Bijari, M., Ardali, G.A.R.: Improvement of auto-regressive integrated moving average models using fuzzy logic and artificial neural networks (ANNs). Neurocomputing 72(4-6), 956–967 (2009) Khemchandani, R., Jayadeva Chandra, S.: Regularized least squares fuzzy support vector regression for financial time series forecasting. Expert Systems with Applications 36(1), 132–138 (2009) Kuo, I.H., Horng, S.J., Kao Chen, Y.H., Run, R.S., Kao, T.W., Chen, R.J., Lai, J.L., Lin, T.L.: Forecasting TAIFEX based on fuzzy time series and particle swarm optimization. Expert Systems with Applications (2009), doi:10.1016/j.eswa.2009.06.102 Kuo, I.H., Horng, S.J., Kao, T.W., Lin, T.L., Lee, C.L., Pan, Y.: An improved method for forecasting enrollments based on fuzzy time series and particle swarm optimization. Expert Systems with Applications 36(3-2), 6108–6117 (2009b) Lai, R.K., Fan, C.Y., Huang, W.H., Chang, P.C.: Evolving and clustering fuzzy decision tree for financial time series data forecasting. Expert Systems with Applications 36, 3761–3773 (2009) Li, D.C., Yeh, C.W., Chang, C.J.: An improved grey-based approach for early manufacturing data forecasting. Computers & Industrial Engineering (2009), doi:10.1016/j.cie.2009.05.005 Lin, C.T., Yang, S.Y.: Forecast of the output value of Taiwan’s opto-electronics industry using the Grey forecasting model. Technological Forecasting & Social Change 70, 177– 186 (2003) Lin, Y.H., Lee, P.C., Chang, T.P.: Adaptive and high-precision grey forecasting model. Expert Systems with Applications 36(6), 9658–9662 (2009) Liu, H.: An integrated fuzzy time series forecasting system. Expert Systems with Applications 36(6), 10045–10053 (2009) Lu, J., Wang, R.: An enhanced fuzzy linear regression model with more flexible spreads. Fuzzy Sets and Systems 160(17), 2505–2523 (2009) Mady, M.T.: Sales forecasting practices of Egyptian public enterprises: survey evidence. International Journal of Forecasting 16(3), 359–368 (2000) Mamlook, R., Badran, O., Abdulhadi, E.: A fuzzy inference model for short-term load forecasting. Energy Policy 137(4), 1239–1248 (2009) Mohr, S.H., Evans, G.M.: Forecasting coal production until 2100. Fuel (2009), doi:10.1016/j.fuel.2009.01.032 Montgomery, D.C., Runger, G.C.: Applied statistics and probability for engineers. John Wiley and Sons, Chichester (2006) Myatt, G.J.: Making sense of data: a practical guide to exploratory data analysis and data mining. John Wiley and Sons, Chichester (2006) Ragulskis, M., Lukoseviciute, K.: Non-uniform attractor embedding for time series forecasting by fuzzy inference systems. Neurocomputing 72(10-12), 2618–2626 (2009) Song, Q., Chissom, B.S.: Forecasting enrollments with fuzzy time series, Part I. Fuzzy Sets and Systems 54, 1–9 (1993a) Song, Q., Chissom, B.S.: Forecasting enrollments with fuzzy time series, Part II. Fuzzy Sets and Systems 62, 1–8 (1994) Song, Q., Chissom, B.S.: Fuzzy time series and its models. Fuzzy Sets and Systems 54, 269–277 (1993b)

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Thury, G., Witt, S.F.: Forecasting industrial production using structural time series models. Omega 26(6), 751–767 (1998) Valckenaers, P., Germain, B.S., Verstraete, P., Brussel, H.V.: Emergent short-term forecasting through ant colony engineering in coordination and control systems. Advanced Engineering Informatics 20(3), 261–278 (2006) Vollmann, T.E.: Manufacturing planning and control systems for supply chain management. McGraw-Hill, New York (2004) Wen, J.C., Huang, K.H., Wen, K.L.: The study of a in GM (1,1) model. Journal of the Chinese Institute of Engineers 23(5), 583–589 (2000) Xiao, Z., Ye, S., Zhong, B., Sun, C.: BP neural network with rough set for short term load forecasting. Expert Systems with Applications 36(1), 273–279 (2009) Xie, N., Liu, S.: Discrete grey forecasting model and its optimization. Applied Mathematical Modelling 33(2), 1173–1186 (2009) Yolcu, U., Egrioglu, E., Uslu, V.R., Basaran, M.A., Aladag, C.H.: A new approach for determining the length of intervals for fuzzy time series. Applied Soft Computing 9(2), 647–651 (2009) Yo-Ping, H., Chih-Hsin, H.: Real-valued genetic algorithms for fuzzy grey prediction system. Fuzzy Sets and Systems 87, 265–276 (1997) Zadeh, L.: Fuzzy sets. Information and Control 8, 338–353 (1965) Zhou, P., Ang, B.W., Poh, K.L.: A trigonometric grey prediction approach to forecasting electricity demand. Energy 31, 2839–2847 (2006)

Chapter 2

Fuzzy Inventory Management Mesut Yavuz*

Abstract. Inventory is one of the most expensive and important assets to many companies, representing as much as 50% of total invested capital. Combined with the fierce pressure to cut costs and the challenge of dealing with various types of uncertainties, researchers and practitioners alike have been motivated to use fuzziness in inventory management. In this chapter, we present economic order quantity, economic production quantity, single period, and periodic review inventory models under fuzziness. All models are demonstrated through numerical examples.

1 Introduction Inventory is one of the most expensive and important assets to many companies, representing as much as 50% of total invested capital. Managers have long recognized that good inventory control is crucial. Hence, inventory control has been a central piece of the industrial engineering and management science disciplines; many production management textbooks (see (Nahmias 2008) for example) provide comprehensive reviews of the classical inventory theory. On one hand, a firm can try to reduce costs by reducing inventory levels. On the other hand, customers become dissatisfied when frequent inventory shortages, i.e., stockouts, occur. As you would expect, cost minimization is the major factor in obtaining this delicate balance. In this chapter, we present key inventory management models under fuzziness. Inventory is any stored resource that is used to satisfy a current or future need. Raw materials, work-in-process, and finished goods are examples of inventory. Inventory levels for finished goods are a direct function of demand. When we determine the demand for an end-product, we can use this information to determine the demand for parts, components, and raw materials required to make that end-product. For example, when we determine the demand for personal computers, we can determine how many motherboards, processors, memory chips, disk drives, etc. are needed to assemble the finished product. The demand for supply items is often referred to as dependent demand. Management of dependent demand is the subject of Mesut Yavuz Shenandoah University, Harry F. Byrd, Jr. School of Business, Winchester, Virginia, U.S.A. C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 25–38. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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the conventional materials requirement planning (MRP) systems presented in Chapter 5. An alternative to MRP is just-in-time (JIT) manufacturing, which advocates pulling materials through a plant, rather than pushing them. JIT manufacturing under fuzziness is discussed in Chapter 6. As these two important topics related to inventory management are discussed in separate chapters, we do not discuss them here but focus on managing inventories of independent demand items. All organizations have some sort of inventory planning and control systems. A bank has a method to control its inventory of cash. A hospital has methods to control blood supplies and other important items. Virtually every production and service organization is concerned with inventory planning and control. In a majority of real-life inventory management problems, decision makers face uncertainty in the key parameters of their model. For example, costs associated with holding inventories and placing an order, cost of a stockout, demand in the planning horizon and production capacity may all be uncertain. Fuzzy logic has been successfully used in various inventory management problems. In this chapter, we focus on single-item, independent demand models and present a review of the recent literature on several key models. The remainder of this chapter is organized as follows. In Section 2 we present the preliminary definitions of fuzzy numbers and operators. In Sections 3 and 4 we present the economic order quantity and the economic production quantity models, respectively. Sections 5 and 6 are devoted to periodic review models. In Section 5, we present the newsboy (single period) model and in Section 6 we present the order up to r model. Section 7 concludes the chapter by discussing its relevance and relationship to other chapters in the book.

2 Preliminaries In this chapter we will model fuzzy variables/parameters as trapezoidal or triangular fuzzy numbers. We note that all the fuzzy variables and parameters we use in this chapter are positive numbers by definition. In Figure 2.1 we depict a trapezoidal number ~ x = ( xa , xb , xc , xd ) and a fuzzy number ~ y = ( ya , yb , yc ) .

μx 1

0

xa xb

xc

xd ya

Fig. 2.1 Trapezoidal and triangular fuzzy numbers

yb

yc

x

Fuzzy Inventory Management

27

Using the function principle of fuzzy arithmetic we define the following operations on two positive trapezoidal fuzzy numbers ~ x = ( xa , xb , xc , xd ) and

~ y = ( ya , yb , yc , yd ) , and a constant k ∈ R . The multiplication of a fuzzy number with a constant:

if k ≥ 0 ⎧ (kxa , kxb , kxc , kxd ), k⊗~ x =⎨ ⎩(−kxd ,−kxc ,−kxb ,−kxa ), if k < 0 The addition of two fuzzy numbers:

~ x⊕~ y = ( x a + y a , xb + y b , x c + y c , x d + y d ) The multiplication of two fuzzy numbers:

~ x⊗~ y = ( x a y a , xb y b , x c y c , x d y d ) The subtraction of two fuzzy numbers:

~ x−~ y = ( x a − y d , xb − y c , x c − y b , x d − y a ) The division of two fuzzy numbers:

~ x/~ y = ( x a / y d , xb / y c , x c / y b , x d / y a ) To defuzzify a trapezoidal fuzzy number we define the following two approaches. Median:

x med =

x a + xb + xc + x d 4

Center of gravity (cog):

xcog =

∫ μ xd ∫μ d x

x

x

x

=

x a + xb + x c + x d x a xb − x c x d + 3 3( x d + xc − xb − x a )

Again, using the function principle of fuzzy arithmetic we define the following operations on two positive triangular fuzzy numbers ~ x = ( xa , xb , xc ) and

~ y = ( ya , yb , yc ) , and a constant k ∈ R . The multiplication of a fuzzy number with a constant:

if k ≥ 0 ⎧ (kxa , kxb , kxc ), k⊗~ x =⎨ ⎩(− kxc ,−kxb ,− kxa ), if k < 0

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M. Yavuz

The addition of two fuzzy numbers:

~ x⊕~ y = ( xa + ya , xb + yb , xc + yc ) The multiplication of two fuzzy numbers:

~ x⊗~ y = ( xa ya , xb yb , xc yc ) The subtraction of two fuzzy numbers:

~ x−~ y = ( xa − yc , xb − yb , xc − ya ) The division of two fuzzy numbers:

~ x/~ y = ( xa / yc , xb / yb , xc / ya ) To defuzzify a triangular fuzzy number we define the center of gravity (cog) approach as follows. Center of gravity (cog):

xcog =

∫ μ xd ∫μ d x

x

x

x

=

xa + xb + xc 3

3 Economic Order Quantity The economic order quantity (EOQ) is the simplest and most fundamental of inventory models. It addresses the important trade-off between ordering and holding costs. The EOQ model dates back to early 1900s (Harris 1915). Let us start with the crisp EOQ model. Annual demand for a product is denoted by D. Placing an order costs $o regardless of the order size, and it costs $h to hold one unit of the product in inventory for a whole year. Let us denote the order quantity with Q. It is assumed that D is known and constant, i.e., the demand is uniform throughout the year. When we assume that lead time is also known and constant, we can easily conclude that the optimal time to receive a shipment is the time when we deplete the current stock. In other words, the inventory level of the item decreases with a constant rate until it reaches zero, when it increases back up. Another key assumption of the EOQ model is instantaneous receipt. That is, each order is received in the exact quantity and at once. This causes the inventory level to jump from zero to Q at the time of the receipt. Inventory level of a product under the EOQ model is shown in Figure 2. Note that stockouts are not allowed. Each order has the same size, Q, thereby requiring D/Q orders to be placed throughout the year. Therefore the annual ordering cost is o × D / Q . The

Fuzzy Inventory Management

29

Fig. 2.2 Inventory level vs. time in the EOQ model

average inventory level is Q/2, and, hence, the annual holding cost is h × Q / 2 . The total inventory holding and ordering cost is

TC = o

D Q +h . Q 2

The optimal order quantity minimizing the total cost is

Q* =

2oD h

(2.1)

Annual demand, D, is rarely known and constant in real life. Therefore, in most mathematical models it is treated as a (probabilistic) random variable. However, cost coefficients o and h, too can be uncertain. Here, we will model them as trapezoidal fuzzy numbers

~ o~ = (oa , ob , oc , od ) and h = (ha , hb , hc , hd ) .

We present three methods to find the economic order quantity differing in the defuzzification operator and defuzzification point in the calculations. The first method is proposed by Park (1987). His method uses the median method to defuzzify

~

~ and h and plugging in o and h for o and h, rethe fuzzy cost parameters o med med spectively, in Eq. (2.1). Resulting Q* value is used as the economic order quantity. The second method is proposed by Vujosevic et al. (1996). Their method uses

~

~ and h . Then, similarly the center of gravity approach in the defuzzification of o to Park’s method, defuzzified cost parameters are plugged into Eq. (2.1) to obtain the economic order quantity. The third method is also due to Vujosevic et al. (1996). In this method, cost parameters are used as fuzzy numbers and fuzzy arithmetic is applied in Eq. (2.1) as follows:

~ Q* =

2o~D ⎛⎜ 2o a D 2ob D 2oc D 2o d D ⎞⎟ . , , , ~ =⎜ hc hb ha ⎟⎠ h ⎝ hd

In the end, the fuzzy EOQ is defuzzified using the center of gravity method.

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Example. The annual demand for a product is 50,000 units. The ordering and holding costs are given with trapezoidal fuzzy numbers (10, 12, 16, 20) and (1.5, 2.0, 2.4, 2.7), respectively. Let us calculate the economic order quantity using all three methods. Method 1 omed = 14.50 hmed = 2.15

Q* =

2 × 14.50 × 50,000 ≅ 821 . 2.15

Q* =

2 × 14.57 × 50,000 ≅ 825 . 2.14

Method 2 ocog = 14.57 hcog = 2.14

Method 3

~ Q* =

2o~D * ~ = (609,707,894,1155) ⇒ Qcog ≅ 848 . h

As we see from this example, the three methods can produce different results, but they are fairly close to each other. For a comparison of the three methods as well as discussion of the differences between probabilistic and fuzzy approaches, we refer the reader to (Hojati 2004). We conclude the section by referring interested readers to several recent and important works on the fuzzy EOQ model. Yao and Chiang (2003) develop a fuzzy EOQ model with triangular fuzzy demand and holding cost. They compare the cost functions obtained by using the centroid and signed distance methods of defuzzification. Pai and Hsu (2003) discuss reorder point problems in a fuzzy environment. A recent line of research has also studied fuzzy EOQ models with fuzzy order quantities. See (Lee and Yao 1999, Yao et al. 2000) for models without backorder and (Wu and Yao 2003) for a model with backorder.

4 Economic Production Quantity The instantaneous receipt assumption of the EOQ model implies that an order is received in full quantity. In other words, production capacity at the supplier is infinite so that an order can be fulfilled in time and at once regardless of its size. In many real life production environments the supplier’s capacity is limited, which is especially true when the items are produced in house. In that case, using the EOQ model would yield sub-optimal results.

Fuzzy Inventory Management

31

The economic production quantity (EPQ) model is an enhancement of the EOQ model in that it takes two additional parameters, namely daily demand rate (d) and daily production rate (p). The daily demand rate is obtained by dividing the annual demand rate into the number of operating days in a year. Note that the daily production capacity must be greater than the daily demand rate (p > d). In the EPQ model, we again find a balance between ordering and holding costs. Since the EPQ is typically associated with items produced in-house, we use the term setup cost instead of ordering cost and denote it with s. The crisp solution of the EPQ model is obtained by minimizing the summation of setup and inventory holding costs and is summarized in the following equation:

Q* =

2sD ⎛ d⎞ h⎜⎜1 − ⎟⎟ p⎠ ⎝

(2.2)

Chang (1999) proposes a method based on the Extension Principle of fuzzy arithmetic and the centroid method of defuzzification to obtain the optimal production quantity. In a later work, Hsieh (2002) proposes an alternative method based on the Function Principle of fuzzy arithmetic and the Graded Mean Integration Representation (Chen and Hsieh 1999) of fuzzy numbers. For sake of simplicity in calculations, we present Hsieh’s method here. Let use denote all parameters of the model with the following trapezoidal fuzzy numbers:

~ D = (d a , d b , d c , d d ) , Setup cost per setup: ~ s = (s a , sb , sc , s d ) , ~ Inventory holding cost per unit per year: h = ( ha , hb , hc , hd ) , ~ Daily demand rate: d = ( d a , d b , d c , d d ) , and Daily production rate: ~ p = ( p a , pb , p c , p d ) . Annual demand:

Total setup and holding cost for ordering Q units at a time is

~ ~ ~ ~ D ~ ⎛ d⎞ Q ⎜ C = s ⊗ ⊕ h ⊗ ⎜1 − ~ ⎟⎟ ⊗ Q p⎠ 2 ⎝

(2.3)

In the total cost equation, order quantity (Q) is the only crisp part. In order to solve for Q, the fuzzy total cost is first defuzzified using the graded mean integration representation. The graded mean integration representation of a fuzzy number

x + 2 xb + 2 xc + x d ~ x = ( x a , xb , xc , x d ) is x gmir = a . 6 Then, the first derivation of the defuzzified total cost is set equal to zero and solved for Q. The optimal production quantity is obtained as follows.

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M. Yavuz

Q* =

2(s a Da + 2s b Db + 2s c Dc + s d Dd ) ⎛ d ⎞ ⎛ ⎛ d ⎞ ⎛ d ⎞ d ⎞ ha ⎜⎜1 − d ⎟⎟ + 2hb ⎜⎜1 − c ⎟⎟ + 2hc ⎜⎜1 − b ⎟⎟ + hd ⎜⎜1 − a ⎟⎟ pa ⎠ pb ⎠ pc ⎠ pd ⎠ ⎝ ⎝ ⎝ ⎝

(2.4)

Example. Consider the following data for a fuzzy optimal production quantity model:

~ D = (2250,2400,2600,2750) , ~ s = (35,40,60,65) , ~ h = (0.35,0.45,0.55,0.65) , ~ d = (9.0,9.6,10.4,11.0) , and ~ p = (27,30,33,36) .

Using Eq.(2.4) the optimal production quantity is found to be Q ≅ 859 . Among recent works on the fuzzy EPQ model, we cite few important ones that incorporate quality and finance considerations into production planning. Chang and Chang (2006) present an elaborative cost structure. Their optimization model then includes a production cost component in addition to setup and holding costs. Chen and Ouyang (2006) study a model that includes shortages and permissible payment delays for deteriorating items. Chen et al. (2007) focus on items with imperfect quality. *

5 Single-Period Models The inventory models we have considered so far are continuous review models that assume a product is demanded with a constant rate throughout the year. There are some products that carry a value only for a limited duration, which can be as short as a day. Newspapers would be the best known example of such products. Noting that the single period inventory model is commonly referred to as the newsboy model, we explain the model using the newspaper example here. A newsboy has to decide how many newspapers to purchase in a given day. He purchases a newspaper for c and sells it for p. (Note that p > c). At the end of the day any remaining newspapers are salvaged for a price s smaller than the purchase price (s < c). If the newsboy knows the demand for the day D, he can maximize his profit by setting his order quantity Q equal to the demand (Q = D). When the demand is uncertain he faces two costs, namely the underage cost and the overage cost. Underage occurs when the demand exceeds the purchase quantity (Q < D). In this case the newsboy sells only Q newspapers, losing the opportunity to sell the remaining D – Q units. Therefore, his opportunity loss is (D – Q) (p – c). Overage occurs when the demand is below the order quantity (Q > D). Then the newsboy sells the remaining Q – D newspaper for s each, thereby losing (Q – D) (c – s).

Fuzzy Inventory Management

33

If p, c and s are known with certainty, i.e., they are crisp numbers and the probability distribution of the demand is known, optimal order quantity Q* is obtained as follows. Let FD be the cumulative distribution function of D. Then Q* that satisfies

FD (Q * ) =

p−c is the optimal order quantity. Using the inverse of the p−s

distribution function, F-1, we summarize this result in Eq. (2.5).

⎛ p−c⎞ ⎟⎟ Q * = FD−1 ⎜⎜ ⎝ p−s⎠

(2.5)

~

Kao and Hsu (2002) consider the problem with fuzzy demand D . Here we will focus on fuzzy demand given with a trapezoidal fuzzy number

~ D = (d a , d b , d c , d d ) .

Total cost is also a fuzzy number:

~ ~ ~ T (Q) = ( p − c) max{0, D − Q} + (c − s ) max{0, Q − D} (2.6) ~ Since the total cost T (Q ) is a fuzzy number, it can be ranked using an existing method for ranking fuzzy numbers. The order quantity yielding the minimum total cost is the optimal order quantity. Kao and Hsu obtain the following classification of the optimal order quantity.

⎧ ⎛ 2( p − c ) ⎞ ⎟⎟(Db − Da ), if ( p − c) < (c − s ) ⎪= Da + ⎜⎜ − p s ⎝ ⎠ ⎪⎪ if ( p − c ) = (c − s) ∈ [ Db , Dc ], Q* ⎨ ⎪ ⎛ 2(c − s ) ⎞ ⎟⎟(Dd − Dc ), if ( p − c ) > (c − s) ⎪ = Dd − ⎜⎜ ⎪⎩ ⎝ p−s ⎠

(2.7)

In Eq.(2.7), a definitive optimal order quantity is given for two cases whereas an optimal interval is given for one case. That particular case is the equality of unit underage and overage costs, and it makes any order quantity in the flat part of the trapezoidal fuzzy demand an optimal solution. Example. Consider a single period inventory problem with a trapezoidal fuzzy

~

demand D = (70,80,120,130) , unit selling price p = 0.85, unit purchase price c = 0.25 and unit salvage value s = 0.05. Here, we see p – c = 0.60 > 0.20 = c – s. Thus, the third part in Eq. (2.7) should be used. The optimal solution is found to be Q* = 125. Suppose that the unit purchase price was 0.45 in this example. Using c = 45 in the problem, we see p – c = 0.40 = 0.40 = c – s. Therefore, the second part of Eq. (2.7) should be used. Consequently, any order size between the interval

Q * ∈ [80,120] is optimal.

34

M. Yavuz

Finally, let us change the unit purchase price to c = 0.55. In this case, we obtain p – c = 0.30 < 0.50 = c – s. Thus, the first part in Eq. (2.7) should be used. The optimal solution is Q* = 77.5, which can be rounded up or down.. We would like to refer the reader to two other important references for more advanced scenarios. See Li et al. (2002) for a model with fuzzy costs and probabilistic demand, and (Duttaet al. 2005) for a model with crisp costs and fuzzy random variable demand.

6 Periodic Review Models A typical production system has many parts and raw materials purchased from outside suppliers. Consequently, inventory managers typically have to watch the inventory levels of many items and order the ones that need to be replenished. Continuous review models require keeping track of inventory levels of all items at all times and can generate orders anytime. This overcomplicates the job of inventory managers. As a practical solution a periodic review system is employed in many companies. In this section, we focus on single-item periodic review models. Periodic review models allow one to place at most one order per item per period. The two most commonly used periodic review models are the “order up to r” and “s-S” models. With an order up to r model, one places an order each period to bring the inventory level up to the target level r. If the inventory level at the time of the review is x, an order of r-x is placed. With an s-S model, in contrast, one may or may not place an order in a given period. If x > s, no order is placed; otherwise an order of S-x units is placed. Here, we focus on the order up to r model. The order up to r model is also known as a “constant order cycle, variable order quantity” model. The variability of the order quantity is a reflection to the variability in demand. In other words, demand is uncertain in these models. Therefore, the problem is about finding a balance between inventory holding and shortage costs. Also, when designing an order up to r system, one has to determine not only the order parameter (target inventory level, r), but also the length of the review period. These factors make the order up to r model significantly more complicated than the models we have seen in the preceding sections. Solving the model typically requires an algorithm. Here we adopt Dey and Chakraborty’s (2009) method. Let T be the time between reviews and L be the lead time. Both T and L are crisp numbers measuring the length of a time period in years. For example, if an item is reviewed twice a year, T = 0.5. Let us denote ordering cost with o, unit holding cost with h and unit shortage cost with s. The ordering cost, o, includes the cost of making a review. r is the target inventory level. All the parameters and variables defined so far are crisp. Demand is often uncertain. In order to capture both types of uncertainty, a recent line of research defines demand as a fuzzy random variable. Let n

fuzzy random annual demand. Let pi (

∑p i =1

i

~ D be the

= 1 ) be the probability associated

Fuzzy Inventory Management

35

~

with fuzzy annual demand Di . Also, suppose that fuzzy annual demands are given

~ Di = ( Di , a , Di ,b , Di ,c ) , i = 1, .., n. ~ The expected fuzzy annual demand is D = ( D a , Db , Dc ) , calculated with

are given with triangular fuzzy numbers

n n n n ~ D = ∑ Di p i = ( ∑ Di , a p i , ∑ Di , b p i , ∑ Di , c p i ) . i =1

i =1

i =1

i =1

For defuzzification of a triangular fuzzy number, we use the graded mean integration representation here. The graded mean integration representation of a trian-

( x + 4 xb + xc ) ~ x is x gmir = a . 6 ~ ~ Also, the demand during lead time is D (L ) = D ⊗ L , and the demand during ~ ~ lead time plus one period is D (L + T ) = D ⊗ (L + T ) . Note that these are both

gular fuzzy number

fuzzy random numbers, and, hence, the expected value and the defuzzification operations defined above can be used on them. The total cost is obtained by adding ordering, holding and shortage costs. Since it is a function of the demand, it is a fuzzy random number, too:

~ ⎛ D(T ) ⎞ s o ~ ~ ~ ⎟ + M D( L + T ) − r C (r , T ) = + h⎜⎜ r − D( L) − ⎟ T 2 ⎠ T ⎝ + ~ Here, M D ( L + T ) − r is the expected shortage in each period.

(

(

)

+

(2.8)

)

The problem is a non-linear optimization problem and its solution can be obtained by employing a search algorithm on r and T. When the value of T is given, the optimal value of r can be calculated. Thus, we will search on T and simultaneously optimize r and T through a linear search. Recall that

~ D ( L + T ) = ( Da ( L + T ), Db ( L + T ), Dc ( L + T )) is a triangular fuzzy

number. Let us define the following functions based on the left and right pieces of the triangle.

L(r ) =

r − Da ( L + T ) Db ( L + T ) − D a ( L + T )

(2.9)

R(r ) =

Dc ( L + T ) − r Dc ( L + T ) − Db ( L + T )

(2.10)

The optimal target inventory level, r*, given T, can be calculated from the following conditions:

36

M. Yavuz

s⎛ 1 2 * ⎞ * ⎜1 − L (r ) ⎟, if Da ( L + T ) ≤ r < Db ( L + T ) h⎝ 2 ⎠ s 2 * T= R (r ), if Db ( L + T ) ≤ r * ≤ Dc ( L + T ) 2h

T=

(2.11)

The search moves with a step size z and terminates when the step size is small enough, i.e., smaller than ε, and taking further steps worsens the solution. The problem has only one constraint; the safety stock cannot be negative:

~ SS = r − D gmir ( L + T ) ≥ 0

(2.12)

Now, we present the search procedure: Step 1 – Set k = 0. Determine z and ε. Choose an initial value for T0. Step 2 – Compute

~ D ( L + T0 ) .

Step 3 – Calculate r* using the appropriate condition in Eq. (2.11). Step 4 – Calculate the safety stock and the total cost. Step 5 – If SS < 0, then multiply T0 by 2 and go to Step 2. Step 6 – Increase k by 1. Step 7 – Set Tk = Tk-1 + z. Step 8 – Compute

~ D ( L + Tk ) , r* using the appropriate condition in Eq.

(2.11), the safety stock, and the total cost. Step 9 – If the current solution is better than the previous one, then go to step 6. Step 10 – Set T* = Tk-1. Step 11 – If | T* - Tk-1|< ε, then stop. Otherwise set T0 = T*, select a smaller step size z and go to Step 2. Example. Dey and Chakraborty (2009) provide the following example from a pharmaceutical company that consumes a certain kind of a chemical. The cost of making a review is $30 and of placing an order is $15 (o = 30 + 15 = 45). The inventory holding cost per unit (kg) is $3 (h = 3). Demands occurring in case of shortage are backordered and the unit penalty of backordering is $25 (s = 25). The demand information is given in Table 1. Table 2.1 Demand information Demand (in kg)

Probability

(500, 600, 700)

0.15

(450, 500, 550)

0.18

(475, 525, 560)

0.25

(480, 500, 520)

0.22

(500, 550, 625)

0.20

The lead time is 10 days (L = 10 / 365 = 0.0274).

Fuzzy Inventory Management

37

Following the procedure described above, and using a step size of 0.05 at first and then 0.01, the iterations given in Table 2 are performed. Table 2.2 Iteration details T (in years)

r

Total cost

0.10

73.07

546.82

0.15

101.51

443.02

0.20

129.84

414.54

0.25

158.06

414.75

0.20

129.84

414.54

0.21

135.49

412.44

0.22

141.14

411.86

0.23

146.79

412.12

As seen from Table 2.2, the optimal period length is 0.22, which is approximately 80 days. Corresponding target inventory level is about 141 kg, and minimum total cost is $412 per year. Another important periodic inventory management problem in the literature is the discrete lot sizing problem, which consists of multiple periods with known demands and aims at minimizing the summation of setup and inventory holding costs with or without backorders. We refer the reader to Buckley et al. (2002) for fuzzy modeling and solution of that problem.

7 Conclusions In this chapter we have presented selected topics in inventory management under fuzziness. We have presented two continuous review models first, namely the economic order quantity and the economic production quantity models, and then two periodic review models, namely the newsboy model and the order up to r model. All the models presented here have been around for a long time and have important applications in real life. Also, the use of fuzziness is helpful in capturing the uncertainty present in the real life. We have to mention that the literature contains many more studies than reviewed here. Some of those works are reviewed in other chapters of this book. In this chapter, we have focused only on single item inventory models. The management of dependent demand items is presented in Chapter 5, and JIT manufacturing under fuzziness is discussed in Chapter 6. Inventory management with multiple items is closely related to capacity planning, which is discussed in Chapter 10. Also, multi-item multi-period inventory models are an important part of supply chain management, which is presented in Chapter 18.

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References Buckley, J.J., Feuring, T., Hayashi, Y.: Solving fuzzy problems in operations research: inventory control. Soft Computing 7, 121–129 (2002) Chang, P.T., Chang, C.H.: An elaborative unit cost structure-based fuzzy economic production quantity model. Mathematical and Computer Modelling 43, 1337–1356 (2006) Chang, S.C.: Fuzzy production inventory for fuzzy product quantity with triangular, fuzzy number. Fuzzy Sets and Systems 107, 37–57 (1999) Chen, L.H., Ouyang, L.Y.: Fuzzy inventory model for deteriorating items with permissible delay in payment. Applied Mathematics and Computation 182, 711–726 (2006) Chen, S.H., Hsieh, C.H.: Graded mean integration representation of generalized fuzzy number. Journal of Chinese Fuzzy Systems 5(2), 1–7 (1999) Chen, S.H., Wang, C.C., Chang, S.M.: Fuzzy economic production quantity model for items with imperfect quality. International Journal of Innovative Computing 3(1), 85–95 (2007) Dey, O., Chakraborty, D.: Fuzzy periodic review system with fuzzy random variable demand. European Journal of Operational Research (2009) Dutta, P., Chakraborty, C., Roy, A.R.: A single-period inventory model with fuzzy random variable demand. Mathematical and Computer Modelling 4, 915–922 (2005) Harris, F.W.: Operations and cost. Factory Management Series, Chicago (1915) Hojati, M.: Bridging the gap between probabilistic and fuzzy-parameter EOQ models. International Journal of Production Economics 91, 215–221 (2004) Hsieh, C.H.: Optimization of fuzzy production inventory models. Information Sciences 146, 29–40 (2002) Kao, C., Hsu, W.K.: A single-period inventory model with fuzzy demand. Computers and Mathematics with Applications 43, 841–848 (2002) Lee, H.M., Yao, J.S.: Economic order quantity in fuzzy sense for inventory without backorder model. Fuzzy Sets and Systems 105, 13–31 (1999) Li, L., Kabadi, S.N., Nair, K.P.K.: Fuzzy models for single-period inventory problem. Fuzzy Sets and Systems 132, 273–289 (2002) Nahmias, S.: Production and operations analysis. McGraw-Hill/Irwin (2008) Pai, P.F., Hsu, M.M.: Continuous review reorder point problems in a fuzzy environment. International Journal of Advanced Manufacturing Technology 22, 436–440 (2003) Park, K.S.: Fuzzy-set theoretic interpretation of economic order quantity. IEEE Transactions on Systems, Man, and Cybernetic SMC 17(6), 1082–1084 (1987) Vujosevic, M., Petrovic, D., Petrovic, R.: EOQ formula when inventory cost is fuzzy. International Journal of Production Economics 45, 499–504 (1996) Wu, K., Yao, J.S.: Fuzzy inventory with backorder for fuzzy order quantity and fuzzy shortage quantity. European Journal of Operational Research 150, 320–352 (2003) Yao, J.S., Chang, S.C., Su, J.S.: Fuzzy inventory without backorder for fuzzy order quantity and fuzzy total demand quantity. Computers & Operations Research 27, 935–962 (2000) Yao, J.S., Chiang, J.: Inventory without backourder with fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance. European Journal of Operational Research 148, 401–409 (2003)

Chapter 3

Fuzzy Material Requirement Planning Josefa Mula and Raúl Poler*

Abstract. In this paper, we mainly discuss the convenience of incorporating fuzziness into material requirement planning (MRP) systems. Then, we formulate a MRP model in fuzziness environments. Customer demands and capacity data are assumed to be fuzzy, which is more appropriate in this model. Then we use a fuzzy method to solve a linear model by converting the constraints into their crisp equivalents. Finally, the proposed models are summarised and compared by highlighting the main scientific characteristics and those of the ideal application setting.

1 Introduction The development of the MRP (Material Requirement Planning) system at the end of the 60’s and the beginning of the 70’s in the Western world was revolutionary in terms of planning and control systems. An MRP system transforms a Master Production Schedule (MPS) into a program with details of the materials and components required to manufacture end products by using the list of materials to that end. MRP is based on two fundamental concepts: gross and net requirement explosions and backward scheduling of requirements. The most significant difference of the former technical stock management techniques in terms of order points or periodic supplies was appropriate demand management which depends on the components and raw materials as opposed to these technical techniques managing them as independent demands. Calculations do not take into account time or capacity availabilities, so the outgoings generated must be analysed to determine feasibility. A time-related non-feasibility situation means having to modify the MPS, whereas a capacity-related non-feasibility situation could be solved by modifying lot procedures. Regarding the time-related dimension, MRP systems generally plan on a periodic basis (using discrete periods of time), although continuous-period systems also exist. However, MRP at the planning level could be considered a pull system as all the requirement calculations are MPS-based which are, at the same time, derived from customer demand or customer order forecasts. Nonetheless, this is not the case of the operative level, where MRP is clearly a push system (Al-Hakim and Jenney 1991). So, MRP-generated programs give rise to production processes to Josefa Mula and Raúl Poler Research Centre on Production Management and Engineering (CIGIP), Universidad Politécnica de Valencia, Spain C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 39–57. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

40

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meet delivery dates. The required parts are processed and are sent on to the next stage until the final assembly has taken place. Readers are referred to Orlicky (1975), Vollmann et al. (2005) and Hopp & Spearman (2001) for a more detailed description of the calculation procedures within an MRP system. The MRP system does not consider any capacity constraint. Basically, all the possible capacity problems must be admitted by appropriately establishing delivery times. This fact was considered to be a substantial disadvantage, and led to continuous progress until the Closed Loop MRP and the MRP II (Manufacturing Resource Planning) systems came about (Vollmann et al. 2005). An MRP II system is made up of a variety of interlinked functions: business planning, sales and operations (production planning), MPS, MRP, CRP and support execution systems for materials and capacity. The results and outputs of these systems are made up of financial reports, such as business plans, purchase commitment reports, estimate of shipments, inventory forecasts in monetary units, etc. The production management module within an MRP II maintains the MRP system as the driving motor of production programs at the operative level. At the tactical level however, it includes functions like Rough Cut Capacity Planning (RCCP), which globally specifies capacity requirements, and the Capacity Requirement Planning (CRP) function for more detailed calculations of the capacity at the operative level. Nonetheless, these two names could be confused as neither function works with finite capacity loads, that is, they do not automatically link the required loads with the available loads, but perform a more or less detailed check to see whether the available capacity is sufficient to make the proposed MPS/MRP feasible. If it is not feasible, then the system generates a series of action messages with which a planner would be able to generate an alternative MPS or adjust the capacities. Finally at the operative level, the plant’s control system performs a thorough control. With time, the MRP II systems have been termed ERP (Enterprise Resource Planning). However, not all the ERP include a production management module based on the MPS, MRP, RCCP and CRP systems. Given their nature, MRP II systems require extensive administrative support. However, the most widespread used system for production planning and control is MRP II (Jonsson and Mattson 2002). The technology with which MRP II systems have been involved has evolved along with the Information and Communication Technologies (TIC). Hendry and Kingsman (1989) highlight the lack of a criterion in the customers’ order acceptance phase. Instead, an MRP II determines the approximate impact that an MPS has on the detailed plan. Basically, it treats the world as if it were determinist, and the possible uncertainties in the environment and/or system have to be absorbed by establishing delivery times which tend to increasingly grow (Ho 1989). Koh et al. (2002) and Dolgi and Prodhon (2007), presented reviews of the existing literature on uncertainty under MRP environments and stressed the need to develop a structure which may identify significant uncertainty, and one which also considers all the possible uncertainties combined. On the other hand, the classic resolution procedures applied to MRP/MRP II environments do not optimise production decisions. Although the MRP system minimises the inventory, it only plans orders when the stock balance is negative. For the purpose of obtaining optimum solutions in relation to minimising costs or maximising profits, Billington

Fuzzy Material Requirement Planning

41

et al. (1983), Escudero and Kamesam (1993), and Louli et al. (2008) studied MRP/MRP II modelling with mathematical programming models. It is necessary to distinguish between flexibility in constraints and goals, and uncertainty of the data or epistemic uncertainty. Flexibility is modelled by fuzzy sets and may reflect the fact that constraints or goals are linguistically formulated, and that their satisfaction is a matter of tolerance and degrees, or fuzziness (Bellman and Zadeh 1970). Epistemic uncertainty concerns the ill-known parameters modelled by fuzzy intervals in the possibility theory setting (Dubois and Prade 1988). This chapter aims to formulate a model for the mid-term production planning problem in a multi-product, multi-level and multi-period manufacturing environment with fuzziness in demand and capacity data. This work uses a fuzzy linear programming approach based on fuzzy constraints to generate an optimal crisp solution. In this context, the survey paper by Baykasoglu and Göcken (2008) shows some possibilities of how fuzziness can be accommodated within linear programming. The main goal is to determine the master production schedule, stock levels, delayed demand, and capacity usage levels over a given planning horizon in such a way as to hedge against the fuzziness of demand and capacity constraints. Therefore, we focus on demands and capacities, and it is clear that uncertainty can be the result of a certain imprecision in satisfying constraints. In Mula et al. (2006, 2007 a, b and 2008), other fuzzy mathematical models have been proposed for the MRP under fuzziness and/or with epistemic uncertainty or lack of knowledge. In this chapter, we first describe the different kinds of uncertainty which may affect constrained MRP systems. Then we propose new fuzzy mathematical programming models for production planning in MRP environments under uncertainty conditions. In the proposed models, customer demand and operational capacity parameters are modelled with the Fuzzy Sets Theory. Finally, the proposed models are summarised and compared by highlighting their main characteristics and those of an ideal application setting.

2 MRP under Fuzziness Whybark and Williams (1976) described two basic forms of uncertainty which can affect an article in an MRP system. The first implies changes during the demand of a particular article, while the second source of uncertainty lies in the scheduled receptions for a given article. They both refer to uncertainty in demand and supply, respectively. These types of uncertainty are also divided into uncertainty in amounts and uncertainty in time periods. Besides, if a constrained MRP system is considered in capacity terms, uncertainty may not only be present in the capacity required, but also in the capacity available. Given the time structure in which these uncertainties affect the system, these capacities may be classified as short- and long-term uncertainties (Subrahmanyam et al. 1994). Short-term uncertainties include variations in the day-to-day process, cancelled or urgent customer orders, breakdowns in work centres, etc. Long-term uncertainties refer to changes in demand, price/cost changes, changes in the longterm production ratio, etc.

42

J. Mula and R. Poler

One of the key uncertainty sources in any production system is end product demand (Gupta and Maranas 2003). If demand fluctuations are not taken into account, unsatisfied customer demand may take place which could lead to loss of market share or to excessive inventory maintenance costs (Perkov and Maranas 1997). A firm may adopt two strategic positions to face uncertainty in demand: (a) being a firm which restructures demand or (b) adapting to this restructuring (Gupta and Maranas 2003). The objective of the first strategy is to restructure the distribution of the demand which is achieved through contracts and agreements with the customer. For example, the firm could offer a supply contract and commit itself to a minimum/maximum amount in exchange for a price discount (Anupindi and Bassok 1999). With the adaptation strategy, the firm does not attempt to influence the level of uncertainty in the market, rather it controls the risk of its assets being exposed, for instance, inventory levels and profit margins, by constantly adapting its operations to meet demand. This research work considers an adaptation to the demand viewpoint. This chapter will deal with the problem of uncertainty in market demand and/or in operational parameters related with production capacity. The uncertainty present in the aforementioned problem will be formalised by modelling uncertainty by different approaches based on the Fuzzy Sets Theory. Uncertainty in either supply or delivery times will be approached in a similar fashion to that which Billington et al. considered (1983), who proposed the use of mathematical formulas to calculate the delivery times needed for resources with capacity constraints or which are constrained, and which leave the remaining delivery times with minimum values. Besides, constrained resources are programmed by considering all the production stages. In this way, maximum priority is given to those products in the production programme which use two or more bottleneck resources. In short, the problem is to simultaneously determine lot sizes, delivery times and capacity usage plans. Nonetheless, a future line of research could also focus on fuzzy sets modelling the uncertainty of delivery times.

3 Linear Programming Formulation This linear programming (LP) model, dubbed as MRPDet and originally proposed by Mula et al. (2006a), has been used as the basis of this work. It is a model for the optimisation of the production planning problem in a capacity-constrained, multi-product, multi-level and multi-period manufacturing environment. The LP model is formulated as follows. The decision variables and parameters of the model are defined in Table 3.1. Minimise z=

I

T

∑ ∑ (cp P i =1 t =1

i

it

R

T

+ cii INVTit + crd i Rd it ) + ∑∑ (ctoc rt Toc rt + ctex rt Tex rt )

(3.1)

r =1 t =1

Subject to I

INVTi ,t −1 + Pi ,t −TSi + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it , j =1

Fuzzy Material Requirement Planning

43

i = 1…I, t = 1…T I

∑ AR i =1

ir

Pit + Toc rt − Tex rt = CAPrt , r = 1…R, t = 1…T RdiT =0, i = 1…I

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0, i = 1…I, r = 1…R, t = 1…T

(3.2) (3.3) (3.4) (3.5)

Table 3.1 Decision variables and model parameters Sets T

Set of periods in the planning horizon (t = 1…T)

I

Set of products (i = 1…I)

J

Set of parent products in the bill of materials (j = 1…J)

R

Set of resources (r = 1…R)

Decision Variables

Data

Pit

Quantity to produce of the prod- dit uct i on period t

Market demand of the product i on period t

INVTit

Inventory of the product i at the αij end of period t

Required quantity of i to produce a unit of the product j

Rdit

Delayed demand of the product i at the end of period t

Tocrt

Undertime hours of the resource TSi r in period t

Lead time of the product I

Texrt

Overtime hours of the resource r INVTi0 in period t

Inventory of the product i in period 0

Objective Function Cost Coefficients

Rdi0

Delayed demand of the product i in period 0

cpi

Variable cost of production of a RPit unit of the product i

Programmed receptions of the product i in period t

cii

Inventory cost of a unit of the product i

crdi

Delayed demand cost of a unit of ARir the product i

Required time of the resource r for unit of production of the product i

ctocrt

Undertime hour cost of the resource r in period t

Available capacity of the resource r in period t

ctexrt

Overtime hour cost of the resource r in period t

Technological Coefficients

CAPrt

Equation (3.1) shows the total costs to be minimised: the costs of inventories cii, the costs of the extra time used by resources, ctexrt, and the costs of the lazy time of resources, ctocrt. MRPDet includes a plan to satisfy the delayed demands that are penalised with a cost, crdi. It is assumed that this cost relates linearly to the number of backlogs in each period.

44

J. Mula and R. Poler

The balance equations for the inventory are provided by the group of constraints (3.2). These equations take into account the demand backlogs which act as a negative inventory. It is important to highlight the consideration of parameter RPit that guarantees the continuity of the MRP throughout the successive requirement explosions carried out in a given planning horizon. Production in each period is limited by the availability of a group of shared resources. Equation (3.3) considers the capacity limits of these resources. Decision variables Tocrt and Texrt are not limited by any established parameter, but are penalised with the corresponding costs in the objective function. This provides the model with the maximum generality possible. The limitation of these variables for specific applications could be easily considered by contemplating whether these limits exceed the solution of the model since they may not be feasible. A constraint has also been added (3.4) to complete the delays in the last period (T) of the planning horizon. The model also contemplates non-negativity constraints (3.5) for the decision variables. Finally, decision variables Pit, INVTit and Rdit will be defined as continuous or integer variables depending on the manufacturing environment to which the model is applied.

4 Fuzzy Linear Programming Formulation 4.1 Model 1 This model aims to determine a precise minimisation decision in order to solve an MRP problem in which the market demand values considered are fuzzy. To determine a precise maximisation (or minimisation) problem, Werners (1987) proposed the following definition: In the objective function f: X Æ ⊕1, R̃ is a fuzzy region (or the solution space), while S(R̃) is the support for this region. The maximization set in the fuzzy region, MR̃(f), is defined by its membership function:

⎧0 ⎪ ⎪ f ( x) − inf f ⎪ S ( R) μ MR( f ) ( x) = ⎨ f − inf f ⎪ sup S (R) S ( R) ⎪ ⎪1 ⎩

if f ( x) ≤ inf f S ( R)

if inf f < f ( x) < sup f S (R)

(3.6)

S ( R)

if sup f ≤ f ( x) S ( R)

The intersection of this maximisation set with the fuzzy decision set may be used to calculate a maximisation decision, x0 , as the solution with the highest degree of membership in this fuzzy set. For Werners however (1987), the planner’s judgment being calibrated while seeking the smallest f value in the feasible region

Fuzzy Material Requirement Planning

45

did not seem reasonable. So a better number is the highest f value that can be obtained in a degree of membership of 1 in the feasible region. This leads to Werner’s definition (1987); see below: Given the objective function f: X Æ ⊕1, R̃ = the feasible fuzzy region, S(R̃) = support of R̃ and R1 = α-cut-off level of R̃ for α =1. The membership function of the objective function given the R̃ solution space is defined as:

⎧0 ⎪ ⎪ f ( x) − sup f ⎪ R1 μG ( x ) = ⎨ sup f − sup f ⎪ S (R) R1 ⎪ ⎪1 ⎩

if f ( x) ≤ sup f R1

if sup f < f ( x) < sup f

(3.7)

S ( R)

R1

if f ( x) ≤ sup f R1

Therefore the membership function corresponding to the functional space is:

⎧ sup μG ( x) ⎪ μG (r ) = ⎨ x∈ f −1 ( r ) ⎪⎩0

if r ∈ R, f −1 (r ) ≠ Ø (3.8)

otherwise

By considering the base MRPDet model, R1 is the region defined as: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it , (3.9) j =1

I

∑ AR

ir

Pit + Toc rt − Tex rt = CAPrt ,

(3.10)

i =1

RdiT =0

(3.11)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.12)

Given the fuzzy model with the non-fuzzy objective function and with imprecise inventory balance constraints (3.14) and (3.15): Minimise I

z=

T

R

T

∑∑ (cpi Pit + cii INVTit + crd i Rd it ) + ∑∑ (ctocr Tocrt + ctexrTexrt ) i =1 t =1

(3.13)

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it ∈ d it (3.14) j =1

46

J. Mula and R. Poler I

− INVTi ,t −1 − Pi ,t −TS − RPit + INVTi ,t + Rd i ,t −1 + ∑ α ij ( Pjt + RPjt ) − Rd it ∈- d it (3.15) j =1

I

∑ AR i =1

ir

Pit + Toc rt − Tex rt = CAPrt

(3.16)

RdiT =0

(3.17)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.18)

Considering the membership function:

⎧1 ⎪ ⎪ B x − di μi ( x) = ⎨1 − i pi ⎪ ⎪⎩1

if Bi x ≤ di if di < Bi x ≤ di + pi

i = 1,..., m + 1

(3.19)

if Bi x > di + pi

To obtain the equivalent deterministic model, first it is necessary to “fuzzify” or transform the objective function into a fuzzy function. This is achieved by obtaining the superior and inferior limits of the optimum values. These limits of the optimum values, f0 and f1, are obtained by resolving the following two linear programming models: FuzzyMRP.1a = MRPDet I

T

∑∑ ( cp P

Minimise z =

i it

i =1 t =1

R

T

+ cii INVTit + crdi Rdit ) + ∑∑ ( ctocrTocrt + ctexrTexrt )

(3.20)

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it

(3.21)

j =1

I

∑ AR

ir

Pit + Toc rt − Tex rt = CAPrt

(3.22)

i =1

RdiT =0

(3.23)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.24)

This model corresponds to the MRPDet deterministic model. Where supR1 f = (z)opt = f1; and FuzzyMRP.1b Minimise I

z=

T

∑∑ (cp P i =1 t =1

i

it

R

T

+ cii INVTit + crd i Rd it ) + ∑∑ (ctocr Toc rt + ctexr Tex rt ) (3.25) r =1 t =1

Fuzzy Material Requirement Planning

47

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it + p 2 it (3.26) j =1

I

∑ AR i =1

ir

Pit + Toc rt − Tex rt = CAPrt

(3.27)

RdiT =0

(3.28)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.29)

Where supS(R̃) f = (z)opt = f0. In this model in Equations (3.25) and (3.26), dit is an estimated value corresponding to the inferior limit of the tolerance interval for the product demand i in period t. On the other hand, p2it represents the maximum extension of dit in the demand tolerance interval. The objective function takes values between f1 and f0, whereas the value of the demand varies between dit and dit + p2it. The membership function of the resulting objective function is:

⎧1 ⎪ ⎪ z − f1 μG ( x ) = ⎨ ⎪ f 0 − f1 ⎪⎩0

if f 0 ≤ z if f1 < z < f 0

(3.30)

if z ≤ f1

Symmetry has been achieved between the constraints and the objective function, so the Zimmermann approach (1976) may be applied to develop the equivalent deterministic model. One advantage that the Zimmermann approach (1976) offers over other methods is that the different combinations of membership functions and aggregation operators become equivalent linear models. The formula of the equivalent deterministic model is as follows: Maximise λ

(3.31)

Subject to: λ(f0 –f1) + I

T

∑∑ (cp P i =1 t =1

i

it

R

T

+ cii INVTit + crd i Rd it ) + ∑∑ (ctoc r Tocrt + ctex r Texrt ) ≤ f0, (3.32) r =1 t =1

I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it + λp 2 it ≤ d it + p 2 it (3.33) j =1

I

− INVTi ,t −1 − Pi ,t −TS − RPit + INVTi ,t + Rdi ,t −1 + ∑ α ij ( Pjt + RPjt ) − Rd it + λ p 2it ≤ − d it + p 2it j =1

(3.34)

48

J. Mula and R. Poler I

∑ AR i =1

ir

Pit + Toc rt − Tex rt = CAPrt

(3.35)

RdiT =0

(3.36)

λ≤1

(3.37)

Pit, INVTit, Rdit, Tocrt, Texrt, λ ≥ 0

(3.38)

4.2 Model 2 This model attempts to formalise the uncertainty which refers to market demand, along with the technological coefficients of the capacity required with fuzzy numbers to determine a precise or non-fuzzy decision for the MRP problem under fuzziness. For this purpose, a deterministic objective function and constraints with righthand side fuzzy terms have been considered. Gasimov and Yenilmez (2002) used this same approach for two other types of problems: (a) linear programming problems with only fuzzy technological coefficients; and (b) linear programming problems in which both technological coefficients and the right-hand side terms are fuzzy numbers. Given the following MRP fuzzy model with a deterministic objective function, ~

and with imprecision in demand,

d it is a term which is independent of the inven~

tory balance constraint (3.40), and AR ir of the capacity constraints also available in the technological coefficients (3.41), then: Minimise: I

T

R

T

z = ∑∑ (cpi Pit + cii INVTit + crd i Rd it ) + ∑∑ (ctoc r Toc rt + ctex r Tex rt ) (3.39) i =1 t =1

r =1 t =1

Subject to: I

~

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it (3.40) j =1

I

~

∑ AR i =1

ir

Pit + Toc rt − Tex rt = CAPrt

(3.41)

RdiT =0

(3.42)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.43)

~

It is assumed that function (3.44).

d it and ÃRir are fuzzy numbers with the linear membership

Fuzzy Material Requirement Planning

⎧1 ⎪ ⎪ x − ARi μi ( x) = ⎨1 − pi ⎪ ⎪⎩0

49

if x ≤ ARi if ARi < x ≤ ARi + pi i = 1,..., m + 1 (3.44) if x > ARi + pi

To develop the equivalent deterministic model, first it is necessary to transform the objective function into a fuzzy function. The inferior and superior limits of the optimum values are obtained by resolving the following linear programming models: FuzzyMRP.2a Minimise I

z1 =

T

∑∑ (cp P i =1 t =1

i

it

R

T

+ cii INVTit + crd i Rd it ) + ∑∑ (ctoc r Toc rt + ctex r Tex rt ) (3.45) r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it (3.46) j =1

I

∑ ( AR

ir

i =1

+ p 4 ir ) Pit + Toc rt − Tex rt = CAPrt RdiT =0

(3.47) (3.48)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.49)

where supS(R̃) f = (z)opt = z1. FuzzyMRP.2b Minimise I

T

R

T

z2 = ∑∑ (cpi Pit + cii INVTit + crd i Rd it ) + ∑∑ (ctocr Toc rt + ctexr Texrt ) (3.50) i =1 t =1

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it + p 2 it (3.51) j =1

I

∑ AR

ir

Pit + Toc rt − Tex rt = CAPrt

(3.52)

i =1

RdiT =0

(3.53)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.54)

where supS(R̃) f = (z)opt = z2.

50

J. Mula and R. Poler

FuzzyMRP.2c Minimise I

T

R

T

z3= ∑∑ ( cpi Pit + cii INVTit + crdi Rdit ) + ∑∑ ( ctocr Tocrt + ctexr Texrt ) (3.55) i =1 t =1

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rdi ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rdit = dit + p 2it

(3.56)

j =1

I

∑ ( AR

+ p 4ir ) Pit + Tocrt − Texrt = CAPrt

ir

i =1

RdiT =0

(3.57) (3.58)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.59)

where supR1 f = (z)opt = z3; and FuzzyMRP.2d = MRPDet Minimise I

T

R

T

z4= ∑∑ ( cp P + ci INVT + crd Rd ) + ∑∑ ( ctoc Toc + ctex Tex ) i it i it i it r rt r rt i =1 t =1

(3.60)

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rdi ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rdit = dit

(3.61)

j =1

I

∑ AR i =1

P + Tocrt − Texrt = CAPrt

ir it

RdiT =0

(3.62) (3.63)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.64)

where supS(R̃) f = (z)opt = z4. If we consider f1 = min (z1, z2, z3, z4) and f0 = max (z1, z2, z3, z4), then the objective function takes values between f1 and f0, while the demand value varies between dit and dit + p2it, and the technological coefficients of the capacity required take values between ARir and ARir + p4it. As in the previous model, the fuzzy set with the optimum values of the objective or goal is (3.65):

⎧1 ⎪ ⎪ z − f1 μG ( x ) = ⎨ ⎪ f 0 − f1 ⎪⎩0

if f 0 ≤ z if f1 < z < f 0 if z ≤ f1

(3.65)

Fuzzy Material Requirement Planning

51

Once again, symmetry has been obtained between the constraints and the objective function, so the Zimmermann approach may be applied (1976). The formula is as follows: Maximise λ

(3.66)

Subject to: λ(f0 –f1) +

I

T

∑∑ ( cp P + ci INVT i it

i =1 t =1

i

it

+ crdi Rdit ) + ∑∑ ( ctocrTocrt + ctexrTexrt ) ≤ f0 (3.67) R

T

r =1 t =1

I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rdit + λ p 2it ≤ dit + p 2it

(3.68)

j =1 I

− INVTi ,t −1 − Pi ,t −TS − RPit + INVTi ,t + Rdi ,t −1 + ∑ α ij ( Pjt + RPjt ) − Rdit + λ p 2it ≤ − dit + p 2it

(3.69)

j =1

I

∑ ( AR i =1

ir

+ λp 4 ir ) Pit + Toc rt − Tex rt = CAPrt

(3.70)

RdiT =0

(3.71)

λ≤1

(3.72)

Pit, INVTit, Rdit, Tocrt, Texrt, λ ≥ 0

(3.73)

It is important to stress that Constraint (3.70) contains the product of variables

λ and Pit, so this is a non-linear programming model. Therefore, the solution of this model requires a specially adopted approach to resolve non-convex optimisation problems. The algorithms proposed by Sakawa and Yano (1985), and by Gasimov (2002), are suitable for solving this type of problems.

4.3 Model 3 For model 3, model 2 may be extended to incorporate the uncertainty referring to the independent term of the available capacity restriction. The model provides a precise or non-fuzzy decision. Given the MRP fuzzy model with a deterministic objective function and with imprecision in both the inventory balance constraint (3.75) and the technological coefficients, and also with the right-hand side term of the available capacity constraint (3.76), then: Minimise I

T

R

T

z = ∑∑ (cpi Pit + cii INVTit + crd i Rd it ) + ∑∑ (ctocr Toc rt + ctexr Tex rt ) i =1 t =1

(3.74)

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = j =1

~

d it (3.75)

52

J. Mula and R. Poler I

~

~

∑ AR i =1

ir

Pit + Toc rt − Tex rt = CAPrt RdiT =0

(3.77)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0 ~

It is assumed that

(3.76)

(3.78)

~

d it , ÃRir and CAPrt are fuzzy numbers with the linear mem-

bership function (3.79).

⎧1 ⎪ ⎪ x − ARi μi ( x) = ⎨1 − pi ⎪ ⎪⎩0

if x ≤ ARi if ARi < x ≤ ARi + pi i = 1,..., m + 1 (3.79) if x > ARi + pi

To develop the equivalent deterministic model, first it is necessary to transform the objective function into a fuzzy function. The inferior and superior limits of the optimum values are obtained by resolving the following linear programming models: FuzzyMRP.3a= FuzzyMRP.2a Minimise z1 =

I

T

∑∑ (cp P i =1 t =1

i

it

R

T

+ cii INVTit + crd i Rd it ) + ∑∑ (ctoc r Toc rt + ctex r Tex rt ) (3.80) r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it (3.81) j =1

I

∑ ( AR i =1

ir

+ p 4 ir ) Pit + Toc rt − Tex rt = CAPrt RdiT =0

(3.82) (3.83)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.84)

This model is the equivalent to that already formulated in the previous FuzzyMRP.2a section. Where supS(R̃) f = (z)opt = z1. FuzzyMRP.3b Minimise I

T

R

T

z2 = ∑∑ (cpi Pit + cii INVTit + crd i Rd it ) + ∑∑ (ctocr Toc rt + ctexr Tex rt ) (3.85) i =1 t =1

r =1 t =1

Fuzzy Material Requirement Planning

53

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it + p 2 it

(3.86)

j =1

I

∑ AR

ir

Pit + Toc rt − Tex rt = CAPrt + p3it

(3.87)

i =1

RdiT =0

(3.88)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.89)

Where supS(R̃) f = (z)opt = z2. FuzzyMRP.3c Minimise I

R

T

T

z3 = ∑∑ (cp P + ci INVT + crd Rd ) + ∑∑ (ctoc Toc + ctex Tex ) r rt r rt i it i it i it i =1 t =1

(3.90)

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it + p 2 it

(3.91)

j =1

I

∑ ( AR

ir

i =1

+ p 4ir ) Pit + Tocrt − Texrt = CAPrt + p3it

(3.92)

RdiT =0

(3.93)

Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0

(3.94)

Where supR1 f = (z)opt = z3; y FuzzyMRP.3d = MRPDet Minimise z4 =

I

T

∑∑ (cp P i =1 t =1

i

it

R

T

+ cii INVTit + crd i Rd it ) + ∑∑ (ctocr Toc rt + ctexr Texrt )

(3.95)

r =1 t =1

Subject to: I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it = d it

(3.96)

j =1

I

∑ AR

ir

Pit + Toc rt − Tex rt = CAPrt

(3.97)

i =1

RdiT =0 Pit, INVTit, Rdit, Tocrt, Texrt ≥ 0 Where supS(R̃) f = (z)opt = z4.

(3.98) (3.99)

54

J. Mula and R. Poler

If we consider f1 = min (z1, z2, z3, z4) and f0 = max (z1, z2, z3, z4), then the objective function takes values between f1 and f0, while the demand value varies between dit and dit + p2it; the technological coefficients of the required capacity take values between ARir and ARir + p4it, and the available capacity takes values between CAPrt and CAPrt + p3rt. As in previous models, the fuzzy set of optimum values of the goal is (3.100):

⎧1 ⎪ ⎪ z − f1 μG ( x ) = ⎨ ⎪ f 0 − f1 ⎪⎩0

if f 0 ≤ z if f1 < z < f 0

(3.100)

if z ≤ f1

Once again, symmetry between the constraints and the objective function has been achieved, so this same approach used may be applied to lead to the former model. Its formula is as follows: Maximise λ

(3.101)

Subject to: λ(f0 –f1) +

I

T

∑∑ (cp P i =1 t =1

i

it

+ cii INVTit + crd i Rd it ) + ∑∑ (ctocr Tocrt + ctexr Texrt ) ≤ f0 (3.102) R

T

r =1 t =1

I

INVTi ,t −1 + Pi ,t −TS + RPit − INVTi ,t − Rd i ,t −1 − ∑ α ij ( Pjt + RPjt ) + Rd it + λp 2 it ≤ d it + p 2 it (3.103) j =1 I

− INVTi ,t −1 − Pi ,t −TS − RPit + INVTi ,t + Rdi ,t −1 + ∑ α ij ( Pjt + RPjt ) − Rdit + λ p 2it ≤ − dit + p 2it

(3.104)

j =1

I

∑ ( AR i =1

ir

+ λp 4 ir ) Pit + Toc rt − Tex rt + λp3 rt ≤ CAPrt + p3 rt

(3.105)

I

− ∑ ( ARir + λp 4 ir ) Pit − Toc rt + Texrt + λp3 rt ≤ −CAPrt + p3 rt

(3.106)

RdiT =0

(3.107)

λ≤1

(3.108)

Pit, INVTit, Rdit, Tocrt, Texrt, λ ≥ 0

(3.109)

i =1

Like the previous model, this is a non-linear programming model. The algorithms proposed by Sakawa and Yano (1985), and by Gasimov (2002), are appropriate to solve this type of non-convex optimisation problems.

5 Summary of All the Models Certain fuzzy data required for the MRP process may be found depending on the manufacturing environment. This chapter considers three mathematical fuzzy

Fuzzy Material Requirement Planning

55

programming models for MRP under fuzziness which deal with different possible combinations of imprecise data, but always from a generic point of view that is not specific for any industrial sector. The main scientific characteristics of the proposed models are summarised in Table 3.2. Table 3.2 Scientific characteristics of fuzzy models. Section Model ObjectiveConstraints Data Membership Operator Solution function Fuzzy No Fuzzy Fuzzy co- function fuzzy constraints efficients 4.1

Model Non1 fuzzy

X

X

4.2

Model Non2 fuzzy

X

X

4.3

Model Non3 fuzzy

X

X

X

Linear

Min

Nonfuzzy

X

Linear

Min

Nonfuzzy

X

Linear

Min

Nonfuzzy

The characteristics that the planner requires to select the appropriate model, the uncertainty-modelled parameters and the data needed to represent them, are all shown in Table 3.3. Table 3.3 Characteristics of fuzzy models. Section

Model

Uncertainty

Formal representation

4.1

Model 1

Demand

[dit, dit + p2it]

4.2

Model 2

Demand

[dit, dit + p2it]

Required capacity

[ARir, ARir + p4ir]

4.3

Model 3

Demand

[dit, dit + p2it]

Required capacity

[ARir, ARir + p4ir]

Available capacity

[CAPrt, CAPrt + p3rt]

6 Conclusions This chapter has presented three fuzzy mathematical models for MRP under fuzziness. The MRPDdet linear programming model formulation, presented in Section 3, is a traditional production planning model with deterministic coefficients and well defined relations. Nonetheless, the rigid requirements included in this model do not precisely characterise real manufacturing environment situations where human criteria are inherent to the assessment of certain values or conditions. The fuzzy models proposed herein have attempted to “bend” this rigidity by making the uncertainty-based perception fuzzier by using different modelling approaches.

56

J. Mula and R. Poler

One outstanding development has been the objective function of the MRPDet model which minimises costs, whereas the proposed fuzzy models consider maximise the complete constraints requirements. So total satisfaction aggregates individual accomplishment measures associated with both the operational cost range and the variations of either the technological coefficients or the independent terms of the constraints. As noted throughout this chapter, one of the fundamental advantages of the Fuzzy Set Theory is its extreme generality which enables it to be applied in different manufacturing environments wherever there is a desire to apply it. It is also necessary to stress that the models considered present a non-fuzzy objective function which involves having to link a precise objective function with a fuzzy solution space. So it is necessary to develop specific, more complex models based on a basic “non-symmetrical” “fuzzy” decision model. Finally, the models proposed in this chapter may be used as constructive blocks in a decision-making help system for MRP under fuzziness.

Acknowledgement This work has been funded by the Spanish Ministry of Science and Technology project: ‘Simulation and evolutionary computation and fuzzy optimization models of transportation and production planning processes in a supply chain. Proposal of collaborative planning supported by multi-agent systems. Integration in a decision system. Applications’ (EVOLUTION) (Ref. DPI2007-65501).

References Al-Hakim, L.A., Jenney, B.W.: MRP, an adaptive approach. International Journal of Production Economics 25, 65–72 (1991) Anupindi, R., Bassok, Y.: Supply contracts with quantity commitments and stochastic demand. In: Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Dordrecht (1999) Baykasoglu, A., Göcken, T.: A review and classification of fuzzy mathematical programs. Journal of Intelligent & Fuzzy Systems 19, 205–229 (2008) Bellman, R., Zadeh, L.: Decision making in a fuzzy environment. Management Science 17(4), 141–164 (1970) Billington, P.J., McClain, J.O., Thomas, L.J.: Mathematical programming approaches to capacity constrained MRP systems: review, formulation and problem reduction. Management Sciences 29(10), 1126–1141 (1983) Dolgui, A., Prodhon, C.: Supply planning under uncertainties in MRP environments: a state of the art. Annual Reviews in Control 31(2), 269–279 (2007) Dubois, D., Prade, H.: Possibility theory. Plenum Press, New York (1988) Escudero, L.F., Kamesam, P.V.: MRP Modelling via scenarios. In: Ciriani, T.A., Leachman, R.C. (eds.) Optimization in Industry. John Wiley and Sons, Chichester (1993) Gasimov, R.N.: Augmented lagrangian duality and nondifferentiable optimization methods in nonconvex programming. Journal of Global Optimization 24(2), 187–203 (2002)

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Gasimov, R.N., Yenilmez, K.: Solving fuzzy linear programming problems with linear membership functions. Turkish Journal of Mathematics 26, 375–396 (2002) Gupta, A., Maranas, C.D.: Managing demand uncertainty in supply chain planning. Computers and Chemical Engineering 24, 2613–2621 (2003) Hendry, L.C., Kingsman, B.G.: Production planning systems and their applicability to make-to-order companies. European Journal of Operational Research 40(1), 1–15 (1989) Ho, C.: Evaluating the impact of operating environments on MRP system nervousness. International Journal of Production Research 27, 1115–1135 (1989) Hopp, W.J., Spearman, M.L.: Factory physics-foundations of manufacturing management. Irwin, McGraw-Hill, New York (2001) Jonsson, P., Mattson, S.A.: The selection and application of material planning methods. Production Planning and Control 13(5), 438–450 (2002) Koh, S.C.L., Saad, S.M., Jones, M.H.: Uncertainty under MRP-planned manufacture: review and categorization. International Journal of Production Research 40(10), 2399– 2421 (2002) Louly, M.A., Dolgui, A., Hnanien, F.: Optimal supply planning in MRP environments for assembly systems with random component procurement times. International Journal of Production Research 46(19), 5441–5467 (2008) Mula, J., Poler, R., García-Sabater, J.P.: Capacity and material requirement planning modelling by comparing deterministic and fuzzy models. International Journal of Production Research 46(20), 5589–5606 (2008) Mula, J., Poler, R., García-Sabater, J.P.: MRP with flexible constraints: A fuzzy mathematical programming approach. Fuzzy Sets and Systems 157(1), 74–97 (2006) Mula, J., Poler, R., García-Sabater, J.P., Lario, F.C.: Models for production planning under uncertainty: A review. International Journal of Production Economics 103(1), 271–285 (2006) Mula, J., Poler, R., Garcia-Sabater, J.P.: Material requirement planning with fuzzy constraints and fuzzy coefficients. Fuzzy Sets and Systems 158(7), 783–793 (2007) Mula, J., Poler, R., Garcia-Sabater, J.P.: Fuzzy production planning model for automobile seat assembling. In: Soft Methods for Integrated Uncertainty Modelling. Springer, Heidelberg (2006) Orlicky, J.: Material requirements planning. McGraw Hill, London (1975) Perkov, S.B., Maranas, C.D.: Multiperiod planning and scheduling of multiproduct batch plants under demand uncertainty. Industrial and Engineering Chemistry Research 36, 48–64 (1997) Sakawa, M., Yano, H.: Interactive decision making for multi-objective linear fractional programming problems with fuzzy parameters. Cybernetics Systems 16, 377–397 (1985) Subrahmayam, S., Pekny, J.F., Reklaitis, G.V.: Design of batch chemical plants under market uncertainty. Industrial and Engineering Chemistry Research 33, 26–88 (1994) Vollmann, T.E., Berry, W.L., Whybark, D.C., Jacobs, F.R.: Manufacturing planning and control for supply chain management. Irwin/McGraw-Hill, Boston (2005) Werners, B.: An interactive fuzzy programming system. Fuzzy Sets and Systems 23, 131– 147 (1987) Whybark, D.C., Williams, J.G.: Material requirements planning under uncertainty. Decision Science 7, 595–607 (1976) Zimmermann, H.J.: Description and optimization of fuzzy systems. International Journal General System 2, 209–215 (1976)

Chapter 4

Fuzziness in JIT and Lean Production Systems Mesut Yavuz*

Abstract. Just-in-time manufacturing has become increasingly popular in the last three decades. Today it reaches far beyond Toyota’s manufacturing plants and is adopted in various industries globally. Lean production, an umbrella term for justin-time, includes several tools that can benefit from fuzziness. In this chapter, we present Kanban card number calculation and two scheduling models. We also demonstrate the models through numerical examples where appropriate.

1 Introduction Toyota is widely credited with the creation of the just-in-time (JIT) manufacturing philosophy. In his seminal book, Monden defines JIT as “to produce the necessary units in the necessary quantities at the necessary time” (Monden 1998). Toyota’s success has spread the JIT system all around the World. Toyota’s production philosophy has given birth to Lean Production and even a broader philosophy of Lean Management. The principles of lean are widely implemented in a gamut of industries including services such as healthcare (see (Santos et al. 2006, Porche 2006) for examples). Elimination of waste is the very heart of lean production. Waste is defined as anything that does not add value. Producing defectives, overproducing, overprocessing, keeping inventories and transporting products unnecessarily are all examples of waste. While it makes perfect sense to eliminate all wastes in a production system, it is a tremendous challenge to be lean and offer a large variety of products. Lean production has led to the development of several different production philosophies including Kanban (Kumar and Panneerselvam 2007), CONWIP (Hopp and Spearman 2008, Framinan et al. 2003), the Theory of Constraints (Goldratt 1988, Rahman 1998, Mabin and Balderstone 2000), Agile Manufacturing (Gunasekaran 1999, Silveira et al. 2001) and Quick Response Manufacturing (Suri 1998). A common idea in these different manufacturing philosophies is to pull materials throughout the manufacturing system, not push them. The most widely used pull system is probably the Kanban system created at Toyota. Kanban is a card based pull system in which production authorizations are sent upstream via Kanban cards. Two alternative approaches, namely one-card and Mesut Yavuz Shenandoah University, Harry F. Byrd, Jr. School of Business, Winchester, Virginia, U.S.A. C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 59–75. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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two-card approaches are presented in Section 3. With a Kanban system one must determine the number of Kanban cards in a way that allows smooth flow of materials with minimum inventory levels. The lean, or JIT, approach has affected the scheduling theory, as well. In the classical scheduling theory one usually minimizes the makespan, flow time, idle time, tardiness or a combination of these objectives. With JIT manufacturing in mind, researchers and practitioners alike have recognized earliness as a type of inefficiency and developed scheduling models with earliness-tardiness objectives. In Sections 4 and 5 we present two JIT scheduling models under fuzziness. Ready times of products, processing times, demands and due dates all can be uncertain in real life. Therefore, there is a great potential to incorporate fuzziness in JIT systems. However, there is a shortage of comprehensive fuzzy tools in the area, thereby making it a candidate for future research. In this chapter we present selected topics in JIT – lean manufacturing under fuzziness and provide numerical examples where appropriate. The remainder of this chapter is organized as follows. In Section 2 we present the preliminary definitions of fuzzy numbers and operators. In Section 3 we discuss how to calculate the number of Kanban cards in two alternative systems and how to incorporate a safety factor into those calculations. Sections 4 and 5 are devoted to JIT scheduling under fuzziness. In Section 4, we present a scheduling model for one-of-a-kind production and in Section 5 we present a discrete scheduling model for repetitive products. Section 6 concludes the chapter by discussing its relevance and relationship to other chapters in the book.

2 Preliminaries In this chapter we will model fuzzy variables/parameters as trapezoidal fuzzy numbers unless specified otherwise. An example of a trapezoidal fuzzy number, ~ x = ( xa , xb , xc , xd ) , is depicted in Figure 4.1.

μx 1

0

xa

xb

xc

xd

x

Fig. 4.1 A trapezoidal fuzzy number

We note that all the fuzzy variables and parameters we use in this chapter are positive numbers by definition. Using the function principle of fuzzy arithmetic we define the following operations on two positive fuzzy numbers ~ x = ( xa , xb , xc , xd ) and ~ y = ( ya , yb , yc , yd ) , and a constant k ∈ R .

Fuzziness in JIT and Lean Production Systems

61

The multiplication of a fuzzy number with a constant:

if k ≥ 0 ⎧ (kx a , kxb , kxc , kx d ), k⊗~ x =⎨ ⎩(− kx d ,−kxc ,− kxb ,− kx a ), if k < 0 The addition of two fuzzy numbers:

~ x⊕~ y = ( x a + y a , xb + y b , xc + y c , x d + y d ) The multiplication of two fuzzy numbers:

~ x⊗~ y = ( x a y a , xb y b , x c y c , x d y d ) The subtraction of two fuzzy numbers:

~ x−~ y = ( x a − y d , xb − y c , x c − y b , x d − y a ) The division of two fuzzy numbers:

~ x/~ y = ( x a / y d , xb / y c , x c / y b , x d / y a ) To defuzzify a trapezoidal fuzzy number we define the following three approaches. Median:

x med =

x a + xb + xc + x d 4

Center of gravity (cog):

xcog =

∫ μ xd ∫μ d x

x

x

x

=

x a + xb + x c + x d x a xb − x c x d + 3 3( x d + xc − xb − x a )

Graded mean integration representation (gmir):

x gmir =

x a + 2 xb + 2 xc + x d 6

3 Determining the Number of Kanban Cards The success of JIT production systems is often credited to pull systems used on the shop floor. Pull systems are simpler than push systems in that they do not require one to plan the entire production system. However, there are still some critical questions that need to be answered in designing a pull system: What size should the buffers between two consecutive production stages be?

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When and how should authorization signals be sent for replenishing the buffer stocks? How should operators know what is needed downstream and what to work on next? The most commonly used pull system is the Kanban system. Kanban is basically a card containing all the information required for production/assembly of a product at each stage. Such information may include the part type, producer workstation/process, consumer workstation/process, container type, container size, etc. In JIT manufacturing systems products are manufactured in standard size batches. The ideal batch size is one, which is commonly idealized as one-pieceflow. Where the ideal one-piece-flow is not possible, a reasonably low batch size is selected for a product and containers of that size are procured for that product. A container typically has a pocket on the outside where a kanban card of that product can be inserted. A kanban card moves between two consecutive production stages, sometimes on a full container, sometimes on an empty container and in some cases separately from the container. The forward move of a kanban card symbolizes the completion of a production step on a product. The backward move of a kanban, on the other hand, symbolizes a job order passed upstream. A kanban system can be implemented with just one type of kanban cards, or with two different kinds. In the following paragraphs, we explain these two systems. One-Card System Suppose the two consecutive workstations, the producer and the consumer workstations, cannot be located right next to each other. In this case, the time it takes to transmit a replenishment order from consumer workstation to the producer workstation becomes important. Note that a workstation can produce for more than one downstream workstation and similarly it can withdraw products from more than one upstream workstation. When there are many workstations and they are distant from each other, it is necessary to create an inbound and an outbound buffer for each workstation. The kanban card used in a single card system is called a withdrawal Kanban (in the literature it is also called a conveyance Kanban or a move Kanban sometimes). A withdrawal kanban, called W-kanban hereafter, is an authorization to move a full container from the outbound buffer of the producing workstation, to the inbound buffer of the consuming workstation. Without a W-kanban, containers cannot be moved downstream. Notice that a W-kanban is not a production authorization, but just a move authorization. So, how should the upstream operation know what to produce and when to produce it? With this single-card system, workstations usually produce according to an order list or a daily schedule, which is possibly generated by an MRP system. However, strict production schedules for supplying processes are not favored in JIT manufacturing because they lack the flexibility needed in pull systems. Thus, the operator at the producing workstation has to pay attention to not overproduce a product. This is achieved by limiting the number of full containers of a product in the outbound buffer.

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The withdrawal of materials between workstations is usually carried out by material handlers. Another important element of the system is the kanban mailboxes, which are used for communication between operators and material handlers. After these definitions, we summarize the single-card system below. Step 1. When the operator of the consuming workstation accesses a full container in his inbound buffer, he separates the W-kanban from the container and places it in a kanban mailbox. Step 2. A material handler takes the W-kanban from the mailbox and takes it to the producing workstation of that product. At this time, if the container has been emptied at the consuming workstation, the material handler takes the W-kanban and the empty container to the producing workstation together. Step 3. The material handler attaches the W-kanban to a full container located at the outbound buffer of the producing station and moves the container to the inbound buffer of the consuming workstation. This simple system is easy to implement and effective. It limits the number of full containers that can accumulate at inbound buffers with the number of Wkanban cards for a product. The number of W-kanbans must be determined in such a way that minimizes the average inventory accumulation at the buffers and minimizes the possibility of starvation of consuming workstations. Here, we explain Toyota’s method (Nicholas 2008). Let D be the demand rate (typically daily demand) and S be the container size, i.e., the number of products in a standard container. Let us denote the withdrawal time with W. The withdrawal time is essentially the cycle time of a W-kanban, and it consists of four parts: time it waits in the mailbox (a), moves to the producing workstation (b), moves back to the consuming workstation (c) and waits in the downstream buffer until the container is accessed and the W-kanban is placed back in the mailbox (d). Here, W = a + b + c + d, and all times are in days (or, the length of the planning horizon used in the demand rate). The number of Wkanbans is then:

KW =

DW . S

The container size is standard and known a-priori and with certainty. The demand rate and withdrawal time, on the other hand, may not be known with certainty. Hence, we use them as fuzzy variables.

~ D = ( Da , Db , Dc , Dd ) be a trapezoidal fuzzy number denoting the de~ mand rate. Also let W = (Wa , Wb , Wc , Wd ) be a trapezoidal fuzzy number repLet

resenting the withdrawal time. The withdrawal time can be calculated by summing up the four times constituting a cycle of a W-kanban:

~

~ , b , c~ and where a Then, we have

~ ~ ~ W = a~ ⊕ b ⊕ c~ ⊕ d ,

~ d are all trapezoidal fuzzy numbers.

~ ~ ~ ~ ~ D ⊗ (a~ ⊕ b ⊕ c~ ⊕ d ) D ⊗W ~ ~ KW = or K W = . S S

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Note that the number of W-kanbans is a trapezoidal fuzzy number, too. At the end, we can de-fuzzify it using the graded mean integration representation. Example. Consider the following data for a single-card kanban system under fuzziness.

~ D = (220,240,260,280) , a~ = (0.05,0.08,0.12,0.15) , ~ b = (0.08,0.10,0.13,0.15) , c~ = (0.09,0.12,0.18,0.21) , ~ d = (0.06,0.08,0.12,0.14) , and S = 20 . ~ ~ ~ ~ ~ We first calculate W = a ⊕ b ⊕ c ⊕ d = (0.28,0.38,0.55,0.65) . Then,

~ ~ D ⊗ W (220,240,260,280) ⊗ (0.28,0.38,0.55,0.65) ~ KW = = S 20 (61.6,91.2,143.0,182.0) ~ KW = 20 ~ K W = (3.08,4.56,7.15,9.10) The mean graded integration representation is

K W , gmir ≅ 5.93 . We need to

round it up to 6 to get an integer number of W-kanbans. Two-Card System Recall that the W-kanbans discussed above authorize withdrawal of full containers, not production. The two-card system we present here builds on the single-card system and uses production Kanban cards to authorize production, in addition. A production kanban, called P-kanban hereafter, is an authorization to produce a full container of a product. Without a P-kanban, operators cannot start production of that product. With this two-card system, workstations only produce to fulfill the demand of the consuming workstations. Therefore, they are not given a daily schedule. Since the operators are not given daily schedules, they have to determine the next product to produce based on the P-kanbans at their workstation. Therefore, each workstation has a P-kanban mailbox. The two-card system is summarized below. Step 1. When the operator of the consuming workstation accesses a full container in his inbound buffer, he separates the W-kanban from the container and places it in a W-kanban mailbox. Step 2. A material handler takes the W-kanban from the mailbox and an empty container from the consuming workstation and takes them to the producing workstation of that product.

Fuzziness in JIT and Lean Production Systems

65

Step 3. The material handler removes the P-kanban from a full container at the outbound buffer of the producing workstation and puts it in the P-kanban mailbox of the workstation. He then attaches the W-kanban to the full container. Step 4. The material handler leaves the empty container at the outbound buffer of the producing station and moves the full container to the inbound buffer of the consuming workstation. Step 5. An operator at the producing workstation takes the P-kanban from the mailbox and attaches it to an empty container for that product, and starts the production. Step 6. The operator produces just enough to fill the empty container and places the now full container at the outbound buffer of his workstation. This system is easy to implement and effective, as well. It limits both the number of full containers that can accumulate at inbound buffers and the number of empty containers that can accumulate at outbound buffers. In order to assure that the system works, there must be one more container than the number of Kanban cards. The number of W-kanbans is determined in the same way as in the one-card system. In addition the number of P-kanbans must be determined. Here, we adopt Toyota’s approach. Let D be the demand rate (typically daily demand) and S be the container size, i.e., the number of products in a standard container. Let us denote the withdrawal time with W. The withdrawal time is essentially the cycle time of a W-kanban, and it consists of four parts: time it waits in the W-kanban mailbox (a), moves to the producing workstation (b), moves back to the consuming workstation (c) and waits in the downstream buffer until the container is accessed and the W-kanban is placed back in the mailbox (d). Here, W = a + b + c + d, and all times are in days (or, the length of the planning horizon used in the demand rate). The number of W-kanbans is then:

KW =

DW . S

In addition, let us denote the production time with P. The production time is essentially the cycle time of a P-kanban, and it consists of four parts: time it waits in the P-kanban mailbox (e), time to process a full batch of the product at the producing workstation (f), time to move a full container to the outbound buffer of the producing workstation (g) and wait time in the outbound buffer until the container is accessed and the P-kanban is placed back in the mailbox (h). Here, P = e + f + g + h, and all times are in days (or, the length of the planning horizon used in the demand rate). The number of P-kanbans is then:

KP =

DP . S

The total number of kanban cards is obtained by summing up the numbers of withdrawal and production kanbans:

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K = Kw + K P =

D(W + P) . S

The container size is standard and known a-priori and with certainty. The demand rate and the withdrawal and production times, on the other hand, may not be known with certainty. Hence, we use them as fuzzy variables.

~ D = ( Da , Db , Dc , Dd ) be a trapezoidal fuzzy number denoting the ~ ~ demand rate. Also let W = (Wa , Wb , Wc , Wd ) and P = ( Pa , Pb , Pc , Pd ) be Let

trapezoidal fuzzy numbers representing the withdrawal and production times, re-

~ ~ ~ ~ ~ ~ ~ W = a~ ⊕ b ⊕ c~ ⊕ d and P = ~ e ⊕ f ⊕ g~ ⊕ h , where a~ , b , ~ ~ ~ c~ , d , ~ e , f , g~ and h are all trapezoidal fuzzy numbers.

spectively.

Then, we have

~ ~ ~ ~ ~ D ⊗W D ⊗ (a~ ⊕ b ⊕ c~ ⊕ d ) ~ ~ KW = or K W = , and S S ~ ~ ~ ~ ~ D⊗P D ⊗ (~ e ⊕ f ⊕ g~ ⊕ h ) ~ ~ KP = or K P = . S S

Note that the numbers of kanbans are trapezoidal fuzzy numbers, too. At the end, we can de-fuzzify them using the graded mean integration representation. Example. Consider the following data for a two-card kanban system under fuzziness.

~ D = (220,240,260,280) , a~ = (0.05,0.08,0.12,0.15) , ~ b = (0.08,0.10,0.13,0.15) , c~ = (0.09,0.12,0.18,0.21) , ~ d = (0.06,0.08,0.12,0.14) , e~ = (0.02,0.03,0.05,0.06) , ~ f = (0.08,0.09,0.11,0.12) , ~ g = (0.01,0.02,0.03,0.04) , ~ h = (0.07,0.08,0.12,0.13) , and S = 20 .

K W , gmir ≅ 5.93 . Similarly, we ~ ~ ~ ~ ~ first calculate P = e ⊕ f ⊕ g ⊕ h = (0.18,0.22,0.31,0.35) . Then, For this example we have already calculated

Fuzziness in JIT and Lean Production Systems

67

~ ~ D ⊗ W (220,240,260,280) ⊗ (0.18,0.22,0.31,0.35) ~ KW = = S 20 (39.6,52.8,80.6,98.0) ~ KW = 20 ~ K W = (1.98,2.64,4.03,4.90) The mean graded integration representation is

K W , gmir ≅ 3.37 . We need to

round it up to 4 to get an integer number of P-kanbans. Altogether, we need 10 kanban cards and 11 containers of this product to run the two-card system. Safety Factor JIT manufacturing requires a stable demand, which is typically assured by production smoothing at the final stage of the manufacturing process. However, it is not always possible to perfectly smooth the demand of downstream operations, and also there are unpredictable disruptions in manufacturing such as machine breakdowns, etc. Hence, it is important to incorporate a safety factor into the calculation of the number of kanban cards. As we have presented above, kanban card numbers are typically rounded up. This rounding already introduces a safety factor into the system. Let us denote the number of kanban cards with K, lead time with L, safety factor with α and the container size with S. The relationship among them is:

K=

KS DL(1 + α ) , or α = − 1. DL S

Here, the lead time (L) refers to withdrawal time (W), production time (P), or the sum of the two (W + P). The number of kanban cards associated with these three cases is the number of W-kanbans (KW), the number of P-kanbans (KP) and the total number of kanban cards (K = KW + KP), respectively. Using the relationship formulated above we can (i) compute the number of cards for a given safety factor, and (ii) calculate the actual value of the safety factor for a given number of cards. When

~

~ ~ D and L are trapezoidal fuzzy numbers, the number of kanban cards

( K ) is also calculated as a trapezoidal fuzzy number and then defuzzified. Simi~ is calculated as a trapezoidal fuzzy number when we are larly, the safety factor α given the number of kanban cards and then defuzzified. The relationship among these fuzzy variables/parameters is:

~ ~ ~ D ⊗ L ⊗ (1 + α ) ~ = KS − 1 . K= , or α ~ ~ S D⊗L Example. Consider the following data for a two-card kanban system under fuzziness.

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~ D = (220,240,260,280) , a~ = (0.05,0.08,0.12,0.15) , ~ b = (0.08,0.10,0.13,0.15) , c~ = (0.09,0.12,0.18,0.21) , ~ d = (0.06,0.08,0.12,0.14) , e~ = (0.02,0.03,0.05,0.06) , ~ f = (0.08,0.09,0.11,0.12) , ~ g = (0.01,0.02,0.03,0.04) , ~ h = (0.07,0.08,0.12,0.13) , and S = 20 . For this example we have already calculated the following:

~ W = (0.28,0.38,0.55,0.65) , and ~ P = (0.18,0.22,0.31,0.35) .

Let us calculate the total number of kanban cards for a safety factor of α

= 0 .1 . ~ ~ ~ We first calculate L = W ⊕ P = (0.46,0.60,0.86,1.00) . We then calculate

~ K as follows:

~ ~ ~ D ⊗ L ⊗ (1 + α ) (220,240,260,280) ⊗ (0.46,0.60,0.86,1.00) ⊗ 1.1 K= = S 20 ~ (111.32,158.40,245.96,308.00) K= 20 ~ K = (5.57,7.92,12.30,15.40) The mean graded integration representation is

K gmir ≅ 10.24 . We need to

round it up to 11 to get an integer number of kanban cards. Let us now look back and calculate the safety for using 11 kanban cards:

KS ~ −1 D⊗L

α~ = ~

12 × 20 −1 (220,240,260,280) ⊗ (0.46,0.60,0.86,1.00) 240 −1 α~ = (101.2,144.0,223.6,280.0) α~ = (0.857,1.073,1.667,2.372) − 1 α~ = (−0.143,0.073,0.667,1.372)

α~ =

Fuzziness in JIT and Lean Production Systems

Finally, the mean graded mean integration representation of α is

69

α gmir ≅ 0.45 .

This result shows an actual safety factor value that is much larger than intended and it is typical due to the rounding up of the number of kanban cards. The number of Kanban cards in a system needs to be changed as input parameters such as demand and cycle times change. This calls for an adaptive or dynamic policy. Chang and Yih (1998) develop a fuzzy rule-based system for the dynamic control of Kanban card numbers. To achieve this, they first extract good examples from historical data and then the fuzzy system is trained to fit the selected data. For further reading on this approach, see (Chang and Yih 1998).

4 One-of-a-kind Production with Fuzzy due Dates JIT systems strive to produce products when they are needed and in the exact quantities needed. An important category of manufacturing systems is one-of-akind manufacturing where each product is designed and produced only once. The non-repetitive character of the demand for products makes scheduling of these systems almost identical to project scheduling. One-of-a-kind products are typically required for larger projects and they are needed at a critical time (due date) not to delay the succeeding steps of such master projects. At the same time, the one of a kind nature of the products bring uncertainties into the system. In this section, we focus on a one-of-a-kind production system with fuzzy due dates. Wang et al. (1999) define two types of fuzzy due dates. The first type is an interval fuzzy number that takes the value of 1 between the lower and upper limits of an interval, and 0 outside the interval. The second type is a trapezoidal fuzzy number given with four critical points, or two intervals. The outer interval defines the due date window of the product, whereas the inner interval defines the mostdesired completion time window for the product. Here, we focus on the trapezoidal approach since it allows a smoother handling of completion times. Suppose there are n products to be produced in a planning horizon of T units. Products are indexed with i = 1, 2, .., n. The planning horizon is divided into T equal-length periods, such as days, and indexed with t = 1, 2, .., T. One-of-a-kind production typically involves design and development of products preceding the production stages. This is incorporated into the model by assigning a release time to each product. Let ri be the release time of product i, i.e., production of i cannot start before ri. Manufacturing of product i takes pi periods. The manufacturing system consists of m stages, indexed by j = 1, 2, .., m. Processing of product i takes up ai,j,k units of capacity at manufacturing stage j, in its kth period of production for each i = 1, 2, .., n, j = 1, 2, .., m and k = 1, 2, .., pi. Total capacity of manufacturing stage k in period t is given with Bj,t for each j = 1, 2, .., m and t = 1, 2, .., T. Let us use the completion time of a product as the decision variable. We define

⎧1, if product i is completed in period t xi ,t = ⎨ otherwise ⎩0,

(4.1)

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as our binary decision variable, which is defined for each i = 1, 2, .., n and t = 1, 2, .., T. Let us also denote the completion time of product i with ci. Note that ci can T

be calculated easily with

ci = ∑ txi ,t for each i = 1, 2, .., n. For sake of simplict =1

ity we will mostly use ci in our explanations. Let

~ di = (di , a , di ,b , di ,c , di , d ) be a trapezoidal fuzzy number representing the

due date for product i. Interval

[d i ,b , di , c ] is the inner interval and it defines the

most desired completion time window for product i. Interval

[d i , a , di , d ] is the

outer interval and it is the allowed due date window for product i. If product i is completed inside the inner interval the customer is completely satisfied, i.e., the due date is met. If the completion time is outside the outer interval, then the customer is completely dissatisfied. In other cases, i.e., with completion time inside the outer interval but outside the inner one, the customer is partially satisfied. Let μi (ci ) denote the customer’s degree of satisfaction for product i. μi (ci ) is a trapezoidal fuzzy number, defined as follows.

0, ci ≤ d i , a or ci ≥ di , d ⎧ ⎪ ci − d i , a , di , a < ci < di ,b ⎪ ⎪ di ,b − d i , a μi (ci ) = ⎨ 1, d i ,b ≤ ci ≤ di , c ⎪ ⎪ d i , d − ci , d i ,c < ci < di , d ⎪d − d i d i c , , ⎩

(4.2)

The following optimization model is developed to maximize the minimum customer satisfaction in the system. Maximize

λ

(4.3)

Subject to:

μi (ci ) ≥ λ , i = 1,2,.., n T

∑ tx

(4.4)

i ,t

= ci , i = 1,2,.., n

(4.5)

∑x

= 1, i = 1,2,.., n

(4.6)

ci ≥ ri + pi − 1, i = 1,2,.., n

(4.7)

t =1

T

t =1

i ,t

Fuzziness in JIT and Lean Production Systems n t + p i −1

∑ ∑a i =1

k =t

x ≤ B j ,t , j = 1,2,.., m, t = 1,2,.., T

71

i , j , k − t +1 i , k

(4.8)

xi ,t ∈{0,1}, i = 1,2,.., n, t = 1,2,.., T

(4.9)

λ ∈ [0,1]

(4.10)

In the model, the objective function (4.3) aims to maximize the minimum level of satisfaction, which is defined in constraint (4.4). Constraint (4.5) ties the completion times to the binary decision variables xi,t. Constraint (4.6) guarantees that each product is assigned one and only one completion time in the planning horizon. Constraint (4.7) assures processing of a product does not start before its release time. Constraint (4.8) makes sure that capacities of production stages are not exceeded. Finally, constraints (4.9) and (4.10) define the decision variables xi,t and λ as binary and continuous variables, respectively. The model presented above is a mixed integer non-linear programming problem. (The non-linearity of the model arises from the definition of the satisfaction degree.) Wang et al. (1999) propose an algorithm to solve this model. Their algorithm is too detailed to be discussed in this chapter, and, hence, we refer the interested reader to Wang et al. (1999) for further reading.

5 Earliness-Tardiness Scheduling with Fuzzy due Dates The classical scheduling theory has primarily considered regular objective functions such as makespan, idle time, tardiness and number of tardy jobs. With these objective functions one implicitly aims at maximizing the efficiency of production resources, i.e., machines, but not the system as a whole. For example, tardiness is considered a type of inefficiency and is desired to be eliminated. However, completing a job earlier than it is needed is inefficient for a number of reasons, such as inventory holding, too. With the rise of JIT manufacturing, earliness has been considered in the objective functions of scheduling models. Earliness-tardiness scheduling models penalize both early and tardy completion of jobs. The coefficients of earliness and tardiness may be product dependent, and they may or may not be symmetric. Earliness-tardiness objectives are usually referred to as “JIT objectives.” However, a JIT scheduling problem does not have to use only an earliness-tardiness objective. In this section, we present a study from apparel manufacturing that uses the minimization of makespan as a second objective function. Wong et al. (2006) consider a real-life production scheduling problem from an apparel manufacturer. The problem arises in a fabric cutting department feeding sewing assembly lines downstream. In this department, fabric cutting jobs belonging to different production orders have to be processed on one of the parallel spreading tables so that demand from downstream sewing lines can be timely fulfilled.

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In this chapter, we focus on the definition of the objective function and use of fuzzy numbers therein. Let J be the number of jobs to be processed, and j be the job index, j = 1, 2, .., J. Let us denote the completion time of job j with cj(σ). The completion time of a job is a function of the job’s processing time as well as the entire schedule (σ). Jobs are received in batches, or orders. Let O be the number of orders received and o be the order index, o = 1, 2, .., O. Let xj,o be a binary variable denoting whether job j is part of order o or not:

⎧1, if job j is in order o x j ,o = ⎨ otherwise ⎩0, A fuzzy due date is given for each order. Let us denote the due date of order o with

~ do = (do , a , do ,b , d o, c , do , d ) . The degree of satisfaction of the completion

time of a job is obtained as follows:

0, c j (σ ) ≤ d o, a or c j (σ ) ≥ d o, d ⎧ ⎪ c (σ ) − d o, a ⎪ j , d o , a < c j (σ ) < d o ,b ~ ⎪ d o,b − do , a μ (c j (σ ), do ) = ⎨ 1, d o ,b ≤ c j (σ ) ≤ d o ,c ⎪ ⎪ d o, d − c j (σ ) d o, c < c j (σ ) < d o , d ⎪ d −d , o ,c ⎩ o,d The overall degree of satisfaction for a schedule is then computed by summing up degree of satisfaction of all jobs: J

O

~

μ JIT (σ ) = ∑∑ μ (c j (σ ), d o ) x j , o j =1 o =1

The second objective is the makespan (T), which is the maximum completion time of all jobs:

T (σ ) = max j {c j (σ )} The degree of satisfaction for an actual makespan is inversely proportional to the makespan. In other words, the shorter we complete all the jobs, the more satisfactory the schedule:

μT (σ ) =

Tt arg et T (σ )

The overall quality of the sequence is then evaluated by aggregating the two as follows:

Fuzziness in JIT and Lean Production Systems

73

μ (σ ) = wJIT μ JIT (σ ) + wT μT (σ ) Here, wJIT and wT are weights assigned to the JIT (earliness-tardiness) and the makespan objectives. The practitioner must be very careful when assigning values to these weights. Since the degree of satisfaction for the JIT objective is the summation of degrees of satisfaction of all jobs, it is a number in the interval [0, J]. The degree of satisfaction for the makespan objective, on the other hand, is a number in the open interval (0, +∞). However, if the target is set correctly, it is expected to be approximately 1. As a general rule, if it is not in the interval [0.5, 2.0], we recommend setting a more realistic target for the makespan. Then, the weights should be defined such that the JIT objective does not outweigh the makespan objective in the aggregated degree of satisfaction. The optimization model of the described problem is a complicated one and is solved using sophisticated algorithms. Again, since they are too detailed to be covered in this chapter, we refer the interested reader to Wong et al. (2006) for further reading. Fuzziness in earliness tardiness scheduling can be incorporated through fuzzy processing times, fuzzy demand, fuzzy due dates, etc. Also, manufacturing system of interest can be a single machine, a flow shop, a job shop or an assembly line environment or even a more complex system. In the literature, we see some recent works on earliness tardiness scheduling under fuzziness. We refer the reader to (Yi and Wang 2003) for an application of soft computing on parallel machine scheduling with batch setup times. For applications of earliness tardiness flow shop scheduling with fuzzy processing times, we refer the reader to (Wu and Gu 2004, Zheng and Gu 2004, Yu and Gu 2007).

6 Conclusions In this chapter we have discussed JIT and lean production under fuzziness and presented selected topics therein. We have presented two alternative approaches in implementing Kanban card systems and obtained the number of Kanban cards under fuzziness, as well as included a safety factor in the calculations. In the second half of the chapter we have focused on scheduling models. We have presented earliness tardiness scheduling with fuzzy due dates for (i) one-of-a-kind products and (ii) repetitive products in the apparel industry. Lean production includes many tools and techniques beyond JIT manufacturing. For example, reduction of setup times, total productive maintenance, statistical process control and workload balancing are all quantitative components of lean, and they all can benefit from fuzziness. We would like to identify one particular area for future research: production smoothing. Production smoothing is referred to as “a cornerstone of the Toyota Production System” by Monden (1998) and has enjoyed a good deal of interest from researchers. However, the vast literature of production smoothing (see Yavuz and Akcali (2007) for a recent survey), to the best of our knowledge, does not include any work under fuzziness. Expressing demand and processing times as fuzzy numbers can benefit many real life production systems.

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In this chapter, we have presented two earliness tardiness scheduling models. A broader review of the scheduling literature under fuzziness is presented in Chapter 8. An important tool in lean production is statistical process control, which is presented in Chapter 19.

References Chang, T.M., Yih, Y.: A fuzzy rule-based approach for dynamic control of kanbans in a generic kanban system. International Journal of Production Research 36(8), 2247–2257 (1998) Framinan, J.M., Gonzales, P.L., Ruiz-Usano, R.: The CONWIP production control system: review and research issues. Production Planning & Control 14(3), 255–265 (2003) Goldratt, E.M.: Computerized shop floor scheduling. International Journal of Production Research 26(3), 443–455 (1988) Gunasekaran, A.: Agile manufacturing: A framework for research and development. International Journal of Production Economics 62, 87–105 (1999) Hopp, W., Spearman, M.L.: Factory Physics. McGraw Hill Higher Education, New York (2008) Kumar, C.S., Panneerselvam, R.: Literature review of JIT-KANBAN system. Journal of Advanced Manufacturing Technology 32, 393–408 (2007) Mabin, V.J., Balderstone, S.J.: The World of the theory of constraints: A review of the international literature. The St. Lucie Press/Apics Series on Constraints Management (2000) Monden, Y.: Toyota production system: An integrated approach to Just-In-Time, 3rd edn. Engineering & Management Press (1998) Nicholas, J.: Competitive manufacturing management. Irwin/McGraw Hill Publishing (2008) Porche, R.A.: Doing more with less: lean thinking and patient safety in health care. Joint Commission Resources, Inc., USA (2006) ( Edited Volume) Rahman, S.U.: Theory of constraints A review of the philosophy and its applications. International Journal of Operations & Production Management 18(4), 336–355 (1998) Santos, J., Wysk, R.A., Torres, J.M.: Improving production with lean thinking. John Wilen & Sons, Inc., New Jersey (2006) Silveira, G.D., Borenstein, D., Fogliatto, F.S.: Mass customization: Literature review and research directions. International Journal of Production Economics 72, 1–13 (2001) Suri, R.: Quick response manufacturing: a companywide approach to reducing lead times. Productivity Press (1998) Wang, W., Wang, D., Ip, W.H.: JIT production planning approach with fuzzy due date for OKP manufacturing systems. International Journal of Production Economics 58, 209– 215 (1999) Wong, W.K., Kwong, C.K., Mok, P.Y., Ip, W.H.: Genetic optimization of JIT operation schedules for fabric-cutting process in apparel manufacture. Journal of Intelligent Manufacturing 17, 341–354 (2006) Wu, C., Gu, X.: A genetic algorithm for flow shop scheduling with fuzzy processing time and due date. In: Proceedings of the 5th World Congress on Intelligent Control, Hangzhou, P.R. China (2004)

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Yavuz, M., Akcali, E.: Production smoothing in just-in-time manufacturing systems: a review of the models and solution approaches. International Journal of Production Research 45(16), 3579–3597 (2007) Yi, Y., Wang, W.: Soft computing for scheduling with batch setup times and earlinesstardiness penalties on parallel machines. Journal of Intelligent Manufacturing 14, 311– 322 (2003) Yu, A., Gu, X.: Application of cultural algorithms to earliness/tardiness flow shop with uncertain processing time. In: Third International Conference on Natural Computation (2007) Zheng, L., Gu, X.: Fuzzy production scheduling in no-wait flow shop to minimize the makespan with E/T constraints using SA. In: Proceedings of the 5th World Congress on Intelligent Control, Hangzhou, P.R. China (2004)

Chapter 5

Fuzzy Lead Time Management Mesut Yavuz*

Abstract. Lead time management is at the very center of production management. In this chapter, we discuss the importance of lead times and effective management thereof. We also present selected work from the literature on lead time management such as lead time estimation, due date bargaining, scheduling and lot streaming.

1 Introduction Lead time is the duration between placing an order and receiving the ordered goods. From a manufacturer’s point of view it is the time between accepting an order and shipping the finished products. As is the case throughout the literature, we will use lead time and throughput time interchangeably in this chapter. In today’s fiercely competitive environment we often witness supply chain to supply chain competition. Managing long supply chains is a major challenge (Jain and Benyoucef 2008) and reducing the lead times in supply chains is a key in improving the performance (De Treville et al. 2004). The lead times need to be carefully planned and communicated (Selcuk et al. 2009). Determining production capacities and scheduling jobs in a manufacturing system are major components of managing lead times. However, lead time management is interrelated with many other areas, and even disciplines. For example, a recent line of research focuses on incorporating lead times in joint productionmarketing models (Upasani and Uzsoy 2008). Lead time management has enjoyed significant interest from academics and practitioners alike both in the traditional production scheduling literature and more modern manufacturing philosophies. Elimination of unnecessary steps in a manufacturing process and thereby reducing the lead time is crucial in lean production (see Santos et al. 2006, for example). Reducing lead times to improve customer satisfaction has even given birth to a novel manufacturing philosophy: Quick Response Manufacturing (Suri 1998). In this chapter we review selected important topics in lead time management. In fact, considering the basic relationship among lead time, inventories and the throughput (Hopp and Spearman 2008), it can be argued that lead time management is crucial in the whole of production engineering and management. Mesut Yavuz Shenandoah University, Harry F. Byrd, Jr. School of Business, Winchester, Virginia, U.S.A. C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 77–94. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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The remainder of this chapter is organized as follows. In Section 2 we present the preliminary definitions of fuzzy numbers and operators. In Section 3 we present lead time estimation and relate it to order fulfillment. In Section 4 we discuss due date bargaining. In Section 5 we present scheduling models with lead time objective function(s) and also discuss lot streaming. Section 8 concludes the chapter by discussing its relevance and relationship to other chapters in the book.

2 Preliminaries In this chapter we will model fuzzy variables/parameters as trapezoidal or triangular fuzzy numbers. We note that all the fuzzy variables and parameters we use in this chapter are positive numbers by definition. In Figure 5.1 we depict a trapezoidal number ~ x = ( xa , xb , xc , xd ) and a fuzzy number ~ y = ( ya , yb , yc ) .

μx 1

0

xa xb

xc

xd ya

yb

yc

x

Fig. 5.1 Trapezoidal and triangular fuzzy numbers

Using the function principle of fuzzy arithmetic we define the following operax = ( xa , xb , xc , xd ) and tions on two positive trapezoidal fuzzy numbers ~

~ y = ( ya , yb , yc , yd ) , and a constant k ∈ R . The multiplication of a fuzzy number with a constant:

if k ≥ 0 ⎧ (kxa , kxb , kxc , kxd ), k⊗~ x =⎨ ⎩(−kxd ,−kxc ,−kxb ,−kxa ), if k < 0 The addition of two fuzzy numbers:

~ x⊕~ y = ( x a + y a , xb + y b , x c + y c , x d + y d ) The multiplication of two fuzzy numbers:

~ x⊗~ y = ( x a y a , xb y b , x c y c , x d y d )

Fuzzy Lead Time Management

79

The subtraction of two fuzzy numbers:

~ x−~ y = ( x a − y d , xb − y c , x c − y b , x d − y a ) The division of two fuzzy numbers:

~ x/~ y = ( x a / y d , xb / y c , x c / y b , x d / y a ) To defuzzify a trapezoidal fuzzy number we define the center of gravity (cog) approach as follows. Center of gravity (cog):

x cog =

∫ μ xd ∫μ d x

x

x

x

=

x a + xb + x c + x d x a xb − x c x d + 3 3( x d + x c − xb − x a )

Again, using the function principle of fuzzy arithmetic we define the following operations on two positive triangular fuzzy numbers ~ x = ( xa , xb , xc ) and

~ y = ( ya , yb , yc ) , and a constant k ∈ R . The multiplication of a fuzzy number with a constant:

if k ≥ 0 ⎧ ( kxa , kxb , kxc ), k⊗~ x =⎨ ⎩( −kxc ,− kxb ,− kxa ), if k < 0 The addition of two fuzzy numbers:

~ x⊕~ y = ( xa + ya , xb + yb , xc + yc ) The multiplication of two fuzzy numbers:

~ x⊗~ y = ( xa ya , xb yb , xc yc ) The subtraction of two fuzzy numbers:

~ x−~ y = ( xa − yc , xb − yb , xc − ya ) The division of two fuzzy numbers:

~ x/~ y = ( xa / yc , xb / yb , xc / ya ) To defuzzify a trapezoidal fuzzy number we define the center of gravity (cog) approach as follows. Center of gravity (cog):

xcog =

∫ μ xd ∫μ d x

x

x

x

=

xa + xb + xc 3

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M. Yavuz

3 Throughput Time and Order Fulfillment In this section we will present two models that consider order fulfillment a project, and, hence, compute lead time as project completion time. The former model is concerned with assembly lines whereas the latter is a more generalized model that considers a supply chain with multiple stages and multiple companies at each stage. Li et al. (2005) study an electronics manufacturing system consisting of surface mount technology pick and place machines, printed circuit board assembly, final assembly and packing lines. They first optimize the facility layout and then calculate the lead time using fuzzy logic. In this section, we present a generalized version of Li et al.’s model. Consider an activity-on-arc network with a start node S and a finish node F. Let n be the number of activities, indexed by i = 1, 2, .., n. Activity duration of activity i is given as a triangular fuzzy number

~ di = (di , a , di ,b , di ,c ) . For sake of clarity let us define ~si = ( si , a , si ,b , si , c ) and ~ fi = ( fi , a , fi ,b , fi ,c ) as the start and finish time of activity i, respectively. Note ~ that both ~ si and fi are triangular fuzzy numbers and the finish time is obtained ~ ~ si ⊕ di ) . by simply adding the activity duration to the start time ( f i = ~ Let us index the nodes of the network with j. Also let A j be the set of arcs pointing to node j. Clearly,

AS = {} .

~

The cumulative processing time up to node j ( t j ) is the fuzzy maximum of completion times of activities (arcs) pointing at j:

~ ~ ~ ~ ~ t j = max i '∈ A j {di '} = (max i '∈ A j {di ', a }, max i '∈ A j {di ',b }, max i '∈ A j {d i ',c })

.

~ ~ With this approach, the throughput time ( T = tF ) can be calculated via a single forward pass over the network. Example. Consider a manufacturing system consisting of five processes (activities). For a given product, the activity times and predecessor relationships are given in Table 5.1. Table 5.1 Data for the example

~

Activity (i)

Duration ( d i )

Predecessors

A

(1.25, 1.40, 1.50)

-

B

(1.18, 1.45, 1.65)

-

C

(3.25, 3.69, 4.02)

A, B

D

(2.55, 2.75, 2.95)

C

E

(2.80, 3.05, 3.33)

D

Fuzzy Lead Time Management

81

Fig. 5.2 Network representation of the example Table 5.2 Summary of calculations

~

Node (j)

Time ( t j )

S

(0.00, 0.00, 0.00)

1

(1.25, 1.45, 1.65)

2

(4.50, 5.14, 5.67)

3

(7.05, 7.89, 8.62)

F

(9.85, 10.94, 11.95)

The network representation of this production system is given in Figure 5.2. The calculations are summarized in Table 5.2.

~

As seen from the table, the throughput time is T = (9.85, 10.94, 11.95) . Using the center of gravity method, we calculate

~ Tcog ≅ 10.91 as the mean through-

put time. Chen and Huang (2006) consider a supply chain setting with multiple stages and multiple production nodes at each stage. They use an activity-on-node representation and calculate the lead time as project completion time using Fuzzy PERT. In the network representation of a supply chain, an arc is drawn from a predecessor node to a successor node. In order to complete the network an artificial start and an artificial finish node are created. Let S be the start node preceding all activities with no predecessor. Similarly, let F be the end node succeeding all activities with no successor. Then, draw an arc from S to each activity with no predecessor as well as an arc from each activity with no successor to F. Let us denote the predecessors of i with Ai .

(

For

each activity i =

)

1,

2,

..,

(

n; we

define

activity duration

)

~ di = (di , a , di ,b , di , c ) , earliest start time ~ s ie = ( s ei ,a , s ei ,b , s ei ,c ) , earliest finish ~e e e e time f i = ( f i ,a , f i ,b , f i ,c ) , latest start time ~ s il = ( s li ,a , s li ,b , s li ,c ) and latest ~l l l l finish time f i = ( f i ,a , f i ,b , f i ,c ) , all as triangular fuzzy numbers.

(

(

)

)

(

)

The lead time is found via a forward pass. During the forward pass, the earliest start time of activity i is calculated as the fuzzy maximum of the earliest finish

82

M. Yavuz

(e~

{ })

= max j∈Ai e~jf , and the earliest finish time of ac~ f = e~ s ⊕ d~ . tivity i is obtained by adding its earliest start time and duration e i i i ~ At the end of the forward pass, we obtain the throughput time ( T ), which is ~ the earliest finish time of the final activity ( T = ~ e Ff ). In order to find the critical

times of its predecessors

s

i

(

)

links in the supply chain we need to perform a reverse (backward) pass. The reverse pass starts with setting the latest finish time of the final activity equal to the

~f

~ =T = ~ e Ff ). During the reverse pass, the latest finish time of

throughput time ( l F

activity i is calculated as the fuzzy minimum of the latest start times of its succes-

(~

{ })

~ = min j |i∈ A j l j s , and the latest start time of activity i is obtained by ~s ~ f ~ subtracting its duration from its latest finish time li = li − di .

sors li

f

(

)

When the reverse pass is completed, we can calculate each activity’s slack (float) by subtracting the earliest start time from the earliest finish time of the activity

(~s = ~l i

i

s

)

− e~i s . Slack times may not be equal to 0 as in the crisp version of

PERT. Instead, we need to defuzzify slacks first. The critical activities in Fuzzy PERT are the activities with a defuzzified slack of 0. The throughput time we obtain is a triangular fuzzy number Given a triangular fuzzy due date

~ T = (Ta , Tb , Tc ) .

~ D = ( Da , Db , Dc ) , Chen and Huang (2006)

propose a method to calculate order fulfillment degree (ρ). Let us define earliness as

~ ~ e~ = D − T = (ea , eb , ec ) = ( Da − Tc , Db − Tb , Dc − Ta ) , and membership

degree of a given number x to this earliness set as follows:

x ≤ ea ⎛ 0, ⎜ x−e a ⎜ , ea < x < eb ⎜ eb − ea x = eb μe ( x) = ⎜ 1 ⎜ e −x ⎜ c , eb < x < ec ⎜ ec − eb ⎜ 0, x ≥ ec ⎝ Then, the order fulfillment degree is defined as follows:

1, ea ≥ 0 ⎛ ⎜ + δ , ea < 0 < ec ρ =⎜ − ⎜δ +δ + ⎜ 0, ec ≤ 0 ⎝

Fuzzy Lead Time Management

Here,

83 0

δ− =

∫

μe ( x)dx

and

δ+ =

x =+∞

∫

x =−∞

μe ( x)dx .

0

Example. Consider a supply chain consisting of three suppliers, three manufacturers, three distributors and a retailer. For a given product, the activity (processing, waiting, shipping, etc.) times and predecessor relationships are given in Table 5.3. Table 5.3 Data for the example

~

Activity (i)

Duration ( d i )

Predecessors

Supplier 1 (S1)

(0,0,0)

-

Supplier 2 (S2)

(3,4,5)

-

Supplier 3 (S3)

(3,4,6)

-

Manufacturer 1 (M1)

(4,6,7)

S1, S2

Manufacturer 2 (M2)

(5,7,8)

S1, S2, S3

Manufacturer 3 (M3)

(4,6,8)

S2, S3

Distributor 1 (D1)

(3,4,6)

M1, M2

Distributor 2 (D2)

(5,7,8)

M1, M2, M3

Distributor 3 (D3)

(5,6,7)

M2, M3

Retailer (R)

(6,7,8)

D1, D2, D3

The network representation of this production system is given in Figure 5.3.

Fig. 5.3 Network representation of the example

Note that in the network given above R id the final activity with no successors, hence, an artificial finish node is not added to the network. The calculations are summarized in Table 5.4.

~

As seen from the table, the throughput time is T = (18, 24, 29) . Using the center of gravity method, we calculate time.

~ Tcog ≅ 23.67 as the mean throughput

84

M. Yavuz

Table 5.4 Summary of calculations

~e fi

~ s il

~l fi

(i)

~ s ie

S1

(0,0,0)

(3,4,5)

(-9,1,12)

(-4,5,15)

(-9,1,12)

S2

(0,0,0)

(3,4,6)

(-10,1,12)

(-4,5,15)

(-10,1,12)

S3

(0,0,0)

(4,6,7)

(-11,0,11)

(-4,6,15)

(-11,0,11)

M1

(3,4,6)

(8,11,14)

(-4,5,15)

(4,12,20)

(-10,1,12)

M2

(4,6,7)

(8,12,15)

(-4,6,15)

(4,12,19)

(-11,0,11)

M3

(4,6,7)

(7,10,13)

(-2,8,16)

(4,12,19)

(-9,2,12)

D1

(8,12,15)

(13,19,23)

(4,12,20)

(12,19,25)

(-11,0,12)

D2

(8,12,15)

(13,18,22)

(5,13,20)

(12,19,25)

(-10,1,12)

~ si

D3

(8,12,15)

(14,19,23)

(4,12,19)

(12,19,25)

(-11,0,11)

R

(14,19,23)

(18,24,29)

(12,19,25)

(18,24,29)

(-11,0,11)

From the slack times, we see that S3, M2, D3 and R have zero defuzzified slack, and, hence, are critical. Suppose the fuzzy due date is

~ D = (25,28,30) . Therefore, the earliness is

~ ~ e~ = D − T = (−4,4,12) . In this case, we first need to calculate

δ− =

0

∫ μ ( x)dx = 1 , and e

x = −∞

δ+ =

x = +∞

∫ μ ( x)dx = 7 . e

0

Consequently, ρ = δ /(δ + δ ) = 7 / 8 . In this section we have presented two approaches based on fuzzy network analysis to calculate throughput time in a production system. In the literature there are a few more complex approaches. We refer the interested reader to Manns and Tonshoff (2005) for neuro-fuzzy approximation of throughput time, and to Alex (2007) for a novel approach in fuzzy point estimation and its application on estimating several important supply chain measures including the throughput time. +

−

+

4 Due Date Bargaining Many manufacturing companies constantly battle to meet all the due dates required by their multiple customers. More often than not, fulfilling all orders on time turns into firefighting. It is therefore desirable to be able to optimize due dates at the time of accepting production orders. When the optimum due date for the manufacturer is unacceptable for a customer, the two then can negotiate. The

Fuzzy Lead Time Management

85

compromise solution may require the customer to pay more and the manufacturer to incur delays (and possibly costs) with other customers’ orders. Wang et al. (1998) address the problem of optimally determining due dates for production job orders based on fuzzy due dates provided by the customers and fuzzy capacities available in the production system for a planning horizon. Their model is simple yet effective, and it allows a negotiation between the customers and the manufacturer. Consider a sufficiently long planning horizon consisting of T unit-length periods t = 1, 2, .., T. Let there be n orders received by the manufacturer, which uses a make-to-order approach and does not hold any finished product inventory. For order i = 1, 2, .., n, let wi be the weight (usually a function of the order’s monetary value) associated with the order and di be the due date desired by the customer. Suppose the manufacturing system consists of m critical resources, i.e., machines, production lines, etc. In period t, available regular capacity of resource j is

B j ,t units. However, if necessary, the capacity can be increased to B'j ,t through overtime. Since overtime is more costly than regular labor, it is not desired. The degree of satisfaction from assigning a workload of x units to resource j in period t,

μB ( x) , is modeled as a fuzzy set as depicted in Figure 5.4 and stated in the j ,t

following equation.

μB ( x) j ,t

1

0

x

B j ,t

'

B j ,t

Fig. 5.4 Degree of satisfaction from workload

⎧ 1, 0 ≤ x ≤ B j ,t ⎪ ' ⎪ B −x μ B j ,t ( x) = ⎨ ' j ,t , B j ,t < x ≤ B 'j ,t ⎪ B j ,t − B j ,t ⎪ 0, B 'j ,t < x ⎩ Actual workload of a resource in a period is a function of completion times assigned to the orders. Let pi , j , t (ci ) be the processing requirement of order i on resource j in period t, given that the completion time or order i is ci. Then, the total

86

M. Yavuz

workload

(processing

requirement)

of

resource

j

in

period

t

is

n

Pj ,t (c) = ∑ pi , j ,t (ci ) , where c represents the entire set of completion times. i =1

The manufacturer determines an optimum due date for each order based on the customers’ desired due dates (d) and the available production capacities. The following optimization model is entirely crisp and finds the manufacturer’s desired '

'

due dates ( d ). Let di be the decision variable representing the manufacturer’s desired due date for order i. n

Minimize

∑w

i

i =1

di' − d i

(5.1)

Subject to:

Pj ,t (d ' ) ≤ B j ,t , j = 1, 2,.., m, t = 1, 2,.., T

(5.2)

d i' ≥ 0, i = 1, 2,.., n

(5.3)

The objective function (5.1) minimizes weighted sum of absolute deviations from customers’ desired due dates. Note that the manufacturer’s desired due dates can be before or after those of the customers’, and, hence, the absolute values are taken. Constraint (5.2) makes sure that regular capacities are not exceeded. Con'*

straint (5.3) defines the decision variables as non-negative variables. Let d i be '

the optimal value of decision variable di . When an optimal solution is obtained to the above model, the manufacturer faces another optimization problem. That is, the problem of simultaneously maximizing the satisfaction degrees of resource workloads and closeness of assigned due dates to the ones desired by the customers. We have already defined the satisfaction degree of resource workloads above. The degree of satisfaction from assigning a deadline of x to order i,

μd (x) , is i

modeled via three fuzzy sets corresponding to three cases as depicted in Figure 5.5 and stated in the following equations. Case 1: d i > d i '*

⎧ di'* − x , di ≤ x ≤ di'* ⎪ '* μ d i ( x) = ⎨ d i − di ⎪ 0, x < di or di'* < x ⎩ Case 2: d i < d i '*

⎧ x − di'* , di'* ≤ x ≤ di ⎪ '* μd i ( x) = ⎨ di − di ⎪ 0, x < di'* or di < x ⎩

Fuzzy Lead Time Management

87

Case 3: d i = d i '*

⎧1,

x = di'* = di

⎩0,

otherwise

μd ( x) = ⎨ i

μd (x) i

1

x

0

d i'*

di

μd (x)

(a)

i

1

x

0

d i'*

di

μd (x)

(b)

i

1

0

x

d i = d i'*

(c) Fig. 5.5 Degree of satisfaction from due date assignments

The following optimization model is based on the satisfaction degrees defined above: Maximize

λ

(5.4)

Subject to:

μ d ( d i ) ≥ λ , i = 1, 2,.., n i

(5.5)

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μ B ( Pj ,t (d )) ≥ λ , j = 1, 2,.., m, t = 1, 2,.., T j ,t

di ≥ 0, i = 1, 2,.., n

(5.6) (5.7)

λ≥0

(5.8)

The objective function (5.4) aims to maximize the minimum satisfaction degree of all due date assignments and resource workloads as assured in constraints (5.5) and (5.6), respectively. Constraints (5.7) and (5.8) define the decision variables as non-negative variables. The crisp optimization model presented earlier finds the ideal due dates from the manufacturer’s perspective, with paying attention to not exceeding regular resource capacities. Clearly, the optimal solution obtained by this model is an extreme solution that minimizes the manufacturer’s cost by not allowing overtime work on any manufacturing resource in any period. Although it aims to minimize an aggregate function of deviations from customer’s desired due dates, those deviations may be significantly large. The fuzzy optimization model just presented aims to find a compromise solution by considering using overtime capacity on the manufacturing resources and bridging the gap between the manufacturer’s and the customers’ desired due dates. Let the compromise due date for order i found after *

solving the fuzzy optimization model be di . *

If the customer owning order i agrees to the due date di it can be considered optimal and fixed. However, it is only natural to believe that the customer will want to negotiate. Wang et al. (1998) present a third optimization model to adjust the due dates of all jobs to accommodate one customer’s request. We do not go into the details of that model here, but refer to Wang et al.’s work (1998). Also, for a multi-customer version of the model, see Wang et al. (1999). We would like to note that re-optimizing all the due dates in bargaining with one or more customers may change the due dates of some orders so much that their owners may want to bargain their due dates, too. In that case the optimization model needs to be updated and solved in an iterative fashion with some new customers added to the bargaining process in each iteration.

5 Scheduling with Lead Time Objectives In macro level production planning problems such as aggregate planning or master production scheduling, one typically assumes that lead time of a product is constant. In other words, it is known with certainty how long it takes for a product to go through the production system. With that approach, it is also implicitly assumed that production capacities are infinite, or workloads are so small that it does not have an impact on lead time. However, the workload of a production resource at a certain time determines whether a new job can be processed at that resource or not, and if it can how long it has to wait to be processed. Therefore, the actual schedule in a production

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system is an important determinant of lead time. In this section, we discuss scheduling with lead time objectives. One of the most common objectives in production scheduling is to minimize the makespan, i.e., the completion time of all jobs on hand. From a project scheduling perspective, makespan is equivalent to project completion time. If all the jobs are related to each other in a special way, such as being orders from the same customer, then the makespan gives us the lead time. Completion time and flow time are the closest scheduling objectives to lead time. Flow time of a job is the difference between the job’s release and completion times. If a job is released at time zero, i.e., available at the beginning of the scheduling horizon, then completion and flow times of that job are identical. An objective function can be obtained from either completion or flow times by aggregating over jobs using total, average, weighted total, weighted average or maximum. In presence of due dates, lead time should be interpreted in relation to the due date of a job, i.e., earliness or tardiness of a job is measured. Again, an objective function aggregates measured values for different jobs by the total, average, weighted total, weighted average or maximum operator. The scheduling theory has been one of the most studied areas of operations research since 1950s. For a more detailed discussion of scheduling objectives as well as manufacturing environments and other considerations we refer the reader to Pinedo (2008). Many scheduling problems consider more than one objective function at a time. For such problems, see T’kindt and Billaut (2006). Many parameters of scheduling models are uncertain in real life. Processing times, setup times, costs, release times, due dates can be all uncertain in a particular production system. Consequently, production scheduling under fuzziness has received a significant amount of interest in the literature (Dubois et al. 2003, Zimmermann 2006). We also refer to Chapter 8 of this book for more reading on fuzzy production scheduling. In this chapter, we focus on fuzzy scheduling models with lead-time related objectives. Sakawa and Kubota (2000) are concerned with a job shop scheduling problem with fuzzy processing times and fuzzy due dates. In their work the fuzzy processing times are triangular fuzzy numbers, and, hence, job completion times are also triangular fuzzy numbers. Let

~ C j = (C j , a , C j ,b , C j , c ) be the fuzzy completion

time of job j, j = 1, 2, .., n. One of the objective functions the authors consider is maximum completion time,

~ max j {C j } , which is also known as makespan as

discussed above. In (Sakawa and Kubota 2000) fuzzy due date of job j is defined with two parameters d j and

d 'j , which can be interpreted as the ideal completion time and

latest acceptable completion time of job j, respectively. If job j is completed before d j , it completely satisfies the due date; the satisfaction degree linearly decreases between d j and after

d 'j .

d 'j ; and the job violates the due date if it is completed

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Two of the three objective functions Sakawa and Kubota consider in their study are based on an agreement index, which is defined as the ratio of the area under the intersection of the fuzzy due date and the fuzzy completion time of a job to the area under its fuzzy completion time. Agreement index of job j, AIj, is demonstrated in Figure 5.6.

1

0

d j C j ,a d 'j C j ,b

C j ,c

x

Fig. 5.6 Fuzzy due date and fuzzy completion time

In the figure, AIj is not directly depicted, but it is the ratio of the area of the small triangle to the area of the big triangle. Sakawa and Kubota (2000) use maximization of average agreement index and maximization of the minimum agreement index as objective functions in addition to minimization of the maximum completion time. The authors develop a genetic algorithm to solve this multiobjective scheduling problem. An important aspect of lead time management is determining the lot sizes and scheduling those lots simultaneously. The idea is to split the order of a product into smaller pieces so that it runs more smoothly through the production system. In the lean production literature this phenomenon is commonly discussed through differentiating between production and transfer lot sizes. In the classical scheduling literature, this is commonly studied under the names lot splitting or lot streaming. For a recent and comprehensive review of lot streaming, we refer the reader to Chang and Chiu (2005). A recent line of research has addressed the lot streaming problem under fuzziness (see Petrovic et al. 2008, and the references therein). The orders are first split into smaller lots, which are called jobs. Then, the jobs are scheduled in a job shop environment. Petrovic et al. (2008) develop a fuzzy rule-based system for fuzzy lot sizing. Their system takes four fuzzy input parameters, namely, order size, static slack of the order, workload on the shop floor and priority of the order, for each order. The output is jointly determined by averaging the individual outputs of possibly multiple active rules. Order size is expressed in linguistic terms through three fuzzy sets: small, medium and large. Similarly, three fuzzy sets are defined for slack of the order: small, medium and large. For the workload only two fuzzy sets are defined: not large and large. For the priority or orders, three priority classes are defined: high, medium and low, or 1, 2 and 3, respectively.

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For order j, a due date d j is given in the problem. Then, depending on the priority class of the order, a latest acceptable completion time

d 'j is defined. For

high priority orders, it is identical to the given due date of the order. In other words, the manufacturer has no flexibility with those orders:

d 'j = d j . Degree of

satisfaction from completing a high priority job, therefore, is either 0 or 1. For the medium and low priority orders, latest acceptable completion times are defined with

d 'j = d j + 2 and d 'j = d j + 7 (in days), respectively. The degree of satis-

faction from an order’s completion time is defined in the same way as in (Sakawa and Kubota 2000) as described above. Once the lot sizes are computed, a fuzzy genetic algorithm is run to schedule the jobs. The model has five objective functions: average weighted tardiness, number of tardy jobs, total setup time, total idle time of machines and total flow time of jobs. Tardiness of a job is captured through the agreement index discussed above. Job j is considered to be on time if AIj=1, i.e., the completion time is completely inside the due date in Figure 5.6. It is considered tardy if AIj 0 , is a finite set of places pictured by circles 2. T = {t1 , t 2 ,..., t n }, s > 0 , is a finite set of transitions pictured by bars, with P ∪T ≠ ∅ and P ∩T = ∅ 3. I : P × T → N , is an input function that defines the set of directed arcs from P to T where N = {0,1,2,...} 4. O : T × P → N , is an output function that defines the set of directed arcs from T to P 5. m : P → N , is a marking whose ith component represents the number of tokens in the ith place. An initial marking is denoted by m0 . The tokens are pictured by dots. A marked PN and its elements are shown in Figure 6.1. The four-tuple ( P, T , I , O ) is called a PN structure that defines a directed graph structure. A PN models system dynamics using tokens and their firing rules. Introducing tokens into places and their flow through transitions make it possible to describe and study the discrete-event dynamic behavior of the PN.

Fig. 6.1 A marked PN and its elements

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Table 6.2 Interpretations of the PN elements (Zhou and Jeng 1998) PN Elements

Interpretation

Places

Resource status, operations and conditions

Transitions

Operations, processes, activities and events

Directed Arcs

Material, resource, information, and control flow direction

Table 6.3 Manufacturing concepts representation in PN models (Zhou and Jeng 1998) Manufacturing Concepts PN Modeling Moving or production lot Weight of directed arcs modeling moving size Number of resources, e.g. The number of tokens in places modeling quantity of the AGVs, machines. corresponding resources Capacity of a workstation The number of tokens in places modeling of its availability Work-in-process

The number of tokens in places modeling the buffers and operations of all machines

Production volume

The number of the tokens in places modeling the counter for or the number of firings of transitions modeling end of a product

Time of an operation, e.g. Time delays associated with the place or transitions modelsetup, processing. ing the operation Conveyance or transpor- Time delays associated with the directed arc, place, or trantation time sition modeling the conveyance or transportation System state

PN marking (plus the timing information for timed PN)

The interpretations of the PN elements are given in Table 6.2. Table 6.3 presents how the PN models can represent the concepts in manufacturing. The behavior of a system is described in terms of the system states and their changes. The system state in a PN model is defined as a marking. Everytime a transition fires a new marking occurs which defines the new system state. A state or a marking in a PN is changed according to the execution rules which include enabling and firing rules. • Enabling Rule: a transition t ∈ T is enabled if and only if m( p ) ≥ I ( p, t ) , ∀p ∈ P • Firing Rule: enabled in a marking m, t fires and results in a new marking m ,

which is described as m , ( p) = m( p) − I ( p, t ) + O( p, t ), ∀p ∈ P .

Marking m , is said to be immediately reachable from marking m. The enabling rule says that if all the input places of transition t have at least the required number of tokens, then t is enabled. The firing rule states that an enabled transition t removes w( p, t ) tokens from each input place p of t, and adds w(t , p) tokens to

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Fig. 6.2 (a)

C. Kahraman and F. Tüysüz

t1

is enabled (b) enabled transition

t1

fires

each output place p of t, where w(t , p) is the weight of the arc from t to p. The execution rule is depicted in Figure 6.2.

3.2 Properties of PNs PNs have some important fundamental properties which allows to identify the presence or absence of the functional properties of the modeled system. They can be classified into two groups, behavioral properties and structural properties. Behavioral properties are the ones which depend on the initial marking, whereas structural properties are the ones which do not depend on the initial marking of a PN. The most important properties are explained below. • Reachability: given a PN Z = ( P, T , I , O, m0 ) , marking m is reachable from marking m0 if there exists a sequence of transition firings that changes m0 to m. Reachability which is a behavioral property is a fundamental basis for analyzing the dynamic properties of any system. In terms of manufacturing systems, this property helps us to analyze whether the system can reach a specific state or perform a functional behavior. • Boundedness: given a PN Z = ( P, T , I , O, m0 ) and its reachability set R, a place p ∈ P is k-bounded if m( p) ≤ B, ∀m ∈ R . In other words, a PN is said to be kbounded if the number of the tokens in each place does not exceed a finite number k for any marking reacheable from m0. A PN is said to be safe if it is 1bounded. Places are usually used to represent storage areas, tools, pallets and AGVs in manufacturing systems. The concept of boundedness of a PN is used to identify the existence of overflows for these resources. • Liveness: a transition t is live if at any marking m ∈ R , there exists a sequence of transitions of which firing reaches a marking enabling t. A PN is live if all its transitions are live. A transition t is dead if at any marking, such that there exists no sequence of transition firings to enable t starting from m. Liveness is an important property for many practical systems since the concept of liveness is related to the absence of deadlocks. The deadlocks occur as a result of inappropriate resource utilization. In manufacturing systems, these resources can be

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machines, robots, AGVs, storage areas or buffers. Therefore, if a PN is live, then there is no deadlock related to such resources. • Conservativeness: a PN Z = ( P, T , I , O, m0 ) is conservative with respect to a vector w if there is a vector w = ( w1 , w2 ,..., wn ) and wi > 0, i = 1,2,..., n , such that wτ m = wτ m0 , ∀m ∈ R where R is the reachability set. Conservativeness states that the weighted sum of tokens remains the same for each marking. In real life systems, the number or amount of resources are limited. Since tokens are usually used to represent resources which are constant, then the number of the tokens in a PN model remains unchanged without regard to the marking change. • Reversibility and Home state: a PN is said to be reversible if for each marking m ∈ R(m0 ) , m0 is reachable from m. Therefore, in a reversible PN it is always possible to return back to the initial marking or state. This property is important in error recovery in manufacturing systems. If a net contains a deadlock, then the net is not reversible. The PN properties and their meanings in manufacturing systems are summarized in Table 6.4. Table 6.4 PN properties and their meanings in manufacturing systems (Zhou and Venkatesh 1999) PA Properties

Meanings in Manufacturing Systems

Reachability

A certain state can be reached from the initial conditions

Boundedness

No capacity such as buffer, storage and workstation overflow

Safeness

Availability of single resource or no request to start an ongoing process

Liveness

Freedom from deadlock and guarantee the possibility of a modeled event, operation, process or activity to be ongoing

Conservativeness

Conservation of nonconsumable resources such as machines, equipments and AGVs

Reversibility

Reinitialization and cyclic behavior

3.3 Analysis of PNs There are two main analysis methods for PNs which are reachability analysis and invariant analysis. The first method is the fundamental analysis method for PNs and can be applied to all classes of PNs. Although invariant analysis method is powerful, it does not include all the information of a general PN. It can be applied only to special subclasses of PNs or special situations, in this study invariant analysis method is omitted and the reachability analysis method is handled. Further details about invariant method can be found in Peterson (1981), Murata (1989). In reachability analysis method, starting from the initial state of the system, it is aimed at deriving all the possible states that the system can reach together with their relationships. The obtained reachability graph enable us to analyze all the

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Fig. 6.3 A PN model and its reachability graph

behavioral properties if the number of the states or markings is finite. In reachability graph the nodes represent markings obtained from initial marking m0 and its successors, and each arc represents a transition firing which changes one marking to another. Figure 6.3 shows a finite state PN model and its reachability graph which represents all the possible markings enumerated. It can be easily seen that the system represented by this PN model is bounded, live and reversible.

4 Modeling of Manufacturing Systems Using PNs Modern manufacturing systems are complex and automation intense systems which include many elements and relations among them. Modeling such systems requires taking into accounts not only the elements but also their relations. The basic relations between the processes and operations can be classified as follows (Zhou and Robbi 1994): • Sequential: when one operation follows another, the places and transitions representing them form a sequential relation in PNs. • Concurrent: when two or more operations are initiated by an event, they form a parallel structure starting with the same transition. Such a situation can be modeled as sequentially connected series of places/transitions in which multiple places are marked simultaneously or multiple transitions are enabled at certain markings • Conflicting: if either of two operations follows an operation, then the transitions form two outputs from the same place.

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• Cyclic: when a sequence of operations follows one after another and completion of the last one launches the first one, then the cyclic structure between these operations is formed. • Mutually Exclusive: mutual exclusion between two processes occurs if they cannot be performed at the same time because of the limited usage of a shared resource. These are the most commonly encountered relations in manufacturing systems. The representation of these basic relations in PN models are given in Figure 6.4.

Fig. 6.4 PN representation examples of basic relations: (a) Sequential (b) Concurrent (c) Conflicting (d) Cyclic (e) Mutually Exclusive

In modeling flexible manufacturing systems by using PNs, some commonly used PN modules can be very helpful in modeling process. These modules can be summarized as follows (Zhou and Jeng 1998): 1. Resource/operation module: it defines a single operation stage that requires a dedicated resource 2. Periodically-maintained resource/operation module: it is the extension of the resource/operation module that requires periodical maintenance. 3. Fault-prone resource/operation module: resource/operation module can be extended to a fault-prone resource by adding a loop to it. 4. Priority module: it describes different routes each of which requires a dedicated resource. 5. Rework module: when a process fails, a part may be required to rework. This module can be modeled by adding a new transition to the resource/operation module which which reroutes the part back.

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Fig. 6.5 Some commonly used PN modules in flexible manufacturing (Zhou and Venkatesh 1999): (a) Resource/operation, (b) Periodically-maintained resource/operation, (c) Faultprone resource/operation, (d) Priority, (e) Rework

Figure 6.5 shows these commonly used PN models for flexible manufacturing systems.

5 Stochastic PNs Although the concept of time was not included in the original work by C. A. Petri (1962), for many practical applications, the addition of time is a necessity. Without an explicit notion of time, it is not possible to conduct temporal performance analysis, i.e. to determine production rate, resource utilization. In modeling time critical systems with PNs such as a FMS, timing and activity durations for analyzing temporal performance and dynamics of the system should be taken into consideration. Merlin (1974) and Ramchandani (1973) were the first who independently extended PNs to include time delays in different ways. Curently the two proposed approaches which are named as firing durations and enabling durations are the basis of

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representing time in PNs. In timed PNs, although time delays can be associated to any transitions, places or arcs, it is often associated to transitions. The reason for this is that transitions represent events in a model and it is more natural to consider events to take time rather than time to be related to conditions, that is, places (Murata 1989, Zhou and Venkatesh 1999, Bowden 2000, Gharbi ve Ioualalen 2002). For detailed information and comparison of approaches associating time to PNs, see Bowden (2000). The most popular extension of PNs which are widely used in the application field of manufacturing systems is stochastic PNs. A stochastic PN (SPN) is a Petri net where each transition is associated with an exponentially distributed random variable that expresses the delay from the enabling to the firing of the transition. Due to the memoryless property of the exponential distribution of firing delays, Molloy (1982) showed that the reachability graph of a bounded SPN is isomorphic to a finite Markov chain. Queueing networks and Markov chains provide flexible, powerful and easy to use tools for modeling and analysis of complex manufacturing systems and are widely used (Al-Jaar and Descrochers 1990). In SPN models, we can explicitly describe the causal relation of uncertain events by using places, transitions, and arcs. Therefore, using SPNs, we can construct the model of a FMS more easily than using the other models. SPNs combine the modeling power of PNs and the analytical tractability of Markov processes for the purpose of performance analysis (Molloy 1982). An ordinary continuous-time stochastic PN is a PN with a set of positive, finite and exponentially distributed firing rates Λ = (λ1 ,..., λm ) , possibly marking dependent, associated with all its transitions. An enabled transition can fire after an exponentially distributed time delay with parameter 1 / λ elapses. Assume that every transition in a PN is associated with an exponentially distributed random delay from the enabling to the firing of the transition. Then the firing time of each transition can be characterized as a firing rate. A stochastic PN Z = ( P, T , I , O, m0 , Λ ) is a six-tuple where 1. P = {p1 , p 2 ,..., pn }, n > 0 and is a finite set of places 2. T = {t1 , t 2 ,..., t s }, s > 0 and is a finite set of transitions with P ∪T ≠ ∅ and P ∩T = ∅ 3. I : P × T → N and is an input function that defines the set of directed arcs from P to T where N = {0,1,2,...} 4. O : T × P → N and is an output function that defines the set of directed arcs from T to P 5. mi : P → N and is a marking whose ith component represents the number of tokens in the ith place. An initial marking is denoted by m0 . 6. Λ : T → R + and is a firing function whose ith component represents the firing rate of the ith transition where λi denotes the firing rate of ti and R + is the set of all positive real numbers.

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In a SPN, when a transition is enabled at marking m, the tokens remain in its input places during the firing time delay. At the end of the firing time, tokens are removed from its input places and deposited in its output places. The number of tokens in the flow depends on the input and output functions. Live and bounded SPNs are isomorphic to continuous-time Markov chains due to the memoryless property of exponential distribution (Molloy 1982). This important property allows for the analysis of SPNs and the derivation of the some important performance measures. The states of the Markov chain are the markings in the reachabillity graph, and the state transition rates are the exponential firing rates of the transitions in the SPN. By solving a system of linear equations representing the Markov chain, performance measures can be computed. The performance analysis of a SPN can be summarized as follows (Murata 1989, Marsan et al. 1995, Zhou ve Venkatesh 1999): After generating the reachability graph R(m0 ) , the Markov process is obtained by assigning each arc with the rate of the corresponding transition. The steadystate probability distribution Π = (π 0 ,π 1,..., π q ) of a SPN is obtained by solving the following linear system q −1

ΠA = 0 ,

∑π

i

=1

(6.1)

i =0

( )q×q

where A = aij

is the transition rate matrix. For i = 0,1,..., q − 1 , A’s ith row

elements, i.e. aij , j = 0,1,..., q − 1 are determined as follows: 1. if j ≠ i , aij is the sum of all outgoing arcs from state mi to m j q −1

2. Since any row elements in A satisfies

∑

q −1

aij = 0 , then aii = −

j =0

∑a

ij

, where

j ≠i

aii represents the sum of firing rates of transitions enabled at mi , i.e. transition rates leaving state mi .

From the steady-state distribution Π and transition firing rates Λ , the required performance indices of the system modeled by the SPN can be obtained. For example,

• The probability of a particular condition: Let B the subset of R(m0 ) satisfying a particular condition. Then the required probability is P( B) =

∑π

i

. For ex-

mi ∈B

ample, if the particular condition is that a machine produces a product, then P(B) is the machine utilization. • The expected vaalue of the number of tokens: let Bij be the subset of R(m0 ) such that at each marking in Bij, the number of tokens in place pi is j. Then the expected value of the number of tokens in a k-bounded place pi is

Manufacturing System Modeling Using Petri Nets

E [m( pi )] =

109

k

∑ jP( B ) . For example, if p represents the availabity of one kind ij

i

j =1

of product, then E [m( pi )] is the average inventory of the product. • The mean number of firings in unit time: let Bj be the subset of R(m0 ) in which a given transition tj is enabled. Then the mean number of firings of tj in unit time is f ij = π i λ ji , where λ ji is the firing rate of tj at marking mi.

∑

mi ∈B j

• The mean system throughput or production rate: assume that a product is produced each time a transition in a subset T’ fires. Then the average throughput rate is g = fj .

∑

t j ∈T '

The limitation of the SPN is that the number of states of the associated Markov chain grows very fast as the complexity of the SPN model increases (Marsan et al. 1984, Marsan et al. 1995). Marsan et al. (1984) introduced the generalized SPNs to reduce the complexity of solving a SPN model in which the number of reachable markings is smaller than that in a topologically identical SPN. A generalized SPN is basically a SPN with transitions that are either timed (to describe the execution of time consuming activities) or immediate (to describe some logical behavior of the model). Timed transitions behave as in SPNs, whereas the immediate transitions have an infinite firing rate and fire in zero time. Although generalized SPNs provide a smaller reachability set and reduce the complexity of analyzing SPNs, they both give similar results. Also, the development of software packages which are currently available both for academic and commercial purposes makes the analysis of SPN models quite easier. More information and different modeling examples of both SPNs and generalized SPNs can be found in Marsan et al. (1995).

6 Fuzzy PNs and a New Fuzzy PN Approach Although both PN theory and fuzzy set theory were introduced in 1960’s, the researches for the use of these together began much later, in early 1990’s. Petri (1987) presented some criticism related to timed and stochastic PNs about the conceptualization of time and chance. In his latter study Petri (1996) presented many axioms, among which the axioms of measurement and control related to time and nets, and emphasized mainly on uncertainty. These studies turned the attention on fuzzy set theory and fuzzy logic (Zadeh 1965, Zadeh 1973) which have been applied successfully in modeling and designing many real world systems in environments of uncertainty and imprecision. There are several approaches that combine fuzzy sets and PN theories, differing not only in the fuzzy tools used but also in the elements of the nets that are fuzzified. A PN structure is a four tuple consisting of places, transitions, tokens and arcs, and theoretically each of these can be fuzzified (Srinivasan and Gracanin

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1993). Today there are some fuzzy PN approaches proposed by some reasearchers which are differently named, but there is not a commonly agreed standard approach. The most common point of these approaches is that they usually focus on the fuzzification of time in PNs. Here we give a brief overview of the important fuzzy PN approaches. Valette et. al. (1989) were the first who presented a brief description of fuzzy time PNs and oulined a procedure for computing fuzzy markings and fuzzy firing dates, in context of imprecise markings for modeling real-time control and monitoring manufacturing systems. This approach is an extension of the crisp time interval of Merlin (1974)’s time PN to a fuzzy time interval since to each transition a fuzzy date or interval is assigned. Another important fuzzy PN approach was introduced by Murata (1996) and called as fuzzy timing high level PNs. This approach defines four fuzzy set theoretic functions of time called fuzzy timestamp, fuzzy enabling time, fuzzy occurence time and fuzzy delay. These fuzzy time functions are defined by triangular or trapezoidal possibility distributions. Although these approaches mainly use simple fuzzy operations, the computation procedure still can be time consuming and sometimes impractical in case of large complex systems. And also as mentioned before there is not still an agreed fuzzy approach for PNs and these approaches need further modifications and developments. For example, Wu (1999) claims that, from algebraic point of view, the fuzzy occurence time of Murata (1996)’s approach is not closed which implies that the fuzzy timestamps of the produced tokens no longer have the initial forms (i.e. the trapezoidal or triangularvpossibility distributions). It seems that there is still a long way for the development and applications of PN theory with fuzzy logic. For some other approaches and applications of fuzzy set theory to PNs, the interested reader can see the researches of Yeung et.al. (1996), Wu (1999), Pedrycz and Camargo (2003), Ding et. al. (2005) and Ding et. al. (2006). Fuzzy PNs and stochastic PNs have been separately used in modeling and analysis of discrete-event dynamic systems, such as FMSs. Analysis and design of such complex systems often involve two kinds of uncertainty: randomness and fuzziness. Randomness models stochastic variability which refers to describing the behavior of the parameters by using probability distribution functions. Fuzziness models measurement imprecision due to linguistic structure or incomplete information. Although the dominating concept to describe uncertainty in modeling is stochastic models which are based on probability, probabilistic models are not suitable to describe all kinds of uncertainty, but only randomness. Especially the imprecision of data which is for example as a result of the limited precision of measuring is not statistical in nature and can not be described by using probability (Viertl and Hareter 2004). To be able to better represent uncertainty, both stochastic or probabilistic variability and imprecision, we present an approach which we name as stochastic PNs with fuzzy parameters for modeling FMSs. This approach is based on the stochastic modeling and analysis of PNs by using fuzzy parameters. We believe that this new approach makes significant contribution to both stochastic PN and fuzzy PN literature.

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The memoryless property of exponential distribution of firing delays is very important since live and bounded SPNs are isomorphic to continuous-time Markov chains due to the memoryless property of exponential distribution (Molloy 1982). In our approach, we describe the exponentially distributed transition firing rates as triangular fuzzy numbers. Since our aim is to take into consideration both randomness and fuzziness, the memoryless property for fuzzy exponential function must be satisfied in order to perform the stochastic analysis of fuzzy parameters. The exponential E (λ ) has density

⎧λ e − λ x , x ≥ 0 f ( x; λ ) = ⎨ ⎩0, otherwise

(6.2)

The mean and variance of E (λ ) is 1 / λ and 1 / λ2 , respectively. The probability statement of the memoryless property of crisp exponential is

[

]

P X ≥ t + τ X ≥ t = P[ X ≥ τ ]

(6.3)

~ ~ If we substitute λ for λ in Eq. (6.2) we obtain the fuzzy exponential, E (λ ) . If μ~ denotes the mean, we find its α -cuts as ∞

~

∫

μ~ = { xλe − λx dx λ ∈ λ (α )}

(6.4)

0

for all α . However, each integral in the above equation equals 1 / λ . Hence ~ μ~ = 1 / λ . If σ~ 2 is the fuzzy variance, then we write down an equation to find its ~ α -cuts and we obtain σ~ 2 = 1 / λ 2 . It can be seen that the fuzzy mean (variance) is the fuzzification of the crisp mean (variance). The conditional probability of fuzzy event A given a fuzzy event B is defined by Zadeh (1968) as ~ ~ P ( A ⋅ B) ~ (6.5) P ( A B) = ~ , P ( B) > 0 P ( B) By using the fuzzy exponential α -cuts of the fuzzy conditional probability, relating to the left side of Equation (3) ∞

[

]

∫τλe

P X ≥ t + τ X ≥ t (α ) = { t∞+

∫ λe t

− λx

− λx

dx ~

λ ∈ λ (α )} dx

(6.6)

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for α ∈ [0,1] . Now the quotient of the integrals in Eq. (6.6) equals, after evaluation, e −λτ , so

[

]

∞

~ P X ≥ t + τ X ≥ t (α ) = { λe −λx dx λ ∈ λ (α )}

∫τ

(6.7)

~ which equals P [X ≥ τ ](α ) . Hence, Equation (6.3) holds for the fuzzy exponential which shows the memoryless property. Our approach is a two stage modeling approach. The first stage is same as the conventional SPN modeling approach. The only difference is that the steady-state distributions are obtained parametrically by using Equation (6.1) and no numeric results are calculated. In other words, each steady-state probability, π i , is described in terms of transition firing rates, as a function of λi . Up to the second stage, the system is a crisp one and describing the stochastic nature of the system. At this stage we represent the transition firing rates, λi , as triangular fuzzy numbers which may depend on the opinions of experts. After replacing the fuzzy numeric values of transition firing rates, by using the fuzzy calculation theory we obtain the α -cuts of the fuzzy steady-state probabilities of the system. As Buckley (2005) states that whenever we use interval arithmetic with α -cuts to compute the functions of fuzzy variables, we may get something larger than that obtained by using the extension principle. To be able to find the α -cuts of the fuzzy-steady probabilities we solve an optimization problem that makes the solution feasible. The procedure to compute the fuzzy-steady state probabilities is as follows: Stage 1 1.1. Model the system using a PN and associate exponential time delays with transitions. 1.2. Generate the reachability graph. Assign each arc with the rate of the corresponding transition. Label all states or markings. 1.3. By using Equation (6.1) find the steady-state probabilities parametrically, in terms of transition firing rates. Stage 2 2.1. Place the transition firing rates described as triangular fuzzy numbers in parametric steady-state probabilities obtained in Step (1.3). 2.2. Compute the fuzzy steady-state probabilities by using Equations (6.4)-(6.9) in terms α -cuts. 2.3. For each fuzzy steady-state probability, π i , find α = o values which gives the largest possible interval. It should be noted that for α = 1 the obtained result is the steady-state distribution of the crisp SPN. 2.4. For each π i , the maximum and minimum value ( α = o value) must be in

the interval [0,1] . If α = o for each π i does satisfy this, the result is feasible. If any of the obtained fuzzy probability does not satisfy this, apply the optimization in the next step.

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2.5. As mentioned before, interval arithmetic with α -cuts method is mainly based on max and min operators which may produce larger intervals. Theoretically, α = o cut of a fuzzy number gives the largest possible interval of values. Since we want to calculate the fuzzy probabilities the largest possible interval is restricted to the interval [0,1] . The problem is finding the min α -cut that satisfies this condition which can be found solving the following optimization problem: Assume that the α -cut representation for the fuzzy steady-state probability is

[

]

π i = π i− (α ), π i+ (α ) , where i = 1,2,..., n and n is the number of the states. Then the structure of the problem is Min( Z ) = α

s.t.

π i+ (α ) ≤ 1 π i− (α ) ≥ 0 0 ≤α ≤1 π i− (α ) ≤ π i+ (α ) In the next section a numerical illustration of the proposed approach is given.

7 A Numerical Example for the Proposed Fuzzy Approach Our proposed approach is applied to a FM cell which was selected from (Zhou and Venkatesh 1999). This FM cell is illustrated in Figure 6.6.

Fig. 6.6 The illustration of the flexible manufacturing cell

Our FM cell consists of two machines (M1 and M2), each of which is served by a dedicated robot (R1 and R2) for loading and unloading, as shown in Figure 6.6. An incoming conveyor carries pallets with raw materials one by one, from which R1 loads M1. An outgoing conveyor takes the finished product, to which R2 unloads M2. There is a buffer with capacity of two intermediate parts between two machines. The system produces a specific type of final parts. Each raw workpiece

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fixtured with one of three available pallets is processed by M1 and then M2. A pallet with a finished product is automatically defixtured, then fixtured with raw material, and finally returns to the incoming conveyor. Now suppose that 1. M1 performs faster M2 does, however subject to failures when it is processing a part. On the average, M1 takes two time units to break down, and a quarter time unit to be repaired. Thus, its average failure and repair rates (1/time unit) are 0.5 and 4 respectively. M2 and the two robots are failure-free. 2. R1’s loading speed is 40 per unit time. The average rate for M1’s processing plus R1’s loading is 5 per unit time. 3. The average rate for M2’s processing plus the related R2’s loading and unloading is 4 per unit time. 4. All the time delays associated with the above operations are exponential. The problem is to find the average utilization of M2, assuming that only one pallet is available. The SPN model of the system is given in Figure 6.7. Table 6.5 gives the explanation and interpretation of the PN elements used in the model.

Fig. 6.7 The stochastic PN model of the flexible manufacturing cell

If two pallets were available, the PN model of the system would contain two tokens in place p1 . As it can be seen in Table 6.5, the redundant information in each marking on the token values in places p7 , p8 , p 9 has been eliminated since p7 holds the same number of tokens as p5 does and each of the other two always has one token. The reachability graph and the Markov chain of the modeled system is given in Figure 6.8.

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Table 6.5 Places, transitions and their firing rates used in the model Places

Interpretation

p1

Pallets with workpieces available

p2

M1 in process

p3

Intermediate parts available for processing at M2

p4

M1 in repair

p5

M1 available

p6

Conveyor slots available

p7

R1 available (redundant from the analysis viewpoint)

p8

M2 available (redundant from the analysis viewpoint)

p9

R2 available (redundant from the analysis viewpoint)

Transitions

Interpretation

t1

R1 loads a part to M1

λ1 = 40

t2

M1 machines and R1 unloads a part

λ2 = 5

t3

R1 loads/unloads and M2 machines a part

λ3 = 4

t4

M1 breaks down

λ4 = 0.5

t5

M1 is repaired

λ5 = 0.5

Firing Rates

Fig. 6.8 The reachability graph and Markov chain of the system

By using Equation (6.1) we obtain the following system of equations:

λ1 ⎛ − λ1 ⎜ 0 λ − ⎜ 2 − λ4 (π 0 , π 1 , π 2 , π 3 )⎜ λ 0 ⎜ 3 ⎜ 0 λ5 ⎝ π 0 + π1 + π 2 + π 3 = 1

0

λ2 − λ3 0

0 ⎞ ⎟

λ4 ⎟

=0 0 ⎟ ⎟ − λ5 ⎟⎠

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The solution of the above system gives the steady-state probabilities parametrically, in terms of transition firing rates, as follows: ⎡π 0 ⎤ ⎡λ2 λ3λ5 / λ ⎤ ⎢π ⎥ ⎢ λ λ λ / λ ⎥ ΠT = ⎢ 1 ⎥ = ⎢ 1 3 5 ⎥ ⎢π 2 ⎥ ⎢ λ1λ2 λ5 / λ ⎥ ⎥ ⎢ ⎥ ⎢ ⎣π 3 ⎦ ⎣ λ1λ3 λ4 / λ ⎦

(6.8)

where λ = λ2 λ3λ5 + λ1λ3λ5 + λ1λ2 λ5 + λ1λ3 λ4 . After obtaining the steady-state probabilities in terms of transition firing rates, in the second stage transition firing rates are represented as triangular fuzzy numbers. Assume that the fuzzy number values of each transition firing rate are as in Table 6.6. Table 6.6 The fuzzified transition firing rates and their α − cut representations Fuzzy

λ value

λ1 = (30 / 40 / 50) λ 2 = ( 4 / 5 / 6) λ3 = (3 / 4 / 5) λ4 = (0.4 / 0.5 / 0.6) λ5 = (0.4 / 0.5 / 0.6)

α − cut representation λ1 = [30 + 10α ;50 − 10α ] λ2 = [4 + α ;6 − α ] λ3 = [3 + α ;5 − α ]

λ4 = [0.4 + 0.1α ;0.6 − 0.1α ] λ5 = [0.4 + 0.1α ;0.6 − 0.1α ]

By placing the fuzzy values of Table 6.6 in the previously obtained parametric steady-state probability representations, Equation (6.8), and applying fuzzy mathematics, the following α − cut representations of fuzzy steady-state probabilities are obtained. ⎡

− 0.1α 3 + 1.7α 2 − 9.6α + 18 ⎤ ⎥ 3 2 3 2 ⎣⎢ − 3.1α + 50.7α − 275.6α + 498 3.1α + 32.1α + 110α + 124.8 ⎦⎥

π 0 (α ) = ⎢

0.1α 3 + 1.1α 2 + 4α + 4.8

;

⎡

− α 3 + 16α 2 − 85α + 150 ⎤ α 3 + 10α 2 + 33α + 36 ; ⎥ 3 2 3 2 ⎣⎢ − 3.1α + 50.7α − 275.6α + 498 3.1α + 32.1α + 110α + 124.8 ⎦⎥

π 1 (α ) = ⎢

⎡

− α 3 + 17α 2 − 96α + 180 ⎤ α 3 + 11α 2 + 40α + 48 ; ⎥ 3 2 3 2 ⎢⎣ − 3.1α + 50.7α − 275.6α + 498 3.1α + 32.1α + 110α + 124.8 ⎥⎦

π 2 (α ) = ⎢ ⎡

− α 3 + 16α 2 − 85α + 150 ⎤ α 3 + 10α 2 + 33α + 36 ; ⎥ 3 2 3 2 ⎣⎢ − 3.1α + 50.7α − 275.6α + 498 3.1α + 32.1α + 110α + 124.8 ⎦⎥

π 3 (α ) = ⎢

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0,902

1,202

0,902

0,515

0,390

0,680

1,202

0,515

0,390

0,296

0,225

0,072

(c)

0,170

0

0,129

0,2

0

α

0,097

0,2

1,442

0,4

1,090

0,4

0,828

0,6

0,632

0,6

0,483

0,8

0,370

0,8

0,284

1

0,218

1

0,166

α1,2

0,127

0,296

(b) 1,2

0,096

0,680

(a)

0,225

0,170

0,129

0,072

0,144

0,109

0,083

0,063

0 0,048

0,2

0 0,037

0,2

0,028

0,4

0,022

0,6

0,4

0,017

0,6

0,013

1 0,8

0,010

1 0,8

0,097

1,2

(d)

Fig. 6.9 The graphical representation of the fuzzy steady-state probabilities, (a) π 0 , (b)

π 1 , (c) π 2 , (d) π 3 The graphics of the fuzzy steady-state probabilities are given in Figures 6.9 (a)-(d). Although for each π i = π i− (α ), π i+ (α ) , the maximum and minimum value

[

]

( α = o value) must be in the interval [0,1] , it can be seen that π 1+ , π 2+ and π 3+ do not satisfy this condition. So we must apply optimization to find the min α -cut that satisfies this condition. Since π 1 = π 3 and π i− (α ) ≥ 0 , the structure of the optimization problem can be reduced to the following: Min( Z ) = α

s.t.

π 0+ (α ) ≤ 1 π 1+ (α ) ≤ 1 π 2+ (α ) ≤ 1 0 ≤α ≤1 π 0+ (α ), π 1+ (α ), π 1+ (α ) ≥ 0

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By making the necessary simplifications the problem is obtained as follows: Min( Z ) = α

s.t.

− 3.2α 3 − 30.4α 2 − 119.6α − 106.8 ≤ 0 − 4.1α 3 − 16.1α 2 − 195α + 25.2 ≤ 0 − 4.1α 3 − 15.1α 2 − 206α + 55.2 ≤ 0 0 ≤α ≤1 The solution for the problem can be found by using software packages such as MATLAB or a spread sheet like Excel. The result of the optimization problem is 0.263. This α value is the one that makes the fuzzy steady-state probabilities feasible. The final fuzzy steady-state probabilities are presented in Table 6.7. Table 6.7 The final fuzzy steady-state probabilities

α = 0 cut π0 π1 π2 π3

[0.014;0.1] [0.106;0.826] [0.138;1] [0.106;0.826]

α = 1 cut 0.037 0.296 0.37 0.296

Note that α = 0 cut value represents the largest interval of probability whereas α = 1 cut value represents the crisp SPN probability. M1’s utilization is determined by the probability that M1 is machining a raw workpiece. This corresponds to the marking m1 at which p2 is marked, or state probability π 1 . Therefore, the expected M1’s utilization is π 1 which is (0.106 / 0.296 / 0.826) .

8 Conclusions There is a growing literature and interest in Petri nets theory, and in the scope of this chapter we presented a review of the important concepts of Petri nets and their application mainly in the area of flexible manufacturing systems. Complex systems such as flexible manufacturing systems exhibit characteristics which are difficult to describe mathematically using conventional tools and methods. Although there are many methods and tools used for modeling and analysis of FMSs such as queueing networks, Markov chains and simulation, Petri nets as a mathematical tool provide obtaining state equations describing system behavior, finding

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algebraic results and developing other mathematical models. With respect to other techniques of graphical system representation like block diagrams or logical trees, Petri nets are particularly more suited to represent in a natural way logical interactions among parts or activities in a system. In modeling point of view, Petri net theory allows the construction of the models amenable both for the effectiveness and efficiency analysis. Due to the graphical nature, ability to describe static and dynamic system characteristics and system uncertainty, and the presence of mathematical analysis techniques, Petri nets form an appropriate conceptual infrastructure for modeling and analysis of flexible manufacturing systems. One of the most important points in modeling real world systems is to be able to represent the uncertainty in the system model. Analysis and design of complex systems often involve two kinds of uncertainty: randomness and fuzziness. Randomness refers to describing the behavior of the parameters by using probability distribution functions. Fuzziness models measurement imprecision due to linguistic structure or incomplete information. Although the dominating concept to describe uncertainty in modeling is stochastic models which are based on probability, probabilistic models are not suitable to describe all kinds of uncertainty, but only randomness. We believe that to be able to better represent both dimensions of uncertainty, models that combine stochasticity and fuzziness should be used together. For this reason, the approach which was explained in detail in this study plays a significant role for the use of fuzzy set theory together with stochastic PNs especially in manufacturing system modeling. For further research, the application of the proposed approach in different areas rather than FMSs and sensitivity analysis for the fuzzy parameters can be considered.

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Venkateswaran, P.R., Bhat, J.: Fuzzy Petri net algoritjm for flexible manufacturing systems. ACSE Journal 6(1), 1–5 (2006) Viertl, R., Hareter, D.: Fuzzy information and stochastics. Iranian Journal of Fuzzy Systems 1(1), 39–52 (2004) Villani, E., Pascal, J.C., Miyagi, P.E., Valette, R.: A Petri net-based object-oriented approach for the modelling of hybrid productive systems. Nonlinear Analysis 62, 1394– 1418 (2005) Wang, L.C., Wu, S.Y.: Modeling with colored timed object-oriented Petri nets for automated manufacturing systems. Computers and Industrial Engineering 34(2), 463–480 (1998) Wang, Z., Zhang, J., Chan, F.T.S.: A hybrid Petri nets model of networked manufacturing systems and its control system architecture. Journal of Manufacturing Technology Management 16(1), 36–52 (2005) Wu, F.: Fuzzy time semirings and fuzzy-timing colored Petri nets. International Journal of Intelligent Systems 14, 747–774 (1999) Wu, N., Zhou, M.C., Li, Z.W.: Resource-oriented Petri net for deadlock avoidance in flexible assembly systems. IEEE Transactions on Man, and Cybernetics-Part A: Systems and Humans 38(1), 56–69 (2008) Yan, H.S., Wang, N.S., Zang, J.G., Cui, X.Y.: Modeling, scheduling and simulation of flexible manufacturing systems using extended stochastic high-level evaluation Petri nets. Robots and Computer-Integrated Manufacturing 14, 121–140 (1998) Yeung, D.S., Liu, J.N.K., Shiu, S.C.K., Fung, G.S.K.: Fuzzy coloured petri nets in modelling flexible manufacturing systems. In: Proceedings of Mexico-USA Collaboration in Intelligent Systems Technologies, pp. 100–107 (1996) Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man, and Cybernetics SMC-3(1), 28–44 (1973) Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) Zadeh, L.A.: Probability measures of fuzzy events. J. Math. Anal. Appl. 23, 421–427 (1968) Zha, X.F., Li, L.L., Zhang, W.J.: A knowledge Petri net model for flexible manufacturing systems and its application for design and verification of FMS controllers. International Journal of Computer Integrated Manufacturing 15(3), 242–264 (2002) Zhang, H., Gu, M.: Modeling job shop scheduling with batches and setup times by timed Petri nets. Mathematical and Computer Modelling 49, 286–294 (2009) Zhou, M.C., Jeng, M.D.: Modeling, analysis, simulation, scheduling, and control of semiconductor manufacturing systems: a Petri net approach. IEEE Transactions on Semiconductor Manufacturing 11(3), 333–357 (1998) Zhou, M.C., DiCesare, F.: Parallel and sequential mutual exclusions for Petri net modeling for manufacturing systems. IEEE Transactions on Robotics and Automation 7(4), 515– 527 (1991) Zhou, M.C., DiCesare, F.: Petri Net Synthesis for Discrete Event Control of Manufacturing Systems. Kluwer Academic, Dordrecht (1993) Zhou, M.C., DiCesare, F., Guo, D.: Modeling and performance analysis of a resource- sharing manufacturing system using stochastic Petri nets. In: Proceedings of the 5th IEEE International Symposium on Intelligent Control, vol. 2(5-7), pp. 1005–1010 (1990) Zhou, M.C., McDermott, K., Patel, P.A.: Petri net synthesis and analysis of a flexible manufacturing system cell. IEEE Transactions on Systems, Man, and Cybernetics 23(2), 523–531 (1993)

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Zhou, M.C., Venkatesh, K.: Modeling, simulation and control of flexible manufacturing systems: a petri net approach. World Scientific, Singapore (1999) Zhou, M.C., DiCesare, F.: A hybrid methodology for synthesis of Petri net models for manufacturing systems. IEEE Trans. Robot. Automa 8(3), 350–361 (1992) Zimmerman, H.J.: Fuzzy set theory and its applications. Kluwer Academic Publishers, Dordrecht (1994) Zimmermann, A., Hommel, G.: Modelling and evaluation of manufacturing systems using dedicated Petri nets. International Journal of Advanced Manufacturing Technology 15, 132–138 (1999) Zimmermann, A., Rodriguez, D., Silva, M.: A two phase optimization method for Petri nets models of manufacturing systems. Journal of Intelligent Manufacturing 12, 409–420 (2001) Zuberek, W.M., Kubiak, W.: Throughput analysis of manufacturing cells using timed Petri nets. In: Proc. IEEE Int. Conf. on Systems, Man and Cybernetics, San Antonio, TX, pp. 1328–1333 (1994) Zuberek, W.M., Kubiak, W.: Timed Petri nets in modeling and analysis of simple schedules for manufacturing cells. Computers and Mathematics with Applications 37, 191–206 (1999)

Chapter 7

Fuzzy Technology in Advanced Manufacturing Systems: A Fuzzy-Neural Approach to Job Remaining Cycle Time Estimation Toly Chen*

Abstract. A self-organization map (SOM)-fuzzy back propagation network (FBPN) approach is proposed in this study for estimating the remaining cycle time of each job in a semiconductor manufacturing factory, which was seldom investigated in the past studies but is a critical task to the semiconductor manufacturing factory. The proposed methodology applies the SOM-FBPN approach to estimate both the cycle time and the step cycle time of a job, and then derives the remaining cycle time with the proportional adjustment approach. For evaluating the effectiveness of the proposed methodology, production simulation is also applied in this study to generate some test data.

1 Introduction The remaining cycle time of a job that is being fabricated in a semiconductor manufacturing factory is the time still required to complete the fabrication on the job (see Fig. 7.1). If the job is just released into the semiconductor manufacturing factory, then the remaining cycle time of the job is equal to its cycle time. In other words, the remaining cycle time is an important attribute (or performance measure) for the work-in-progress (WIP) in the semiconductor manufacturing factory. Estimating the remaining cycle time for every job in a semiconductor manufacturing factory is a critical task to the semiconductor manufacturing factory because it has to respond to customers’ queries about the progress of their orders in the factory, and to make sure that the jobs of an order can be outputted in time to deliver the order on time. Otherwise, some efforts should be made to accelerate the progress of the jobs. However, though many of the previous studies have been devoted to estimating the cycle time of a job that will be released into a semiconductor manufacturing factory, few of them investigated the general case of estimating the remaining cycle time of a job that is in progress. Besides, estimating the remaining cycle time of a job is even more difficult. For example, only few data are readily available to estimating the remaining cycle time of a job. Most data collected from a semiconductor manufacturing factory for similar purposes cover either the entire Toly Chen Department of Industrial Engineering and Systems Management, Feng Chia University C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 125–142. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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step # 1 2 3 4 released

……

k

step cycle time

n outputted

remaining cycle time

Fig. 7.1 The concept of the remaining cycle time

processing route or the whole factory, and might not be suitable for estimating the remaining cycle time of a job that has only a few more steps to go through. Conversely, there are also factors facilitating the estimating of the remaining cycle time of a job. For example, the remaining cycle time is always shorter than the cycle time, and therefore the variation accumulated during the remaining fabrication process is less. This fact makes the remaining cycle time easier to estimate. Let the time that has passed by since the release of a job that has undergone several steps be called the step cycle time of the job, then the following relationship holds for the three time measures: the step cycle time + the remaining cycle time = the cycle time. There are therefore various ways to estimate the remaining cycle time of the job. In this paper, we applies Chen’s self-organization map (SOM)-fuzzy back propagation network (FBPN) approach (Chen 2008) to estimate both the cycle time and the step cycle time of a job, and then derives the remaining cycle time of the job with the proportional adjustment approach. The remainder of this paper is organized as follows. Section 2 reviews some existing approaches for job cycle time estimation. Subsequently in Section 3, the SOM-FBPN approach is applied to estimate the step cycle time of a job. The remaining cycle time of the job is then derived with the proportional adjustment approach. To evaluate the effectiveness of the proposed methodology, production simulation (PS) is applied in Section 4 to simulate a semiconductor shop floor environment and to generate some test data. Then the proposed methodology and several existing approaches are all applied to the test data in Section 5. Based on the analysis results, some points are made. Finally, the concluding remarks and some directions for future research are given in Section 6.

2 Existing Job Cycle Time Estimation Approaches There are six major approaches commonly applied to estimate the cycle time of a job in a semiconductor manufacturing factory: multiple-factor linear regression (MFLR), PS, back propagation networks (BPN), case-based reasoning (CBR), fuzzy modeling methods, and hybrid approaches (Chen 2006). Among the six approaches, MFLR is the easiest, quickest, and most prevalent in practical applications. The major disadvantage of MFLR is the lack of forecasting accuracy (Chen 2003). Conversely, a huge amount of data and lengthy simulation time required are

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two disadvantages of PS. Nevertheless, PS is the most accurate job cycle time estimation approach if the simulation model is highly valid and is continuously updated. Considering both effectiveness (forecasting accuracy) and efficiency (execution time), Chang et al. (2005), Chang and Hsieh (2003), and Hsu and Sha (2004) all forecasted the cycle time of a job in a semiconductor factory with a BPN having a single hidden layer. Compared with MFLR approaches, the average estimation accuracy measured with root mean squared error (RMSE) was considerably improved with these BPNs. For example, an improvement of about 40% in RMSE was achieved in Chang et al. (2005). On the other hand, much less time and fewer data are required to generate a cycle-time forecast with a BPN than with PS. Recently, Chen (2007) incorporated the releasing plan of the semiconductor factory into a BPN, and constructed a “look-ahead” BPN for the same purpose, which led to an average reduction of 12% in RMSE. Chang et al. (2001) proposed a k-nearest-neighbors based CBR approach which outperformed the BPN approach in forecasting accuracy. In one case, the advantage was up to 27%. Chang et al. (2005) modified the first step (i.e. partitioning the range of each input variable into several fuzzy intervals) of the fuzzy modeling method proposed by Wang and Mendel (1992), called the WM method, with a simple genetic algorithm (GA) and proposed the evolving fuzzy rule (EFR) approach to estimate the cycle time of a job in a semiconductor factory. Their EFR approach outperformed CBR and BPN in cycle time estimation accuracy. Chen (2003) constructed a FBPN that incorporated expert opinions to form inputs of the FBPN. Chen’s FBPN was a hybrid approach (fuzzy modeling and BPN) and surpassed the crisp BPN especially with respect to efficiency. Another hybrid approach was proposed in Chang and Liao (2006) by combining SOM and WM, in which a job was classified using a SOM before estimating the cycle time of the job with WM. Chen (2007) constructed a look-ahead k-means (kM)-FBPN for the same purpose, and discussed in detail the effects of using different look-ahead functions. The rationale for combining kM or other classifiers and FBPN for job cycle time estimation is explained as follows. Theoretically, a well-trained BPN or FBPN (without being stuck to local minima) with a good selected topology can successfully map any complex distribution. However, job cycle time estimation is a much more complicated problem, and the results of many previous studies have shown the incapability of BPN or FBPN in solving such a problem. One reason is that there might be multiple complex distributions to model, and these distributions might be quite different (even for the same product type and priority). For example, when the workload level (in the semiconductor factory or on the processing route or before bottlenecks) is stable, the cycle time of a job basically follows the well-known Little’s law (Little 1961), and the cycle time of the job can be easily estimated. Conversely, if the workload level fluctuates or keeps going up (or down), estimating the cycle time of a job becomes much more difficult. For this reason, classifying jobs under different circumstances seems to be a reasonable treatment. More recently, Chen (2007) proposed the look-ahead SOM-FBPN approach for job cycle time estimation in a semiconductor factory. Besides, a set of fuzzy inference rules were also developed to evaluate the achievability of a cycle time forecast. There are several studies that suggested the hybrid use of SOM and BPN

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or FBPN and listed the advantages of their joint use. For example, Tiwari and Roy (2002) classified castings with a SOM into some casting families, and then applied a BPN to predict the shrinkages for the castings within a casting family. With this system, casting design modifications could be performed in a more flexible and intelligent manner. In Chandiramani et al. (2004), a SOM was adopted to classify the processes in a multiprocessor system. According to the classification result, the structure of a BPN was modified and then the BPN was applied to predict the completion time of a process. The hybrid SOM-BPN approach achieved a good performance in predicting accuracy. Chiang (1998) developed a hybrid neural network model for handwritten word recognition, in which at first the word image was segmented into many primitive segments with a SOM, and then the segments were matched to the strings in the lexicon. The confidence of matching was assigned by applying a BPN. With the aid of the hybrid neural network, the word recognition rate was considerably improved (with an increase of 10.4%). Subsequently, Chen et al. (2008) added a selective allowance to the cycle time estimated using the look-ahead SOM-FBPN approach to determine the internal due date. Further, Chen (2008) showed that the suitability of the SOM and FBPN combination for the data could be improved by feeding back the estimation error by the FBPN to adjust the classification results of the SOM. For embodying the uncertainty of job classification, Chen (2007) used fuzzy c-means (FCM) instead. In Chen (2007), some BPNs formed an ensemble, and the estimated cycle times from these BPNs were aggregated with another BPN. Subsequently in Chen (2008), the ensemble of FBPNs was formed, and the efficiency was considerably improved. Except few studies in which the historical data of a real semiconductor factory were collected, most studies in this field used simulated data. Besides, a majority of the previous studies discussed the fixed product-mix cases. Recently, Chen (2008) proposed the multiple-bucket approach based on the concept of the look-ahead functions to deal with the dynamic product-mix cases. On the other hand, most existing approaches are focused on a full-scale semiconductor factory, while Chen (2008) modified the SOM-FBPN approach with multiple buckets and partial normalization to consider a ramping-up semiconductor factory. The results of these studies showed that: 1. Pre-classifying a job was beneficial to the estimation accuracy of the cycle time of the job. 2. Taking the future releasing plan of the shop floor into account, i.e. look-ahead, was also helpful. 3. Forming ensembles did improve the estimation accuracy further.

3 Methodology Variables that are used in the proposed methodology are defined: 1. 2. 3. 4.

Rn: the release time of job n. LS n: the job size of job n. Un: the average factory utilization at Rn. Qn: the total queue length on the processing route of job n at Rn.

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5. BQn: the total queue length before bottlenecks at Rn. 6. FQn: the total queue length in the whole factory at Rn. 7. WIPn: the factory WIP at Rn. 8.

Dn(i ) : the delay of the i-th recently completed job, i = 1 ~ 3.

9.

Bn( j ) : the j-th bucket of job n, j = 1 ~ m.

10.CTEn: the cycle time estimate of job n. 11.CTn: the cycle time (actual value) of job n. 12. SCTEnj : the step cycle time estimate of job n at step j. 13. SCTnj : the step cycle time (actual value) of job n at step j. 14. RCTE nj : the remaining cycle time estimate of job n at step j. 15. RCTnj : the remaining cycle time (actual value) of job n at step j. Obviously,

CTn = SCTnj + RCTnj .

(7.1)

(− ) , (×) : fuzzy addition, subtraction, and multiplication, respectively. ~ A fuzzy variable X is derived by multiplying the importance of its crisp version X

16.(+),

that is expressed with a fuzzy value to X. There are three steps in applying the SOM-FBPN approach to estimate the remaining cycle time of a job. At first, the procedure of applying SOM in forming inputs to the FBPN is detailed. Every job fed into the FBPN is called an example. Examples are pre-classified into different categories before they are fed into the FBPN with SOM. The structure of SOM is 10*10, and the number of output nodes is 100. Let xn denotes the eight-dimensional feature vector (Un, Qn, BQn, FQn, WIPn,

Dn(1) , Dn( 2) ,

Dn( 3) ) corresponding to job n. The feature vectors of all jobs are fed into an SOM network with the following learning algorithm: 1. Set the number of output nodes and the number of input nodes. Initialize the learning rate, the neighborhood size, and the number of iterations. 2. Initialize the weights (wij) randomly where i = 1~p and p stands for the number of output nodes; j = 1~8. 3. (Iteration) Provide an input vector to the network. 4. Find the output node (winner) based on the similarity between the input vector and the weight vector. For an input vector xn, the winning unit can be determined by distance || xn – wc || = min || x n − wi || , where wi is the weight vector of the i

i-th unit and the index c refers to the winning unit. 5. Update the weight vector of the winner node using Kohonen's learning rule (Chang and Hsieh 2003). wi(t + 1) = wi(t) + α(t) (xn – wi), for each i ∈ Nc(t),

(7.2)

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where t is the discrete-time index of the variables; the factor α(t) ∈ [0, 1] is a scalar that defines the relative size of the learning step; Nc(t) specifies the neighborhood around the winner in the map array. 6. Stop if the number of iterations has been completed. Otherwise, go to step 3. After the training procedure is accomplished, a labeling process is realized. The distribution of labeled (categorized) output nodes of SOM after the labeling process is drawn (see Fig.7.2). According to this figure, merged or isolated clusters can be visually analyzed, and the number of categories can be clarified. The Neural Network Toolbox of MATLAB 2006a is applied to construct the SOM. An example program is provided in Fig.7.3. After classification, examples of different categories are then learned with different FBPNs but with the same topology. The procedure for determining the parameter values of FBPNs is described below. The configuration of the FBPN is established as follows: 1. Inputs: eight parameters associated with the n-th example/job including Un, Qn, BQn, FQn, WIPn, and

Dn( r ) (r = 1~3). These parameters have to be normalized

so that their values fall within [0, 1]. Then some production execution/control experts are requested to express their beliefs (in linguistic terms) about the importance of each input parameter in predicting the cycle time or step cycle time of a job. Linguistic assessments for each input parameter are converted into several pre-specified fuzzy numbers that are demonstrated in Fig.7.4. The subjective importance of an input parameter is then obtained by averaging the corresponding fuzzy numbers of the linguistic replies for the input parameter by all experts. The subjective importance of each input parameter is then obtained by averaging the corresponding fuzzy numbers of the linguistic replies for the input parameter by all experts. For example, if five supervisors are requested to assess the importance of input parameter “Un” in estimating the cycle time of a job, and the replies are {“Very Important”, “Important”, “Very Important”, “Moderate”, “Important”}, respectively. Then the subjective importance of input parameter “Un” is calculated by averaging the corresponding fuzzy numbers as follows: Subjective importance of “Un”

1 = ((0.7, 1, 1) (+)(0.5, 0.7, 1) (+)(0.7, 1, 1) (+)(0.2, 0.5, 0.8) (+)(0.5, 0.7, 1)) 5

= (0.52, 0.78, 0.96) .

The subjective importance for an input parameter is multiplied to the normalized value of the input parameter. After such a treatment, all inputs to the FBPN become triangular fuzzy numbers (TFNs). For example, if Un = 0.75, then the following fuzzy value is inputted to the FBPN instead of 0.75:

~ U n = 0.75 ⋅ (0.52, 0.78, 0.96) = (0.39, 0.59, 0.72). The fuzzy arithmetic for TFNs is therefore applied to deal with all calculations involved in training the FBPN.

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Fig. 7.2 A SOM example

% The raw data of jobs U=[U1 U2 ... Un]; Q=[Q1 Q2 ... Qn]; ...; D3=[D31 D32 ... D3n]; RD=[U; Q; ...; ER]; % normalization U2=(U-min(U))/(max(U)-min(U)); Q2=(Q-min(Q))/(max(Q)-min(Q)); ...; D32=(D3-min(D3))/(max(D3)-min(D3)); ND=[U2; Q2; ...; D32]; % The SOM configuration net=newsom([0 1; 0 1; 0 1; 0 1; 0 1; 0 1; 0 1; 0 1],[10 10]); % Start training net=train(net,ND); plot(ND(1,:),ND(2,:),'.g','markersize',20) hold on plotsom(net.iw{1,1},net.layers{1}.distances) hold off

Fig. 7.3 An example SOM program

1. Single hidden layer: Generally one or two hidden layers are more beneficial for the convergence property of the FBPN. 2. Number of neurons in the hidden layer: the same as that in the input layer. Such a treatment has been adopted by many studies (e.g. Barman 1998, Chandiramani et al. 2004, Chang and Chen 2006). 3. Output: the (normalized) cycle time or step cycle time estimate of the example. 4. Network learning rule: Delta rule. 5. Transformation function: Log-Sigmoid function,

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f ( x) =

1 . 1 + e−x

(7.3)

6. Learning rate (η): 0.01~1.0. 7. Batch learning.

Fig. 7.4 Linguistic terms used in assessing the importance of an input parameter

The procedure for determining the parameter values is now described. After pre-classification, a portion of the adopted examples in each category is fed as “training examples” into the FBPN to determine the parameter values for the category. Two phases are involved at the training stage. At first, in the forward phase, inputs are multiplied with weights, summated, and transferred to the hidden layer. Then activated signals are outputted from the hidden layer as:

~ h j = ( h j1 , h j 2 , h j 3 ) =

1 1+ e

−n~ hj

=(

1 1+ e

− n hj1

1

,

1+ e

− n hj2

1

,

1+ e

− n hj3

)

(7.4)

where

~ ~ n~ hj = ( n hj1 , n hj2 , n hj3 ) = I jh ( − )θ jh = ( I hj1 − θ hj3 , I hj2 − θ hj2 , I hj3 − θ hj1 )

(7.5)

~ ~ h (×) ~ I jh = ( I hj1 , I hj2 , I hj3 ) = ∑ w x(i ) ij all i

≅ ( ∑ min( wijh1x(i )1, wijh3 x(i )3 ) , all i

∑ wijh 2 x(i )2 , all i

∑ max(wijh1x(i)1, all i

wijh3 x(i )3 )) .

(7.6)

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~ h j ’s are also transferred to the output layer with the same procedure. Finally, the output of the FBPN is generated as:

o~ = (o1 , o2 , o3 ) =

1 1+ e

− n~ o

1 1 1 =( ) , , o o − n1 − n2 − n3o 1+ e 1+ e 1+ e

(7.7)

where

~ ~ n~ o = ( n1o , n 2o , n 3o ) = I o ( − )θ o = ( I1o − θ 3o , I 2o − θ 2o , I 3o − θ1o ) (7.8)

~ I o = ( I 1o , I 2o , I 3o ) =

~

∑ w~ oj (×)h j all j

≅ ( ∑ min(woj1h j1, woj3h j 3 ), all j

∑ woj2h j 2 , ∑ max(woj1h j1,

woj3h j 3 )) (7.9)

all j

all j

To improve the practical applicability of the FBPN and to facilitate the comparisons ~ is defuzzified according with conventional techniques, the fuzzy-valued output o to the centroid-of-area (COA) formula:

o + 2o2 + o3 o = COA(o~ ) = 1 4

(7.10)

Then the output o is compared with the normalized actual cycle time or step cycle time a, for which RMSE is calculated:

∑ RMSE =

(o − a ) 2

all examples

number of examples

.

(7.11)

Subsequently in the backward phase, the deviation between o and a is propagated backward, and the error terms of neurons in the output and hidden layers can be calculated respectively as:

δ o = o(1 − o)( a − o), ~ ~ ~ ~ oδ o δ jh = (δ hj1 ,δ hj2 ,δ hj3 ) = h j (×)(1 − h j )(×) w j ≅ (min(min(h j1(1 − h j 3 ) woj1, h j 3 (1 − h j1) woj1)δ o , max(h j 3 (1 − h j1) woj3 , h j1(1 − h j 3 ) woj3 )δ o ),

h j 2 (1 − h j 2 ) woj2δ o , max(min(h j1(1 − h j 3 ) woj1 ,

(7.12)

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h j 3 (1 − h j1 ) w oj1 )δ o , max(h j 3 (1 − h j1 ) w oj3 , h j1 (1 − h j 3 ) woj3 )δ o ))

(7.13)

Based on them, adjustments that should be made to the connection weights and thresholds can be obtained as:

~ o = ( Δw o , Δw o , Δw o ) =ηδ o h~ Δw j j1 j2 j3 j = ηδ o (min( h j1 , h j 3 ), h j 2 , max( h j1 , h j 3 ))

(7.14)

~ h = ( Δwh , Δwh , Δwh ) = ηδ~ h (×) ~ Δw xi ij ij1 ij 2 ij 3 j ≅ η (min(δ hj1 x i1 , δ hj1 x i 3 , δ hj3 x i1 , δ hj3 x i 3 ) , δ hj2 x i 2 , max(δ hj1xi1, δ hj1xi 3 , δ hj3 xi1, δ hj3 xi 3 ))

(7.15)

(7.16) Δθ o = −ηδ o ~ ~ Δθ jh = ( Δθ hj1 , Δθ hj2 , Δθ hj3 ) = −ηδ jh = ( −ηδ hj3 , − ηδ hj2 , − ηδ hj1 ) (7.17) To accelerate convergence, a momentum can be added to the learning expressions. For example,

~ o = ηδ o h~ + α ( w ~ o (t ) − w ~ o (t − 1)) Δw j j j j

(7.18)

Theoretically, network-learning stops when RMSE falls below a pre-specified level, or the improvement in RMSE becomes negligible with more epochs, or a large number of epochs have already been run. Then test examples are fed into the FBPN to evaluate the accuracy of the network that is also measured with RMSE. The learning of the FBPN is according to completely the same mechanism as that of a typical BPN. Therefore, the same network convergence behavior can be observed. The only difference is that all calculations are performed with TFNs instead of crisp values. An example is demonstrated in Fig.7.5, in which the convergence processes of BPN and FBPN trained with the same data set are compared. The convergence patterns are the same. Besides, the FBPN appears to start with a considerably smaller value of the initial RMSE, and reaches the minimum RMSE with much fewer epochs than the crisp BPN does. However, the accumulation of fuzziness during the training process continuously increases the lower bound, upper bound, ~ (and those of many other fuzzy parameand spread of the fuzzy-valued output o ters), and might prevent RMSE from converging to its minimal value. An example is given in Fig.7.6 to demonstrate the accumulation of fuzziness with fuzzy operations. Conversely, the centers of some fuzzy parameters are becoming smaller and smaller because of network learning. It is possible that a fuzzy parameter becomes invalid in the sense that the lower bound higher than the center. To deal with this

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problem, the lower and upper bounds of all fuzzy numbers in the FBPN will no longer be modified if Chen’s index (Chandiramani et al. 2004) converges to a minimal value:

∑ min((o1 − a ) 2 ,

α

( o3 − a ) 2 ) + (1 − α )

all examples

number of examples

∑ max((o1 − a ) 2 ,

(o3 − a ) 2 )

all examples

(7.19)

number of examples 0 < α < 1.

RMSE

When a new job is released into the factory, the eight parameters (except LSn) associated with the new job are recorded and compared with those of each category 160 140 120 100 80 60 40 20 0

BPN FBPN

0

200

400

600

800

1000

1200

Iteration

Fig. 7.5 The convergence processes of BPN and FBPN

μX ~ 1 A

~ B

~ ~ A(×) B

0.8 0.6 0.4 0.2 X

0 1

4

7 10

Fig. 7.6 The accumulation of fuzziness with fuzzy operations

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center. Then the FBPN with the parameters of the nearest category center is applied to estimate the cycle time (or step cycle time) of the new job. The SOM-FBPN approach introduced above is applied to estimate both the cycle time and the step cycle time of a job when it is released into the semiconductor manufacturing factory. Namely, there will be two groups of FBPNs to estimate the cycle time and the step cycle time of a job, respectively. A product in a semiconductor manufacturing factory usually has up to hundreds of steps to undergo, and we can estimate the step cycle time for each step. After that, the remaining cycle time can be derived in the following way: RCTEnj = (CTEn – SCTnj) * (1 + SCTnj / SCTEnj)

(7.20)

4 PS for Generating Test Data In real situations, the historical data of each job is only partially available in the factory. Further, some information of the previous jobs such as Qn, BQn, and FQn is not easy to be obtained on the shop floor. Therefore, a simulation model is often built for the manufacturing process of a real semiconductor manufacturing factory (Barman 1998, Chang and Hsieh 2003, Chang et al. 2001, Chang et al. 2005, Chang and Liao 2006, Chen 2003, Chen 2006). Then, such information can be derived from the shop floor status collected from the simulation model (Chang et al. 2001). To generate some test examples, a simulation program coded using Microsoft Visual Basic 6.0 is constructed to simulate a semiconductor manufacturing environment with the following assumptions: 1. The distributions of the times between the adjacent machine breakdowns are exponential. 2. The distribution of the time required to repair a machine is uniform. 3. The percentages of jobs with different product types in the factory are predetermined. As a result, this study is only focused on fixed-product-mix cases. 4. The percentages of jobs with different priorities released into the factory are controlled. 5. Jobs are sequenced on each machine first by their priorities, then by the first-in-first-out (FIFO) policy. Such a sequencing policy is not uncommon in many foundry factories. 6. A job has equal chances to be processed on each alternative machine/head available at a step. 7. A job cannot proceed to the next step until the fabrication on its every wafer has been finished. 8. No preemption is allowed. The basic configuration of the simulated semiconductor manufacturing factory is the same as a real-world semiconductor manufacturing factory which is located in the Science Park of Hsin-Chu, Taiwan, R.O.C. Assumptions (1)~(2), and (5)~(8) are commonly adopted in related studies (e.g. Chang and Hsieh 2003, Chang et al. 2001, Chang et al. 2005, Chang and Liao 2006, Chen 2003, Chen 2006), while assumptions (3)~(4) are made to simplify the situation. There are five products (labeled as A~E)

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in the simulated f semiconductor manufacturing factory. A fixed product mix is assumed. The percentages of these products in the factory’s product mix are assumed to be 35%, 24%, 17%, 15%, and 9%, respectively. The simulated factory has a monthly capacity of 20,000 pieces of wafers and is expected to be fully utilized (utilization = 100%). Jobs are uniformly (every a fixed interval) released into the factory, and have a standard size of 24 wafers per job. The mean inter-release time of jobs into the factory can be obtained as (30.5 * 24) / (20000 / 24) = 0.88 hours. Three types of priorities (normal, hot, and super hot) are randomly assigned to jobs. The percentages of jobs with these priorities released into the factory are restricted to be approximately 60%, 30%, and 10%, respectively. Each product has 150~200 steps and 6~9 reentrances to the most bottleneck machine. The singular production characteristic “reentry” of the semiconductor industry is clearly reflected in the simulation model. It also shows the difficulty for the production planning and scheduling staff to provide an accurate due-date for the product with such a complicated routing. Totally 102 machines (including alternative machines) are used to process single-wafer or batch operations in the factory. Thirty replications of the simulation are successively run. The time required for each simulation replication is about 15 minute on a PC with 256MB RAM and Athlon™ 64 Processor 3000+ CPU. A horizon of twenty-four months is simulated. The maximal cycle time is less than three months. Therefore, four months and an initial WIP status (obtained from a pilot simulation run) seemed to be sufficient to drive the simulation into a steady state. The statistical data were collected starting at the end of the fourth month. For each replication, data of 30 jobs are collected and classified by their product types and priorities. Totally, data of 900 jobs can be collected as training and testing examples. Among them, 2/3 (600 jobs, including all product types and priorities) are used to train the network, and the other 1/3 (300 jobs) are reserved for testing. A traced report was generated every simulation run for verifying the simulation model. The average cycle times have also been compared with the actual values for validating the simulation model.

5 Experimental Results and Discussions To evaluate the effectiveness of the proposed methodology and to make comparison with some existing approaches – MFLR, BPN, FBPN, CBR, and FCM-FBPN, all these methods were applied to some test cases containing the data of jobs with two major product types (A and B) and various priorities. Because there might be up to hundreds of steps for each job to undergo, estimating the remaining cycle time at every step was laborious, and therefore only the remaining cycle times at four representative steps were estimated. Their numbers were the 20%, 40%, 60%, and 80% percentiles in all. For example, if a job has 120 steps to undergo, then the remaining times at the 24th, 48th, 72nd, and 96th steps will be estimated in the experiment. The minimal RMSEs achieved by applying these approaches to different cases were recorded and compared in Table 7.1 ~ Table 7.4, respectively. Note that the minimal RMSEs have been converted back to the un-normalized values to be more meaningful practically. In the BPN or FBPN, there was one hidden layer with 9~18 nodes, depending on the results of a preliminary analysis for establishing the best configuration. In the

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proposed methodology, firstly jobs were classified with SOM. Subsequently, examples of different categories were then learned with different FBPNs but with the same topology. The convergence condition was established as either the improvement in the RMSE becomes less than 0.001 with one more epoch, or 75000 epochs have already been run. Table 7.1 Comparisons of the RMSEs of various approaches (the 20%-th step)

RMSE

MFLR BPN

(hours)

FBPN

CBR

FCM-FBPN

The proposed methodology

A(normal)

48

31(-35%) 30(-37%) 42(-12%) 15(-68%)

15(-68%)

A(hot)

30

21(-30%) 18(-38%) 24(-19%) 9(-71%)

9(-71%)

A(super hot)

23

17(-29%) 16(-33%) 21(-8%)

5(-79%)

5(-79%)

B(normal)

67

42(-37%) 42(-37%) 62(-7%)

12(-82%)

12(-82%)

14(-70%)

B(hot)

47

31(-33%) 31(-33%) 44(-5%)

B(super hot)

16

11(-30%) 11(-33%) 14(-13%) 4(-73%)

14(-70%) 5(-70%)

Table 7.2 Comparisons of the RMSEs of various approaches (the 40%-th step)

RMSE

MFLC BPN

FBPN

CBR

FCM-FBPN

The proposed methodology

A(normal)

76

59(-23%)

57(-25%)

69(-10%)

27(-64%)

28(-63%)

A(hot)

52

38(-27%)

33(-36%)

39(-25%)

15(-71%)

14(-72%)

A(super hot)

39

30(-23%)

28(-28%)

35(-11%)

8(-80%)

8(-80%)

B(normal)

106

80(-25%)

80(-25%)

101(-5%)

21(-80%)

21(-80%)

B(hot)

74

59(-21%)

59(-21%)

72(-3%)

25(-66%)

23(-68%)

B(super hot)

33

20(-39%)

19(-42%)

23(-31%)

7(-78%)

6(-81%)

(hours)

Table 7.3 Comparisons of the RMSEs of various approaches (the 60%-th step)

RMSE

MFLC

BPN

FBPN

CBR

FCM-FBPN

The proposed methodology

A(normal)

93

71(-24%)

70(-25%)

84(-10%)

36(-61%)

33(-64%)

A(hot)

64

46(-28%)

40(-38%)

48(-25%)

19(-70%)

17(-73%)

A(super hot) 58

36(-38%)

34(-41%)

43(-26%)

10(-82%)

11(-81%)

B(normal)

130

97(-26%)

97(-26%)

124(-5%)

27(-79%)

29(-78%)

B(hot)

90

71(-21%)

71(-21%)

88(-3%)

33(-63%)

32(-65%)

B(super hot) 46

24(-47%)

23(-50%)

28(-39%)

9(-80%)

8(-82%)

(hours)

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Table 7.4 Comparisons of the RMSEs of various approaches (the 80%-th step)

RMSE

MFLC BPN

FBPN

CBR

FCM-FBPN

The proposed methodology

A(normal)

115

98(-15%)

95(-18%)

105(-8%)

48(-58%)

47(-59%)

A(hot)

73

62(-15%)

55(-25%)

60(-17%)

26(-65%)

25(-65%)

A(super hot) 59

49(-17%)

46(-22%)

54(-9%)

13(-77%)

13(-79%)

B(normal)

161

134(-16%)

134(-16%)

156(-3%)

36(-77%)

36(-78%)

B(hot)

112

98(-12%)

98(-12%)

111(-1%)

44(-60%)

43(-61%)

31(-40%)

31(-40%)

35(-32%)

12(-77%)

11(-79%)

(hours)

B(super hot) 52

MFLR was adopted as the comparison basis, and the percentage of improvement in the minimal RMSE by applying another approach is enclosed in parentheses following the performance measure. The optimal value of parameter k in the CBR approach was equal to the value that minimized the RMSE (Barman 1998). According to experimental results, the following points are made: 1. Experimental results revealed that the remaining cycle time of a latter step was more difficult to estimate. 2. The same phenomenon could also be observed if the proposed methodology was applied instead. However, the difference was smaller. 3. As stated in many previous studies, the cycle times of jobs with higher priorities were easier to estimate. 4. From the effectiveness viewpoint, the estimation accuracy (measured with the RMSE) of the proposed methodology was significantly better than those of most approaches in most cases and at all of the four steps. However, the advantage declined as the fabrication proceeded. 5. The estimation accuracy of proposed methodology slightly surpassed that of the FCM-FBPN approach. On the other hand, the efficiency of the proposed methodology was much better than that of the FCM-FBPN approach.

6 Conclusions and Directions for Future Research To enhance the accuracy and efficiency of estimating the remaining cycle time for each job in a semiconductor manufacturing factory, which was seldom investigated in the past studies but is a critical task for the semiconductor manufacturing factory, the SOM-FBPN approach is applied in this study to estimate both the cycle time and step cycle time of a job. Subsequently, the proportional adjustment approach is applied to derive the remaining cycle time of the job based on the estimated cycle time and step cycle time. To evaluate the effectiveness of the proposed methodology, PS is also applied in this study to generate some test data. Then the proposed methodology and several existing approaches are all applied to the test data in. Based on the experimental results:

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1. The estimation accuracy (measured with the RMSE) of the proposed methodology was significantly better than those of most approaches in most cases. 2. By using SOM as the classifier, the number of job categories did not need to be presumed. Besides, the efficiency was also greatly enhanced. 3. The proportional adjustment approach applied in this study did improve the performance of deriving the remaining cycle time. To further evaluate the effectiveness and efficiency of the proposed methodology, it has to be applied to factory models of different scales, especially a full-scale actual semiconductor manufacturing factory. Besides, we can also collect the data suitable for estimating the remaining cycle time and then estimate it directly. These constitute some directions for future research.

Acknowledgement This paper is supported by the National Science Council of R.O.C.

References Barman, S.: The impact of priority rule combinations on lateness and tardiness. IIE Transactions 30, 495–504 (1998) Chandiramani, V., Jayaseelan, R., Nathan, V.S.L., Priya, K.S.: A neural network approach to process assignment in multiprocessor systems based on the execution time. In: Proceedings of International Conference on Intelligent Sensing and Information Processing (ICISIP 2004), Chennai, India, pp. 332–335 (2004) Chang, P.C., Chen, L.Y.: A hybrid regulation system by evolving CBR with GA for a twin laser measuring system. International Journal of Advanced Manufacturing Technology (2006), doi:10.1007/s00170-005-0286-4 Chang, P.C., Hsieh, J.C.: A neural networks approach for due-date assignment in a wafer fabrication factory. International Journal of Industrial Engineering 10(1), 55–61 (2003) Chang, P.C., Hsieh, J.C., Liao, T.W.: A case-based reasoning approach for due date assignment in a wafer fabrication factory. In: Proceedings of the International Conference on Case-Based Reasoning (ICCBR 2001), Vancouver, British Columbia, Canada (2001) Chang, P.C., Hsieh, J.C., Liao, T.W.: Evolving fuzzy rules for due-date assignment problem in semiconductor manufacturing factory. Journal of Intelligent Manufacturing 16, 549–557 (2005) Chang, P.C., Liao, T.W.: Combining SOM and fuzzy rule base for flow time prediction in semiconductor manufacturing factory. Applied Soft Computing 6, 198–206 (2006) Chang, P.C., Liu, C.H., Wang, Y.W.: A hybrid model by clustering and evolving fuzzy rules for sale forecasting in printed circuit board industry. Decision Support Systems 42(3), 1254–1269 (2006) Chang, P.C., Lai, C.Y.: A hybrid system combining self-organizing maps with case-based reasoning in wholesaler’s new-release book forecasting. Expert Systems with Applications 29(1), 183–192 (2005) Chen, T.: A fuzzy back propagation network for output time prediction in a wafer fab. Applied Soft Computing 2(3), 211–222 (2003)

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Chen, T.: A fuzzy mid-term single-fab production planning model. Journal of Intelligent Manufacturing 14, 273–285 (2003) Chen, T.: A hybrid SOM-BPN approach to lot output time prediction in a wafer fab. Neural Processing Letters 24(3), 271–288 (2006) Chen, T.: A look-ahead fuzzy back propagation network for lot output time series prediction in a wafer fab. In: King, I., Wang, J., Chan, L.-W., Wang, D. (eds.) ICONIP 2006. LNCS, vol. 4234, pp. 974–982. Springer, Heidelberg (2006) Chen, T.: A fuzzy–neural approach for estimating the monthly output of a semiconductor manufacturing factory. International Journal of Advanced Manufacturing Technology (2007) (in press) Chen, T.: A hybrid look-ahead SOM-FBPN and FIR system for wafer lot output time prediction and achievability evaluation. International Journal of Advanced Manufacturing Technology 35, 575–586 (2007) Chen, T.: An intelligent hybrid system for wafer lot output time prediction. Advanced Engineering Informatics 21, 55–65 (2007) Chen, T.: Incorporating fuzzy c-means and a back-propagation network ensemble to job completion time prediction in a semiconductor fabrication factory. Fuzzy Sets and Systems 158, 2153–2168 (2007) Chen, T.: Predicting wafer lot output time with a hybrid FCM-FBPN approach. IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics 37(4), 784–793 (2007) Chen, T.: A fuzzy-neural and multiple-bucket approach for estimating lot cycle time in a wafer fab with dynamic product mix. Computers & Industrial Engineering 55, 423–438 (2008) Chen, T.: A hybrid fuzzy-neural approach to job completion time prediction in a semiconductor fabrication factory. Neurocomputing 71, 3193–3201 (2008) Chen, T.: A SOM-FBPN-ensemble approach with error feedback to adjust classification for wafer-lot completion time prediction. International Journal of Advanced Manufacturing Technology 37(7-8), 782–792 (2008) Chen, T., Jeang, A., Wang, Y.C.: A hybrid neural network and selective allowance approach for internal due date assignment in a wafer fabrication plant. International Journal of Advanced Manufacturing Technology 36, 570–581 (2008) Chen, T., Wang, M.J.J.: A fuzzy production planning model for a semiconductor fabrication plant. International Journal of Operations and Quantitative Management 4(3), 277–292 (1998) Chen, T., Wang, Y.C., Tsai, H.R.: Lot cycle time prediction in a ramping-up semiconductor manufacturing factory with a SOM-FBPN-ensemble approach with multiple buckets and partial normalization. International Journal of Advanced Manufacturing Technology (2008) (in press) Chen, T., Wang, Y.C., Wu, H.C.: A fuzzy-neural approach for remaining cycle time estimation in a semiconductor manufacturing factory – a simulation study. International Journal of Innovative Computing, Information and Control (2008) (in press) Chiang, J.H.: A hybrid neural network model in handwritten word recognition. Neural Networks 11, 337–346 (1998) Chung, S.H., Yang, M.H., Cheng, C.M.: The design of due date assignment model and the determination of flow time control parameters for the wafer fabrication factories. IEEE Transactions on Components, Packaging, and Manufacturing Technology – Part C 20(4), 278–287 (1997)

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Foster, W.R., Gollopy, F., Ungar, L.H.: Neural network forecasting of short, noisy time series. Computers in Chemical Engineering 16(4), 293–297 (1992) Goldberg, D.E.: Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading (1989) Gordon, V., Proth, J.M., Chu, C.: A survey of the state-of-the-art of common due date assignment and scheduling research. European Journal of Operational Research 139, 1–25 (2002) Haller, M., Peikert, A., Thoma, J.: Cycle time management during production ramp-up. Robotics and Computer Integrated Manufacturing 19, 183–188 (2003) Hsu, S.Y., Sha, D.Y.: Due date assignment in wafer fabrication using artificial neural network. International Journal of Advanced Manufacturing Technology 23(9-10), 768–775 (2004) Hung, Y.F., Chang, C.B.: Dispatching rules using flow time predictions for semiconductor wafer fabrications. In: Proceedings of the 5th Annual International Conference on Industrial Engineering Theory Applications and Practice, Taiwan (2001) Ishibuchi, H., Nozaki, K., Tanaka, H.: Distributed representation of fuzzy rules and its application to pattern classification. Fuzzy Sets and Systems 52(1), 21–32 (1992) Jiang, Y., Zhou, Z.H.: SOM ensemble-based image segmentation. Neural Processing Letters 20, 171–178 (2004) Ku, K.C., Kao, H.P., Gurumurthy, C.K.: A collaborative model for capacity expansion of the IC foundry industry. International Journal of Advanced Manufacturing Technology 34(3-4), 252–261 (2007) Lin, C.Y.: Shop floor scheduling of semiconductor wafer fabrication using real-time feedback control and prediction. Ph.D. Dissertation Engineering-Industrial Engineering and Operations Research University of California, Berkeley (1996) Little, J.D.C.: A proof of the queueing formula L = λ W. Operations Research 9, 383–387 (1961) Pai, P.F., Lee, C.E., Su, T.H.: A daily production model for wafer fabrication. International Journal of Advanced Manufacturing Technology 23(1-2), 58–63 (2004) Piramuthu, S.: Theory and methodology – financial credit-risk evaluation with neural and neuralfuzzy systems. European Journal of Operational Research 112, 310–321 (1991) Ragatz, G.L., Mabert, V.A.: A simulation analysis of due date assignment. Journal of Operations Management 5, 27–39 (1984) Tiwari, M.K., Roy, D.: Minimization of internal shrinkage in casting using synthesis of neural network. International Journal of Smart Engineering System Design 4, 205–214 (2002) Vig, M.M., Dooley, K.J.: Dynamic rules for due-date assignment. International Journal of Production Research 29(7), 1361–1377 (1991) Wang, C.N., Wang, C.H.: A simulated model for cycle time reduction by acquiring optimal lot size in semiconductor manufacturing. International Journal of Advanced Manufacturing Technology 34(9-10), 1008–1015 (2007) Wang, L.X., Mendel, J.M.: Generating fuzzy rules by learning from examples. IEEE Transactions on Systems, Man, and Cybernetics 22(6), 1414–1427 (1992) Weeks, J.K.: A simulation study of predictable due-dates. Management Science 25, 363–373 (1979) Xie, X.L., Beni, G.: A validity measure for fuzzy clustering. IEEE Transactions of Pattern Analysis and Machine Intelligence 13, 841–847 (1991)

Chapter 8

Fuzzy Project Scheduling Na¨ım Yalaoui, Fr´ed´eric Dugardin, Farouk Yalaoui, Lionel Amodeo, and Halim Mahdi

1 Introduction This book lights out the different improvement of the recent history of fuzzy logic. The present chapter deals with the connections that exist between fuzzy logic and production scheduling. Production scheduling is a part of operational research which relies on combinatorial optimization solved by discrete methods. This large area covers several well-known combinatorial problems: vehicle routing problem (in which several vehicle must visit customers at once), scheduling problem [18] (explained in section 2 of this chapter), bin-packing problem (where piece must be placed in a rectangle), assignment problem (where piece must be assign to machine while optimizing a criterion). These short number of both theoretical and practical problems are persistent in numerous technical areas: transportation (flights, trucks, ships, auto-guided-vehicles), shop scheduling, surgery operating theater, layout of warehouse, landing/takeoff runway scheduling, timetable. To solve these multiple problems, operational research applies two main principles: exact methods which provide the absolute best solution but solve only small sized problems, and approximated methods which provide only good solutions but solve near real life sized problems. The second category provides various methods divided into: problem dedicated methods called heuristics and general method called metaheuristics. Even if many metaheuristics exist and every year new ones appear, some of them are leading the literature: Genetic Algorithm (GA) [22], Ant Colony System (ACS) [14], Na¨ım Yalaoui, Fr´ed´eric Dugardin, Farouk Yalaoui, and Lionel Amodeo Institut Charles Delaunay, Universit´e de Technologie de Troyes, FRE CNRS 2848 12 rue Marie Curie, 10000 Troyes, France e-mail: [email protected] Na¨ım Yalaoui and Halim Mahdi Caillau Company, 28, rue Ernest Renan 92130 Issy-les-Moulineaux, France C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 143–170. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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Simulated Annealing (SA) [38], Tabu Search (TS) [21], Particle Swarm Optimization [52][31]. Although metaheuristic are numerous the basic concepts are common: the goal of this method is to explore the searching space using global memory mechanism with ”interesting solutions” and selection method. The main idea of this chapter is the introduction of fuzzy logic in order to improve the solutions found by metaheuristic. Indeed fuzzy logic may improve the classic metaheuristic by introducing dynamic characteristic into the latter. The rest of this chapter is organized as follows: section 2 introduces the main principles of scheduling problem, then it shows the existing connections between fuzzy logic and production scheduling. Section 3 deals with the definition of reentrant scheduling problem. This one solved using Fuzzy Logic and Genetic Algorithm. section 4 develops the single objective optimization, whereas section 5 deals with multiobjective one. The last part of this chapter presents the conclusion and resume the contribution of the fuzzy logic in the production scheudling.

2 State of the Art Firstly, we recall the main principle of scheduling problem and its connection with the fuzzy logic. Scheduling problem is an well-known area of the OR domain. This problem consists in organizing tasks to be processed on one or several machines. To explain the scheduling technique, some its common concepts area is defined in this part [51] as job, resource, constraint and criterion. A job is a task (product for example in the production scheduling). A job is defined by a set of informations such as release date (from which the job is available for processing), processing time, completion time and due date. A job is usually composed of several successive operations. Resources are production means (machines in production scheduling, for example) available to perform the jobs. Resources can be renewable or consumable. A resource is called renewable if it becomes available after its use and can be of two types: disjunctive or cumulative. A resource is called consumable if it is available after its use. A constraint is a necessary condition for achievement of the jobs. Many constraints exist in this area as: preemption (each which has began can be interrupted until the end of the operation), splitting (a job can be divided into smaller ones). The criterion is the objective to be optimized. Many criteria exist concerning production scheduling as: minimization of the makespan (required duration to achieve all jobs), minimization of the total tardiness (sum of the tardiness for all jobs), minimization of the sum of the completion times of all jobs. Several criteria can be optimized at once or combined in a weighted sum if necessary. Since fuzzy sets theory has been introduced more than 20 years ago, it has had an unexpected growth.

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Coffin and Taylor [10] presented a multiple criteria model for scheduling. This model uses fuzzy logic and a standard beam search. The fuzzy beam search model is applied via a twenty project examples problem using three primary goals: maximizing expected profit, maximizing average success probability for the portfolio, and minimizing the makespan of the portfolio. The quality of the solutions is provided trough a comparison between the results from the model and those generated by a complete enumeration procedure. The study shows that the fuzzy beam search model is able to generate high quality solutions, using very small beam widths, moreover it needs less computational efforts than exact methods. Bugnon et al. [2] described an approach based on a fuzzy rules controller which can be dynamically adapted following different perturbations in a shop. This method provides a dynamic approach to solve efficiently real-time scheduling problems. Slany [47] developed a method which combines repair-based methods and fuzzy constraints. This algorithm can solve real-world multi-criteria decision making, especially scheduling problems. This method reaches a compromise between different criteria, moreover it asses priorities among fuzzy constraints. The results obtained from a steel making application indicate the efficiency of the proposed approach compared to constructive non-fuzzy methods in terms of modeling and performance. Grabot et al. [23] suggested in there paper a Decision Support System (DSS) that helps to take into account the different functions of production activity control (PAC), and completes the schedule module of PAC. In order to provide an efficient help, the key point is certainly express the implicit workshop objectives. Fuzzy logic aspects is hidden transparent for the end user by using linguistic labels and modifiers in the front end of the application. The implementation of this DSS in a real workshop is realized and the first tests have shown the importance of expert knowledge in the workshop management. Chan et al. [4] showed that scheduling of manufacturing system is made up of several decision points at which a decision rule should be applied: for example, the selection of an operation among several alternatives. A simple operation-selection rule, such as SNQ (Shortest Number of jobs in Queue), always blindly pursues a single objective, despite more than one objective are important. To fix this problem, the authors introduced an intelligent approach to a multi-objective real-time alternative operation. Each alternative operation is evaluated, and its contribution to system performance using membership functions is calculated. The proposed method is easy to apply in a simulation model. They mentioned that the objective set used in this paper is not exhaustive, but is a significant generic issue in machine selection. The results of the proposed method show a good improvement in some performance measures, such as net profit and average lead time. Allet [1] interested in representing a practical case of a pharmaceutical company in 2003. This problem was already treated in the classical case

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(fixed data and strict constraints) and modeled as a generalized job shop problem. In this model, several aspects of this problem were neglected (existence of preference relations on the possible values of due dates and delay between successive operations of the same job). In this work, a new model is proposed, with use of fuzzy logic and flexibility on delay and due dates. A new method generalizing the previous ones is developed. The former considers the existence of the preference relation on each of the two parameters. Suhail and Khan [48] studied the development procedure of a fuzzy control system (FCS) for production processes. The latter are perfectly balanced over time but need to be controlled due to the randomness involved by their functioning. The authors try to control it dynamically with FCS system in different configuration and in taking into account the randomness operator. In this section, a state of the art have been presented. This one allow us to see the differents work done on the scheduling problem under fuzzy control. The different works cited bellow show that the fuzzy logic controller can be used in different ways such as in the multiobjective aproach, to choose the creterias. As resumed, the FLC can be considred as a tool for decision support. In the next section, another side of scheduling problem will be presented which is a reentrant scheduling problem. The problem will be described, and solution method to solve it using a fuzzy logic controller will be presented and compared with a standard ones.

3 Reentrant Scheduling This section is devoted to a variant of scheduling problem which is the reentrant one. This reentrant scheduling problem is increasingly studied in the literature especially in the electronics field. In smart card production for example, each product have to be checked one time or more to ensure his efficiency. The reentrant side of this scheduling appears when the products have to be processed two times or more on the system.

3.1 Problem Description This section presents the description of the reentrant scheduling problem. Especially, this one is about reentrant hybrid flowshop (RFS). The system illustrated by figure 8.1 is composed of M stages. Each stage j consists in Mj parallel machines. The input of the system is realized by a set of jobs called orders o. There are no orders to be processed. Each order is divided into several independent batches depending on its size. We have two types of orders (as well as batches): the first is treated once on the stages (single-pass orders), second type have to visit twice or more the stages (double-pass orders). The same designation is applied to batches.

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Single−pass lots

Double−pass lots

Stage 2

M

Stage 3

Stage 4

M

M

M

M

M

M

M

M

M

M

M

M

Fig. 8.1 Flow shop hybrid reentrant: 4 stages, (4,2,4,3) machines

In this problem, the objective is to minimize total tardiness. The latter is the sum of the tardiness of each job defined by the duration that exists between completion time and due date. Each machine can process only one batch at once. We note that the treatment of a batch can be considered as a task (or a job), and the treatment of a batch on each stage can be considered as an operation. It is assumed in this study, as for Choi et al. [8], that: • • • •

All orders are available at time t = 0 (no release date) No batch could be preempted (but orders can be preempted); There is no failure of machine. There is no setup time between batches.

We consider a problem with static scheduling. We assume that orders incoming during the horizon h are scheduled before end of this horizon. We use the third assumption, when the machines are not down in general, otherwise, if so, the repair time is considerably smaller (regular preventive maintenance). An order is considered finished when all batches of this order are finished.

3.2 Context in Reentrant Scheduling Recently significance of Reentrant Flow Shop (RFS) scheduling problem has grown up. The first study on reentrant system is realized by Graves et al. [24]: the authors presented a production of electrical circuits as RFS, where the objective is to minimize the average processing time for a given rate of production. The reader can refer to the survey of Gupta and Sivakumar [26] to get a review on reentrant scheduling problems. This review is updated hereafter with some of the most important advances in this field. This short review recalls firstly the papers which concern the makespan minimization and then the ones which concern the total tardiness minimization. Kubiak et al. [33] examined the scheduling of reentrant lines to minimize the sum of the completion times. They considered a class of reentrants lines in which jobs take the path of M1 , M2 , M1 , M3 , . . . , M1 , Mm and M1 , they demonstrated that the rule SPT (Shortest-Processing-Time) is optimal if

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some restrictive conditions are kept. Hwang and Sun [28] interested on flow shop problem with two machines and reentrant lines and repair time which is sequence dependent for minimizing the makespan (maximum completion time). Demirkol and Uzsoy [12] proposed a decomposition method to minimize the maximum tardiness for the RFS with repair time which depends on the sequence. Pan and Chen [43] studied the RFS. Their aim is to minimize makespan and the average total completion time of tasks. They proposed linear program, and heuristics based on active scheduling without due date. Drobouchevitch and Strusevich [15] developed an algorithm to solve the two machines reentrant flow shop scheduling problem with makespan minimization. They guaranteed performance in worst case for a ratio of (4/3), and then it reduces the best ratio bound of (3/2). Wang et al. [49] suggested the scheduling of workshop composed of reentrant lines where each task passes firstly on a machine called primary machine, after it pass on the other machines with a given sequence, and finally it returns to the primary machine at the last operation. The objective function of their problem is to minimize the makespan, they proved that when the number of machines is equal to two, there is an optimal scheduling that can be splitted into three segments, where the first and third segments are scheduled by Johnson’s rule [29] and the second segment is scheduled arbitrarily. From this result, they proposed three approximation algorithms and they compared them with the Branch-and-Bound optimization algorithm. Kooll and Pot [32] studied the minimization of the workload of reentrant lines, with exponential service time and orders preemptive policies. They used a numerical algorithm called power series algorithm, they obtained policies for systems with eight queues. Choi et al. [9] presented a scheduling algorithms for two-stage re-entrant hybrid flow shops for minimizing makespan under the maximum allowable due dates. Chen et al. [7] studied the reentrant (RFS) flow shop problem using hybrid tabu search to minimize the makespan. Dugardin et al. [20] solved reentrant lines scheduling. They proposed a multi-objective algorithm called L-NSGA based on Lorenz dominance. The objective is the minimization of the cycle time and the maximization of the bottleneck utilization rate. The model used is a queuing system where the queues are managed by priority rules. They also compared in [16][19] their method with a multiobjective ant colony system (MOACS). Few studies are applied to the RFS scheduling problem minimizing total (or maximum) tardiness (or lateness). Ovacik and Uzsoy [42] suggested heuristics to solve RFS scheduling problem and to minimize the maximum tardiness. Bertel and Billaut [3] suggested a genetic algorithm to minimize the weighted number of tardy jobs concerning the hybrid flow shop with recirculation. Chen et al.[7] carried out a case study of a scheduling problem on integrated circuits multi-stages with reintroduction, they have presented three algorithms. Dugardin et al. developed a multiobjective method [17] to solve a reentrant hybrid flow shop problem which involves total tardiness.

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Table 8.1 Recent works about reentrant shops Year 2004 2005 2006 2007 2008 -

author Bertel and Billaut Pan and Chen Wang et al. Gupta and Sivakumar Miragliotta and Perona M¨ onch et al. Choi et al. Chen M¨ onch et al. Chen et al. Hsieh et al. Kang et al. Pearn et al. Manikas and Chang Yang et al. Choi et al. Chen et al.

reference [3] [44] [50] [25] [37] [39] [8] [5] [40] [7] [27] [30] [45] [36] [53] [6] [6]

Objectives  min wj Tj 2 min Cmax min Cmax 3 different 3 min W.I.P.  min  Tj min Tj min C max min Tj min Cmax 3 different  min Tj min workload weighted sum min Cmax min Cmax min Cmax

Method G.A. M.I.P. global heuristic heuristic and simulation decentralized heuristic distributed heuristic Heuristic, list scheduling B&B G.A. Tabu Search Scheduling policy 3 phases algorithm Heuristic G.A. B&B B & B and Heuristic modular algorithm

 :  cj is the sum of the jobs completion time 2 : wj Tj is the weighted sum of the tardy jobs 3 : Work In Progress 4  : Tj sum of the jobs tardiness 1

Choi et al. [8] have studied the problem of scheduling orders with reentrant batches in a hybrid flow shop to minimize the total tardiness. Finally table 8.1 summarizes the recent results of the literature concerning reentrant scheduling problems.

4 Single Objective Optimization In this section, the reentrant hybrid flow shop scheduling problem illustrated by figure 8.1 is solved. The objective is to minimize the total tardiness of the batches. We solve the problem using a method which involves Genetic Algorithm guided by Fuzzy Logic Controllers (FLC-GA) and we improve a heuristic called OMN1 and solve our problem with it. Finally we compare all of this results to get efficiency of the different methods.

4.1 Fuzzy Genetic Algorithm The basic methods presented hereafter is a Genetic Algorithm (GA) Since the efficiency of this method is firmly established, we just recalled here that

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GA is a population based metaheuristic which combined different crossover and mutation of chromosomes in order to select efficient solutions for several problems Encoding chromosome Each chromosome in the population defines a solution for the scheduling problem. Each chromosome is composed of the sequence of tasks at each stage. Figure 8.2 shows a chromosome which defines the scheduling of 3 tasks on three stages. The size (number of genes) of a chromosome is the product of the number of tasks multiplied by the number of stages. Each gene of a chromosome contains an integer number that represents the index of the task. stage1 stage2 stage3 21 3 21 3 12 3 Fig. 8.2 Chromosome encoding

Crossover The crossover operator consists to create two children by crossing one or more genes from two parents. For solving our problem, a LX procedure has been applied in each stage. The crossover is applied in each iteration reguarding the updated value of his probability operator for crossing pc . This probability parameter is initially set to: pc = 0.9. The update is made under the FLC (fuzzy logic control). Mutation The objective of the mutation is to prevent the algorithm from being trapped untimely in a local optimum. Two points are randomly selected on the

stage1 stage2 stage3 P arent 1 2 1 3 2 1 3 1 3 2 P arent 2 1 3 2 2 3 1 3 2 1 Fig. 8.3 Parents : select point for crossing stage1 stage2 stage3 Child 1 1 3 2 2 3 1 3 2 1 Child 2 2 1 3 2 1 3 1 3 2 Fig. 8.4 Children : new chromosome

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chromosome. In our case, random mutation is generated at each stage of the chromosome to disturb solutions. As the crossover operator the mutation operator happens with a probability of pm which is also under the FLC. This parameter is set to: pm = 0.1, at the beginning of each algorithm. Stopping criteria In the literature two types of stopping criteria exist: either the number of generations is fixed, or the search process stops when the best objective function is not improved for a fixed number of generations. In this study the two criteria are combined. The hard limit concerning the number of generations is set to: Ng en = 2000 generations. Fuzzy logic As the previous section shows, the genetic algorithm (GA) used here has two probability parameters: the probability of mutation denoted by pm and the probability of crossover denoted by pc which are usually constant. In their paper Lau et al. [34] introduced a Fuzzy Logic Controller (FLC) which aimed to set proper parameter values at each iteration of the GA. In the present section, the FLC works is explained, hereafter this explanation is illustrated by an example. Decision table Membership function of f Membership function of d f(t−1) f(t−2) d(t−1)

FLC

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Fig. 8.5 Description of FLC which controls pc

The fuzzy logic was introduced by Lotfi Zadeh [54], firstly as data processing and then used to control system. It consists in introducing noise trough membership function and making a decision by using a table. The functioning is explained below. The FLC is made up of 3 steps: fuzzyfication, decision making and defuzzyfication. In our algorithm two different FLCs exist: one dedicated to the probability of mutation pm and another one dedicated to the probability of crossover pc . The functioning of the first FLC is described here, as this can be easily adapted to the FLC which concerns pm .

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NLR

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The FLC which controls pc value is described by figure 8.5. Some informations and parameters are considered: the iteration denoted by t, the probability of crossover is denoted by pc (t) at this iteration and by pc (t − 1) and pc (t − 2) at the two former iterations. Parameter pc is calculated by pc (t) = pc (t − 1) + Δpc , FLC provides the value of Δpc by using the value of the difference between f (t − 1) and f (t − 2) which are the average value of the objective function (in this section the total tardiness) at the iteration t − 1 and t − 2 respectively. More over FLC uses the parameter d(t) which is the sum of the hamming distance between each individuals of the entire population (see the paper of Lau et al. [34] for more details). A value of pc (t − 1) = 0.5 is considered in the following example. The aim is here to provide the value of p(t) trough the FLC. The first step in the FLC is the fuzzyfication of the variables d and f (t − 1)−f (t−2) trough the membership functions described by figure 8.6 and 8.7. The different part (NLR, . . . , PLR) are called linguistic terms. This is a mean to described the objective function (here the value of the total tardiness). For example, if a large value of the difference between f (t − 1) and f (t − 2) is considered and f (t − 2) > f (t − 1) then the membership function will provides a value in NL (negative large) or NLR (negative larger), . . . All of the linguistic terms are defined in table 8.2 and concerning d(t) the linguistic terms (VS, . . . , VL) are in table 8.3.

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Fig. 8.8 Exemple of fuzzyfication

Concerning the membership function of f (t − 1) − f (t − 2), it must be noticed that a coefficient α exists. This coefficient is settled here for scaling the function to fit the difference f (t − 1) − f (t − 2). Here a scale of α is equal to 1000. An example is provided here: if the difference f (t − 1) − f (t − 2) = f1 = −3300, it means that the sum of the total tardiness over the entire population at iteration (t − 1) is 3300 less than this sum at iteration (t − 2). Then the membership function of figure 8.8 is used, and the value f1 is placed: the function provides us NLR with a proportion of 0.3 and NL with a proportion of 0.7. The same is done with d(t) = d1 = 0.27 which provides S with a proportion of 0.4 and SS with 0.6. After the fuzzyfication comes the decision making. This step is achieved by the decision table denoted by table 8.4 (respectively with table 8.5 concerning pm ). This table allows to make a decision with by using the value of the member ship function of (f (t − 1) − f (t − 2)) (NLR, . . . , PLR) and that of d(t) (VS, . . . , VL). It must be stressed here that this two tables are in conflict with each other. Indeed, table 8.4 means that if the population is very different concerning the decision space (f (t − 1) − f (t − 2) is NLR) as well as concerning the searching space (d(t) is VL) the decision table increases the intensification in providing higher value for Δpc (indeed Δpc is set to PLR). Concerning table 8.5 if the same values are considered this table decreases the value of pm by providing NLR for Δpm . The example of the previous paragraph to explain how using the decision table (depicted by table 8.4) is resumed as follow: Several possibilities exists: NLR(0.3) or NL(0.7) and S(0.4) of SS(0.6). The first possibility is NLR(0.3) and S(0.4), knowing that the combination of NLR and S provides PS, moreover the minimum value between 0.3 and 0.4 is kept which is 0.3. Then PS(0.3) is kept. The 3 remaining cases are: N LR(0.3)⊗SS(0.6) → P S(0.3), then N L(0.7)⊗ S(0.4) → Z(0.4) and finally N L(0.7) ⊗ SS(0.6) → P S(0.6). The final step is the defuzzyfication: this is done by using the membership function of figure 8.6. Indeed depending of the value of parameter α this

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Fig. 8.9 Exemple of defuzzyfication Table 8.2 Linguistic terms for pc , pm , Δpc and Δpm [34] linguistic term meaning NLR Negative larger NL Negative large NM Negative medium NS Negative small Z Zero PS Positive small PM Positive medium PL Positive large PLR Positive larger

Table 8.3 Linguistic terms for d [34] linguistic term VS S SS LM M UM SL L VL

meaning Very small Small Slightly small Lower medium Medium Upper medium Slightly large Large Very large

membership function concerns f (t − 1) − f (t − 2), Δpc or Δpm . Here the aim is to compute the Δpc then α = 0.02 (this value is determined after several experiments). The defuzzyfication begin with the sum of the maximum value for each linguistic term as NLR, NL, . . . , PLR. Following the former example, the next step is to compute the sum of P S(0.3), P S(0.3), Z(0.4), P S(0.6). The result is shown in figure 8.9. Then the last step is to compute the value of Xg the abscissa of the gravity center

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Table 8.4 Decision Table pc [34] d(t − 1) fa NLR NL NM NS Z PS PM PL VL PLR PLR PL PL PM PM PS PS L PLR PL PL PM PM PS PS Z SL PL PL PM PM PS PS Z NS UM PL PM PM PS PS Z NS NS M PM PM PS PS Z NS NS NM LM PM PS PS Z NS NS NM NM SS PS PS Z NS NS NM NM NL S PS Z NS NS NM NM NL NL VS Z NS NS NM NM NL NL NLR

PLR Z NS NS NM NM NL NL NLR NLR

Table 8.5 Decision Table pm [34] d(t − 1) fa NLR NL NM NS Z PS PM PL VL NLR NLR NL NL NM NM NS NS L NLR NL NL NM NM NS NS Z SL NL NL NM NM NS NS Z PS UM NL NM NM NS NS Z PS PS M NM NM NS NS Z PS PS PM LM NM NS NS Z PS PS PM PM SS NS NS Z PS PS PM PM PL S NS Z PS PS PM PM PL PL VS Z PS PS PM PM PL PL PLR

PLR Z PS PS PM PM PL PL PLR PLR

of the shaded area of figure 8.9. Let x be the horizontal axis of figure 8.9 (from −5α to 5α), let h(x) be height of the shaded area of figure 8.9 (for example h(−5α) = 0 and h(α) = 0.6), the abscissa Xg is computed by the following equation: 

5α

Xg =

xh(x)dx x=−5α

In this example Xg is equal to 0.011, then Δpc = 0.011. The final step is to compute the value of pc (t) concerning the iteration t from the value of pc (t−1) and Δpc . The following equation is considered : pc (t) = pc (t−1)+Δpc. When a value of pc (t − 1) = 0.5 is considered, then pc (t − 1) = 0.5 + 0.011 = 0.511. Finally it must be noticed that hard limit concerning pc and pm have been set: pc ∈ {0.5, . . . , 1.0} and pm ∈ {0, . . . , 0.25}. Main principles of the FLC mechanism have been illustrated, for more details the reader can refers to the works of Lau et al. [34].

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4.2 Heuristics In this section we provide heuristics to solve this problem. First we recall definition of OMN1 heuristics from Choi et al. [8]. Then we propose an heuristic which improves the former one. OMN1 of Choi, Kim and Lee [8] In this algorithm, orders priorities are determined at each stage using a constructive method as NEH algorithm of Nawaz et al. [41]. We note that the batches of the same order have the same priority at stage, but it could have different priorities depending on the stage. Thus, in this algorithm a set of orders priorities is determined at each stage. Sets of orders priorities (sequences) are determined at each stage, starting with the first one. The orders sequences are obtained as explained hereafter. At first, a set of orders priorities is determined in the first stage with a constructive approach. In this one, an order is selected and inserted in the best position of scheduled set of priorities. On this stage, the initial sequence is selected using the Earliest Due Date (EDD) rule. Secondly, to select a better position to insert an order, the sequence inculding subset and the reste of the partial sequence priorities is evaluated. The same one are considered in all the succeeding stages. With these sets of orders priorities given, a complete scheduling of batches using list-scheduling method is obtained, and the total tardiness of the the batches are caculated. The reentrant ones have a highest priority than nonreentrant. Also, the priorities of reentrant batches are equal to its priorities of last stage during its first passages. These priorities of reentrant batches are kept with all stages. OMN1.M1 Agorithm This algorithm is similar to OMN1 algorithm proposed by Choi et al. [8]. In the inserting step of an order in the best position in partial sequence, the numerical computation of orders due date is different. In our algorithm, we calculate due dates for each order at each stage with the formula below: di,j = di,j+1 − Pi,j+1 , ∀i = 1, ..., no , ∀j = 1, ..., M With : i j di,j Pi,j

: : : :

Index orders Index stages Due date of order i at stage j Processing time of order i at stage j

(8.1)

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4.3 Performance Evaluation In real industry the EDD rule (the earliest due date) is used to schedule the studied system. The evaluation of algorithms is made on randomly generated problems. The generator is designed to fit to the real systems. As in the study of Choi et al. [3], we generate 324 instances : four problems for each of all combinations of three levels of number of orders (10, 15 and 20), of three levels of number of stage (4, 6 and 8), and nine levels for the parameters used to generate due dates. In the tested problems, flexible due dates is controlled by two parameters: T and R, called the due date factor and the range of due date respectively. The values of T are as follow 0.1, 0.1, 0.1, 0.1, 0.1, 0.3, 0.3, 0.3, 0.5 and for R, the following values have been chosed 0.8, 1.0, 1.2, 1.4, 1.6, 0.8, 1.0, 1.2, 0.8 the values of R and T are mixed to obtain 81 couples. The due dates are generated by: R R ), P (1 − T + )) 2 2 where DU (x, y) is uniform discret distribution in [x, y]. DU (P (1 − T −

P = maxj {mini {

j−1 

Pil } +

l=1

nb m  1  . Pij + mini { Pil }} Mj i=1 l=j+1

P is the lower bound of makespan and nb is the total number of batches to be processed. The number of machines (Mj ) at each stage and the number of batches of each order are generated by DU (1; 10) and DU (5; 15) respectively. All algorithms are coded with C++, a laptop have been used with a processor Intel Duo Core with 2.00 GHz frequency. The performances of algorithms are shown by a relative measure, called Relative Deviation Index (RDI) and the number of cases, where each algorithm obtains the best solution (NBS) as in the work of Choi et al.[8]. For each problem, RDI for an algorithm is defined as: RDI =

(Sa − Sb ) (Sw − Sb )

Where: Sa The value of objective function obtained by algorithm Sb The best objective function among those obtained by all algorithms tested Sw The worst objective function among those obtained by all algorithms tested If Sw = Sb RDI of all algorithms are set to 0 for the problem. The Table 8.6 shows the different results obtained by the four methods such as FLC-GA, GA, OMN1.M1, OMN1 and EDD. The results obtained by each method are summarized by its Related deviation index and the number of best solutions.

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Table 8.6 Algorithm performances Algorithm RDI average FLC-GA 0.012 GA 0.019 OMN1.M1 0.526 OMN1 0.746 EDD 0.842

N BS (%) 214 (66%) 194 (59%) 4 (1.2%) 1 (0.3%) 1 (0.3%)

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This table shows that FLC-GA method gives best number of best solutions with 214 solutions which represent 66%. It shows also that GA is not far from the FLC-GA with a number of best solution equal to 194 (59%). With 1.2% and 0.3%, we can see that the OMN1.M1 and OMN1 which generally gives a very good solutions are completely dominated by the ones obtained by the FLC-Genetic Algorithm and the Genetic Algorithm. To compare the efficiency of the FLC-GA compared to the standard GA, different side of the solution obtained by the two algorithms has been studied such as the evolution of the objective function and the diversity of the population under generation evolution. Figure 8.10 shows that the solution of FLC-GA improves those of GA (the evolution of the objective function). Indeed, the FLC-GA find better results at the final generation (2000). In the Figure 8.11, 8.12, and 8.13, the quality of the population of the FLCGA and that of the GA have been compared, by providing all solution of the

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population at generation 0 (initial one), 1000 and 2000 (final one). Figure 8.11 shows that FLC-GA becomes better population than GA one: many solutions found by FLC-GA have lower criterion than solutions found by GA. In the middle of the execution, the GA made up for its difference with FLC-GA even if FLC-GA has a short advantage. Finally at last iteration FLC-GA provides several interesting solutions whereas GA provides only few interesting solutions:: FLC-GA continues to explore the searching space where as GA is stopped. The hybrid reentrant flow shop scheduling problem has been solved where the aim is to minimize the total tardiness. The comparison between the five methods presented (FLC-GA, GA, OMN1, OMN1.M1, and EDD) shows that the genetic algorithm under fuzzy logic control are better than all the rest than the other methods. Opposite to the standard genetic algorithm, the FLC-GA adjust its parameters dynamically with updating the probabilities of crossover and mutation regarding on the quality of solutions obtained through the generations. The heuristics (OMN1, OMN1.M1, EDD) are generally efficient to solve this type of problem but dominated in this study by the genetic algorithms. In the next section, the efficiency of FLC is tested on a multi-objective genetic algorithm. The same problem will be solved with the two objectives which are the minimization of the total tardiness and the makespan.

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5 Multiobjective Scheduling Optimization In this section, multiobjective optimization of a reentrant hybrid flowshop is described. The main difference with section 4 is that multiple criteria are optimized at once. This area of operational research uses special optimization methods as multiobjective genetic algorithms (SPEA2 [56], NSGA2 [11], LNSGA [20]), Multiple ants colony system [19], or exact methods as k-PPM from [13]. A promising method is used here to solved a scheduling problem: it consists of genetic algorithm controlled by fuzzy logic, the aim here is to compare results obtained by usual multiobjective genetic algorithm NSGA2 from Deb et al. [11]. The system is described at section 5.1, then the methods used is depicted in the section 5.2 and finally the results are discussed in section 5.3.

5.1 Description of the Problem The problem studied here is connected with the problem of section 3.1. The main difference is that in this part machines are preceded by a queue with several priority rules. Moreover the processing times are stochastic. The system studied here is an hybrid flow shop: each product must undergoes successively trough stage E1 then trough stage E2 . There are two different stages, E1 and E2 composed with M1 and M2 machine respectively. Finally this shop is reentrant: each product must be processed a number of L time at each stage. Description is completed by these notations : there is a number of N jobs to be processed in this reentrant hybrid flow shop, each job i ∈ {1, . . . , N } has a processing time pi generated by an exponential distribution. Finally the system has indirect scheduling: each machine of each stage has a separate input queue. This queue is managed by one priority rule among the eight ones of table 8.7. The type of scheduling leads to the coding strategy of the figure 8.14. The system is evaluated trough simulation. Actually a discrete event simulation module has been developed: it provides the two objectives for each solution. Each simulation has an horizon H of H = 4 hours. It is assumed that this horizon is large enough to process all the submitted jobs. A difficulty has arisen because some events of the system are stochastic then the evaluation can fluctuate from an evaluation to another one. This leads us to evaluate several times all the obtained solutions. On account of these multiple evaluations, a compromise must be considered between the computational time and the precision of the evaluation. The computational time of each replication is Trepli = 0.015 sec. Each evaluation is called replication and the number of replication is denoted by Nrepli . Here 100 000 replications have been done and this shows that a number of Nrepli = 60 replication is enough.

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Table 8.7 Sequence rules [20] gi Rule 0 LOGistical time MINimization 1 First In First Out 2 Last In First Out 3 First Buffered First Served 4 Last Buffered First Served 5 Shortest Remaining Processing Time 6 High priority Order Transition 7 Fluctuation Smoothing of the Mean Cycle Time (FSMCT∗ ) ∗

Definition The product corresponding to the previously installed operation on the unit is chosen. The first arrived product is served first. The last arrived product is served first. The product with the smallest cycle number in the cell is served first. The product with the highest cycle number in the cell is served first. The product with the smallest processing time before the end of the process is served first. The product with the highest priority is served first. The product with the smallest quantity αi + ψi , is served first (where αi is the arrival time of the product in the system and ψi is the processing time on the machine).

: FSMCT is defined by Lu et al. [35]

246 3154 M1 M2 Fig. 8.14 Example of solution when M1 = 3 and M2 = 4 [19]

5.2 Algorithms Fuzzy Logic Controlled Genetic Algorithm The method used here is based on a well known multiobjective genetic algorithm from Deb et al. [11] called NSGA2. This type of algorithm provides competitive results compared with other metaheuristics but the tuning of the parameters is difficult and often empiric. Recently Lau et al.[34] used Fuzzy Logic Controller to guide the search ability of a genetic algorithm. The contribution here is to get the impact of such controller on a NSGA2 algorithm. The principles of Fuzzy Logic Controllers (FLC) are described in the works of [34] and we have recalled main lines of this methods in previous section. Membership functions and decision table used here, are described in the section 4.1. NSGA2 The NSGA2 will not be described here (see [11] for more details). Parameters settings concerning both algorithms are provided : size of the population is sizepop = 100, the number of replications is nrepli = 60 whereas the number

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of generations is ngen = 80 (it is short because of the computational time needed to realize one replication). Finally the probability of crossover pc and the probability of mutation pm are controlled by Fuzzy Logic Controllers. At the beginning of each experiment probability parameters are initialized to pc = 1.0 and pm = 0.1.

5.3 Experiments and Results In multiobjective optimization a recurring problem is to compare two solutions and in a wider range two set of solutions. Recently the literature shows several measure and we have choose two measures to distinguish between two fronts in all of the situation encountered. Firstly we have chosen the μd -distance based on the paper of Dugardin et al. [19] and that of Zitzler and Thiele [55]. We provide some notations: let be two Pareto optimal fronts F1 and F2 provided by the different methods, F1 consists of n1 solutions and F2 consists of n2 solutions. Then we define μd -distance by: n2 1 i=0 di n2 μ ¯d =  2 (f1max − f1min ) + (f2max − f2min )2 Where, di represents the distance between one point i ∈ F1 and its orthogonal projection on F2 (see[46]), di is positive if F1 is above F2 , negative otherwise. In addition f1max = max{f1 (x)|x ∈ F1 ∪F2 }, f1min = min{f1 (x)|x ∈ F1 ∪F2 } and f2max = max{f2 (x)|x ∈ F1 ∪ F2 }, f2min = min{f2 (x)|x ∈ F1 ∪ F2 }.

Fig. 8.15 Example of D diagonal [19]

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 Therefore we get D = (f1max − f1min )2 + (f2max − f2min )2 where D is the cross of the corresponding rectangle - see figure 8.15. We provide a second measure C1 which represents the Zitzler measure of the first member of the column (A in A/B comparison) C2 represents the other (B). C1 represents the proportion of solutions obtained by A which are dominated by at least one solution obtained by B whereas C2 measures the proportion of solutions obtained by B dominated by at least one solution of A. Finally the number of solutions ns in the optimal fronts is mentioned. NSGA2 and FLC-NSGA2 are compared in table 8.8. Tests have been realized on 24 different configurations of the system: each configuration is denoted by (n,M1 ,M2 ,L). The different values are n ∈ {2, 5, 8, 15}, M1 ∈ {1, 3}, M2 ∈ {1, 3, 6} and L = 6, finally each processing time is generated through a uniform distribution in [15; 30]. Each configuration has been executed 10 times with 10 different instances. The nf value shows the number of fronts compared. In fact due to the C1 /C2 measure, the number of solutions in each Table 8.8 Comparison between NSGA2 and FLC - NSGA2 n 2 5 8 15 2 5 8 15 2 5 8 15

C1 , ns1 C2 , ns2

M1 1 1 1 1 3 3 3 3 3 3 3 3

2 5 8 15 2 5 8 15 2 5 8 15 and and

M2 1 1 1 1 3 3 3 3 6 6 6 6

L 3 3 3 3 3 3 3 3 3 3 3 3

nf μ ¯d (10−4 ) σμ (10−4 ) 1 4 0 6 2 2 2 12 12 8 2 2 1 -12 0 3 23 17 6 48 101 1 6 0 1 -16 0 2 2 7 4 4 3 6 -9 23

1 1 5 1 3 3 1 1 5 5 1 1 1 5 4 -1 1 1 5 4 3 3 5 0 3 4 3 3 5 3 4 3 3 5 5 0 3 3 5 3 3 6 5 0 2 -2 3 6 5 2 -2 3 6 5 4 1 3 6 5 3 t1 concern FLC-NSGA2. t2 concern NSGA2.

0 3 3 2 1 5 4 0 1 4 4 0

C1 0.50 0.15 0.00 0.01 0.10 0.08 0.08 0.00 0.00 0.08 0.15 0.00

C2 0.78 0.91 1.00 0.87 0.89 0.88 0.88 0.92 0.86 0.92 0.93 0.87

ns1 9.0 11.8 9.0 9.3 9.0 8.3 8.8 12.0 7.0 11.0 11.5 10.0

ns2 10.0 11.7 10.0 10.0 10.0 8.7 8.8 13.0 7.0 10.5 10.8 9.5

t1 22 135 160 316 17 89 136 290 33 61 221 322

t2 16 119 125 271 17 73 115 242 22 56 183 278

0.00 0.00 0.05 0.00 0.00 0.00 0.15 0.03 0.02 0.05 0.00 0.23

1.00 0.95 0.91 0.91 0.71 0.86 0.90 0.85 0.87 0.90 0.94 0.90

9.0 10.4 9.8 8.3 9.0 7.7 8.4 9.0 7.8 8.5 7.0 9.7

9.0 9.0 10.3 8.3 9.0 8.0 8.8 9.7 8.9 8.0 7.5 9.7

55 447 276 1275 58 291 294 1350 102 136 296 1091

53 385 208 1023 51 237 284 1215 113 135 284 1057

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utilization rate

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Fig. 8.16 Example of non dominated fronts obtained by FLC-NSGA2 and SPEA2

front must be not as much different as 15%. Then some instances do not allow to generate the C1 /C2 . Then there are only nf fronts that are considered to compute the value of C1 /C2 as well as μ ¯d . Table 8.8 shows that FLC-NSGA2 is competitive with NSGA2. The former outperforms the algorithm of Deb as C1 /C2 measure shows: that only 16 % of the overall tested configurations present dominated solutions which is very few. Morover μd > 0 on the major part of the tested configuration which seems that FLC-NSGA2 outperforms NSGA2. Computationnal time of FLC - NSGA2 is higher than NSGA2 ones with no more than 10 %. This is due to the computational of fuzzyfication and especially defuzzyfication with gravity center method which is time consuming. We present an example of the non dominated (best found) solutions provided by the two methods: FLC-SNGA2 and NSGA2. The results are illustrated by figure 8.16: the solutions provided by FLC-NSGA2 dominates solutions provided by NSGA2, even if the solution are close this is a promising way to improve algorithm (trough the modification of the membership functions). In order to get the population management of the two algorithms, we have depicted the entire population provided by the two algorithms. Figure 8.17 provides the best and the poorest front found by NSGA2 and FLC-NSGA2 whereas figure 8.18 shows these two fronts at the 100th generation (end of execution of algorithms). One can see that the two algorithms provide very different solutions at the beginning of the execution whereas near from the end of the execution the solutions seems to be concentrated in a short area

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0.72

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0.71

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0.69 FLC−NSGA2 (non dominated) NSGA2 (non dominated) FLC−NSGA2 (dominated) NSGA2 (dominated)

0.685

0.68 5.6

5.8

6

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6.6

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Fig. 8.17 Population of the algorithms at the 10th generation

0.725

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0.69

0.685 5.6

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6

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Fig. 8.18 Population of the algorithms at the 100th generation

of the searching space. It can be seen as in the single objective section, that FLC-NSGA2 continues to improve the results at the end of the algorithm whereas NSGA2 seems to stay in a local optimum.

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6 Conclusion This chapter presents the existing connections between scheduling and fuzzy logic. At first, the basics of scheduling are recalled and a review of recent works concerning reentrant scheduling problems is proposed. Afterward the contributions of fuzzy logic concerning scheduling are detailed. An application of fuzzy logic combined with genetic algorithm is presented. Actually this method controls a genetic algorithm with Fuzzy Logic Controller trough the settings of mutation and crossover parameters. This method allows to dynamically adapt the ratio between intensification and diversification during the execution of the algorithm. This chapter presents two different sides of the reentrant hybrid flow-shop scheduling problem. On the one hand the latter is solved by a deterministic as well as mono objective approach, on the other hand this problem is solved using stochastic and multiobjective methods. Our conclusion is that fuzzy can improve results of genetic algorithm, moreover it must be noticed that the choice of membership function is a central point to improve in this application. Finally this Fuzzy Controller mechanism must be tested on other metaheuristics, especially with ant colony system which are well known to be hard to setup. In a wider range, the coupling of metaheuristic with fuzzy logic is a interesting way of improvement, especially when the former are hard to tune up. Indeed, this can be a solution when the number of parameters is growing, with technique such as Ant Colony System or Particle Swarm optimization. Finally these methods can be interesting due to the variations of the parameters during the execution of the algorithm. This behavior can be adapted to speed up the search of the algorithm during the first iterations of the algorithm.

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Chapter 9

Interval PERT and Its Fuzzy Extension Didier Dubois, J´erˆome Fortin, and Pawel Zieli´ nski

1 Introduction In project or production management, an activity network is classically defined by a set of tasks (activities) and a set of precedence constraints expressing which tasks cannot start before others are completed. When there are no resource constraints, we can display the network as a directed acyclic graph. With such a network the goal is to find critical activities, and to determine optimal starting times of activities, so as to minimize the makespan. The first step is to determine the earliest ending time of the project. This problem was posed in the fifties, in the framework of project management, by Malcolm et al. [32] and the basic underlying graph-theoretic approach, called Project Evaluation and Review Technique, is now popularized under the acronym PERT. The determination of critical activities is carried out via the so-called critical path method (Kelley [29]). The usual assumption in scheduling is that the duration of each task is precisely known, so that solving Didier Dubois Universit´e Paul Sabatier, 118 route Narbonne, 31062 Toulouse Cedex 4, France e-mail: [email protected] J´erˆ ome Fortin LIRMM, Universit´e Montpellier 2 and CNRS, 161 rue Ada, 34392 Montpellier Cedex 5 France e-mail: [email protected] Pawel Zieli´ nski Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland e-mail: [email protected] 

The work was partially supported by Polish Committee for Scientific Research, grant N N111 1464 33.

C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 171–199. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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the PERT problem is rather simple. However, in project management, the durations of tasks are seldom precisely known in advance, at the time when the plan of the project is designed. Detailed specifications of the methods and resources involved for the realization of activities are often not available when the tentative plan is made up. This difficulty has been noticed very early by the authors that introduced the PERT approach. They proposed to model the duration of tasks by probability distributions, and tried to estimate the mean value and standard deviation of earliest starting times of activities. Since then, there has been an extensive literature on probabilistic PERT (see Adlakha and Kulkarni [1] and Elmaghraby [18] for a bibliography and recent views). Even if the task duration times are independent random variables, it is admitted that the problem of finding the distribution of the ending time of a project is intractable, due to the dependencies induced by the topology of the network [25]. Another difficulty, not always pointed out, is the possible lack of statistical data validating the choice of activity duration distributions. Even if statistical data are available, they may be partially inadequate because each project takes place in a specific environment, and is not the exact replica of past projects. In the simplest form of uncertainty representation, activity duration times are specified as closed intervals. Assigning some interval to an activity duration means that the actual duration of this activity will take some value within the interval but it is not possible at present to predict which one. Strangely enough, the PERT analysis with ill-known processing times modeled by simple intervals does not seem to have received much attention in the literature. The overwhelming part of the literature devoted to this topic adopts an orthodox stochastic approach, thus leading to the afore-mentioned very complex problem that is still partially unsolved to-date. To the best of the authors’ knowledge, the interval-valued PERT analysis seems to have existed only as a special case of so-called fuzzy PERT, in which ill-known durations are modeled by fuzzy intervals [4, 8, 37, 34, 33, 36, 38, 26, 27]. Interval uncertainty may indeed be considered as poorly expressive. Experts may also know likely values of duration times. Then, fuzzy intervals may be useful. A fuzzy duration time can be easily defined by an expert as a triangular fuzzy interval whose support is defined by all the possible duration times, and the core is the most likely one. Resorting to fuzzy set and possibility theory [15] for the modelling of ill-known task duration times may help building a tradeoff between the lack of expressive power of mere intervals and the computational difficulties of stochastic scheduling techniques. Fuzzy set theory also allows for the specification of local preferences [9]. This kind of methodology is not yet so common in operational research, even if quite a few works in fuzzy PERT and other types of fuzzy scheduling methods have been around for more than two decades (Dubois and Prade [13], Prade [37], Chanas and Kamburowski [4]), that is, quite early in the development of fuzzy set theory. Apart from the book by Loostma [31], overviews on various aspects of fuzzy scheduling can be found in a recent edited

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volume (Slowi´ nski and Hapke [39]) and in papers by Chanas and Kuchta [5] on graph-theoretic aspects, Werners [42] on fuzzy project management and Turksen [40] on fuzzy rule-based production management. An abundant bibliography on fuzzy set applications in production management is supplied in Guiffrida and Nagi [24]. However, many previous approaches to the fuzzy PERT problem are ad hoc. This chapter puts together various existing partial results on finding intervals containing possible values of earliest and latest starting dates and floats of tasks and criticality evaluation of tasks [6, 7, 11, 12, 22, 43], thus providing a complete solution to the PERT scheduling problem under the interval-based representation of uncertainty. We extend these results to the fuzzy-valued case by exploiting a very recent notion of gradual number, which provides a new outlook on fuzzy intervals [10, 17]. This allows us to apply the interval algorithms to PERT with fuzzy durations whose membership functions are possibility distributions for values of unknown task duration times. In consequence, we give some algorithms for determining the degree of necessary optimality of a task and a possibility distribution for values of latest starting dates. Similarly, one can obtain algorithms for determining a possibility distribution for values of earliest starting dates and the gradual number representing the fuzzy upper bound of the possibility distribution of floats (see [21]).

2 From Interval to Possibilistic Uncertainty In practice, precise values of parameters (data) may be unknown. In this section, we discuss two uncertainty representations, namely, by means of a closed intervals and by means of fuzzy intervals in the setting of possibility theory. We lay bare the link between interval and possibilistic uncertainty.

2.1 Possibilistic Framework One of the simplest form of uncertainty representations is to assume that values of unknown parameter a may fall within a given range specified by a closed interval [a− , a+ ]. It means that the actual value of this parameter will take some value within the interval A = [a− , a+ ] but it is not possible at present to predict which one. A fuzzy set, in particular a fuzzy interval, allows us to express the uncertainty connected with an ill-known parameter in a more sophisticated manner. A fuzzy set (see [15]) A˜ is defined by means of a reference set V together with mapping μA˜ from V into [0, 1], called a membership function. The value ˜ of μA˜ (v), v ∈ V, is called the degree of membership of v in the fuzzy set A. λ ˜ ˜ ˜ A λ-cut, λ ∈ (0, 1], of A is a classical set, i.e. A = {v ∈ V : μA˜ (v) ≥ λ}. Aλ is the set of all elements that have a membership degree greater or equal to λ. The cuts of A˜ form a family of nested sets, i.e if λ1 ≥ λ2 , then A˜λ1 ⊆ A˜λ2 .

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The support of A˜ is the set {v : μA˜ (v) > 0} and it will be denoted as A˜0 . A fuzzy set in R, whose membership function is normal, quasi-concave and upper semi-continuous is called a fuzzy interval (see [15]). We will additionally assume that the support of a fuzzy interval is bounded. It can be shown [15] that if A˜ is a fuzzy interval, then A˜λ is a closed interval for every λ ∈ [0, 1]. We can thus represent a fuzzy interval A˜ as a family of cuts A˜λ , λ ∈ [0, 1]. One can obtain the membership function μA˜ from the family of λ-cuts in the following way: μA˜ (v) = sup{λ ∈ [0, 1] : v ∈ A˜λ } and μA˜ (v) = 0 if v ∈ / A˜0 . − + A classical closed interval A = [a , a ] is a special case of a fuzzy one with membership function μA (v) = 1 if v ∈ A and μA (v) = 0 otherwise. In this case we have Aλ = [a− , a+ ] for all λ ∈ [0, 1]. Very popular and convenient in applications is fuzzy interval of the L-R type, denoted as (a− , a+ , α, β)L−R (see, e.g. [15]). Its membership function is of the following form: ⎧ 1 for v ∈ [a− , a+ ], ⎪ ⎪ ⎨ a− −v  − μA˜ (v) = L α  for v ≤ a , ⎪ ⎪ ⎩ R v−a+ for v ≥ a+ , β where L and R are continuous and nonincreasing functions, defined on [0, +∞). The parameters α and β are nonnegative real numbers. If L(v) = R(v) = max{0, 1−v} and a− = a+ , then we obtain a triangular fuzzy interval, which is shortly denoted by triple (a, α, β). Let us now recall the possibilistic interpretation of a fuzzy set. Possibility theory [15] is an approach to handle incomplete information and it relies on two dual measures: possibility and necessity, to express plausibility and certainty of events. Both measures are built from a possibility distribution. be attached with a single-valued variable a. The memberLet a fuzzy set A ship function μA is understood as a possibility distribution, πa = μA , which describes the set of more or less plausible, mutually exclusive values of the variable a. It plays a role similar to a probability density, while it can encodes a family of probability functions [16]. The value of πa (v) represents the possibility degree of the assignment a = v, i.e. Π(a = v) = πa (v) = μA˜ (v), where Π(a = v) is the possibility of the event that a will take the value of v. In particular, πa (v) = 0 means that a = v is impossible and πa (v) > 0 means that a = v is plausible, that is, not surprising. Equivalently, it means that the value of a belongs to a λ-cut interval A˜λ with confidence (or degree of necessity) 1 − λ. A detailed interpretation of the possibility distribution and some methods of obtaining it from the possessed knowledge are described in [15]. A degree of possibility can be viewed as the upper bound of a degree of probability [16].

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2.2 Fuzzy Interval as an Interval of Gradual Numbers Classical intervals model uncertainty in a Boolean way: a value in the interval is possible and a value outside is impossible. The idea of fuzziness is to move from the Boolean way to a gradual one. Hence fuzziness makes the boundaries of the interval softer and thus making uncertainty gradual. In order to model the essence of graduality without uncertainty, the concept of a gradual number has been recently proposed. Following the notation of [21], a gradual number is defined as follows: A gradual real number (a gradual number for short) r˜ is defined by an assignment function Ar˜ from (0, 1] to R. A gradual number can be seen as a number parametrized by a value λ ∈ (0, 1]. The arithmetic operations on gradual numbers are defined by operations on their assignment functions. For instance, if r˜ and s˜ are two gradual numbers, then the sum of r˜ and s˜ is defined by summing their assignment functions, that is ∀λ ∈ (0, 1] Ar˜+˜s (λ) = Ar˜(λ) + As˜(λ). The set of gradual numbers is the set of all functions from (0, 1] to R. It forms a commutative group according to addition with identity ˜ 0 (A˜0 (λ) = 0, ∀λ ∈ (0, 1], i.e. the standard zero). Indeed, the gradual number r) = ˜0. r˜ has an inverse −˜ r under the addition: A−˜r (λ) = −Ar˜(λ), and r˜ + (−˜ The subtraction, maximum and minimum are straightforwardly defined: (a) subtraction: Ar˜−˜s (λ) = Ar˜(λ) − As˜(λ), (b) maximum: max(Ar˜, As˜)(λ) = max(Ar˜(λ), As˜(λ)), (c) minimum: min(Ar˜, As˜)(λ) = min(Ar˜(λ), As˜(λ)). The product and quotient of gradual numbers can be defined in a similar manner. It is worth pointing out that most algebraic properties of real numbers are preserved for gradual numbers, contrary to the case of fuzzy intervals. However, contrary to real numbers, the maximum operation on gradual numbers is not selective: in general, max(Ar˜, As˜) = Ar˜ or As˜. For λ in some sub–ranges of (0, 1], max(Ar˜, As˜)(λ) = Ar˜(λ) and max(Ar˜, As˜)(λ) = As˜(λ) in the complementary range. Based on the notion of gradual number, one can describe a fuzzy interval A˜ by an ordered pair of two gradual numbers [˜ a− , a ˜+ ], where a ˜− is a gradual + ˜ ˜ lower bound of A and a ˜ is a gradual upper bound of A. In order to ensure the ˜+ must satisfy the following well known shape of a fuzzy interval, a ˜− and a properties: (a) Aa˜− is increasing function, (b) Aa˜+ is decreasing function, (c) Aa˜− ≤ Aa˜+ , i.e Aa˜− (1) ≤ Aa˜+ (1). Therefore, a fuzzy interval is an interval of gradual numbers bounded by a ˜− + and a ˜ (see Fig. 9.1). Selecting a gradual number from this interval boils down to picking one element in each λ-cut. ˜+ ] which verifies the three preceding conditions describes a A pair [˜ a− , a fuzzy interval with membership function:

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a ˜−

a ˜+

v Fig. 9.1 The left and right bounds of fuzzy interval A˜ (in bold)

⎧ sup{λ | Aa˜− (λ) ≤ v} ⎪ ⎪ ⎪ ⎨ 1 μA˜ (v) = ⎪ sup{λ | Aa˜+ (λ) ≥ v} ⎪ ⎪ ⎩ 0

if v ∈ Aa˜− ((0, 1]) if Aa˜− (1) ≤ v ≤ Aa˜+ (1) if v ∈ Aa˜+ ((0, 1]) otherwise

Conversely, an upper semi-continuous membership function μA˜ (v) of a fuzzy interval can be described by a pair of gradual numbers [˜ a− , a ˜+ ] defined as follows: Aa˜− : (0, 1] → R λ→

Aa˜− (λ) = inf{v | μA˜ (v) ≥ λ}, Aa˜+ : (0, 1] → R λ → Aa˜+ (λ) = sup{v | μA˜ (v) ≥ λ}. In particular, every fuzzy interval of the L-R type A˜ = (a− , a+ , α, β)L−R can ˜+ ]. If the shape functions be described by a pair of gradual numbers [˜ a− , a are invertible, the assignment functions are given by the following equations: Aa˜− (λ) = a− − L−1 (λ)α and Aa˜+ (λ) = a+ + R−1 (λ)β.

(9.1)

To be consistent with the notion of λ-cuts, we extend the domains of assignment functions Aa˜− and Aa˜+ to interval [0, 1] such that [Aa˜− (0), Aa˜+ (0)] = A˜0 . For a deeper discussion on gradual numbers, we refer the reader to [10, 17, 21].

3 From Classical to Interval PERT: The Basic Setting In this section, we briefly recall the classical PERT model. Then we formally introduce the interval-valued PERT model, and the results instrumental in devising algorithms for solving the interval PERT.

3.1 The Classical PERT Model An activity network is classically defined as a set of activities (or tasks) with given duration times, related to each other by means of precedence

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constraints. When there are no resource constraints, it can be represented by a directed, connected and acyclic graph G =< V, A >, where V is the set of nodes (events), |V | = n, and A ⊆ V × V is the set of arcs (activities), |A| = m. We use the activity-on-arc convention. The set V = {1, 2, . . . , n} is labeled in such a way that i < j for each activity (i, j) ∈ A. Activity duration times dij (the weights of the arcs) (i, j) ∈ A are known. Two nodes 1 and n are distinguished as the initial node (source) and final node (sink), respectively, (no activity enters 1 and no activity leaves n). Of major concern is the minimization of the ending time of the last task, also called the makespan of the network. Of interest to the project manager are earliest starting dates, latest starting dates and floats of activities. The critical activities have zero float. The essence of the PERT method are two recurrence formulae, forward and backward recursions. The earliest starting dates of events k ∈ V and tasks (k, l) ∈ A, denoted by estk and estkl respectively, are determined by means of the forward recursion:

0 if k = 1, (9.2) estk = max (estj + djk ) otherwise, j∈P red(k)

estkl = estk ,

(9.3)

where P red(i) refers to the set of nodes that immediately precede node i ∈ V . The earliest starting date estk of event k is the length of a longest path from the beginning of the project, represented by node 1, to node k. We arbitrarily fix the starting time of project to est1 = 0. Of course the earliest starting date of a task (k, l) is equal to the earliest starting date of event k. The earliest ending time of the project is the earliest starting date of the last event n. In order to ensure the minimal duration of the project, the latest date of event n, denoted by lstn , is equal to its earliest starting date, lstn = estn . The latest starting date lstk of event k is the latest time to complete tasks that end at k without delaying the end of the project and thus it is the latest completion date of tasks ending at k. Difference lstn − lstk is the length of a longest path from node k to node n. Hence, the latest starting date of task (k, l), denoted by lstkl , is equal to lstl − dkl . The latest starting dates of events and tasks can be found by means of the backward recursion:

estn if k = n, (9.4) lstk = min (lstl − dkl ) otherwise, l∈Succ(k)

lstkl = lstl − dkl ,

(9.5)

where Succ(i) refers to the set of nodes that immediately follow node i ∈ V . The float of task (k, l) which represents the length of the time window for the beginning of the execution of the task is the difference between the latest starting date and the earliest starting date:

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fkl = lstkl − estkl .

(9.6)

The PERT method computes the earliest starting dates, latest starting dates and floats of tasks in O(m + n) time. One can also determine, simultaneously, critical tasks and critical paths and thus a subgraph consisting of all critical tasks and paths.

3.2 The Interval-Valued PERT Model Suppose that we only know that values of duration times dij of activities + (i, j) ∈ A (the weight of arcs) will fall within closed intervals Dij = [d− ij , dij ], − dij ≥ 0. This means that neither do we know the exact duration times of tasks, nor can we set them precisely. Now, depending on the actual duration of each task (that we do not precisely know), several earliest starting dates, latest starting dates and floats can be considered. We assume that the task durations are unrelated to one another. A vector Ω = (dij )(i,j)∈A , dij ∈ Dij , that represents an assignment of duration times dij to task (i, j) ∈ A is called a configuration [2]. Thus every configuration expresses a realization of the duration times. We denote by C the set of all the configurations, i.e. + C = ×(i,j)∈A [d− ij , dij ]. The duration of task (i, j) in configuration Ω is denoted by dij (Ω), dij (Ω) ∈ Dij . Among the configurations of C we distinguish the + extreme ones, which belong to ×(i,j)∈A {d− ij , dij }. Let B ⊆ A be a given subset + of activities. We define the extreme configuration ΩB the configuration where + all activities (i, j) ∈ B have duration times dij and all the remaining activities − have duration times d− ij . Similarly, we define the extreme configuration ΩB the − configuration where all activities (i, j) ∈ B have duration times dij and all the remaining activities have duration times d+ ij . These extreme configurations will play a crucial role in further considerations. Using configurations, the possible values ESTkl for the earliest starting date estkl , the possible values LSTkl for the latest starting date lstkl and the possible values Fkl for the float fkl form closed intervals denoted ESTkl = + − + − + [est− kl , estkl ], LSTkl = [lstkl , lstkl ], and Fkl = [fkl , fkl ]. Their bounds can be rigorously defined as follows [12, 19]: + est− kl = min estkl (Ω), estkl = max estkl (Ω),

(9.7)

+ lst− kl = min lstkl (Ω), lstkl = max lstkl (Ω),

(9.8)

− + fkl = min fkl (Ω), fkl = max fkl (Ω),

(9.9)

Ω∈C

Ω∈C

Ω∈C

Ω∈C

Ω∈C

Ω∈C

where estkl (Ω), lstkl (Ω) and fkl (Ω) denote the earliest and the latest starting date and the float of task (k, l) in configuration Ω, respectively. The above model was first formulated by Buckley [2] and the mentioned

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problems of computing intervals of possible values of project characteristics were completely solved in [22]. Criticality in interval-valued problems is defined as follows [3] : a task (k, l) ∈ A (resp. a path p ∈ P ) is possibly critical if there exists a configuration Ω ∈ C in which (k, l) (resp. p) is critical in the usual sense. A task (k, l) ∈ A (resp. a path p ∈ P ) is necessarily critical if (k, l) (resp. p) is critical in the usual sense in all configurations Ω ∈ C . There are relations between the criticality of a task and its possible values of floats and the earliest and latest starting dates, namely, task (k, l) is possibly (resp. necessarily) critical if and − + = 0 (resp. fkl = 0). If task (k, l) is necessarily critical then ESTkl = only if fkl LSTkl . The converse statement is false. In fact, when ESTkl = LSTkl , we can only ensure that (k, l) is possibly critical. Many intuitions learned on the classical PERT model fail in the interval version. For example, when duration times are precisely set, critical tasks always form a critical path. But in the interval case, possibility critical tasks do not always form a possibly critical path. Moreover, there may be no necessarily critical tasks, and, if some exist, they may be isolated and may not form a necessarily critical path.

3.3 The Complexity of Interval PERT In order to compute the intervals of possible values of earliest starting dates, latest starting dates and floats (see (9.7)-(9.9)), the same PERT algorithm has been traditionally used (see formulae: (9.3)-(9.6)), the only difference being the use of the interval arithmetic instead of the classical arithmetic. The classical PERT uses the addition, the subtraction, the maximum and the minimum. Their interval counterparts are defined as follows: A +I B = [a− + b− , a+ + b+ ], A −I B = [a− − b+ , a+ − b− ], minI {A, B} = [min{a− , b− }, min{a+ , b+ }], maxI {A, B} = [max{a− , b− }, max{a+ , b+ }], where A = [a− , a+ ] and B = [b− , b+ ] are intervals such that a− ≤ a+ and b− ≤ b+ (see [35]). For such a straightforward extension of the PERT algorithm, it turns out the forward recursion correctly computes the intervals of possible earliest + starting dates, ESTij = [est− ij , estij ], (i, j) ∈ A, [4, 14, 23]. Indeed, one can obtain all the lower bounds of the earliest starting dates of tasks by applying − and all the forward recursion (formulae: (9.2), (9.3)) for configuration ΩA the upper bounds of earliest starting dates of tasks by applying the forward + . But the backward recursion (formulae: (9.4), recursion for configuration ΩA (9.5)) fails to compute the set of possible latest starting dates [33, 34, 37] and in consequence floats can no longer be recovered from this procedure. Let us

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illustrate the point that the backward recursion can not be used to compute the latest starting dates and the floats. Consider the one activity network with duration time d12 ∈ D12 = [0, 1]. Its earliest starting date is EST1 = EST12 = [0, 0]. The earliest ending time of the project is ill-known: EST2 = EST1 +I D12 = [0, 1]. Thus the backward recursion yields the following information EST2 −I D12 = [0, 1] −I [0, 1] = [−1, 1] on latest starting dates LST1 and LST12 , which is of course false since the latest starting dates of event 1 and activity (1, 2) are always null in order to ensure a minimal project duration, LST1 = LST12 = [0, 0] ⊂ EST2 −I D12 = [−1, 1]. This error comes from the fact that the variable which represents the duration appears two times in the computations: once in the forward recursion, and once in the backward one. Thus, making use of the interval computations with linked variables gives an overestimation of the exact result. For example, for all x ∈ [0, 1] x − x = 0, but [0, 1] −I [0, 1] = [−1, 1]. Some authors tried to repair this propagation error in several ways: one classical approach is the use of a non-standard interval arithmetic [27, 28, 38]. For example, some authors use for subtraction an “inverse” of the interval addition [27]. This operation is defined for fuzzy intervals, but its interval counterpart can be expressed as a Hukuhara difference: [a− , a+ ] − [b− , b+ ] = [a− − b− , a+ − b+ ]. The use of this particular arithmetic does not lead to correct results, be it for computing latest starting dates, or floats of tasks. In particular, the results may not be well formed intervals. For example, the difference between the intervals [0, 1] and [0, 2] would lead to [0, −1] which is not an interval. Thus, techniques of using non-standard interval arithmetic do not yet issue the correct values. In [36], symbolic computations on variable duration times were suggested. However, this technique is unwieldy and highly combinatorial. Therefore, the classical PERT method cannot be directly adapted to the interval valued problem. It is worth pointing out that the problems of finding intervals containing earliest and latest starting times and floats of tasks and of evaluating criticality (possible and necessary) of tasks were first completely solved when a network is series-parallel (see [41] for a definition). Fargier et al. [19] proposed an O(n) algorithm for computing bounds on possible values of latest starting dates and bounds on possible values of floats of a task and O(n) algorithms for evaluating possible and necessary criticality of a task in a series-parallel network. For such networks, the configurations of duration times for which latest starting times and floats are minimal or maximal are easy to guess. Therefore, applying algorithms provided in [19] to each task in a network for computing bounds on floats and bounds on latest starting dates of all tasks leads to methods that require O(n2 ) time (note that for series-parallel graphs m = O(n)). In [44], the time for computing bounds on floats and bounds on latest starting dates and for evaluating the possible and necessary criticality for one task in a series-parallel network has been reduced to O(log n) by applying a more clever data structure. Therefore, computing bounds on floats

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and bounds on latest starting dates and evaluating the possible and necessary criticality of all tasks in a series-parallel network requires O(n log n) time. The complexity of algorithms for determining the criticality of tasks, the optimal intervals containing their latest starting times, and their floats in general acyclic networks was studied in [6, 7, 11, 12, 22, 43]. It turns out that the only strongly NP-hard problem is the one of finding the minimal float [7], which is closely related to asserting the possible criticality of a task. The latter is strongly NP-complete [6] and remains NP-complete even in acyclic planar network of node degree 3 [7]. On the contrary, computing the bounds of the possible earliest starting dates, latest starting dates, the upper bound of the float and determining necessary criticality of tasks are polynomially solvable problems. We introduce some additional notations. – SU CC(i, j) (resp. P RED(i, j)) denotes the set of all arcs that come after (resp. before) (i, j) ∈ A, and SU CC(j) (resp. P RED(j)) stands for the set of all nodes that come after (resp. before) j ∈ V . – P is the set of all paths in G from node 1 to node n. – P1,(k,l),n is the set of paths in G from node 1 to node n going through task (k, l) and P(k,l),n represents the set of paths in G from node k to node n going through task (k, l). – G(i, j) is the subgraph of G composed of nodes succeeding i and preceding j. – G(dij = d) is the graph G in which duration of task (i, j) is replaced by d.

4 The Path Enumeration Approach to Solving Interval-Valued PERT In this section, we give two algorithms based on some properties in which paths play a key-role. The first one, proposed in [11], computes the maximal latest starting date, the minimal and the maximal float of each task of a network in one execution. We call it the path algorithm. The second algorithm only computes the minimal latest starting date in polynomial time. It is called the polynomial path algorithm.

4.1 The Path Algorithm First we state a result establishing that the maximal latest starting date and the minimal and the maximal float are attained on extreme configurations in which all task duration times are set to their minimal values but on a path from node 1 to node n on which the duration times are set to their maximal values.

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Proposition 1 ([11]). Let (k, l) ∈ A be a task of G. There exists paths p1 , p2 , p3 ∈ P such that Ωp+1 maximizes lstkl (.), Ωp+2 minimizes fkl (.) and Ωp+3 maximizes fkl (.). The key to construct the path algorithm (Algorithm 1), proposed in [11], is Proposition 1. The idea of the algorithm consists in performing the classical PERT analysis for each configuration Ωp+ such that p is a path in network G from 1 to n. Clearly, the number of tested configurations is potentially exponential, but in practice the algorithm runs fast on realistic problems (see [11, 22]). Algorithm 1. [Path Algorithm] A calculation of the maximal latest starting date, the minimal and the maximal float of each task in network G Input: A network G, interval duration times Duv , (u, v) ∈ A. Output: The maximal latest starting date, the minimal and the maximal float of each task in G. begin foreach task (k, l) ∈ A do + − lst+ kl ← 0; fkl ← 0; fkl ← +∞;

foreach path p ∈ P do Use the classical PERT to compute the latest starting date and the float of each task of G in Ωp+ ; foreach task (k, l) ∈ A do + + + if lst+ kl < lstkl (Ωp ) then lstkl ← lstkl (Ωp ); + + + + if fkl < fkl (Ωp ) then fkl ← fkl (Ωp ); − − > fkl (Ωp+ ) then fkl ← fkl (Ωp+ ); if fkl end

4.2 The Polynomial Path Algorithm We first recall a result given in [11], that describes the form of configurations where the minimal latest starting date of a given task (k, l) in a network G is attained. Proposition 2 ([11]). Let (k, l) ∈ A be a task of network G. There exists a path pkl ∈ P(k,l),n that induces the extreme configuration such that lst− kl = lstkl (Ωp+kl ). It is easily seen that optimal path pkl ∈ P(k,l),n , which induces configuration Ωp+kl , is a longest path from k to n through l in Ωp+kl . This path can be recursively constructed. Suppose that a path plu ∈ P(l,u),n which in+ duces configuration Ωp+lu such that lst− lu = lstlu (Ωplu ) is known for each node u ∈ Succ(l). Then one can construct optimal path pkl from one of paths plu , where u ∈ Succ(l).

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Algorithm 2. [Polynomial path algorithm] Calculation of the minima of latest starting dates of all tasks in a network Input: A network G, interval duration times Duv , (u, v) ∈ A. Output: The minima of latest starting dates of all the tasks in G. begin V ← V ∪ {n + 1}; A ← A ∪ {(n, n + 1)}; Dnn+1 ← [0, 0]; − ) each (i, j) ∈ A by the classical PERT; Compute estij (ΩA − − − ∗ ); lnn+1 ← estnn+1 (ΩA ); lnn+1 ← 0; lstnn+1 ← estnn+1 (ΩA foreach (k, l) such that k ← n − 1 downto 1 do Let (l, v) be a task such that v ∈ Succ(l); lkl ← d+ kl + llv ; − ∗ ) + lkl > llv then if estkl (ΩA − ∗ lkl ← estkl (ΩA ) + lkl ; − lst− kl ← estkl (ΩA ); else ∗ ∗ lkl ← llv ; − ∗ − lkl ; lstkl ← lkl foreach (l, u) such that u ∈ Succ(l) \ {v} do − ∗ if estkl (ΩA ) + d+ kl + llu > llu then ∗ − + lkl ← estkl (ΩA ) + dkl + llu ; lkl ← d+ kl + llu ; − ← estkl (ΩA ); lst− kl else − ∗ if llu − d+ kl − llu < lstkl then ∗ ∗ lkl ← llu ; lkl ← d+ kl + llu ; ∗ lst− kl ← lkl − lkl ; end

Proposition 3 ([22]). Let (k, l) be a task of network G and let plu ∈ P(l,u),n − + be a path such that lst− lu = lstlu (Ωplu ), where u ∈ Succ(l). Then lstkl = + minu∈Succ(l) lstkl (Ω{(k,l)}∪plu ). From Proposition 3, we immediately deduce a polynomial algorithm for computing the minimal latest starting date of each task. Namely, the algorithm recursively finds a path pkl for which configuration Ωp+kl minimizes lstkl (Ω) by + means of paths plu such that lst− lu = lstlu (Ωplu ), where u ∈ Succ(l), starting from nodes in P red(n). Its overall running time is O(m2 (m+n)). Fortunately, it is possible to reduce the complexity of the algorithm to O(m2 ). It is enough to store the length of a longest path p∗ ∈ P in optimal configuration Ωp+lu for ∗ , and the length of path plu in Ωp+lu denoted by llu , for (l, u), denoted by llu ∗ every u ∈ Succ(l). Obviously, lst− lu = llu − llu . Let (l, v) be a task such that − + lstkl = lstkl (Ω{(k,l)}∪plv ) (such task always exists, see Proposition 3). The + for (k, l) equals length lkl of path pkl in optimal configuration Ω{(k,l)}∪p lv

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+ ∗ d+ kl + llv . Let p ∈ P be a longest path in Ω{(k,l)}∪plv . We consider two cases: + ∗ of p∗ equals est− (i) if (k, l) ∈ p∗ then the length lkl kl + dkl + llv (note that − − + − − estkl = estkl (ΩA ) = estkl (Ω{(k,l)}∪plv )). Hence lstkl = estkl . (ii) if (k, l) ∈ / p∗ − ∗ ∗ ∗ then lkl = llv and lstkl = lkl − lkl . The above remarks allow us to construct an O(m2 ) algorithm (see Algorithm 2).

5 A Constructive Approach to Solving Interval-Valued PERT In this section, we present polynomial algorithms for asserting necessary criticality of tasks, and computing the optimal intervals containing their latest starting dates, as well as the maximal float. These algorithms are constructive in the sense that they select appropriate values of task duration times in a step by step manner and directly construct one configuration (and just one) for which the extremum of the quantity of concern is attained. Thus, the computation of some bound comes down to computing the classical PERT problem on the constructed configuration. This type of technique, which yields polynomial complexity algorithms, cannot be applied to deciding possible criticality nor computing minimal float values, as the latter problem is NP-hard.

5.1 Deciding Necessary Criticality We first give an algorithm which can decide in polynomial time if a given task (k, l) is necessarily critical under the assumption that the duration times of the predecessors of task (k, l) are precisely known. The following propositions provide the theoretical basis for constructing the algorithm. Proposition 4 ([43]). A task (k, l) ∈ A is not necessarily critical in G if and only if (k, l) is not critical in an extreme configuration in which the duration of (k, l) is at its lower bound and all tasks from set A\(SU CC(k, l)∪ P RED(k, l) ∪ {(k, l)}) have duration times set to their upper bounds. Now under the assumption that tasks preceding (k, l) have precise duration times, we can set the ones of tasks succeeding (k, l) at precise values while maintaining the status of (k, l) in terms of necessary criticality. It yields a configuration where (k, l) is critical if and only if it is necessarily critical in the interval-valued network. These duration times are given by Proposition 5. Proposition 5 ([22]). Let (k, l) ∈ A be a distinguished task, and (i, j) be a task such that (i, j) ∈ SU CC(k, l). Assume that every task (u, v) ∈ P RED(i, j) has precise duration. If (k, l) is critical in the subnetwork G(1, i), then the following conditions are equivalent: (i) (k, l) is necessarily critical in G, (ii) (k, l) is necessarily critical in G(dij = d− ij ).

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If (k, l) is not critical in G(1, i), then the following conditions are equivalent: (i) (k, l) is necessarily critical in G, (ii) (k, l) is necessarily critical in G(dij = d+ ij ). Propositions 4 and 5 lead us to Algorithm 3 for asserting the necessary criticality of a given task (k, l) in a network in which all tasks that precede (k, l) have precise durations. The algorithm works as follows: first all duration times of tasks neither succeeding nor preceding (k, l) are set to their upper bounds and the earliest starting times of events not succeeding event l are computed by the classical PERT. Then the algorithm tests if (k, l) is G(u,v) the float of (k, l) critical in the subnetwork G(1, l). We denote by fkl computed in the network G(u, v). Precise duration times are fixed for tasks immediately succeeding node l. This step is repeated recursively for network G(1, l + 1), . . . , G(1, n − 1). At the end of this process, all duration times are precisely set, and (k, l) is necessarily critical in the network with the interval durations if and only if (k, l) is critical in the network with the fixed G(1,i) and testing if (k, l) is critical in G(1, i) can be durations. Computing fkl done in constant time, since we already know estj for all j ∈ P red(i), and so Algorithm 3 runs in O(m). Propositions 4 and 5 have counterparts for asserting possible criticality. This leads to an O(m) algorithm for asserting the possible criticality of tasks

Algorithm 3. Asserting whether a task is necessarily critical when duration times of predecessors are precisely known Input: A network G, task (k, l), interval duration times Duv , (u, v) ∈ A and for every task in P RED(k, l) the duration is precisely given. G(1,n) G(1,n) = 0, (k, l) is necessarily critical in G; and if fkl > 0, Output: If fkl (k, l) is not necessarily critical in G. /* Initialization */ foreach (u, v) ∈ A \ (SU CC(k, l) ∪ P RED(k, l) ∪ {(k, l)}) do duv ← d+ uv ; dkl ← d− ; kl /* Computation */ Compute esti of events i ∈ V \ (SU CC(l) ∪ {l}) by the classical PERT in the partially instantiated configuration; for i ← l to n − 1 such that i ∈ SU CC(l) ∪ {l} do G(1,i) Compute fkl ; G(1,i) if fkl = 0 /* (k, l) critical in G(1, i)*/ then foreach j ∈ Succ(i) do dij ← d− ij else foreach j ∈ Succ(i) do dij ← d+ ij G(1,n)

Compute fkl ; G(1,n) ; return fkl

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whose predecessors have deterministic durations [43]. It suffices to swap du+ ration times d− uv and duv in Propositions 4 and 5 and Algorithm 3. We now present an algorithm for evaluating the necessary criticality of a fixed task (k, l) ∈ A in network G with interval duration times, without any restriction. The key to the algorithm lies in Proposition 6 that enable a network with interval duration times to be replaced by another network with precise durations for tasks preceding a fixed (k, l), in such a way that (k, l) is necessarily critical in the former if and only if it is necessarily critical in the latter. Proposition 6 ([22]). Let (k, l) ∈ A be a distinguished task, and (i, j) be a task such that (i, j) ∈ P RED(k, l). If (k, l) is necessarily critical in G(j, n), then the following conditions are equivalent: (i) (k, l) is necessarily critical in G, (ii) (k, l) is necessarily critical in G(dij = d− ij ). If (k, l) is not necessarily critical in G(j, n), then the following conditions are equivalent: (i) (k, l) is necessarily critical in G, (ii) (k, l) is necessarily critical in G(dij = d+ ij ). We are now in a position to give an algorithm (Algorithm 4) for asserting the necessary criticality of a prescribed task in a general acyclic network. At each step of the algorithm, tasks between j and k have precise duration times (so Algorithm 3 can be invoked), and Algorithm 4 assigns precise duration times to tasks preceding j, while preserving the criticality of task (k, l). Since Algorithm 3 runs in O(m), Algorithm 4 requires O(mn) time. Unfortunately, Proposition 6 can not be adapted to the study of possible criticality, and asserting if a task is possibly critical, in general, is strongly NPcomplete [6]. So, while these results are instrumental for asserting necessary criticality and computing the maximal float, the same approach can neither be applied to assert possible criticality nor to compute the minimal float in the general case.

5.2 Computing Optimal Bounds on Latest Starting Dates Let us present a polynomial algorithm, proposed in [43], that computes the maximal latest starting date. Let us recall the following simple but important result that allows us to reduce the set of configurations C . Proposition 7 ([12]). The minimal upper bound on latest starting dates lst+ kl of task (k, l) in G is attained on an extreme configuration in which the duration of (k, l) is at its lower bound and all tasks that do not belong to set SU CC(k, l) have duration times at their upper bounds.

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Algorithm 4. Asserting necessary criticality of task (k, l) Input: A network G =< V, A >, task (k, l), interval duration times Duv , (u, v) ∈ A. G(1,n) G(1,n) = 0, (k, l) is necessarily critical in G; and if fkl > 0, Output: If fkl (k, l) is not necessarily critical in G. /* Initialization */ foreach (u, v) ∈ A \ (SU CC(k, l) ∪ P RED(k, l) ∪ {(k, l)}) do duv ← d+ uv ; dkl ← d− ; kl /* Computation */ for j ← k downto 2 such that j ∈ P RED(k) ∪ {k} do G(j,n) fkl ← Algorithm 3 with G(j, n) and without the initialization; G(j,n) = 0 then if fkl foreach i ∈ P red(j) do dij ← d− ij else foreach i ∈ P red(j) do dij ← d+ ij G(1,n)

fkl ← Algorithm 3 with G(1, n) and without the initialization; G(1,n) ; return fkl

The main idea of the algorithm for determining lst+ kl of a given task (k, l) ∈ A is based on Lemma 1. It consists in determining the minimal nonnegative ∗ that, added to the lower bound of the duration interval of a real number fkl specified task (k, l), makes it necessarily critical. ∗ Lemma 1 ([43]). Let fkl be the minimal nonnegative real number such that + + ∗ ∗ (k, l) is necessarily critical with a duration d− kl + fkl . Then lstkl = estkl + fkl .

An algorithm for determining the maximal latest starting date (Algorithm 5) proceeds as follows. First the duration times of tasks not succeeding (k, l) are set to their maximal values, and the duration of (k, l) is set to d− kl (see Proposition 7). From this point on, it computes the maximal earliest starting date of (k, l). Then it tests if (k, l) is necessarily critical in the partially instantiated configuration by calling Algorithm 3. During this call, the algorithm G(1,i) determined by Algorithm 3. If (k, l) is not necessarily retains all values fkl G(1,i) critical, the smallest fkl , previously computed, is the minimal duration δ to be added to dkl . Then, the algorithm adds this δ to dkl and repeats the previous steps. Since Algorithm 5 makes at most n calls to Algorithm 3, its complexity is O(nm). There exist counterparts to Proposition 7, Lemma 1 and Algorithm 5 for minimal latest starting dates. They are obtained by symmetry: it is sufficient + − − to change upper bounds by lower ones (d+ uv becomes duv , estkl becomes estkl ) − + and the lower bounds by the upper ones (dkl becomes dkl ). The running time of the algorithm is also O(nm) [43].

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Algorithm 5. given task

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Computation of the maximal latest starting date of a

Input: A network G, task (k, l), interval duration times Duv , (u, v) ∈ A. Output: The maximal latest starting date lst− kl . /* Initialization */ foreach (u, v) ∈ A \ (SU CC(k, l) ∪ {(k, l)}) do duv ← d+ uv ; ; dkl ← d− kl Compute est+ kl by the classical PERT; Δ ← 0; /* Computation */ G(1,n) ← Algorithm 3 without the initialization; fkl G(1,i) ’s computed by Algorithm 3 */ /* we can retain fkl while fkl (1, n) > 0 do G(1,i) G(1,i) δ ← mini∈SU CC(l) {fkl |fkl = 0}; Δ ← Δ + δ; dkl ← d+ kl + Δ; G(1,n) fkl ← Algorithm 3 without the initialization; G(1,i) ’s computed by Algorithm 3 */ /* we can retain fkl return est+ kl + Δ

5.3 Computing the Maximal Float + The key idea to compute the maximal float fkl of a given task (k, l) ∈ A is to increase step by step the duration time of (k, l) from dkl = d− kl until (k, l) becomes necessarily critical. One can prove that the maximal float is equal to the overall increment of d− kl [22]: ∗ Proposition 8. Let fkl be the minimal nonnegative real number such that + ∗ ∗ (k, l) is necessarily critical in G(dkl = d− kl + fkl ). Then fkl = fkl .

A sketch of an algorithm, based on Proposition 8 that computes the maximal float is as follows [22]: for each node i preceding (k, l) it computes the maximal +G(j,n) +G(1,n) float fkl in G(j, n). If fkl in G(1, n) is not 0, then the minimum of the computed positive maxima of float values is added to the duration d− kl of (k, l), and the algorithm is run again from the beginning until (k, l) becomes necessary critical in G. The sum of added values to the duration d− kl is then the maximal float of (k, l). The total running time is O(n3 m).

6 Fuzzy Interval-Valued PERT In this section, we give a rigorous possibilistic interpretation of the fuzzyinterval PERT, which is a direct extension of interval-valued PERT and provide some algorithms for computing degrees of necessary criticality and

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fuzzy intervals for latest starting times. These algorithms extend those of the interval-valued case by means of the notion of gradual numbers.

6.1 Fuzzy-Valued PERT in the Setting of Possibility Theory We now present a possibilistic formalization of the PERT problem with un˜ ij , (i, j) ∈ A. A memberknown task durations modeled by fuzzy intervals D ˜ ship function of Dij is regarded as a possibility distribution for the values of an unknown task duration dij , i.e. πdij = μD˜ ij , (i, j) ∈ A. Thus the possibility degree of the assignment dij = v is Π(dij = v) = πdij (v) = μD˜ ij (v). Let Ω = (vij )(i,j)∈A ∈ Rm be a configuration of the task durations. The configuration Ω represents a state of the world in which dij = vij , for all (i, j) ∈ A. It defines an instance of the PERT problem with the deterministic task duration times (vij )(i,j)∈A . Assuming that duration times are unrelated to one another, the degree of possibility of configuration Ω = (vij )(i,j)∈A is obtained by the following joint possibility distribution on configurations ˜ ij , (i, j) ∈ A (see [9, 12]): induced by D π(Ω) = Π(∧(i,j)∈A [dij = vij ]) = min Π(dij = vij ) = min μD ij (vij ). (i,j)∈A

(i,j)∈A

Hence, the possibility distributions that represent more or less plausible values of earliest starting dates estkl and latest starting dates lstkl and of floats fkl of task (k, l) ∈ A are defined in the following way: Π(estkl = v) = μEST  kl (v) = Π(lstkl = v) = μLST  kl (v) = Π(fkl = v) = μF˜kl (v) =

sup {Ω: estkl (Ω)=v}

sup {Ω: lstkl (Ω)=v}

sup {Ω: fkl (Ω)=v}

π(Ω),

π(Ω),

π(Ω),

(9.10) (9.11) (9.12)

where Π(estkl = v), Π(lstkl = v) and Π(fkl = v) stand for the possibility degree that estkl = v, lstkl = v and fkl = v, respectively. The degrees of possibility and necessity that a task (k, l) ∈ A is critical are defined as follows (see [3])): Π((k, l) critical) =

sup {Ω:(k,l) critical in Ω}

π(Ω),

N((k, l) critical) = 1 − Π((k, l) not critical) = inf (1 − π(Ω)).

(9.13) (9.14)

{Ω:(k,l) not critical in Ω}

Π((k, l) is critical) = α means that there exists at least one configuration Ω such that π(Ω) = α and task (k, l) is critical in Ω. N((k, l) is critical) = α

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means that for all configurations Ω such that π(Ω) > 1 − α, task (k, l) is critical in Ω. The degrees of possibility and necessity that a path is critical can be similarly defined. We now show that, similarly to the interval-valued case (see Section 3.2), we can express the degrees of possible and necessary criticality of a task in terms of fuzzy float. Since the statement “(k, l) is critical in Ω” is equivalent to the condition fkl (Ω) = 0, we get the following relationships between criticality degrees (9.13), (9.14) and fuzzy float (9.12): Π((k, l) critical) =

sup {Ω:fkl (Ω)=0}

N((k, l) critical) = 1 −

π(Ω) = Π(fkl = 0) = μF˜kl (0),

sup {Ω:fkl (Ω)>0}

π(Ω) = N(fkl = 0) = 1 − sup μF˜kl (v). v>0

6.2 Fuzzy Extreme Configurations The basic idea is to consider a fuzzy interval representing an ill-known duration time as a regular interval with fuzzy bounds. We start with introducing ˜ = (d˜ij )(i,j)∈A of gradthe notion of a fuzzy configuration, which is a vector Ω ual numbers that represents an assignment of gradual duration times to tasks ˜ is a gradual duration time in fuzzy configuration Ω. ˜ (i, j) ∈ A. Now d˜ij (Ω) Suppose now that for every duration time dij , (i, j) ∈ A, there is a given a ˜ ij = [d˜− , d˜+ ], where d˜− is a gradual lower bound and d˜+ is a fuzzy interval D ij ij ij ij ˜ ij . Now the fuzzy earliest starting date EST  kl and gradual upper bound of D  kl and the fuzzy float F˜kl of task (k, l) ∈ A the fuzzy latest starting date LST are fuzzy intervals which can be also described by pairs of gradual numbers, − +  − , lst  + ] and F˜kl = [f˜− , f˜+ ]. It is  kl = [lst  kl = [est  kl , est  kl ], LST namely EST kl kl kl kl − + easy to verify that f˜kl (λ) is increasing according to λ and f˜kl (λ) is decreasing. We thus have

− (λ) = 0, 0 if ∀λ, f˜kl Π((k, l) critical) = μF˜kl (0) = − ˜ sup{λ | fkl (λ) = 0} otherwise. A similar reasoning leads to the following equality:

+ 0 if ∀λ, f˜kl (λ) = 0, N((k, l) critical) = + ˜ 1 − inf{λ | fkl (λ) = 0} otherwise.

(9.15)

In order to apply the interval methods, given in Sections 4 and 5, to the fuzzy interval computations, we need to replace extreme configurations by fuzzy extreme configurations, i.e. tuples of gradual numbers which belong to ˜+ ×(i,j)∈A {d˜− ij , dij }. We must also extend the standard PERT procedure to the case when duration times are gradual numbers.

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6.3 Gradual PERT A network G =< V, A > is given. All assumptions on the network are the same as in the deterministic case except for activity duration times which are modeled by means of gradual numbers d˜ij , (i, j) ∈ A. A gradual duration time d˜ij is defined by assignment function Ad˜ij parametrized by the value of λ ∈ [0, 1] (see Section 2.2). Ad˜ij (λ) is the value of the duration time of (i, j) in the scenario parametrized by λ. For simplicity of the notation we will write d˜ij (λ) instead of Ad˜ij (λ). This problem can be seen as an optimization problem on a family of scenarios: for a given λ, d˜ij (λ) is a precise duration, and so configuration (the set of task durations) at degree λ is a particular scenario. This parametric approach can model deterministic dependencies among tasks durations, allowing only one degree of freedom. This problem formulation leads to a PERT algorithm with gradual numbers that is instrumental to extend the interval-valued PERT algorithms to the case of fuzzy intervals. Fig. 9.2a shows a simple project, and Fig. 9.2b presents gradual durations of tasks of the project. d˜13 (λ) = 2 for every λ means that the duration of task (1, 3) is 2 in each scenario. One can observe that the durations of tasks (1, 2) and (2, 3) are correlated, and the more time the task (1, 2) will require, the faster the task (2, 3) will be executed. Computing earliest starting dates, latest starting dates and floats of tasks is easy. It suffices to run the classical PERT method, using formulae (9.2)-(9.6), where +, −, max and min are replaced by the four operations on gradual numbers. This is due to the fact that all algebraic properties needed to apply the PERT method are preserved by gradual numbers (see Section 2.2). A gradual counterpart of the PERT method is given in the form of Algorithm 6. The forward recursion that computes gradual earliest starting dates is implemented in lines 1-5. Lines 6-11 implement the backward recursion which computes gradual latest starting dates and gradual floats. Of course, Algorithm 6 outputs gradual results. Fig. 9.3 shows the computed gradual earliest starting date, latest starting date and float of task (2, 3) of Fig. 9.2a.. It means that for λ ∈ [ 13 , 1] task (2, 3) is critical, but it is not for λ ∈ [0, 13 ). In general, a result in the gradual setting is obtained from several optimal paths corresponding to various sub-intervals of membership values. The change of optimal paths form which optimal values of quantities of (a)

(b) 2

d˜12

d˜23

λ

λ

1

1

0.5 3

1

0.5

d˜12

λ 1

d˜23

d˜13 0.5

d˜13 d˜12 (λ)

d˜23 (λ)

Fig. 9.2 (a) A network project (b) Its Gradual task durations

d˜13 (λ)

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Algorithm 6. Gradual PERT

1 2 3 4 5 6 7 8 9 10 11

Input: A network G =< V, A >, gradual duration times d˜ij , (i, j) ∈ A.  ij , the gradual latest starting Output: The gradual earliest starting date est  ij and the gradual float f˜ij of each task (i, j) of the network. date lst i ← ˜ 0; for i ← 1 to n do est for i ← 1 to n − 1 do foreach j ∈ Succ(i) do  i;  ij ← est est  j , est  i + d˜ij );  estj ← max(est  i ← est  n; for i ← 1 to n do lst for j ← n downto 2 do foreach i ∈ P red(j) do  j − d˜ij ;  ij ← lst lst  ij − est  ij ; f˜ij ← lst   j − d˜ij );  lsti ← min(lsti , lst

λ

λ

1 0.5

λ

1

 23 est  23 (λ) est

0.5

1  23 lst

0.5

 23 (λ) lst

f˜23 f˜23 (λ)

Fig. 9.3 The gradual earliest starting date, latest starting date and float of task (2, 3) for the network project shown in Fig. 9.2a

interest are attained can be observed via kinks in the resulting piecewise linear membership functions.

6.4 Solving Fuzzy Interval PERT The concept of a gradual number allows us to extend naturally all the results from Sections 4 and 5 to the fuzzy-valued case. The idea is to use the gradual PERT algorithm explained earlier, and fuzzy extreme configurations. In consequence we obtain a method for computing the degree of necessity that a task is critical (see (9.14)) and methods for calculating the fuzzy latest starting date of a task (see (9.11)). In the same way, one can obtain algorithms for determining the upper bound of the fuzzy float (the right profile of possibility distribution of floats) of a task (see (9.12)) and the fuzzy earliest starting date of a task (see (9.10)).

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We can directly extend the Path Algorithm (Algorithm 1) to compute fuzzy bounds of floats and upper bound of latest starting dates of tasks. To do this, we use gradual bounds and the gradual PERT algorithm instead of the classic one in Algorithm 1. All other algorithms can also be extended by taking into account a fuzzy counterpart of the interval PERT. These extensions are not as obvious as the Path Algorithm. In the remainder of this chapter, we provide algorithms for evaluating necessary criticality of a task and for computing the gradual bounds of fuzzy latest starting dates of a task. Computing the degree of necessary criticality of a task We now present methods for computing the degree of necessary criticality of a task. First under assumption that the duration times of predecessors of the task are given by gradual numbers. Then we provide an algorithm without any restrictions. In the interval case (see Section 5) one can precisely set durations of all tasks in a network without changing the necessary criticality of a specified task (k, l). In the same manner, using gradual numbers instead of real numbers, one can determine the degree of necessary criticality of task (k, l) in the fuzzy case. From Proposition 5 (resp. Proposition 6) it follows that in order to fix a gradual duration of task (i, j) one has to test if task (k, l) is critical in subnetwork G(1, i) (resp. G(j, n)). Note that a task can be critical for some values of λ and not critical for others (see Fig. 9.3). Thus, a gradual duration time assigned to task (i, j) may involve both upper and lower gradual ˜ ij = [d˜− , d˜+ ]. Therefore, Algorithm 7 for evaluating the bounds of duration D ij ij necessary criticality of a task when durations of its predecessors are gradual numbers is a direct extension of Algorithm 3. Lines 1-2 of Algorithm 7 set gradual durations of some tasks of the network according to Proposition 4. Lines 3-8 assign gradual values to tasks succeeding (k, l), depending on the criticality of (k, l), with respect to Proposition 5. Constructing gradual duration d˜ij (lines 5 and 8) consists in determining sequences 0 = λ0 < λ1 < . . . < λr = 1 such that for λ ∈ [λp , λp+1 ], p = 1, . . . , r − 1, d˜ij (λ) is either lower gradual bound d˜− ij (λ) or upper gradual + ˜ bound dij (λ). This can be done easily if duration times d˜ij are are trapezoidal or triangular fuzzy intervals (their shape functions are linear) or if d˜ij are fuzzy intervals of the L-L type (their right and left shape functions are the same) since function L can be then easily linearized (see (9.1) for the assignment functions). Thus, all computed gradual durations are piecewise linear. Note that operations +, −, max and min preserve the piecewise linearity. This remark applies to Algorithm 6 and all algorithms presented in this section. To do these operations on gradual numbers, we can use methods proposed G(1,n) that provides an inforin [30]. Algorithm 7 outputs gradual number f˜kl mation about the necessary criticality of (k, l) in network G in which duration G(1,n) (λ) = 0 if times of predecessors of (k, l) are gradual numbers. Namely, f˜kl and only if N((k, l) critical) ≥ 1 − λ in G according to (9.15). This algorithm,

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Algorithm 7. Evaluating the necessary criticality of a task when durations of its predecessors are gradual numbers

1 2 3 4 5 6 7 8 9 10

Input: A network G, distinguished task (k, l), fuzzy interval durations ˜ ij = [d˜− , d˜+ ], (i, j) ∈ A and for every task (i, j) ∈ P RED(k, l) D ij ij duration d˜ij is a gradual number. G(1,n) describing the necessary criticality of (k, l) Output: Gradual number f˜kl G(1,n) (λ) = 0 then (k, l) is necessarily critical for λ; for λ ∈ [0, 1]; if f˜kl otherwise is not necessarily critical for λ. /* Initialization */ foreach (u, v) ∈ A \ (SU CC(k, l) ∪ P RED(k, l) ∪ {(k, l)}) do d˜uv ← d˜+ uv ; − ˜ ˜ dkl ← dkl ; /* Computation */ for i ← l to n − 1 such that i ∈ SU CC(l) ∪ {l} do G(1,i) ← Algorithm 6 in G(1, i); f˜kl G(1,i) foreach λ ∈ [0, 1] such that f˜kl (λ) = 0 do ˜ foreach j ∈ Succ(i) do dij (λ) ← d˜− ij (λ); G(1,i) foreach λ ∈ [0, 1] such that f˜kl (λ) > 0 do ˜ foreach j ∈ Succ(i) do dij (λ) ← d˜+ ij (λ); G(1,n) ← Algorithm 6 in G(1, n); f˜kl G(1,n) return f˜kl ;

similarly as its interval counterpart, is useful in the algorithm for evaluating the necessary criticality in a network without restrictions (Algorithm 8). It is easily seen that Algorithm 8 is a direct extension of Algorithm 4. It recursively assigns gradual duration times to tasks preceding node k, based on G(1,n) G(1,n) such that f˜kl (λ) = 0 if Proposition 6 and outputs gradual number f˜kl G(1,n) and only if N((k, l) critical) ≥ 1 − λ in G according to (9.15) (if f˜kl (λ) = G(1,n) + ˜ ˜ 0 then fkl (λ) = fkl (λ)). In order to compute the degree of necessary criticality of (k, l), we need to determine the minimal value of λ such that G(1,n) f˜kl (λ) = 0. Determining the bounds of fuzzy latest starting dates  kl of task We now consider the problem of fuzzy latest starting date LST (k, l) ∈ A that is a possibility distribution of more or less plausible values of latest starting dates (9.11). This problem reduces to determining grad− +  of fuzzy interval LST  and lst  kl . We give only an algorithm ual bounds lst kl

kl

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Algorithm 8. Evaluating the necessary criticality of a task

1 2 3 4 5 6 7 8 9 10

Input: A network G, distinguished task (k, l), fuzzy interval durations ˜ ij = [d˜− , d˜+ ], (i, j) ∈ A. D ij ij G(1,n) Output: Gradual number f˜kl describing the necessary criticality of (k, l) G(1,n) for λ ∈ [0, 1]; if f˜kl (λ) = 0 then (k, l) is necessarily critical for λ; otherwise is not necessarily critical for λ. /* Initialization */ foreach (u, v) ∈ A \ (SU CC(k, l) ∪ P RED(k, l) ∪ {(k, l)}) do d˜uv ← d˜+ uv ; − ˜ ˜ dkl ← dkl ; /* Computation*/ for j ← k downto 2 such that i ∈ P RED(k) ∪ {k} do G(j,n) ← Algorithm 7 with G(j, n) without the initialization; f˜kl G(j,n) foreach λ ∈ [0, 1] such that f˜kl (λ) = 0 do ˜ foreach i ∈ P red(j) do dij (λ) ← d˜− ij (λ); G(j,n) foreach λ ∈ [0, 1] such that f˜kl (λ) > 0 do ˜ foreach i ∈ P red(j) do dij (λ) ← d˜+ ij (λ); G(1,n) ← Algorithm 7 with G(1, n) without the initialization; f˜kl G(1,n) return f˜kl ;

+

 which is an extension of Algorithm 5 to (Algorithm 9) for determining lst kl −  is symmetrical. The the fuzzy case. An algorithm for the lower bound lst kl idea of the algorithm is to make (k, l) necessarily critical for all λ ∈ [0, 1] by ˜ to d˜− (see Lemma 1 in the interval case). The adding a gradual duration Δ kl algorithm proceeds as follows. The partial fuzzy extreme configuration is first constructed according to duration times of tasks Proposition 7 (lines 1-2). It +  of fuzzy earliest starting date of (k, l) determines gradual upper bound est by Algorithm 6 in the partial configuration (line 3). Then it receives an inforG(1,n) mation f˜kl about necessary criticality of (k, l) in G for λ ∈ [0, 1] by calling Algorithm 7 (line 5). If there exists λ for which (k, l) is not necessarily critical (line 6), the algorithm constructs gradual duration δ˜ that must be added to d˜kl in order to make (k, l) necessarily critical in at least one more subnetwork G(i,n) (λ) = 0 (lines 7-8). The G(i, n) for all possibility degree λ such that f˜kl G(1,i) ˜ algorithm makes use of all retained fkl ’s determined by Algorithm 7. It repeats the steps (lines 7-12) until (k, l) becomes necessarily critical in G for +  is the sum of est  + and the total all λ ∈ [0, 1]. The upper gradual bound lst kl ˜ increase Δ. The same approach applies to compute the upper bound of fuzzy floats of a task (for more details on this algorithm, we refer the reader to [20]).

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Algorithm 9. Determining the upper gradual bound of fuzzy latest starting dates of (k, l)

1 2 3 4 5 6 7 8 9 10 11 12

13

Input: A network G, distinguished task (k, l), fuzzy interval durations ˜ ij = [d˜− , d˜+ ], (i, j) ∈ A. D ij ij + Output: The upper gradual bound lst kl of fuzzy latest starting dates of (k, l). foreach (u, v) ∈ A \ (SU CC(k, l) ∪ {(k, l)}) do d˜uv ← d˜+ uv ; − ˜ ˜ dkl ← dkl ; + Compute est kl by Algorithm 6; ˜ ← 0; ˜ Δ G(1,n) ← Algorithm 7 without the initialization; f˜kl G(1,i) /* we can retain f˜kl ’s computed by Algorithm 7 */ G(1,n) ˜ ˜ do while fkl = 0 δ˜ ← ˜ 0; G(1,n) foreach λ ∈ [0, 1] such that f˜kl (λ) = 0 do G(i,n) G(i,n) ˜ (λ) | i ∈ SU CC(l), f˜ (λ) = 0}; δ(λ) ← min{f˜ kl

kl

˜ ˜←Δ ˜ + δ; Δ ˜ d˜kl ← d˜kl + Δ; G(1,n) ˜ ← Algorithm 7 without the initialization; fkl G(1,i) /* we can retain f˜kl ’s computed by Algorithm 7 */

˜ + + Δ return est

7 Conclusion In this chapter, we have shown a complete solution to the criticality analysis in activity networks with uncertain task durations modeled by means of closed intervals. We have provided a set of algorithms for determining the criticality of tasks, the optimal intervals containing their earliest starting dates, latest starting dates, and their floats. The only strongly NP-hard problem is the one of finding the minimal float [7], which is closely related to asserting the possible criticality of a task that turned out to be strongly NP-complete [6]. All other problems turn out to be polynomial. We have extended the presented results to the criticality analysis of activity networks with imprecise duration times of tasks modeled by fuzzy intervals, whose membership functions are regarded as possibility distributions for the values of the unknown duration times. This extension has been done by exploiting the notion of gradual numbers that allows us to extend the interval algorithms to solve the fuzzy-valued PERT. Finally, we have obtained, among others, algorithms for determining the degree of necessary optimality of a task and a possibility distribution for values of latest starting dates of a task.

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References 1. Adlakha, V.G., Kulkarni, V.G.: A classified bibliography of research on stochastic PERT networks: 1966- 1987. INFOR 27, 272–296 (1989) 2. Buckley, J.J.: Fuzzy PERT. In: Evans, G., Karwowski, W., Wilhelm, M. (eds.) Applications of Fuzzy Set Methodologies in Industrial Engineering, pp. 103– 114. Elsevier, Amsterdam (1989) 3. Chanas, S., Dubois, D., Zieli´ nski, P.: On the sure criticality of tasks in activity networks with imprecise durations. IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics 34, 393–407 (2002) 4. Chanas, S., Kamburowski, J.: The use of fuzzy variables in PERT. Fuzzy Set Systems 5, 1–19 (1981) 5. Chanas, S., Kuchta, D.: Discrete fuzzy optimization. In: Slowi´ nski, R. (ed.) Fuzzy Sets in Decision Analysis, Operation Research and Statistics. The Handbooks of Fuzzy Sets Series, pp. 249–280. Kluwer Academic Publishers, Dordrecht (1998) 6. Chanas, S., Zieli´ nski, P.: The computational complexity of the criticality problems in a network with interval activity times. European Journal of Operational Research 136, 541–550 (2002) 7. Chanas, S., Zieli´ nski, P.: On the hardness of evaluating criticality of activities in planar network with duration intervals. Operation Research Letters 31, 53–59 (2003) 8. Dubois, D.: Mod`eles math´ematiques de l’impr´ecis et de l’incertain en vue d’applications aux techniques d’aide ` a la d´ecision. Th`ese d’´etat de l’Universit´e Scientifique et M´edicale de Grenoble et de l’Institut National Politechnique de Grenoble (1983) 9. Dubois, D., Fargier, H., Fortemps, P.: Fuzzy scheduling: Modelling flexible constraints vs. coping with incomplete knowledge. European Journal of Operational Research 147, 231–252 (2003) 10. Dubois, D., Fargier, H., Fortin, J.: A generalized vertex method for computing with fuzzy intervals. In: Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 541–546 (2004) 11. Dubois, D., Fargier, H., Fortin, J.: Computational methods for determining the latest starting times and floats of tasks in interval-valued activity networks. Journal of Intelligent Manufacturing 16, 407–422 (2005) 12. Dubois, D., Fargier, H., Galvagnon, V.: On latest starting times and floats in activity networks with ill-known durations. European Journal of Operational Research 147, 266–280 (2003) 13. Dubois, D., Prade, H.: Algorithmes de plus courts chemins pour traiter des donnees floues. RAIRO-Recherche Op´erationnelle/Operations Research 12, 212– 227 (1978) 14. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980) 15. Dubois, D., Prade, H.: Possibility theory: an approach to computerized processing of uncertainty. Plenum Press, New York (1988) 16. Dubois, D., Prade, H.: When upper probabilities are possibility measures. Fuzzy Sets and Systems 49, 65–74 (1992) 17. Dubois, D., Prade, H.: Gradual elements in a fuzzy set. Soft Computing - A Fusion of Foundations, Methodologies and Applications 12, 165–175 (2008)

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18. Elmaghraby, S.E.: On criticality and sensitivity in activity networks. European Journal of Operational Research 127, 220–238 (2000) 19. Fargier, H., Galvagnon, V., Dubois, D.: Fuzzy PERT in series-parallel graphs. In: 9th IEEE International Conference on Fuzzy Systems, San Antonio, TX, pp. 717–722 (2000) 20. Fortin, J., Dubois, D.: Solving Fuzzy PERT using gradual real numbers. In: Penserini, L., Peppas, P., Perini, A. (eds.) Starting AI Researcher’s Symposium (STAIRS), Riva del Garda, Italy. Frontiers in Artificial Intelligence and Applications, vol. 142, pp. 196–207. IOS Press, Amsterdam (2006), http://www.iospress.nl/ 21. Fortin, J., Dubois, D., Fargier, H.: Gradual numbers and their application to fuzzy interval analysis. IEEE Transactions on Fuzzy Systems 16, 388–402 (2008) 22. Fortin, J., Zieli´ nski, P., Dubois, D., Fargier, H.: Interval analysis in scheduling. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 226–240. Springer, Heidelberg (2005) 23. Gazdik, I.: Fuzzy network planning. IEEE Transactions on Reliability 32, 304– 313 (1983) 24. Guiffrida, A.L., Nagi, R.: Fuzzy set theory applications in production management research: A literature survey. Journal of Intelligent Manufacturing 9, 39–56 (1998) 25. Hagstrom, J.N.: Computational complexity of pert problems. Networks 18, 139– 147 (1988) 26. Hapke, M., Jaszkiewicz, A., Slowi´ nski, R.: Fuzzy project scheduling system for software development. Fuzzy Sets and Systems 67, 101–107 (1994) 27. Hapke, M., Slowi´ nski, R.: Fuzzy priority heuristics for project scheduling. Fuzzy Sets and Systems 86, 291–299 (1996) 28. Kaufmann, A., Gupta, M.M.: Fuzzy Mathematical Models in Engineering and Management Science. North-Holland, Amsterdam (1991) 29. Kelley, J.E.: Critical path planning and scheduling - mathematical basis. Operations Research 9, 296–320 (1961) 30. Kerre, E., Steyaert, H., Parys, F.V., Baekeland, R.: Implementation of piecewise linear fuzzy quantities. International Journal of Intelligent Systems 10, 1049– 1059 (1995) 31. Loostma, F.A.: Fuzzy logic for planning and decision-making. Kluwer Academic Publishers, Dordrecht (1997) 32. Malcolm, D.G., Roseboom, J.H., Clark, C.E., Fazar, W.: Application of a Technique for Research and Development Program Evaluation. Operations Research 7, 646–669 (1959) 33. McCahon, C.S.: Using PERT as an Approximation of Fuzzy Project-Network Analysis. IEEE Transactions on Engineering Management 40, 146–153 (1993) 34. McCahon, C.S., Lee Stanley, E.: Project network analysis with fuzzy activity times. Computers and Mathematics with Applications 15, 829–838 (1988) 35. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979) 36. Nasution, S.H.: Fuzzy critical path method. IEEE Transactions on Systems, Man, and Cybernetics 24, 48–57 (1994) 37. Prade, H.: Using fuzzy sets theory in a scheduling problem: a case study. Fuzzy Sets and Systems 2, 153–165 (1979) 38. Rommelfanger, H.: Network analysis and information flow in fuzzy environment. Fuzzy Sets and Systems 67, 119–128 (1994)

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39. Slowi´ nski, R., Hapke, M. (eds.): Scheduling under Fuzziness. Physica-Verlag, A Springer-Verlag Co., Heidelberg (1999) 40. Turksen, I.B., Zarandi, M.H.F.: Production Planning and Scheduling: Fuzzy and Crisp Approaches. In: Zimmerman, H.J. (ed.) Practical Applications of Fuzzy Technology. The Handbooks of Fuzzy Sets Series, pp. 479–529. Kluwer Academic Publishers, Dordrecht (1999) 41. Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series parallel digraphs. SIAM Journal on Computing 11, 298–313 (1982) 42. Werners, B., Weber, R.: Decision and Planning in Research and Development. In: Zimmerman, H.J. (ed.) Practical Applications of Fuzzy Technology. The Handbooks of Fuzzy Sets Series, pp. 445–478. Kluwer Academic Publishers, Dordrecht (1999) 43. Zieli´ nski, P.: On computing the latest starting times and floats of activities in a network with imprecise durations. Fuzzy Sets and Systems 150, 53–76 (2005) 44. Zieli´ nski, P.: Efficient Computation of Project Characteristics in a SeriesParallel Activity Network with Interval Durations. In: Della Riccia, G., Dubois, D., Kruse, R., Lenz, H.J. (eds.) Decision Theory and Multi-Agent Planning. CISM Courses and Lectures, vol. 482, pp. 111–130. Springer, Wien, New York (2006)

Chapter 10

Fuzziness in Supply Chain Management Péter Majlender*

Abstract. In this chapter, we shall present some theoretic and practical aspects of employing fuzzy logic and possibility theory in Supply Chain Management (SCM). We will present a wide point of view to our topics by introducing basic concepts of supply chain management and fuzzy logic without requiring any background knowledge from the reader in these areas. First, we will generally present the topic of supply chain management along with some of its classical methods that can be used to work with problems in that area. Then, we will introduce fuzzy logic to supply chain management by incorporating possibility distributions in the mathematical models. Doing the mathematical formulations we will also present methods for dealing with the bullwhip effect using possibility distributions.

1 Introduction Supply Chain Management (SCM) is the management of a network of interconnected businesses that collectively aim at creating some product or service package for their end customers. In general, SCM deals with all movement and storage of raw materials, work-in-progress inventory, and finished goods as taken from point-of-origin to point-of-consumption. From managerial point of view, SCM embraces activities of planning and managing the processes of sourcing, procurement, conversion, and logistics. From operations research point of view, SCM involves both supply and demand management within a business unit and across a network of companies that build up the supply chain. More recently, a new term Extended Enterprise emerged for representing a loosely coupled, self-organizing network of businesses that cooperate to offer and provide some product or service. More importantly, SCM also includes coordination and collaboration with other partners in the network. They can be suppliers, intermediaries, third-party service providers, and customers. As we will see shortly, implementing a sound collaboration scheme is a major issue is managing a supply chain. When business Péter Majlender Department of Management and Organization, Hanken School of Economics, Casa Academica, Perhonkatu 6B, FI-00100 Helsinki, Finland Department of Information Technologies, Åbo Akademi University, Joukahaisenkatu 3-5 A, FI-20520 Turku, Finland C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 201–247. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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units in the network are not centrally coordinated, information sharing becomes a problematic part in the overall management. In particular, when units in the supply chain try to maximize their profits independently of each other and without sharing adequate amount of information with their partners about their individual observations on the market situation, the overall cost of maintaining the supply chain will increase. This will eventually drive the individual profits of the business units in the network towards the negative territories as well. This phenomenon is called the bullwhip effect. In essence, it says that an independent business unit in the supply chain can only optimize its costs if it addresses information sharing with its partners in the network. It does not matter, how smart methods it uses, how reliable forecast it has, and how rationally it behaves, if there is no communication in the supply chain, the rational behaviors of the individual units to maximize their own profits will drive costs up. We will take a closer look at this phenomenon later in this chapter. In essence, supply chain execution is the management and coordination of the movement of materials, information, and funds across the supply chain. A fundamental feature here is that the flow is bidirectional. In general, SCM poses complex problems to businesses, which can only be addressed efficiently enough if considered on the level of the organization as a whole. Usually, SCM is incorporated in powerful software systems (such as SAP) that are responsible for the day-to-day operations of businesses. Among other things, these systems are used for the execution of supply chain transactions, management of supplier relationships, and control of the associated business processes. The techniques used to manage problems related to the supply chain vary depending on the company’s profile and its management’s strategy (and knowledge). Usually, methods from operations research / management science are widely applied along with probabilistic methods and stochastic processes. We shall review some of the basic problems and their solutions. The standard approach is to consider all (i.e. in practice most of) the possible events and factors that can occur in the future, and can cause disruption or some other problem in the supply chain. Using this technique, referred to as Supply Chain Event Management (SCEM), we can create and analyze scenarios, and simulate future flows in the supply chain to plan and design solutions for our business.

2 Problems for Supply Chain Management In general, we need to address the following problems when working with supply chain management: 1. Establishment of the distribution network. In this area we focus on the optimal selection of the numbers and locations of suppliers, production facilities, distribution centers, warehouses, transshipment hubs, and customers. When solving these problems, we usually employ methods and techniques from operations research / management science. Quite often, we end up having to solve a linear programming (LP) or integer programming (IP) problem. Furthermore, if there is some ambiguity or uncertainty in the initial data, we usually use stochastic programming techniques.

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2. Implementation of the distribution strategy. In this area we aim to answer questions related to the operating control (particularly including delivery scheme, mode of transportation, replenishment strategy, and transportation control) of the supply chain. 3. Setting up protocol for information sharing. In this area we seek to find methods for integrating information and other processes through the supply chain. In order to have a better coordination between the partners in the supply chain, we have to share valuable information between the individual business units. Our primary focus has to be the integration of data concerning demand signals, forecasts, inventory, and transportation. Failing to carry out this particular activity can lead up to the development of the so-called bullwhip effect, where businesses in the supply chain have to spend large amount of money on inventory costs just to pay for their business partners’ strategic manipulations. 4. Implementing inventory management. In this area we control processes that manage the quantity and location of the whole inventory (including all raw materials, work-in-progress products, and finished goods). 5. Management of cash flow transactions. In this area we focus on the issues of implementing the actual payment terms and methodologies for exchanging funds between entities in the supply chain. We note that this task is not as straightforward as it first looks, since usually payments follow their own schedules and do not correspond in time with the actual exchange of goods or services. In addition, several businesses operate with negative working capital, which makes their cash-flow transactions even more complicated. The above activities have to be coordinated well together in order to achieve least total logistics cost. It is imperative to consider the problems on the business-level or even on the level of the whole supply chain, i.e. to take a systems approach, when planning logistical activities.

3 Using Operation Research in Supply Chain Management In the following we shall review some important topics of Network Optimization that can successfully be applied in designing the distribution network configuration for supply chains. For an extensive review of the most applicable methods in this area, the reader can consult (Jensen and Bard 2003). Network optimization problems are special cases of linear programming (LP) models. In general, it is important to identify problems that can be modeled as networks, because (i) network representations make the optimization models easier to visualize and explain, and more importantly (ii) very effective algorithms are available for solving network-related problems. Several types of network optimization problems exist that are worth considering when working with supply chain management. The transportation problem (TP) and assignment problem (AP) both aim to capture some particular features of supply chain management problems. Each of them represents a special case of the pure minimum cost flow problem (PMCFP), which has an attractive feature concerning the integrity of its optimal solution. The general minimum cost flow

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Fig. 10.1 Example of a classical distribution network

problem (GMCFP) extends the pure minimum cost flow model by introducing the concept of spoilage (of goods) into its framework. In the following we shall briefly review and analyze all of these problems from mathematical point of view. Before getting into the details, let us recall some of the basic concepts of networks as used in the environment of supply chains. When explaining the terminology, we will refer to its origin, i.e. the framework of the classical distribution networks (see Fig. 10.1). The nodes denote the places where products have to be transported from and delivered to. The arcs denote the possibilities of transportation between the nodes, and arc flow refers to the actual amount of transportation on an arc. Each arc has a tail, the node where it originates, and a head, the node where it terminates1. There are upper bounds and lower bounds specified for each arc, and we aim at finding solutions where all arc flows are within the range of the lower and upper bounds on each arc. The cost on an arc represents the amount of money that is incurred when transporting one unit of products on that arc, and the gain on an arc defines the rate of spoilage of goods that occurs when transporting on that arc2. The external flow of a node refers to the case when supply or demand is present at the node. Feasible flow represents an overall scheme of transportation in the network such 1

Thus, every arc defines the possibility of transportation in one direction. If transportation is allowed between two nodes in both directions then we define two parallel arcs between those nodes with opposite directions. 2 There is a historical reason why we call this quantity gain (and not loss). In general, when transporting on an arc we can both gain additional units (gij > 1) and lose some of the original units (0 < gij < 1). The case when units are generated due to transportation on an arc may seem to be counter-intuitive, but it has a natural interpretation and application in the field of finance.

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that all demand is satisfied with each arc flow being within its lower and upper bounds. Optimal flow is a feasible flow that has minimum cost.

3.1 Transportation Problem We define the transportation problem (TP) on a bipartite network, where arcs only go from supply nodes to destination nodes. For easier analysis of the problem, we introduce dummy nodes for sources or destinations in order to make the total supply be equal to the total demand (see Fig.10.2).

Fig. 10.2 Example of minimum cost flow network as a transportation problem

The linear programming (LP) formulation of the transportation problem with m number of sources and n number of destinations is given by (Jensen and Bard 2003) m

min

n

∑∑ c i =1 j =1 n

s.t.

∑x j =1 m

∑x i =1

ij

x ij

ij

= si ,

i = 1, K , m,

ij

= dj,

j = 1, K , n,

xij ≥ 0,

i = 1, K , m, j = 1, K , n,

where si denotes the supply at source node i, dj is the demand at destination node j, cij gives the cost of transportation from source i to destination j, and xij represents the amount of units transported from source i to destination j, i = 1, …, m, j = 1, …, n.

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Remark 1. An important property of the transportation problem comes from its network structure, and it states that if si, i = 1, …, m, and dj, j = 1, …, n, are all integers then there exists an optimal integer solution as well. Furthermore, when solving the problem with the simplex method, we will always get an integer solution as optimal solution.

3.2 Assignment Problem The assignment problem (AP) can be considered as a particular case of the transportation problem. In general, here we seek to find an optimal (minimum cost) assignment of some tasks to some objects. In a supply chain, this can practically mean of finding a one-to-one matching between batch of goods (loads) and their possible destinations (shipping). In this case the number of sources and destinations are the same (m = n), and we set all supplies and demands to 1 (si = di = 1, i = 1, …, n). The problem itself can be formulated as the following LP problem (Jensen and Bard 2003) n

min

n

∑∑ c i =1 j =1 n

s.t.

ij

xij

∑x

ij

= 1,

i = 1, K , n,

∑x

ij

= 1,

j = 1, K , n,

j =1 n

i =1

xij ≥ 0,

i, j = 1, K , n.

Here, xij represents the decision variable of taking load i to destination j. From the structure of the problem (see Remark 1) we have that there is an optimal solution for which xij* ∈ {0, 1}, i, j = 1, …, n, and hence we indeed obtain the optimal assignment with xij* = 1 (xij* = 0) meaning that we take (do not take) the whole load i to destination j. Remark 2. We can appreciate the integrity of the optimal solution to the AP if we try to solve it as an integer programming (IP) problem. From computational point of view, solving the AP as an IP problem, we would encounter executing an algorithm of O(2n) steps, which, for larger n (n >> 30) would not be computationally feasible.

3.3 Minimum Cost Flow Problems In the following we shall formulate the pure and general forms of the minimum cost flow problem. As we will see, they indeed present an extension of TP (and AP). Typically, the minimum cost flow problems aim at finding optimal logistics to distributions problems (see Fig.10.1). We have a number of production facilities or warehouses that store a particular commodity, and have a number of customers waiting for their demand on that

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commodity. Let us denote the supply of each source node i by si, and the demand of each destination node j by dj. Supplies and demands at nodes are collectively referred to as external flows. The possible transportation routes are represented by a network, where nodes are the places of supply, demand, or transshipment. Let us denote the set of nodes by N. Shipping links are denoted by arcs, A ⊆ N × N = {(i, j ) | i, j ∈ N },

where each arc (i, j) ∈ A has some fixed lower bound lij and upper bound uij that limit the flow on that arc, and has a cost of transportation per unit of product cij. The problem is to determine the optimal shipping plan that minimizes the overall transportation cost (see Fig.10.1). The only difference between the pure and general forms of the minimum cost flow problem is the consideration of conservation of flows on arcs. In PMCFP flow is preserved on any arc, while in GMCFP flows can have gains or losses on some arcs. Network flow problems can readily be formulated as LP problems, where the decision variables are the flow variables xij that define the amount of products transported from node i to node j. In order to give a sound mathematical formulation to the distribution network model, let us introduce the following notions. Let G = (N, A) denote the graph representing the distribution network with set of nodes N and set of arcs A. Let us define the forward star of a node as the set of all arcs that originate (have their tails) in that node FS(i ) = {(i, j ) | (i, j ) ∈ A}, ∀i ∈ N .

Furthermore, let the reverse star of a node be defined as the set of all arcs that terminate (have their heads) in that node RS(i ) = {( j , i ) | ( j , i ) ∈ A}, ∀i ∈ N .

Then the flow balance constraint at node i in the pure minimum cost flow model can be formulated by

∑x

ij ( i , j )∈FS( i )

−

∑x

ji ( j ,i )∈RS ( i )

= bi ,

where bi denotes the external flow, i.e. the supply or demand, at node i. In particular, at supply nodes we have external inflow to the network with bi = si > 0 (i.e. supplies are positive), and at demand nodes there is external outflow from the network with bi = –di < 0 (i.e. demands are negative). In the general minimum cost flow model we consider the possibility of gaining (or losing) a certain amount of product on each arc. For an arc (i, j) ∈ A the gain is denoted by gij, where gij ≥ 0, and it is defined as the ratio of the amount of goods arrived at node j and the amount of goods transported from node i. For instance, if gij = 0.85 then 85% of the products transported from node i will eventually arrive at node j. The flow balance constraint at node i in the general minimum cost flow model can be formulated as

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∑x

ij ( i , j )∈FS( i )

−

∑g

ji ( j ,i )∈RS( i )

x ji = bi ,

where bi stands for the external flow at node i. Let us introduce decision variables xij for representing the flow on arc (i, j) ∈ A. Then, mathematically formulating all constraints we can present the LP formulation of the PMCFP is as follows (Jensen and Bard 2003) min

∑c x ∑x − ∑x ij

ij

( i , j )∈ A

s.t.

ij ( i , j )∈FS( i )

ji ( j ,i )∈RS ( i )

= bi ,

l ij ≤ xij ≤ u ij ,

∀i ∈ N , ∀(i, j ) ∈ A.

Analogously, by extending the pure model with the notion of gain we can formulate the LP model of the GMCFP as (Jensen and Bard 2003) min

∑c x ∑x − ∑g ( i , j )∈ A

s.t.

ij ( i , j )∈FS( i )

ij

ij

( j ,i )∈RS ( i )

ji

x ji = bi ,

l ij ≤ xij ≤ u ij ,

∀i ∈ N , ∀(i, j ) ∈ A.

Remark 3. In case of gij = 1 for all (i, j) ∈ A then the GMCFP gives the PMCFP. The property formulated in the following theorem plays a significant role in various applications of the pure minimum cost flow problem (including applications closely related to supply chain management) (Jensen and Bard 2003). Theorem 1. If the coefficients of the pure minimum cost flow problem bi, lij, and uij all take integer values, i.e. bi ∈ Z for all i ∈ N and lij, uij ∈ Z for all (i, j) ∈ A, then there exists an optimal solution xij* with integer values only, i.e. xij* ∈ Z for all (i, j) ∈ A. Remark 4. Technically, this theorem states that in case of bi ∈ Z for all i ∈ N and lij, uij ∈ Z for all (i, j) ∈ A, the extreme points of the feasible region of the pure minimum cost flow problem have integer values. Consequently, using the simplex method for solving the PMCFP we definitely have integer optimal solution, provided that we have an optimal solution. Notice that we cannot say anything about the integrity of the optimal solutions to the generalized minimum cost flow problem. On the other hand, Theorem 1 plays a crucial role when formulating particular cases of network problems, including the transportation problem (see Remark 1) and the assignment problem (see Remark 2).

4 Introduction to Fuzzy Sets and Possibility Distributions The theory of possibility distributions presents a powerful tool for describing, manipulating, and analyzing non-stochastic uncertainties. Fuzzy sets were introduced

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by Zadeh in 1965 to represent and manipulate data and information with non-statistical uncertainty (Zadeh 1965). The theory of fuzzy sets was specifically designed to mathematically model uncertainty and vagueness, and to provide formalized tools for dealing with imprecision intrinsic to many practical problems. The core idea behind the introduction of fuzzy logic was based on the notion of infinite-valued logic and the mathematical development of fuzzy set theory. In Zadeh (1965) formally introduced the theory of fuzzy sets by extending the notion of membership function of sets such that it operates over the range of real numbers [0, 1] (instead of the two-element set {0, 1}), and developed fuzzy logic by proposing new set theoretic operations on membership functions for the calculus of logic. Viewing in this way, we can see that the theory of fuzzy logic is in principle an extension of the classical (Boolean) logic. In classical set theory, a subset A of a set X can be defined by its characteristic function χA as a mapping from the elements of X to the elements of the set {0, 1}

χ A : X → {0,1}. This mapping can also be viewed as a set of ordered pairs, with exactly one ordered pair standing for each element of X. The first element of the ordered pair is an element of the set X, and the second element is an element of the set {0, 1}. Value zero is used to represent non-membership, and value one is used to represent membership. In this environment, the truth of the statement “x is in A” is determined by the ordered pair (x, χA(x)). The statement is true if the second element of the ordered pair is 1, and the statement is false if it is 0. Similarly, a fuzzy subset A of a set X can be defined as a set of ordered pairs, each with the first element from X, and the second element from the interval [0, 1], with exactly one ordered pair standing for each element of X. This set of ordered pairs defines a mapping μA between elements of the set X and values in the interval [0, 1]. Value zero is used to represent complete non-membership, value one is used to represent complete membership, and values in between are used to denote intermediate degrees of membership. Set X is called the universe (of discourse) for the fuzzy subset A. Frequently, the mapping μA is referred to as the membership function of A. In this environment, the degree to which the statement “x is in A” is true is determined by finding the ordered pair (x, μA(x)). The degree of truth of the statement is the second element of the ordered pair. In the following we shall present a brief introduction to fuzzy sets in a mathematically formal way.

4.1 Fuzzy Sets Definition 1. Let X be a nonempty set. A fuzzy set A in X is defined by its membership function

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μ A : X → [0,1], where μA(x) is interpreted as the degree of membership of element x in fuzzy set A for each x ∈ X. It is clear that A is completely determined by the set of tuples A = {(x, μA(x)) | x ∈ X}. Frequently, we write A(x) instead of μA(x). The family of all fuzzy sets in X is denoted by Y (X). Example 1. The membership function of the fuzzy set “close to 1” on the real numbers R can be defined as A( x) = e −ϑ ( x −1) , 2

where ϑ ≥ 0 is a positive real number (see Fig.10.3).

Fig. 10.3 A membership function for “x is close to 1”

Example 2. Let us assume that we are operating a warehouse in a supply chain. We order blocks of cast steel of some standard size from our suppliers, several medium-size companies, store the blocks in our warehouse, and sell them to car maker factories. Concerning the current market situation (which implies both the supply and demand of steel in our business) we might want to characterize when we consider our inventory low. Aiming at operating our warehouse at minimum cost, we have to balance a trade-off with the following two conflicting criteria: (i) to generate high profit we need to sell (and thus order too) as many blocks of cast steel as possible; and (ii) to minimize our inventory costs we need to have as little amount of steel standing idle in our warehouse as possible. The reason of why these criteria are conflicting is that in practice our order seldom arrives on time and with the exact quantity we specified, and our business partners that we supply can often change their orders. In the end, we can easily end up in a situation where we received all of our orders from our suppliers, but cannot deliver the products to our customers, because in the meanwhile they changed their orders. Identifying an optimal level of operating inventory is crucial. We can call this optimal level low inventory. As a rule of thumb, we can follow the following simple method in our inventory management. Until we reach the state of low inventory, we fulfill our customers’ needs using our inventory (without ordering from suppliers). However, once we have reached the level of low inventory, we start the replenishment.

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Low can be represented as a fuzzy set on the universe of the amounts of cast steel blocks, and it depends on the market situation, the size of our business, and our readiness to fulfill our customers’ orders3. For instance, low can roughly be interpreted as follows (see Fig.10.4): • Below 100 blocks of cast steel inventory is considered low, and independent of the stored amount there is no sign of ineffective inventory management. However, depending on the actual stockpile we start the replenishment. • Between 100 and 150 blocks an increase in the amount of stock induces some sign of ineffective inventory. Furthermore, a decrease in the amount of inventory calls for a preparation to some replenishment. • Between 150 and 200 blocks an increase in stockpile generates warning about inefficient inventory. Moreover, a decrease in the amount of stock indicates a future need of replenishment. • Beyond 200 blocks of cast steel inventory is considered as inefficient, and no replenishment should take place.

Fig. 10.4 Membership function of “low” inventory

In the following we will present the basic notions of fuzzy sets, and then introduce the concept of fuzzy number. In our theoretical development as well as practical applications fuzzy numbers will play fundamental roles in representing nonstochastic uncertainties. Furthermore, they can be considered as “building bricks” of fuzzy sets through fuzzy set theoretic operations union, intersection, and negation (see Majlender 2004, for more details).

4.2 Fuzzy Numbers Definition 2. Let A be a fuzzy set in X. The support of A, denoted by supp(A), is defined as a (classical) subset of X whose elements all have nonzero membership grades in A, i.e. supp( A) = {x ∈ X | A( x) > 0}. 3

In case we consider low inventory as warehouse with too few products, it can happen that we are unable to fulfill a larger order from our customers.

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Definition 3. A fuzzy set A of a classical set X is called normal if there exists x ∈ X such that A(x) = 1. Otherwise A is called subnormal. Definition 4. A γ-level set of a fuzzy set A of X is a classical set defined by

[A]γ

if γ > 0, ⎧{x ∈ X | A( x) ≥ γ } =⎨ ⎩cl{x ∈ X | A( x) > 0} if γ = 0,

where cl denotes the closure of a set (see Fig.10.5).

Fig. 10.5 A γ-level set of a fuzzy set with Gaussian membership function

Definition 5. A fuzzy set A of X is called convex if [A]γ is a convex subset of X for all γ ∈ [0, 1]. In many real-world situations we are only able to characterize numeric information imprecisely. For example, concerning SCM applications, we might use terms such as about 150 blocks of cast iron, near one percent of our orders will not be fulfilled, or the demand for our product is essentially bigger than 500 units. These are examples of what are called fuzzy numbers. Definition 6. A fuzzy number A is a fuzzy set on the real line R with a normal, (fuzzy) convex, and continuous membership function of bounded support. The family of all fuzzy numbers is denoted by Y . Let A ∈ Y be a fuzzy number. Then [A]γ is a compact (closed and bounded) subset of R for all γ ∈ [0, 1]. Let us introduce the notations a1(γ) = min[A]γ and a2(γ) = max[A]γ for γ ∈ [0, 1]. That is, let a1(γ) denote the left-hand side and a2(γ) denote the right-hand side of the γ-level set of A for any γ ∈ [0, 1]. Then, we can use the notation

[A]γ = [a1 (γ ), a2 (γ )],

∀γ ∈ [0,1].

Notice that in this case we represented a fuzzy number by a set of intervals. The following theorem can readily be proven.

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Theorem 2. Let A ∈ Y be a fuzzy number with [A]γ = [a1(γ), a2(γ)], γ ∈ [0, 1]. Then the left-hand side function a1: [0, 1] → R is monotone increasing, and the right-hand side function a2: [0, 1] → R is monotone decreasing. Furthermore, for any 0 ≤ γ ≤ δ ≤ 1 [A]γ ⊇ [A]δ holds. In general, fuzzy numbers can be considered as possibility distributions representing non-statistical uncertainty (Dubois and Prade 1988). In the following, we shall introduce two special types of fuzzy numbers that are of particular importance in practical decision support systems. Due to their straightforward interpretability and simple mathematics, they can readily be incorporated in the frameworks of most applications. Definition 7. A fuzzy set A is called triangular fuzzy number with peak (or center) a ∈ R, left width α ≥ 0, and right width β ≥ 0, if its membership function is of the following form (see Fig.10.6) ⎧ ⎪1 + ⎪⎪1 A( x) = ⎨ ⎪1 − ⎪ ⎪⎩0

x−a

α x−a

β

if a − α < x < a, if x = a, if a < x < a + β , otherwise.

Here we use the notation A = (a, α, β). We can easily verify that

[A]γ = [a − (1 − γ )α , a + (1 − γ )β ],

∀γ ∈ [0,1].

In particular, the support of A is the open interval (a – α, a + β). A triangular fuzzy number with center a can be seen as a fuzzy quantity “x is approximately equal to a”.

Fig. 10.6 Triangular fuzzy number

Definition 8. A fuzzy set A is called trapezoidal fuzzy number with tolerance interval [a, b], left width α ≥ 0, and right width β ≥ 0, if its membership function is of the following form (see Fig.10.7)

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⎧ ⎪1 + ⎪⎪1 A( x) = ⎨ ⎪1 − ⎪ ⎪⎩0

x−a

α x −b

β

if a − α < x < a, if a ≤ x ≤ b, if b < x < b + β , otherwise.

Here we use the notation A = (a, b, α, β). It can easily be shown that

[A]γ = [a − (1 − γ )α , b + (1 − γ )β ],

∀γ ∈ [0,1].

Especially, the support of A is the open interval (a – α, b + β). A trapezoidal fuzzy number with tolerance interval [a, b] can be seen as a fuzzy quantity “x is approximately between a and b”.

Fig. 10.7 Trapezoidal fuzzy number

Obviously, a trapezoidal fuzzy number with a tolerance interval that only includes a single point converts to a triangular fuzzy number, i.e. (a, a, α, β) = (a, α, β). A triangular or trapezoidal fuzzy number whose right and left widths are equal (α = β) is called symmetric. For a symmetric triangular fuzzy number we can use the notation (a, α) = (a, α, β). In the following, we will present some fundamental notions that we may encounter when working with fuzzy sets in applications.

4.3 Operations on Fuzzy Sets Definition 9. Let A and B be two fuzzy sets of a classical set X. Then A is called a subset of B, denoted by A ⊆ B, if A(x) ≤ B(x) holds for all x ∈ X. Furthermore, if A(x) < B(x) holds for some x ∈ X, then A is called a proper subset of B. If A, B ∈ Y are two fuzzy numbers, then they represent some non-stochastic uncertainty on the real line. In this case A ⊆ B implies that A has more information content, and thus involves less uncertainty than B does (see Fig.10.8).

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Fig. 10.8 A is a subset of B

Definition 10. Let A and B be two fuzzy sets of a classical set X. A and B are said to be equal, denoted by A = B, if A ⊆ B and B ⊆ A both hold. Obviously, A = B if and only if A(x) = B(x) for all x ∈ X. The following definition mathematically formulates the basic set theoretic operations on fuzzy sets. Definition 11. Let A and B be two fuzzy sets of a classical set X. Then the intersection of A and B is defined by ( A ∩ B)( x) = min{ A( x), B ( x)}, ∀x ∈ X ;

The union of A and B is defined by ( A ∪ B)( x) = max{ A( x), B( x)}, ∀x ∈ X ;

and the complement of A is defined by (¬A)( x) = 1 − A( x), ∀x ∈ X .

Remark 5. The fuzzy set theoretic operations can be viewed as extensions of the classical set theoretic concepts, since they reduce to their usual meaning when the fuzzy sets only take membership degrees from {0, 1}. Definition 12. Let A ∈ Y be a fuzzy number. If supp(A) = {x0} then A is called a fuzzy point. Clearly, if A is a fuzzy point with supp(A) = {x0}, then A does not represent any uncertainty. In this case [A]γ = [x0, x0] = {x0} for all γ ∈ [0, 1]. When every fuzzy parameter in a fuzzy system becomes a fuzzy point, then the system will reduce to a classical system (that works with crisp numbers). In the following, we will present the mathematical definitions of the basic arithmetic operations on fuzzy numbers.

4.4 Arithmetic Operations on Fuzzy Numbers Zadeh (1965) introduced the sup-min extension principle for the development of mathematics on fuzzy sets.

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Definition 13. Let A1, …, An ∈ Y be fuzzy numbers, and let f : Rn → R be a continuous function. Then we can extend function f to the case of fuzzy numbers via the sup-min extension principle by

f ( A1 , K , An )( y ) =

sup f ( x1 ,K, xn ) = y

min{ A1 ( x1 ), K , An ( xn )}, ∀y ∈ R.

For practical applications of the extension principle we can use the following result, which was proven by Nguyen (1978): Theorem 3. Let A1, …, An ∈ Y be fuzzy numbers, and let f : Rn → R be a continuous function. Then

[ f ( A1 , K , An )]γ

= { f ( x1 , K , xn ) | x1 ∈ [ A1 ] , K , xn ∈ [ An ] }, ∀γ ∈ [0,1]. γ

γ

Focusing on the specific deployment of fuzzy numbers in the framework of SCM, let us present some basic arithmetic operations on fuzzy numbers under the supmin extension principle. Theorem 4. Let A, B ∈ Y be fuzzy numbers with [A]γ = [a1(γ), a2(γ)] and [B]γ = [b1(γ), b2(γ)], γ ∈ [0, 1], and let λ ∈ R be a constant. Then we have

[A + B]γ = [a1 (γ ) + b1 (γ ), a2 (γ ) + b2 (γ )], [A − B]γ = [a1 (γ ) − b2 (γ ), a2 (γ ) − b1 (γ )], and

[λA]γ

⎧[λa (γ ), λa2 (γ )] if λ ≥ 0, =⎨ 1 ⎩[λa2 (γ ), λa1 (γ )] if λ < 0.

The following property is widely used in soft intelligent systems. It states that certain arithmetic operations preserve the shape of linear membership functions of fuzzy numbers.

Remark 6. The addition, subtraction, and scalar multiplication of fuzzy numbers of trapezoidal (triangular) form result in a trapezoidal (triangular) fuzzy number as well. More precisely, let A = (a1, a2, α1, α2) and B = (b1, b2, β1, β2) be two trapezoidal fuzzy numbers, and let λ ∈ R be a fixed number. Then (a1 , a 2 , α1 , α 2 ) + (b1 , b2 , β1 , β 2 ) = (a1 + b1 , a2 + b2 , α1 + β1 , α 2 + β 2 ) (a1 , a 2 , α1 , α 2 ) − (b1 , b2 , β1 , β 2 ) = (a1 − b2 , a2 − b1 , α 1 + β 2 , α 2 + β1 ) ⎧(λa1 , λa2 , λα 1 , λα 2 ) if λ ≥ 0, ⎩(λa2 , λa1 , λ α 2 , λ α 1 ) if λ < 0.

λ (a1 , a2 , α1 , α 2 ) = ⎨

4.5 Normative Measures on Possibility Distributions In what follows, we shall formulate the notions of weighted possibilistic mean and variance of fuzzy numbers. In practical applications, where we use possibility

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distributions, we may want to characterize a distribution with a single value that can stand as its representative value. In addition, we may also want to characterize the degree of dispersion of a possibility distribution to quantify how much it is scattered about its representative value. Looking at this approach from a probabilistic point of view, we can comprehend that we need to develop a theoretical framework for possibility distributions that works in a similar manner as elementary descriptive statistics works for probability distributions. Following the approach of Fullér and Majlender (2003), we will present a development of a normative framework for possibility distributions.

Definition 14. A function f:[0, 1] → R is called a weighting function if it is nonnegative, monotone increasing, and normalized on the unit interval by 1

∫ f (γ )dγ = 1. 0

We will use weighting functions to assign different (case-dependent) degrees of importance to different γ-level sets of possibility distributions.

Definition 15. Let A ∈ Y be a fuzzy number with [A]γ = [a1(γ), a2(γ)], γ ∈ [0, 1], and let f be a weighting function. Then the weighted possibilistic mean value of A (with respect to weighting function f) is defined by 1

E f ( A) = ∫

0

a1 (γ ) + a2 (γ ) f (γ )dγ . 2

We can see that Ef(A) is actually the f-weighted average of the arithmetic means of the γ-level sets; that is, the weight of the arithmetic mean of a1(γ) and a2(γ) is f(γ). We can readily prove that Ef : Y → R is a linear operator with respect to addition and scalar multiplication of fuzzy numbers as defined by the sup-min extension principle:

Theorem 5. Let f be a weighting function, and let A, B ∈ Y be two fuzzy numbers and λ, μ ∈ R. Then the following relationship holds E f (λA + μB) = λE f ( A) + μE f ( B),

where addition and scalar multiplication of fuzzy numbers is defined by the supmin extension principle.

Definition 16. Let A ∈ Y be a fuzzy number with [A]γ = [a1(γ), a2(γ)], γ ∈ [0, 1], and let f be a weighting function. Then the weighted possibilistic variance of A (with respect to weighting function f) is defined by 2 2 1 ⎡⎛ a1 (γ ) + a2 (γ ) a1 (γ ) + a 2 (γ ) ⎞ ⎤ ⎞ ⎛ ( ) ( ) + − − a γ a γ ⎟ ⎜ ⎜ ⎟ ⎢ ⎥ f (γ )dγ 1 2 02 2 2 ⎠ ⎦⎥ ⎠ ⎝ ⎣⎢⎝ 1

Var f ( A) = ∫

1⎛ a (γ ) − a (γ ) ⎞ 1 = ∫⎜ 2 ⎟ f (γ )dγ . 0 2 ⎝ ⎠ 2

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We can see that Varf(A) is nothing else but the f-weighted average of the squared deviations between the arithmetic mean and the endpoints of the γ-level sets of A. The following theorem can readily be proven, and it shows an important relationship between the notions of subset and variance of fuzzy numbers.

Theorem 6. Let A, B ∈ Y be fuzzy numbers with A ⊆ B. Then Varf(A) ≤ Varf(B) holds. Furthermore, if f > 0 is a positive function and A is a proper subset of B, then Varf(A) < Varf(B). Remark 7. When characterizing real-world quantities with possibility distributions, subsethood can always be considered as a representation of a “stronger restriction” on the possible values. In the following, let us compute the weighted possibilistic mean value and variance of trapezoidal fuzzy number under some special weighting functions. Let A = (a, b, α, β) be a fuzzy number of trapezoidal form, and let us choose weighting function as f(γ) = p1(γ) = 2γ, γ ∈ [0, 1]. Then [A]γ = [a – (1 – γ)α, b + (1 – γ)β], γ ∈ [0, 1], and we can compute the simple level-weighted possibilistic mean and variance of A as E p1 ( A) =

a + b β −α + , 2 6 (b − a ) 2 (b − a )(α + β ) (α + β ) 2 ⎛ b − a α + β ⎞ (α + β ) 2 + + =⎜ + , ⎟ + 4 6 24 6 ⎠ 72 ⎝ 2 2

Varp1 ( A) =

which was introduced by Carlsson and Fullér (2001). Furthermore, selecting f(γ) = p0(γ) = 1, γ ∈ [0, 1], we obtain the unweighted mean and variance of A as E p0 ( A) =

a + b β −α + , 2 4 (b − a ) 2 (b − a )(α + β ) (α + β ) 2 ⎛ b − a α + β ⎞ (α + β ) 2 . + + =⎜ + ⎟ + 4 4 12 4 ⎠ 48 ⎝ 2 2

Varp0 ( A) =

This special case of the weighted mean, where all γ-level sets of a fuzzy number have equal weights, was suggested by Goetschel and Voxman for defining an ordering on fuzzy numbers (Goetschel and Voxman 1986). In general, let us set f(γ) = pn(γ) = (n + 1)γ n, γ ∈ [0, 1], where n ≥ 0 is a fixed constant. Then we can formulate the power-weighted possibilistic mean and variance of A as E pn ( A) =

β −α a+b + , 2 2(n + 2)

Varpn ( A) =

(b − a ) 2 (b − a )(α + β ) (α + β ) 2 + + . 4 2(n + 2) 2(n + 2)(n + 3)

This particular class of weighted functions was introduced by Fullér and Majlender (2003).

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4.6 Measure of Possibility To conclude this section, let us consider the following formulations for computing measure of possibility. We will draw on these definitions later when introducing network optimization models in fuzzy environment. For A, B ∈ Y we define the truth value of the statement “A is equal to B” by Pos( A = B ) = sup{ A( x) ∩ B ( x) | x ∈ R} = ( A − B )(0),

where ∩ stands for intersection of fuzzy sets4 (see Carlsson and Fullér 2002, for details). For A, B ∈ Y we compute the truth value of the statement “A is less than or equal to B” by Pos( A ≤ B) = sup{ A( x) ∩ B ( y ) | x, y ∈ R , x ≤ y} = sup( A − B)( z ) z ≤0

(see Carlsson and Fullér 2002, for details). In what follows, we shall apply fuzzy numbers in the classical network optimization problems, and develop an enhanced modeling methodology for solving problems related to supply chain management and logistics.

5 Introducing Fuzziness in Network Optimization When establishing or operating a logistic network for supply chain management, we usually face the problem of correctly estimating the associated parameters. For instance, setting the future (or even present) supplies and demands of all business units in a supply chain network can be very difficult. Furthermore, estimating the capacities of to-be-built warehouses and production facilities (which should incorporate future market demand for the products stored and produced there) involves great uncertainty and is usually based on some “educated guesses”. However, these quantities can naturally be characterized by possibility distributions, i.e. fuzzy numbers. Our experiences in real-world applications show that managers and executives responsible for corporate logistic network operations prefer using linguistic characterization of numeric quantities (e.g. for supplies, demands, and capacities) that can be represented by fuzzy numbers. Moreover, due to the simple (yet powerful enough) properties of the triangular and trapezoidal fuzzy numbers, senior management tends to be very comfortable with using these fuzzy quantities in their decision making. In the following, we will incorporate the notion of trapezoidal fuzzy numbers into the framework of classical network optimization. Focusing on the models we 4

Notice that if A = B (as fuzzy sets) then Pos(A = B) = 1.

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discussed above (TP and AP, which belong to the PMCFP, or more generally the GMCFP, as special cases), we will present an enhanced soft decision support method for logistic network optimization. Viewing the models in a unified framework, where we use the observation that certain constraints in their LP are of type equality constraint or inequality constraint, we suggest an extended approach to solving SCM problems based on fuzzy optimization.

5.1 Possibilistic Linear Equality Systems Modeling real-world problems mathematically, we often have to find a solution to a linear equality system that has the form ai1 x1 + L + ain xn = bi , i = 1, K , m,

where aij, bi, xj ∈ R are real numbers, i = 1, …, m, j = 1, …, n. In general, this system of equations belongs to the class of ill-posed problems, because a small change (called perturbation) in the parameters aij and bi can cause a large deviation in the solution x* = {xj*}j = 1, …, n to the system. A possibilistic linear equality system (PLES) with fuzzy numbers is of the form ~ a~i1 x1 + L + a~in xn = bi , i = 1, K , m, where ãij, b̃i ∈ Y are fuzzy quantities, xj ∈ R are real numbers, i = 1, …, m, j = 1, …, n, and the addition and multiplication by real number of fuzzy quantities are defined by the sup-min extension principle (Carlsson and Fullér 2002). Furthermore, we consider each equation of the system in a possibilistic sense. Let us denote the degree of satisfaction of the i-th equation in the system at point x = {xj}j = 1, …, n ∈ Rn by μi(x), i.e. ~ μi (x) = Pos(a~i1 x1 + L + a~in xn = bi ). Following the methodology of Bellman and Zadeh (1970) the fuzzy solution (or the fuzzy set of feasible solutions) of the possibilistic linear equality system above can be viewed as the intersection of fuzzy sets μi(x), i = 1, …, m, i.e.

μ (x) = min{μ1 (x), K , μ m (x)}. We can introduce a measure of consistency for the PLES in the following way

μ * = sup{μ (x) | x ∈ R n }. Let X* be the set of points x* ∈ Rn for which μ(x) reaches its maximum (provided it exists), i.e. X * = {x* ∈ R n | μ (x * ) = μ * }.

If X* ≠ ∅ and x* ∈ X*, then x* is called a maximizing (or best) solution to the PLES. Working with linear equality systems with fuzzy quantities, we always aim

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at finding a maximizing solution. In this case, following the approach of Negoita (1981), we need to solve the following optimization problem: max

λ

s.t.

μi ( x1 , K , xn ) ≥ λ ,

x = ( x1 , K , xn ) ∈ R ,

i = 1, K , m,

n

λ ∈ [0,1].

In general, finding the solutions to this problem requires the use of nonlinear programming techniques, and could be difficult. However, if the fuzzy numbers in our original system of linear equalities are of trapezoidal form, then this problem turns into a mathematical programming problem, where all constraints only consist of stepwise rational functions. Carlsson and Fullér (2002) summarized several important results concerning the stability properties of possibilistic linear equality systems. If certain conditions hold for the membership functions of the fuzzy numbers in the system, we can be sure that small changes in the fuzzy coefficients of the PLES imply small changes in the optimal solution as well. These results are based on the analysis of the original PLES and its perturbed version (where the fuzzy coefficients are changed up to some degree). In particular, several important results exist that formulate stability properties of possibilistic linear equality systems, where the coefficients are fuzzy numbers of trapezoidal forms (see Carlsson and Fullér 2002, for a good survey). In practice, stability of a PLES means that approximate (non-precise) estimates of the fuzzy quantities does not invalidate our results. Since the optimal solution to a stable PLES is robust (up to some degree), we can be certain that small changes in the fuzzy coefficients of the system will approximately keep the optimal solution. Before concluding this section, let us formulate the degree of satisfaction of each equation in the PLES above, when all parameters are estimated by symmetric trapezoidal fuzzy numbers of the forms ~ a~ij = (aij − ϑ , aij + ϑ , α , α ), bi = (bi − ϑ , bi + ϑ , δ i , δ i ), where ϑ, α, δi ≥ 0 are parameters that fuzzify the classical linear equality system, i = 1, …, m, j = 1, …, n. Then our original possibilistic linear equality system turns into (ai1 − ϑ , ai1 + ϑ , α , α ) x1 + L + (ain − ϑ , ain + ϑ , α , α ) xn = (bi − ϑ , bi + ϑ , δ i , δ i ),

i = 1, …, m. From the properties of fuzzy arithmetic operations on trapezoidal fuzzy numbers we obtain ~ a~i1 x1 + L + a~in xn − bi =

( a ,x i

− bi − ϑ ( x + 1), a i , x − bi + ϑ ( x + 1), α x + δ i , α x + δ i ),

where 〈ai, x〉 = ai1x1 + ⋅⋅⋅ + ainxn for i = 1, …, m, and ||x|| = ||x||1 = |x1| + ⋅⋅⋅ + |xn|. Thus, using the definition of the measure of possibility we find the degree of satisfaction of the i-th equation of the PLES at point x as

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~

~

μi (x) = Pos(a~i1 x1 + L + a~in xn = bi ) = (a~i1 x1 + L + a~in xn − bi )(0) =

⎧1 if a i , x − bi ≤ ϑ ( x + 1), ⎪ a i , x − bi − ϑ ( x + 1) ⎪ if ϑ ( x + 1) < a i , x − bi < ϑ ( x + 1) + α x + δ i , ⎨1 − α x + δi ⎪ ⎪0 if a i , x − bi ≥ ϑ ( x + 1) + α x + δ i . ⎩

Especially, if we estimate the parameters of the PLES by symmetric triangular possibility distributions, then we have ϑ = 0, and thus ⎧ a i , x − bi ⎪1 − μ i (x) = ⎨ α x + δi ⎪0 ⎩

if a i , x − bi < α x + δ i , if a i , x − bi ≥ α x + δ i .

5.2 Flexible Linear Programming The conventional model of linear programming (LP) can be stated as n

∑c x

min

j

j

j =1

n

∑a

s.t.

j =1

ij

x j ≤ bi

i = 1, K , m.

In many practical problems (and this is especially true for problems belonging to logistics and supply chain management), instead of minimizing the objective function c1x1 + ⋅⋅⋅ + cnxn it may be sufficient to determine a solution x = {xj}j = 1, …, n such that n

∑c x j =1

j

n

∑a j =1

ij

j

≤d

x j ≤ bi , i = 1, K , m,

where d is some predefined aspiration level. In this way, we “convert” the objective function into a restriction (constraint). Now let us assume that all parameters in this system are fuzzy quantities, and are described by symmetric trapezoidal fuzzy numbers. Then the following flexible (or fuzzy) linear programming (FLP) problem can be obtained by replacing the crisp parameters cj, d, aij, and bi with symmetric trapezoidal fuzzy numbers ~ ~ c~j = a~0 j = (a0 j − ϑ , a0 j + ϑ , α , α ), d = b0 = (b0 − ϑ , b0 + ϑ , δ 0 , δ 0 ), ~ a~ = (a − ϑ , a + ϑ , α , α ), b = (b − ϑ , b + ϑ , δ , δ ), ij

ij

ij

i

i

i

i

i

i = 1, …, m, j = 1, …, n, respectively (Carlsson and Fullér 2002): (ai1 − ϑ , ai1 + ϑ , α , α ) x1 + L + (ain − ϑ , ain + ϑ , α , α ) xn ≤ (bi − ϑ , bi + ϑ , δ i , δ i ),

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i = 0, 1, …, m. Here, δ0 is interpreted as the tolerance level for the objective function, and δi is called the tolerance level for the i-th constraint, i = 1, …, m. The parameters ϑ ≥ 0 and α ≥ 0 that we universally use to fuzzify the objective function as well as the left-hand sides of the constraints in our LP model will guarantee the stability property of the solution to the FLP problem under small changes in the coefficients aij and bi. Moreover, if δi = α holds for any i = 0, 1, …, n, then the stability property of the fuzzy solutions to the FLP system does not depend on ϑ (Fullér 1989). Let us denote the degree of satisfaction of the i-th constraint at point x = {xj}j = n 1, …, n ∈ R by μi(x), i.e. ~ μi (x) = Pos(a~i1 x1 + L + a~in xn ≤ bi ), i = 0,1, K , m. Then we define the (fuzzy) solution to the FLP problem as a fuzzy set on Rn with membership function of the form

μ (x) = min{μ 0 (x), μ1 (x), K , μ m (x)}. Furthermore, the maximizing solution x* to the FLP problem satisfies the equation

μ (x* ) = max μ (x). x∈R n

Now the left-hand side of each constraint in the FLP model is a symmetric trapezoidal fuzzy number, and ~ a~i1 x1 + L + a~in xn − bi =

( a ,x i

− bi − ϑ ( x + 1), a i , x − bi + ϑ ( x + 1), α x + δ i , α x + δ i ),

where 〈ai, x〉 = ai1x1 + ⋅⋅⋅ + ainxn for i = 0, 1, …, m, and ||x|| = ||x||1 = |x1| + ⋅⋅⋅ + |xn|. Hence, using the definition of the measure of possibility we can formulate the degree of satisfaction of the i-th restriction of the FLP problem at point x as ~ ~ μi (x) = Pos(a~i1 x1 + L + a~in xn ≤ bi ) = sup(a~i1 x1 + L + a~in xn − bi )( z ) z ≤0

⎧1 if a i , x − bi ≤ ϑ ( x + 1), ⎪ a x ϑ x , b 1 ( ) − − + ⎪ i if ϑ ( x + 1) < a i , x − bi < ϑ ( x + 1) + α x + δ i , = ⎨1 − i α x + δi ⎪ ⎪⎩0 if a i , x − bi ≥ ϑ ( x + 1) + α x + δ i .

In particular, if we characterize the parameters of the FLP problem by symmetric triangular possibility distributions, then we have ϑ = 0, and thus ⎧1 ⎪ a , x − bi ⎪ μi (x) = ⎨1 − i α x + δi ⎪ ⎪⎩0

if a i , x ≤ bi , if bi < a i , x < bi + α x + δ i , if a i , x ≥ bi + α x + δ i .

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Furthermore, in the extreme case when α = 0 but δi > 0, we get a linear membership function for μi, i.e. the Zimmermann principle (1978). Indeed, for α = 0 we have the following system for the FLP problem (ai1 ,0) x1 + L + (ain ,0) xn ≤ (bi , δ i ), i = 0,1, K , m,

where we can compute the degree of satisfaction of the i-th restriction at x ∈ Rn as ⎧1 ⎪ a , x − bi ⎪ μi (x) = ⎨1 − i δi ⎪ 0 ⎩⎪

if a i , x ≤ bi , if bi < a i , x < bi + δ i , if a i , x ≥ bi + δ i ,

for i = 0, 1, … , m. If α = 0 then μi can easily be interpreted as follows: (i) if for some x ∈ Rn the value of 〈ai, x〉 is less than or equal to bi then x satisfies the i-th constraint with maximal conceivable degree one; (ii) if bi < 〈ai, x〉 < bi + δi then x is not feasible in classical sense, but the decision maker can still tolerate the violation of the crisp constraint, and accept x as a solution with some positive degree; however, the bigger the violation is, the less the degree of acceptance becomes (and this relationship is linear); and (iii) if 〈ai, x〉 ≥ bi + δi then the decision maker no longer tolerates the violation of the crisp constraint, and thus μi(x) = 0 (see Fig.10.9).

Fig. 10.9 Degree of satisfaction μi(x) as a fuzzy set when α = 0

Sensitivity analysis of FLP problems with crisp parameters and soft constraints was first considered in (Hamacher et al. 1978), where a functional relationship between changes of parameters of the right-hand sides of the constraints and those of the optimal value of the objective function was derived for almost all conceivable cases. In (Tanaka et al. 1986) a special type of FLP problem with symmetric triangular fuzzy numbers was formulated, and the value of information was discussed through sensitivity analysis. Using the theoretical foundations we presented above, in the following we shall introduce various possibilistic models for network optimization that can successfully be applied in the areas of supply chain management and logistics.

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6 Possibilistic Approach to Network Optimization In this section, we shall present a possibilistic approach to SCM by employing fuzzy numbers in classical network optimization problems. In practice, when we work with a network optimization problem, we usually face a situation where we can only estimate certain quantities (e.g. supplies, demands, and costs) in the network with uncertainty. Since the parameters (even the structure) of a network can change rapidly and significantly in real-world problems, we cannot carry out our estimations by using statistics or some well-chosen probability distributions. Simply, we do not know the exact values of certain quantities in the network. However, we can use possibility distributions, i.e. fuzzy numbers, to characterize those quantities, and thus solve our (fuzzy) network optimization problem. In this case, we actually provide imprecise estimates for the parameters of the network, and find a solution that is good enough for us. Concerning the complexity of practical network optimization problems in the area of SCM, we can comprehend that theoretically optimal (or even close to it) solutions are unattainable. We are either not capable of formulating an exact mathematical model of the real situation, or using estimates that are rough and thus almost certainly incorrect. However, incorporating possibility distributions in the decision making of network-related SCM problems has a significant advantage: due to the stability properties of FLP problems, we can be sure that estimates that are approximately correct will give approximately optimal solutions; furthermore, the more exact estimates we can generate, the better (more optimal) solutions we will arrive at. Hence, based on the classical models of network optimization, we suggest the use of the following methods for designing and operating supply chains and logistic networks. We should be aware of the fact that the application of fuzzy logic in the environments of SCM and logistics has been attracting much attention by both academic researchers and practitioners. For various novel approaches to this field as well as in-depth analyses of its methods, the reader can consult fundamental works such as Chanas et al. 1984, and Chanas and Kuchta 1998, (fuzzy transportation problem), and Chanas et al. 1995, and Shih and Lee 1999 (fuzzy minimum cost flow problem).

6.1 Transportation Problem Referring to the classical TP we discussed in the beginning of this chapter, let us assume that the supplies, demands, and costs are characterized by fuzzy numbers of trapezoidal forms ~ si = ( si − ϑ , si + ϑ , σ iL , σ iR ), i = 1, K , m, ~ d j = (d j − ϑ , d j + ϑ , δ jL , δ jR ), j = 1, K , n, c~ij = (cij − ϑ , cij + ϑ , α ijL , α ijR ), i = 1, K , m, j = 1, K , n,

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where σiL and σiR are the tolerance levels for the constraint on the i-th supply, i = 1, …, m, δjL and δjR are the tolerance levels for the constraint on the j-th demand, j = 1, …, n, and αijL and αijR are the tolerance levels for the cost estimates cij, i = 1, …, m, j = 1, …, n. The parameter ϑ ≥ 0 that uniformly fuzzifies the classical TP will assure the stability property of the solution to the FLP problem under small changes in the coefficients si, dj, and cij. Then the fuzzy linear programming formulation of the transportation problem is given by m

n

∑∑ c~ x i =1 j =1 n

~

∑1

j =1 m

(i)

~( j)

∑1 i =1

ij

ij

≤ a~,

xij = ~ si ,

i = 1, K , m,

~ xij = d j ,

j = 1, K , n,

xij ≥ 0,

i = 1, K , m, j = 1, K , n,

i.e. m

n

∑∑ (c i =1 j =1 n

ij

− ϑ , cij + ϑ , α ijL , α ijR ) xij ≤ (a − ϑ , a + ϑ , α L , α R ),

∑ (1 − ϑ ,1 + ϑ , σ

L i

∑ (1 − ϑ ,1 + ϑ , δ

L j

j =1 m

i =1

, σ iR ) xij = ( si − ϑ , si + ϑ , σ iL , σ iR ),

i = 1, K , m,

, δ jR ) xij = (d j − ϑ , d j + ϑ , δ jL , δ jR ),

j = 1, K , n,

xij ≥ 0,

i = 1, K , m, j = 1, K , n,

where ã = (a – ϑ, a + ϑ, αL, αR) is a trapezoidal fuzzy number that stands for a fuzzy aspiration level, and 1̃(i) = (1 – ϑ, 1 + ϑ, σiL, σiR) and 1̃(j) = (1 – ϑ, 1 + ϑ, δjL, δjR) are representations of the crisp number 1 by trapezoidal fuzzy numbers for the constraints on the i-th supply and j-th demand, respectively. Here, since m

n

∑∑ c~ x ij

i =1 j =1

( c, x n

~

ij

− a~ =

)

− a − ϑ ( x + 1), c, x − a + ϑ ( x + 1), α L , x + α R , α R , x + α L ,

∑1

xij − ~ si =

(x

− si − ϑ x (i ) + 1 , x (i ) − si + ϑ x ( i ) + 1 , σ iL x ( i ) + σ iR , σ iR x ( i ) + σ iL ,

j =1

m

(i )

(i)

~( j)

∑1 i =1

(x

(

( j)

)

(

)

)

~ xij − d j =

(

)

(

)

)

− d j − ϑ x ( j ) + 1 , x ( j ) − d j + ϑ x ( j ) + 1 , δ jL x ( j ) + δ jR , δ jR x ( j ) + δ jL ,

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where c = {cij}i = 1, …, m, j = 1, …, n, αL = {αijL}i = 1, …, m, j = 1, …, n, αR = {αijR}i = 1, …, m, j = (j) 1, …, n, x = {xij}i = 1, …, m, j = 1, …, n, and x(i) = {xij}j = 1, …, n, and x = {xij}i = 1, …, m, we find for the degree of satisfaction of the objective function ⎛

m

n

⎞

⎛

⎠

⎝

m

⎞

n

μ 0 (x) = Pos⎜⎜ ∑∑ c~ij xij ≤ a~ ⎟⎟ = sup⎜⎜ ∑∑ c~ij xij − a~ ⎟⎟( z ) z ≤0 i =1 j =1 i =1 j =1 ⎝

⎠

⎧ if c, x − a ≤ ϑ ( x + 1), ⎪1 c, x − a − ϑ ( x + 1) ⎪ if ϑ ( x + 1) < c, x − a < ϑ ( x + 1) + α L , x + α R , = ⎨1 − L R α , x α + ⎪ ⎪0 if c, x − a ≥ ϑ ( x + 1) + α L , x + α R , ⎩

The degree of satisfaction of the restriction on the i-th supply ⎛

n

~

⎞

⎛

⎠

⎝

n

⎞

~

μ ( i ) (x) = Pos⎜⎜ ∑ 1(i ) xij = ~si ⎟⎟ = ⎜⎜ ∑ 1( i ) xij − ~si ⎟⎟(0) j =1 j =1 ⎝

⎧1 ⎪ x ( i ) − si − ϑ x ( i ) + 1 ⎪ ⎪1 − σ iL x (i ) + σ iR ⎪ =⎨ x ( i ) − si + ϑ x ( i ) + 1 ⎪ ⎪1 + σ iR x (i ) + σ iL ⎪ ⎩⎪0

(

)

(

)

⎠

(

)

(

)

if − ϑ x ( i ) + 1 ≤ x ( i ) − si ≤ ϑ x ( i ) + 1 ,

(

)

if 0 < x (i ) − si − ϑ x ( i ) + 1 < σ iL x ( i ) + σ iR ,

(

)

(

)

if − σ iR x (i ) + σ iL < x ( i ) − si + ϑ x ( i ) + 1 < 0, otherwise,

i = 1, …, m, and the degree of satisfaction of the restriction on the j-th demand ⎛

n

⎝

j =1

~

~ ⎞

⎛

n

⎠

⎝

j =1

~ ⎞

~

μ ( j ) (x) = Pos⎜⎜ ∑ 1 ( j ) xij = d j ⎟⎟ = ⎜⎜ ∑ 1 ( j ) xij − d j ⎟⎟(0) ⎧1 ⎪ x( j) ⎪ ⎪1 − ⎪ =⎨ x( j) ⎪ ⎪1 + ⎪ ⎪⎩0

(

− d j −ϑ x

δ x L j

( j)

( j)

+δ

R j

(

)

+1

)

− d j + ϑ x( j) + 1

δ x R j

( j)

+δ

L j

(

if − ϑ x

( j)

)

⎠

(

)

+ 1 ≤ x( j) − d j ≤ ϑ x( j) + 1 ,

(

)

if 0 < x ( j ) − d j − ϑ x ( j ) + 1 < δ jL x ( j ) + δ jR ,

(

)

(

)

if − δ jR x ( j ) + δ jL < x ( j ) − d j + ϑ x ( j ) + 1 < 0, otherwise,

j = 1, …, n. Furthermore, we seek to find a solution x* = {xij*}i = 1, …, m, j = 1, …, n for which the function defined by

{μ0 (x), μ(i) (x), μ ( j ) (x)} μ (x) = i =min 1,K, m j =1,K, n

attains its maximum value, i.e.

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P. Majlender

μ (x* ) = max μ (x) = max i =min {μ 0 (x), μ(i ) (x), μ ( j ) (x)}. 1,K,m x∈R m×n

x∈R m×n

j =1,K,n

6.2 Assignment Problem From the fuzzy TP, we can formally obtain the fuzzy version of the assignment problem by setting m = n, and choosing the following possibility distributions for all (fuzzy) supplies and demands: ~ ~ ~ ~ si = 1(i ) = (1 − ϑ ,1 + ϑ , σ iL , σ iR ), d i = 1 ( i ) = (1 − ϑ ,1 + ϑ , δ iL , δ iR ), i = 1, K , n. The fuzzy LP formulation as well as the degrees of satisfaction of the objective function and the constraints of the assignment problem can readily be obtained by taking the fuzzy formulation of the TP above and substituting si = 1̃(i) and di = 1̃(i) for i = 1, …, n. That is, the fuzzy formulation of the AP can be written as n

n

∑∑ c~ x i =1 j =1 n

~

∑1

j =1 n

(i )

~( j)

∑1 i =1

ij

ij

≤ a~,

~ xij = 1( i ) ,

i = 1, K , n,

~ xij = 1 ( j ) ,

j = 1, K , n,

xij ≥ 0,

i, j = 1, K , n,

or n

n

∑∑ (c i =1 j =1 n

ij

− ϑ , cij + ϑ , α ijL , α ijR ) xij ≤ (a − ϑ , a + ϑ , α L , α R ),

∑ (1 − ϑ ,1 + ϑ , σ

L i

, σ iR ) xij = (1 − ϑ ,1 + ϑ , σ iL , σ iR ),

i = 1, K , n,

∑ (1 − ϑ ,1 + ϑ , δ

L j

, δ jR ) xij = (1 − ϑ ,1 + ϑ , δ jL , δ jR ),

j = 1, K , n,

j =1 n

i =1

xij ≥ 0,

i, j = 1, K , n,

where ã = (a – ϑ, a + ϑ, αL, αR) stands for a fuzzy aspiration level that is characterized by a trapezoidal fuzzy number. Then using the vector notations we introduced above, we can compute the degrees of satisfaction as ⎛

n

n

⎞

⎛

⎠

⎝

n

n

⎞

μ 0 (x) = Pos⎜⎜ ∑∑ c~ij xij ≤ a~ ⎟⎟ = sup⎜⎜ ∑∑ c~ij xij − a~ ⎟⎟( z ) z ≤0 i =1 j =1 i =1 j =1 ⎝

⎠

⎧ if c, x − a ≤ ϑ ( x + 1), ⎪1 c, x − a − ϑ ( x + 1) ⎪ = ⎨1 − if ϑ ( x + 1) < c, x − a < ϑ ( x + 1) + α L , x + α R , L R α ,x +α ⎪ ⎪0 if c, x − a ≥ ϑ ( x + 1) + α L , x + α R , ⎩

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229

and ⎛

n

⎝

j =1

~

~ ⎞

⎛

n

⎠

⎝

j =1

~ ⎞

~

μ ( i ) (x) = Pos⎜⎜ ∑ 1(i ) xij = 1( i ) ⎟⎟ = ⎜⎜ ∑ 1( i ) xij − 1( i ) ⎟⎟(0) ⎧1 ⎪ x (i ) − 1 − ϑ x ( i ) + 1 ⎪ ⎪1 − σ iL x (i ) + σ iR ⎪ =⎨ x (i ) − 1 + ϑ x (i) + 1 ⎪ ⎪1 + σ iR x ( i ) + σ iL ⎪ ⎪⎩0

(

)

(

)

(

)

⎠

(

)

if − ϑ x ( i ) + 1 ≤ x ( i ) − 1 ≤ ϑ x (i ) + 1 ,

(

)

if 0 < x ( i ) − 1 − ϑ x ( i ) + 1 < σ iL x ( i ) + σ iR ,

(

)

(

)

if − σ iR x ( i ) + σ iL < x (i ) − 1 + ϑ x (i ) + 1 < 0, otherwise,

i = 1, …, n, and ⎛

n

⎝

j =1

~

~

⎞

⎛

n

⎠

⎝

j =1

~

⎞

~

μ ( j ) (x) = Pos⎜⎜ ∑ 1 ( j ) xij = 1 ( j ) ⎟⎟ = ⎜⎜ ∑ 1 ( j ) xij − 1 ( j ) ⎟⎟(0) ⎧1 ⎪ x( j) − 1 − ϑ x( j) + 1 ⎪ 1 − ⎪ ⎪ δ jL x ( j ) + δ jR =⎨ x( j) − 1 + ϑ x( j) + 1 ⎪ 1 + ⎪ δ jR x ( j ) + δ jL ⎪ ⎪⎩0

(

)

(

)

(

if − ϑ x

( j)

⎠

)

+1 ≤ x

(

( j)

(

)

− 1 ≤ ϑ x( j) + 1 ,

)

if 0 < x ( j ) − 1 − ϑ x ( j ) + 1 < δ jL x ( j ) + δ jR ,

(

)

(

)

if − δ jR x ( j ) + δ jL < x ( j ) − 1 + ϑ x ( j ) + 1 < 0, otherwise,

j = 1, …, n. Furthermore, we aim at finding a solution x* ∈ Rn×n for which

μ (x* ) = max min {μ 0 (x), μ ( i ) (x), μ ( i ) (x)}. x∈R n×n i =1,K,n

Remark 8. The fuzzy version of the assignment problem can be of particular use in applications of SCM, when we aim at finding cost-effective solutions to tasks related to • setting up delivery plans (where we decide which shipment goes to which customer), and • developing a system of production sites (where we select the site of production for each of our product) under uncertainty. Establishing a new delivery plan or system of production facilities involves great uncertainty and is usually considered as a very risky process. Major part of the risk comes from the fact that we are unable to provide accurate (or even approximately correct) estimations of the real supplies, demands, and costs that we will have in the network. In fact, it is simply impossible to provide accurate forecasts of these quantities by using statistics or probabilistic approaches. Usually, there is no adequate statistics available for these types of tasks, because companies and businesses only reorganize their systems of delivery plans and production sites very seldom. They cost a lot of resources, and after the new

230

P. Majlender

system is set up by contracts with other business partners, any change or alteration can cause either some damage in reputation or substantial money expenditure for the company. Thus, once a system has been established, it is usually considered as final, and no initiation of change (unless inevitable) is supported by the executive management. However, we can circumvent or overrule probabilistic estimations by using opinions of experts. Collecting information about similar events that they have knowledge of being happened before, we can formulate their estimates as possibility distributions. Furthermore, as we can readily represent quantitative knowledge of experts by trapezoidal (or triangular) fuzzy numbers, we can indeed utilize the fuzzy formulation of the assignment problem we presented above in real-world situations. Incorporating fuzzy numbers in the decision-making will equip us with the power of flexibility: even if we do not know or are very uncertain about the correctness of our estimations of the network parameters, we can still obtain solutions that we consider good enough. Working with several experts, we can naturally obtain different optimal solutions to our fuzzy AP, wherefrom we can pick the one that seems to be a suitable solution for our business.

Remark 9. In general, the fuzzy version of the AP does not possess the property of integrity. If we work with non-degenerate (non-crisp) fuzzy numbers, our fuzzy AP will not yield integer optimal solutions 5 . Instead, we will always have a fuzzy set ⎧ ⎫ X = {(x, μ (x)) | x ∈ R n×n } = ⎨⎛⎜ x, min {μ 0 (x), μ (i ) (x), μ ( i ) (x) ⎞⎟ x ∈ R n×n ⎬ 1 , , K = i n ⎝ ⎠ ⎩ ⎭

as fuzzy solution (also called as the fuzzy set of feasible solutions), and we aim to find a point x* ∈ Rn×n that has maximal membership degree in that fuzzy set. We call this point as a maximizing (or best) solution to the fuzzy AP. The fact that a maximizing solution is not necessarily integer-valued vector can be explained by the nature of flexibility that fuzzification brings to the AP: when we introduce approximate values in the objective function and the constraints, we can no longer obtain exact (crisp) values for optimal (or even feasible) solutions. This phenomenon can be comprehended as propagation of uncertainty: if input values are imprecise, output values generally cannot be exact. In the framework of AP, the lack of integrity of the maximizing solutions means that we will have mixed (instead of pure) assignments. For instance, we will consider delivery plans where we supply one costumer using various shipments, or set up a system of production facilities where one type of product is produced in various locations. These features can clearly add some flexibility in our decision-making method. Overseeing the possible alternatives X with their degrees of satisfaction (with respect to the problem), 5

For the same reasons, the fuzzy versions of TP as well as PMCFP (that we will discuss shortly) do not have the integrity property either. However, in those cases the lack of integer-valued solutions does not reduce the applicability of the fuzzy version as compared to the classical framework. For the classical AP, losing the integrity property in the fuzzy formulation requires some explanation.

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after some analysis we can always select a feasible solution that is integer-valued (if it is any). With this particular property, we can see how fuzzy formulation enhances the classical version of AP.

6.3 Minimum Cost Flow Problems Let us consider now the two versions of the minimum cost flow problem: the PMCFP and the GMCFP. We have learned that the TP as well as the AP are special cases of the PMCFP. Using the general notations for the mathematical formulation of the distribution network problem that we presented when discussing the classical PMCFP and GMCFP, let us develop the fuzzy versions of the pure and then the general forms of the minimum cost flow models. Let us assume that the costs cij, the lower bounds lij, and the upper bounds uij on all arcs (i, j) ∈ A, as well as the external flows bi at all nodes i ∈ N are represented by trapezoidal fuzzy numbers of the forms c~ = (c − ϑ , c + ϑ , α L , α R ), (i, j ) ∈ A, ij

ij

ij

ij

ij

~ lij = (lij − ϑ , lij + ϑ , λijL , λijR ), (i, j ) ∈ A, u~ij = (uij − ϑ , uij + ϑ ,ν ijL ,ν ijR ), (i, j ) ∈ A, ~ bi = (bi − ϑ , bi + ϑ , β iL , β iR ), i ∈ N , where for each arc (i, j) ∈ A, αijL and αijR are the tolerance levels for the cost estimates cij, λijL and λijR stand for the tolerance levels for the lower bound estimates lij, νijL and νijR denote the tolerance levels for the upper bound estimates uij, and for each node i ∈ N, βiL and βiR are the tolerance levels for the external flow estimates. The parameter ϑ ≥ 0 that uniformly fuzzifies the classical minimum cost flow model will guarantee the stability property of the solutions to the fuzzy version under small changes in the coefficients cij, lij, uij, and bi. In this environment we can formulate the fuzzy linear programming model of the PMCFP as c~ x ≤ a~,

∑

ij

( i , j )∈A

~ ∑ 1(i ) xij −

( i , j )∈FS( i )

ij

~

∑1

(i ) ( j ,i )∈RS ( i )

~ x ji = bi ,

~ ~ lij ≤ 1( i , j ) xij ≤ u~ij ,

∀i ∈ N , ∀(i, j ) ∈ A,

that is,

∑ (c

ij

− ϑ , cij + ϑ , α ijL , α ijR ) xij ≤ (a − ϑ , a + ϑ , α L , α R ),

( i , j )∈A

∑ (1 − ϑ ,1 + ϑ , β

( i , j )∈FS( i )

L i

, β iR ) xij −

= (bi − ϑ , bi + ϑ , β , β ), L i

R i

∑ (1 − ϑ ,1 + ϑ , β

( j ,i )∈RS( i )

∀i ∈ N ,

(lij − ϑ , lij + ϑ , λ , λ ) ≤ (1 − ϑ ,1 + ϑ , κ ijL , κ ijR ) xij L ij

R ij

≤ (uij − ϑ , uij + ϑ ,ν ijL ,ν ijR ),

∀(i, j ) ∈ A,

L i

, β iR ) x ji

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P. Majlender

where ã = (a – ϑ, a + ϑ, αL, αR) is a trapezoidal fuzzy number that represents a fuzzy aspiration level, 1̃(i) = (1 – ϑ, 1 + ϑ, βiL, βiR) is a representation of the crisp number 1 by trapezoidal fuzzy number for the constraint on the external flow at node i for i ∈ N, and 1̃(i, j) = (1 – ϑ, 1 + ϑ, κijL, κijR) is a fuzzy characterization of the crisp number 1 by trapezoidal fuzzy number for the constraint on the (lower and upper) bounds of flow on arc (i, j) for (i, j) ∈ A. Concerning the practical applicability of the possibilistic framework of the PMCFP, we shall assume that xij ≥ 0 for any (i, j) ∈ A. We can naturally enforce this property in fuzzy environment by requiring that the support of each possibility distribution representing the lower bound on an arc is part of the set of positive real numbers, i.e. supp(l̃ij) ⊆ R+ = {x ∈ R | x > 0} for (i, j) ∈ A. However, since in our representation each l̃ij is a trapezoidal fuzzy number, this condition actually means that lij – ϑ – λijL ≥ 0 holds for any (i, j) ∈ A. Using the properties of the basic arithmetic operations on trapezoidal fuzzy numbers, we can compute c~ x − a~ =

∑

ij

ij

( i , j )∈A

( c, x

)

− a − ϑ ( x + 1), c, x − a + ϑ ( x + 1), α L , x + α R , α R , x + α L , ~ ~ ~ ∑ 1(i ) xij − ∑ 1(i) x ji − bi =

( i , j )∈FS( i )

(x

( j ,i )∈RS ( i )

(

)

− x RS(i ) − bi − ϑ x FS( i ) + x RS(i ) + 1 ,

FS( i )

(

) (x

x FS( i ) − x RS( i ) − bi + ϑ x FS( i ) + x RS(i ) + 1 ,

β x FS( i ) + β L i

R i

(x

RS ( i )

)

+ 1 , β x FS( i ) + β R i

L i

~ ~ lij − 1(i , j ) xij =

(l

~ 1(i , j )

RS( i )

))

+1 ,

)

− xij − ϑ ( xij + 1), lij − xij + ϑ ( xij + 1), λijL + κ ijR xij , λijR + κ ijL xij , x − u~ =

ij

(x

ij

ij

ij

)

− uij − ϑ ( xij + 1), xij − uij + ϑ ( xij + 1), κ ijL xij + ν ijR , κ ijR xij + ν ijL ,

where c = {cij}(i, j) ∈ A, αL = {αijL}(i, j) ∈ A, αR = {αijR}(i, j) ∈ A, and x = {xij}(i, j) ∈ A, xFS(i) = {xij}(i, j) ∈ FS(i), xRS(i) = {xji}(j, i) ∈ RS(i), and where we define 〈c, x〉 = Σ(i, j) ∈ A cij xij and ||x|| = ||x||1 = Σ(i, j) ∈ A | xij| = Σ(i, j) ∈ A xij. Thus, we can formulate the degree of satisfaction of the objective function as ⎛

μ 0 (x) = Pos⎜⎜

∑ c~ x

⎝ (i , j )∈A

ij

ij

⎞ ⎞ ⎛ ≤ a~ ⎟⎟ = sup⎜⎜ ∑ c~ij xij − a~ ⎟⎟( z ) ⎠ z≤0 ⎝ (i , j )∈A ⎠

⎧ if c, x − a ≤ ϑ ( x + 1), ⎪1 c, x − a − ϑ ( x + 1) ⎪ = ⎨1 − if ϑ ( x + 1) < c, x − a < ϑ ( x + 1) + α L , x + α R , L R α ,x +α ⎪ ⎪0 if c, x − a ≥ ϑ ( x + 1) + α L , x + α R , ⎩

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233

The degree of satisfaction of the constraint on the external flow at node i as ⎛

μ ( i ) (x) = Pos⎜⎜ ⎝

~

∑1

(i) ( i , j )∈FS( i )

xij −

~⎞ ⎛ ~⎞ ~ ~ x ji = bi ⎟⎟ = ⎜⎜ ∑ 1( i ) xij − ∑ 1( i ) x ji − bi ⎟⎟(0) = ( j ,i )∈RS ( i ) ⎠ ⎠ ⎝ ( i , j )∈FS(i )

~

∑1

(i ) ( j ,i )∈RS ( i )

(

)

⎧1 if x FS( i ) − x RS( i ) − bi ≤ ϑ x FS( i ) + x RS( i ) + 1 , ⎪ x FS( i ) − x RS( i ) − bi − ϑ x FS(i ) + x RS( i ) + 1 ⎪ ⎪1 − β iL x FS( i ) + β iR x RS(i ) + 1 ⎪ ⎪ ⎪ if 0 < x FS( i ) − x RS( i ) − bi − ϑ x FS(i ) + x RS( i ) + 1 ⎪ ⎪ < β iL x FS(i ) + β iR x RS( i ) + 1 , ⎨ x FS(i ) − x RS(i ) − bi + ϑ x FS(i ) + x RS( i ) + 1 ⎪ ⎪1 + β iR x FS(i ) + β iL x RS( i ) + 1 ⎪ ⎪ R L ⎪ if − β i x FS( i ) + β i x RS(i ) + 1 ⎪ < x FS(i ) − x RS(i ) − bi + ϑ x FS(i ) + x RS( i ) + 1 < 0, ⎪ ⎪⎩0 otherwise,

( (

(

(

(

(

( (

)) (

)

)

)

)

)

)

)

i ∈ N, and the degree of satisfaction of the constraint on the bounds of flow on arc (i, j) as

(~

)

(~

)

(~

)

μ ( i , j ) (x) = Pos lij ≤ 1( i , j ) xij ≤ u~ij = min ⎧⎨sup lij − 1(i , j ) xij ( z ), sup 1(i , j ) xij − u~ij ( z )⎫⎬, ~

⎩ z ≤0

~

z ≤0

where

(

~ ~ sup lij − 1(i , j ) xij z ≤0

)

⎧1 ⎪ ⎪ lij − xij − ϑ ( xij + 1) = ⎨1 − λijL + κ ijR xij ⎪ ⎪⎩0

if lij − xij ≤ ϑ ( xij + 1), if ϑ ( xij + 1) < lij − xij < ϑ ( xij + 1) + λijL + κ ijR xij , if lij − xij ≥ ϑ ( xij + 1) + λijL + κ ijR xij ,

and

(

~ sup 1(i , j ) xij − u~ij z ≤0

)

⎧1 ⎪ ⎪ xij − uij − ϑ ( xij + 1) = ⎨1 − κ ijL xij + ν ijR ⎪ ⎪0 ⎩

if xij − uij ≤ ϑ ( xij + 1), if ϑ ( xij + 1) < xij − uij < ϑ ( xij + 1) + κ ijL xij + ν ijR , if xij − uij ≥ ϑ ( xij + 1) + κ ijL xij + ν ijR ,

⎭

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P. Majlender

(i, j) ∈ A. Notice that here we drew upon our assumption of xij ≥ 0, (i, j) ∈ A. Furthermore, in these circumstances we seek to find a solution x* = {xij*}(i, j) ∈ A ∈ R|A| for which the function defined by

{μ0 (x), μ (i ) (x), μ (i, j ) (x)} μ (x) = min i∈N ( i , j )∈A

attains its maximum value, i.e.

{μ0 (x), μ (i ) (x), μ(i, j ) (x)}. μ (x* ) = max μ (x) = max min i∈N x∈R | A|

x∈R| A|

( i , j )∈A

Remark 10. Let us assume that we characterize all parameters in our fuzzy PMCFP by symmetric trapezoidal fuzzy numbers that have the same left and right widths. In this case we uniformly approximate all quantities of the model with the same type of possibility distribution. In fact, the possibility distributions will merely be translations of each other. That is, let us assume that α = αijL = αijR = λijL = λijR = νijL = νijR = κijL = κijR for all (i, j) ∈ A, and α = βiL = βiR for all i ∈ N. Then we find for the degree of satisfaction of the objective function ⎧1 if c, x − a ≤ ϑ ( x + 1), ⎪ c, x − a − ϑ ( x + 1) ⎪ if ϑ ( x + 1) < c, x − a < (α + ϑ )( x + 1), μ 0 (x) = ⎨1 − α ( x + 1) ⎪ ⎪⎩0 if c, x − a ≥ (α + ϑ )( x + 1),

The degree of satisfaction of the constraint on the external flow at node i

μ ( i ) ( x)

(

)

⎧1 if x FS( i ) − x RS( i ) − bi ≤ ϑ x FS( i ) + x RS( i ) + 1 , ⎪ x FS( i ) − x RS( i ) − bi − ϑ x FS( i ) + x RS( i ) + 1 ⎪ ⎪1 − α x FS(i ) + x RS(i ) + 1 ⎪ ⎪ ⎪ = ⎨ if 0 < x FS(i ) − x RS(i ) − bi − ϑ x FS(i ) + x RS(i ) + 1 < α x FS(i ) + x RS(i ) + 1 , x FS( i ) − x RS( i ) − bi + ϑ x FS(i ) + x RS(i ) + 1 ⎪ ⎪1 + α x FS(i ) + x RS(i ) + 1 ⎪ ⎪ ⎪ if − α x FS(i ) + x RS(i ) + 1 < x FS( i ) − x RS(i ) − bi + ϑ x FS( i ) + x RS( i ) + 1 < 0, ⎪⎩0 otherwise,

(

(

(

(

(

)

(

)

)

)

)

) ( (

)

)

i ∈ N, and the degree of satisfaction of the constraint on the lower and upper bounds of flow on arc (i, j)

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235

⎧1 if max{lij − xij , xij − u ij } ≤ ϑ ( xij + 1), ⎪ max{l − x , x − u } − ϑ ( x + 1) ij ij ij ij ij ⎪⎪1 − α ( xij + 1) μ ( i , j ) ( x) = ⎨ ⎪ if ϑ ( x + 1) < max{l − x , x − u } < (α + ϑ )( x + 1), ij ij ij ij ij ij ⎪ ⎪⎩0 if max{lij − xij , xij − uij } ≥ (α + ϑ )( xij + 1),

(i, j) ∈ A. Now let us extend our framework, and formulate the general form of the minimum cost flow problem in a possibilistic environment by incorporating the notion of gain. Considering the formulation of the classical GMCFP, let us assume that the gain on each arc is represented by a trapezoidal fuzzy number of the form g~ = ( g − ϑ , g + ϑ , β L , β R ), (i, j ) ∈ A, ij

ij

ij

i

i

where β and β represent the tolerance levels for the estimation of the gain on arc (i, j). Then using the notations we introduced for the fuzzy PMCFP, we can state the fuzzy LP formulation of the GMCFP as c~ x ≤ a~, L i

R i

∑

~

∑1

(i ) ( i , j )∈FS( i )

ij

( i , j )∈A

xij −

ij

∑ g~

( j ,i )∈RS ( i )

ji

~ x ji = bi ,

∀i ∈ N ,

~ ~ lij ≤ 1( i , j ) xij ≤ u~ij ,

∀(i, j ) ∈ A,

that is,

∑ (c

ij

− ϑ , cij + ϑ , α ijL , α ijR ) xij ≤ (a − ϑ , a + ϑ , α L , α R ),

( i , j )∈A

∑ (1 − ϑ ,1 + ϑ , β

( i , j )∈FS( i )

L i

, β iR ) xij −

= (bi − ϑ , bi + ϑ , β iL , β iR ),

∑ (g

( j ,i )∈RS( i )

ji

− ϑ , g ji + ϑ , β iL , β iR ) x ji

∀i ∈ N ,

(lij − ϑ , lij + ϑ , λ , λ ) ≤ (1 − ϑ ,1 + ϑ , κ ijL , κ ijR ) xij L ij

R ij

≤ (uij − ϑ , uij + ϑ ,ν ijL ,ν ijR ),

∀(i, j ) ∈ A,

where ã = (a – ϑ, a + ϑ, αL, αR) is a trapezoidal fuzzy number representing a fuzzy aspiration level. We only need to analyze the constraints on the external flows, since all other restrictions remain the same as compared to the PMCFP. From the properties of the basic arithmetic operations on trapezoidal fuzzy numbers we find ~ ~ ∑ 1(i ) xij − ∑ g~ ji x ji − bi = ( i , j )∈FS( i )

(x

FS( i )

( j ,i )∈RS ( i )

(

)

− g RS( i ) , x RS( i ) − bi − ϑ x FS( i ) + x RS(i ) + 1 ,

(

)

x FS( i ) − g RS(i ) , x RS(i ) − bi + ϑ x FS( i ) + x RS(i ) + 1 ,

(

)

(

))

β iL x FS( i ) + β iR x RS(i ) + 1 , β iR x FS( i ) + β iL x RS(i ) + 1 , where gRS(i) = {gji}(j, i) ∈ RS(i), and 〈gRS(i), xRS(i)〉 = Σ(j, i) ∈ RS(i) gji xji.

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P. Majlender

Hence, we can formulate the degree of satisfaction of the constraint on the external flow at node i as ⎛

μ ( i ) (x) = Pos⎜⎜ ⎝

~

∑1

(i ) ( i , j )∈FS( i )

xij −

∑ g~

( j ,i )∈RS ( i )

ji

~⎞ ⎛ ~⎞ ~ x ji = bi ⎟⎟ = ⎜⎜ ∑ 1( i ) xij − ∑ g~ ji x ji − bi ⎟⎟(0) ( j ,i )∈RS ( i ) ⎠ ⎝ (i , j )∈FS(i ) ⎠

(

)

⎧1 if x FS(i ) − g RS(i ) , x RS( i ) − bi ≤ ϑ x FS( i ) + x RS(i ) + 1 , ⎪ x FS( i ) − g RS(i ) , x RS(i ) − bi − ϑ x FS(i ) + x RS( i ) + 1 ⎪ ⎪1 − β iL x FS( i ) + β iR x RS( i ) + 1 ⎪ ⎪ ⎪ if 0 < x FS(i ) − g RS( i ) , x RS( i ) − bi − ϑ x FS( i ) + x RS( i ) + 1 ⎪ ⎪ < β iL x FS( i ) + β iR x RS( i ) + 1 , =⎨ x FS(i ) − g RS( i ) , x RS( i ) − bi + ϑ x FS(i ) + x RS( i ) + 1 ⎪ ⎪1 + β iR x FS( i ) + β iL x RS(i ) + 1 ⎪ ⎪ R L ⎪ if − β i x FS(i ) + β i x RS(i ) + 1 ⎪ < x FS( i ) − g RS( i ) , x RS(i ) − bi + ϑ x FS(i ) + x RS( i ) + 1 < 0, ⎪ ⎪⎩0 otherwise,

(

(

(

(

(

(

(

))

)

(

)

)

(

)

)

)

)

i ∈ N.

Remark 11. If g̃ji = 1̃(i) for all (j, i) ∈ RS(i), i ∈ N, then the fuzzy formulation of the GMCFP reduces to the fuzzy PMCFP. In the following we shall focus on a different area of supply chain management, and formulate the problem of bullwhip effect. As we will see, the theory of possibility distributions represents a powerful tool for dealing with the bullwhip effect.

7 The Bullwhip Effect in Supply Chain Management In this section we will briefly discuss one of the biggest difficulties in the smooth operation of supply chains: the bullwhip effect. In our framework we will consider a series of companies in a supply channel, where each company orders the products from its immediate supplier (see Fig.10.10). As has been researched and pointed out in the industry, the orders of retailers usually do not coincide with their actual sales. The bullwhip effect refers to the phenomenon, where (i) orders placed to the supplier tend to have larger variance than sales to the buyer (demand distortion), and (ii) this distortion propagates upstream on the supply chain in an amplified form (variance amplification) (Carlsson and Fullér, 2000). The bullwhip effect started to become the focus of systematic theoretical work not long before. The first articles to report on such research results in a more systematic manner were (Lee et al. 1997). Since then, these papers have become

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237

Fig. 10.10 Structure of a supply channel and its key mechanisms

fundamentals of the research on the bullwhip effect, and provided bases for several theoretical and practical decision support systems for supply chain management. The major cause behind the bullwhip effect seems to be the fact that the variance of the estimates or the forecasts concerning customer demand amplifies as the orders move up in the supply chain from the customer, through retailers and wholesalers, to the producer of the product (see Fig.10.10). A number of case studies as well as our consultation with industry show that the bullwhip effect generates several negative effects that are responsible for significant inefficiencies. They include: 1. Excessive capital expenditures for inventory throughout the supply chain, as all business units (producers, wholesalers, retailers, distributors, and logistics operators) need to insure themselves against the variations. 2. Poor customer service, as some part of the supply chain can run out of products due to the variations in the supply and insufficient means for coping with those variations. 3. Lost revenues due to shortages that have been caused by the variations. As a consequence, (since revenues and thus profits are lower) the productivity of invested capital in operations becomes substandard. 4. Decision-makers can easily misinterpret the market situation, when they react to the fluctuations in demand by making investment decisions for changing inventory capacities to meet peak demands. These decisions might be based on misguided information, as peak demands can usually be eliminated by reorganizing the supply chain. 5. Demand variations imply variations in the logistic chain, which can cause serious fluctuations in the planned use of transportation capacity. This will lead to sub-optimal transportation schemes that will inevitably increase transportation costs. 6. Demand fluctuations caused by the bullwhip effect are often responsible for missed production schedules that could otherwise be avoided. In these cases only the inefficiencies in the supply chain get visibly widespread without significant change in the real demand. There are some studies, which have identified four key reasons for the formation of the bullwhip effect (see Lee et al. 1997, for an in-depth analysis). They include (i) the updating of demand forecasts, (ii) order batching, (iii) price fluctuations, and (iv) rationing and shortage gaming. Empirical evidence suggests that the

238

P. Majlender

updating of demand forecasts is one of the major sources of the bullwhip effect. In the following we will discuss some characteristic details of each of these four causes (Carlsson and Fullér 2000): • The updating of demand forecasts: Usually, all units in the supply chain build their estimates and forecasts on the historical demand patterns of their immediate customers. In this way, only the retailers build on the actual demand patterns of the end customers of the supply chain; the other business units adjust to fluctuations in the ordering policies of those following them in the supply chain. This feature will imply a particular effect: if all units in a supply chain react to fluctuations with smoothing techniques (by introducing safety stocks for instance), the fluctuations will amplify up through the supply chain, and increase the overall bullwhip effect. • Order batching: There are two different forms of order batching that appear in industry: (i) periodic ordering and (ii) push ordering. Periodic ordering takes place when we build batches of individual orders to save costs, since processing orders frequently can be (and usually is) expensive. However, it is clear that employing periodic ordering we essentially destroy customer demand patterns (unless our orders indeed follow customer demand in each period). Notice that the reasons behind periodic ordering are rational; however, when carried out in practice, it will inevitably amplify variability and contribute to the bullwhip effect. On the other hand, push ordering is a different industrial phenomenon: it happens when producers, aiming at fulfilling their quarterly or annual plans, convince their customers (with price discounts and other benefits) to order higher amounts of their products at the end of the quarter or year. However, as can easily be comprehended, this behavior leads up to the amplification of the variance of customer orders between the end and beginning of quarters and years, and adds significant power to the evolvement of the bullwhip effect. • Price fluctuations: The producers initiate and control the price fluctuations. By shifting the prices the producers try to attract end customers to buy their products in larger quantities. The behavior of customers is rational and thus predictable: to make optimal use of their resources (money) they will use opportunities when unit prices of products decrease. The implication of this behavior is that buying patterns will no longer reflect consumption patterns: customers buy in quantities that do not reflect their current needs. In particular, this will contribute to the amplification of the bullwhip effect. • Rationing and shortage gaming: This phenomenon is a strategy of customers towards their suppliers in the case demand exceeds supply. If a rational customer once faced a situation where its orders met rationed deliveries by its supplier (due to shortages it had), the customer will exaggerate its real demand when it starts to worry about supply not covering demand. Due to this type of strategic manipulation, the bullwhip effect can draw supply chain management out of control. In addition, if customers are allowed to cancel orders from suppliers (they find that their real demand is satisfied), the rationing and shortage gaming can be the biggest hurdle in making the overall supply chain cost

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effective (or even manageable). The gaming leaves little or no information on real demand, and will destroy the demand patterns of customers. Following the elaboration of (Carlsson and Fullér 2000), three possible approaches seem to generate solutions to the problem of counteracting the bullwhip effect. They are the following: 1. Find some means (tools as well as policies) to share information about the real demand of end customers of the supply chain. This issue essentially involves finding and deploying some information technology (IT) system that is appropriate for all business units in the distribution channel. 2. Coordinate business units of the supply chain by using techniques for channel alignment. Harmonizing pricing, transportation planning, and inventory management among units of the supply chain can create significant value-added to the overall distribution system. Note, however, that some (or even all) of these activities might be made illegal by anti-trust legislations in certain environments. This issue can be addressed by initiating (and later maintaining) negotiations about possible coordination between business units. In practice, we usually start aligning one particular activity (transportation planning or inventory management), and later on we gradually widen our cooperation to explore possible interactions with other elements as well. 3. Improve operational efficiency by reducing (inventory as well as transportation) costs and shortening lead times. Following this approach we need to find operational inefficiencies in certain strategic business units (SBUs) that play crucial roles in the supply chain, and develop methods to reduce their costs and shorten their lead times. Furthermore, we also need to explore the possibilities that these solutions can be generalized to other business units in the supply chain. One of the most challenging tasks in eliminating the bullwhip effect can be to address as many issues of the three approaches above as possible. In practice, we aim to combine their elements to form a strategy program for both our business and (as a result of cooperation) the entire supply chain, which combines (and determines) actions from each actor for the added benefits of all participants in the supply chain. In the following we shall take a closer look at one of the first mathematical models that was designed to work with the bullwhip effect. Building the model we will also obtain a standard mathematical explanation for the development of the bullwhip effect and its possible remedies. In later sections we will use this fundamental model to develop the basic version of our possibilistic decision support system for taming the bullwhip effect.

7.1 A Standard Model for the Bullwhip Effect Lee et al. (1997) focused their study on the demand information flow, and worked out a theoretical framework for studying the effects of systematic information

240

P. Majlender

distortion as information works its way through the supply chain. They simplify the context for their theoretical work by defining an idealized situation. They start with a multiple period inventory system operated under a periodic review policy. The following assumptions are included: 1. 2. 3. 4.

past demands are not used for forecasting; re-supply is infinite with a fixed lead time; there is no fixed order cost; and purchase cost of the product is stationary over time.

If the demand is stationary then the standard optimal result for this type of inventory system is to order up to some constant amount S. The optimal order quantity in each period is exactly the same as the demand of the previous period. This means that the orders and demands have the same variance, and therefore there is no bullwhip effect. This particular idealized situation is useful as a starting point, since it gives a good basis for elaborating on the consequences that distortion of information can imply in the supply chain. Since variability is the indicator of the bullwhip effect, we aim at measuring the notion of information distortion in terms of variance of demands we observe from our customer and variance of orders we place to our supplier. By relaxing one of the assumptions 1-4 at a time, it is possible to produce the bullwhip effect. In the following we shall present the methodology of demand signal processing, the driving engine behind the model for the bullwhip effect, as introduced in Lee et al. 1997.

7.2 Demand Signal Processing In this section let us focus on the retailer-wholesaler relationship in general. We note that this framework can also be applied to the wholesaler-distributor or the distributor-producer relationship in a straightforward manner. Now we consider a multiple period inventory model, where demand is non-stationary over time, and demand forecasts are updated from observed demand. Let us assume that in one period the retailer gets a demand that is much higher than usual. The retailer will interpret this phenomenon as a signal for higher demand in the future, so it gets the demand forecasts for future periods adjusted, and will place a larger order to the wholesaler. Since the demand is non-stationary, the optimal policy of ordering up to amount of S also gets non-stationary. A further consequence of this situation is that the variance of the orders starts growing, which will imply the development of the bullwhip effect. If the lead time between the point of order and the point of delivery is long, then uncertainty (about the delivery) increases, thus the retailer (being rational) will add a “safety margin” to S. This issue will further increase the variance and enhance the bullwhip effect. Following the framework of Lee et al. 1997 we simplify the context even further by focusing on a single-item, multiple period inventory system. By doing this we will be able to work out the exact details of the bullwhip model. The timing of the events is as follows:

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1. At the beginning of period t we make a decision to order a quantity zt. This particular point in time is called the “decision point” for period t. 2. Next, the goods we ordered ν periods before arrive. 3. Finally, we realize the real demand, and use the available inventory to meet the demand. We put excess demand on backlog. Let St denote the amount in stock and that on order (including those in transit) after decision about zt has been made for period t. Following Lee et al. we assume that the retailer faces serially correlated demands that follow the process Dt = d + ρDt −1 + ut , t ∈ N,

where Dt is the demand in period t, d is a fixed parameter that estimates the basic demand in the retailer’s business, ρ is a constant representing a correlation coefficient between demands in adjacent periods with –1 < ρ < 1, and ut denotes independent random variables that are normally distributed with mean zero and variance σ 2, i.e. ut ∼ N(0, σ 2) for t ∈ N, and Cov(ut, us) = 0 for t ≠ s. Here, we assume that σ 2 is significantly smaller than d, so the probability of a negative demand is negligible. The existence of parameter d for the basic demand, which is a constant in the process, is reasonable (though can be challenged): in most of the cases in the materials and chemicals markets (and especially in heavy industry) producers as well as wholesalers can expect to have some “granted demand”. However, it is generally agreed to be very difficult to estimate over time. In our case the introduction of d is merely technical, namely to avoid negative demand in the process. After formulating the cost minimization problem we can state the following theorem (Lee et al. 1997):

Theorem 7. In the environment we set up above, the following properties hold: 1. If 0 < ρ < 1 then the variance of retail orders are strictly larger than the variance of retail sales, i.e. Var( zt ) > Var( Dt −1 ), ∀t ∈ N.

2. If 0 < ρ < 1 then the larger the replenishment lead time ν is, the larger the variance of orders becomes, i.e. Var(zt) is strictly increasing in ν for t ∈ N. The following relationship plays a key role in computing the optimal amount of order, and it is used to prove Theorem 7: ν +1

z t*+1 = S t +1 − S t + Dt = ( Dt − Dt −1 )∑ ρ i + Dt = i =1

ρ (1 − ρ ν +1 ) ( Dt − Dt −1 ) + Dt , 1− ρ

where zt+1 denotes the optimal amount of order in time period t + 1 for t ∈ N. Hence, we obtain *

Var( zt*+1 ) = Var( Dt ) +

ρ (1 − ρ ν +1 ) 2(1 − ρ ν + 2 ) × > Var( Dt ), 1− ρ 1− ρ2

which proves both parts of our theorem.

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P. Majlender

In particular, if ν = 0, i.e. the goods ordered arrive immediately, then these relationships collapse into z t*+1 = ρ ( Dt − Dt −1 ) + Dt = (1 + ρ ) Dt − ρDt −1

and Var( zt*+1 ) = Var( Dt ) + 2 ρ ,

respectively. The optimal order quantity is an optimal ordering policy that sheds some light on the evolvement of the bullwhip effect. We can see that the bullwhip effect gets started by rational decision making; in fact, it is actually initiated by decision makers aiming to find the best (optimal) solutions for their companies. As a consequence, we cannot avoid the bullwhip effect by simply changing the ordering policy, since it is difficult to motivate educated decision makers to act in a counter-rational way. We need to analyze other means to motivate members of the supply chain to tame the bullwhip effect. Concerning practical real-world applications, our experience shows that producers (especially in the materials and chemicals markets) can easily counteract or significantly decrease the bullwhip effect by forming alliances with the retailers and/or the end-customers. For instance, it could be an important initial step in the coordination of the members of the supply chain to set up a distributed (networkbased) system with forecasting tools that would continuously update the market situation and the real demand estimates. Despite the fact that this solution appears to be obvious, producers usually do not take this approach seriously enough, and thus many supply chain systems do not involve alliances of this kind. The deployment of this type of intelligent system can also become more complicated due to anti-trust legislations in certain business areas and geographical regions. For example, due to this reason wholesalers might not be able to join the alliance and be part of the system. The theoretical results of demand signal processing can be summarized as follows: • The bullwhip effect is initiated and driven by forecasts and optimal ordering policy on the individual level, which usually changes frequently and abruptly. • The bullwhip effect increases in the supply chain when there is missing information about the real demand patterns. • The bullwhip effect is further amplified by long delivery times, uncertainties about the precision of the deliveries (with respect to both time and quantity), and random delivery disturbances. • Developing a system for sharing information among members of the supply chain can (and usually does) reduce the bullwhip effect.

7.3 Fuzzy Version of the Standard Bullwhip Model As the optimal ordering policy in the standard model is responsible for driving and amplifying the bullwhip effect, we follow the framework of Carlsson and Fullér

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(2000), and suggest a methodology for determining optimal ordering policy, where orders are imprecise. In our development the data can implicitly contain vague information about future (optimal) orders. As senior managers pointed out in our corporate studies, the consideration and theoretical involvement of imprecision in an intelligent decision support system for optimal ordering can create a significant value-added to the supply chain management of the company. Mathematically, we can include imprecision in a model by representing its notion by non-statistical uncertainty distributions. In our case this essentially means that orders can be intervals, and the actors in the supply chain are allowed to make their orders more precise as the time of delivery gets closer. In the following we shall develop the fundamentals of a possibility-based model for optimal ordering policy by replacing the crisp orders by fuzzy numbers. The driver behind the model for optimal order quantity is demand signal processing. Thus, let us develop the mathematical model for demand signal processing with possibility distributions. Let us consider the standard model for the bullwhip effect (Lee et al. 1997), and replace the probability distributions by possibility distributions. Then we have for the optimal ordering policy (Carlsson and Fullér 2000) ~ ~ ~ ρ (1 − ρ ν +1 ) ~ ~ ~ ~ z t*+1 = S t +1 − S t + Dt = ( Dt − Dt −1 ) + Dt , t ∈ N, 1− ρ

where D̃t and D̃t–1, the demands for periods t and t – 1, are defined by possibility distributions, i.e. fuzzy numbers. Therefore, the optimal amount of order z̃t+1* is computed as possibility distribution as well. Notice that in this way z̃t+1* implicitly involves the imprecision and vagueness of information about former demands. Using the definition of weighted possibilistic variance we can compute that 2

⎛ ρ (1 − ρ ν +1 ) ⎞ ~ ~ ~ ~ ⎟⎟ Var f ( Dt − Dt −1 ) + Var f ( Dt ) ≥ Var f ( Dt ) Var f ( ~ zt*+1 ) ≥ ⎜⎜ ⎝ 1− ρ ⎠

holds for any weighting function f. Thus, a simple adaption of the probabilistic model in possibilistic environment does not reduce the bullwhip effect. However, we can show that by including better and better estimates of future sales in period t with Dt̃ ∈ Y , we are actually able to reduce the variance of zt̃ +1* by replacing the basic rule for ordering with an adjusted rule

⎛ ρ (1 − ρ ν +1 ) ~ ~ ~ ~⎞ ~ ~ z t(+i1) = ~ zt*+1 ∩ Dt(+i1) = ⎜⎜ ( Dt − Dt −1 ) + Dt ⎟⎟ ∩ Dt(+i 1) , 1 − ρ ⎝ ⎠ where D̃t+1(i) stands for an estimation of the real demand in period t + 1, D̃t+1, such that the series {D̃t+1(i)}i = 1, 2, … represents a more and more accurate estimations of D̃t+1 for any t ∈ N. Representing the degree of accuracy of estimations by the Hausdorff metric H: Y × Y → R+, we can mathematically formulate this particular property as ~ ~ ~ ~ j ≤ i ⇒ H ( Dt +1 , Dt(+j1) ) ≥ H ( Dt +1 , Dt(+i1) ), where

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H ( A, B) = sup max{| a1 (γ ) − b1 (γ ) |, | a2 (γ ) − b2 (γ ) |} γ ∈[0 ,1]

for any A, B ∈ Y with [A]γ = [a1(γ), a2(γ)] and [B]γ = [b1(γ), b2(γ)], γ ∈ [0, 1]. Using the method of recurrent rule adjustments we described above, we are indeed able to decrease the bullwhip effect. From basic properties of the intersection operator on fuzzy sets we get

⎛ ρ (1 − ρ ν +1 ) ~ ~ ~⎞ ~ ~ z t(+i1) = ⎜⎜ ( Dt − Dt −1 ) + Dt ⎟⎟ ∩ Dt(+i 1) ⊂ ~ zt*+1 , ⎝ 1− ρ ⎠ which, according to Theorem 6, implies Var f ( ~ zt(+i1) ) ≤ Var f ( ~ zt*+1 ).

That is, the variance of the optimal order z̃t+1(i) as computed by the adjusted rule is getting smaller and smaller, as the demand estimate D̃t+1(i) is getting sharper and sharper. In the case ν = 0 we have ~ ~ ~ ~ z t(+i1) = ((1 + ρ ) Dt − ρDt −1 ) ∩ Dt(+i 1) , which implies ~ ~ ~ Var f ( ~ zt(+i1) ) ≤ min{Var f ((1 + ρ ) Dt − ρDt −1 ), Var f ( Dt(+i 1) )}

for any weighting function f and t ∈ N. Especially, in the limit case ρ → 0 we get ~ ~ lim Var f (~zt(+i1) ) ≤ min{Var f ( Dt ), Var f ( Dt(+i 1) )}, ρ →0

which implies that the bullwhip effect is eliminated.

Remark 12. The appropriate construction of the series of possibility distributions {Dt̃ +1(i)}i = 1, 2, … is crucial. There is no general rule how we can derive “good” estimates of future demands and sales. However, we managed to identify some principles in setting up systems of future sales forecasts based on specific industrial environments and market situations. Once identified, the knowledge of creating distributions {D̃t+1(i)}i = 1, 2, … for all t ∈ N can become a best practice solution for a company, which should only be shared with business partners through an advanced information system (that applies an elaborate policy for information sharing in the supply chain). For a good starting point in implementing an effective possibility-based method for future demand estimation we can refer to the areas of fuzzy logic controllers (see (Carlsson and Fullér 2000) and group decision support systems (see Fedrizzi et al. 1999). Notice that both Varf(z̃t+1(i)) and Varf(z̃t+1*) are increasing functions of the replenishment lead time ν for any weighting function f. Moreover, if f > 0 is strictly

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Fig. 10.11 Simulation model for demand signal processing based on trapezoidal fuzzy numbers (screenshot)

Fig. 10.12 Possibilistic simulation for demand signal processing (screenshot)

positive on the unit interval [0, 1] then Varf(z̃t+1*) is strictly increasing in ν. This particular property is preserved from the probabilistic model, and shows the similarity between the stochastic and possibilistic approaches to the methodology of optimal order quantity.

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We have implemented this particular model, and successfully applied in industrial environments (see Fig. 10.11 and 10.12). Consulting with corporate partners we have also developed several enhanced versions of this model, which we tailored for specific industrial environments and market situations. The areas of applications were mainly in the materials and chemicals markets, and embraced industries like metals, and paper and forest products.

8 Conclusions In this chapter, we formulated some theoretic and practical aspects of employing fuzzy logic and possibility theory in the area of Supply Chain Management (SCM). We generally presented the topic of supply chain management along with some of its classical methods that we could use to work with problems in that area. We introduced fuzzy logic to supply chain management by incorporating possibility distributions in the mathematical models. Doing the mathematical formulations we also discussed methods for dealing with the bullwhip effect using possibility distributions.

References Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Management Sciences 17, 141–164 (1970) Carlsson, C., Fullér, R.: Soft computing and the bullwhip effect. Economics & Complexity 2, 1–26 (2000) Carlsson, C., Fullér, R.: On possibilistic mean value and variance of fuzzy numbers. Fuzzy Sets and Systems 122, 315–326 (2001) Carlsson, C., Fullér, R.: Fuzzy reasoning in decision making and optimization. Studies in Fuzziness and Soft Computing Series, vol. 82. Springer, Heidelberg (2002) Chanas, S., Delgado, M., Verdegay, J.L., Vila, M.A.: Fuzzy optimal flow on imprecise structures. European Journal of Operational Research 83, 568–580 (1995) Chanas, S., Kolodziejczyk, W., Machaj, A.: A fuzzy approach to the transportation problem. Fuzzy Sets and Systems 13, 211–221 (1984) Chanas, S., Kuchta, D.: Fuzzy integer transportation problem. Fuzzy Sets and Systems 98, 291–298 (1998) Dubois, D., Prade, H.: Possibility theory. Plenum Press, New York (1988) Fedrizzi, M., Fedrizzi, M., Marques Pereira, R.A.: Soft consensus dynamics in group decision making. In: Mayor, G., Suñer, J. (eds.) Proceedings of the EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain, pp. 17–20 (1999) Fullér, R.: On T-sum of fuzzy numbers. BUSEFAL 39, 24–29 (1989) Fullér, R., Majlender, P.: On weighted possibilistic mean and variance of fuzzy numbers. Fuzzy Sets and Systems 136, 363–374 (2003) Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets and Systems 18, 31–43 (1986) Hamacher, H., Leberling, H., Zimmermann, H.J.: Sensitivity analysis in fuzzy linear programming. Fuzzy Sets and Systems 1, 269–281 (1978)

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Jensen, P.A., Bard, J.F.: Operations research models and methods. John Wiley and Sons, Chichester (2003) Lee, H.L., Padmanabhan, V., Whang, S.: Information distortion in a supply chain: The bullwhip effect. Management Science 43, 546–558 (1997) Lee, H.L., Padmanabhan, V., Whang, S.: The bullwhip effect in supply chains. Sloan Management Review Spring, 93–102 (1997) Majlender, P.: A normative approach to possibility theory and soft decision support. University Doctorate Dissertation TUCS Dissertations No. 54, Turku Centre for Computer Science. Åbo Akademi University, Åbo (2004) Negoita, C.V.: Fuzzy systems. Abacus Press (1981) Nguyen, H.T.: A note on the extension principle for fuzzy sets. Journal of Mathematical Analysis and Applications 64, 369–380 (1978) Shih, H.S., Lee, E.S.: Fuzzy multi-level minimum cost flow problems. Fuzzy Sets and Systems 107, 159–176 (1999) Tanaka, H., Ichihashi, H., Asai, K.: A value of information in FLP problems via sensitivity analysis. Fuzzy Sets and Systems 18, 119–129 (1986) Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1, 45–55 (1978)

Chapter 11

Fuzzy Simulation and Optimization of Production and Logistic Systems L. Dymowa and P. Sevastjanov1

Abstract. The basic paradigm for simulation of production and logistic systems is the probabilistic approach to describing the real world uncertainty. However, in many cases we do not have the information that would be precise enough to build corresponding probabilistic models or there are some human factors preventing doing so. In such situations, the mathematical tools of fuzzy sets theory may be successfully used. It seems that a simple and natural way to do this is the replacement of probability densities with appropriate fuzzy intervals and the use of fuzzy arithmetic to build adequate fuzzy models. Since the models of production and logistic systems are usually used for the optimization of simulated processes, the problem of fuzzy optimization arises. The problems of simulation and optimization in the fuzzy setting can be solved with use of fuzzy and interval arithmetic, but there are some inherent problems in formulation of basic mathematical operations on fuzzy and interval objects. The more important of them (especially for the fuzzy optimization) is the interval and fuzzy objects comparison. The paper presents a new method for crisp and fuzzy interval comparison based on the probabilistic approach. The use of this method in the synthesis with α -cut representation of fuzzy values and usuall interval arithmetic rules, makes it possible to develope an affective approach to fuzzy simulation and optimization. This approach is illustrated by the examples of fuzzy simulation of linear production line and logistic system and by the example of fuzzy solution of optimal goods distribution problem. The results obtained with use of the proposed approach are compared with those obtained using Monte-Carlo method.

1 Introduction In traditional modeling and optimization of economic systems, all natural uncertainties are usually interpreted in probabilistic sense. However, in practice this does not always correspond to the nature of uncertainties that often appear as the effects of subjective estimations. In many cases it is impossible or unnecessary to describe the behavior of modeled system with precision that may be obtained by means of the exact probability densities. Indeed, often there is no money or time to L. Dymowa and P. Sevastjanov Institute of Comp. & Information Sci., Czestochowa University of Technology, Dabrowskiego 73, 42-200 Czestochowa, Poland C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 249–277. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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provide the detailed statistical analysis of the modeled system. Sometimes we deal with the system that has not been fully developed and completed. The list of practical situations, when it seems better to avoid the traditional probability interpretation of uncertainty is very long. Mainly because of this, during two last decades the growing interest of researchers in the alternative uncertainty representations has been observing. In the fields of manufacturing and economic systems modeling, the tools of fuzzy sets theory were mainly used as an alternative or complementation to the traditional probability approach. In this Chapter, we present an approach based directly on using fuzzy intervals instead of probability distributions in the numerical modeling and optimization procedures. Fuzzy arithmetic has been used as the main mathematical tool for building fuzzy models in the framework of approach presented in this Chapter. Although fuzzy arithmetic seems to be a well-developed and well-formalized branch of fuzzy sets theory, there are some problems of its practical implementation, e.g., the problem of fuzzy values comparison. Besides, some problems of methodological nature should be discussed before describing the fuzzy modeling and optimization approaches. Therefore, the rest of Chapter is organized as follows. In Section 2, the basic mathematical tools which we have used for interval and fuzzy modeling and optimization including a new method for interval and fuzzy values comparison are presented. Section 3 presents the two-criteria fuzzy optimization method based on probabilistic approach to interval and fuzzy values comparison. In Section 4, the developed methods are used for the fuzzy simulation of linear production line and logistic system and for the optimization of distributor’s decisions in the fuzzy setting. Section 5 concludes with some remarks.

2 Mathematical Tools In this Section, the basic concept of an approach to incorporate an uncertainty into simulation and optimization models and the methods for operation on interval and fuzzy objects are presented.

2.1 The Basic Concept In practice, the following typical situation can be seen: the probability distributions of some well studied parameters are known and there are all the necessary prerequisites for considering them in future as unchangeable; the values of other parameters can be presented only in the form of fuzzy value (Zadeh 1965) or crisp intervals (Moore 1966, Jaulin et al. 2001) on the base of objective information and subjective expert estimations available. It is well known that fuzzy and crisp intervals possess smaller accuracy of description than probability distribution.

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In such a situation, we are faced with the problem of using jointly the specific types of uncertainty which require principally different mathematical methods to operate. It is clear that in practice the accuracy of modeling results will be predetermined by the accuracy of the most vague parameters of the model. Hence, artificial transformation of fuzzy or crisp intervals into probabilities distribution will only provide the illusion of accuracy and is not a good practice for methodological reasons, since it does not allow us to obtain the true presentation of resulting uncertainties. Obviously, in such a situation, the transformation of probability distributions into fuzzy intervals seems more methodologically justified. As it is shown by Yager (1980), the fundamental difference between the probability distribution and the membership function of fuzzy interval is of informational nature. So, to define a probability distribution f(x), it is necessary to know the value of relation f(a)/f(b) for each two a and b from the support set. In contrast, for mathematical formalization of the membership function µ(x), the only information needed is the qualitative information. It is enough to know that a is more possible (realizable, acceptable, probable) than b. It is worth noting that the reduction of requirements to the available information makes it possible to increase greatly the constructive capabilities of fuzzy set theory, particularly to build relatively simple and practically useful arithmetic for handling fuzzy intervals. In many practical situations, it is reasonable even to sacrifice the accuracy of the probability distributions available, transforming them into fuzzy intervals so that an effective and constructive model can be built. There are many different methods for building the membership functions on the base of the probability distributions proposed in the literature. We have no aim to review them here, but it is worthy to emphasize that in any case, the good transformation should save the most information contained in the initial probability distribution. For this purpose, we propose a simple numerical transformation procedure. The method is based on converting the confidence intervals of probability distribution into the so-called α -cuts of fuzzy set (Kaufmann and Gupta 1985). At first, we choose the set of probabilities P1, . . . , Pn and corresponding confidence intervals [x1, x2]Pi ,i = 1, . . . , n, using the considered probability distribution f(x). Of course, the number n reflects good our demands for accuracy of the representation of f(x) by finite sets {Pi}, {[x1, x2]Pi} in the considered problem. Further, we get a set of α -cuts using simple normalization procedure α i = Pi/Pn, i = 1 to n. Finally, we transform the confidence intervals to the corresponding set of crisp α -cut intervals by mapping illustrated in Fig. 11.1. It is easy to see that the transformation saves all the quantitative information about locations and widths of confidence intervals as well as qualitative information about initial probabilities. It is clear that the greater the number of α -cuts, the more exact result of transformation may be obtained. Some researchers, e.g., Dubois and Prade, make a distinction between fuzzy intervals and fuzzy numbers depending on the multiplicity or uniqueness of modal values. Here we shall use the term "fuzzy number" in its most general sense. A fuzzy number may be viewed as an elastic constraint acting on a certain variable which is only known to lie "around" a certain value. It generalizes both concepts of real number and closed interval (Deneux 2000).

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x μ (x ) 1

α - c u ts

0 .6 0 .3 0

x

Fig. 11.1 The transformation of the probability distribution f(x) into membership function µ(x).

So in all cases when we write “fuzzy interval” this may be treated as fuzzy value as well.

2.2 Applied Interval Analysis as the Basis of Fuzzy Arithmetic The next problem is the practical realization of fuzzy arithmetic rules. They are based on the extension principle introduced by Zadeh (1965). The general formulation of this principle uses an arbitrary t-norm. Let A, B, Z be fuzzy numbers (intervals) and @ ∈ {+, -, *, /} be an arithmetical operation. Then Z = A@B = {z = x@y, µ(z) =

max t ( μ ( x), μ ( y )), x ∈ A, y ∈ B }

(11.1)

z

As emphasized by Zimmermann and Zysno (1980), the choosing of concrete realization of t-norm is rather an application dependent problem, but the three main t-norms are usually used in practice: t(µ(x),µ(y)) = min(µ(x),µ(y))

(11.2)

1 (µ(x) + µ(y)) 2

(11.3)

t(µ(x),µ(y)) =

t(µ(x),µ(y)) = µ(x)µ(y)

(11.4)

An alternative approach to implementation of fuzzy arithmetic is based on the -cuts presentation of fuzzy numbers (Kaufmann and Gupta 1985). So, if A is a fuzzy number, then

α

A =

U α Aα α

,

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where

αAα

fuzzy subset

is the fuzzy subset x ∈ U, µ A(x)

253

≥ α and Aα is the support set of

αAα , U is the universe of discourse. It was proved that if A and B

are fuzzy numbers (intervals), then all the operations on them may be presented as operations on the set of crisp intervals corresponding to their α -cuts:

( A @ B)α = Aα @ Bα

(11.5)

Of course, the direct α -cut representation of fuzzy number seems as a rough one in comparison with the generalized expression (11.1). But for the practical numerical realization of representation (11.1) we have to use the discretization of the supports of considered fuzzy numbers A, B and Z if we deal with non-trivial forms of µ(x), µ(y). It is well known for the practicians that any direct discretization of (11.1) leads to the wrong results (Piegat 1999, Sevastjanov and Rog 2003). To illustrate, let us consider the example adopted from (Piegat 1999 ). Let A = “near 5” and B = “near 7” be fuzzy numbers represented by the corresponding finite fuzzy sets (see Table 11.1). Table 11.1 Discrete representation of the test fuzzy numbers A and B µA(xi)

0

0.33

0.66

1.0

0.5

0

xi

2

3

4

5

6

7

µB(yi)

0

0.5

1.0

0.66

0.33

0

yi

5

6

7

8

9

10

Then for multiplication result shown in Fig. 11.2.

A ⋅ B in the case of t(µ(x),µ(y)) = µ(x)µ(y) we get the

Fig. 11.2 Multiplication of two fuzzy numbers (see Table 11.1): I—using the definitions (11.1) and (11.4), II—using the α -cuts (11.5).

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We can see that the use of general definition (11.1) does not provide the convex resulting fuzzy number in the case of multiplying fuzzy numbers, whereas we have no problems when using an approach based on α -cuts representation of fuzzy numbers. It seems natural that the results of use of general definition (11.1) may be improved using the more detailed discretization, but regrettable, it is hard to do this. Consider the case of addition of fuzzy numbers. The results of adding two trapezoidal fuzzy numbers A and B, presented by quadruples A = {1, 3, 4, 5}, B = {7, 8, 8.5, 9} are shown in Fig. 11.3 for the two approaches examined. For the first approach to the formulation of fuzzy arithmetic rules based on the expression (11.1), the definition (11.2) was used. It is clear that in the case of addition, the combination of the basic definition (11.1) and definition (11.2) provides the result which may be qualified as rather senseless. It is worthy to emphasize here that such unacceptable results were obtained using all the definitions (11.2)–(11.4) for all arithmetical operations, but there is no problem if we use the α-cut presentation for operations on fuzzy intervals. More details, examples and analysis may be found in (Piegat 1999, Sevastjanov and Rog 2003). However, for our purposes it is quite enough to treat the result presented above as only the empirical fact, which wishes that there are some difficult problems when using the discretization of general expression (11.1) and that such problem can be easily eliminated by use of α-cut representation for fuzzy arithmetic rules. Hence, α -cut representation of fuzzy numbers (intervals) and operations on them can be accepted as the basic concept for fuzzy modeling of the real-world processes. Since in the case of α -cut presentation, fuzzy arithmetic is based on crisp interval arithmetic rules, the basic definitions of applied interval analysis should be

Fig. 11.3 Addition of two fuzzy numbers A = {1, 3, 4, 5}, B = {7, 8, 8.5, 9}: I—using the definitions (11.1) and (11.2), II—using the α -cuts (11.5).

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presented too. There are some definitions of interval arithmetic (see Moore 1966, Jaulin et al. 2001), but in practical applications the so-called “naive” form proved to be the best. According to it, if A = [a1, a2] and B = [b1, b2] are crisp intervals, then Z = A@B = {z = x@y, x ∈ A, y ∈ B}

(11.6)

As the direct consequence of the basic definition (11.6) the following expressions were obtained: A + B = [a1 + b1, a2 + b2], A- B = [a1 - b2, a2 - b1], A ⋅ B = [min(a1b1, a2b2, a1b2, a2b1), max(a1b1, a2b2, a1b2, a2b1)], A/B= [a1, a2]

⎡1 1⎤ ⎢ , ⎥ , 0∉ B . ⎣ b2 b1 ⎦

Of course, there are many internal problems within applied interval analysis, e.g., the division by zero-containing interval, but in general, it can be considered as the good mathematical tool for modeling under the conditions of uncertainty. As the natural consequence of the assumed basic concept, the method for fuzzy values comparison should be developed on the basis of crisp interval comparison.

2.3 Interval and Fuzzy Values Comparison Theoretically, intervals can only be partially ordered and hence cannot be compared in ordinary sense. However, if intervals are used in applications, the comparison of them becomes necessary. There are different approaches to interval (and fuzzy numbers) comparison proposed in the literature. To compare intervals, usually the quantitative indices are used (see reviews by Wang and Kerre 2001 and Sevastjanov 2007) . Wang et al. (2005) noted that most of the proposed interval comparison methods are “totally based on the midpoints of interval numbers”. Therefore, the authors developed a simple heuristic method which (at least explicitly) makes no use of the midpoints. For intervals B = [b1 , b2 ] ,

A = [a1 , a2 ] , the degree of possibility of B ≥ A is defined by Wang et al. max{0, b2 − a1}− max{0, b1 − a2 } (2005) as follows: P ( B ≥ A) = . The a2 − a1 + b2 − b1 similar expressions were proposed earlier by Facchinetti et al. (1998) and by Xu and Da (2002). Xu and Chen (2008) showed that the expressions proposed by Facchinetti et al. (1998), by Wang et al. (2005) and by Xu and Da (2002) are equivalent. The main limitation of these approaches is the lack of the separate interval equality relation since for A = B , i.e., a1 = b1 , a2 = b2 they provide

P ( B ≥ A) = 0.5, P ( B ≤ A) = 0.5 . It is important to note that interval equality is usually considered as an impossible relation (Wadman et al. 1994), identity (Moore 1966) or only in conjunction with interval inequality (Yager and Detyniecki 2000). Nevertheless, according to the classical interpretation (Moore 1966), any

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A is completely defined by its bounds ( A = [a1 , a2 ] ). In other words, we can treat an interval A as a mathematical object defined by the pair a1 , a2 . Therefore, if we meet two such objects A and B with equal bounds ( a1 = b1 , a2 = b2 ) we can say they are equal objects. Let us introduce a measure (possibility or probability) m( A@B ) ∈ [0.1] ( @ ∈ {>, =, B ) = 0 , m( A < B ) = 0 for equal A and B and m( A = B ) = 0 for the completely different A and B when their intersection is

interval

empty. This reasoning coincides with the definitions of equality index introduced by Dubois et al. (2000) and Bustince et al. (2007). The equality index is interpreted as a reasonable measure of the degree of equality of two sets. Let A and B be some sets, e.g., intervals or fuzzy sets. Then the equality index is equal to 1 if and only if A = B and is equal to 0 if A and B have disjoint supports. A separate equality relation is considered as an integral part of a full set of interval valued fuzzy relations and the detailed mathematical analysis of these relations is presented by Bustince et al. (2000). Nevertheless, the discussed result ( P ( B ≥ A) = 0.5, P ( B ≤ A) = 0.5 when

a1 = b1 , a2 = b2 ) obtained by Facchinetti et al. (1998), Wang et al. (2005) , Xu and Da (2002) and some other authors make sense if we look at the problem from another point of view. Nguyen et al. (2004) proposed the interpretation of this result observing (see also Sevastjanov 2007) that in practice, after measurements of some parameters a and b , we only have intervals A and B containing the possible real values of a and b . If the corresponding two intervals intersect, then none of the parameters is guaranteed to be higher than another. A natural idea is therefore to choose an interval for which the probability that a ≤ b is greater than the probability that a ≥ b . As it is shown by Nguyen et al. (2004) , such a reasoning leads to the expression equivalent to that proposed by Xu and Da (2002):

P ( B ≥ A) =

b2 − a1 if b2 ≥ a1 and 0 else. a2 − a1 + b2 − b1

Although this expression leads to

P ( B ≥ A) → 0.5, P ( B ≥ A) → 0.5 when

a1 → b1 , a2 → b2 , the result is completely justified in context of the above reasoning. It is easy to see that only possible real valued parameters ranging in corresponding intervals are compared, not intervals. It is shown by Nguyen et al. (2004) that there are many real-life situations when such reasoning is valid. Nevertheless, in practice, there may be situations when intervals, e.g., such as A = [0,1000.1] and B = [0,1000.2] , from common sense, should be considered rather as equal ones since the small difference may be coursed by inaccuracy of measurements. Obviously, in such situations even intuitively, we

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feel that P ( A = B ) > P ( A < B ) and the relation P ( A ≤ B ) seems to be not satisfactory. Therefore, to avoid possible misunderstanding, we emphasize here that we prefer to use the methods for comparison of intervals treated as mathematical objects defined completely by their bounds. Such approach to interval comparison is not a purely mathematical concept, it originates from realworld problems of simulation and optimization in the interval and fuzzy setting (Sevastjanov and Rog 2003, 2006). This concept is natural in the framework of the interval analysis and leads to the reasoning presented by Sevastjanov (2007) as follows: “An important point is that in the framework of the conventional interval analysis, the relation A > B for the overlapping intervals shown in Fig.11.4 is senseless. Moore (1966) first postulated that if a1 < b1 and a2 < b2 then B > A

A > B is impossible since there are no real values a ∈ A such that a > b2 . In other words, there are no arguments in favor of A > B in this case. Indeed, if a2 = b2 then an interval A cannot be greater than B until a1 < b1 . It is clear that any other assumption should be in contradiction with and the relation

common sense. On the other hand, as we deal with overlapping intervals, there is a common area, where events a > b ( a ∈ A , b ∈ B ) take place. Of course, if these events can be considered as arguments in favor of A > B , we are in a conflict with the basics of interval analysis and common sense. The source of this contradiction is the assumption (not obvious) that relations between the particular a ∈ [b1 , a2 ] and b ∈ [b1 , a2 ] may be used to compare intervals as a whole. Since this assumption leads to the incorrect conclusion (we can not ignore common sense), it is wrong in context of interval analysis. Thus, the relation A > B for overlapping intervals shown in Fig. 11.4 is senseless, but there are no such strong reasons to exclude the possibility (in some extent) of events A = B and A < B ”. In a similar way, it was shown by Sevastjanov (2007) that in the inclusion case (see Fig.11.4) there are arguments in favor of B > A , A > B and B = A. Using the above reasoning and the so called probabilistic approach to the interval comparison (see review of the methods based on this approach in (Sevastjanov and Rog 2006), two sets of expressions containing the separate interval equality relation were introduced by Sevastjanov (2007). There are two different Overlapping case

Inclusion case

B A b1

a1

B A

a2 b2

x

Fig. 11.4 The examples of interval relations.

a1

b1

a2

b2

x

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possible assumptions concerned with conditional probabilities in the framework of this method which provide two sets of interval relations referred to as ``weak'' and ``strong'' relations. We present here only the ``strong'' relations (the probabilities of B > A , B < A and B = A ) as they are a particular case of the more general approach based on DST (Sevastjanov, 2007) . For the overlapping intervals (Fig. 11.4):

P ( B < A) = 0, P( B = A) =

(a2 − b1 )2 , (a2 − a1 )(b2 − b1 )

P( B > A) = 1 − P( B = A). (11.7)

In the inclusion case (Fig. 11.4):

P ( B < A) =

a1 − b1 a −a b −a , P ( B = A) = 2 1 , P ( B > A) = 2 1 . b2 − b1 b2 − b1 b2 − b1

(11.8)

Observe that in all cases P ( A < B ) + P ( A = B ) + P ( A > B ) = 1 . Nevertheless, the existence of two different results gained from mutually exclusive assumptions (“weak” and “strong”) may be considered as a drawback of the proposed approach. This ambiguity is treated by Sevastjanov (2007) as a consequence of the limited ability of a purely probabilistic approach to deal with such objects as intervals and fuzzy numbers. The problem is that the probability theory allows us to represent only uncertainty, whereas interval and fuzzy objects are inherently characterized by imprecision and ambiguity. To solve this problem, the use of DST is proposed (Sevastjanov,2007) . The set of fuzzy inteval relations can be obtained using the above crisp interval relations and α -cut representation of compered fuzzy values.

~ ~ A and B be fuzzy intervals (numbers) on X with corresponding membership ~ ~ functions μA(x), μB(x):X→[0,1]. We can represent A and B by the sets of α-cuts ~ ~ A= U Aα , B = U Bα , where Aα ={x ∈X:μA(x)≥α}, Bα ={x ∈X:μB(x)≥α} Let

α

α

~

~

are the crisp intervals. Then all fuzzy interval relations A rel B , rel ∈ {} may be presented by the set of α-level relations

~ ~ A rel B = UAα rel Bα . α

Since Aα and Bα are crisp intervals, the probability

Pα ( Bα > Aα ) for each pair

Aα and Bα can be calculated in the way described above. The set of the probabilities Pα (α∈(0,1]) may be treated as the support of fuzzy subset ⎫ α ~ ~ ⎧ P( B > A) = ⎨ ⎬, ⎩ Pα ( Bα > Aα ) ⎭

Fuzzy Simulation and Optimization of Production and Logistic Systems

where the values of

~ ~ P( B > A).

α

259

denotes the grade of membership to the fuzzy interval

~ ~

In this way, the fuzzy subset P( B = A) may also be easily obtained. The resulting “fuzzy probabilities” can be used directly. For instance, let

~ A,

~ ~ ~ ~ ~ ~ B , C be fuzzy intervals and P( A> B ),P( A > C ) be fuzzy intervals expressing ~ ~ ~ ~ the probabilities A > B and A > C , respectively. Hence the probability ~ ~ ~ ~ P(P( A> B )>P( A > C )) has a sense of probability's comparison and is expressed

in the form of fuzzy interval as well. Such fuzzy calculations may be useful at the intermediate stages of analysis, since they preserve the fuzzy information avail-

~ ~

~ ~

able. It can be shown that in any case P( A> B )+P( A= B ) =”near 1" (overlapping

~ ~

~ ~

~ ~

case), and P( A> B )+P( A= B )+P( A< B ) =”near 1" (inclusion case), where “near 1" is a fuzzy number symmetrical in relation to 1. It is worth noting here that the main properties of probability are remained in the introduced operations, but in the fuzzy sense. However, a detailed discussion of these questions is out of the scope of this paper. In practice, the real value indices are sometimes needed for fuzzy interval ordering. For this purpose, some characteristic numbers of fuzzy set could be used. But it seems more natural to use the defuzzification, which for a discrete set of α cuts can be presented as follows:

~ ~ P ( B > A) =

α ⋅ Pα ( Bα ∑ α α ∑ α

> Aα ) .

(11.9)

The last expression indicates that the contribution of α -cut to the overall probability estimation is rising along with the rise in its number.

3 Optimization with Interval and Fuzzy Cost Function In practice, there are many optimization problems formulated using imprecise parameters. Frequently, such parameters may be considered as intervals or fuzzy numbers. As a consequence, the optimization tasks with interval or fuzzy interval cost function of real arguments are usually obtained. It is worthy to note that above mentioned problems differ essentially from the so-called global interval optimization when real value objective functions are used (Hansen 1992, Jaulin et al. 2001). In addition, the cost function often can not be performed by a set of analytical expressions, e.g., when it is presented as the result of some numerical calculations with use of corresponding algorithm. Of course, in such cases, it is impossible to use gradient methods for the optimization, but different kinds of numerical direct search methods may be successively applied.

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It is important to note here that the nature of intervals and fuzzy value is that there is no universal method for their comparison: the choice of appropriate method is a context dependent problem. For example, let us compare two financial projects represented by their profits estimated as the intervals A and B (see Fig.11.5). Using the method described in previous Section, we conclude that project B seems to be the best one since B>A, but prudent investor may reject the project B from the consideration at all since it can result in negative profits, i.e. , in bankruptcy.

Fig. 11.5 Interval profits of investment.

Therefore, when dealing with the optimization problem formulated using imprecise parameters (intervals or fuzzy numbers) we need to compare the values of corresponding interval of fuzzy cost functions taking into account the specificity of analyzed problem. The procedure of direct numerical solution of optimization task can generally be presented as a sequence of searching steps. On each step we try to get a smaller/greater objective function’s value than on previous one. It is clear that in such cases, we are faced with a problem of consecutive interval or fuzzy interval function reduction. Unfortunately, even in the case of successful solution of the problem there are no any guarantees that the result will be obtained with minimal uncertainty. It is easy to see that in our case such uncertainty can be in a natural way presented by the width of interval or fuzzy interval representing the value of minimized/maximized objective function. In a nutshell, in the considered case we deal with two local criteria. In general, simultaneous satisfaction such criteria in a maximal degree is rather impossible. Thereby, the problem is to build a compromise criterion, which may be performed as an aggregation of local criteria. To solve this problem, at first we have to built the local criterion for quantitative assessment of the degree to which one interval - crisp or fuzzy - is greater than another one and the local criterion based on the widths of compared intervals or fuzzy intervals representing the degreee of uncertainty of such interval comparison.

3.1 Two-Objective Approach to Interval Comparison As noted earlier, at each step of direct numerical optimization there are at least two main local criteria which reflect our intention to minimize/maximize the objective function and simultaneously to minimize the uncertainty of the obtained result. Obviously, in our case, the criterion of interval objective function

Fuzzy Simulation and Optimization of Production and Logistic Systems

261

minimization/maximization may be presented using probabilistic approach described in previous Sections. On the other hand, local criterion of uncertainty minimization may be in a natural way presented by the relation of widths of compared intervals. Let's consider the local criteria of interval comparison that can be introduced as the mathematical formalization of the above inexact reasoning. Let A and B be compared crisp or fuzzy intervals. As the first criterion, it is possible to accept directly the probability that one of compared intervals is less (in the case of minimization) than another one:

μ pA = P( A < B), μ pB = P( B < A), μ pAB = P( A = B).

(11.10)

The method for calculation of such probabilities is described above (Sections 2). To define the second criterion, the relations of intervals widths are considered:

xwA =

wA wB , xwB = , max( wA , wB ) max( wA , wB )

(11.11)

wA , wB are the widths of intervals A and B , respectively. A B Parameters xw , xw may be used to introduce the criteria that explicitly reflect

where

our intention to decrease the uncertainty (the width of interval objective function) on the successive stages of numerical optimization procedure:

⎧1 − x wA ⎩ 0

μ wA = ⎨

⎧1 − xwB if xwA < 1, μ wB = ⎨ otherwise ⎩ 0

if x wB < 1, otherwise

(11.12)

Obviously, in such a case, for the general essesment of the degree to which

A < B it is necessary to use the pair of criteria μ PA , μ wA , otherwise for the essesment of the degree to which

B < A the local criteria μ PB , μ wB chould be

considered. It is easy to see, that there may be some situations when, e.g. , on a certain stage of optimization we get

μ PA > 0.5

and

μ wA = 0 .

In other words, in such

cases the width of the smaller (in the probabilistic sense) interval A is greater than the width of the interval B . It is clear that to continue the optimization process in such cases, we are compelled to recognize that A < B . Therefore, the satisfaction the local criteria

μ wA

and

μ wB

in the optimization tasks may be rather

desirable, but it is not necessary. Actually, this means that local criterion introduced to estimate directly the uncertainty of the result of optimization using the width of the target function can rather be used to supplement the basic probabilistic criterion which in implicit way also implies the uncertainty.

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The second problem is the aggregation of local criteria to some generalized criterion taking into account their ranks. Now, three main approaches to aggregation are usually used:

D1 = min( μ1 ( x1 ) 1 , μ 2 ( x2 ) 2 ,K , μ n ( xn ) n ), r

D2 =

r

r

1 n ∑(ri μi ( xi )), n i =1 n

D3 = ∏μi ( xi ) i , r

i =1

where

ri are the ranks of local criteria μi . It can be shown that many other forms

of aggregation (Hauke 1999, Choi and Oh 2000, Shih and Lee 2000) are only some combinations of the basic aggregation modes: D1 , D2 , D3 . As proved by. Yager (1979) shown that general criterion D1 reflects better the preferences on the set of local criteria expressed by the ranks. Nevertheless, there are a lot of cases, when such criterion does not entirely conform to imagination of decision makers about the sense of optimization (Dubois and Koenig 1991). On the other hand, criteria D2 , D3 have usually undesirable property to compensate the small values of the part of criteria by large values of others. Since no approach to build a general criterion can claim to be uniformly the best obe, we prefer (from practical point of view) to choose the additional form D2 . Consider some additional resons for such a choice. As noted earlier, the cases when the local criterion μ w is equal to zero where

μp > 0

may occur. Since in such situation we get

D1 = 0 , D3 = 0 and only

D2 > 0 , the additive form of general criterion should be recognized as the best one in such cases. In addition, it is worth emphasizing that the compensatory property of additive criterion play an important role in our case and completely corresponds to the sense of optimization task. Therefore, the general criteria for evaluation of interval inequality degree may be presented as follows:

D A< B =

(

)

DB < A

(

)

D A= B where

1 rp μ pA + rw μ wA , 2 1 = rp μ pB + rw μ wB , 2 1 = rp μ pAB + rw max μ wA , μ wB , 2

(

(

))

rp , rw are the ranks or parameters of relative importance of local criteria.

Fuzzy Simulation and Optimization of Production and Logistic Systems

Of course, there are no problems to find the ranks

263

rp , rw in such a simple case

of only two local criteria, but conventional constraint

(rp + rw )/2 = 1 must be

fulfilled. It is easy to see that in any case 0 ≤ D A< B , DB < A , D A= B ≤ 1 . Let us describe roughly the main features of possible numerical algorithms based on the proposed approach. In the procedure of optimization, while using, '

e.g., the direct random search methods, in each n s step of an algorithm some interval B characterizing an interval cost function is obtained. If A is an interval value of a cost function in the next possible step, then in the case of minimization we can qualify it as a good step when A < B . The problem is to estimate the generalized degree of possibility of A < B . For this purpose, the general criteria DA< B , DB < A can be used. Of course, if DA< B < DB < A then B < A and otherwise B > A . Thus, in each step of optimization we have a small local twocriteria optimization task. We note that the similar approach has also been used to build the general criterion D A= B , which can be useful for mathematical formalization of interval equality constraints. Of course, the fuzzy extension of two-objective comparison can be easily obtained with use of α -cut representation of fuzzy intervals. So,

if

~ ~ D(B > A) =

~ ~ A and B ∑α ⋅ Dα (Bα > Aα ) α

α ∑ α

are

fuzzy

intervals(numbers)

then

.

3.2 The Illustrative Examples To illustrate the developed technique of optimization, let's consider the specially constructed interval double-extreme discontinuous function:

⎧ [4.05,4.95] − 0.5 x − 0.04, x < 0, ⎪− x ⎪ F ( x) = [ f , f ]⎨ ⎪[4.275,4.725] + 0.5 x, x > 0. ⎪ x ⎩ The introduced interval test function F (x ) has two local minimums at the points x = −3 and x = 3 . It can be seen (see Fig. 11.6 and Table 11.2) that at the point x = −3 we have the greater width of interval

[ f , f ] and the lower mean value of f on this interval than at the point x = 3 . Hence, we have a

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L. Dymowa and P. Sevastjanov

Fig. 11.6 An interval double-extreme discontinuous test function

Table 11.2 Interval test function

F (x)

Characteristics of local extreme Interval value of

F (x)

Mean value of interval Width of interval

F (x)

F (x)

F (x) .

in the points of its local extremes.

x = −3 [2.81,3.11]

x=3 [2.93,3.08]

2.96

3

0.3

0.15

typical situation when we have to choose the point of global optimum under condition of controversial local criteria of the cost function minimization and the result’s certainty maximization presented by the width of this function. Of course, there is no need to use any complicated numerical method to obtain the results presented in Fig. 11.6 and Table 11.2. However, such methods surely should be employed if the general criteria D A< B , DB < A described in previous subsection are used to aggregate our controversial local criteria. It is clear that because of objective function discontinuity the gradient methods can not be employed in our case. Therefore, to avoid the problem of interval and especially fuzzy function derivation in the optimization procedure, the interval and fuzzy generalizations of one of the direct random search methods (Luus and Jaakola, 1973) had been developed using the crisp and fuzzy interval comparison methodology described above. We do not intend to describe here this method with all details, but it seems reasonable to note that its algorithm based mainly on the object-oriented programming. It is worth noting that especially object-oriented methodology allows us to extend directly the usual numerical optimization algorithms changing real valued parameters by corresponding objects representing crisp or fuzzy intervals. It should be noted that in our case, the minimization of F (x ) in fact is substituted with the procedure of maximization of global

Fuzzy Simulation and Optimization of Production and Logistic Systems Table 11.3 The results of global minimization of interval test function Characteristics of global minimum

rp = 2

Interval value of

265

F (x) .

rp = 1.25

rp = 0.4

rw = 0

r p = 1.3 rw = 0.7 rw = 0.75

rw = 1.6

[2.81,3.11]

[2.815,3.107]

[2.926,3.074]

[2.944,3.081]

F (x) The mean of interval F (x)

2.96

2.961

3.0

3.012

Width of interval

0.3

0.292

0.148

0.137

−3

− 3.08

3.045

3.286

F (x) Points of global minimum

Fig. 11.7 The dependence of the test function value in the points of global minimum on ranks of local criteria.

criterion D . The results of global minimization of interval test function F (x ) for different ranking parameters are collected in Table 11.3 and Fig. 11.7. The testing has shown (see Table 11.3) that with increasing of the relative importance of “width” criterion (rising rw ), at first only the slight growth of mean values of interval test function are observed simultaneously with the reduction of its width in points of minimum. But in interval rw ∈ [0.7,0.75] the crisp jump (see Fig. 11.7) takes place and as the result of it, another point of global minimum is reached. It can be seen that in this point ( x = 3.045 ) we have somewhat greater mean value of our interval test function than in the previous global minimum point ( x = −3.08 ), but almost twice smaller width. Of course, this is the consequence of our preferences expressed by ranks rp , rw of used local criteria.

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The obtained results are in a good compliance with our intuition. In should be emphasized that offered approach allowsus to solve the described problems using introduced quantitative measures for estimating the degrees of interval/fuzzy interval equality and inequality. One of important features of the developed method is that there is an opportunity to estimate separately the degrees of equalities and inequalities. Of course, in each step of direct numerical minimization procedure, there is no need of high degree of objective function reduction, because usually it is enough to state the fact of such a reduction only. On the other hand, equality or inequality constraints can require the different quantitative degrees of “flexibility” of their performance. In practice, some constraints (frequently, equality ones) can require almost absolute performance. In these cases we require the execution of corresponding interval relations with probability 0.95 - 0.99 . For other constraints (inequalities) usually there are not such strict requirements and the probabilities of only a little more than 0.5 may be used. Our experience has shown that the described features of proposed method allow us to create a very convenient and flexible approach to the solution of real world optimization problems. Of course, there are no problems to generalize the method to the case of fuzzy intervals using α -cuts technique. To estimate the efficiency of the proposed numerical method, two additional examples are considered. In both cases, for the simplicity we assume rp = rw = 1 . We use the Rosenbrock's function

f = c( x2 − x12 ) 2 + (1 − x1 ) 2

(11.13)

as the base for interval and fuzzy interval extensions, since it is the most widely used test for numerical methods of optimization. In practical optimization, we often deal with interval or fuzzy interval objective function. It is worthy to note that these functions have real valued arguments. In such cases, the problem is to find the minimum/maximum of the interval or fuzzy interval function. So if

F ( x) = [ F ( x), F ( x)] is an interval function, the aim is

to find the real vector

x , providing an extreme of interval function

[ F ( x), F ( x)] directly. To avoid the problems of interval and especially fuzzy function derivation, the interval and fuzzy generalizations of one of the direct random search methods (Luus and Jaakola, 1973) had been developed. To obtain the test interval function, the initial Rosenbrock's function (11.13) has been extended and expressed in the following interval form:

F ( x1 , x2 ) = [ F ( x1 , x2 ), F ( x1 , x2 )] = [c − cα , c + cα ]( x2 − x12 ) 2 + (1 − x1 ) 2 (11.14) where α is a real number parameter determining the width of interval. The curves of equal values of function (11.14) are represented in Figure 11.8.

Fuzzy Simulation and Optimization of Production and Logistic Systems

267

Fig. 11.8 Interval extension of Rosenbrock's function (11.14) in the case of

α = 0.05

Table 11.4 The results of tests (the case of interval function (11.14) with

c = 100 ,

c = 100 )

α = 0.005

α = 0.01

α = 0.05

[ F , F ]in

[58.2, 58.8]

[57.9, 59.0]

[55.7, 61.3]

[ F + F ]in /2

58.5

58.5

58.5

[ F − F ]in

0.563

1.125

5.625

Initial (starting) point:

x1in = −0.5 , x2in = −0.5

The point of minimum

[ F , F ]min

−3 [5.43,5.44] ⋅10 −3 [4.67,4.72] ⋅ 10

[3.31,3.46] ⋅ 10 −3

[ F + F ]min /2

5.44 ⋅10−3

4.69 ⋅10−3

3.39 ⋅10−3

[ F − F ]min

0.01⋅10−3

0.05 ⋅10−3

0.15 ⋅10−3

x1min

1.022

1.015

0.986

x2min

1.044

1.029

0.971

The results of the tests are summarized in Table 11.4. It is interesting to note that the same level of accuracy as in the case of real number function was achieved by using the same number of steps of our algorithm (3900 in our case). It

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L. Dymowa and P. Sevastjanov min

min

is easy to see that the results of optimization ( x1 , x 2 ) only slightly depend on the width of interval representing the minimized function. The next example is the minimization of the fuzzy function. For the sake of simplicity, the case of trapezoidal fuzzy function is considered. Let us represent our base function (11.13) in fuzzy extended form:

F ( x1 , x2 ) = [ F1 ( x1 , x2 ), F2 ( x1 , x2 ), F3 ( x1 , x2 ), F4 ( x1 , x2 )] = = [c − cα , c − cα/2, c + cα/2, c + cα ]( x 2 − x12 ) 2 + (1 − x1 ) 2 , (11.15) where

F1 , F2 , F3 , F4 are the left support, left core, right core, right support of

trapezoidal fuzzy interval (number), respectively, and α is the real parameter determining the form of fuzzy interval. The fuzzy generalization of the direct random search method has been used to minimize the function (11.15). The same values of parameter α and the initial point as in the case of interval function (11.14) were used. After the same number of steps of the algorithm as in the last case (3900), the results shown in Table 11.5 were obtained. Table 11.5 The results of tests (the case of fuzzy function (11.15) with

α =0.005

α = 0.01

α = 0.05

x1min

0.992

0.989

0.997

x2min

0.982

0.977

0.992

c = 100 )

It can be seen that the results obtained in the fuzzy case are also good. These simple examples prove the practical validity of the developed method for crisp and fuzzy interval comparison and its ability to generate effective numerical implementation.

4 Applications 4.1 The Fuzzy Simulation of Production and Logistic Systems The mathematical tools presented in the previous sections may be used to build the fuzzy models of a wide range of real-world processes. However, to estimate the efficiency and accuracy of fuzzy modeling it seems reasonable to consider such examples that can be directly compared with the results of the conventional simulation method based on the Monte-Carlo approach. Therefore, the examples of fuzzy modeling of the production and logistic system are presented in this subsection.

Fuzzy Simulation and Optimization of Production and Logistic Systems w1,

Storehouse x1 1

w2,

Processing x2 unit 1

w3,

Processing unit 2 x3

x1z Buffer box 1

269

Processing x4 unit 3

x2z Buffer box 2

Storehouse 2

x3z Buffer box 3

Fig. 11.9 The scheme of production line.

Let us consider the simplest production line (Fig. 11.9), where

xi stands for

wi represents the capacities of the corresponding processing units, whereas xiz denotes inputs of corresponding buffer boxes. The capacities wi are assumed to be fuzzy numbers. Hence, if the production time t is a real value then the value of xi may be interpreted as a fuzzy quantity of semi-finished products manufactured at the time t and xiz may be interpreted as fuzzy quantity of semi-finished product stored in i th buffer boxes at the same

inputs,

time. Since a actual output of each processing unit depends on the capacities of other processing units in production line, the deterministic fuzzy model of the considered system may be presented as follows: x1 = tw1

if (P( w2 < w1 ) > 0 ) x1z = t ( w1 − w2 ) else x1z = 0, if (P( w2 < w1 ) > 0 ) x 2 = tw2 if (P( w1 < w2 ) > 0 )

else x 2 = tw1 ,

if (P ( w2 < w1 ) > 0)x 2 z = t ( w1 − w3 ) else x 2 z = 0

* 1.25

else

if (P ( w3 < w2 ) > 0 )x 2 z = t ( w2 − w3 ) else x 2 z = 0

if (P( w1 < w2 ) > 0 ) else

if (P ( w3 < w1 ) > 0) x3 = tw3

else x3 = tw1

if (P ( w3 < w1 ) > 0) x3 = tw3

else x3 = tw2 ,

x 4 = x3 ,

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where P ( wi < w j ) is the probability of wi < w j in terms of fuzzy interval comparison (see Section 2). To build this fuzzy model, the special software has been developed using mathematical tools presented in Section 2. In order to estimate the efficiency of the proposed approach, the results of fuzzy modeling were compared with the those obtained using conventional simulation based on Monte-Carlo approach. When imlementing the Monte-Carlo approach, the capacities wi assumed to be normally distributed with the following parameters of the probability distributions:

where

w1 = 0.04,

σ 1 = 0.0024, w2 = 0.03,

w3 = 0.05,

σ 3 = 0.003,

σ 2 = 0.0018,

wi denotes the expectation value, σ i ia a standard deviations. To obtain

more precise simulation results, 1000 numerical experiments were carried out for processing time t =1000. The described above probability distributions for wi were used to build the corresponding fuzzy intervals needed for fuzzy modeling. For the transformation of probability distributions into fuzzy intervals, the method described in Section 2 has been used. According to this method, all the α -cuts of fuzzy interval correspond to the confidence intervals of probability distribution. For example, in our case:

α = 0 we assume wi 0 = wi − dw0 , wi + dw0 , where dw0 = 3σ i , for others α ∈ (0,1]

[

[

for level

]

]

wiα = wi − dwα , wi + dwα , where dwα = σ i − 2 ln (α ) . Evidently, in such a case, the bottom cut ( α = 0 ) of the obtained fuzzy number corresponds to confidence interval of 99.7% probability. The results of fuzzy modeling and Monte-Carlo simulation are presented in Fig. 11.10. It can be seen that fuzzy model generates the results which can be interpreted as pessimistic ones since they are characterized by the greater uncertainty. On the other hand, such results are the guaranteed estimation because of the inherent property of fuzzy modeling which consists in operating on the bounds of fuzzy intervals available. It's worth noting that it is not possible to obtain guaranteed estimations by means of the conventional Monte-Carlo simulation due to probability nature of such kind of modeling. Let us consider the simple logistic system for supplying the plant with raw material from three mines by trucks (see Fig. 11.11). The uncertain parameters of considered system are: track loading time tl , track unloading time t ul , speed of tracks

v , capacity of mines a1, a2, a3, repair time t re (for the not complicated

cases when a driver can reject the breakage on way without assistance).

Fuzzy Simulation and Optimization of Production and Logistic Systems

271

I

II

Fig. 11.10 The number of produced units ( x4 ) for processing time model, II – Monte-Carlo simulation.

t = 1000 : I- fuzzy

mine №1 17km truck Capacity - 30 t/h. 18km Plant

truck

mine №2 №2

Capacities t/h

10

truck 20km mine №3

Fig. 11.11 Example of logistic system

To perform a more realistic example, the values of parameters were adopted from the normative literature in the field of road-building where they had been presented in the form of normal or uniform probability distributions. When using usual Monte-Carlo simulation, about 1000 workdays were simulated to obtain the result with sufficient accuracy. We have built the fuzzy model in two steps. At first, the simple deterministic real valued balance model has been developed. This model can be represented by the set of equations such as

ti = 2

Si + t l + t ul , i = 1,2,3, v

(11.16)

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L. Dymowa and P. Sevastjanov

where i is the mine’s number,

S i is the fixed distance between mine i and plant,

t i is the time, which is necessary for loading, carriage, unloading and returning of truck to the mine i. The whole model seems to be rather cumbersome one to present it here. However, it is quite enough to consider the only Eq. (11.16) to explain what we did. In accordance with fuzzy extension principle, we substitute the uncertain parameters with their fuzzy presentations in all equations like Eq. (11.16) and logical expressions too. We have used the transformation of probability distribution to fuzzy number described in Section 2 to get such fuzzy representations of uncertain parameters. All calculations with obtained fuzzy model were made using mathematical tools described in Section 2. One of the fuzzy modeling result in comparison with the corresponding result of Monte-Carlo simulation is presented in Fig. 11.12. f(τ), μ(τ)

I

1 0.8 0.6 0.4 II 0.2 0 τ Fig. 11.12 The waiting time for truck unloading: I - fuzzy model, II – Monte-Carlo simulation.

As it follows from Fig. 11.12, the results of fuzzy modeling may be close enough to those obtained using conventional Monte-Carlo simulation.

4.2 The Optimization of Distributor’s Decisions in the Fuzzy Setting The problem of distributor’s decisions optimization can be formulated as the generalization of the classical transportation problem. In this Chapter, we present further development of the methods for solution of these problems in the case of fuzzy coefficients and generalize an approach proposed by Dymowa et al. (2003, 2004). Isermann (1979) introduced an algorithm for solving classical transportation problem, which provides effective solutions. Ringuest and Rinks (1987) proposed two iterative algorithms for solving linear, multicriterial transportation problem. Similar solution has been proposed by Bit et al. (1992). Das et al. (1999)

Fuzzy Simulation and Optimization of Production and Logistic Systems

273

solved this problem in the case of interval uncertainty of transporting costs. In works by Chanas et al. (1993), Chanas and Kuchta (1996), an approach based on interval and fuzzy coefficients had been elaborated. Further development of this approach is made by Waiel and Abd El-Wahed (2001). All the above mentioned works introduce the restrictions on forms of membership functions. This makes it possible to transform the initial fuzzy linear programming problem into the set of usual linear programming tasks using well defined analytic procedures. However, in practice the membership functions, which represent uncertain parameters of models may have considerably complicated forms. In such cases, a numerical approach is needed. The main technical problem when building a numerical fuzzy optimization algorithm, is to compare fuzzy values. To decide this problem, we use the probabilistic approach described in Section 2 of this Chapter. The proposed approach allows us to accomplish the direct fuzzy extension of classical numerical simplex method with its implementation using the tools of object-oriented programming. In the proposed approach, we not only minimize the transportation costs, but in addition we maximize the distributor's profits under the same conditions. The distributor deals with M wholesalers and N consumers (see Fig.11.13). Let ai,, i=1 to M, be the maximal quantities of goods that can be proposed by wholesalers and bj, j=1 to N, be the maximal goods requirements of consumers. In accordance with the signed contracts, distributor must buy at least pi goods units at price of ti monetary units for unit of goods from each ith wholesaler and to sell at least qj goods units at price of sj monetary units for unit of goods to each jth consumer. The total transportation cost of delivering one goods unit from ith wholesaler to jth consumer is denoted as cij. wholesaler 1 wholesaler 2

consumer 1

DISTRIBUTOR

...

wholesaler M

consumer 2

...

consumer N

Fig. 11.13 The scheme of distributor’s activity

There are reduced prices ki for distributor if he/she buy the greater quantities of goods than stipulated in contract (pi) and also the reduced prices rj for consumers if they buy the good quantities grater than contracted qj . The problem is to find the optimal goods quantities xij (i=1,...,М; j=1,...,N) delivering from ith wholesaler to jth consumer maximizing the distributor’s total benefit D under constraints. Assuming that all above mentioned parameters are fuzzy ones, resulting optimization task has been formulated as:

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L. Dymowa and P. Sevastjanov

Dˆ = N

∑ xˆ ≤ aˆ ij

j =1

N

∑ xˆ j =1

where

ij

≥

pˆ

i

i

∑ ∑ ( zˆ M

N

i =1

j =1

ij

∗ xˆ ij ) → max

(i = 1..M ),

M

∑ xˆ ≤ bˆ ij

i =1

(i = 1..M ),

M

∑ xˆ ≥ qˆ i =1

ij

j

j

(11.17) ( j = 1..N ),

( j = 1..N ),

(11.18) (11.19)

zˆij = rˆj − kˆi − cˆij (i=1,...,М; j=1,...,N) and Dˆ , zˆ ij , aˆ , bˆ , qˆ , pˆ are

fuzzy values. To solve the problem (11.17)-(11.19), the numerical method based on the α-cut representation of fuzzy numbers and probabilistic approach to the interval and fuzzy interval comparison has been developed. The direct fuzzy extension of usual simplex method is used. The use of object-programming tools makes it possible to get the results of fuzzy optimization, i.e., xˆ ij in the form of fuzzy values as well. To estimate the effectiveness of the proposed method, the results of fuzzy optimization were compared with those obtained from (11.17)-(11.19) when all the uncertain parameters were considered as normally distributed random values. Of course, in the last case we have used Monte-Carlo simulation and the parameters in (11.17)-(11.19) were considered as randomly chosen (according to the Gauss distribution) h real values. To make the results we get using the fuzzy and probability approaches comparable, the simple method for transformation frequency distributions into fuzzy numbers without lost of useful information has been used to achieve the comparability of uncertain initial data in the fuzzy and random cases. This method is described in Section 2. The standard Monte-Carlo procedure has been used for the realization of probability approach to the description of uncertain parameters of the optimization problem (11.17)-(11.19). In fact, for each randomly selected set of real valued parameters of problem (11.17)(11.19), we solve the usual linear programming problem. Numerical example To compare the results of fuzzy programming with those obtained when using the Monte-Carlo method, all the uncertain parameters previously were performed by Gaussian frequency distributions. The averages of them are presented in Table 11.6. For the sake of simplicity, all standard deviations σ were accepted to be equal to 10 i.m. Table 11.6 The average values of Gaussian distributions of uncertain parameters. a1=460

b1=410

p1=440

q1=390

t1=600 s1=1000 k1=590 r1=990

a2= 460

b2=510

p2=440

q2=490

t2=491 s2=1130 k2=480 r2=1100

a3= 610

b3=610

p3=590

q3=590

t3=581 s3=1197 k3=570 r3=1180

c11=100

c12=30

c13=100

c31=120 c32=148 c33=11

c21=110

c22=36

c23=405

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The results we have obtained using the fuzzy optimization method and MonteCarlo method (usual linear programming with real valued, but random parameters) are presented in Fig. 11.14-11.16 for the case M=N=3, where the final frequency distributions F are drown by dotted lines, fuzzy numbers µ are drown by continuous lines. F(x12),μ (x12)

F(x11),μ (x11)

1

1

0.5

0.5

0

200

300

400

500 x11

Fig. 11.14 Frequency distribution F and fuzzy number µ of optimized x11

100

0

100

200

300

400

500 x12

Fig. 11.15 Frequency distribution F and fuzzy number µ of optimized x12

It is easy to see that fuzzy approach provide us some more wider fuzzy intervals than Monte–Carlo simulation. It is interesting that using probabilistic method we can get even two-extreme results whereas fuzzy approach always provide us the results without ambiguity. It is worth noting that probabilistic method demands too mach of random steps (about 100 000 000) to obtain the smooth frequency distribution of the resulting benefit D. Thus, it seems rather senseless to use this method in practice. F(D),μ (D)

1

2

1

3 0.5

0 500000

600000 700000

800000

900000 1000000

D

Fig. 11.16 Frequency distribution F and fuzzy number µ of optimized benefit D: 1 - MonteCarlo method for 10 000 random steps; 2 - Monte-Carlo method for 100 000 000 random steps; 3 - fuzzy approach.

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5 Conclusion The set of methods of conventional interval arythmetic are supplement with a new probabilistic approch to interval and fuzzy values comparison and two-objective interval and fuzzy values comparison.It is shown that mathematical tools of applied interval analysis with such supplementation may be directly used for fuzzy simulation and optimization in the fuzzy setting.The developed approach is illustrated by the examples of fuzzy simulation of linear production line and logistic system and by the example of fuzzy solution of optimal goods distribution problem. The results obtained with use of the proposed approach are compared with those obtained using Monte-Carlo simulation and the advantages of the developed methods are performed.

References Bit, A.K., Biswal, M.P., Alam, S.S.: Fuzzy programming approach to multicriteria decision making transportation problem. Fuzzy Sets and Systems 50, 135–142 (1992) Bustince, H., Burillo, P.: Mathematical analysis of interval-valued fuzzy relations: Application to approximate reasoning. Fuzzy Sets and Systems 113, 205–219 (2000) Bustince, H., Pagola, M., Barrenechea, E.: Construction of fuzzy indices from fuzzy DIsubsethood measures: Application to the global comparison of images. Information Sciences 177, 906–929 (2007) Chanas, S., Kuchta, D.: Multiobjective programming in optimization of the Interval Objective Functions: a generalized approach. European Journal of Operational Research 94, 594–598 (1996) Chanas, S., Delgado, M., Verdegay, J.L., Vila, M.A.: Interval and fuzzy extensions of classical transportation problems. Transportation Planning Technology 17, 203–218 (1993) Choi, D.Y., Oh, K.W.: Asa and its application to multi-criteria decision making. Fuzzy Sets and Systems 114, 89–102 (2000) Das, S.K., Goswami, A., Alam, S.S.: Multiobjective transportation problem with interval cost, source and destination parameters. European Journal of Operational Research 117, 100–112 (1999) Deneux, T.: Modeling vague beliefs using fuzzy-valued belief structures. Fuzzy Sets and Systems 116, 167–199 (2000) Dubois, D., Koenig, J.L.: Social choice axioms for fuzzy set aggregation. Fuzzy Sets and Systems 43, 257–274 (1991) Dubois, D., Ostasiewicz, W., Prade, H.: Fuzzy sets: history and basic notions. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets. Kluwer, Boston (2000) Dymowa, L., Dolata, M.: The transportation problem under probabilistic and fuzzy uncertainties. Operations Research and Decisions 4, 23–31 (2003) Dymowa, L., Dolata, M., Sewastianow, P.: The fuzzy decision of transportation problem. In: International conference on Modeling and Simulation, Minsk, Belarus, pp. 308–311 (2004) Facchinetti, G., Ricci, R.G., Muzzioli, S.: Note on ranking fuzzy triangular numbers. International Journal of Intelligent Systems 13, 613–622 (1998) Hansen, E.R.: Global optimization using interval analyses. Marcel Dkker, NewYork (1992)

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Hauke, W.: Using Yager’s t-norms for aggregation of fuzzy intervals. Fuzzy Sets and Systems 101, 59–65 (1999) Isermann, H.: The enumeration of all efficient solution for a linear multiple-objective transportation problem. Naval Research Logistics Quarterly 26, 123–139 (1979) Jaulin, L., Kieffir, M., Didrit, O., Walter, E.: Applied interval analysis. Springer, Heidelberg (2001) Kaufmann, A., Gupta, M.: Introduction to fuzzy arithmetic-theory and applications. Van Nostrand Reinhold, New York (1985) Luus, R., Jaakola, T.: Optimization by direct search and systematic reduction of size of search region. AIChE J. 19, 760–766 (1973) Moore, R.E.: Interval analysis. Prentice-Hall, Englewood Cliffs (1966) Nguyen, H.T., Kreinovich, V., Longpre, L.: Dirty pages of logarithm tables lifetime of the universe and (subjective) probabilities on finite and infinite intervals. Reliable Computing 10, 83–106 (2004) Piegat, A.: Fuzzy modelling and control. EXIT, Warszawa (1999) ( in Polish) Ringuest, J.L., Rinks, D.B.: Interactive solutions for the linear multiobjective transportation problem. European Journal of Operational Research 32, 96–106 (1987) Sevastjanov, P., Rog, P.: Fuzzy modeling of manufacturing and logistic systems. Mathematics and Computers in Simulation 63, 569–585 (2003) Sevastjanov, P., Rog, P.: Two-objective method for crisp and fuzzy interval comparison in optimization. Computers and Operations Research 33, 115–131 (2006) Sevastjanov, P.: Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster–Shafer theory. Information Sciences 177, 4645–4661 (2007) Shih, H.S., Lee, E.S.: Compensatory fuzzy multiple level decision making. Fuzzy Sets and Systems 114, 71–87 (2000) Wadman, D., Schneider, M., Schnaider, E.: On the use of interval mathematics in fuzzy expert system. International Journal of Intelligent Systems 9, 241–259 (1994) Waiel, F., El-Wahed, A.: A multi-objective transportation problem under fuzziness. Fuzzy Sets and Systems 117, 27–33 (2001) Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (I) (II). Fuzzy Sets and Systems 112, 387–405 (2001) Wang, Y.M., Yang, J.B., Xu, D.L.: A preference aggregation method through the estimation of utility intervals. Computers and Operations Research 32, 2027–2049 (2005) Xu, Z., Chen, J.: Some models for deriving the priority weights from interval fuzzy preference relations. European Journal of Operational Research 184, 266–280 (2008) Xu, Z., Da, Q.: The uncertain OWA operator. International Journal of Intelligent Systems 17, 569–575 (2002) Yager, R.R.: Multiple objective decision-making using fuzzy sets. Int. J. Man-Mach. Sfud. 9(14), 375–382 (1979) Yager, R.R., Detyniecki, M.: Ranking fuzzy numbers using weighted valuations. International Journal of Uncertainty, Fuzziness and Knowledge based Systems 8, 573–591 (2000) Yager, R.R.: A foundation for a theory of possibility. J. Cybernet. 10, 177–204 (1980) Zadeh, L.A.: Fuzzy Sets. Inf. Contr. 8, 338–358 (1965) Zimmermann, H.J., Zysno, P.: Latest connectives in human decision making. Fuzzy Sets Systems 4, 37–51 (1980)

Chapter 12

Fuzzy Investment Planning and Analyses in Production Systems Cengiz Kahraman and A. Çağrı Tolga1

Abstract. Investment planning is a part of investment analysis that contains real investments, such as machines, lands, a new plant, a new ERP system implementation etc. Investment analysis concerns evaluation and comparison of the investment projects. In the planning phase timing is the important point to execute the project. A production system is an aggregation of equipment, people and procedures to perform the manufacturing operations of a company. Production systems can be divided into categories named facilities and manufacturing support systems. The facilities of the production system consist of the factory, the equipment, and the way the equipment is organized. In this chapter, the components of investment planning are given. Then, the fuzziness of the investment is presented. Fuzzy present worth, fuzzy annual worth, fuzzy rate of return analysis, fuzzy B/C ratio, fuzzy replacement analysis and fuzzy payback period techniques -performed in this chapter- are fuzzy investment analysis techniques. At the last section application of these techniques are subjected.

1 Introduction In its broadest sense, investment signifies the surrender of current money for future money. Time and risk are the two different characteristics of investment. Surrender is certain and takes place in present. Prize comes in the future and its amount is uncertain. In some cases for example in government bonds the time element surpasses. In other cases risk is the dominant attribute (for example, call options on common stocks) (Sharpe et al. 1999). In again others, both risk and time are important (for example stocks). Though almost every human activity concerns risk, it is surprising how little consensus there is about how to define risk. For clarifying the minds about “risk”, the term uncertainty has to be defined. Uncertainty is the lack of complete certainty, that is, the existence of more than one possibility (Hubbard 2007). Cengiz Kahraman Department of Industrial Engineering, İstanbul Technical University, 34367 Maçka İstanbul, Turkey A. Çağrı Tolga Department of Industrial Engineering, Galatasaray University, 34357 Ortaköy İstanbul, Turkey C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 279–298. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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Accordingly, risk is a state of uncertainty where some of the possibilities involve a loss, catastrophe, or other undesirable outcome. Mostly a distinction between savings and investment is made. Saving refers to preserving money for future use, typically by putting it on deposit; this is distinct from investment where there is an element of risk. With savings, your principal typically remains constant and earns interest or dividends. However, depositing your savings to mutual funds, options, collectibles, precious metals, or real estate contains risk and so it is investment. A metaphor can be made between savings and potential energy. Also another metaphor between investment and kinetic energy should be made after. Investments can go up or down in value and may or may not pay interest or dividends. Each example for investment involves the sacrifice of something now for the prospect of something later (Park et al. 1990). To be more understandable, examples about real life have to be given: Renergy Firm invests millions of dollars in a wind energy project. The president of the company may invest hundreds of thousands of dollars by buying shares of a newly formed company that manufactures steel towers. Financial manager in the company may invest several thousand dollars in derivatives of renewable energy firm index. The production manager considers investing in a new residence supported with renewable energy sources. The examples concerning investment are often classified in two categories: financial and real investments. A financial investment is one in which the investor allocates his or her resources to some form of financial instrument, such as stocks or bonds (Park et al. 1990). On the other hand, real investments involve some kind of tangible assets, such as a new plant, R&D laboratory, machines, real estates, lands, etc. These cases are sufficiently tangible to be considered real investments. But where do the resources come from to pay for the land and the construction of the real estate? Some may come from savings of individual. The rest of the resources may be provided by banking loan. This chapter is about real (project) investments as contrasted with financial investments. As stated above, time, risk and uncertainty are the main components of investment. Uncertainty is reminiscent of vagueness because of its nature. When it is difficult to explain the data crisp way or there is lack of data, fuzzy logic is set in to respond these inadequacies. A production system is a collection of people, equipment, and procedures organized to perform the manufacturing operations of a company (or other organization) (Groover 2008). Production systems can be divided into categories named facilities and manufacturing support systems. The facilities of the production system consist of the factory, the equipment, and the way the equipment is organized. The set of the procedures used by the company to manage production and to solve the technical and logistics problems encountered in ordering materials, moving the work through the factory, and the ensuring that products meet quality are called manufacturing support systems. In this chapter, at the following sections, investment planning will be employed. The components of investment planning concerning technical, financial and economical analysis will track the preceding section. Then, “why the investment environment is fuzzy?” topic will be discussed at the next section. Broad information about fuzzy logic is the subject of that section. Fuzzy investment

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techniques and applications sections are the keynote of this chapter. After these sections a brief conclusion will be made to summarize the topics in that chapter.

2 What Is Investment Planning? Planning is the organizational process of creating and maintaining a plan. This thought process is essential to the creation and refinement of a plan, or integration of it with other plans, i.e., it combines forecasting of developments with the preparation of scenarios of how to react to them. In other words, planning is preparing a sequence of action steps to achieve some specific goal. If you do it effectively, you can reduce much the necessary time and effort of achieving the goal. The process of thinking about the activities or procedures requires establishing a desired goal on some scale. It is a fundamental property of intelligent behavior. When following a plan, you can always see how much you have progressed towards your project goal and how far you are from your destination. Knowing where you are is essential for making good decisions on where to go or what to do next. Investment planning often confused with financial planning. Financial planning is the long-term process of wisely managing someone’s or organization’s finances, so she/he/it can achieve her/his/its goals, while at the same time negotiating the financial barriers that inevitably arise in every stage of planning. While financial planning is a process used to achieve major goals of life, investment planning forms a part of financial planning which deals with building a strategic investment portfolio keeping in mind one’s future needs and risk bearing exercise. Investment planning deals with real investments, such as machines, lands, a new plant, a new ERP system implementation. Holistic investment analysis approach for a real investment is the main subject of investment planning. Investment analysis concerns evaluation and comparison of the investment projects. Many techniques are developed for this analysis; however in this chapter six of them will be investigated in a fuzzy manner, such as fuzzy present worth, fuzzy annual worth, fuzzy rate of return etc. The earliest and most systematic treatment of the central issue of investment planning is named the timing of investment. Timing decision is important for real investments. To make right timing, at first, someone has to know the components of investment planning. After understanding these components such as technical, financial, and economical analysis, founding the exact plans, timing for real investments is as easy as winking.

3 The Components of Investment Planning Generally in the planning of investment projects, at first the economical analysis is scoped out, then technical analysis is made. Finally, financial analysis comes for computing the financial efficiency of these projects. However, these studies necessitate each other. In fact, as one might have needed technical analysis data in the economical and financial analysis processes, on the other hand; technical analysis

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might be replaced according to the financial analysis data. This fact induces interpenetration of the studies from the technical, financial and economical point of view. In another words, a holistic point of view is needed to evaluate the investment projects. Although this study covers the financial analysis of the investment project like investing in production systems, also technical and economical side of the investment project has to be reviewed.

3.1 Economical Analysis In the evaluation and determination of working procedure of investment projects the following studies should be composed: Expected developments in the branch of business, sharing of market and pricing, choosing the capacity and location of the investment, benefits of the project from the economical view. However, in production systems investments, all of these studies could be adapted to the micro scale. In the production systems, forecasts will depend upon the demand of the product and the size of the market. There are many forecasting techniques in the investigation of the developments of the branch of business. Classification of these techniques could be cited as direct and indirect forecasting techniques. Market share of the product and pricing are the main essentials of the investment project. The insufficient market share or inconvenient demand states could affect the future of the project (Güvemli 1979). Producer’s cost price is the first data in determining the price of the product. Although, competition conditions, the other competitor’s cost price, capacity are the other factors of the pricing. The benefits of the project from the economical view depend on the resources constraints. In this chapter the evaluations will be made from the entrepreneur’s view. In this evaluation process the present worth (PW), equal annual worth (EAW), rate of return analysis, B/C ratio, and etc. analyses will be employed.

3.2 Technical Analysis In the technical analysis procedure, main subjects are choosing the location of the investment, determining the technology of the system, deciding the capacity, selection of the machines and equipments, fixing the necessitates of the employees, and computing the cost price of the investment. However in the production systems, except the selection of the location of the investment, all of these subjects have to be investigated while analyzing the technical conditions. Decision makers have to be very careful about the selection of the right technology of the production system investment, also choosing the right machines and equipments and capacity have to be the main occupation of the decision makers. While computing the manufacturing cost, reaching and utilizing the full capacity time, general administration expenses, costs of sales, depreciation have to be also calculated.

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3.3 Financial Analysis In the components of investment planning frame, thus far, the predicted sales in periods were determined, and then in unison with these studies, computation of the cost price of the investment was displayed. In the financial analysis, these mentioned studies are converted to balance sheet, income statement and if needed they are put into another financial statement. With no doubt, in this analysis, one can turn back into economical and technical analysis and if needed one can refresh them. The financial analysis is accurate if the previous analyses are exact. Because of this, the uncertainties about technical and economical analyses have to be disambiguated. The financial analysis of the investment project is approximately the same with economical analysis. Especially, profit and depreciation computations of financial analysis are benefited from the economical analysis. However, the profitability of the project and the financial structure have major role in financial analysis.

4 Fuzziness of Investment The conventional investment analysis methods such as present worth (PW), equal annual worth (EAW), rate of return analysis, B/C ratio, etc. analyses are based on the exact numbers. Essentially, however, it is not practical to expect that certain data, such as the future cash flows and discount rates, are known exactly in advance. Usually, based on past experience or educated guesses, decision makers modify vague data to fit certain conventional decision-making models. The modified data consist of vagueness such as approximately between 3% to 5 % or around $15,000—in linguistic terms. For instance according to the various conditions, the initial cost of an advanced manufacturing system may lie between $150.000 and $200.000. Fuzzy set theory, first introduced by Zadeh (1965) can be used in the uncertain economic decision environment to deal with the vagueness. For these inputs fuzzy numbers can be used, such as cash amounts and interest rates in the future, for conventional decision-making models (Chan et al. 2000). In investment analysis, the selection of alternatives problem is common and fuzzy set theory was applied extensively to solve that problem. Ward (1985) developed fuzzy present worth analysis by introducing trapezoidal cash flow amounts. Buckley (1987) proposed a fuzzy capital budgeting theory in the mathematics of finance. Kaufmann and Gupta (1988) used the fuzzy number to the discount rate. They derived a fuzzy present worth method for investment alternative selection. Kuchta (2001) considers net present value as a quadratic 0-1 programming problem in a fuzzy form. Kahraman et al. (2002) derived the formulas for the analyses of fuzzy present value, fuzzy equivalent uniform annual value, fuzzy final value, fuzzy benefit-cost ratio and fuzzy payback period in capital budgeting. Dimitrovski and Matos (2008) gave the techniques for comparing and ordering fuzzy numbers of independent numbers and dependent numbers and they examined fuzzy case with partial correlation.

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4.1 Fuzzy Ranking In every evaluation methods ordering is the key factor due to its importance in decision making. Fuzzy numbers have to be ranked just because of this fact. The nature of fuzzy numbers drives us to use different methods for ranking according to the cases. There are many numbers of methods that are devised to rank mutually exclusive projects such as Chang's method (1981), Jain's method (1976), Dubois and Prade's method (1983), Yager's method (1980), Baas and Kwakernaak's method (1977). In this chapter, an area-based approach proposed by Kahraman and Tolga (Kahraman and Tolga 2009) for ranking fuzzy numbers is used. An index that measures the possibility of one fuzzy number being greater than another will be determined. That preference index will be illustrated by I (ω ) ∈ [0,1] and it is determined by Eq. (12.1):

0 ⎧ ⎪ (a4 − b1 )2 ⎪ ⎪ (b2 − b1 − a3 + a4 ) ⎪ ⎪ (a4 + a3 − a2 − a1 ) + (b4 + b3 − b2 − b1 ) ⎪ a4 + a3 − b2 − b1 ⎪ I (ω ) = ⎨ a a a ( + − 3 2 − a1 ) + (b4 + b3 − b2 − b1 ) ⎪ 4 ⎪ (a2 − b3 ) 2 ⎪ (a4 + a3 − b2 − b1 ) − (b4 − b3 + a2 − a1 ) ⎪ ⎪ ( a + a − a − a ) + (b + b − b − b ) 3 2 1 4 3 2 1 ⎪ 4 ⎪⎩ 1

,

b1 ≥ a4

, b2 ≥ a3 , b1 < a4 , b3 ≥ a2 , b2 < a3

(12.1)

, b3 < a2 , b4 > a1 ,

b4 ≤ a1

The fuzzy preference relation ( PKT ) of the fuzzy numbers will be determined as following: ⎧ A% f B% if I ( w) ∈ (0.5,1] ⎪ (12.2) PKT ( A% , B% ) = ⎨ A% = B% if I ( w) = 0.5 ⎪ B% f A% if I ( w) ∈ [0, 0.5) ⎩ Calculation of index I (ω ) is the key factor in our method. Two different trapezoidal fuzzy numbers are illustrated in Fig. 12.1. As it is seen, the area that is not overlapping is named S lfavor for the left side and S rfavor for the right side. S joint is the intersection area of these fuzzy numbers. SA and SB are the areas of the fuzzy numbers à and B̃ consecutively. If the outcome of Eq. (12.1) is larger than 0.5, this means that the fuzzy number % A is preferred to B% . For example, in Fig. 12.2 let the fuzzy numbers A% and B% take the following values, respectively: (2, 3, 4, 8) and (1, 5, 6, 7). Then, from Eq. (12.1) we find the preference index: I (ω ) = 0.438 . Therefore, we can infer that the fuzzy number A% is smaller than the fuzzy number B% with a possibility of 0.438.

Fuzzy Investment Planning and Analyses in Production Systems μ(x)

S

l favor

285

S joint A%

B%

1.0

S rfavor x

0.0

b1

a 1 b2

a2

a3

b3

b4

a4

Fig. 12.1 Comparison of fuzzy numbers μ(x)

S lfavor

S joint

S rfavor

% A

B% 1.0

x

0.0

a1

b1

b2

b3

a2

a3

a4

b4

Fig. 12.2 An example for the fourth row of Eq. (12.1)

5 Fuzzy Investment Analysis Techniques Several fuzzy investment analysis techniques can be employed to measure the efficiency of a project uniquely or some alternatives. However, the fuzzy cash flows of these alternatives have to be recognized and determined. These techniques are illustrated at the following chapters. Each of them can be applied to any given problem. But, relying on the particular problem and associated data, there can be certain advantages to apply one technique or another. These advantages related to different techniques will be given at the next chapters. Also, the following chapters deal with present worth, annual worth, rate of return analysis, B/C ratio, replacement analysis, payback period techniques in their fuzzy form.

5.1 Fuzzy Present Worth In some special problems revenues may be fixed and alternatives may differ in their costs. In such cases it is easier to consider only the costs in fuzzy present worth (FPW) analysis technique where they are taken as positive. Definitely, the best alternative is the one that has the lowest FPW.

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Likewise, in some special problems the costs may be fixed and alternatives may differ in their revenues only. In such cases it is easier to omit the costs in present worth analysis and implement the analysis as usual. In their paper, Kahraman et al. (2002) developed the formulas for the analyses of fuzzy present value; fuzzy equivalent uniform annual value, fuzzy future value, fuzzy benefit-cost ratio, and fuzzy payback period. Rebiasz (2007) discoursed on the issue of project risk appraisal, mixing fuzzy and probabilistic models. Dimitrovski and Matos (2008) gave the techniques for comparing and ordering fuzzy numbers of independent numbers and dependent numbers and they examined fuzzy case with partial correlation. Kuchta (2008b) presented the fuzzy net present value maximization as an objective in project selection problems, applications to the valuation of projects with future options, interpretation of fuzzy present value and fuzzy net present value as objective in optimization problems from the industry. The fuzzy net present worth (present value) for alternative k, FNPWk, that has a general cash flow diagram shown on Fig. 12.3, is the sum of all its annual cash flows, discounted to the present time: n

A% kj

j =1

(1 + i ) j

FPNWk = A% k 0 ⊕ ∑

(12.3)

where A%kj is the fuzzy net cash flow for alternative k at the end of period j; n is the number of years (periods) in the planning horizon; i is the discount rate.

0

1

2

3

…

n-1

n

t

Fig. 12.3 Cash flow diagram

Alternatives are ranked in accordance with their FPNW with the best one having the highest FPNW. To be attractive the corresponding alternative has to be bigger than or equal zero. Also the corresponding alternative’s rate of return should be bigger than the discount rate. Otherwise, the money is better spent elsewhere in the economical system. Ranking these fuzzy alternatives is made with the ranking approach explained in the previous section. Either triangular or trapezoidal fuzzy numbers can be ordered with this effortless ranking method. Example: Assume that there are two projects that have initial costs A% f 0 = (10000,15000, 20000) and A% s 0 = (12000,15000,18000) which are named

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first and second in subscripts. The fuzzy annual cash flows of these two alternative for two year period are given as: A% f 1 = (17000,18500,19750) , A% s1 = (11000,14250,19500) ,

A% f 2 = (9500,14250,19100) ,

and

A% s 2 = (14800,18500, 21250) . Assume the discount rate is 9% per year. Rank the alternatives. 2 A% fj FPNW f = A% f 0 ⊕ ∑ j j =1 (1 + 0, 09) ⎛ 17000 18500 19750 ⎞ ⎛ 9500 14250 19100 ⎞ = (−10000, −15000, −20000) ⊕ ⎜ , , , , ⎟⊕⎜ 2 2 2 ⎟ ⎝ 1.09 1.09 1.09 ⎠ ⎝ 1.09 1.09 1.09 ⎠ = (3592,13966, 22512)

⎛ 11000 14250 19500 ⎞ FPNWs = (−12000, −15000, −18000) ⊕ ⎜ , , ⎟⊕ ⎝ 1.09 1.09 1.09 ⎠ ⎛ 14800 18500 21250 ⎞ = (4549,13644, 23776) , , ⎜ 2 2 2 ⎟ ⎝ 1.09 1.09 1.09 ⎠ Anyone can simply compute I ( w ) = 0.504 from Eq. (12.1) which indicates FPNW f f FPNWs .

5.2 Fuzzy Rate of Return Analysis In this section, at first crisp rate of return analysis is presented. Then fuzzy rate of return analysis will be discussed. The definition of fuzzy internal rate of return (IRR) is ambiguous and requires discussion. The use of fuzzy numbers when projects are classified on the basis of their IRR value will also be shown. Ward (1985) considers the Eq. (12.4) and states that such an expression cannot be applied to the fuzzy case because the left side of the Eq. (12.4) is fuzzy, 0 in the right hand side is a crisp value and equality is impossible. Hence, the Eq. (12.4) is senseless from the fuzzy viewpoint (Sewastjanow and Dymowa 2008). Kuchta (2000) proposed a method for fuzzy IRR estimation where α-cut representation of fuzzy numbers was used. The IRR of a project is defined as the rate of interest that equates the present value (PV) of the entire series of cash flows to zero. The IRR of a project is such a number E for which the following equality is fulfilled (Kuchta 2008a):

NPV (P ) = CF0 +

n

∑ PV (CIF ) = 0 i

(12.4)

fori = 1,..., n

(12.5)

i =1

where PV (CIFi ) =

CIFi

(1 + E )i

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CIFi is cash in flow for period i. The IRR is thus the cost of capital at which the project in question has NPV equal to zero. For this section the following definition of fuzzy number will be used: ~ A fuzzy number, denoted by A = a1 , a 2 , a3 , a 4 , f1A , f 2A , where f1A ( x ) is de-

(

)

fined and increasing for x ∈ (a1 , a 2 ) , f 2A ( x ) is defined and decreasing for x ∈ (a3 , a 4 ) , both function are continuous and transform their domain intervals

into the interval [0,1] , is defined as the family of closed intervals for α ∈ [0,1] : −1 −1 α Aα = ⎡⎢( f1 A ) (α ) , ( f 2A ) (α ) ⎤⎥ = ⎡⎣ A , Aα ⎤⎦ ⎣ ⎦

(12.6)

Using the definition above, the fuzzy IRR for a project with fuzzy parameters can be defined as below: Let P be a project with fuzzy parameters. Then its fuzzy IRR is the family of intervals E% = ⎡⎣ E α , E α ⎤⎦ : α ∈ [ 0,1] such that:

{

}

n

(

CF 0 + ∑ PV CIF i α

i =1

α

n

(

α

α

CF 0 + ∑ PV CIF i i =1

) = 0, where PV (CIF ) = α i

) = 0, where PV (CIF ) = α i

α

(

CIF i

1+ E

α

)

i

(12.7)

α

(

CIF i

1+ E

α

)

i

(12.8)

5.3 Fuzzy B/C Ratio Analysis Benefit-Cost Analysis is an economic tool to aid social decision-making, and is typically used by governments to evaluate the desirability of a given intervention in markets. The aim is to gauge the efficiency of the intervention relative to the status quo. The costs and benefits of the impacts of an intervention are evaluated in terms of the public's willingness to pay for them (benefits) or willingness to pay to avoid them (costs). Inputs are typically measured in terms of opportunity costs the value in their best alternative use. The guiding principle is to list all of the parties affected by an intervention, and place a monetary value of the effect it has on their welfare as it would be valued by them (Kahraman and Kaya 2008). The benefit-cost ratio can be defined as the ratio of the equivalent value of benefits to the equivalent value of costs. The equivalent values can be present values, annual values, or future values. The benefit-cost ratio (BCR) is formulated as

BCR =

B C

(12.9)

where B represents the equivalent value of the benefits associated with the project and C represents the project's net cost (Blank and Tarquin 1998). A B/C ratio

Fuzzy Investment Planning and Analyses in Production Systems

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greater than or equal to 1.0 indicates that the project evaluated is economically advantageous. In B/C analyses, costs are not preceded by a minus sign. The objective to be maximized behind the B/C ratio is to select the alternative with the largest net present value or with the largest net equivalent uniform annual value, because B/C ratios are obtained from the equations necessary to conduct an analysis on the incremental benefits and costs. Suppose that there are two mutually exclusive alternatives. In this case, for the incremental BCR analysis ignoring drawbacks the following ratios must be used:

ΔB2−1 ΔPVB2−1 = ΔC 2−1 ΔPVC 2−1

(12.10)

ΔB2−1 ΔEUAB2−1 = ΔC 2−1 ΔEUAC 2−1

(12.11)

or

where ΔB2−1 is the incremental benefit of Alternative 2 relative to Alternative 1,

ΔC 2−1 is the incremental cost of Alternative 2 relative to Alternative 1, ΔPVB2−1 is the incremental present value of benefits of Alternative 2 relative to Alternative1, ΔPVC 2−1 is the incremental present value of costs of Alternative 2 relative to Alternative 1, ΔEUAB2−1 is the incremental equivalent uniform annual benefits of Alternative 2 relative to Alternative 1 and ΔEUAC 2−1 is the incremental equivalent uniform annual costs of Alternative 2 relative to Alternative 2. Thus, the concept of B/C ratio includes the advantages of both NPV and net EUAV analyses. Because it does not require to use a common multiple of the alternative lives (then B/C ratio based on equivalent uniform annual cash flow is used) and it is a more understandable technique relative to rate of return analysis for many financial managers, B/C analysis can be preferred to the other techniques such as present value analysis, future value analysis, rate of return analysis. In the case of fuzziness, the steps of the fuzzy B/C analysis in discrete compounding form are given in the following: Step 1: Calculate the overall fuzzy measure of benefit-to-cost ratio and eliminate the alternatives that have ⎛ ~ ~ ⎜ B C =⎜ ⎜ ⎝

∑ ∑

n

t =0 n

t =0

( ) ∑ , C ( ) (1 + r ( ) ) ∑ Btl ( y ) 1 + r r ( y ) l y t

−t

r y −t

n

t =0 n

t =0

( ) C ( ) (1 + r ( ) ) Btr ( y ) 1 + r l ( y ) l y t

−t

r y −t

⎞ ⎟ ~ ⎟p1 ⎟ ⎠

(12.12)

where r̃ is the fuzzy interest rate and r(y) and l(y) are the right and left side representations of the fuzzy interest rates and 1̃ is (1, 1, 1), and n is the crisp life cycle. Step 2: Assign the alternative that has the lowest initial investment cost as the defender and the next lowest acceptable alternative as the challenger. Step 3: Determine the incremental benefits and the incremental costs between the challenger and the defender.

290

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~ ~ Step 4: Calculate the ΔB ΔC ratio, assuming that the largest possible value for the cash in year t of the alternative with the lowest initial investment cost is less than the least possible value for the cash in year t of the alternative with the nextlowest initial investment cost. The fuzzy incremental BCR is ~ ⎛ ΔB ⎜ ~ =⎜ ΔC ⎜ ⎝

∑ (B ( ) − B ( ) )(1 + r ( ) ) , ∑ (B ( ) − B ( ) )(1 + r ( ) ) ∑ (C ( ) − C ( ) )(1 + r ( ) ) ∑ (C ( ) − C ( ) )(1 + r ( ) ) n

t =0 n

t =0

l y 2t

r y 1t

r y 2t

l y 1t

r y −t

n

l y −t

t =0 n

t =0

r y 2t

l y 1t

l y 2t

r y 1t

l y −t r y −t

⎞ ⎟ ⎟ ⎟ ⎠

(12.13)

~ ~ If ΔB ΔC is equal or greater than (1, 1, 1), Alternative 2 is preferred. Operations on fuzzy numbers were given at the preceding section. Sometimes this method may produce results that are not included at these operations. In such a case defuzzification has to be made. Through the centroid method usually gives the right results, in this chapter this method will be preferred. In the case of a regu~ ~ lar annuity, the fuzzy B C ratio of a single investment alternative is

(

)

(

~ ~ ⎛ A l ( y )γ n, r r ( y ) A r ( y )γ n, r l ( y ) B C = ⎜⎜ , C r(y) C l(y) ⎝

)⎞⎟

(12.14)

⎟ ⎠

where C̃ is the first cost and à is the net annual benefit, and

((

)

)

γ (n, r ) = (1 + r )n − 1 (1 + r )n r .

~ ~ The ΔB ΔC ratio in the case of a regular annuity is

(

)(

)(

)(

l(y) − A1r ( y ) γ n, r r ( y ) A2r ( y ) − A1l ( y ) γ n, r l ( y ) ~ ~ ⎛ A , ΔB ΔC = ⎜ 2 ⎜ C 2r ( y ) − C1l ( y ) C 2l ( y ) − C1r ( y ) ⎝

)⎞⎟ ⎟ ⎠

(12.15)

Step 5: Repeat steps 3 and 4 until only one alternative is left, thus the optimal alternative is obtained. The cash-flow set {At = A : t = 1,2,..., n} consisting of n cash flows, each of the same amounts A, at times 1, 2,…, n with no cash flow at time zero, is called the equal-payment series. An older name for it is the uniform series, and it has been called an annuity, since one of the meanings of “annuity” is a set of fixed payments for a specified number of years. To find the fuzzy present value of a ~ ~ ~ ~ regular annuity At = A : t = 1,2,..., n . The membership function μ x Pn for Pn

{

}

is determined by

( )

( )

~ ~ f ni y Pn = f i y A γ (n, f 3−i ( y ~ r )) for i=1,2 and γ (n, r ) =

( )

(12.16)

(1 − (1 + r ) ) . Both à and r̃ are positive fuzzy numbers. f (.) −n

1

r and f2(.) shows the left and right representations of the fuzzy numbers, respectively.

Fuzzy Investment Planning and Analyses in Production Systems

291

~ ~ In the case of a regular annuity, the fuzzy B C ratio may be calculated as in the following: ~ ~ The fuzzy B C ratio of a single investment alternative is

(

)

(

~ ~ ⎛ A l ( y )γ n, r r ( y ) A r ( y )γ n, r l ( y ) B C = ⎜⎜ , FC r ( y ) FC l ( y ) ⎝

)⎞⎟

(12.17)

⎟ ⎠

where FC̃ is the first cost and à is the net annual benefit. ~ ~ The ΔB ΔC ratio in the case of a regular annuity is

(

)(

)(

)(

l(y) r(y) r(y) A2r ( y ) − A1l ( y ) γ n, r l ( y ) ~ ~ ⎛ A − A1 γ n, r , ΔB ΔC = ⎜ 2 ⎜ FC 2r ( y ) − FC1l ( y ) FC 2l ( y ) − FC1r ( y ) ⎝

)⎞⎟ ⎟ ⎠

(12.18)

Up to this point, we assumed that the alternatives had equal lives. When the alternatives have life cycles different from the analysis period, a common multiple of the alternative lives (CMALs) is calculated for the analysis period. Many times, a CMALs for the analysis period hardly seems realistic (CMALs (7, 13) =91 years). Instead of an analysis based on present value method, it is appropriate to comp are the annual cash flows computed for alternatives based on their own service ~ ~ ~ ~ lives. In the case of unequal lives, the following fuzzy B C and ΔB ΔC ratios will be used:

( ) ( ) ⎞⎟ ( ) ( ) ⎟⎠ ⎛ ⎛ PVB ( ) β (n, r ( ) ) − PVB ( ) β (n, r ( ) ) ⎞ ⎞ ⎜⎜ ⎟, ⎟ ⎜ ⎜ PVC ( ) β (n, r ( ) ) − PVC ( ) β (n, r ( ) ) ⎟ ⎟ ~ ~ ⎝ ⎠⎟ ΔB ΔC = ⎜ ⎜ ⎛ PVB ( ) β (n, r ( ) ) − PVB ( ) β (n, r ( ) ) ⎞ ⎟ ⎟⎟ ⎜⎜ ⎜ ⎜ PVC ( ) β (n, r ( ) ) − PVC ( ) β (n, r ( ) ) ⎟ ⎟ ⎠⎠ ⎝⎝ ~ ~ ⎛ PVB l ( y ) β n, r l ( y ) PVB r ( y ) β n, r r ( y ) , B C =⎜ ⎜ PVC r ( y ) β n, r r ( y ) PVC l ( y ) β n, r l ( y ) ⎝ l y 2 r y 2

l y

r y 2 l y 2

r y

r y

l y

r y 1 l y 1

r y

l y 1 r y 1

l y

(12.19)

l y

(12.20)

r y

where PVB is the present value of benefits, PVC the present value of costs and ⎛ (1 + n )n i ⎞ ⎟. β (n, r ) = ⎜ ⎜ (1 + r )n − 1 ⎟ ⎝ ⎠

(

)

5.4 Fuzzy Payback Period Analysis The payback period method involves the determination of the length of time required to recover the initial cost of investment based on a zero interest rate ignoring the time value of money or a certain interest rate recognizing the time value of money. Let C j 0 denote the initial cost of investment alternative j, and R jt denote

292

C. Kahraman and A.Ç. Tolga

the net revenue received from investment j during period t. Assuming no other negative net cash flows occur, the smallest value of m j ignoring the time value of money such that: mj

∑R

jt

≥ C j0

(12.21)

t =1

or the smallest value of m j recognizing the time value of money such that mj

∑R

jt

(1 + r )−t

≥ C j0

(12.22)

t =1

define the payback period for the investment j. The investment alternative having the smallest payback period is the preferred alternative. In the case of fuzziness, the smallest value of m j ignoring the time value of money such that ⎛ mj ⎜ r1 jt , ⎜ ⎝ t =1

∑

mj

∑

mj

r2 jt ,

t =1

∑ t =1

⎞ r3 jt ⎟ ≥ C1 j 0 , C 2 j 0 , C 3 j 0 ⎟ ⎠

(

)

(12.23)

and the smallest value of m j recognizing the time value of money such that

(

⎛ mj l( y) ⎜⎜ ∑ R jt ⎝ t =1

(1 + r ) ), ∑ ( R r ( y) t

mj

t =1

r ( y) jt

(1 + r ) ) ⎟⎟ ≥ ( ( C l ( y) t

⎞ ⎠

2 j0

− C1 j 0 ) y + C1 j 0 , ( C2 j 0 − C3 j 0 ) y + C3 j 0

)

(12.24) define the payback period for investment j, where rkjt is the k.th parameter of a triangular fuzzy R jt ; C kj 0 is the k.th parameter of a triangular fuzzy C j 0 ; R ljt( y ) is the left representation of a triangular fuzzy R jt ; R rjt( y ) is the right representation of a triangular fuzzy R jt . If it is assumed that the discount rate changes from

(

one period to another, 1 + r r ( y )

∏ (1 + r ) , respectively. t

t ′=1

)

t

(

and 1 + r l ( y )

)

t

will be

∏ (1 + r ) t

t ′=1

r( y) t t′

and

l ( y) t t′

Ranking is made with the method offered in the fuzziness of the investment section.

6 Applications A production firm desires to choose the best automated system part to utilize in their production system. Three single-station automated cells part alternatives are being considered for use. Alternative A involves picking items manually from floor space.

Fuzzy Investment Planning and Analyses in Production Systems

293

With alternative B, items are picked manually from a horizontal carousel conveyor; hence, essentially no walking is required by the order picker. With alternative C, items are picked automatically by an automatic item retrieval robot. The alternatives have different space and labor requirements. Likewise, they have different acquisition, maintenance, and operating costs. For the study period of 10 years, the fuzzy estimates for the alternatives are given as follows: Table 12.1 Fuzzy values for single-station automated cells part selection problem. Ã(1000×$)

B̃(1000×$)

C(̃1000×$)

Initial cost:

(1,000; 1,150; 1,300)

(1,200; 1,300; 1,400)

(1,700; 1,800; 1,900)

Salvage value:

(700; 750; 800)

(800; 850; 900)

(900; 950; 1,000)

Annual receipts:

(625; 700; 775)

(700; 750; 800)

(750; 800; 850)

Annual disbursements: (375; 425; 475)

(350; 375; 400)

(225; 275; 325)

Based on a MARR of %8, determine which alternative is preferred, using the methods given below.

6.1 Fuzzy Present Worth The equations were given in the appertaining section. In this part, one of alternatives’ explanations will be made, and then the other alternatives’ solution will be presented. 10 A% Aj FPNWA = A% A0 ⊕ ∑ j j =1 (1 + 0.08) ⎛ 150, 000 275, 000 400, 000 ⎞ = (−1, 300, 000; −1,150, 000; −1, 000, 000) ⊕ ⎜ , , ⎟⊕ 1.08 1.08 ⎠ ⎝ 1.08 ⎛ 150, 000 275, 000 400, 000 ⎞ ⎛ 700, 000 750, 000 800, 000 ⎞ , , , , K⊕ ⎜ ⎟ ⎟⊕⎜ 10 1.0810 1.0810 ⎠ ⎝ 1.0810 1.0810 1.0810 ⎠ ⎝ 1.08 = (30, 747;1, 042, 668; 2, 054, 587), FPNWB = (983, 579;1, 609, 995; 2, 236, 411) , FPNWC = (1, 368, 659; 2,162, 827; 2, 956, 994) . Ranking these fuzzy alternatives is made with the ranking approach explained in the previous section. From Eq. (12.1) we find the preference index: I wFPNW A f FPNW B = 0.214 . Therefore, we can infer that FPNW A is smaller than

(

)

FPNW B with a possibility of 0.214. Likewise, from Eq. (12.1) it is obvious that FPNW B is smaller than FPNWC with a possibility of 0.187 and FPNW A is smaller than FPNWC with a possibility of 0.072. In consequence, the ranking of ~ ~ ~ these alternatives can be presented as C f B f A .

294

C. Kahraman and A.Ç. Tolga

Fig. 12.4 FPNW values of alternatives.

6.2 Fuzzy B/C Ratio Analysis In this fuzzy evaluation method, the steps those were explained in the previous section will be applied. At the first step, all the alternatives’ overall fuzzy measures of benefit-to-cost ratios are going to be calculated as seen below: Fuzzy present value of benefit includes all discounted annual receipts and salvage value. Furthermore, fuzzy present value of cost contains initial cost and annual disbursements. ⎛ 700,000 750,000 800,000 ⎞ ⎛ 625,000 700,000 775,000 ⎞ ~ ⎟⊕⎜ ⎟K BA = ⎜ ; ; ; ; ⎜ (1.08)10 (1.08)10 (1.08)10 ⎟ ⎜ (1.08)10 (1.08)10 (1.08)10 ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ 625,000 700,000 775,000 ⎞ ⎜ ⎟ = (4,518,036;5,044,452;5,570,868) ; ; ⎜ (1.08)1 (1.08)1 (1.08)1 ⎟ ⎝ ⎠ with same discount rate fuzzy present value of cost can be found: ~ C A = (3,516,281;4,001,785;4,487,289) ~ ~ B A C A = (1.24148;1.26055;1.28498)

(

)

With Eq. (12.1), it has to be checked the superiority of this ratio according to (1,1,1). I wB~ C~ f ~1 = 1.000 result can be easily computed from this equation and A A ~ ~ ~ it shows that B A C A f 1 . In consequence, alternative A is accepted. With the same processes ratios of the alternatives and their preference index values are given successively: ~ ~ B B C B = (1.41648;1.42188;1.42808) , I wB~ C~ f ~1 = 1.000 , B B ~ ~ BC CC = (1.51117;1.59332;1.69777) , I wB~ C~ f ~1 = 1.000 .

( (

C

C

) )

Now, we have to determine the incremental benefits and the incremental costs between the alternatives. First, incremental analysis between alternative A and B is going to be made. Then incremental analyses between alternative B and C, and

Fuzzy Investment Planning and Analyses in Production Systems

295

Table 12.2 Fuzzy benefit and cost values of alternatives A and B. Ã

B̃

Benefits

(4,518,036; 5,044,452; 5,570,868)

(5,067,612; 5,426,276; 5,784,939)

Costs

(3,546,281; 4,001,785; 4,487,289)

(3,548,528; 3,816,281; 4,084,032)

between alternative A and C will follow. Benefits and costs values of alternatives A and B are given in Table 12.2. ~ ~ As written in theoretical section, here, ΔB ΔC division produces insignificant result. Defuzzified values, ΔB and ΔC , obtained from incremental analysis between alternatives A and B are as follows: ΔB = 381,823.42 , ΔC = 185,504.07 and the ratio ΔB B − A ΔC B − A = 2.0583 can be found easily. Though this ratio is greater that 1, one can say that alternative B is preferred. With same analysis, the incremental analysis between alternative C and A produces the ratio ΔBC − A ΔCC − A = 2.1420 which means alternative C is preferred. Finally, the incremental analysis between alternative C and B produces the ratio ΔBC − B ΔCC − B = 2.2328 which means alternative C is preferred. From ~ ~ ~ these analyses we can find ranking as C f B f A which is same with FPNW analysis.

6.3 Fuzzy Payback Period Analysis However the assumptions of payback period analysis do not reflect the real applications very well, it is mostly used in practical life because of its easily application. Fuzzy payback period analysis’ formulations were given in the related section. In this section, the numeric application of these formulations to the singlestation automated cells part selection problem will be investigated. For alternative A, annual net cash flow can be determined as (250,000; 275,000; 300,000). As the logic of payback period analysis drives us to sum up annually from the beginning, discount beginning from year 1 has to be applied. As anyone can see from Table 12.3, the payback period is between year 5 and year 6. As in crisp analysis, fuzzy payback period analysis is the same except compari~ son between fuzzy values and 0 . This comparison is made with the Eq. (12.1). The computation is applied all the values beginning from year 1 till the fuzzy val~ ues overbear the fuzzy number 0 . From Eq. (12.1) anyone can calculate ~ ~ ⎛ ⎞ I ⎜ w ~ ~ ⎟ = 0.313 value for year 5 which means CPNV5 p 0 . With the same ⎝ CNPV5 f 0 ⎠

⎛ ⎞ calculation for year 6, I ⎜ w ~ ~ ⎟ = 0.852 value can be computed that means CNPV f 0 6 ⎝ ⎠

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Table 12.3 Fuzzy Payback Period Analysis for alternative A. MARR: 12%

Alternative Ã

A0

(1,000,000; 1,150,000; 1,300,000)

Salvage Value

(700,000; 750,000; 800,000)

Annual receipts

(625,000; 700,000; 775,000)

Annual disbursements

(375,000; 425,000; 475,000)

Net CF

(250,000; 275,000; 300,000)

0

Cumulative NPV (-1,300,000; -1,150,000; 1,000,000)

1

(231481.5; 254926.6; 277777.8)

(-1,068,519; -895,370; -722,222)

2

(214334.7; 235768.2; 257201.6)

(-854,184; -659,602; -465,021)

3

(198458.1; 218303.9; 238149.7)

(-655,726; -441,129; -226,871)

4

(183757.5; 202133.2; 220509.0)

(-471,968; -239,165; -6,361.9)

5

(170145.8; 187160.4; 204175.0)

(-301,822; -52,004; 197,813) < 0

6

(157542.4; 173296.6; 189050.9)

(-144,280; 121,291.9; 386,863.9) >

7

(145872.6; 160459.9; 175047.1)

8

(135067.2; 148573.9; 162080.7)

9

(125062.2; 137568.5; 150074.7)

10

(115798.4; 127378.2; 138958.0)

~

~ 0

~ ~ CPNV6 f 0 . For the calculation of crisp year value we have to defuzzify cumulative net present values by centroid index method. Exact year value n=5.3 is found by interpolation between year 5 and year 6. Similarly, exact year values for alternatives B and C could be computed. The same interpolation method is used after defuzzifications. For alternative B, nB equals 1.94. For alternative C, nC equals 2.59. Hereby the shortest payback is con~ ~ ~ venient, the ordering can be found like: B f C f A which is different from fuzzy net present worth and fuzzy benefit-cost ratio methods.

7 Conclusions In this chapter, fuzzy investment analysis in production system is discussed. The main topic of the investment analysis is the evaluation and comparison of the production systems investments. As discussed earlier, production systems consist of coexistent parts; people, procedures and equipments. Production systems are divided into two categories named facilities and manufacturing support systems and

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they are also discussed in this chapter. Then the components of investment planning (economical, technical and financial analyses) are investigated. Fuzziness of the investment followed the investment planning section. Fuzzy investment analysis techniques which consist of fuzzy present worth, fuzzy annual worth, fuzzy rate of return analysis, fuzzy B/C ratio, fuzzy replacement analysis and fuzzy payback period techniques are performed to evaluate or to choose the investment project alternatives. Then applications of three basic techniques to case problems are presented at the last section.

References Baas, S.M., Kwakernaak, H.: Rating and ranking multiple aspect alternatives using fuzzy sets. Automatica 13, 47–58 (1977) Blank, L.T., Tarquin, A.J.: Engineering economy, 4th edn. McGraw Hill Inc., New York (1998) Buckley, J.J.: The fuzzy mathematics of finance. Fuzzy Sets and Systems 21, 257–273 (1987) Chan, F.T.S., Chan, M.H., Tang, N.K.H.: Evaluation for technology selection. Journal of Material Technology 107, 330–337 (2000) Chang, W.: Ranking of fuzzy utilities with triangular membership functions. In: Proceedings of the International Conference of Policy Anal. and Inf. Systems, pp. 263–272 (1981) Chen, S.J., Hwang, C.L.: Fuzzy Multiple Attribute Decision Making Methods and Application. Springer, New York (1992) Chiu, C.Y., Park, C.S.: Fuzzy cash flow analysis using present worth criterion. The Engineering Economist 39(2), 113–138 (1994) Dimitrovski, A., Matos, M.: Fuzzy present worth analysis with correlated and uncorrelated cash flows. In: Kahraman, C. (ed.) Fuzzy Engineering Economics with Applications. Springer, Heidelberg (2008) Dubois, D., Prade, H.: Ranking fuzzy numbers in the settings of possibility theory. Information Sciences 30, 183–224 (1983) Groover, M.G.: Automation, production systems, and computer-integrated manufacturing, 3rd edn. Pearson Education Inc., New Jersey (2008) Güvemli, O.: Yatırım projelerinin düzenlenmesi ve değerlendirilmesi. Çağlayan Kitabevi, Istanbul (1979) (in Turkish) Hubbard, D.: How to measure anything: finding the value of intangibles in business. John Wiley & Sons, New Jersey (2007) Jain, R.: Decision-making in the presence of fuzzy variables. IEEE Transactions on Systems, Man, and Cybernetics 6, 693–703 (1976) Kahraman, C., Kaya, İ.: Fuzzy benefit/cost analysis and applications. In: Kahraman, C. (ed.) Fuzzy Engineering Economics with Applications. Springer, Heidelberg (2008) Kahraman, C., Ruan, D., Tolga, E.: Capital budgeting techniques using discounted fuzzy versus probabilistic cash flows. Information Sciences 142(1-4), 57–76 (2002) Kahraman, C., Tolga, A.C.: An alternative ranking approach and its usage in multicriteria decision making. International Journal of Computer Intelligence Systems 2(3), 219–235 (2009)

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Kaufmann, A., Gupta, M.M.: Fuzzy mathematical models in engineering and management science. Elsevier, Amsterdam (1988) Kuchta, D.: Fuzzy capital budgeting. Fuzzy Sets and Systems 111, 367–385 (2000) Kuchta, D.: A fuzzy model for R&D project selection with benefit, outcome and resource interactions. The Engineering Economist 46(3), 164–180 (2001) Kuchta, D.: Fuzzy rate of return analysis and applications. In: Kahraman, C. (ed.) Fuzzy Engineering Economics with Applications. Springer, Heidelberg (2008a) Kuchta, D.: Optimization with fuzzy present worth analysis and applications. In: Kahraman, C. (ed.) Fuzzy Engineering Economics with Applications. Springer, Heidelberg (2008b) Liou, T.S., Chen, C.W.: Fuzzy decision analysis for alternative selection using a fuzzy annual worth criterion. The Engineering Economist 51, 19–34 (2007) Park, C.S., Sharp-Bette, G.P.: Advanced engineering economics. John Wiley and Sons Inc., New York (1990) Rebiasz, B.: Fuzziness and randomness in investment project risk appraisal. Computers & Operations Research 34(1), 199–210 (2007) Sewastjanow, P., Dymowa, L.: On the fuzzy internal rate of return. In: Kahraman, C. (ed.) Fuzzy Engineering Economics with Applications. Springer, Heidelberg (2008) Sharpe, W.F., Alexander, G.J., Bailey, J.V.: Investments, 6th edn. Prentice Hall, New Jersey (1999) Ward, T.L.: Discounted fuzzy cash flow analysis. In: Proceedings of 1985 Fall Industrial Engineering Conference, Institute of Industrial Engineers, pp. 476–481 (1985) Yager, R.R.: On choosing between fuzzy subsets. Cybernetics 9, 151–154 (1980) Zadeh, L.A.: Fuzzy sets. Information and Control 8(3), 338–353 (1965)

Chapter 13

Fuzzy Production and Operations Budgeting and Control Dorota Kuchta1

Abstract. Basic information about crisp production and operations budgeting is given, as well as selected information about fuzzy numbers. Then some ideas of how to use the fuzzy approach in production and operations budgeting and control are presented. Several numerical examples illustrate the reasoning.

1 Introduction Budgeting means in fact planning and it always refers to the future. And the future is always uncertain. That is why we never have complete information to prepare our plans, they are always based on assumptions. Putting crisp numbers in our quantitative plans almost always means putting just one number from among many ones which according to our incomplete knowledge seem possible (an average? a randomly selected number?). The fuzzy approach offers the possibility to put into our plans the whole knowledge we have, the incomplete knowledge, sometimes based on intution and experience, but still a knowledge, which gets lost in the crisp (traditional) approach to budgeting. In the present paper we will present the idea of how the fuzzy approach can be used in production and operations budgeting. First of all, we will present basic, selected information about crisp budgeting. It is a vast area, thus we had to make a choice. Also as far as the information about fuzzy arithmetic is concerned, it is limited to what is really essential to follow the last section of the paper, where fuzzy production and operations budgeting is discussed.

2 Selected Information about Crisp Production and Operations Budgeting and Control In this section we will present some information about crisp production budgeting and budget control, having in mind that almost all the crisp values discussed here may need to be fuzzyfied – which will be the topic of Section 4. This is because budgeting means planning the future – and the future is always to some extent Dorota Kuchta Institute of Organization and Management, Wroclaw University of Technology, Poland C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 299–327. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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uncertain. Thus planning with crisp values is always biased with a high error possibility. Nevertheless, before going over to fuzzy production planning and control, we have to know how this is done in the crisp case.

2.1 Production and Operations Budgeting in the Context of General Budgeting Budget can be defined as “quantitative model of the expected consequences of the organisation’s short-term operating activities” (Atkinson et al. 1995). The production budget or production plan (we will consider the two terms as synonyms) is a part of the so called master (or general) budget of the whole company, which can be illustrated by the following scheme:

Fig. 13.1 Elements of budgeting (Atkinson et al. 1995)

As we can see, the input for the production plan is constituted by the information about how much the company is going (will be able) to sale in the period in consideration, what it has and wants to have as inventory (among others of the products it manufactures) and how much it is able to manufacture in the considered period, taking into consideration both the material and human capacities of the company. The outputs are the needs in terms of material resources and working force, as well as in terms of other expenses. Those needs, thus the output of the production plan, can be summarized as production cost – as the usage in the

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production process of material resources, of working force and of all kinds of services, expressed in monetary terms, constitute by definition the cost of production. Thus, we can illustrate the production budgeting in the following simplified way:

Fig. 13.2 Simplified illustration of production budgeting

Thus we can assume that the output of production budgeting is production cost. We will distinguish between the direct cost – which will mean the cost which can be linked to individual products, and the indirect cost, which is the common cost, concerning all the products together. If the budgeted cost is not crisp, if it is fuzzy, which may happen in case the input to the production plan is fuzzy (see Section 4), this fuzziness may be than transmitted to the output of the whole budgeting process (Fig.13.1), thus also to the projected financial statements of the company. Some information about fuzzy financial statements can be found in (Rhys 1992) and (Kuchta 2000, 2001). Let us now pass to some selected details concerning the process of production budgeting. Before doing so one remark: budgeting periods are usually one year or six months, we will consider in our examples a much shorter budgeting period – one week, in order to simplify the examples.

2.2 Constructing Production and Operations Budgets It is not easy to identify all the necessary input mentioned in Fig. 13.2, this is even more difficult than the computation of the output. Thus in Section 2.2.1 we will present selected models which can be used to determine both the input and the output from Fig. 13.2. 2.2.1 General Models The absolutely most basic input for the production planning is the information about the number of products to be produced. Let us assume that the company manufactures N types of products, and let us denote by np i , i=1,…,N, the units number of the i-th product to be manufactured in the budgeting period. The values of np i , i=1,…,N have to be found on the basis of the market demand, of the current state and the of the required end state of the inventory of

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finished goods and of the capacities of the company in the budgeted period. Having those data, it is wise to find optimal values of np i , i=1,…,N, in the sense of the criterion or criteria that are most important to us. Here the whole vast domain of linear (one- and multicriteria) programming should be exploited. We will just present two models to give a general idea how the values of np i should be determined. In each case the model has to be adopted to the very situation, and sometimes it will be easier go determine the values of np i without any models, just on the basis of the market information – in such cases these values should be very often expressed in the form of fuzzy numbers (Section 3). We can apply linear programming models to find out which values of np i , i=1,…,N, would give us the highest overall profit, the lowest production etc., while all the requirements and constraints are fulfilled. The requirements and constraints are the bounds for the market demand for individual products, the upper bound for the usage of machine hours, labour force, materials etc, the inventory input and requirements etc. Let us consider the following example situation: Model 1 (Budgeting input) Each unit of the i-th product requires

MCH i machine hours (one machine is used), MTi material units (one material type is used) and LBH i labour hours

(one labour force type is used). In the budgeting period there are upper bounds on the usage of those resources, thus we have at our disposal in total MCHB machine hours, MTB material units and LBHB labour hours. The beginning state of the inventory of each product equals BINVi i=1,…,N, the required state of the inven-

EINVi i=1,…,N. The expected demand for the individual products is at the most Di . The expected selling price of one unit of each product unit in the budgeting period equals Pi , i=1,…,N. We want to find the values of np i , i=1,…,N such that all the bounds and requirements tory at the end of the budgeting period equals

are met and the total sales value is the highest possible. We can than formulate the following model, which can be than solved by e.g. the well known simplex algorithm, giving us the required values of np i , i=1,…,N that should be manufactured in the budgeting period. N

∑ Pi ⋅ npi → max i N

∑ MCH i ⋅ npi ≤ MCHB i

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N

∑ MTi ⋅ npi ≤ MTB i N

∑ LBH i ⋅ npi ≤ LBHB i

BINVi + npi − EINVi ≤ Di More general models, enabling to model more complex situations, e.g. multiperiod budgeting, are presented e.g. in (Gen et al. 1992). For multiperiod budgeting we can also use transportation models, like the following one: Model M2 (Budgeting input) (Evans et al. 1987) Let is consider P budgeting periods in a company which manufacture one product type (N=1). We are searching for each period the number of product units to be

np1p , p=1,…,P. The maximal demand for the product in the p respective periods is supposed to be known ( D1 , p=1,…,P). We also suppose to manufactured, thus

know the manufacturing cost of each unit of the product in the respective periods1,

MANC1p (p=1,…,P). What is more, we assume to know the maximal p production capacities (in products units, MCAP1 (p=1,…,P)) in each budgeting

denoted as

period, which in some periods may be smaller than the demand, in some bigger2 – in such cases we may plan to manufacture earlier the products destined to be sold later and held them in the inventory in between. To do so, we have to be able to estimate the cost of keeping one unit of the product in the inventory between two consecutive budgeting periods p and p+1, , p=1,…,P-1, let us denote it as

IC1p ,

p=1,…,P-13. Then we can construct a transportation model4, which, when we apply the corresponding transportation algorithm (e.g. Anderson 2007), will allow us to find such values of

np1p , p=1,…,P for which the total manufacturing and in-

ventory cost will be minimal. The model takes the following form:

1

2

3

4

The unitary manufacturing cost may differ from period to period even for identical products, e.g. because of fixed cost, which, even they do not change from period to period in total, when allocated (divided between) to product units, do change if the number of units manufactured change. The production capacity depends on many factors, e.g. on the year period (a traditional holiday period or not); The reasons for these cost to change from period to period is similar to those responsible for the change of unitary manufacturing cost. Here we do not talk about transportation, although some transportation cost may be included in the manufacturing or inventory cost, but the model – as well as the corresponding algorithm – holds such a name traditionally, because of its background.

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Table 13.1 Transportation table for Model 2 p=1 p=1

p=2 1 1

… 1 1+

MANC MANC

…

IC11

p=P-1

p=P 1 1+

MANC

MANC11 +

P −2

P −1

∑ IC1p

∑ IC1p

p =1

p=2

•

MANC12

…

p =1

MANC12 + MANC12 + P −2

∑ IC1p

p=2

…

…

…

p=P-1

•

•

…

MCAP12

P −1

∑ IC1p

…

MCAP11

p =2

…

… P −1 1

MANC

… P −1 + 1

MANC

MCAP1P −1

IC1P −1 p=P

•

•

…

•

MANC1P

D11

D12

…

D1P −1

D1P

MCAP1P

The individual elements (row p=s and column p=r) represent the cost of the following event: a product unit will be manufactured in period s and sold in period r. If sr, the event mentioned above is not possible, which is expressed by the black circle in the corresponding box5 p

Applying the transportation algorithm to the above model, we get the np1 , p=1,…,P for which the total cost of manufacturing and inventory is minimal. The model can of course be generalized to the case of N>1. Even if we already know the number of units of individual products to be manufactured, in order to begin with production budgeting we have to know all the data about the production process. These data comprise among many others: 1. that linked to the individual products (i=1,…,N): •

mqil - the quantity of the l-th material needed for the production of one unit of

the i-th product, l=1,…,M; • ap il - the acquisition price of one unit of the l-th material needed for the production of one unit of the i-th product; 5

We can still consider such an event, if it is physically possible (i.e. the customer would accept it) for a product unit to cover the demand from an earlier budgeting period, possibly with a penalty for the delay added to the manufacturing cost.

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•

lqis - the number of direct labour hours of the s-th labour force needed for the

•

production of one unit of the i-th product, s=1,…,R; lris - the wage rate per hour of the s-th labour force used for the production of

one unit of the i-th product, s=1,…,R; • mriw - the rate for one hour of the processing of the i-th product on the w-th machine, w=1,…,T • pt iw - the processing time of one unit of the i-th product on the w-th machine, •

w=1,…,T; nbi - number of production batches of the i-th product

•

lbci - the cost of launching one production batch of the i-th product 2.

that common for all the products:

• Mks – the makespan, i.e. the length of the period in which the production of all the units of the all the products on all the machines treated together is started and ended • ilrh – indirect labour rate, thus the rate per one hour of the labour which is liked to both products at the same time, e.g. the wage rate for one hour of the makespan. • Dep – the depreciation value6 for the whole budgeting period. Not only the number of units to be manufactured, but also some other input data from the above list has to be calculated, and sometimes requires more complex calculation algorithms or will have to be determined in an intuitive way, possibly as fuzzy numbers. In the above list we have the makespan, which should be calculated (in order to be optimised) using special algorithms, so that it is as short as possible. The choice of the algorithm will depend on the properties of the production process. In the literature (Evans et a. 1987, Gładysz and Kuchta 2003, Roseaux 1985, 1991) several cases are distinguished • The flow shop – the case each item of each product has to be processed on each of the machines, but the order in which the items go through the machines is prescribed and the same for all the product types (there are distinguished two subcases: one in which each item, having left one machine, has to enter the next one immediately, and the other one, when the items can wait in a queue to enter the next machine) • The job shop – the case in which the order of the machines is prescribed, but may be different for each product type • The open shop – the case when each item has to be processed on several machines, but the order in which the items will enter on the machines does not matter.

6

Depreciation is the value by which the worth of the machines and other assets is diminished in the budgeting period.

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• Having this kind of information, together with the processing times

pt iw and

the number of batches

nbi - we can - using corresponding algorithms – deter-

mine the optimal schedule according to which individual items should be processed on the machines and the optimal makespan (more details will be given in Example 1). Having such or similar data as the above list shows, be can calculate the output required in Fig. 13.2, thus the planned cost linked to the production in the budgeting period. Model M3 (budgeting output) The direct production cost DC is composed of the following elements (for i=1,…,N) • Material direct cost:

M

mtdci = np i ⋅ ∑ mq il ap il l =1

• Labour direct cost:

R

ldci = np i ⋅ ∑ lqis lris s =1

• Machine direct cost: mchdci • Batches launching cost:

T

= npi ⋅ ∑ pt iw mriw w =1

bci = nbi ⋅ lbci

The indirect production cost IC will be composed of (it is determined for the whole production of all the products, it may be later allocated between individual product types, in order to calculate the unitary cost of the products, but basically the indirect cost concerns several products at the same time – (Atkinson et al. 2001)): • Deprecation cost DEC (calculated in the accounting system of each company) • Indirect labour cost: ILR=Mks·Ilrh The total production cost TC will than be equal to DC+IC. Of course, the models used in praxis can be and are usually more complex, they comprise other types of cost, however, their idea is similar. There is only one aspect of budgeting which is not taken into account in the above mode and which is important and should be considered: the cost of unused capacity (CUC). For example, the depreciation – like many other costs in real cases, above all those wages which are fixed in each budgeting period independently of the actual time the respective persons are actually needed and occupied – is incurred for the whole budgeting period, although the machines may not be used all this time. The cost linked to this unused time is considered to be important, especially because it makes the products which are actually manufactured more expensive per unit

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(the cost of unused capacity should later be allocated to the manufactured products and the customer has to pay for it if the company wants to be profitable) and is also calculated and analysed, which will be illustrated in the example further on. 2.2.2 Numerical Example Let us consider the following example: Example 1 Let us assume that a company manufactures 5 types of products (N=5) and that all the information we posses has led to the conclusion that the number of units we should manufacture in the budgeting period are as follows: np1 = 4, np 2 = 4, np 3 = 4, np 4 = 4, np 5 = 2 . Let us suppose there is just one material type and one work force type needed (M=R=1, with ap11 = ap 21 = ap 31 = ap 41 = ap 51 = 1$ and

lr11 = lr21 = lr31 = lr41 = lr51 = 1$ )

and

we

have:

mq11 = mq 21 = mq31 = mq 41 = mq51 = 1 , lq11 = lq 21 = lq 31 = mq 41 = mq 51 = 1 .

Each of the products has to be processed on two machines M1 and M2: first on machine M1, than on M2, each product, having left M1, can wait to be processed on M2. We know the processing times on both machines: pt11 = 1, pt12 = 2 ,

pt 21 = 2,5, pt 22 = 1 , pt 31 = 1, pt 32 = 1 , pt 41 = 1, pt 42 = 2 , pt 51 = 1, pt 52 = 1 . What is more, the person responsible for the machines operations is paid only for the time when at least one machine is in operation, the wage rate per hour (ilr ) is 5$, and the weekly depreciation (Dep) of both machines is equal to 50$. It is planned that each product will be produced in two equal batches ( nb1 = nb2 = nb3 = nb4 = nb5 = 2 ), so that the customer can receive the ordered goods as soon as possible. The cost of launching a new butch ( lbci ) equals 10$ for i=1,2,.3,4,5. There is also a cost rate for each hour each machine is actually used, it is planned to be the same one for each product unit: mr11 = mr21 = mr31 = mr41 = mr51 = 1 and mr12 = mr22 = mr32 = mr42 = mr52 = 2 . The maximal working time in the week in the company is 40 hours. In order to work off the budget (and more exactly its indirect cost part) we have to know how much time both machines will be actually used. Assuming two equal batches for each product, we apply the algorithm corresponding to the given situation7 and get the following optimal (as far as the makespan (Mks) is concerned) schedule:

7

It is a flow shop with two machines in which the product items can wait between leaving one machine and entering another one – in such a case the Johnson algorithm (Evans et al. 1987) gives the optimal solution.

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Fig. 13.3 Planned schedule for Example 1.

The numbers at the top of each rectangular are the numbers of the products, the numbers below are the numbers of the batches. It follows that the production is planned to fill almost the maximal working time of the week, as it is planned to take 35 hours (Mks=35) and this is the optimal makespan for the given data. Now, knowing the schedule according to which the machines are supposed to work we are in position to elaborate the budget The production budget will be as follows: I)

Direct cost (DC) i) Material cost:

mtdc1 = mtdc 2 = mtdc 3 = mtdc 4 = 4

·1·1$=4$,

mtdc 5 = 2·1·1$=2$ ii) Labour cost: ldc1 = ldc 2 = ldc 3 = ldc 4 = 4·1·1$=4$, iii) Machines

direct

cost:

ldc5 = 2·1·1$=2$

mchdc1 = 4·(1·1$+2·2$)=20$, mchdc3 = 4·(2·1$+1·2$)=16$,

mchdc 2 = 4·(1·1$+2,5·2$)=24$, mchdc 4 = 4·(1·1$+2·2$)=20$, mchdc5 = 2·(1·1$+1·2$)=6$ iv) Batch start cost: bc1 = bc 2 = bc3 = bc 4 = bc5 = 2·10$=20$ DC=222$

II) Indirect cost (IC): i) Depreciation (DEP) 50$ ii) Labour cost of the responsible of the machines (ILR): 35·5$=175$ IC=225$ III) TC=447$ The cost of unused capacity (CUC) is the cost of the deprecation allocated to the five hours the machines will not be working, thus CUC=6,25$. It would have also comprised the cost of the person responsible for the machines if this person had been paid for all the 40 hours of the week. However, in our case this person is paid only for the time when at least one of the machines is working, so this person does not cause any CUC. We should not consider the idle times of the two

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machines within the planned makespan as unused capacity, because it is difficult to avoid such periods, and certainty we cannot do it simply through increasing the manufactured number of product items – while such strategy will surely help to avoid the CUC of depreciation. Let us now pass to the problem of controlling the production budget – because it is actually normal that reality is never identical to what we have planned. We will use the following convention: if a symbol x stands for a budgeted (planned) value, the symbol will stand for the actual (realised) value.

2.3 Production and Operations Budgets Control 2.3.1 General Models

When it comes to the budget control, we have to find out what the difference -TC is. In this place we have to clarify one terminology problem. Traditionally, the planned (budgeted) values are usually subtracted for the actual ones, which in case of cost gives a positive value in a negative case – the actual cost are higher than the planned ones. We will call such a variance a negative one, although mathematically it is positive. And we will call positive a variance which mathematically is negative – the actual cost are lower than the planned ones. Thus, at the end of the budgeting period, we have to find out for the variance -TC whether it is negative significant and if so, why it has come into being and what should be improved in order not to have it again in the future. It is also important to use the motivating role of the budgets and to check the achievement degree of individual goals of various organisational units and individual persons, always having in mind how the goals were set. If the were set as realistic goals, thus if while budgeting all the parties involved agreed that the values in question were possible to attain, then each negative variance has to be analysed thoroughly. However, sometimes the goals are set as ambitious goals, in fact in the very moment unrealistic, seen more as long term goals for a far away future. In such cases only relatively high variances will have to be analysed seriously – so that people are not blamed for things it has been clear from the beginning they cannot achieve. Thus, a high negative variance -TC has to be disaggregated into such variances which can be analysed in such a way that exact reasons and responsibilities of the high value of -TC are identified. This means above all that the individual items of the budget, thus those in the list from Section 2.2.1, have to be analysed separately. Some general approaches to the analysis of variances have been summarised or proposed by Kloock and Schiller (1997) and Pollard (1986). Kloock and Schiller (1997) consider an arbitrary cost position in the budget, let us call it CPj (j=1,…BP), where BP is the number of cost positions in the budget, and take into account all the factors influencing it, let us denote them

F jk , k = 1,..., B j ,

j=1,…BP, where B j is the number of factors influencing the j-th budget cost

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D. Kuchta k

position. The values < F j > may of course differ for the values that we have the equality

(

CPj = CPj F j1 , F j2 ..., F j

Bj

)

F jk . We can say

Then Kloock and Schiller (1997) discuss and criticize various approaches to the analysis of the variances (for j=1,…BP):

(

)− CP (F , F ,..., F )

VAR j = CPj F j1 , F j2 ..., F j

Bj

1 j

j

Bj j

2 j

(13.1)

They propose to disaggregate formula (13.1) in the following way:

( )+ R

VAR j = VAR j (F j1 ) + VAR j (F j2 ) + ... + VAR j F j

Bj

(13.2)

j

where R j is a so called joint variance, which should be analysed separately, but it shows rather the joint influence of more than one factor and it is more informative to analyse

( )

VAR j F jk , k = 1,..., B j , which show the influence of only one fac-

tor and is defined as follows:

( )

(

)− CP ( F

VAR j F jk = CPj F j1 , F j2 ,..., F jk ,..., F j

Bj

Thus, in the formula for

j

1 j

Bj

, F j2 ,..., F jk ,...., F j

) (13.3)

( )

VAR j F jk the values of all the influencing factors are

set to the actual values except for the k-th factor, which is present in the minuend with its actual value and in the subtrahend with its budgeted value. Kloock and Schiller (1997) also mention that sometimes it might be useful for the decision maker to consider another definition of the partial variances, i.e. the following one:

( )

(

Bj

k 1 2 VAR j F jk = CPj F j , F j ,..., F j ,..., F j

) − CP (F , F j

1 j

2 j

Bj

,..., F jk ,..., F j

) (13.4)

Kloock and Schiller (1997) give some advice when to use the one and the other approach, we will concentrate on approach (13.3)8. The idea of both approaches is the same: we want to analyse variances with respect to just one influencing factor, as when we have a variance influenced by more than one factor, it is not clear which factor has “behaved” well and which one not nor what should be improved and what does not need to. Let us apply the above approach to our budgeting scheme (Section 2.2.1). The first budget cost positions are material direct cost values of individual products, thus mtdci , i=1,…,N. Each of them is influenced by 2M+1 factors: the number of items of the product in question ( np i ), the acquisition price of the l-th 8

There is one rather strong reason for us to choose approach from (13.3) – the corresponding formula contains only one planned value, thus – when we consider the fuzzy approach in Section 3 – it will contain only one fuzzy value, which allows us to avoid some serious practical problems linked to arithmetical operations on fuzzy arguments, discussed in Section 3.

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material type for the i-th product ( ap il ) and the quantity of l-th material needed for one item of the i-th product ( mq il ) The analysis of the variances

VAR(mtdci ) = mtdci − mtdci , i=1,…,N,

will be carried out on the basis of the following formula:

VAR(mtdci ) = VAR(mtnp i ) + ∑ VAR(ap il ) + ∑ VAR(mqil ) + R(mtdc i ) (13.5) where

M

M

l =1

l =1

R(mtdci ) will usually be neglected and:

M M M (13.6) VAR(mtnpi ) = np ⋅ ∑ mq il ap il − np i ⋅ ∑ mq il ap il = ( np i − np i ) ⋅ ∑ mq il ap il i l =1

l =1

l =1

This variance will allow us to asses the influence of the difference between the planned quantity of product items and the actual quantity on the total budget variance. We can also consider:

( ap

VAR(apil ) = npi mqil

− api1 ) , l=1,…,M

i1

These variances make visible the influence on the total budget variance of the difference in the purchase price of the individual materials. In an analogous way we get the variances making clear what size of budget problem was caused by the difference in the unitary material usage of individual products:

VAR(mqil ) = npi apil

( mq

i1

− mqi1 ) , l=1,…,M

(13.7)

In an analogous way we get the variances to be analysed in case of the direct labours cost. We consider the following variances: R

R

s =1

s =1

VAR(ldci ) = VAR(lnpi ) + ∑ VAR (lris ) + ∑ VAR(lq is ) + R (ldci ) (13.8) where R (ldci ) will usually be neglected,

VAR(lnpi ) = ( npi − np i ) ⋅ ∑ lqis lris R

(13.9)

s =1

will show the effect of the difference in the number of units produced,

VAR(lris ) = npi lqis

( lr

is

− lris ) , s=1,..,R

(13.10)

will emphasize the influence of the problem of the difference in the payment of the human resources executing direct work on individual products and

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D. Kuchta

VAR(lqis ) = npi lris

( lq

− lqis )

is

(13.11)

will stress the influence of the labour usage (work efficiency) of individual resources working at one unit of the product in question. While analysing machines direct cost, we have:

VAR(mchdci ) = VAR(mchnpi ) +

∑ VAR( pt ip ) + ∑VAR(mrip ) + R (mchdci ) T

T

p =1

p =1

where again we will have analogous interpretation of the corresponding variances. They will show us respectively: the influence of the difference in the number of units produced, of the difference in the processing time on individual machines needed by the individual products units and of the difference of the per hour cost of using the machine for one unit of individual products:

VAR(mchnpi ) = ( np i − npi )∑ pt ip mrp T

(13.12)

p =1

VAR ( pt ip ) = np i mrp

VAR (mrip ) = np i pt ip

( pt ( mr

ip

p

) − mr ) , p=1,…,T − pt ip , p=1,…,T p

(13.13) (13.14)

The cost linked to launching new batches depends for each product on only two factors (Model 3). Thus we can write

VAR(bci ) = VAR(nbi ) + VAR(lbci ) + R(bci ) where, as usual, the last component will usually be neglected, and the other two are defined as follows, stressing respectively the problem of the number of batches in which each product should have been and was manufactured and that of the cost of launching a new batch for each individual product.

VAR(nbi ) = lbci VAR(lbci ) = nbi

( nb ( lbc

i

− nbi ) , i=1,…,N

(13.15)

i

− lbci ) , i=1,…,N

(13.16)

As far as the indirect cost are concerned, we can assume the depreciation is a constant, although in some special cases9 this would not be true. There may also exist some indirect cost subject to change as a function of certain parameters, like the labour cost of the responsible of the machines (ILR) in Example 1, which is the function of the makespan and of hour rate ilr. In this case we would have: 9

E.g. one depreciation method calculates the depreciation of a machine for the budgeting period as a linear function of the number of hours the machine operates – in such a case the depreciation cost would change accordingly to the number of hours the machine actually operated. In some cases in is even advisable to consider a fuzzy depreciation (Rhys 1992).

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VAR(ILR ) = VAR(Mks ) + VAR(ilr ) + R(ILR ) where usually only the first two variances would be analyzed and we would have:

( Mks VAR(ilr ) = Mks ( ilr

VAR(Mks ) = ilr

− Mks )

(13.17)

− ilr )

(13.18)

The variances (13.9)-(13.18) allow to analyse, at the end of the budgeting period, the reasons for a negative value of variance -TC . The variance TC is of course important, however, when we calculate it and check at the end of the budgeting period and we find out it is too big, it is too late. It would be good, especially in cases where it is really important to keep the production cost as low as possible, to have a warning system, showing us in the course of the budgeting period that there is a danger that the variance -TC we will have at the end of the budgeting period will be unacceptable. To order to have such a system, it may be good to use a method which is actually destined for project management, but may also be useful in controlling the normal production process – as each budgeting period is to some extent different and special, which are features of projects. The method is called Earned Value (Kuchta 2005). In order to apply the method, we have to be able at the end of the t-th time unit in the course of the budgeting period, to calculate or estimate three groups of values: 1. for each i=1,…,N the number of units actually processed so far on each of the t

machines, w=1,…,T, denoted as npi , w 2. for each cost position CPj (j=1,…BP) being a direct cost, the planned (budgt

eted) cost of manufacturing npi , w items (i=1,…,N), denoted10

BCWPjt (we

consider the total cost, do not distinguish between the machines); 3. for each cost position CPj (j=1,…BP) being a direct cost, the actual cost of t

manufacturing npi , w

items (i=1,…,N), denoted11

ACWPjt (again, we con-

sider only the total cost). t

Values npi , w

and

ACWPjt (i=1,…N, j=1,…,BP for such j that CPj corre-

sponds to a direct cost) will always be available without any problems. The estimation of

BCWPjt may require some work, because it involves the answer to the

question: “how much should have cost the work we have actually done according to our budget, to our plan?”. This answer will not always be easy, especially if the 10

11

The abbreviation BCWP comes from the name “Budgeted Cost of Work Performed” used in the Earned Value method. ACWP - “Actual Cost of Work Performed”.

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actual schedule differs strongly from the planned one, but this answer is really essential for the budget control. We will illustrate this approach in the following example. Having estimated

BCWPjt (j=1,…,BP for such j that CPj corresponds to

a direct cost), we can analyse the variances

ACWPjt - BCWPjt . They will tell us

how much more/less expensive the work we have already done has been with respect to the plan (we do not take here into account changes in the number of units manufactured with respect to the plan, we look only at the work that has been actually executed). If this difference is negative and big, it is a warning that something is wrong with the realisation of the budget. We can also calculate – still ACWPjt imitating the Earned Value method for projects - values CPj , BCWPjt which can be good estimators of the final values CPj , which are still unknown. Using the same approach, we can estimate the future – thus those from the end of the budgeting period – values of the variances (13.9)-(13.18) and try to do something if it seems that they are going to be too big. The application of the Earned Value method to the production budget control may of course cause some problems, as projects have a quite different nature, but still it seems that it may bring many advantages, giving a warning if something is wrong with the cost earlier than at the end of the budgeting period. Details will be illustrated in the example in the following section. 2.3.2 Numerical Example

Let us consider again Example 1. Once the budgeting week in question has passed, we have the actual values of the production process. Let us list only those values on which the budget was based which were different than in the budget – the other ones remained untouched: < mq11 >= 2 , < lq31 >= 0,5 , < ap11 >= 0,5$ , < lr31 >= 3$ , < pt 52 >= 2 ,

< np3 >= 5 , < nb2 >= 1 . The actual schedule according to which the work was executed was as follows:

Fig. 13.4 Realised schedule for Example 1.

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The realized budget is as follows: I)

Direct cost i) Material

< mtdc1 >= 4 ⋅ 2 ⋅ 0,5 = 4$ , < mtdc 2 >= 4 ⋅ 1 ⋅ 0,5 = 2$ , < mtdc3 >= 5 ⋅ 1 ⋅ 0,5 = 2,5$ , < mtdc 4 >= 4 ⋅ 1 ⋅ 0,5 = 2$ , < mtdc 5 >= 2 ⋅ 1 ⋅ 0,5 = 1$ ii) Labour cost: < ldc1 >= 4 ⋅ 1 ⋅ 3 = 12$ , < ldc 2 >= 4 ⋅ 1 ⋅ 3 = 12$ , < ldc 3 >= 5 ⋅ 0,5 ⋅ 3 = 7,5$ , < ldc 4 >= 4 ⋅ 1 ⋅ 3 = 12$ , < ldc 5 >= 2 ⋅ 1 ⋅ 3 = 6$ iii) Machines direct cost: < mchdc1 >= mchdc1 , < mchdc 2 >= mchdc 2 < mchdc3 >= 5·(2·1$+1·2$)=20$, < mchdc 4 >= mchdc 4 , < mchdc5 >= 2·(1·1$+2·2$)=10$ iv) Batch start cost: < bc1 >=< bc 3 >=< bc 4 >=< bc 5 >= 2·10$=20$, < bc 2 >= 10$ cost:

=245$

II) Indirect cost: i) Depreciation (): 50$ ii) Labour cost of the responsible of the machines (): 37·5$=185$ =235$ II) =480$ Now we can proceed to the analysis of variances. The total budget has been exceeded, both in direct and indirect cost. It is now the time to look for reasons. Let us start with the direct cost analysis. We will give the values of variances (13.9)(13.18) together with an interpretation, which will not be decisive and unequivocal, because the interpretation of variances has always to be connected with the analysis of the context and the very situation, can never be based purely on numbers – but we have to confine ourselves to numbers and try to imagine various scenarios. Variance name

VAR(ap11 ) VAR(mq11 )

Variance Variance Interpretation Value -4$

Positive: the cheaper material bought for the production of the first product diminished the total cost by 4$

2$

Negative: the less economical use of the material in the production of the first product increased the material cost. The reason may be the fact that a cheaper material has been bought – in such a case the preceding variance would be only apparently positive.

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D. Kuchta

VAR(nb2 )

-10$

VAR(lq31 )

Positive: less batches used in the production of the second product diminished the total cost by 10 $. There may however arise the question whether the client satisfaction had not suffered – as ha may have had to wait longer for the delivery.

-6$

VAR(lr31 )

Positive: an apparently more efficient work in the production of the third product diminished the total cost by 6$.

4$

Negative: the direct labour force used in the production was more expensive than planned, which increase the cost by 4$. This may be the reason for the higher efficiency (maybe a higher qualified work force was used) shown by the preceding variance.

VAR(lnp5 ) = VAR(mtnp5 ) 3$

Negative: the number of units of product 5 actually manufactured was higher than planned, which increased both the direct material as well as the direct labour cost.

VAR(mchnp5 )

15$

VAR( pt52 )

Negative: the increase of the number of units manufactured increased also the machine direct cost of product 5.

10$

Negative: the machine direct cost of product 5 was also increased because the unitary time needed by each item of the product 5 on machine 2 was higher than planned

As far as the indirect cost is concerned, we have the negative variance

VAR(Mks ) =2, cause by the longer makespan than planned. Let us now illustrate

how the Earned Value method could be applied in the considered example. Let us consider the end of the fifth hour. We have finished the production of product 5 on both machines and of product 1 on machine 1 (on machine 2 no entire unit of product 1 has been finished, so we consider no processing of product 1 on machine 2 has been executed). If, for sake of simplicity, we take into account just the N

∑ VAR(mchnpi ) , we will have:

ACWPj5 =45 (this is the actual direct cost of ma-

i =1

chine usage incurred so far) and BCWPj5 =35 – this is the cost of machine usage we should have – according to the plan - incurred for the processing of product 5 on both machines and of product 1 on machine 1. We can thus see (without even taking into account that the number of units in case of the product 5 differs from the plan) that we work we have done so far has led to a much higher cost than planned. The important thing is that we can notice it at the end of the fifth hour, whereas the planned makespan is 35 hours, thus we know about the problem long before the budgeting period is finished. In the next section, before we make the passage to fuzzy budgeting, we will present basic information about fuzzy numbers.

3 Selected Information about Fuzzy Numbers Here we will consider only triangular or one-sided fuzzy numbers (Dubois and Prade 1987, Kaufmann and Gupta 1985). A triangular fuzzy number

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~ A = (a1 , a 2 , a3 ) , where a1 , a 2 , a3 are such real numbers that a1 < a 2 < a3 , will represent a quantity which – according to the knowledge or belief of the decision maker in the very moment – will take on the value a 2 with the possibility degree 1, values smaller than or equal

a1 and greater than or equal a3 with the

possibility degree 0, and the other values with the possibility degree between 0 and 1 – the smaller, the greater the distance from a 2 is, while the corresponding function is linear. This can be represented by the following picture:

Fig. 13.5 A triangular fuzzy number

~ A = (a1 , a 2 , a3 ) and its possibility degree function

Triangular fuzzy numbers will be used to represent magnitudes which are not completely known in the decision moment and their values will become known

only in the future. In fact, a triangular fuzzy number A = (a1 , a 2 , a 3 ) is inseparably linked to the possibility degree function, represented by a triangle and unequivocally defined be the numbers a1 , a 2 , a 3 . One sided fuzzy numbers can be divided into left hand and right hand fuzzy numbers. A right hand fuzzy number and a left hand fuzzy number will be denoted

~

as A = (a1 , a 2 ) 12, where a1 , a 2 are such real numbers that a1 < a 2 . We will use one sided fuzzy numbers not to represent incomplete knowledge, but rather to represent the preferences of the decision maker. Thus a right hand fuzzy number represents the information that values greater or equal a 2 satisfy the decision

~

maker to the highest degree (which is degree 1), values smaller than a1 are for him or her completely unsatisfactory, an the satisfaction with the values form the interval (a1 , a 2 ) increases in a linear way. This is represented by the following picture: 12

The same notation for both, we will indicate each time which case in considered.

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D. Kuchta

Fig. 13.6 A right hand fuzzy number

~ A = (a1 , a 2 )

and its satisfaction degree function.

A left hand fuzzy number represents the inverse preference of the decision maker:

Fig. 13.7 A left hand fuzzy number

~ A = (a1 , a 2 )

and its satisfaction degree function.

Of course, it is possible to consider generalizations of the triangular and one sided fuzzy numbers: e.g. trapezoidal fuzzy numbers or fuzzy numbers with nonlinear possibility or satisfaction function. The extension of the considerations presented here to this more general case would be straightforward. We will need arithmetical operations on triangular fuzzy numbers. Definition 1 (Dubois and Prade 1987, Kaufmann and Gupta 1985): Let us con-

~

sider two triangular fuzzy numbers A Then we have: a) b)

~ = (a1 , a 2 , a3 ) and B = (b1 , b2 , b3 ) .

(a1 , a2 , a3 ) + (b1 , b2 , b3 ) = (a1 + b1 , a2 + b2 , a3 + b3 ) (a1 , a2 , a3 ) · (b1 , b2 , b3 ) = (min (a1 ⋅ b1 , a1 ⋅ b3 , a3 ⋅ b1 , a3 ⋅ b3 ), a 2 ⋅ b2 , max(a1 ⋅ b1 , a1 ⋅ b3 , a3 ⋅ b1 , a3 ⋅ b3 ))

Fuzzy Production and Operations Budgeting and Control

c) d)

e)

f)

319

(a1 , a2 , a3 ) - (b1 , b2 , b3 ) = (a1 − b3 , a2 − b2 , a3 − b1 ) (a1 , a2 , a3 ) ⊗ (b1 , b2 , b3 ) = (min(a1 ⋅ b3 , a3 ⋅ b1 ), a 2 ⋅ b2 , max(a1 ⋅ b3 , a3 ⋅ b1 )) (a1 , a2 , a3 ) • (b1 , b2 , b3 ) = (min(a1 ⋅ b1 , a3 ⋅ b3 ), a 2 ⋅ b2 , max(a1 ⋅ b1 , a3 ⋅ b3 )) (a1 , a2 , a3 ) - (b1 , b2 , b3 ) = (a1 − b1 , a 2 − b2 , a3 − b3 ) ,

defined

if

a1 − b1 < a 2 − b2 < a3 − b3 . Operations between a triangular fuzzy number A = (a1 , a 2 , a3 ) and a crisp number b can be defined according to the same definition, simply assuming

~

~ B = (b1 , b2 , b3 ) =(b,b,b).

Items a), b) and c) of the above definition define the respective operations on fuzzy numbers in such a way that they allow a positive value of the possibility degree for all the combinations of numbers from the interval a1 , a 3 and b1 ,b3 . This makes the resulting fuzzy number sometimes very “broad”, which may well correspond to reality - if really both magnitudes, represented by fuzzy numbers

[

]

[

]

~ ~ A = (a1 , a 2 , a3 ) and B = (b1 , b2 , b3 ) , are independent. But sometimes they

are not, and in such a case items b) or c) of the above definition lead to unnecessarily broad fuzzy numbers, which do not correspond to reality and make any decision more difficult, because they show a positive possibility degree for values which for sure will not occur. E.g. in the extreme case, if we consider the subtraction (a1 , a 2 , a3 ) - (a1 , a 2 , a3 ) , where (a1 , a 2 , a3 ) represents a certain quantity, the result of the subtraction will always be 0 (because in reality one concrete value will be taken on), but formula c) from the above definition will falsify the result, leading to a proper triangular fuzzy number, which shows a positive possibility degree for values which will never occur in the future. Thus, if there is some kind of dependency between the actual realizations of fuzzy numbers

~ ~ A = (a1 , a 2 , a3 ) and B = (b1 , b2 , b3 ) , we may consider using modified opera-

tions, like those from items d) and e) (modified multiplication) and f) (modified subtraction). Multiplication d) may be used e.g. while we multiply the number of units of a good by its unitary price, and we think that if we buy a smaller quantity we will get a higher price and if we buy a higher quantity we will get a lower

= (a1 , a 2 , a3 ) will ~ always by accompanied by smaller realization of B = (b1 , b2 , b3 ) , we have to ~

price. If for some reason we think smaller realizations of A

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D. Kuchta

use modified multiplication and modified subtraction from items e) and f). In the above mentioned case of the subtraction (a1 , a 2 , a3 ) - (a1 , a 2 , a3 ) formula f) has to be used. Then we have to mention the vast area of comparing fuzzy number with fuzzy numbers and fuzzy numbers with crisp numbers (Sevastianov 2007). There are lots of methods of doing this, their choice depend on the preferences of the decision maker. We will present just examples. Definition 2 (Kaufmann and Gupta 1985): Let us consider two fuzzy numbers

~ ~ ~ A = (a1 , a 2 , a3 ) and B = (b1 , b2 , b3 ) , where A is a triangular fuzzy number ~ and B a triangular fuzzy number or a one sided fuzzy number. Let us call the tri~ ~ ~ angle formed by the possibility function of A Fig A and its surface Sur A ,

()

()

analogous notation will be applied to the figure formed by the satisfaction function

~

~

of a right hand fuzzy number and the axis. The agreement index of A with B ,

(~ ~ )

denoted Agr A, B , is defined as:

( ( ) ( )) ( ( ))

~ ~ ~ ~ Sur Fig A ∩ Fig B Agr A, B = ~ Sur Fig A ~ Definition 3. Let us consider a triangular fuzzy number A = (a1 , a 2 , a 3 ) and a ~ ~ real number s. The degree to which A exceeds s, defined as Deg A ≥ s , will

(

)

(

be defined in the following way:

( ()

)

)

~ Sur Fig A ∩ {x ∈ ℜ : x ≥ s} ~ Deg A ≥ s = ~ Sur Fig A

(

)

( ( ))

Examples of the application of both above definitions will be given in the following.

4 Fuzzy Production and Operations Budgeting In this last section of the paper we will present briefly some ideas why and how to use the fuzzy approach in the production and operations budgeting.

4.1 Justification of the Use of the Fuzzy Approach in Production and Operations Budgeting We may need the fuzzy approach in production budgeting for the following reasons: • Budgeting is based to a great extent on numbers from the future, which are not fully known in the budgeting moment and may be subject to change. We may have some knowledge and intuition about their possible values and their possibility degrees, but we cannot say yet which value will actually occur. This may

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321

concern magnitudes dependent both on internal (machines and workers efficiency etc.) and external (e.g. linked to market conditions) factors. • Budgeting is to a great extent a compromise between various goals of various actors in the company. We can use one sided fuzzy numbers to model the preferences (satisfaction) of various actors with various values and search for a compromising solution (see e.g. model (19)). • Budgeting plays a motivating role: it sets goals and the budget control should check their fulfilment and contribute to a better and more efficient work in the future, but stepwise – too ambitious goals may have a destructive effect. The goals can be set in the form of the variances values which should be achieved. In some cases (very good divisions, very ambitious workers) the ideal variance 0 can be given as a goal, in other cases a less ambitions variance value should be given. However, also a good division and good workers may have in some cases problems with achieving a very ambitious goal – there may always occur some technical problems or other unexpected events. That’s is why – in order to motivate and not to depress workers - Omer et al. (1995) use fuzzy numbers to model goal variances: in case of a good division a fuzzy variance “close to 0 and rather positive” would be set as a goal, in other cases a slightly negative fuzzy variances would constitute the goal. Also in the case of the good division negative variance would be admitted, although regarded as less possible – and this softening of goals can be achieved thanks to the fuzzy approach. • In one special budgeting method, the so called zero based budgeting13, we do not take into account the information from the past and the given production process and system, but question all this, trying to build everything from the beginning – avoiding in this way committing old errors and opening ourselves to new ways and possibilities. In this budgeting approach everything is uncertain, everything should be decided as a compromise – and here the fuzzy approach is more than adequate (Buckley 1987).

4.2 Constructing Fuzzy Production and Operations Budgets Generally, if we allow fuzzy numbers in the budgeting process, the procedure described in section 2.2 should be repeated, setting triangular fuzzy numbers for those values which are not completely known or – if different budgeting scenarios are to be considered – expressing preferences by one sided fuzzy numbers. In the next section we will give just some examples of the models which can be used in this context, as fuzzy optimization, fuzzy scheduling etc. are vast fields, where many various model and algorithms exist and can be used, especially while determining the input for the budgeting process (Fig. 13.2). 4.2.1 General Models

As far as Model 1 (and its numerous generalizations) is concerned, we can introduce fuzziness in the individual coefficients or in the preference modelling. As a 13

We do not present this method here because of lack of space.

322

D. Kuchta

result, we get the number of units of individual products to be produced fulfilling various fuzzy conditions and relations – each fuzzy model has its own interpretation and has to be constructed in cooperation with the decision maker. For example, Chanas (1983) considers the following linear programming problem: N

~

∑ Pi ⋅ npi → P i =1

N

~

∑ MCH i ⋅ npi ≤ MCHB i N

~

∑ MTi ⋅ npi ≤ MT B

(13.19)

i N

~

∑ LBH i ⋅ npi ≤ LBHB , i

~ where P is a right hand fuzzy number representing the preferences concerning ~ ~ ~ the sales and MC HB , MT B , LB HB are left hand fuzzy numbers, representing the preferences concerning the usage of resources. The solution are such numbers npi , i=1,..,N, which constitute a compromise between attaining a relatively high sales value and having to guarantee relatively few resources. An example will be given in the following section. In the above model one sided fuzzy numbers are used. In other models triangular fuzzy numbers may represent the usage of resources by each product unit, the unitary profit etc., thus the coefficients on the left hand side of model (19). Various fuzzy models used in production planning in order to find out how many items of each product type should be manufactured can be found e.g. in: (Chanas and Kuchta 1994, 1996b, 1998b, 1999), (Gen at al. 1992), (Kuchta 2003, 2005a, 2008), (Wang and Liang 2004), (Wang and Fang 2001). Dolata (2009) proposes a version of the simplex algorithm for linear programming problems with fuzzy coefficients (e.g. triangular fuzzy numbers). The application of the transportation model to the determination of the number of units to be manufactured (Model 2) using triangular fuzzy numbers (representp

p

p

ing either D1 , p=1,…,P or MANC1 and MCAP1 (p=1,…,P)) , is presented e.g. in (Chanas and Kuchta 1996a, 1998a) and (Dolata 2009). Methods of determining the fuzzy makespan in various types of operations scheduling problems when the processing times on individual machines are fuzzy are presented in various papers, e.g. (Chanas et al. 1999), (Gladysz and Kuchta 2003), (Konno and Ishii 2000), (Sakawa and Kubota 2000). As far as Model 3 is concerned, the one which gives us the output required in Fig. 13.2, we simply have to apply to partially crisp and partially fuzzy input data

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(the latter represented by triangular fuzzy numbers – this will be the case for those data whose exact values is not yet known in the planning moment) operations from Definition 1, paying a special attention to the problem of dependency and the above mentioned choice of operation type. As a result, we will get the cost types from Model 3 in the form of triangular fuzzy numbers. Let us consider one cost position from Model 3 - for the other cost positions considerations are analogous. Let us take into account direct material direct cost. We may have with the following case: M

m~ t dci = n~ p i ⋅ ∑ mq~il a~ p il

(13.20)

l =1

where all the factors are triangular fuzzy numbers (thus only incompletely known in the planning moment), with independent realisations (high number of production units may be combined both with low and high unitary prices of the materials etc.). However, it may happen that some of the factors in (13.20) will be crisp (e.g. if we get the number of units to be produced from solving a model like model (13.19) and if there is some dependency between the individual factors – e.g. a higher purchase price of the material will imply a lower unitary usage of the material or a higher number of units to be produced will imply a lower purchase price. In such a case formula (13.20) will have to be modified: some of the factors may be crisp and we may have to choose another type of multiplication from Definition 1. In either case, however, we will get a fuzzy value (a triangular fuzzy number) ~ for the direct material cost: m t dc i , representing all the possible realisations of this cost together with the possibility degree of their occurrence. 4.2.2 Numerical Example

Let us start with an example for model (13.19). The example is taken from (Chanas 1983).

~ = 3 , P2 = 4 , P3 = 4 , P = (1600, 1750) (a right hand fuzzy ~ number), MCH 1 = 6 , MCH 2 = 3 , MCH 3 = 4 , MCHB = (1200, 1300) (a ~ left had fuzzy number), MT1 = 5 , MT2 = 4 , MT3 = 5 , MT B = (1550, 1700) Let N=3, P1

(a left had fuzzy number). The other coefficients from model (13.19) are equal to 0. Then we get the compromising input for the budgeting process, taking into account both the preferences concerning the sales, which are contradictory with the preferences concerning the guaranteed resources: np1 = 0, np2 = 416, np3 = 0 . For such production numbers and the price of the material equal to (3,4,5) - the only factor in our example which will be fuzzy, because it is unknown and independent of the company, as the materials are purchased from an external supplier, and the company thinks it can control the other elements – we will get m~ t dc 2 =(1248,1664,2080).

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4.3 Production and Operations Fuzzy Budgets Control In case of fuzzy budgeting the individual planned cost positions will be triangular fuzzy numbers. In the control phase they will have to be compared with crisp actual values, but not only – also with goals and preferences of the decision maker. Also during the budgeting period it would be good to have a fuzzy warning system, a fuzzy variant of the Earned Value method (Kuchta 2005). The three approaches will be briefly presented in the following. 4.3.1 General Models

For each variance (13.9)-(13.18), as well as for the total variance -TC, we may get a fuzzy value, a triangular fuzzy number in our case. Let us denote it with

(

)

~ ~ V . For this variance we can calculate Deg V ≥ 0 - which will measure the de-

gree of the negativity of the variance. The closer to 0 it is, the better. We can also have at our disposal a goal: the decision maker may have given a

~

left hand fuzzy number G expressing his preferences or satisfaction with various

~

variance values. G may also be a triangular fuzzy number, expressing to the expectations of the management concerning the respective variance (Omer et al.

(~ ~ )

1995). In each case, we will calculate Agr V , G , measuring to which extent the satisfaction of the decision maker, the expectations or the goal have been achieved. The closer to 1, the better. In the course of the budgeting period we can calculate the fuzzy number ex-

(~ )

~

pressing the future, expected value of V . Calculating Deg V ≥ 0

(

)

and/or

~ ~ Agr V , G before these values are a reality, gives us a chance to act on certain

factors (those for which we can see that the corresponding variance (13.9)-(13.18) is going to unacceptable if we do not act) and prevent the negative variance from occurring. 4.3.2 Numerical Example

~

Let G1 be a left handed fuzzy number (1,2) – it expresses the fact that the decision maker accepts fully variances smaller than 1 and partially those smaller than 2 and greater than 1. So for him a slightly negative variance is acceptable – this is

~

an example of a less ambitious goal. Let G 2 be a triangular fuzzy number (-1,1,2). It also expresses the goal for the variance: it says that the decision maker will not be surprised too much if the variance is between -1 and 2, but according to his knowledge the most possible values of the variance are those close to 1. Let

~

us now suppose that the variance is V =(-1,1,3). Then we calculate

Fuzzy Production and Operations Budgeting and Control

( (

325

) )

~ ~ Agr V , G1 =0,75 ~ ~ Agr V , G2 =0,625 Thus, the goal has been achieved to a rather high degree and also the agreement with the expectations of the decision maker is rather high, however, it could be closer to 1.

(~ )

We can also calculate Deg V ≥ 0 - it is equal to 0,75. It is rather high, with-

~

~

out the goal G1 or the expectations of the decision maker expressed in G 2 we might evaluate it negatively. But the decision maker thinks in this case negative variances are possible and acceptable and he should only motivate the workers

~

(maybe through a more ambitious goal G1 or through more ambitious expecta-

~

(~ )

tions G 2 in the future) to attain a lower Deg V ≥ 0 .

5 Conclusions and Further Research Fuzzy production and operations budgeting is vast subject and it has only been treated here very briefly. However, we think that these few reflections on the subject will give all the persons responsible for production and operations budgeting some ideas of how to use the fuzzy approach in the praxis to get the most advantage of it. The paper may also give some new ideas for the research. Not all the theoretical questions have been solved yet, e.g. it is still open to develop a “production” version of the fuzzy Earned Value method, originally destined for projects budget control. This paper only proposes this direction of research, without giving any ultimate solution.

References Anderson, D.R.: Quantitative methods for business. Cengage Learning (Thompson) (2007) Atkinson, A., Banker, R., Kaplan, R.S., Young, M.: Management accounting. PrenticeHall, Englewood Cliffs (1995) Buckley, J.J.: The fuzzy mathematics of finance. Fuzzy Sets and Systems 21, 257–273 (1987) Chanas, S.: The use of parametric programming in fuzzy linear programming. Fuzzy Sets and Systems 11, 243–252 (1983) Chanas, S., Kasperski, A., Kuchta, D.: Fuzzy flow shop sequencing problem - a possibility approach. In: Chanas, S., Kasperski, A., Kuchta, D. (eds.) W: EuroFUSE 1999. The Fourth Meeting of the Euro Working Group on Fuzzy Sets and SIC 1999, pp. 65–69 (1999) Chanas, S., Kuchta, D.: Linear programming problem with fuzzy coefficients in the objective function. In: Delgado, M., et al. (eds.) Fuzzy optimization recent advances. PhysicaVerlag, Heidelberg (1994)

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Chanas, S., Kuchta, D.: A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. Fuzzy Sets and Systems 82(3), 299–305 (1996a) Chanas, S., Kuchta, D.: Multiobjective programming in optimization of interval objective functions - a generalized approach. European Journal of Operational Research 94(3), 594–598 (1996b) Chanas, S., Kuchta, D.: Fuzzy integer transportation problem. Fuzzy Sets and Systems 98(3), 291–298 (1998a) Chanas, S., Kuchta, D.: Discrete fuzzy optimization. In: Słowiński, R. (ed.) Fuzzy sets in decision analysis, operations research and statistics. Kluwer Academic, Dordrecht (1998b) Chanas, S., Kuchta, D.: Linear programming with words. In: Zadeh, L.A., Kacprzyk, J. (eds.) Computing with words in information/intelligent systems, Heidelberg, New York (1999) Dolata, M.: Zastosowanie podejścia obiektowego do rozszerzenia metody simplex w rozwiązywaniu rozmytego logistycznego zagadnienia dystrybutora, Rozprawa doktorska (Application of object approach to an extension of the simplex method to solve the fuzzy logistic distribution problem, a PhD dissertation, in Polish). IBS PAN, Warszawa (2009) Dubois, D., Prade, H.: Fuzzy numbers: an overview. In: Bezdek, J. (ed.) Analysis of fuzzy information. CRC Press, Boca Raton (1987) Evans, J.R., Anderson, D.R., Sweeney, D.J., Williams, T.A.: Applied production & operations management. West Publishing Company (1987) Gen, M., Tsujimura, Y., Ida, K.: Method for solving multiobjective aggregate production planning problem with fuzzy parameters. Computers & Industrial Engineering 23, 117– 120 (1992) Gładysz, B., Kuchta, D.: The two-machine open shop problem with fuzzy processing times. Badania Operacyjne i Decyzyjne 4, 43–47 (2003) Kaufmann, A., Gupta, M.M.: Introduction to fuzzy theory and application. Van Nostrand Reinhold, New York (1985) Kloock, J., Schiller, U.: Marginal costing: cost budgeting and cost variance analysis. Management Accounting Research 8(3), 299–323 (1997) Konno, T., Ishii, H.: An open shop scheduling problem with fuzzy allowable time and fuzzy resource constraint. Fuzzy Sets and Systems 109(1), 141–147 (2000) Kuchta, D.: A step towards fuzzy accounting. Badania Operacyjne i Decyzyjne 1, 75–84 (2000) Kuchta, D.: Miękka matematyka w zarządzaniu: Zastosowanie liczb przedziałowych i rozmytych w rachunkowości zarządczej (Soft mathematics in management. Use of interval and fuzzy numbers in managerial accounting – a monograph in Polish), Oficyna Wydaw. PWroc, Wrocław (2001) Kuchta, D.: A generalisation of a solution concept for the linear programming problem with interval coefficients. Badania Operacyjne i Decyzyjne 4, 115–123 (2003) Kuchta, D.: Fuzzy solution of the linear programming problem with interval coefficients in the constraints. Badania Operacyjne i Decyzyjne 3/4, 35–42 (2005a) Kuchta, D.: Fuzzyfication of the earned value method. WSEAS Transactions on Systems 4(12), 2222–2229 (2005b) Kuchta, D.: A modification of a solution concept of the linear programming problem with interval coefficients in the constraints. Central European Journal of Operations Research 16(3), 307–316 (2008)

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Omer, K., de Korvin, A., Leavins, J.R.: Determining the extent of budgetary slack: a fuzzy set approach. In: Siegel, P.H., de Korvin, A., Omer, K. (eds.) Applications of fuzzy logic and the theory of evidence to accounting. Studies in Managerial Accounting, vol. 3. JAI Press, Greenwich (1995) Pollard, W.B.: Teaching standard costs: A look at textbook differences in overhead variance analysis. Journal of Accounting Education 4(1), 211–220 (1986) Rhys, H.: A note on the fuzzy nature of depreciation. Working Papers in Accounting and Finance Department of Accounting Aberyst University of Wales (1992) Rhys, H.: Fuzzy financial statements. Working Papers in Accounting and Finance Department of Accounting Aberyst University of Wales (1997) Roseaux - a collective name: Exercises et problèmes résolus de recherche opérationnelle. t.3, Masson (1985) Roseaux - a collective name: Exercises et problèmes résolus de recherche opérationnelle. t.1, Masson (1991) Sakawa, M., Kubota, R.: Fuzzy programming for multiobjective job shop scheduling with fuzzy processing time and fuzzy due date through genetic algorithms. European Journal of Operational Research 120(2), 393–407 (2000) Sevastianov, P.: Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster-Shafer theory. Information Sciences 177, 4645–4661 (2007) Wang, R.C., Liang, T.F.: Application of fuzzy multi-objective linear programming to aggregate production planning. Computers & Industrial Engineering 46(1), 17–41 (2004) Wang, R.C., Fang, H.H.: Aggregate production planning with multiple objectives in a fuzzy environment. European Journal of Operational Research 133(3), 521–536 (2001)

Chapter14

Fuzzy Location Selection Techniques Cengiz Kahraman, Selcuk Cebi, and Fatih Tuysuz*

Abstract. Facility location has an important role in terms of firms’ strategic planning in the management science and operational research. One of the most common facility location problems is the location selection problem which is the most ancient but has been still a current problem for organizations. Plants, warehouses, retail outlets, terminals, storage yards, distribution centers etc. are typical facilities that must be located strategically since the location selection problem influences organizations’ strategic competitive position in terms of operating cost, transportation cost, delivery speed performance, and organization’s flexibility to compete in the marketplace. In this chapter, different approaches and techniques used in location selection problems are presented. In the scope of this chapter, we focus on the fuzzy multi-criteria decision making methods. Especially, in the literature, fuzzy AHP and fuzzy TOPSIS are presented since they are the most widely used ones. Furthermore, it is the first time a framework based on the fuzzy information axiom is proposed for facility location selection.

1 Introduction A general facility location problem includes a set of spatial resource allocation problems in which one or more service facilities to serve spatiality distributed set of demands which have to be located according to one or several objective functions depending on the interaction between demand and facilities (Drezner and Hamacher 2004). In the management science and operational research, facility location has an important role for strategic planning such that locating facilities, designing the supply chain, selecting suppliers, defining the physical and information flows along a supply chain are critical parts of strategic planning. Following topics illustrate the relationship between facility location and supply chain design Cengiz Kahraman Department of Industrial Engineering, Istanbul Technical University, Macka 34367, Istanbul, Turkey Selcuk Cebi Department of Industrial Engineering, Karadeniz Technical University, 61080, Trabzon, Turkey Fatih Tuysuz T.C. Beykent University, Department of Industrial Engineering, Ayazağa, İstanbul, Turkey C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 329–358. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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problems in different situations (Thanh et al. 2008) , (i) when a company extends its activity to new geographical areas, it has to locate facilities and design a new part of its supply chain; (ii) if there is a limited production capacity, the strategic decision is given between opening new facilities or enlarging the existing; (iii) in the case of an obsolete facility, the decision is given to renovate it, close it, or replace it by a new one, (iv) the merger of two companies requires the decision of merging the supply chains too, etc. One of the most common facility location problems is the location selection problem which is the most ancient but has been still a current problem for organizations. Plants, warehouses, retail outlets, terminals, storage yards, distribution centers etc. are typical facilities that must be located strategically since the location selection problem for an investment facility of an organization influences organizations’ strategic competitive position in terms of operating cost, transportation cost, delivery speed performance, and organization’s flexibility to compete in the marketplace. Therefore, the final decision of a facility location must contribute to the success of corporate strategic plans for financing, marketing, human resource, and production objectives (Mount 1990). So, the location selection problem is one of the critical decisions in supply chain design and management in order to maximize supply chain performance and profitability. The selection of a facility location among alternative locations is a multicriteria decision-making problem including both quantitative and qualitative criteria. The quantitative criteria include the costs of land and buildings, the inbound and outbound transportation costs, and the raw materials supply quantity. The qualitative criteria include the closeness to suppliers and retailers, the government policies, the environment factors, the quality of life, the availability of required technical labor, and the availability of utilities (Chuang 2001). The conventional approaches to facility location problem tend to be less effective in dealing with the imprecise or vagueness nature of the linguistic assessment (Kahraman et al. 2003). Since the measurements of qualitative criteria in facility location problem include imprecise or vagueness, the fuzzy set theory, proposed by Zadeh (1965) has been used to solve the facility location selection problem in the literature. The fuzzy set theory has been proposed in order to cope with vagueness or uncertainty in human thinking process. The rest of this chapter is organized as follows; the next section presents the facility location and location selection literature. Section 3, summarizes the facility location techniques. In section 4, recent fuzzy approaches to the facility location selection problem are presented illustrative examples are given for each fuzzy approach. Finally, conclusion and suggestions for future studies are given in the last section.

2 Literature Review Facility location problems have been studied in various forms for a long time within Operation Research area. Numerous papers and books have been published up to now. Moreover, there have been a number of review papers on facility location research area, which classify the studies in different perspectives (Kolse and

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Drexl 2005, Şahin and Süral 2007, Revelle et al. 2008, Melo et al. 2009, etc.). In this section, an overview of the selected studies published in the literature within the last decade is presented. We present facility location problems in two tables. Table 14.1 presents the general facility location problems and solution algorithms whereas Table 14.2 gives the facility location selection problems and solution algorithms. Table 14.1 consists of three columns; problem type, problem attributes, and method type. Problem types consist of seven main topics; p-median, p-center, capacitated facility location problem (CFLP), uncapacitated facility location problem (UCFLP), single source capacitated facility location problem (SSCFLP), fix charged problem, and other special cases. Problem attributes are related to modeling of the problem in stochastic or deterministic structure. When there is an uncertainty or unknown variable in the problem, stochastic modeling is selected for the solution of the problem. Then, method types are given in the last column of the table; linear programming (LP), integer programming (IP), mix integer programming (MIP), branch and bound algorithm (BB), dynamic programming (DP), Lagrange relaxation (LR), ant colony (AC), genetic algorithm (GA), tabu search (TS), and approximation algorithm (AA). Table 14.2 also consists of three main columns; type of evaluation, solution algorithms, and application area. The following figures are derived from Table 14.1. In Figure 14.1, the types of problems are presented for the facility location problems. According to Figure 14.1, p-median problem and CFLP are the most encountered problems in the literature.

Fig. 14.1 Types of problems

Fig. 14.2 Solution methods

The solution algorithms are given in Figure 14.2. According to Figure 14.2, Lagrange relaxation is the most used solution algorithm for the facility location problems. Figure 14.3 depicts the frequency-years graph of the papers. In Table 14.2, facility location selection studies are presented in the literature. Since the facility location problem is a multicriteria decision making (MCDM) problem, MCDM methods heve been provided for the solution of the problem. The evaluations of the qualitative criteria are often imprecisely defined. Therefore, in the facility location problem, the conventional approaches tend to be less effective in dealing with the imprecise or vague nature of the linguistic assessment.

Authors

Hindi and Pienkosz

Holmberg et al.

Piersma

Al-Sultan et.al

Avella and Sforza

Hindi and Pienkosz

Daskin and Owen

Tragantalerngsak et al.

&KL\RVKLDQG*DOYmR

Hinojosa et al.

Sherali and Park

Khuller et al

Khuller an Sussmann

Nozick

Avella and Sassano

Avella and Sassano

Year

1999

1999

1999

1999

1999

1999

1999

2000

2000

2000

2000

2000

2000

2001

2001

2001

p-Median









p-Centered

 







9 







9  

UCFLP

 9

  

  

 9

  

 9

  

9 

  

  



































LP





IP

9 

9 

9 

9 

9 













MIP













9 

9 















9 







9 9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 





Stochastic

9 



















Other

  9 

  

  

9   







9 

9 







9 



SSCFLP

 9

Fix Charged

9   





CFLP

9 



9 







9 









Deterministic

BB













































9 











9 



Method Type

DP

Attribute





GA



 

























9 



9 



9 

9 













TS











9 

9





9









Other

 



















9

9 







9 



















AA

9 9 













9 





9 9 

9 

9 

LR

Problem Type

AC

Table 14.1 Problem types and solution methodologies for facility location in crisp set

332 C. Kahraman, S. Cebi, and F. Tuysuz

Melkote and Daskin

Melkote and Daskin

Jaramillo et al.

Sun and Gu

Alfieri

Baldacci et al

Berman

Daskin et al.

Farias

Diaz and Fernandez

Garcia-Lopez et al.

Salhi

9  Agarwal and Procopiuc  9 Bespamyatnikh et al. 9 9 Konemann et al.  9

Shimizu and Wada

2001

2001

2002

2002

2002

2002

2002

2002

2002

2002

2002

2002

2003

2002

2002

2002

Hansen et al









 





9 



9 



9 

9 



9 







9 

Authors

p-Median

2001

p-Centered

Year

CFLP

Fix Charged

SSCFLP

UCFLP



9 9 

  

Other

  











  

  

  

  

  

 9

9 

















  9 

  

9   



















9   

9   



9   

9   9 





Stochastic



































LP

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 





















9 9 



BB









































































































9 

9 

9 













9 





9 





9 









9 

9 



MIP

9 









IP

9 9 

9 

9 

9 

9 

9 

Deterministic

Method Type

DP

Attribute

LR

Problem Type

AC

Table 14.1 (continued)

GA

















AA



















9





9



Other





















9

















9



9 









9 











9 

TS

9 

















Fuzzy Location Selection Techniques 333

Alp et al.

Harkness and ReVelle

Jayaraman and Ross

Kalcsics et al.

Huang, R. Li

Marianov et al

Thanh et al.

Bischoff et al.

Puerto et al.

Meng et al.

 Albareda-Sambola et al.  Du et al. 

Kuban et al.

Engin et al.

Zaferanieh et al.

2003

2003

2003

2008

2008

2008

2009

2009

2009

2009

2009

2009

2009

2009

Ortega, F., Wolsey

2003



p-Centered



 







CFLP

































9 



SSCFLP

UCFLP

  

9 

  

  

  

9 

  



 9

9   

9   

9   



 





Stochastic



















9 





9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9  

LP

9 

9 



9 



9 









9 

9 

  9 

  

9   





Other

9 9 

Fix Charged

9   





9 9 





9 



Authors

2003

p-Median

Year

Deterministic

IP





MIP



























































9 





9 















































9 







9 















9 

















9 

BB

9 









9 





Method Type

DP

Attribute

LR

Problem Type

AC

Table 14.1 (continued)

GA

































TS





AA





































9





9



Other







9

9

9

9 

9 

9 















9 





334 C. Kahraman, S. Cebi, and F. Tuysuz

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Fig. 14.3 Frequency-years graph of the papers including the facility location problem

There has been an increasing interest for fuzzy sets to be used for the facility location problem in the literature (Kahraman et al. 2003). A fuzzy decision making method under multiple criteria integrates the various linguistic assessments and weights to evaluate the location suitability and determine the best selection (Chen 2001). In Table 14.2, the column Evaluation of Criteria represents whether the study is crisp or fuzzy. The column Solution Algorithms includes the methods such as Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS), Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), Simple Additive Weighting (SAW), Mathematical Programming (MP) and other methods such as PROMETHEE, Quality Function Deployment (QFD), etc. According to Table 14.2, in recent years, the fuzzy set theory has been preferred to the crisp set theory and AHP is the most used MCDM method for the location selection problem (Figure 14.4).

Fig. 14.4 Frequency-years graph: the preferred sets for facility location selection problems

3 Location Selection Techniques: A classification Location analysis refers to the modeling, formulation and solution of class problems that can be defined as placing facilities in some given space. There are many different approaches and techniques used in location selection. The use of which depends on the type of the problem on hand. For example, the factory location problems may involve a different set of criteria or objectives when compared to the retail location problems. Facility location problems are solved to minimize the total cost of serving all customers whereas retail location problems focus on how to identify a location that can maximize the number of the customers. The

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Table 14.2 Methods and algorithms for facility location selection problem

classification of the techniques and models used in location analysis can be done in a variety of ways. For different classificition of techniques and models, see Owen and Daskin (1998), Hale and Moberg (2003), Cheng and Li (2004), Klose and Drexl (2005), Revelle and Eiselt (2005), Revelle et al. (2008). The formulation and solution of each technique and model vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. We classify the literature on location analysis into two main categories as follows: (1) Mathematical Programming Models (Network Models, Continuous Models, Discrete Models), (2) Multi-criteria Methods (Simple Additive Weighting, AHP, ANP, TOPSIS)

3.1 Mathematical Programming Models Mathematical models generally use linear programming, goal programming and dynamic programming to find the optimal location. The aim of these models is to maximize the utilization of the sources and minimize the overall cost and/or maximize the profit. Mathematical programming models can be classified as static and deterministic models and dynamic and stochastic models (Owen and Daskin 1998, Cheng and Li 2004). Mathematical models consist of network model, continuous model, and discrete models. Network models assume that the location problem is embedded in a network consisting of links and nodes. The demands or the possible location of facilities occur on the nodes as well as edges of the network. Distances in network location

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problems are measured on the network itself, typically as the shortest route on the network of arcs connecting the two points. Network models can be either continuous or discrete location problems. Continuous location models have two important assumptions: the first, the solution space is continuous, that is, facilities can be located on any point in the plane while demands are often taken as being at discrete locations. The second, distance is measured with a suitable metric, typically the Manhattan (right-angle) distance metric or Euclidean (straight-line) distance metric are used. Continuous location models require calculating the coordinates ( x, y ) in the plane for p facilities. The objective is to minimize the sum of the distances between the p facilities and the m demand points. Continuous models can be applied in limited contexts in which it is possible to locate facilities anywhere in the space being considered (ReVelle et al. 2008). The placement of a helicopter for trauma rescue or the location of an ambulance along a highway is two examples of continuous models. These models are usually planar problems and tend to be nonlinear optimization problems. Discrete location models assume that there is a discrete set of demands and a discrete set of candidate locations. In discrete models, the facilities can only be located only at a limited number of suitable points on the plane or network. Due to the restricted location set, the problem is somewhat easier which allows for more realism in the models. These problems are often network problems and generally formulated as integer or mixed integer programming problems which are NP-hard type. Discrete location problems are the most widely used ones in practical applications. Retail location problems can be given as examples of this type of models. These models are based on some assumptions and many extensions of these models with some modifications have been developed for case specific conditions. Although the contexts in which these models are situated may differ, their main features are always the same: a space including a metric, customers or demand points whose locations in the given space are known, and facilities whose locations have to be determined according to some objective function (Revelle et al. 2008).

3.2 Multi-criteria Methods The appropriateness of a location depends on the factors that are selected and evaluated together with their effects on organization’s objectives and operations. The mathematical models and their extensions usually take into consideration a few factors and one or two objectives. Since the mathematical models seek optimal solutions as the number of the alternatives and objectives increase, the complexity of the problem increases which results a long solution procedure difficult to solve and may sometimes give no feasible solution. Due to the reason that location selection problem encompasses several independent criteria and each criterion has numerous detailed subcriteria, location selection is a typical MCDM problem which requires both quantitative and qualitative factors to be evaluated. MCDM is the approach dealing with the ranking and selection of one or more location sites from the alternatives. Recent studies mainly focus on the use of multiple crtiteria

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methods for location selection. In this section we present some of the most commonly used multiple criteria evaluation methods for location selection decisions. 3.2.1 Simple Additive Weighting Simple additive weighting (SAW) method is the best known and widely used MCDM. In this method, a score is obtained by adding the contributions of each criterion. Since criteria can be in different measurement units, normalization is applied for addition among criteria. Linear normalization is most often used with the SAW method. The value of each alternative in the SAW method is calculated as n

V ( Ai ) = Vi = ∑ w j v j ( xij ) , i = 1,..., m

(14.1)

j =1

where V ( Ai ) is the value function of alternative Ai , and w j and v j are the weight and value functions of criteria X j respectively. After the linear normalization, the index value of each alternative Ai can be rewritten as n

Vi = ∑ w j rij ,

i = 1,..., m

(14.2)

j =1

where rij is the comparable scale of xij obtained by the normalization process. In this method, first the weights for each criterion are assigned to reflect its relative importance to the location selection decision for which a 1-10 point scale can be used. Then the criteria are ranked in order of importance. The final weights are obtained by normalizing the sum of the points or scores to one. The underlying assumption of this method is that the criteria are preferentially independent which means that the contribution of each criterion to the total score is independent of other criteria values. Also SAW method presumes the weights are proportional to the relative value of a unit change in each criterion’s value function. 3.2.2 Analytic Hierarchy Process and Analytic Network Process Both AHP and ANP were developed and introduced by Saaty. ANP (Saaty 1996) is a general form of AHP (Saaty 1980). AHP and ANP mainly focus on how to solve decision problems with uncertainty and with multiple criteria characteristics by decomposing a complex problem into a hierarchy. They both incorporate the evaluations of all decision makers into a final decision, without having to elicit their utility functions on subjective and objective criteria, by pairwise comparisons of alternatives. The ANP is a comprehensive decision making technique that captures the outcome of dependence and feedback within and between clusters of elements. Whereas AHP represents a framework based on a unidirectional hierarchical relationship, ANP permits more complex interrelationships among decision levels and attributes (Saaty 1996). Let C1 , C2 ,..., Cn denote the set of elements, while aij represents a quantified judgement on a pair of elements Ci and C j . The relative importance of two

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elements is rated using a scale with the values 1, 3, 5, 7 and 9, where 1 refers to “equally important”, 3 denotes “slightly more important”, 5 equals “strongly more important”, 7 represents “demosrably more important” and 9 means “absolutely more important”. This produces a n × n square matrix A as follows: C1 ⎡ 1 a12 C2 ⎢⎢1 / a12 1 A = ⎡⎣ aij ⎤⎦ = . ⎢ ... ... ⎢ Cn ⎣1 / a1n 1 / a2 n

... a1n ⎤ ... a2 n ⎥⎥ ... ... ⎥ ⎥ ... 1 ⎦

(14.3)

where aij = 1 when i = j , and aij = 1 / a ji for i, j = 1, 2,..., n . In A, the problem becomes one of assigning to the n elements C1 , C2 ,..., Cn a set of numerical weights W1 ,W2 ,...,Wn that reflect the recorded judgments. If A is a consistency matrix, the relations between weights Wi and judgments aij are simply given by Wi / W j = aij and C1 ⎡ w1 / w1 C2 ⎢⎢ w2 / w1 A= . ⎢ ... ⎢ Cn ⎣ wn / w1

w1 / w2 ... w1 / wn ⎤ w2 / w2 ... w2 / wn ⎥⎥ ... ... ... ⎥ ⎥ wn / w2 ... wn / wn ⎦

Saaty (1990) suggests that the largest eigenvalue n

Wj

j =1

Wi

λmax = ∑ aij

(14.4)

λ max would be (14.5)

If A is a consistency matrix, eigenvector X can be calculated as ( A − λmax I ) X = 0

(14.6)

In logical consistency check step, consistency index CI and consistency ratio CR are used to verify the consistency of the comparison matrix. CI and CR are calculated as follows: CI = (

λmax − n n −1

) and CR =

CI RI

(14.7)

where RI represents the average consistency index over numerous entries of same order reciprocal matrices. If CR ≤ 0.1 the matrix is accepted as consistent, otherwise the evaluation procedure is repeated to improve consistency until CR ≤ 0.1 . ANP has four basic steps which are deconstructing a problem into a complete set of network models, generating pairwise comparisons to estimate priorities at each level, building a supermatrix to represent the influence priority of elements and making decisions based on the supermatrix. As it can be seen the first two steps are similar to the AHP method. The main difference is that, in ANP method, the supermatrix concept which resembles the Markov chain process is used

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(Saaty 1996). This supermatrix handles the interdependence characteristics among elements and components. To obtain global priorities in a system with interdependent influences, the local priority vectors are added to the appropriate columns of a matrix, known as a supermatrix. A supermatrix is actually a partioned matrix, in which each matrix segment represents a relationship between the two nodes in a system (Meade and Sarkis 1999). In ANP method, first the relative importance weight of each criterion from pairwise comparisons is entered into the unweighted supermatrix. To determine the final local priorities to the global priorities, the limit supermatrix can be obtained by raising the weighted supermatrix to its powers. In the limiting case, each column is the same and this obtained matrix is the limit supermatrix for the criteria. For further information about the AHP and the ANP methods, see Saaty (1980, 1996, 2005). 3.2.3 TOPSIS Method The basic principle of the technique for order preference by similarity to ideal solution (TOPSIS) which was proposed by Hwang and Yoon (1981) is that the chosen alternative should have the shortest distance from the positive ideal-solution and the longest distance from the negative-ideal solution (Yoon and Hwang, 1995). The TOPSIS method can be summarized as follows: Let A1 , A2 ,..., Aj be the j different alternatives. For alternative A j the rating of the ith aspect is denoted by f ij which is the value of the ith criterion function for the alternative A j , and n is the number of the criteria. Then Step 1. Calculate the normalized decision matrix: The normalized value rij for which vector normalization is used is calculated as: fij

rij =

,

n

∑ j =1

j = 1, 2,..., J and i = 1,2,..., n

(14.8)

fij2

Step 2. Calculate the wighted normalized decision matrix: the weighted normalized value v ij is calculated as: vij = wi ∗ rij

j = 1, 2,..., J and i = 1,2,..., n

(14.9)

n

where

wi is the weight of the ith attribute or criterion, and ∑ wi = 1 . i =1

Step 3. Determine the ideal and negative solution: the in terms of the weighted normalized values:

{ = {v ,..., v } = {(min v

∗

A and the A − are defined

A∗ = {v1∗ ,..., vi∗ } = (max vij i ∈ I ' ),(min vij i ∈ I '' )} j

A−

− 1

− i

j

ij

j

i ∈ I ' ),(max vij i ∈ I '' )} j

where I ' is a set of benefit criteria and I '' is a set of cost criteria.

(14.10) (14.11)

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Step 4. Calculate the seperation measures using the n-dimensional Euclidean distance. The seperation of the each alternative from the positive-ideal solution is given as n

∑ (v

D∗j =

ij

i =1

− vi∗ )2

j = 1, 2,..., J

(14.12)

Similarly, the separation from the negative-ideal solution is given as: D −j =

n

∑ (v

ij

i =1

− vi− )2

j = 1, 2,..., J

(14.13)

Step 5. Calculate the relative closeness to the positive-ideal solution: the relative closeness of the alternative A j is defined as C ∗j =

D −j ∗ j

D + D −j

j = 1, 2,..., J

(14.14)

Step 6. Rank the preference order: choose an alternative with the maximum C ∗j or rank the alternatives according to C ∗j in descending order.

4 Fuzzy Approaches to Facility Location Selection The location selection includes both quantitative and qualitative criteria. In most cases, the values for the qualitative criteria are often imprecisely defined for the decision makers. In evaluating the qualitative criteria, the desired value and the weight of importance for the criteria are usually defined in linguistic terms. The conventional approaches tend to be less effective in dealing with the imprecision or vagueness nature of the linguistic assessment. Due to this reason, there has been an increasing interest for fuzzy sets to be used for the location selection problem in recent years (Yager 1982). In this subsection, fuzzy AHP, fuzzy TOPSIS and fuzzy Information Axiom are discussed. The following numerical example is taken into consideration by each method in order to illustrate the methods (Cebi 2007). The problem is that a construction firm is looking for a suitable location for its public housing project to maximize its profit obtained by selling each flat. The criteria, unit selling price (C1), unit purchasing price (C2), and selling velocity of flats (demand) (C3), the characters of ground on which a building rests (C4), zoning status (C5), and transportation opportunity (C6) have an important role on the final decision. Except for C2, the rest of the criteria are benefit.

4.1 Fuzzy AHP Method There are many fuzzy AHP methods proposed by various authors (see Demirel et al. 2008 for further information). These methods are systematic approaches to the

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alternative selection and justification problem by using the concepts of fuzzy set theory and hierarchical structure analysis. Decision-makers usually find that it is more confident to give interval judgments than fixed value judgments. This is because usually he/she is unable to explicit about his/her preferences due to the fuzzy nature of the comparison process (Kahraman et al. 2003). The earliest work in fuzzy AHP appeared in van Laarhoven and Pedrycz (1983), which compared fuzzy ratios described by triangular membership functions. Buckley (1985) determines fuzzy priorities of comparison ratios whose membership functions trapezoidal. Chang (1996) introduces a new approach for handling fuzzy AHP, with the use of triangular fuzzy numbers for pairwise comparison scale of fuzzy AHP, and the use of the extent analysis method for the synthetic extent values of the pairwise comparisons. Here we will give Buckley’s (1985) fuzzy AHP approach in detail. Step 1. Consult the decision maker, and obtain the comparison matrix A whose elements are p% ij = ( aij , bij , cij ) where all i and j are triangular fuzzy numbers. Step 2. The fuzzy weights

~ are calculated as follows. The geometric mean of w i

fuzzy comparison value of attribute i to each attribute can be found as 1

⎡ n ⎤n r%i = ⎢∏ p% ij ⎥ , for all i ⎣ j =1 ⎦

(14.15)

then the fuzzy weight w% i of the ith attribute indicated by a triangular fuzzy number is calculated as −1

⎡ n ⎤ w% i = r%i ⊗ ⎢ ∑ r%j ⎥ = ( wil , wim , wiu ) ⎣ j =1 ⎦

(14.16)

Eqs. 14.15-14.16 are repeated for obtaining the fuzzy performance values of alternatives. Step 3. Defuzzyfication of the results. Finally the fuzzy priority weights are converted into crisp values by using the center of area method as follows wi =

w% i n

∑ w% j

=

wil + wim + wiu

j =1

n

∑ w% j

(14.17)

j =1

Step 4. The aggregation of the fuzzy weights and fuzzy performance values. The fuzzy priority weight of each alternative is calculated as follows n

Pi = ∑ w j rij , ∀i j =1

4.1.1 A Numerical Example to Fuzzy AHP Method The hierarchy of the problem is presented in Figure 14.5.

(4.18)

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Fig. 14.5 Problem hieararchy

Step 1. The evaluation of the criteria and alternatives are presented by Table 14.3 and Table 14.4. The mean of the labels which is used in the tables are presented by linguistic terms in Table 14.5. Table 14.3 Pairwise comparisons for criteria

C1

C1

C2

C3

-

Es

1/ES

Eq

Wk

Vs

-

1/Vs

Vs

Vs

Ab

-

Vs

Vs

Vs

-

Es

Vs

-

Vs

C2 C3 C4

C4

C5

C5

C6

C6

-

Table 14.4 Pairwise comparisons of alternatives with respect to criteria

Table 14.5 Linguistic scale for weight matrix (Hsieh et al. 2004) Linguistic scales

Scale of fuzzy number

(1,1,3)

Equally important

(Eq)

(1,3,5)

Weakly important

(Wk)

(3,5,7)

Essentially important

(Es)

(5,7,9)

Very strongly important

(Vs)

(7,9,9)

Absolutely important

(Ab)

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Step 2. Before the calculation procedure, the evaluations given in Table 14.3-14.4 are covered into triangular fuzzy numbers. For the transformation, Table 14.5 is used (Hsieh et al. 2004). An example is given for the criteria in Table 14.6. Table 14.6 Triangular fuzzy numbers for the evaluation of Table 14.3

The fuzzy weights

wi for the criteria are calculated as follows:

1 1 1 ⎡ ⎤ z%1 = ⎢(1 × 3 × 0.14 × 1 × 1 × 5) 6 , (1 × 5 × 0.2 × 3 × 1 × 7) 6 , (1 × 7 × 0.33 × 3 × 5 × 9) 6 ⎥ ⎣ ⎦ z%1 = (1.14, 1.66, 2.61)

The calculated z%i values for the other criteria are; ~ z 2 =(1.19, 1.53, 1.91),

~ z 3 =(3.51, 4.79, 5.98), ~z 4 =(0.63, 0.95, 1.17), ~ z 5 = (0.35, 0.46, 0.70), ~ z 6 =(0.16, 0.19, 0.25). The fuzzy weights are calculated as follows; −1

⎡ n ⎤ w% 1 = z%1 ⊗ ⎢∑ z% j ⎥ = (1.14 × 0.08, 1.66 × 0.1, 2.61 × 0.14 ) ⎣ j =1 ⎦ w% 1 = (0.09, 0.17, 0.37)

~ values for the other criteria are; w ~ = (0.09, 0.16, 0.27), The calculated w 1 2

~ = (0.28, 0.50, 0.86), w ~ = (0.03, 0.05, 0.10), w ~ = ~ = (0.05, 0.10, 0.17), w w 3 5 6 4 (0.01, 0.02, 0.04). The weights of criteria are given in Table 14.7. Step 3. The aggregation of the fuzzy weights and fuzzy performance scores is done (Table 14.7).The order of the alternatives is as follows; A2>A1>A3 according to Table 14.7. So the best alternative is A2 Table 14.7 The weights of criteria and scores of alternatives Criteria C1

C2 C3 C4 C5 C6

Weights 0.19 0.2 0.5 0.1 0.1 0

Total Score

A1

0.21 0.5 0.3 0.4 0.2 0.5 0.32

A2

0.33 0.4 0.6 0.5 0.3 0.4 0.47

A3

0.46 0.1 0.2 0.1 0.5 0.1 0.21

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4.2 Fuzzy TOPSIS Method There are some different fuzzy TOPSIS methods developed in the literature. A comparison of the fuzzy TOPSIS methods in the literature is given in Table 14.8 which includes the computational differences among the methods. In addition to the fuzzy TOPSIS approaches presented in Table 14.8, Kahraman et al. (2008) proposed a new fuzzy TOPSIS method called as fuzzy hierarchical TOPSIS which can be used for solving multi-attribute hierarchical problems. The proposed approach combines the hierarchical structure of AHP method, which provides a great superiority in multi-attribute problems, together with the TOPSIS method. Table 14.8 A Comparison of Fuzzy TOPSIS Methods

In this chapter, we present Chen’s (2000) fuzzy TOPSIS method. The methodology consists of six main steps which are listed below: Step 1. Decision makers evaluate the importance of criteria and the ratings of alternatives with respect to the criteria. Step 2. If there are any linguistic terms which are used in evaluation phase, they must be transformed into fuzzy triangular numbers (TFNs). Then rating of

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alternatives ( x%ij ) and the importance of the criteria ( w% j ) is calculated by Equation 14.19 and Equation 14.20. 1 1 ( x%ij + x%ij2 + ... + x%ijK ) , x%ij = (aij , bij , cij ) K

14.19)

1 1 ( w% j + w% 2j + ... + w% Kj ) , w% j = ( w j1 , w j 2 , w j 3 ) K

(14.20)

x%ij = w% j =

where K is the number of decision makers. Step 3. In this step, evaluation values are normalized because of two different scales. To avoid the complicated normalization formula used in classical TOPSIS, the linear scale transformation is used to obtain normalized fuzzy decision matrix

~

denoted by R . R% = [ r%ij ]mxn

⎛a b c ⎞ ⇒ r%ij = ⎜ ij* , ij* , ij* ⎟ ⎜ cj cj cj ⎟ ⎝ ⎠

(14.21)

where c*j = max cij if the criterion is benefit. Otherwise, if the criterion is cost, the i

following equation is used.

⎛ a −j a −j a −j ⎞ ~ rij = ⎜ , , ⎟ ⎜ cij bij aij ⎟ ⎝ ⎠ where

(14.22)

a −j = min aij i

Step 4. The weighted normalized fuzzy decision matrix is constructed as follows: V% = [v%ij ]mxn

v%ij = r%ij .w% j

i = 1, 2,L, m

j = 1, 2,L, n

(14.23)

Step 5. Then, the distances ( d i* , d i− ) of each alternative from fuzzy positive-ideal solution (FPIS, A*) and fuzzy negative-ideal solution (FNIS, A-) are calculated, respectively. A* = (v1* , v2* ,...., vn* ) where v*j = (1, 1, 1) A− = (v1− , v2− ,...., vn− ) where v −j = (0, 0, 0) n

n

j =1

j =1

d i* = ∑ d (v%ij , v% *j ) i = 1, 2,L, m di− = ∑ d (v%ij , v% −j ) i = 1, 2,L , m

(14.24) (14.25)

*

−

Step 6. A closeness coefficient ( CCi ) is calculated by using d i and d i in Eq. 14.26. CCi =

di− d + di− * i

i = 1, 2,L, m

(14.26)

Obviously, if an alternative Ai is closer to the FPIS (A*) and farther from FNIS (A ), CCi approaches to 1. Hence, the alternatives are ranked via CCi. The alternative that has the biggest CCi value is the best in all for our goal. -

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4.2.1 A Numerical Example to Fuzzy TOPSIS Step 1. In the presented numerical example, a decision maker evaluates the alternatives with respect to the related criteria. The evaluation values are presented in Table 14.9. Table 14.9 Assessments of alternatives C1 ($)

C2($)

C3

C4

C5

C6

A1

~1000

~2000

M

M

5

VH

A2

~2000

~1750

VH

H

6

H

A3

~2500

~1000

L

VL

8

L

Step 2. In Table 14.9, linguistic terms, approximate numbers and precise numbers are used during the evaluation procedure. So, these values are transformed into triangular fuzzy numbers. Table 14.10 is used for linguistic terms and Table 14.11 presents the triangular fuzzy numbers for evaluation values. Table 14.10 Linguistic scale Linguistic

Triangular Fuzzy Numbers

Terms

Label

Lower

Upper

Middle

Very Low

VL

0

0

0.25

Low

L

0

0.25

0.5

Medium

M

0.25

0.5

0.75

High

H

0.5

0.75

1

Very High

VH

0.75

1

1

Table 14.11 Triangular fuzzy numbers for expert’s assessment C1 L A1

M

C2 U

L

M

C3 U

L

M

900 1000 1100 1800 2000 2200 0.25 0.5

C4 U

L

M

0.75 0.25 0.5

A2 1800 2000 2200 1600 1750 1900 0.75 1 1 A3 2250 2500 2750 900 1000 1100 0 0.25 0.5

0.5 0

C5 U

C6

L M U

L

0.75 5

5

5 0.75

0.75 1 6 0 0.25 8

6 8

6 8

M

U

1

1

0.5 0.75 1 0 0.25 0.5

Step 4. For the weight of the criteria, the following values are taken into consideration; 0.19, 0.16, 0.49, 0.09, 0.05, 0.02. Hence, the weighted normalized fuzzy decision matrix is given in Table 14.12. Table 14.12 Weighted normalized fuzzy decision matrix

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C. Kahraman, S. Cebi, and F. Tuysuz *

−

Step 5. Then, the distances ( d i , d i ) of each alternative from fuzzy positive-ideal solution (Table 14.13) and fuzzy negative-ideal solution (Table 14.14) are calculated, respectively. Table 14.13 The distances of each alternative from fuzzy positive-ideal solution C1

C2

C3

C4

C5

C6

A*

A1

0.93

0.93

0.76

0.91

0.97

0.98

5.48

A2

0.86

0.92

0.55

0.87

0.96

0.99

5.15

A3

0.83

0.86

0.88

0.99

0.95

1.00

5.50

Table 14.14 The distances of each alternative from negative-ideal solution C1

C2

C3

C4

C5

C6

A-

A1

0.07

0.07

0.26

0.10

0.03

0.02

0.55

A2

0.14

0.08

0.45

0.14

0.04

0.02

0.87

A3

0.17

0.15

0.16

0.03

0.05

0.01

0.56 *

Step 6. Closeness coefficient ( CCi ) of alternatives calculated by using d i and d i− in Eq. 14.26 are 0.092, 0.144, 0.092, respectively. The order of the alternatives is as follows; A2>A1=A3. So the best alternative is A2

4.3 Fuzzy Information Axiom Information axiom is the second axiom of Axiomatic Design (AD) methodology which has been proposed to compose a scientific and systematic basis that provides structure to design process for engineers. The primarily goal of AD is to provide a thinking process to create a new design and/or to improve the existing design (Suh 2005). The first axiom, independence axiom, is related to the independence of functional requirements (FRs) which are defined as the minimum set of independent requirements that characterizes the design goals (Suh 2001). The information axiom is a conventional method and facilitates the selection of proper alternative that has minimum information content (Suh 1990). In other words, information axiom is used to select the best alternative when there is more than one design that satisfies independence axiom. The information axiom is symbolized by the information content (I) that is related to probability of satisfying the design goals (Suh 1990). I is calculated by I = log 2

1 pi

(14.27)

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349

where pi is the probability of achieving a given FR and pi is calculated by Eq. 14.28. If there is more than one FR, information content is obtained as follows (Suh 1990); common area system area

(14.28)

I system = − log 2 p( m )

(14.29)

pi =

m

I system = − log 2 (∏ pi )

(14.30)

i =1

I system = −∑i =1 log 2 pi = ∑i =1 log 2 (1 / pi ) m

m

(14.31)

In the recent studies, Kulak and Kahraman (2005a, 2005b) extended the information axiom under fuzzy environment and they used the developed methodology for multicriteria decision making problems under fuzzy environment. They used triangular fuzzy numbers to depict design goal and properties of the alternatives. Figure 14.6 illustrates the calculation procedure of information content with triangular fuzzy numbers. Both system and design ranges consist of triangular fuzzy numbers. So, information content is calculated by I = log 2

TFN of System Range Common Area

(14.32)

Fig. 14.6 The common area of system and design ranges

In the literature information axiom has been applied to various decision making problems (Kulak and Kahraman 2005a, 2005b, Kulak et al. 2005, Kulak 2005, Coelho and Mourão 2007, Celik et al. 2009a, 2009b, 2009c, Kahraman and Cebi, 2009). The proposed algorithm consists of following steps (Kahraman and Cebi 2009), Step 1. Identify potential decision criteria in the literature and possible alternatives.

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Step 2. Determine the weights of the criteria. The Buckley’s AHP procedure is used to calculate the weights of the criteria Step 3. Construct the decision matrix. The experts are required to provide their judgments on the basis of their knowledge and expertise for each factor. Step 4. Define FRs. In the basis of information axiom method, the most important factor is the definition of FRs and they can be defined by group members via brain storming method. The FRs are identified for three different cases (Kahraman and Cebi 2009). Thus, these cases let us use IA methodology according to the characteristic of the problem. The definition of FRs characterizes the type of the problem and affects the solution of the problem. Step 5. Calculate I values. In this step, the calculation procedure is realized. The calculation procedure is divided into two phases. The first one is the fuzzy phase and the other is the crisp phase. If the evaluation of an alternative is fuzzy, I is calculated as follows (Kahraman and Cebi 2009): Case 1. Exact value problems if there is no intersection ⎧infinitive, ⎪ I =⎨ TFN of System Range , otherwise ⎪log 2 Common Area ⎩

(14.33)

Case 2. Expected value problems ⎧ ⎪0, α ≤ a, β ≤ b for benefit criteria ⎪ ⎪infinitive, c ≤ α for benefit criteria ⎪ I = ⎨0, β ≥ b, θ ≥ c for cost criteria ⎪infinitive, a > θ for cost criteria ⎪ TFN of System Range ⎪ , otherwise ⎪ log 2 Common Area ⎩

(14.34)

Case 3. Ranking value problems:Ideal FR definition is used in this case. To rank the alternatives, the limits of FRs are chosen for benefit attributes for α=0, µ(α)=0 and for β=θ=Xmax (maximum upper value of the alternative), µ(θ)=1 and for cost attributes for α=β=0, µ(α)=1 and for θ=Xmax, µ(θ)=0. This definition is named as ideal FR (IFR). For the calculation of I values, Eq. (14.33) is used (Kahraman and Cebi 2009). If there are objective criteria in the evaluation of an alternative like Si = {xi | ∀x ∈ R} , I is calculated by Eq. (14.35), ⎧ xi − α ⎪⎪ θ − α , 1 I = log 2 , μ ( x) = ⎨ μ ( xi ) ⎪θ − xi , ⎪⎩ θ − α

for benefit attributes

(14.35) for cos t attributes

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Step 6. Select the best alternative. In this step, information contents (I) and total information contents (It) are calculated. After the calculation procedure, the alternative which has the minimum It value is selected as the best alternative. 4.3.1 A Numerical Example to Fuzzy Information Axiom Step 1. The potential criterion was given in Section 4. Step 2. The criteria weights determined by Buckley’s AHP procedure given in Section 4.1.1 are used. Step 3. The evaluation values presented in Table 14.9 and the triangular fuzzy numbers given in Table 14.11 are used. Step 4. To rank the alternatives, ideal FR definition is used. To make the calculation procedure easy, the evaluation values for C1, C2, and C5 are normalized. So, the ideal FRs are as follow; (0,0,1) for C2 (cost criterion) and (0, 1,1) for the others. Step 5. The calculated I values are presented in Table 14.15. Table 14.15 The calculated I values Criteria

C1

C2

C3

C4

C5

Weights

0.19

0.16

0.49

0.09

0.05

C6 0.02

A1

0.750

0.331

0.447

0.447

1.379

0.000

A2

0.112

0.448

0.000

0.100

1.222

0.100

A3

0.012

2.573

1.322

2.322

1.000

1.322

Step 6. The weighted information contents and total values are presented in Table 14.16. The order of the alternatives is as follow A2>A1>A3 according to Table 14.16. So the best location alternative is A2. Table 14.16 The weighted I values Criteria

C1

C2

C3

C4

C5

C6

Total

Weights

0.19

0.16

0.49

0.09

0.05

0.02

I

A1

0.14

0.05

0.22

0.04

0.07

0.00

0.52

A2

0.02

0.07

0.00

0.01

0.06

0.00

0.17

A3

0.00

0.41

0.65

0.21

0.05

0.03

1.35

5 Conclusions and Suggestions for Further Research Facility location decisions play an important role in organizations’ strategic plans and directly affect the overall success of the firm. Although the facility location

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decisions usually depend on the type of the business and the sector, today it is commonly accepted that location selection has important strategic implications since usually it involves long-term commitment of resources. The appropriateness of a location mostly depends on the considered factors that are evaluated with respect to their effects on organization’s objectives and operations. Since there is a growing literature and interest in location science, in the scope of this chapter, approaches and techniques for facility location selection problems are taken into consideration. From literature review it is concluded that MCDM methods have been widely used and applied for the location selection problem. Moreover, fuzzy MCDM methods have been increasingly used day by day. Therefore, this chapter mainly focuses on the fuzzy multi-criteria decision making methods. Especially, fuzzy AHP and fuzzy TOPSIS are presented since they are the most widely used ones. Furthermore, it is the first time a framework based on the fuzzy information axiom is proposed for facility location selection. The main contribution of the proposed approach is that it provides a great flexibility for decision makers. They have opportunities to define decision goals by functional requirements so that any alternative which does not satisfy the decision makers’ aim is eliminated. Due to the mentioned reasons addressing the importance of a facility location selection decision, it is suggested that one or more methods which are mentioned above should be used for deriving a final decision. For further research, a sensitivity analysis can be added into multi-criteria decision making methods in order to better analyze the reliability of the results.

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Chapter 15

Fuzziness in Materials Flow and Plant Layout Kuldip Singh Sangwan*

Abstract. This chapter presents a multicriteria heuristic model to solve the plant layout problem taking into consideration the quantitative and qualitative factors affecting plant layout in a fuzzy environment. The fuzzy approach integrated with a multicriteria decision making method, Analytical Hierarchy Process (AHP), has been used in the model. An attempt has been made to make the model practical by taking into consideration the product demand, transfer batch size and multiple non-consecutive visits of parts to the same facility.

1 Introduction Plant layout deals with the arrangement of the most valuable assets of an organization i.e., facilities; so that man, machine and material can work together efficiently and safely. A facility in this context is a physical entity such as a machine tool, a workcentre, a manufacturing cell, a welding shop, a department, a warehouse, etc., which facilitates the performance of a job. The basic objective of the plant layout is to ensure a smooth flow of material and people through the system. Effective plant layouts: • • • • • • • • •

Minimize material handling cost Minimize manufacturing cycle time Minimize capital investment Minimize inventory Utilize space and labour efficiently Provide employee convenience, safety and comfort Eliminate bottlenecks Provide visual controls Maintain flexibility in changing conditions

Plant layout affects everyone in an organization. Top management is interested in plant layout as layout involves long term commitment and requires substantial irreversible investment in terms of money and efforts. Middle management is interested in plant layout as it affects output and has significant impact on cost and Kuldip Singh Sangwan Mechanical Engineering, Birla Institute of Technology and Science, Pilani, India C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 359–380. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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efficiency of operations. Finally, workers are interested in layout as it affects their output, earning, efficiency and comfort. The plant layout is of tremendous importance for effective utilization of facilities and cost saving as in manufacturing it has been emphasized that 20 – 50 % of the total operating costs are attributed to the material handling and layout related costs (Tompkins and White 1984, Sule 1988). Use of effective methods for plant layout can reduce the costs at least by 30% (Chiang and Kouvelis 1996), thus leading to tremendous savings. In fact, material processing time in comparison to total production time is usually less than 10% for an unautomated plant. Moreover, a large number of accidents or damage to the part happen during material flow and material flow accounts for 60% wage bill (Matson et al 1992). It is for this reason that the unnecessary material flow should be eliminated during plant layout. Layout is designed to facilitate the flow of the product from raw material to the finished product. Layout design starts with the collection of data about the product to be produced (Muther 1961). This data can be quantitative, such as flow relationships (material flow, equipment flow, people flow, information flow) and/or qualitative, such as control requirements, process relationships, environmental relationships, etc.; which are non-quantifiable and must be considered in making transition from pure model to practicable solution (Shang 1993). Some of the qulitative factors are positive which require proximity of facilities, while others are negative factors, which require the facilities to be farthest apart from each other. It is also important that quantitative approach should not be dismissed in favour of totally qualitative approach, as the quantitative aspect of the facilities problem cannot be reckoned with accuracy through intuition alone (Francis and White 1974). Further, quantitative data have an element of vagueness or uncertainty in them (Evans et al. 1987, Dweiri and Meier 1996). Traditional layout techniques treat these inputs as exact. In this chapter fuzzy set theory is used, which provides a framework for modeling vague systems and allows for the treatment of uncertainty to derive closeness rating. This chapter aims to provide a multicriteria approach to the plant layout wherein both quantitative and qualitative factors for the design of layout will be considered simultaneously. The qualitative issues will be addressed subjectively and systematically in a strict mathematical sense; at the same time, quantitative factors will be dealt objectively and analytically. Practically, all qualitative factors shall not have equal importance. Therefore, a combination of the Analytic Hierarchy Process (AHP) and fuzzy logic will be used to assign different weights to the qualitative factors between each pair of facilities to make the layouts more practical. The operation sequence, multiple non-consecutive visits to the same facility, product demand and transfer batch size have been considered, which are essential to make the layout models pragmatic (Sangwan and Kodali 2009). Furthermore, to improve the application of the model, the flow data computation has been integrated in the proposed model which otherwise is a cumbersome process and leads to many errors (Sangwan and Kodali 2003). Section 2 of the chapter gives the relevant review of different approaches to the plant layout problem. Section 3 highlights the reasons for fuzziness in material flow and plant layout. Section 4 introduces the fuzzy logic and AHP used in the chapter. The proposed model and computational results are illustrated in section 5

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and 6, respectively. Section 7 provides the application and salient features of the proposed model. The conclusions and scope for further improvements are provided in the last section.

2 Literature Review Based on the layout objective, the literature on plant layout can be broadly classified into three categories: quantitative approach, qualitative approach and multicriteria approach. The quantitative approach of the plant layout is to minimize the material handling cost. The distance travelled by the parts, i.e, flow between the facilities is used as a surrogate measure to measure the material handling cost. One of the most widely known and widely written facilities layout tool, CRAFT (Computerized Relative Allocation of Facilities Technique), presented by Armour and Buffa (1963), is based on the objective of minimization of cost of travel between the facilities. However, models based on quantitative objectives have not been accepted by the researchers to solve the real world problems (Shang 2003, Evans et al 1987) as the input data required by these models is to be exact but in real world this data is uncertain and vague as explained in next section. In plant layout, there remain a number of non-quantifiable questions, which must be considered in making the transition from a pure model to a practicable solution (Shang 2003). The qualitative approach to layout design typically employs REL (relationship) charts. Two earliest and widely used layout tools based on qualitative approach are CORELAP (Computerized Relationship Layout Planning) developed by Lee and Moore (1967) and ALDEP (Automated Layout Design Program) developed by Seehof and Evans (1967). REL charts give the desirability of having a pair of facilities adjacent to each other. The adjacency rating usually based on letters (AAbsolute importance; E–Essential importance; I–Important; O–Ordinary importance; U–Unimportant; X – Negative importance) are converted into numerical data. In this chapter the conversion of letter score into numerical data is done by using Dutta and Sahu (1982) scale (A=6, E=5, I=4, O=3, U=2, X=1). In qualitative approach the objective is to maximize the closeness rating by considering the qualitative factors such as safety, flexibility, noise, dirt, odor, etc. The main problem faced in qualitative approach is the method of scoring (Dutta and Sahu 1982). The scoring is based on pre-assigned numerical values for different closeness ratings but does not consider the flow data. Francis and White (1974) has also pointed out that facility layout problem can not be reckoned with accuracy through intuition alone and quantitative approach should not be dismissed in favour of a completely qualitative approach. This generated the interest of researchers towards multicriteria approach. It is necessary to satisfy multiple objectives such as the overall integration of all the functions, minimum material movement, smooth workflow, employee satisfaction, safety, flexibility etc. for the better design of layouts. Rosenblatt (1979), Dutta and Sahu (1982), Fortenberry and Cox (1985), Urban (1987), Shang (1993), Raoot and Rakshit (1993), Sangwan and Kodali (2003), Sangwan and Kodali (2006), Raman et al (2009) and Sangwan and Kodali (2009) have developed

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models based on the multicriteria approach. In this chapter, a multicriteria model is developed which considers the flow data as well as safety and noise factors for plant layout.

3 Fuzziness in Materials Flow and Plant Layout Flexibility is one of the most crucial parameters for market survival in today’s manufacturing environment as there is a shift from traditional mass production method to a new paradigm of mass customization, which requires adequate manufacturing flexibility at an organization. In addition to traditional manufacturing flexibilities of volume, product mix, expansion, operation, product, process, routing, labour, machine, material handling; another kind of flexibility called FLF (Facilities Layout Flexibility) has to be considered and improved if an organization wants to improve their manufacturing flexibility and hence the productivity (Raman et al 2009). FLF is defined as the ability of a layout to effectively withstand various changes that arise from unceasing transformations in customers’ requirements and the enterprises’ internal disturbances in terms of cost and time (Yang and Peters 1998). Various authors have felt that there is a lot of impreciseness and vagueness in the quantitative flow data considered for the design of layouts (Raman et al 2009, Kulturel-Konak 2007, Deb and Bhattacharyya 2003, Sethi and Sethi 1990, Evans et al. 1987, Dweiri and Meier 1996), which otherwise is treated as exact. However, it is important to note that the flow data is based on the forecasts which typically made several years ahead. This forecast data includes only estimates about the amount of various types of material flow and cost of moving the various types and size of material which may not be exact when plant becomes operational. Following can be summarized as some of the reasons, in modern manufacturing environment, for the fuzziness in flow data: • • • • • •

Functional interrelationships between the factors are not well defined There may exist more than one route for processing Uncertainty in demand and product mix Equipment breakdowns, rejects, reworks and queuing delays Improved facilities requiring high utilization catering to different operations Flexibility factors generally cannot be measured precisely and modeled mathematically • General vagueness in defining flexibility The fuzzy set theory is one approach to deal with the imprecise and vague data (Karwoski and Mital 1986). This provided the motivation for the present research. In this chapter fuzzy logic based methodology is proposed which utilizes the expert’ knowledge and the inconsistency in expert’ decision making is reduced by using AHP.

4 Overview of Fuzzy Logic and Analytical Hierarchy Process Fuzzy set theory was introduced by Zadeh (1965) to deal with vague, imprecise and uncertain problems. The lack of data is the reason for uncertainty in many

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daily problems. Fuzzy set theory has been used as a modeling tool for complex systems that are hard to define precisely but can be controlled and operated by humans. Humans can make decisions in the absence of clearly defined boundaries based on expertise and general knowledge of the task of the system. The human actions are based on the IF-THEN rules, which are developed over the years of knowledge and experience. Basic concepts of fuzzy theory are presented in subsections. More detailed discussion on fuzzy theory can be found in (Zimmermann 1987, Mamdani and Gains 1981, Schmucker 1984, Lee 1990, Klir et al. 1997).

4.1 Definition A collection of objects U has a fuzzy set A described by a membership function µ A that takes values in the interval (0,1), µ A: U → (0,1). Thus A can be represented as: A ={(µ A (u)/u) | u ε U}. The degree that u belongs to U is the membership function µ A(u).

4.2 Fuzzy Linguistic Variables Linguistic variables take on values that are words in natural language, while numerical variables use numbers as values. Since words are usually less precise than numbers, linguistic variables therefore provide a method to characterize complex systems that are ill defined when it comes to a standard numerical description (Zadeh, 1975). A linguistic variable is defined by the name of the variable x and the set term P(x) of the linguistic values of x with each value a fuzzy number defined on U. For example, if a qualitative factor is a linguistic variable, then its term set P(qualitative factor)={Very High, High, Medium, Low, Very Low}, where each term is characterized by a fuzzy set in a universe of discourse U=(0,1).

4.3 Fuzzy Control Fuzzy set theory is very useful in modeling complex and vague systems. It depicts the control actions of the operators when they can only describe their actions using natural language. Fuzzy set theory is a tool that transforms this linguistic control strategy into mathematical control method. Fuzzy control was first introduced by Mamdani (1974). It has been successfully applied to many areas including plant layout (Wilhelm et al. 1987).

4.4 Analytic Hierarchy Process (AHP) AHP is a decision tool for dealing with complex, unstructured and multiple– attribute decisions (Satty, 1980). It uses a pairwise comparison of attributes in the decision-making process. This comparison is called the importance intensity of the attributes affecting the decision. This comparison technique is useful in this

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research for finding factor weights of each attribute affecting the decision. The decision maker assigns an importance intensity number from 1 to 9, which represents the preference of each reason with respect to others. These numbers represent the weight factors of the reasons involved in the decision-making process. Scale 1 reflects equal weightage and 9 reflects absolute importance as shown in Table 15.1 below: Table 15.1 Scale of relative importance for AHP Intensity

Definition

Explanation

1

Equal Importance

Two activities contribute equally to the objective

3

Weak importance of one over the other

5

Essential or strong

7

Very strong

9

Absolute importance

2,4,6,8

Intermediate values

Experience and judgment slightly favour one another Experience and judgment slightly favour one another An activity is strongly importance favoured and its dominance is demonstrated in practice The evidence favouring one activity over another is of the highest degree When compromise is needed

5 Proposed Model for the Design of Plant Layout Using Fuzzy Logic and AHP This section provides the mathematical formulation and the proposed multicriteria heuristic to solve the plant layout problem under fuzziness. The operation sequence, multiple nonconsecutive visits to the same facility, product demand and the transfer batch size has been considered in the proposed model.

5.1 Mathematical Formulation of the Problem The mathematical model given in this section deals with the maximization of the closeness rating between the facilities. The problem is modeled to assign n facilities to n locations. Notations: Indices: i, j : indices for facilities, i, j = 1, …, n (i≠j) k : index for parts, k = 1, …, P p, q : indices for facility locations, p, q = 1, …, n (p≠q)

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Parameters: : production volume of part k for a given planning horizon Dk Bk : transfer batch size of part k Ski : operation number for the operation done on part k using facility i Fijk: flow between facilities i and j for part k (Fijk = Dk/Bk) Variable:

⎧1, if facility i is assigned to location p eip = ⎨ ⎩0, otherwise The objective function is: Maximize n

n

n

n

z = ∑ ∑ ∑ ∑ c ij eip e jq i =1 j =1 p =1q =1

(15.1)

where: cij = Fuzzy closeness rating value when facilities i and j are neighbours with common boundary as determined using fuzzy logic and AHP Subject to: n

∑e

ip

=1

∀p = 1, ...,n

(15.2)

= 1 ∀i = 1, ..., n

(15.3)

i =1

n

∑e

ip

p =1

Constraint (15.2) insures that each location can be assigned to one facility only and constraint (15.3) insures that each facility can be assigned to one location only.

5.2 Development of the Proposed Heuristic for the Plant Layout Step 1: Input Number of facilities (n); number of products to be manufactured/processed with their operation sequence, demand and transfer batch size; number of qualitative factors affecting layout design, relationship matrix (qualitative data) for each pair of facilities and the desired relative importance of each factor affecting plant layout for each pair of facilities. Step 2: Computation of flow data Compute the flow data (fij)for the each pair of facilities required to manufacture/process all products using Eq. (15.4) and then normalize this value using Eq. (15.5). The normalization is necessatiated because the qualitative factors are on a scale of 0-100 and the quantitative factor (flow data) vary depending upon the product demand and transfer batch size. The normalization brings all the factors on the same scale of 0-100.

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n

P

f ij = ∑ ∑ ∑ Fijk a ijk

(15.4)

i =1 j =1 k =1

f ijnor =

f ij f ijmax

× 100

(15.5)

1, if | Ski − S kj | =1& S ki , S kj ≠ 0 where, a k = ⎧⎨ ij ⎩0, otherwise Step 3: Determination of factor weights for each pair of facilities using AHP as below (step i to step ix): Step i) Form matrix [X]n1 where its element

xn =

n

n ∏ a nk where n is the number

k =1

of factors affecting plant layout. Step ii) Compute principal vectors/weight factors [P]n1 where its element pn = xn xn

∑

Step iii) Compute [A]nn[P]n1 = [F]n1 Step iv) Compute [Z]n1 where its element zn = fn/pn Step v) Find the maximum eigenvalue (λmax= ∑ z n n ), which is the average Step vi) Find the consistency index (CI)= (λmax-n)/(n-1) Step vii) Find the random index (RI) from Table 15.2 for number of factors (n) used in the decision making process. Step viii) Find the consistency ratio (CR)=(CI)/(RI). Any value of CR less than or equal to 0.1 is considered an acceptable ratio of consistency. Step ix) If consistency ratio is greater than 0.1, change the relative importance and repeat step i to step vii, otherwise stop. Step 4: Fuzzification of factor weights Define the membership function and linguistic variables for the factor weights (see Figure 15.1) and accordingly fuzzify the factor weights (linguistic variables and membership values) for each pair of facilities. The membership function used here is triangular and the five linguistic variables used are very low, low, medium, high and very high. Step 5: Fuzzification of factors affecting plant layout Define the membership function and the linguistic variables for factors affecting the plant layout (see Figure 15.2). The membership functions are developed using expert’s knowledge, interviews of involved people and/or past history of plant layout. However in this chapter the values of the qualitative factors are chosen arbitrarily on a scale of 0 - 100. The membership function used is trapazoidal. In the actual design process, the designer has to collect the data and define the proper membership function for various factors. Accordingly, the linguistic variables and membership values for all factors for each pair of facilities are computed.

Fuzziness in Materials Flow and Plant Layout

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Table 15.2 The Random Index (RI) for factors used in the decision-making for AHP N

1

2

3

4

5

6

7

8

9

10

11

12

RI

0

0

0.58

0.9

1.12

1.24

1.32

1.41

1.45

1.49

1.51

1.58

Fig. 15.1 Membership function for factor weights

Fig. 15.2 Membership function for factors affecting plant layout

Step 6: Establish the decision making logic or decision rules. These rules usually take the form of IF-THEN rules. These rules imitate the designer’s decision and are conveniently tabulated in look-up tables (see Table 15.3, Table 15.4, and Table 15.5). Table 15.3 Decision rule for flow data (FD) FD/WF

Very Low Low Medium High Very High

Very Low U

U

O

O

I

Low

U

O

O

I

E

Medium

O

O

I

E

E

High

O

I

E

E

A

E

E

A

A

Very High I

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K.S. Sangwan

Table 15.4 Decision rule for safety rating (QF1) QF1/WF

Very Low Low Medium High Very High

Very Low U

U

U

O

O

Low

U

U

O

O

I

Medium

U

O

O

I

E

High

O

O

I

E

A

I

E

A

A

Very High O

Table 15.5 Decision rule for safety rating (QF2) QF2/WF

Very Low Low Medium High Very High

Very Low U

U

O

I

I

Low

U

O

I

I

E

Medium

O

I

I

E

E

High

I

I

E

E

A

Very High I

E

E

A

A

Table Note: A-Absolute importance; E–Essential importance; I–Important; O– Ordinary importance; U–Unimportant Step 7: Use the Minimum operator (Mamdani and Assilian 1975, Mamdani 1976) the membership function of the closeness rating for each decision is the minimum value of the input variable’s membership function - as shown below: label label label μ closeness rating = Minimum {μ input value ........μ input value } 1

2

Step 8: Defuzzification Find the crisp (exact) values of the closeness rating (defuzzification) by centre of area method (Lee 1990) as shown below:

∑ μR × R g

R0 =

r

∑ μR

g

r

Where : Ro = the final crisp rating of the activity r = the rules used in the activity R = the numerical rating of the activity of the rule μ Rg = membership value of the activity for rule Step 9: Repeat steps 2 to 8 for all pair of facilities to generate fuzzy closeness rating matrix for plant layout. Step 10: Input initial layout and compute the objective function value (z) using Eq. (1).

Fuzziness in Materials Flow and Plant Layout

369

Fig. 15.3 Flowchart of the proposed multicriteria heuristic model using fuzzy logic and AHP

Step 11: Generate different layouts by pairwise exchange of facilities and retain the layout with minimum objective value as given below in sub-steps: Step 11.1: Set i=0 Step 11.2: j=i+1 Step 11.3: Exchange i and j, compute znew

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K.S. Sangwan

Step 11.4: If znew ≥ zold, update znew = znew, layoutnew = layoutnew else znew = zold, layoutnew = layoutold Step 11.5: If j < n, j = j+1 and go to Step 11.3 else if i< n-1, set i=i+1and go to Step 11.2 else go to Step 11.6 Step 11.6: Print layout and znew Step 12: Stop The flow diagram of the proposed model is given in Figure 15.3 for a quick lookup of the procedure adopted in the proposed model. Computational results The computer program for the proposed model, coded in C language, was run on a Dell OPTIPLEX 330(Core Duo 3GHz processor) PC. The proposed model has been validated by solving three examples. Example 1 To validate the proposed model an example of assigning five facilities to five locations has been considered. The operation sequence, demand and transfer batch size of the four parts to be manufactured are given in Table 15.6. Three factors – one quantitative (flow data) and two qualitative (safety and noise) factors – are considered for the layout design. Values for these two factors, for each pair of facilities, on a scale of 0-100 are given in Table 15.7 along with the intensity importance for all the three factors considered for the layout design. The computer program was run seven times for different initial layouts as heuristics are sensitive to the initial solution. The computational results for each step of the proposed model are illustrated below: Table 15.6 Operation sequence, demand and transfer batch size for Example 1 Parts

Operation number (machine no.)

Demand

Transfer Batch size

1

1(1), 2(5), 3(2), 4(3)

100

25

2

1(5), 2(4), 3(3)

80

40

3

1(2), 2(3), 3(5)

80

20

4

1(5), 2(3), 3(2)

80

20

Step 1: Input number of facilities (six) and number of products manufactured/processed (four), operation sequence, demand, transfer batch size, qualitative factor values and relative importance of factors given in Table 15.6 and Table 15.7. Step 2: Flow data computation Compute the flow data between each pair of facilities by using Eq. (15.4), e.g., between pair 1-5 only part 1 is transferred. Therefore, f1-5=100/25=4. Similarly, f3-4=2, f3-5=4+4=8 and f4-5=2. Normalize these values for f2-3=4+4+4=12, f2-5=4, each pair of facilities using Eq. (15.5):

Fuzziness in Materials Flow and Plant Layout

f1nor −5 =

371

4 × 100 = 33 12

Step 3: Factor weight computation using AHP Again consider facility pair 1-5. Table 15.8 illustrates a sample calculation of weight factors using AHP for this facility pair. The factor weights for FD, QF1 and QF2 are 0.60, 0.17 and 0.23 respectively. The designer’s decision in this case is consistent since the consistency ratio is less than 0.1. The factor weights determined using AHP for factors and all pair of facilities are shown in Table 15.9. Table 15.7 Flow data (computed), safety rating, noise rating and relative intensity importance of factors for example 1 Intensity Importance of factors

Relationship data Facility pair

Flow Data Safety Rating Noise Rating FD Over FD Over QF1 Over (FD) (QF1) (QF2) QF1 QF2 QF2

1-2

0

0

0

5

2

1

1-3

0

0

72

1

1

1

1-4

0

10

35

2

5

1

1-5

33

20

22

3

1

1/3

2-3

100

50

0

3

2

1/3

2-4

0

0

0

3

2

1/3

2-5

33

70

52

1

1

1

3-4

17

80

26

1

1

1

3-5

67

30

35

1

1

1

4-5

17

45

90

1

1

1

Table 15.8 Sample calculation of factor weights for facility pair 1-5 using AHP k/l

1

2

3

X

Y

P

1

1

3

1

3

1.44

0.43

2

1/3

1

1/3

1/9

0.48

0.14

3

1

3

1

3

1.44

0.43

¦3.36

Step 4: Fuzzification of factor weights The weight factor 0.43 belongs to fuzzy subset High with a membership value of 0.866 (see Figure 15.1). Similarly, 0.14 belongs to fuzzy subset Low with membership values of 0.933. Step 5: Fuzzification of factors FD, QF1 and QF2 have values of 33, 20 and 22 respectively for facility pair 1-5. The linguistic variables and membership values for these factors are Low with 0.7, Low with 1.0 and Low with 1.0 respectively (see Figure 15.2).

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K.S. Sangwan

Table 15.9 Weight factors determined using AHP for example 1 Weight factors (WF) Facility pair FD over QF1 FD over QF2 QF1 over QF2 1-2

0.60

0.17

0.23

1-3

0.33

0.33

0.33

1-4

0.60

0.23

0.17

1-5

0.43

0.14

0.43

2-3

0.52

0.14

0.33

2-4

0.52

0.14

0.33

2-5

0.33

0.33

0.33

3-4

0.33

0.33

0.33

3-5

0.33

0.33

0.33

4-5

0.33

0.33

0.33

Table 15.10 Fuzzy closeness rating matrix for Example 1 Activity

1

2

3

4

5

1

-

3.06

3.33

3.23

3.25

2

-

-

3.68

2.58

3.69

3

-

-

-

4.03

4.00

4

-

-

-

-

3.69

5

-

-

-

-

-

Table 15.11 Initial layout, final layout and final objective value for example 1

S. No.

Initial Layout

Final layout

Final objective value

1

12345

53214

18.1973

2

34512

34125

18.3316

3

45123

15243

18.2028

4

54123

15243

18.2028

5

13524

43152

18.4313

6

24135

35241

18.2028

7

32154

53214

18.1973

Fuzziness in Materials Flow and Plant Layout

373

Table 15.12 Operation sequence, demand and transfer batch size for Example 2 Parts Operation number (machine no.) Demand Batch size 1

1(4), 2(2), 3(1)

80

40

2

1(2), 2(4), 3(1)

50

25

3

1(4), 2(5), 3(6), 4(7), 5(3)

150

25

4

1(3), 2(6), 3(7)

40

20

5

1(4), 2(5), 3(2)

20

10

6

1(5), 2(6), 3(7)

150

25

7

1(2), 2(1), 3(4), 4(3)

150

25

Step 6: Development of IF-THEN decision rules When fuzzification process is completed for all pairs of facilities, IF-THEN decision rules are developed. The IF-THEN rules for the facility pair 1-5 framed using Table 15.3, Table 15.4, and Table 15.5 are: Rule 1: IF FD is Low and its weight factor is High THEN the rating is ‘I’ Rule 2: IF QF1 is Low and its weight factor is Low THEN the rating is ‘U’ Rule 3: IF QF2 is Low and its weight factor is High THEN the rating is ‘I’ Step 7: Application of IF-THEN rules using minimum operator Using the minimum operator; Rule1 results in rating of ‘I’ with membership value of 0.7 [Minimum {0.866, 0.7}]. Similarly, Rule 2 results in rating of ‘U’ with membership value of 0.933, and Rule 3 results in rating of ‘I’ with membership value of 0.866. Step 8: Defuzzification The crisp value for activity 1-5 using the Centre of Area (COA) method is: 4 × 0.7 + 2 × 0.933 + 4 × 0.866 = 3.25 0.7 + 0.933 + 0.866

Step 9: This process is repeated for all pair of facilities and subsequently the fuzzy closeness rating matrix generated is shown in Table 15.10. Step 10: The final results with different initial layouts are illustrated in Table 15.11. Example 2 This is an example of assigning seven facilities to seven locations in a plant. The operation sequence, demand and transfer batch size of the parts to be manufactured in these facilities is given in Table 15.12. The computed flow data, safety rating, noise rating and relative intensity importance of these factors for each pair of facilities is given in Table 15.13. The fuzzy closeness rating matrix generated for this example is shown in Table 15.14. The final results are presented in Table 15.15.

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K.S. Sangwan

Table 15.13 Flow data (computed), safety rating, noise rating and relative intensity importance of factors for Example 2 Intensity Importance of factors

Relationship data Activity

Flow Data Safety Rating Noise Rating FD Over FD Over QF1 Over (FD) (QF1) (QF2) QF1 QF2 QF2

1-2

57

0

20

5

2

1

1-3

0

0

32

1

1

1

1-4

57

0

55

2

5

1

1-5

0

0

72

3

1

1/3

1-6

0

0

0

3

2

1/3

1-7

0

0

0

3

2

1/3

2-3

0

0

0

1

1

1

2-4

29

26

35

1

1

1

2-5

14

85

0

1

1

1

2-6

0

72

0

1

1

1

2-7

0

31

0

1

1

1

3-4

43

55

75

5

2

1

3-5

0

0

23

2

5

1

3-6

14

0

29

3

1

1/3

3-7

43

0

0

4

1

1/4

4-5

57

0

35

1

1

1

4-6

0

76

0

1

1

1

4-7

0

0

0

1

1

1

5-6

86

50

95

3

1

1/5

5-7

0

0

0

3

2

1/3

6-7

100

0

82

1

1

1

Table 15.14 Fuzzy closeness rating matrix for example 2 Activity

1

1

-

2 3 4 5 6 7

2

3

4

5

6

7

3.84 3.00 4.20 3.25

2.58 2.58

-

2.67 3.24 3.72

3.33 3.00

-

2.97 3.81

4.50 3.20 -

3.57

3.56 2.67

-

4.99 2.58 -

4.00 -

Fuzziness in Materials Flow and Plant Layout

375

Table 15.15 Initial layout, final layout and final objective value for example 2

S. No.

Initial Layout

Final layout

Final objective value

1

1234567

3426517

30.3561

2

2143657

4563127

30.2867

3

3412567

4123567

31.1181

4

3421657

1432567

30.7850

5

4561237

1253467

30.6572

6

5463712

6513427

30.1617

7

6754321

1564372

31.1862

Table 15.16 Operation sequence, demand and transfer batch size for Example 3 Parts

Operation number (machine no.)

Demand

1

1(5), 2(1), 3(3), 4(6), 5(8)

20

Batch size 20

2

1(2), 2(8), 3(7)

200

50

3

1(6), 2(3), 3(0), 4(1), 5(5)

30

15

4

1(7), 2(8), 3(2)

20

5

5

1(3), 2(6), 3(5), 4(5), 5(1)

50

25

6

1(2), 2(4), 3(1), 4(2)

50

10

7

1(3), 2(1), 3(5)

100

20

8

1(2), 2(7), 3(8), 4(4)

50

25

9

1(6), 2(4), 3(1), 4(5)

40

20

Example 3 This is an example of assigning eight facilities to eight locations in a plant. The operation sequence, demand and transfer batch size of the nine parts to be manufactured/processed in these eight facilities is given in Table 15.16. The computed flow data, noise rating, safety rating and relative intensity importance of these factors for each pair of facilities is given in Table 15.17. The fuzzy closeness rating matrix generated for this example is shown in Table 15.18. The final results are is given in Table 15.19.

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K.S. Sangwan

Table 15.17 Flow data (computed), safety rating, noise rating and relative intensity importance of factors for Example 3 Intensity Importance of factors

Relationship data Activity

Flow Data Safety Rating Noise Rating FD Over (FD) (QF1) (QF2) QF1

FD Over QF1 Over QF2 QF2

1-2

0

65

12

5

2

1

1-3

50

70

18

1

1

1

1-4

58

80

38

2

5

1

1-5

100

45

0

3

1

1/3

1-6

0

5

75

3

2

1/3

1-7

0

8

0

3

2

1/3

1-8

0

0

0

1

1

1

2-3

0

0

0

1

1

1

2-4

12

0

0

1

1

1

2-5

0

0

0

1

1

1

2-6

0

0

0

1

1

1

2-7

17

18

0

5

2

1

2-8

67

27

75

2

5

1

3-4

0

0

0

3

1

1/3

3-5

0

0

0

4

1

1/3

3-6

42

0

0

1

1

1

3-7

0

0

0

1

1

1

3-8

0

0

0

1

1

1

4-5

0

0

65

3

1

1/5

4-6

17

19

50

3

2

1/3

4-7

0

0

0

1

1

1

4-8

17

88

22

5

2

1

5-6

0

0

0

5

2

1

5-7

0

0

0

1

1

1

5-8

0

0

0

1

1

1

6-7

0

0

15

1

1

1

6-8

8

55

0

3

1

1/3

7-8

83

65

0

5

2

1

Fuzziness in Materials Flow and Plant Layout

377

Table 15.18 Fuzzy closeness rating matrix for Example 3 Activity 1

2

1

3.42 4.00 5.00 4.29 3.03 2.58 2.67

-

2

-

3

3

5

6

7

8

2.67 2.67 2.67 2.67 3.30 4.73 -

4

4

2.97 3.08 3.00 2.67 2.67 -

5

3.37 3.17 2.67 4.33 -

6

3.06 2.67 2.67 -

7

2.62 3.39 -

8

4.25 -

Table 15.19 Initial layout, final layout and final objective value for Example 3

S. No.

Initial Layout

Final layout

Final objective value

1

12345678

41382675

36.7337

2

21436587

65741382

37.3263

3

56781234

35714628

36.3926

4

87563412

78264135

36.9800

5

45678123

36718254

35.6900

6

45123678

31584627

37.2043

7

18762345

28714635

37.9800

6 Application of the Proposed Model The proposed model is highly useful in the layout of plants which are based on the clustering of similar facilities/machines together for job shop production.The merits of the proposed application of fuzzy logic and AHP can be judged by looking at the illustrative example 1 results. The relationship data for facility pairs 1-2 and 2-4 for all the factors is zero (Table 15.7). However, a look at the fuzzy closeness rating in Table 15.10 shows a closeness rating of 3.06 and 2.58 respectively for the two facility pairs. These non zero values occur because fuzzy logic does not cut the data in a dichotomous way. Without fuzzy logic application, the closeness ratings for these pair of facilities would be zero and suppose if the values in the input data were one each instead of zeroes, then non-fuzzy approach would have given non-zero values in the final closeness rating whereas there is only a small difference between 0 and 1 on the scale of 0-100. The difference in closeness rating for the considered facility pairs is because the relative importance of the three factors is different for the two pairs and the expert’s decisions are different for

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different factors (Tables 15.3 – 15.5). This proves the merit of the proposed model under fuzzy environment. The proposed model is a general model for the layout of any plant which is not strictly based on the operation sequence of parts (product layout). The proposed model is equally applicable in service operations layouts such as hospitals, universities, libraries, post offices, ports, etc. Another application of the proposed model can be in the defence sector, e.g., the layout of a composite army garrison. In garrison layout, the ammunition store and the field garage should be as close as possible (quantitative factor) whereas the ammunition store should be farthest from the mess (community kitchen) and the hospital (qualitative factor). In defense applications the flow data may not be as important as qualitative data like safety and security. Some of the salient features of the proposed model are: • It considers the quantitative as well as qualitative factors for the plant layout under fuzzy environment. • It gives the importance to the designer/experts in rating the factors affecting plant layout problem. • It checks the consistency of the designer/experts rating. • It considers practical inputs of product demand, transfer batch size, operation sequence and multiple non consecutive visits to the same facility. However, the proposed model considers facilities of equal area and provides only block layouts. A close look at the final results (Tables 15.11, 15.15 and 15.19) shows that the difference in the final objective value is very small. This is an inherent problem of the closeness rating approach.

7 Conclusions and Suggestions for Further Research This chapter presents a multicriteria heuristic model which considers both quantitative as well as qualitative factors for the plant layout under fuzzy environment. The qualitative factors have been addressed subjectively and systematically in a strict mathematical sense and at the same time quantitative factors have been dealt objectively and analytically. The chapter also presents the reasons for fuzziness in material flow in today’s manufacturing environment. The proposed model is validated by solving three plant layout problems taking into consideration practical inputs of product demand, transfer batch size, operation sequence and multiple non-consecutive visits to the same facility. The proposed model can be further improved by the application of artificial intelligence techniques such as genetic algorithms, simulated annealing algorithms, tabu search, neural networks, etc. to improve the application of the model to layout large number of facilities as layout is a NP hard problem. Further research is required to consider the non-equal facilities with pick up and drop points for material flow. There is need to develop a better closeness scoring method so that the difference among the different layouts is reflected in the final objective values.

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Sangwan, K.S., Kodali, R.: Multicriteria heuristic model for design of layouts for cellular manufacturing systems. J. Institution of Engineers (India) 84, 23–29 (2003) Sangwan, K.S., Kodali, R.: Multicriteria heuristic model for design of facilities layout usıng fuzzy logıc and AHP. Int. J. Ind. Eng.Theory Appl. Pract. 13, 364–373 (2006) Sangwan, K.S., Kodali, R.: FUGEN: a tool for the design of layouts for cellular manufacturing systems. Int. J.Services and Operations Management 5, 595–616 (2009) Satty, T.L.: The analytical hierarchy process. McGraw-Hill, New York (1980) Schmucker, K.J.: Fuzzy sets. In: Natural language computations, and risk analysis. Computer Science Press, Rockville (1984) Seehof, J.M., Evans, W.O.: Automated layout design. J. Ind. Eng. 18, 690–695 (1967) Sethi, A.K., Sethi, S.P.: Flexibility in manufacturing: a survey. International Journal of Flexible Manufacturing Systems 2, 289–328 (1990) Shang, S.J.: Multicriteria facility layout problem: an ıntegrated approach. European Journal of Operational Research 66, 291–304 (1993) Sule, D.R.: Manufacturing facilities. PWS-Kent, New York (1998) Tompkins, J.A., White, J.A.: Facilities layout planning. John Wiley and Sons, New York (1984) Urban, T.L.: A multiple criteria model for the facilities layout problem. International Journal of Production Research 25, 1805–1812 (1987) Wilhelm, M.R., Karwowski, W., Evans, G.W.: A fuzzy set approach to layout analysis. International Journal of Production Research 25, 1431–1450 (1987) Yang, T., Peters, B.A.: Flexible machine layout design for dynamic and uncertain production environments. European Journal of Operational Research 108, 49–64 (1998) Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) Zadeh, L.A.: The concept of linguistic variables and its applications to approximate reasoning. Information sciences 8, 199–249 (1975) Zimmermann, H.J.: Fuzzy sets, decision making and expert systems. Kluwer Academic Publications, Boston (1987)

Chapter 16

Fuzzy Cognitive Maps for Human Reliability Analysis in Production Systems Massimo Bertolini and Maurizio Bevilacqua1

Abstract. Cognitive maps provide a graphical and mathematical representation of an individual’s system of beliefs: a cognitive map shows the paths taken, including the alternatives, to reach a destination. With the current increasing need for efficiency of both plant and human operator, fuzzy cognitive maps (FCM) have proved to be able to provide a valid help in assessing the most critical factors for operators in managing and controlling production plants. A FCM represents a technique that corresponds closely to the way humans perceive it; they are easily understandable, even by a non-professional audience and each parameter has a perceivable meaning. FCMs are also an excellent means to study a production process and obtain useful indications on the consequences which can be determined by the variation of one or more variables in the system examined. They can provide an interesting solution to the issue of assessing the factors which are considered to affect the operator’s reliability. In this chapter fuzzy cognitive maps will be investigated for human reliability in production systems.

1 Introduction Cognitive maps were introduced in the early 70s, thanks to the studies conducted by Axelrod (1976) and followed up by other authors. The foundations for the development of this new approach were laid in the context of research on decisionmaking, inherent in the activities of international politics, with a view to analyzing top-level political decisions that generally coincided with major international crises (Codara 1998). In principle, the complexity of such decisions and their structural uncertainty mean that the decision-makers cognitive processes have an essential role, and that is why it is fundamental to reconstruct such people’s decision-making processes, analyzing the operations by means of which they reconstruct and explain the world around them. Massimo Bertolini Department of Industrial Engineering - University of Parma, Viale G.P. Usberti 181/A - 43100 Parma - Italy Maurizio Bevilacqua Dipartimento di Energetica - Università Politecnica delle Marche - Via Brecce Bianche - 60100 Ancona, Italy C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 381–415. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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Cognitive maps provide a graphical and mathematical representation of an individual’s system of beliefs: a cognitive map shows the paths taken, including the alternatives, to reach a destination. Individuals use cognitive maps every time they are able to think of several alternative ways to achieve their objectives. The map describes how individuals perceive a given situation or problem and offers the decision makers a simplified representation of the outside world. According to Axelrod (Axelrod 1976), in order to use this representation, individuals must have certain beliefs that correlate their possible choices with the potential outcomes. These beliefs take the form of single causal relationships between concepts, and combining them gives rise to concatenations that link the alternatives identified to the predicted outcomes. Cognitive maps are constructed by considering two fundamental elements: the concepts and the causal relationships. The concepts represent the variables in the system, while the causal relationships, which connect the concepts together, represent the causal dependences of the variables chosen to describe the system. According to Codara (1998), cognitive maps (and therefore also the fuzzy cognitive maps discussed below) can be used for various purposes, including: • to reconstruct the premises behind the behaviour of a given agent, to understand the reasons for their decisions and for the actions the take, highlighting any distortions and limits in their representation of the situation (explanatory function); • to predict future decisions and actions, or the reasons that a given agent will use to justify any new occurrences (prediction function); • to help decision-makers ponder over their representation of a given situation in order to ascertain its adequacy and possibly prompt the introduction of any necessary changes (reflective function); • to generate a more accurate description of a difficult situation (strategic function). Once decision-makers have become acquainted with the technique needed to construct and analyze the maps, they will be able to proceed independently in drawing up their representation of a given situation, using the map as a tool to support their decision-making process every time they are faced with very complex and uncertain situations. We could conclude that using such maps not only improves our understanding of the cognitive processes behind the reaching of a decision, but may also contribute to generating a significant improvement in the decision-making processes of a given single or collective subject. Understanding the reasons why other agents with which we interact have reached a given decision, predicting what their future decisions may be, and developing a more complete representation of a problem cannot fail to concur towards improving our performance. If we then go on to analyse the functions of the maps, it becomes evident, for instance, that a better understanding of the methods individuals use to construct simplified representations of the world can offer an essential contribution to our prediction of their decisions.

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2 Fuzzy Cognitive Maps With the introduction of cognitive maps, what Axelrod (1976) aimed to do was to grasp the causal assertions that individuals make in relation to a given domain, and to use these assertions to analyze the potential effects of any alternatives, in the political or economic context, for instance. As mentioned previously, a cognitive map only has two fundamental elements, i.e. concepts, that represent the variables in the system, and causal beliefs (or relationships), that represent the causal relations existing between the variables in the system. The variables (or concepts) may be: • continuous variables, e.g. the amount of something; • ordinal variables, e.g. more or less of something; • dichotomous variables, demonstrating the existence or nonexistence of something. Axelrod (1976) suggested two different types of cognitive map: functional cognitive maps, that simply enable the representation of an individual’s system of beliefs; and weighted cognitive maps, characterized by the fact that a sign is attributed to each relationship to represent its causal sense of influence (positive or negative). The causal relationships connecting the variables to one another may thus be negative or positive. The variables that determine a change are termed cause variables, while those that undergo the change are effect variables. The following causal claim shows the differences existing between the two different notions of cause variable and effect variable (Kosko 1994): “the ability to respond to customers’ demands increases the quality of customer service.”

The cause variable “ability to respond to customers’ demands” positively affects (i.e. it increases) the effect variable “quality of customer service”. If the link is positive, as in the example, an increase or reduction in a cause variable induces a change in the same sense in the effect variable (increase for increase, reduction for reduction). If the link is negative, then the change incurred in the effect variable will be in the opposite sense (an increase in the cause variable determines a reduction in the effect variable and vice versa). Figure 16.1 shows a graphic representation of a cognitive map in which the variables (A, B, etc...) are represented as the nodes in a graph and the causal relationships as arrows pointing between the nodes in the graph. A path between two variables, A and D, in a cognitive map is a sequence of all the nodes linked by an arrow extending from the first node (variable A) to the second, another arrow from the second node to the third, and so on, until we come to the arrow leading from the last but one to the last node, D, along the path (Axelrod 1976). According to Axelrod (1976), the analysis of a cognitive map begins with the manipulation of the signs of the causal paths to determine the sense of the effect induced by changes to the cause variable. The following rules, proposed by

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B

+

+

D

A

-

C

+

E

Fig. 16.1 Example of a cognitive map

Axelrod (1976), apply to a binary system (where the weights of the causal relations can only be + and -), but they can also be extended to more complex systems: Rule 1 The indirect effect of a path from a cause variable x to an effect variable y, called I(x, y), is positive if the path has an even number of negative causal relationships and negative if it has an odd number of negative relationships; otherwise it is indeterminate. The indirect effect is consequently defined as the multiplication of the signs of the causal relationships forming the path from the cause variable to the effect variable. So, for instance, the indirect effect of the variable A on the variable D along the path P(A, C, E, D) in Figure 16.1 is positive (+). Rule 2 The total effect of a cause variable x on an effect variable y, called T(x, y) is the sum of all the indirect effects along all possible paths connecting x to y. So, in Figure 16.1 the total effect of A on D is the sum of the two indirect effects I1 along the path P1(A, C, E, D) and I2 along the path P2(A, B, D); the first is positive and so is the second, so T(A, B) is positive. In weighted cognitive maps, the causal relationships are assessed by means of a numerical assessment, or a function, that measures the amount of the causal influence of the cause variables on the effect variables. Using these maps, we can obtain more information about the knowledge domain and reduce the problem of in determination, i.e. the impossibility to define the total effect as the sum of the indirect effects in cases where the indirect effects have not been calculated. From here, the natural evolution of a cognitive map is represented by a fuzzy cognitive map (FCM), that can be seen as a map delineating a causal image that links facts and other things, and that processes them as values, policies and goals. This enables predictions to be made on how complex events interact and take place Kosko (1985). As Kosko explained (1985, 1992, 1994) we might debate the way in which the map was obtained from Kissinger’s article, or have doubts about the content of the article itself. We might wish to add some concepts, or to remove others, to include

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some more arrows (causal connections) or eliminate others, to reverse the signs, or to modify the weights of the influences. All this can be done, but once it has been fine-adjusted, the FCM is able to predict consequences that can be verified by placing them in relation to the data. The FCM serves to visualize a rough image of the “world” for us to use, but the most important thing is that, behind the FCM, there is a pure mathematical schema that can be processed with the aid of a computer. This mathematical model (that will be discussed in more detail later on) writes the status, or a snapshot, of the FCM as a list of numbers, like a vector. The binary vector (0, 1, 1, 0) means that, at the time of its measurement, an FCM with four nodes has only the second and third nodes switched on. Fractions (i.e. values coming between 0 and 1) measure the degree to which the nodes are activated, so the mathematical schema writes a large square matrix of numbers: if there are n nodes in an FCM, the mathematical schema interprets it as a square matrix, n x n. Kosko (1985) developed FCMs starting from Axelrod’s cognitive maps; the original cognitive maps permitted no fuzzy nodes or fuzzy arrows, or rules, nor did they allow for feedback, i.e. the arrows could not form a closed circle, as they do in the real world. In other words, Axelrod’s cognitive map was not a dynamic system. Fuzzy cognitive maps, on the other hand, are based on feedback: developing the cognitive maps as if they were neural systems can make them dynamic, so that they turn and reverberate like bidirectional associative memory until they become stable, converging towards a situation of equilibrium (Codara 1998). An example of the dynamic nature of a cognitive map can come from observing the FCM developed by Kosko (1994) to establish how bad weather influences daily driving speeds on a Californian highway (Figure 16.2).

Fig. 16.2 Example of a fuzzy cognitive map developed to analyze driving risks

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The arrows, or fuzzy rules, have non-quantitative weights, such as “usually” and “a little”. The FCM has two small feedback envelopes, two short cycles: • in the first feedback cycle, traffic congestion on the highway increases the road accidents “quite a lot”, and the latter “often” increase the traffic congestion on the highway; • in the second cycle, road accidents increase the frequency of police patrols on the road “quite a lot “; but a greater frequency of the police patrols helps to “reduce” the road incidents. The feedback lies in just this: a flow of information, of influences and causalities, that propagates in two directions. As mentioned earlier, the fuzzy cognitive maps introduced by Kosko eliminate the problem of the indeterminacy of the total effect. The development and analysis of this tool relies on the theory of fuzzy sets Zadeh (1965, 1973). Fuzzy logic was introduced for the first time by Zadeh in 1965 and considers fuzzy sets as mathematical tools for representing the imprecision, or vagueness with which people provide and interpret information on the world around them. Fuzzy sets are characterized by a membership function and Zadeh (1965) defined them as follows: • a fuzzy set A of a universe of discourse U is characterized by a membership function μA(x): U → (0, 1) that associates with each element y of U a number μA(x) in the interval (0, 1) that represents the degree of membership of y in the set A. Different approaches have been proposed and developed for specifying the fuzzy weights of the causal relationships in a cognitive map. The first is to ask a group of experts on the system/process in question to assign a real number in the interval (0, 1) to each relationship, then combine the opinions by means of an averaging operation (Taber 1991, 1994). For example, if a causal relationship is assessed by three experts as (0.3, 0.6, 0.6), then the weight of the relationship becomes the mean of the three, i.e. 0.5. Taber (1991) suggested a method for combining several fuzzy cognitive maps prepared by different experts, weighting the knowledge domains described according to the credibility of each expert. We might wonder why it is better to collect several FCMs prepared by different experts. The answer could come from the law of large numbers: if several experts independently prepare their fuzzy cognitive maps of the same domain, we can reasonably assume that combining these single maps together will give us a global FCM that is potentially more significant than a single map because the information is drawn from a variety of sources, making any errors increasingly less significant. To make the combination of several cognitive maps prepared by different people more effective, however, we need to assign a weight to the degree of credibility of a given expert in relation to the topic being examined. Taber’s approach (Taber and Siegel 1987) consisted in having an FCM prepared separately by several experts and then obtaining a linear combination of the maps according to the following expression:

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NE

Fw = ∑ Fi ⋅ w i

(16.1)

i =1

where: Fw is the averaged weight of the global cognitive map; i=1, …, NE are the various experts contributing to the construction of the cognitive map; Fi are the weights of the causal connections on each expert’s cognitive map; and wi represents the weight of their credibility. The end result is the creation of an FCM in which the weights of the causal relations are obtained as a linear combination of the weights of the single cognitive maps weighted on the basis of the credibility of each expert. Zhang et al. (1989, 1992) also recommended considering the real numbers as membership functions of a membership function of a fuzzy set in a fuzzy schema, but the approach that directly and numerically attributes a weight to the causal relations is an extremely difficult approach for the experts. This is the main reason why many authors (Bowles and Pelàez 1995b, 1996, Kosko 1985, 1992, 1994, Zhang and Chen 1989, Zhang et al. 1992) prefer to use partially ordinal linguistic variables, such as weak < moderate < strong, to weight the influence of causal relationships. Another method for automatically constructing cognitive maps - and therefore both for identifying the causal relationships and for assigning their degree of causal influence - was proposed by Schneider et al. (1998). In this work, the authors present a technique for automatically constructing fuzzy cognitive maps based on the use of numerical vectors of data on the system variables to analyze. The method consists essentially in seeking the degree of similitude between two variables (represented by two numerical vectors), and thereby seeking to establish whether the relationship between the variables is direct or inverse, using a fuzzy export system tool (FEST) to determine the causality between these variables. Going into more detail, the authors claim that an analysis of the literature reveals several problems relating to the use/construction of fuzzy cognitive maps: • Taber and Siengel (1987) said that the simultaneous presence of an expert system and of other systems enables high levels of knowledge to be achieved, which are needed considering that the cognitive maps contain a high degree of experience. These authors suggest a method for assessing an expert’s credibility, and consequently the credibility of each map, in the method used to define the construction of collective maps, as described above; • for the preparation of collective cognitive maps, based on the work by Taber (1991, 1994) we need to weight the matrix of the connections according to the credibility of each expert. Schneider and co-workers (Schneider et al. 1998). pose several questions on this aspect, e.g. Why should we use the sum, and not other mathematical operators, to combine the knowledge of several experts? Why is the combination based on the sum and not on the minimum or maximum operator? How can we measure the experts’ credibility?

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To answer these and other questions, they recommend their own technique for automatically constructing an FCM. In particular, each concept is represented as a numerical vector (V) consisting of n numbers, where each element of the vector vi constitutes a measure of the concept. Since three types of causal relationship between the concepts are possible (a positive, a negative or no causal relationship), three parameters must be considered every time the value of Fij is assigned automatically. At this point, we need to convert the numerical vector into a fuzzy set and then compare the different fuzzy sets with one another to obtain the matrix of the connections (power and sign). The method proposed for converting a numerical vector into a fuzzy set is as follows: seek the maximum value in V and set μ=1, thus:

MAX {V(v i ) : i = 1, ..., n} = v i ⇒ (μ vi = 1)

(16.2)

seek the minimum value in V and set μ=0, thus :

MIN {V(v i ) : i = 1, ..., n} = v i ⇒ (μ vi = 0 )

(16.3)

project all the other elements i of the vector (vi) in the interval [0, 1] proportionally, so that:

μ vi =

v i − MIN(V ) MAX(V ) − MIN(V )

(16.4)

Instead of setting the value μ=1 for the maximum and μ=0 for the minimum, we can ask the experts to provide an upper threshold value and a lower threshold value, αU and αL, such that:

∀ v i (v i ≥ α U ) ⇒ (μ vi = 1)

∀ v i (v i ≤ α L ) ⇒ (μ vi = 0)

(16.5)

The next step consists in seeking the degree of linkage between the two variables and the sign of the causal relationship. All this is done by relying on the concept of distance between two vectors that, in the case in point, are vectors represented by degrees of membership in fuzzy sets (16.5). Given two vectors, V1 and V2, the distance between two corresponding elements belonging to the two vectors will be given by:

d i = μ1 (vi ) − μ 2 (vi )

(16.6)

and the mean distance between the two vectors, AD can be obtained from (16.7) or from (16.8): n

AD =

∑d i =1

n

i

(16.7)

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AD =

n

∑ (d ) i =1

2

i

389

(16.8)

By calculating the mean distance between two vectors, we can define the degree of similitude S as:

S = 1 - AD

(16.9)

The closer S comes to 1, the more the two vectors will resemble one another. To understand the diversity between the two vectors, we need to configure the distance in the case of an inverse relationship between the two vectors, V1 and V2:

id i = μ1 (vi ) − (1 − μ 2 (vi ))

(16.10)

and calculate the mean distance, again using (16.7) or (16.8), substituting idi instead of di. Finally, to obtain the sign of the causal relationships and the power of each causal relationship, we need to calculate the mean distance of each variable from every other variable, assuming a perfect similitude - and therefore using (16.6) in the formula for the mean distance, and a perfect dissimilitude - and thus using (16.10) as a measure of the distance. The greater of the two will be chosen as the sign and weight of the causal relationship between the two concepts. Various ideas have been advanced as concerns the sign of the causal relationships. Kosko, in particular (1985, 1994) said that - given the difficulty of processing knowledge in the presence of negative causal relationships - the links with negative causal relationships can be eliminated. To do so, it becomes necessary to substitute each link Ci→Cj with the link Ci→~Cj, where Ci and Cj are respectively the cause variable and the effect variable, and ~Cj represents the negation (or the complement) of Cj. All this, however, gives rise to a considerable workload, both in the reading of the FCM and in its subsequent processing. Some researchers (Bowles and Pelàez 1995a, 1995b, 1996, Kosko 1985, Warren 1995) have suggested that the experts’ opinion is essential for determining the variables needed to describe the system being studied, the existence of causal relationships and their direction, and the polarity and power of the causal links. Other studies on fuzzy cognitive maps have focused on considering the temporal variations according to a preset clock of the degree of activation of the concepts (Park and Kim 1995). Margaritis and Tsadiras (1997, 1999) introduced fuzzy cognitive maps integrated with neural networks, called certainty neuron fuzzy cognitive maps (CNFCM), to increase the capacity of representation and processing of FCMs, and they presented an application on a dynamic system. Cognitive maps were used in association with simulations to present different possible future scenarios and consequently intended to be of help in the strategic setting to decide between possible alternatives (Fu 1991, Kardaras and Karakostas 1999, Mahmood and Soon 1991, Paradice 1992); as a Decision Support Systems Tool (DSST) (Higgins et al. 1998, Warren 1995); in the reliability setting, particularly in the application of the FMECA methodology to complex systems (Bowles

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and Pelàez, 1995a, 1995b, 1996); and to model an intelligent system controller (Groumpos and Stylios 1999a, 1999b, 2000). Decision making is becoming increasingly complicated every day, and FCMs make it possible to provide helpful support, enabling an analysis of the impact of changes to projects, so as to be able to identify the best solutions and to provide answers to questions such as: • what alternatives are there for arriving at a given objective? • which of the alternatives is preferred by the designers, also in relation to its feasibility? • what will be the effects of a change imposed by a given alternative on the organization as a whole? • which particular variables will change, in what way and to what degree? • what will be the consequence of the change (increment vis-à-vis reduction) made to a variable? The causal reasoning needed to analyze a fuzzy cognitive map follows the rules outlined below, starting in particular by considering what Axelrod (1976) first said concerning the analysis of the map with the aid of a numerical analysis. We need to represent the concepts by means of a state vector and the causal relationships between the concepts as a relational matrix. A concept is activated by setting its vector element as 1: the vector’s activation consists in making the concept change and, in so doing, seeing how this change influences the other concepts and the system as a whole. The new state vector, which will show the effect of the activation of the concept, will be calculated from the product between the old vector state and the relational matrix (Bowles and Pelàez, 1995a, 1995b, 1996) according to the formula (16.11).

[C ...C ] 1

n new

⎡R ⎢ 11 = C1...C n ⋅ ⎢ ... old ⎢R ⎣ n1

[

]

... R ij ...

R 1n ⎤ ⎥ ... ⎥ R nn ⎥⎦

(16.11)

where Ci identifies the i-th concept, and Rij is the relationship from the concept i to the concept j. The calculation is terminated when we reach a limit vector (e.g. when we arrive at C ...C ) or when we arrive at a limit cycle = C ...C

[

] [ (e.g. when we obtain that [C ...C ] 1

n new 1

1

n new

]

n old

is equal to a previous vector), or after a

preset number of iterations. With an analysis of this type, the values of the concepts in the vectors can only be 0 or 1 (i.e. Axelrod’s binary logic), where the value 1 indicates that the concept is activated, and 0 that it is deactivated. We can use the same causal analysis in a trivalent logic, in which case the concepts can acquire the values {-1,0,1}, where we indicate a negative effect as –1, no effect as 0, and a positive effect as 1. An example of a numerical analysis with the corresponding cognitive map is given in Figure 16.3.

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Fig. 16.3 Example of the analysis of a cognitive map (Bowles and Pelàez 1996)

[C

A

CB CC CD

]

new

[

= CA

⎡0 ⎢0 C B CC C D ⋅ ⎢ old ⎢0 ⎢⎣0

]

1 0 0 −1

−1 0 0 1

− 1⎤ − 1⎥ ⎥ 1⎥ 0 ⎥⎦

Activation of A: [1 0 0 0]

1st multiplication: [0 1 - 1 - 1] Thresholding: [1 1 - 1 - 1]

2nd multiplication: [0 2 - 2 - 3]

Thresholding: [0 1 - 1 - 1] Limit vector For fuzzy cognitive maps, we still use the rules introduced by Axelrod, as modified by Kosko (1985), both for the development of the map and for the analysis of the indirect and total effects. Kosko defines the k-th indirect effect Ik, between two concepts Ci and Cj, using the minimum operator: Ik(Ci, Cj) = min {e(Cp, Cp+1)} where e(Cp, Cp+1) is the weight of the causal relationship between the two concepts Cp, Cp+1, p and (p+1) are near, indices of all the concepts forming the pathway that leads from Ci to Cj. The total effect T(Ci, Cj), is defined using the maximum operator instead: T(Ci, Cj) = max Ik(Ci, Cj) for k=1, …, n all the indirect effects between the concepts Ci and Cj. To calculate the k-th indirect effect Ik(Ci, Cj) along a causal pathway Пk(Ci, Cj), where i ≤ k ≤ n, and n is the number of pathways between two concepts Ci and Cj, we must (Kardaras and Karakostas 1999): • ensure the existence of the smallest causal pathway Пk(Ci, Cj), that enables the propagation of causality from Ci to Cj;

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• specify the sign Sk that shows the polarity of the k-th indirect effect Ik(Ci, Cj); Sk depends on the number of negative signs in the relationships along the pathway Пk, where 1 ≤ l ≤ m, l indicates the relationship along the pathway and m is the number of relationships in Пk. More precisely, the sign Sk of an indirect effect Ik(Ci, Cj) is positive (+) if the number of relationships with a negative sign on the pathway Пk(Ci, Cj) is zero. If not, the polarity will be negative; • determine the degree of membership μk in the fuzzy set, which indicates the existence of a causality of the k-th indirect effect, where μk depends on the weight μkl of each relationship along the pathway Пk(Ci, Cj). More precisely, μk = min (μkl), where 1 ≤ l ≤ m, l indicates the relationship on the pathway and m the number of relationships. The degree of membership, μk of the indirect effect, Ik, is not defined if the smallest relationship in the k-th pathway is undefined. The effect with the maximum membership, ΔΠ(Ci, Cj), of all the indirect effects Ik(Ci, Cj) between the two concepts Ci and Cj, follows from the determination of all the indirect effects. ΔΠ(Ci, Cj) is determined with respect to the degree of membership of all the indirect effects between the cause variables and the effect variables. It is defined as follows: ΔΠ(Ci, Cj) = max (μk), where 1≤k ≤n, and n is the number of pathways. Finally, ΔΠ(Ci, Cj) takes the sign of the strongest pathway.

Fig. 16.4 Example of a fuzzy cognitive map for calculating the indirect and total effects (Kosko 1985).

Following the above-explained rules, we can give a small example of a calculation of the indirect effects and total effects. If we consider the part of map shown in Fig. 16.4, there are three causal pathways linking the concept C1 to the concept C5, i.e. (C1, C3, C5), (C1, C3, C4, C5), and (C1, C2, C4, C5). Along the three pathways, the three indirect effects of C1 on the concept C5 are:

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I1(C1, C5)= min(e13, e35)= min (much, a lot)= much I2(C1, C5)= some I3(C1, C5)= some while the total effect of C1 on the concept C5 is: T(C1, C5) = max { I1(C1, C5), I2(C1, C5), I3(C1, C5)} = max{much, some, some}=much In conclusion, C1 can impart a “much” causality on the concept C5. As we can see, Kosko relies on an algebra that operates only in the field of the positive values. The negative effects, according to Kosko (1985) should be converted into positive ones by converting the consequent variable into its opposite (logical). In this way, the number of the concepts is doubled, because if the original variable is linked to others by positive relationships, it must nonetheless remain on the map. This gives rise to a redundant representation of the situation and a greater complexity. To overcome this computational difficulty, Kardaras and Karakostas (1999) recommend a series of procedural steps to calculate the indirect effect and the total effect on an FCM, as explained above.

3 Human Reliability Analysis 3.1 Introduction Having examined how fuzzy cognitive maps work and the logic behind them, in order to continue the study we need to seek the factors that might potentially crop up in relation to the presence of human beings in production systems. In particular, the research focuses on the aspects of human behaviour and, more in general, of the human environment, that can influence human reliability. By reliability, we mean here the capacity of a production system to function according to its design principles, without interruptions or unexpected events. In this sense, a scheduled stoppage for servicing, for instance, is not a sign of unreliability – quite the reverse, it contributes to improving the system’s reliability. A system is reliable when it functions smoothly, reducing to a minimum any contingencies or emergencies that might facilitate the occurrence of incidents, near-incidents, nearinjuries, minor injuries or severe injuries (Berra and Prestipino 1983). The concept of reliability provides the framework for explaining what a great deal of field research in specific occupational settings has highlighted, and that is the tendency of anomalies to be triggered one after the other, forming chains of incidents. When the state of a system has deteriorated, i.e. when its reliability declines, it becomes unstable and demands frequent action on the part of human operator to cope with anomalies. From the reliability perspective, incidents and injuries are not isolated events, but the by-products and warning signs of the general functional conditions of a system. Ergonomics not only analyses injuries, i.e. situations in which operators experience physical harm, but also closes the gap between human and technical factors, linking the two in the common framework of a system’s reliability.

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Before the advent of the reliability perspective, safety studies considered injuries separately from incidents. The human component in the system was considered separately from the technical component, although they interacted continuously in real life. This separation meant that injuries were assumed have a human cause, while for breakdowns the tendency was to seek technical causes. This approach made the functioning of the system and its reliability invisible. When we come to deal with complex technological systems (such as aircraft, ships, air traffic control rooms, chemical plants and energy generation systems), that operate in hazardous environments, it becomes fundamental to consider the interactions between humans and control systems. As concerns the human factors and the man-machine interactions, these systems can be formally represented in much the same way. In man-machine systems, all interaction between a machine and its operator is always seen in a realistic context, that is characterized by the real plant, with its interfacing and control systems, and the working environment in which the interaction takes place. As for the causes of errors, it is assumed that exogenous factors are the main source of influence on behaviour, although it is well-known that endogenous factors - i.e. aspects of individual personalities - are at least as important. But while, for the exogenous factors, we can identify specific families of factors and transmission thresholds, that influence groups of people and operators in certain technological domains, in the case of endogenous factors it is very difficult and too situation-specific to study single human beings when we want to conduct design studies or safety analyses. The presence and influence of these factors must be acknowledged, however, and considered in the safety study somehow, by introducing causal errors in the iteration process, for instance. In other words, the endogenous factors are too aleatory and personal for them to be considered as constituent elements of an analytical technique applied to the design state or to a safety study, and can only be considered by means of aleatory expressions. To assess the exogenous factors that influence human behaviour it is essential for the safety analyst or designer to be able to define a series of specific causes that can influence behaviour and corresponding threshold values that trigger an inadequate behaviour. Generally speaking, the exogenous factors that have an influence on human behaviour are identified in a series of generic categories, such as the surrounding physical environment, communications, the equipment or interfaces, or procedures, the workplace or the organizational culture. Thus, given a system to analyze and a reference classification, we need to identify physical quantities and socio-technical-environmental variables that can prompt human error. Examples of such variables include physical variables such as ambient pressure, elevation and temperature, time distributions, and also logical and environmental variables such as noise, interfaces and alarms. The types of error, i.e. the form that an operator’s erroneous actions may take, which are generically classifiable, must also be converted into significant expressions of erroneous behaviour in relation to the system being studied for them to be useful for the purpose of reliability and safety studies. Defining the forms taken by the errors, for each cognitive function, implicates a massive reduction and

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in-depth simplification in the representation of an operator’s possible actions and expressions. This is one of the forms of simplification introduced by the model-tosimulation step. Examples of these simplifications can be seen in (Berra and Prestipino 1983, Melchers and Steward 1997): • the limitation of possible premature actions, defined as an expression of an erroneous action; • the limitation of the duration of the action; • the consideration of a known, limited number of repetitions of the action, identifying in advance which actions and how many times they can be repeated.

3.2 The Process for Identifying Factors To be able to characterize the data and the factors associated with a model of human behaviour, the safety analyst has to acquire a thorough understanding of the occupational domain where the activities to control and manage the system being studied are conducted. This understanding is obtained by means of three specific investigations i.e. (Melchers and Steward 1997) • a number of visits to the plant and control room for discussions and to get the operators to complete some questionnaires; • a review of all the information contained in the safety reports and the plant control and protection system design documents; • a study of the procedures and of the operators’ tasks relating to plant control both in normal conditions and in the event of incidents or, in a word, a job analysis. Job analysis developed as of the fifties in an attempt to formalize human behaviour as series of elementary, simple components, but it is no longer considered sufficient for describing a human being’s actual tasks and real expressions. There has been a change in the role of operators, who nowadays have become more supervisors than handlers of control systems, so the approach that is most often used is an analysis of the cognitive tasks, which focuses on the mental and decisionmaking processes and which is also supported by suitable questionnaires and interviews in the workplace. The formalization of an application or procedure for the analysis of manmachine interactions represents an essential methodological step in providing the designer or safety analyst with the means for implementing models and simulations of physical systems and human behaviour. It enables the study and the design of control and emergency procedures, and an assessment of the efficacy of safety measures and protection systems once they have been included in the system (Berra and Prestipino 1983, Melchers and Steward 1997). In this scenario, the whole object of research is to produce the principal factors influencing reliability to enable them to be considered as concepts and to construct a fuzzy cognitive map around them. The methodology for identifying human factors has taken two types of investigation into consideration: retrospective investigations and prospective investigations.

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A retrospective investigation consists in assessing the events that have involved man-machine interactions, such as incidents or hazardous events, seeking to identify the primary causes that contributed to the incidental sequence, and the reasons why the human being and the machine behaved in a given way, the final object being to contribute to the development of measures for preventing incidents. A prospective investigation involves predicting and assessing the consequences and risks relating to incidental sequences at various levels of severity, prompted by different trigger events and man-machine interactions, with a view to contributing to the development of control, protection and emergency systems, or to the analysis of a plant’s reliability and safety. This enables the design and analysis of protection systems, emergency systems, procedures and alarms that enable a system to be managed in transient incidental or operational conditions. Starting from models and the related simulations, and with the support of appropriate data and parameters, prospective investigations can be used to study the consequences of certain incidental conditions both initially and at the boundaries of the manmachine system, and to evaluate the quality of the plant’s protection and safety systems. Being structured on events that have actually happened, for which the sequence of interactions is known and only needs to be interpreted in research terms, retrospective analyses demand an understanding of the working environment and an accurate study of the events that actually took place. But retrospective analyses can provide important results as concerns the factors inherent in the man-machine relationship, as we can see from the literature review that was conducted. The first factors to be identified when we study methodologies for improving the workplace concern the general conditions in which workers do their jobs and fulfil their responsibilities. It is pointless to conduct analyses intended to improve manufacturing methods or general working procedures if, for instance, the lighting is so poor that workers have to constantly strain their eyes to see what they are doing, or if the air is so warm, damp or dense with noxious fumes that keep having to go outside for a breath of fresh air. Inadequate working conditions are not only anti-economical, they also directly influence the behaviour of the people in the workplace. The first requirement for any workplace must be cleanliness, maintaining a hygienic and healthy, and consequently also tidy environment. This enables many potential workplace incidents to be avoided. If passageways are not kept clear of materials and other hindrances, then time has to be spent shifting these items in order to be able to move about. A fundamental issue, as mentioned previously, concerns lighting. Good lighting in the workplace speeds up production, it is essential to health and safety, and to the workers’ efficiency. Without it, the strain on the workers’ eyesight would increase, and so would the incidents in the workplace and production areas, so productivity would also deteriorate progressively. Especially in industries where the work in hand demands a high precision, inadequate lighting can severely reduce the workers’ productivity as a result of errors in the manufacturing process. The adequacy of the lighting depends on both its quality and its quantity. The factors that determine its quality include glow, diffusion, direction, uniformity of distribution, the colour

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of the light source, and brightness. Glow has a negative influence on the workers’ eyes and/or production in general. Direct glow can be contained by reducing the luminosity of light sources, increasing the luminosity in the areas surrounding the light sources, or increasing the angle between the light source and the line of vision. Windows that directly let in the glow of sunlight can be obscured or screened; central lighting fixtures can be arranged above the normal line of vision and the quantity and intensity of the lighting can be suitably limited. Generally speaking, natural light is preferable to artificial light but, where it is insufficient, it is a good idea to reinforce or replace it with appropriate artificial lighting to enable proper system maintenance so as to keep it constantly fully efficient. Another aspect relating to the hygienic-sanitary aspects of the environment and worker health concerns air quality. Like excessively high or low temperature, an inadequate ventilation reduces worker productivity as a result of the numerous diseases and disorders that these conditions can cause. The three main aspects for classifying ambient air include: • air exchange rate; • air temperature; • air humidity. The combination of these three factors enables us to calculate a condition of environmental well-being. Ventilation may be natural or forced, or may result from a combination of the two methods. Air conditioning is useful to combat excessive temperatures, be they too high or too low. Noise is also an important factor that influences a worker’s physical and psychic efficiency. It is frequently a cause of fatigue, irritability and a decline in productivity. Constant exposure to excessive noise can also cause permanent hearing disorders. Obviously no workers can do their jobs properly if they do not have enough space to work in, place their tools and utensils, and move without bumping into colleagues or equipment, or stocks of material. Standing for a long time to do a job is one of the main causes of fatigue and general malaise, so seating should be provided to enable workers to do a job sitting down or, where this is impossible, breaks must be permitted or scheduled so that workers can rest and recover. Preventing fatigue generally promotes the physical efficiency of the workers. In addition to these apparently tangible aspects, however, there are others that are not so obvious, such as mental workload or mental fatigue. When this phenomenon was first studied in the 70s, many researchers thought they could measure the phenomenon as precisely as they measured physical loads. As it turned out, associating the idea of loads and physical fatigue became misleading when applied to the human mind. This issue was studied in air traffic controllers (one of the professional figures that suffer most from the burden of their mental workload): as their load increased, which was measured in terms of the number of aircraft they had to monitor, the air traffic controllers tended to change their control strategy, i.e. they reorganized their system for treating the information and the style of their response.

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So far, we have considered problems relating to working conditions from the point of view of the environment and fatigue, but the machinery and tools involved can also have features that are not well suited to the worker using them. Difficult movements are often needed to handle controls (machine ergonomics) and, during the design stages, the manufacturers’ attention should focus on avoiding these shortcomings. For instance, the accessibility of control panels or dials should be ascertained, wherever possible, directly at the plant where they are to be installed in order to improve the conditions of the workers. Once operators have been placed in the best possible conditions to do their job, we must not neglect the quantity of tasks that their job entails, or rather their complexity and the speed with which they must be completed. It is indispensable for operators to have received the necessary training for their job, but this alone is not enough, because they might find themselves in a situation in which the production process does not give them enough time to do a job properly, or the complexity of some tasks may demand not only a basic training (that is taken for granted) but also sufficient experience in the field to enable operators to cope with or prevent any failures or stoppages in the production process. This last aspect is also related to the issue of information and communication, which can clearly represent a weak link. Being able to review previously completed work and past experiences can be an important advantages from the point of view of occupational safety. If information is difficult to access, or insufficient, or the timing of communications between departments is wrong, this can often be a cause of errors (Berra and Prestipino 1983, Melchers and Steward 1997, Petersen 1996).

3.3 Literature Review to Identify Factors To further analyze the factors that directly or indirectly influence human operator reliability in the conduction of industrial plants, a literature review was conducted on studies in this field. The work that has contributed most to the identification of these factors is definitely the analysis of performance shaping factors (PSF) published by Toriizuka (2001) concerning such a high-risk scenario as a nuclear power plant. This enabled us to obtain some preliminary indications, dividing the factors into categories and establishing the relationships for each of them. The outcome enabled us to clarify which factors were most important within each category in relation to three different aspects, i.e. human reliability, workload and work efficiency. Another source that enabled us to identify factors that tend to influence operator reliability in the conduction of industrial plant was a study by Chadwell, Leverenz and Rose (1999). By means of an investigation on approximately 130 incidents that happened at oil refineries in several states in the USA, they established which factors contributed directly to each incident. Then they prepared a database with a view to pinpointing the most recurrent errors with a view to their prevention. Research on human errors and the measurement of performance in control rooms at nuclear power plant led Kecklund and Svenson (1997) to perform a study

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on 98 plants, selecting potential anomalies, their causes, and types of behaviour that need to be avoided for a proper operation of the plant. An article by Gordon (1998) drew attention to how human factors can cause incidents in the refining industry. Referring to an analysis of 25 incidents reported by the industries, and placing the focus on the identification of their causes led to a reduction in the likelihood of errors. Various other studies enabled the identification of human factors, though they did not consider the relationships between them, and the literature review as a whole enabled us to draw up a table containing all the human factors identified. 3.3.1 Study on Performance Shaping Factor Human behaviour is modelled and assessed using various tools. The one proposed in the work by Toriizuka (2001) consisted in an identification process and analysis that prompted direct actions to improve job performance. Identifying and analyzing performance shaping factors (PSF) enabled a study on the impact of various factors with a view to optimizing workload, work efficiency or workplace comfort in terms of human reliability. A study of this type is consequently fundamental to industry, partly because - up until its publication - few efforts had been made to study these human factors and establish their importance. In 1996, Yukimachi and Toriizuka conducted just such a study to analysis potential human errors and propose a model for assessing human behaviour in servicing procedures. The first step to develop the methodology proposed by Toriizuka (2001) was to identify the factors to consider, including only the most important, leaving aside all those factors relating to education, and to manual and managerial procedures, because we only want to consider the factors that have a direct influence on the way workers do their job. In the paper by Toriizuka (2001), the importance of each factor was assessed by a team of various experts, who identified eight categories in which they included 38 factors. Each factor was weighted by considering it from the points of view of work efficiency, workload and human reliability. In relation to these three aspects, each factor was initially attributed three values for its importance on a scale from 0 (the least important) to 5 (the most important) (see Table 16.1). To increase the reliability of these opinions, the authors collected the opinions of 13 experts on the importance of each factor relating to human reliability, and of eight experts for the other two aspects (work efficiency and workload). Once the main factors had been identified, the opinions of the experts were sought again to weight the relationships existing between the various factors and the degree with which one factor influences all the others. This procedure was restricted to the eight categories identified (i.e. judgmental load, physical load, mental load, information and confirmation, indication and communication, machinery or tools, environment, work space) in Table 16.1. As an example, Figure 16.5 shows the relationships existing between the factors identified by the experts belonging to the “physical load” category. We can see that the “a precision task” factor had quite a weak influence (0,10) on the “heavy labour” factor, whereas it had a stronger influence on the factor “a complex task which needs experienced skill”.

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Table 16.1 PSF and categories resulting from the expert opinions Toriizuka (2001). Category

PSF

Not only knowledge but also an interpretation or a judgment based on experience is needed Judgmental load A prediction or keen insight is needed A concrete standard for judgment is dubious Long continuous working time A precision task Physical load A complex task which needs experienced skill Only one part of the body is exerted strongly Heavy labor Quick response is required because of time pressure Dual duty, punctuality and perfect task result are required An error during task leads to great plant damage Task is difficult to redo, correct or interrupt Mental load Continuity of similar tasks Task progresses very slowly Many factors exist which distract workers’ attention Many additional tasks, such as recording data Information and Difficult to get feedback of work result confirmation Difficult to grasp state of working process Incorrect or lack of indication or communication Unsuitable indication or communication Indication and communication Unsuitable timing of indication or communication Difficult to listen because of work place noise Complex operation of machinery and tools Too many kinds of tools Difficult to operate crane or hoist Machinery or tools Unsuitable tools, materials, or machinery Difficult to control or operate machinery and tools Unsuitable temperature and humidity Loud noise Environment Large vibration Dark lighting Incomplete arrangement of work place Narrow and inconvenient work space Dangerous work space such as narrow scaffolding Individual work space overlaps with others’ Work space Difficult to approach work object Demands difficult movement or posture Protective clothing or safety devices are obstructive

HR

Importance WE WL

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Fig. 16.5 Example of relationships between factors in the “physical load” category Toriizuka (2001).

Fig. 16.6 Example of how the PSF is weighted for a case with three factors Toriizuka (2001).

Using the method proposed by Toriizuka (2001), once we have identified all the relationships between the factors in each category, we can calculate the value of each PSF. This is established from the relationship links the importance of the factors that interact with the one taken into consideration, and their degrees of influence, based on the arrangement shown in Figure 16.6. Thus, having defined all the weights for each PSF, we can arrange the factors on a scale of importance, paying particular attention to those at the top of the list. 3.3.2 Study on Contribution of Human Factors to Incidents in the Petroleum Refining Industry Another study, by Chadwell et al. (1999), enabled a description of the contribution of the human factors inherent in incidents that take place in industries in the oil refining sector. These results emerged from the documentation collected on 130 incidents recorded over a period of eight years of production.

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Clearly, the importance of human errors can acquire a different weight depending on their potential effects, so the aim of the study conducted by these researchers was to produce a characteristic model to develop and apply to the dataset so as to be able to identify the most recurrent human factors in the incidents recorded. To obtain this information and create the database of the incidents that had occurred, the authors conducted a series of studies exploiting various sources, including: • • • • •

local and regional daily papers; Internet; sector-specific journals; reports from auditing agencies; safety updates.

The refineries were also contacted directly, but this type of data collection was only partially successful. In the end, the authors were able to collect material on 136 incidents. After completing the data collection phase, the authors divided the factors into categories on three levels: on the first level they included three categories: ease of design, procedures, and management systems. Alongside each factor they gave an example of a possible situation in which an incident occurred that could be traced back to said factor, as shown in Table 16.2. When they came up against an incident caused by more than one factor, it was attributed a value that was distributed between the two categories concerned. For instance, if an incident had been caused by the factor “ease of design/group/definition” and by “management systems/test/initial”, a weight of 0.5 could be attributed to each of the two factors, whereas if incidents had been caused by only one factor then the factor concerned would be given a weight corresponding to 1. On developing the analysis, the authors arrived at a classification based on the importance of each single factor. Briefly, the incidents were attributable mainly to group errors. In particular, while 19% of the human errors were of a causal nature, the other 81% could be attributed to specific human factors, such as procedures (the most common cause), the management system (the second cause in order of importance). Continuing this distribution on all the levels, the authors created the layout shown in Table 16.2 for the set of incidents analyzed, which could be brought down, in percentage terms, to the factors listed in the table. 3.3.3 Other Human Reliability Assessment (HRA) Studies There are various studies in the literature on the assessment of human reliability that provide information on which factors are important in defining reliability, and a number of these papers are briefly mentioned below. From these analyses, we can derive the factors to consider in an analysis using FCM. The results are collected in Table 16.3. The work by Stewart and Melchers (1997) focuses on studying the risks and likelihood of occurrence of the factors that most strongly influence HRA. In

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Table 16.2 characteristic cause-contribution factors drawn from the article by Chadwell et al. (1999) Contribution Factor

Training

Communication

Scheduling

Respons Ops/Maint e to Safe Work Upset

Management Systems

Procedural

Environment

Controls

Facility Design

Equipment

Example Situation

Random Human Error Equipment Failure (No. Human Error)

Labelling Access Operability Layout Uniqueness Labelling Mode Involvement Displays Feedback Noise Level

Visibility Lighting Content Identification Format

Mislabelled, or not labelled at all Hard to reach or access Difficult to operate/change position Confusing/inconsistent arrangement Several components look alike Mislabelled, or not at all Manual operation; many manual steps Operator detached from process Unclear/complex/non-representational None, or potentially misleading Area where hearing protection required Extremes in temperature, humidity, precipitation, wind, etc. Often foggy ort other visibility limitations Inadequate lighting for task Incomplete/too general/out of date Ambiguous device/action identification Confusing/inconsistent; difficult to read

Aids

Task sequence done by memory

Alarms

Many simultaneous or false alarms

Coverage

Operator not always present

Time

Inadequate time to respond

Preparedness

No drills/simulation of scenario

Last-Resort Overtime Consistency No of tasks Task freq. Intensity

Shutdown discouraged or unsafe Extreme enough to affect performance Inconsistent shift rotations/schedules Tasks required exceed time available Very infrequent; lack of experience Differing tasks in rapid succession Inadequate communications between shifts on plant status No/poor communication between control and field operators Little or no supervisory checks No distinction between alarms in areas or types Little or no job specific training Over-due or non-existent Little or no training on procedural or process changes Human error with no contributing human factors Equipment failure with no contributing human factors

Climate

Shift Changes Field/Control Supervision Emergency Initial Refresher Safety Awareness

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addition to suggesting various approaches to the problem, the researchers concentrate on drawing a distinction between the different types of human error and to seeking their causes. Berra and Prestipino (1983) conducted an analysis on workers’ conditions, paying special attention to studying job timing and methods, productivity, and job organization in relation to human beings, going on to suggest a variety of factors that influence the human reliability issues. The studies conducted by Kirwan (1997a, 1997b) aimed to analyze the validity of three methods - the Technique for Human Error Rate Prediction (THERP), the Human Error Assessment and Reduction Technique (HEART), and the Justification of Human Error Data Information (JHEDI), used in Ukraine to predict human performance in high-risk industries. The techniques were used to determine the risks relating to human error and to establish the likelihood of each error occurring, or to identify each type of failure. The crucial point of approaches of this type lies in the reliability of the analysis and of the results produced, but they enable us to obtain useful information on the factors influencing HR. In a second phase, the same author Kirwan (1997a) focused on assessing human performance at a nuclear chemical plant, presenting a new approach to the problem based on the Human Reliability Management System (HRMS). The author claimed that it was possible to overcome the drawbacks of the JHEDI technique, in that the HRMS succeeded in providing more detail and was able to suggest mechanisms for reducing the incidence of errors. While the HRMS was applied to 20 high-risk scenarios, the JHEDI enabled the probability of a large number of human errors in low-risk scenarios to be identified. A new procedure proposed by Hauptmanns et al. (2001) enabled an assessment and optimization of the man-machine relationship. This procedure, called the Human Error Assessment and Optimizing System (HEROS) was based on fuzzy theory and adopted fuzzy linguistic states in the assessment of human factors. The 30 states were expressed in terms of fuzzy numbers or ranges that allow for mathematical operations. The HEROS facilitates the analytical procedure in the assessment of human factors that is indispensable to the optimization of the manmachine system. This technique was applied to the assessment of the tasks performed by personnel working at a nuclear power plant. Allen et al. (2000) developed a method for use in the aeronautical industry, called the Maintenance Error Decision Aid (MEDA), to determine the factors that contribute to errors and the corrective action designed to prevent the recurrence of similar errors. It is clearly essential to consider the human factors in an approach of this type, since they can have a fundamental role in incidents and failures. We must also mention the publication by Latorella and Prabhu (2000), again referring to the aeronautical industry, in which they identified the anomalies that occurred in 14 airplane accidents attributable to human error. Shorrock and Kirwan (2002) also focused on the same sector, developing a tool for identifying human error in air traffic control. The technique is called TRACE and is based on cognitive aspects of human behaviour and their surroundings, with a view to containing errors.

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The calculation of the likelihood of errors was the object of a study by Park and Jung (1996), that enabled a Human Error Probability (or nominal HEP) to be determined. This approach is essential when the effects of performance shaping factors (PSF) are not taken into account. The study presents a variant of the nominal HEP based on a combination of effects relating to a small set of human factors. The aim of the article by Higgins et al. (1998) was to list and classify human errors and their effects on system performance, so it proved a valid aid in the search for elements that influence HR. These authors organized various factors into categories, such as “human operator reliability” and “man machine system efficiency”. The case study by Edland et al. (1996) refers to a nuclear power plant, and its purpose was to analyze the process and identify the various types of error, including human errors; they thus defined two types of the latter, one relating directly to the person and the other to the surrounding environment. Soresen (2002) began to introduce the concept of safety starting from nuclear power plants, but later expanding the area of interest to all the sectors defined as hazardous, where complex technologies are used and an error can have severe consequences, and the author listed a series of human factors that must be taken into account. The article by Kirwan et al. (1996) reports on a two-year study on Human Reliability Assessment (HRA) at a nuclear power plant using the probabilistic safety assessment (PSA) approach. The results demonstrate that an improvement was obtained in the process by defining, analyzing and assessing the job, thereby reducing the chances of error. Together with Kirwan et al. (1997) considered the same approach to the study of HRA, and also provided a historical picture of the assessment of human behaviour. According to Strater and Bubb (1999), the main problem to consider is always how to collect the data and the fallout of inadequate information on the assessment of human reliability. So the first issue that they considered was the accurate analysis of human behaviour in relation to the process, then they implemented an application on 165 reactors, assessing the influence of the human operator. In the maritime transportation of petroleum, the damage that can be caused by any anomalous behaviour can be very severe for humans and for the whole ecosystem, so the study conducted by Harrald et al. (1998) aimed to reduce errors as far as possible: after collecting information by interviewing various experts in the sector, the author conducted simulations on the data collected with a view to predicting the potential risks. The same stance was adopted by Basra and Kirwan (1998) in their qualitative and quantitative assessment on the prevention of errors at nuclear power plants, and they developed an approach based on a Computerized Human Error Data Base (CORE-DATA). Cacciabue (2000) emphasized the fact that the impact of human factors can differ, depending on the complexity of the plant where the people work. That is why the importance attributable to this aspect also differs, though it certainly cannot be ignored, and several techniques are described for a precise risk analysis, such as the Human-Machine Interaction (HMI) model.

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Table 16.3 Human factors obtained from literature review CATEGORY

Environment

Work Space

Machinery or Tools

Physical Load

Mental Load

Judgmental Load

SYMBOL

DESCRIPTION

E1

Unsuitable temperature and humidity

E2

Loud noise

E3

Large vibration

E4

Dark lighting

WS1

Incomplete arrangement of work place

WS2

Dangerous work space such as narrow scaffolding

WS3

Individual work space overlaps with others

WS4

Difficult to approach work object

WS5

Demands difficult movement or posture

WS6

Protective clothing or safety devices are obstructive

MT1

Complex operation of machinery and tools

MT2

Too many kinds of tools

MT3

Difficult to operate crane or hoist

MT4

Unsuitable tools, materials, or machinery

MT5

Difficult to control or operate machinery and tools

PL1

Long continuous working time

PL2

A precision task

PL3

A complex task which needs experience skill

PL4

Only one part of the body is exerted strongly

PL5

Heavy labor

ML1

Quick response is required because of time pressure

ML2

Dual duty, punctuality and perfect task result are required

ML3

An error during task leads to great plant damage

ML4

Task is difficult to redo, correct or interrupt

ML5

Continuity of similar tasks

ML6

Task progresses very slowly

ML7

Many factors exist which distract workers ’attention

JL1

Not only knowledge but also an interpretation or a judgment based on experience is needed

JL2 Information and Confir- I1 mation I2 Indication and Commu- IC1 nication IC2

A prediction or keen insight is needed Difficult to get feedback of work result Difficult to grasp state of working process Incorrect / Unsuitable or lack of indication or communication Unsuitable timing of indication or communication

Ayers and Kleiner (2000) conducted a study on human reliability as seen from the economic standpoint, identifying how human errors can mean additional costs, and reporting precise data from the National Safety Council.

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Also in shipping, human efficiency is fundamental to the avoidance of incidents, and so is the instrumentation and the human being’s aptitude for interacting with it. Goulielmos and Tzannatos (1997) studied the problem of man-machine interaction in this sector, listing a series of important human factors. It is essential to study human behaviour in the building sector, too, for the purpose of error prevention, and this was the focus of a study conducted by Atkinson (1999), who collected details on 23 incidents in the construction industry, enabling a number of important human factors to be identified. In the light of the above, we pinpointed various factors that influence human reliability, as discussed in the literature and included in the following table, divided by category and the studies in which they were mentioned. The table thus brings together a wide range of factors subsequently used to construct a fuzzy cognitive map. The framework of factors in Table 16.3 is the result of bibliographic analysis. The variables which allow to describe a productive system in terms of influences on the operator’s reliability are divided into eight macro-categories, as suggested by Toriizuka (2001), for a total of 37 factors.

3.4 Application of Fuzzy Cognitive Maps to Factor Analysis After formalizing the framework with the data obtained from the scientific literature, cognitive maps were applied to a specific production setting with a view to establishing which system variable was most influential in relation to the reliability of the human operator. Nearly every industrial plant must be designed and managed to ensure optimal boundary conditions from the reliability standpoint, and this makes it necessary to create an environment in which the characteristic variables are as near ideal as possible. The particular nature of the tool used to conduct our analysis lies in that the analysis on the influence of each variable used to describe the environment considers both the direct effects of a given variable on human reliability and the indirect effects deriving from other system variables. A panel of experts was created, comprising 25 people, the dual purpose of which was to validate the framework drawn from the literature and to develop a fuzzy cognitive map based on said framework. The panel included: academics from the University of Parma; designers, builders, and installers of plants for producing vegetable preserves; and production, safety and quality managers at vegetable preserve production plants. The number of panel members (25) may seem rather large at first glance, but it stemmed from the application of the Delphi technique (Linstone and Turoff 1975) used to conduct the research. In particular, this method demands a group of at least 20 people to reduce the mutual influences due to differences of opinion. The Delphi technique is a structured process that studies a complex or ill-defined problem with the aid of a panel of experts. The method has proved appropriate for this type of research because it enables different individual opinions to be obtained from a structured group during a simple communication process (Delbecq et al. 1975). The Delphi approach is particularly indicated in the case of complex, interdisciplinary problems involving a large number of variables and new concepts (Meredith et al. 1989).

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In particular, the panel members included: • academics from the Department of Industrial Engineering at the University of Parma, whose research areas mainly concerned procurement management, production planning and control, distribution and supply chain management; • technicians working in the design, construction and installation of plants for the production of vegetable preserves at major plant manufacturing companies with registered offices in Parma; • managers responsible for production, safety and quality at the production plants of major industries in the vegetable preserves sector in Parma. This group achieved the necessary balance between the different skills involved in the decision-making process (Akkermans 1999). The panel worked for approx. 3 weeks, and sessions were planned on a four round Delphi process. A moderator was responsible for collecting the panel’s results, conducting the various steps of the research. First of all, the panel members were asked to validate the framework defined on the strength of a critical literature review. Each member was asked individually and anonymously to confirm the presence of the factors identified, possibly combining together any factors they considered redundant, eliminating variables they considered scarcely significant, or adding any missing variables. Once the panel had agreed on the framework as applied to the industrial sector in question, each panel member was asked, again individually and anonymously, to establish the existence and the related signs of all the causal connections between the various concepts adopted to describe the system, as shown in Figure 16.7 and Table 16.4. The functional cognitive map, reported in Table 16.4, was constructed on the basis of the panel’s convergence concerning the causal relationships. Then, to obtain a fuzzy cognitive map, the panel members were asked to express a literal opinion on the degree of causal influence between the variables. Using fuzzy logic

Fig. 16.7 Excerpt of an FCM for the vegetable preserves sector

E1 E2 E3 E4 WS1 WS2 WS3 WS4 WS5 WS6 MT1 MT2 MT3 MT4 MT5 PL1 PL2 PL3 PL4 PL5 ML1 ML2 ML3 ML4 ML5 ML6 ML7 JL1 JL2 I1 I2 IC1 IC2 HR

E1 + +

+

+

+ +

+

+

+ + +

E2 +

+ + + + -

+

+ + + + -

+

+

+

+

+ + + + +

+ + +

+ + + + + +

+ + + +

+

E3 + +

E4 +

+

+

-

+ + -

WS1 + + +

+ +

+

+

+

+ + + + + +

+ + + -

WS2 + + -

+

+ + + +

+ + + + +

+

+ +

WS3 + + + + + + + + + + + + -

+ + + + + + + + + + + +

+ + + + +

WS4 +

+

+ + +

+

+

+ + +

WS5 + + -

+ + + + +

+ + +

+

+ + + + + + +

+ + + + + + +

WS6 + + -

+

+ + +

+ + + + + +

+ + + + + +

+ + + + + +

+

MT1 + + -

+ +

+ +

-

+

+

+

+

+ +

+ + + +

MT2 + + + + + + + + + + + + + + + + + + +

+

+ + -

+ +

+ + + -

MT3 + + + + + + + + + +

+

+ + +

+

+

+ +

+

+ +

MT4 + + + + + + + + + + +

+ + +

+ +

+ + + + +

+

MT5 + + + + -

+

+ + + + + +

+ +

+

+ + + + + + + + + + + + +

PL1 + + + + + + + + + + + + + + + -

+ + + + + + + + + + + + + +

PL2 + + + + -

+ + + + + + +

+ +

+ + + + + + + + + + + +

+ + + + + + + + + + + + + -

+ + + + + + + + + + + + + + + +

PL3

Table 16. 4 Functional cognitive maps (CM) for food industry.

PL4 + + + + + + + + + + + + +

+ + + + + + +

+ +

+ + +

+ +

PL5 -

+

+ + -

+

+ +

+

+

+

+

+ +

+ + +

ML1 + + + + + + + + + +

+

+

+ + + + + +

+ -

+

+

ML2 +

+

+ + + + + + + -

+ + + +

+

+ -

+

+ + +

ML3 + + + +

+ + + + +

+ +

+ + +

+

+

ML4 + + + + +

+ + -

+ + +

+ + + + + + +

+ +

+

+

+

+

ML5 + + + + +

+ + +

+ + + + + + + + + + + + + + + + + +

+ + +

ML6 + +

+ + + + -

+ + + +

-

+

+ + + + +

+

ML7 + + + +

+ + -

+ + -

+ +

+ +

-

JL1 + + + + + +

+ + + + + + -

+ + + + + + + + + + + + + + +

+

+

JL2 + + + + +

+ + + + + + +

+ + + + + + + + + + + + + + + +

+

I1 + + + +

+ + + -

-

+ + +

+ + + + + +

+

+

+ + +

I2 + + +

+ + + +

-

+

+ -

+ + +

+ +

+

+

+ + +

IC1 + -

+ + + + + + +

+

-

+

+ + + +

+ +

IC2 -

+ + + + + + + +

+

+

+ +

+ +

+ +

HR + -

Fuzzy Cognitive Maps for Human Reliability Analysis in Production Systems 409

enables a relatively straightforward, strict treatment of linguistic variables, given the ambiguity and inaccuracy inherent in the experts’ opinions. In fact, the linguistic variables used were translated into isosceles triangular fuzzy numbers that were subsequently defuzzified and the resulting crisps could be used as the input for the FCM. The scale of opinions on the proposed assessment parameters was represented by five elements in order to balance a good assessment accuracy with the need to facilitate the identification of the experts’ answers. Their opinions could be: “not important” - “low” - “moderate” - “high” - “very important”. These literal terms identified the central value of the fuzzy triangular numbers associated with

0

0

0

WS2

E4

0

-0.320

0.32 0

0.08 0.04 0.28 0

0.24 0

0

0.04 0.12 0

0 0

ML10 ML20 ML30

0

0

0

0

0

0.12 0.12 0 0 0 0.08 0.12 0

ML70 JL1 0 JL2 0 0 0

I1 I2

0.08 0

0.12 0

0

0

HR 0

0

0.08 0

IC2 0.12 0.12 0.12 0

0

0.08 0

0

0.08 0

0.12 0

0

0

0

0

0

0

0

0

0

0

-0.360

0.56 0

0

0

0

0.20 0.04 0

0

0

0

0

0.20 0

0

0

0

0

0

0

0

0

0

ML7

-0.080

0

0

0

0

-0.60

-0.48

0

0.28 -0.28-0.040.04 0.04 0.04 0

0

0.28 0.20 0.08 0.04 0

0

0

0

-0.36

-0.24

-0.32

0.04 -0.32

0.20 0.04 -0.28

-0.56

0.04 0.04 -0.52 0.36 0.36 0.20 0.12 0.04 0

0

0.08 0.12 0.04 0.04 0.24 0.24 -0.44

0

0

0.40 0.24 0

0

0

0

0

0

0

0.08 0

-0.040

0

0

0

0

0

-0.52

-0.28

-0.24

-0.32

0.40 0.44 0

0.08 0.08 -0.40

0.04 -0.040.24 0

0.16 0.60 0.72 0.32 0.36 0

-0.08-0.12-0.080.08 0.04 0.04 -0.24-0.24-0.12-0.200.04 -0.040.16 0.16 0.20 0.16 0.16 0.08 0

0.16 0.16 0.04 0.04 -0.36-0.360

-0.56

0.48 -0.68 0.16 0.28 0.16 0.04 0.04 0.40 0.60 0.32 0.40 0.24 0

0.28 0.16 0.16 0.40 0.24 0

0.24 0.28 -0.32

0.36 0.28 0.24 -0.24

-0.04-0.04-0.04-0.04-0.28 0.08 0.12 0.08 0.16 0.32 0.32 0

0

0.04 -0.12-0.12-0.12-0.12-0.24

-0.12-0.12-0.12-0.12-0.12-0.08-0.20

-0.120.24 0.24 0.08 0.08 0.08 0.08 -0.36

0.12 0.04 0.28 0.28 0.04 0.04 0.44 0.24 -0.56

0.08 0.08 0.16 -0.040.12 0.40 0.44 0.32 0

0.16 -0.080

0

0.24 0.16 0.04 0.32 0.32 0.08 0.04 0.08 0.08 -0.44

0.16 0.32 0.08 0

0.04 0.28 0.20 0.24 -0.040.40 0.40 -0.12-0.120.12 0.12 -0.32

0.04 0.24 0.08 -0.08-0.120.28 0.20 -0.16-0.16-0.120

0

0.04 0.08 0.16 -0.20-0.120.52 0.40 0.12 0.12 0.12 0 0

-0.36

-0.40

0.20 0.12 0.16 -0.08-0.08-0.120.12 -0.44

0.16 0.08 0.08 0.12 0.32 0.36 0.16 0.12 0

0.32 0.12 0

0.20 0.16 0.12 0.12 0.28 0.24 0.16 0.16 0.12 0.08 -0.32

0.12 0.16 0.04 0

0.20 0

IC1

0.08 0.08 0.20 0.24 -0.72

0.12 0.12 -0.08-0.080.24 0.20 0.12 0.12 0

0

0.12 0.12 -0.080

0.08 0.12 0

0.12 0.12 0.08 0.04 0.20 0.16 0.24 0.28 0.08 0.16 0

0.16 0.24 0.08 0.20 0.20 0.08 0.24 0.36 0.24 0.40 0.32 0

0

0.20 0.08 0.20 0.04 0

0

-0.120.08 -0.080.16 0.28 0.16 -0.040.04 0.12 -0.080

0.12 0.24 0.12 0.08 0.24 0.24 0.24 0.20 0

JL1

0.28 0.20 0.12 0.20 0.12 0

-0.080.04 0.48 -0.080.12 0.24 0.16 -0.08-0.120.20 0.20 0

0.08 -0.08-0.040.16 0

I2

0.08 0.04 0.04 0.04 0.12 0.12 0.08 0.04 0

0

0.12 0.32 -0.080.04 0.08 0.12 0.44 0.20 0.08 0.28 0.40 0.32 0.16 0.24 0 0.16 0.12 0.08 0

0

0

IC2

0.16 0.08 0.04 0.08 0.04 0.04 -0.64

JL2

-0.04-0.040.04 0.04 0

0

0

0.52 0.28 0.04 0

0

0.08 0.08 0.16 0.20 0

0.08 0.12 0.12 0.08 0.40 0.28 0.04 0

0

0

I1

0.08 0.04 0.08 0.12 -0.08-0.040.16 0.16 0.04 0

0.16 0.12 0

0.12 0.12 0.12 0.28 0.20 0.08 0

0.04 0.08 0.12 0.24 0.28 0.12 0

0.04 0

0.28 0

ML6

0.28 0.32 0.04 0

0

HR

0.08 0.16 0.32 0.20 0.04 0.24 0.20 0.16 0.16 -0.12-0.08-0.24 0

0.24 0.08 0

0.08 0.16 0

0.16 0

0.08 0.04 0.08 0.08 0.24 0.24 0.44 0.32 0

0.08 0

0.16 0.08 0

0.12 0

ML3

0.64 0.20 0.08 0.04 0.12 0.08 -0.080.32 0

0.24 0.40 0.32 0

0.04 0.24 0.36 0.16 0

0

0

0.12 0.12 0

0

0.24 0.28 0.32 0.32 0.08 0.24 0.12 0

0.16 0

0

0.08 -0.120.12 0.12 0.04 0.08 0

0

0

ML5

0.24 0.20 0

ML4

0.04 0.08 0

0

0.32 0.08 0

0

0.08 0

0.20 0.32 0.24 0.24 0.16 0

0.12 0.08 0.08 0.04 0.08 0.12 0

0.08 0

0.08 0.12 0

0.12 0.08 0

0

0.12 0

0.04 0

0.12 0

0.04 0.04 0.24 0.04 0.12 0.32 0.12 0.08 0

0.28 0.20 0.08 0.24 0.28 0.08 0.32 0

0.16 0.12 0

0.08 0.04 0

0.08 0.12 0

0

0.12 0.16 0.04 0.04 0

0.08 -0.040.16 0.08 0.08 0

0.12 0

0.04 0.12 0

0

0

ML1

0.16 0

0.08 0

0.04 0.24 0.32 0.20 0.28 0.16 0

0

0.28 0.04 0

0.08 0.08 0.04 0.16 0 0

0.08 0.28 0.32 0.04 0.28 0

0.12 0

0.04 0.28 0.12 0.20 0 0

0

0.16 0

0.36 0.40 0.20 0.40 0.16 0

-0.280

0.28 0

0.20 0.56 0.24 0.08 0.16 0.16 0

0

-0.52-0.360

0.28 0.44 0

0.60 0.12 0.36 0.20 0.20 0.36 0.28 0.16 0.24 0.40 0.24 0.40 0.48 0

0.08 0

0.08 0

0.04 0

0.12 0

0.08 0

-0.080.12 0

0.08 0

0

0

IC1 0.12 0.24 0.12 0

-0.12-0.080

0

-0.080

0

0.12 0

0.08 0.56 0

0

0

0.08 0

0

0

0 0

MT4

0.12 0.24 0.48 0.20 0.28 0.28 0.24 0.08 0.20 0.08 0.12 0.28 0.16 0.28 0.08 0

0

0

0

0

0

0

0.08 0.08 0

0

0

0 0

0

0

0.16 0

ML60

0.16 0

0

0

0

0

0

ML50

ML40.20 0

0 0

PL5 0

0

PL4 0

0

0.20 0.16 0.04 0.04 0.04 0.36 0

PL3 0

0.16 0

0

0

0

PL2 0

0

0.08 0.36 0

0

0.32 0

0

0.16 0.32 0.36 0.16 0.24 0.44 0

PL1 0.12 0.12 0.12 0.12 0

0

-0.040

MT50

0

0

0

0

0.60 0.12 0

PL5

0.68 0.04 0.12 0

ML2

0.16 0.12 0.04 0.40 0.28 0.28 0.32 0.12 0.16 -0.08-0.080

0.04 0.20 0.08 0.08 0.04 0

0.28 0

0

0.24 0.56 0.04 0.16 0.32 0.04 0

0.20 0.16 0

MT5

0.28 0.40 0

PL2

0.36 0.28 0.16 0.12 0.08 0.28 0.40 0.20 0.32 0.32 0

0.40 0.20 0

0

0

0

-0.44-0.160.08 -0.080

0.48 0.12 0.12 0.08 0

MT40.04 0.40 0.24 0

0

MT3

0.36 0.12 0.04 0

-0.040.08 0.04 0.24 0.12 0.16 0.16 0

0.16 0.16 0

MT1

MT30.16 0.08 0.16 0.08 0

MT20

MT10.20 0.12 0

WS4

0.08 0.24 0.28 0.08 0

0

WS60

0

0

0

WS50 0

WS5

0.36 0

PL3

0.20 0.44 0.48 0.44 0.32 0.08 0.24 0.56 0.28 0.08 0.56 0.36 0

-0.040.24 0.24 0.28 0

0

MT2

0.52 0.40 0.36 0.36 0.08 0.08 0

0.04 0.04 0.04 0.44 0.32 0

PL1

0.36 0.28 0.20 0.08 0.04 0.12 0.28 0.36 0

WS6

0.44 0.60 0.28 0.44 0.44 0

-0.120

0.24 0

0.32 0.12 0.52 0

WS40.24 0.16 0

WS30

WS3

0.24 0.20 0.32 0

WS1

WS20.24 0.28 0.12 0.12 0.48 0

WS10.28 0.32 0.20 0

E4 0.12 0

0.52 0

E3 0

0.32 0

0

E3

0.16 0

0

E1

E2 0

E2

E1 0

PL4

Table 16.5 Fuzzy cognitive maps for food industry.

410 M. Bertolini and M. Bevilacqua

the experts’ opinion. To allow for any ambiguity and uncertainty the expert may have in formulating an opinion, each value on this scale was associated with an attribute so as to form, together with the value, a “compound opinion”. These attributes were: “certainly”, “probably”, and “roughly”. The ambiguity expressed by the attributes will be translated into FCMs by means of the amplitude of their base, i.e. the uncertainty expressed by the attribute goes to determine the amplitude of the base of the FCM associated with the opinion. The FCMs defined by each expert were composed so as to have a single opinion for each causal connection: this was done by simply averaging the FCMs, attributing to each expert the same degree of importance (Taber 1994). The “aggregate opinion” thus obtained for each criterion then has to be defuzzified before it can be used in the process of evolution of the fuzzy cognitive map. So an assessment was conducted on the degree of causal influence of each variable on the top event variable (human reliability), then the FCM could be

Fuzzy Cognitive Maps for Human Reliability Analysis in Production Systems

411

represented analytically and the indirect effect and total effect of each variable considered could be determined. The results obtained at the end of this process of evolution of the FCM are given in Table 16.5. Taber (1991) suggested a relation to unify different judgments of experts, based on the credibility weight of each expert. On the basis of the law of large numbers, suggested that a large number of independent experts focusing on the same problem tend to produce a stable edge way. 3.4.1 Results and Discussion In order to analyze the factors affecting human reliability in industrial plants, it is necessary to conduct the numerical analysis of FCMs. As sketched above, there are two alternative approaches to effort numerical analysis of FCMs: on one side, the traditional approach proposed by Kosko (1985) and widely applied in literature (Bowles and Pelàez 1995a, 1995b, Kardaras and Karakostas 1999). In case of “big” FCM (characterised by many concepts), the limit of the traditional approach is connect to the high simulation time. This is not a problem for the study here presented, because the FCM developed is a DSS that allows to discover which factors are the most important in order to improve human reliability, and therefore decide where to concentrate resources in order to improve the work environment. If FCM is used as control system simulation time is an important factor to guarantee a rapid response of controller. To illustrate how simulate using CM here reported, Table 16.5 reports the inference process for the concept E2 (“loud noise”) of the case study. To conduct a numerical analysis of FCM it is necessary perform an iteration process: Table 16.5 reports the iteration process for concept E2, and the total effect calculate is equal to -0.72. The total effects for each factor should be evaluated to determine the influence’s on human reliability. The total effect is calculate use the min-max inference approach and date necessary are reported in Table 16.6. Table 16.6 Inference process results for concept E2.

412

M. Bertolini and M. Bevilacqua

Table 16.7 Ranking of the main significant factors based on total effect in the field of vegetable preserves. SYMBOL

TOTAL EFFECT

DESCRIPTION

E2

-0.72

Loud noise

IC1

-0.68

Incorrect or lack of indication or communication

E1

-0.64

Unsuitable temperature and humidity

E4

-0.60

Dark lighting

ML3

-0.58

An error during task leads to great plant damage

ML7

-0.58

Many factors exist which distract workers ’attention

IC2

-0.56

Unsuitable indication or communication

MT4

-0.56

Unsuitable tools, materials, or machinery

WS1

-0.56

Incomplete arrangement of work place

WS2

-0.54

Dangerous work space such as narrow scaffolding

E3

-0.52

Large vibration

PL1

-0.52

Long continuous working time

PL5

-0.52

Heavy labor

WS3

-0.50

Individual work space overlaps with others

ML5

-0.48

Continuity of similar tasks

ML2

-0.44

Dual duty, punctuality and perfect task result are required

ML6

-0.44

Task progresses very slowly

MT3

-0.42

Difficult to operate crane or hoist

MT5

-0.40

Difficult to control or operate machinery and tools

The results obtained at the end of the FCM evolution process (i.e. each total effect) are shown in Table 16.7. All total effects are negative, as an increase of the variable being examined (i.e. for concept E2, “loud noise” total effect is equal to -0.72) entails a decrease in the operator’s reliability. The two most significant factors are “loud noise” and “incorrect or lack of indication or communication”, respectively. The two macro-categories “environment” and “indication and communication” emerge as the most important categories as regards the causal influence on the top event human reliability. In particular, when taking into consideration the categories “environment” and “work space”, it can be immediately noted that they include the factors which are the most significant in improving the operator’s reliability.

5 Conclusion With the current increasing need for efficiency of both plant and human operator, FCMs have proved to be able to provide a valid help in assessing the most critical factors for operators in managing and controlling production plants. A FCM represents a technique that corresponds closely to the way humans perceive it; they are easily understandable, even by a non-professional audience and

Fuzzy Cognitive Maps for Human Reliability Analysis in Production Systems

413

each parameter has a perceivable meaning. FCMs are also an excellent means to study a production process and obtain useful indications on the consequences which can be determined by the variation of one or more variables in the system examined. They can provide an interesting solution to the issue of assessing the factors which are considered to affect the operator’s reliability. From a study carried out in a specific productive sector, such as the one of vegetable preserves, it clearly emerges that the system’s variables which most affect human reliability belong to the categories “indication and communication”, “environment” and “work space”. The FCM developed can be easily modified to include new/different factors, and if its behaviour is different than expected, it is usually easy to find which factor should be modified and how. The FCM proposed can be used to analyse, simulate, test the influence of parameters and predict the behaviour of the system. These results can be a valuable aid to the heads of safety and production, who can thus direct resources by improving few selected factors knowing for certain that this will increase safety and reliability of the human operator.

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Codara, L.: Le mappe cognitive. Carrocci Editore, Roma (1998) Delbecq, A.L., Van de Ven, A., Gustafson, D.H.: Group techniques for program planning. In: A guide to nominal group of Delphi processes. Scott Foresman, Glenview (1975) Edland, A., Keclund, L., Svenson, O.J., Wedin, P.: Safety barrier function analysis in a process industry: A nuclear power application. International Journal of Industrial Ergonomics 17(3), 275–284 (1996) Fu, L.: CAUSIM: a rule-based causal simulation system. Simulation 56(4) (1991) Gordon, R.P.E.: The contribution of human factors to accidents in the offshore oil industry. Reliability Engineering & System Safety 61(1-2), 95–108 (1998) Goulielmos, A., Tzannatos, E.: The man-machine interface and its impact on shipping safety. Disaster Prevention and Management: An International Journal 6(2), 107–117 (1997) Groumpos, P.P., Stylios, C.D.: A soft computing approach for modelling the supervisor of manufacturing systems. Journal of Intelligent and Robotic Systems 26, 389–403 (1999a) Groumpos, P.P., Stylios, C.D.: Fuzzy cognitive maps: a model for intelligent supervisory control system. Computers in Industry 39, 229–238 (1999b) Groumpos, P.P., Stylios, C.D.: Modelling supervisory control systems using fuzzy cognitive maps. Chaos, Solitons and Fractals 11, 329–336 (2000) Harrald, J.R., Mazzucchi, T.A., Spahn, J., Van Dorp, R., Merrick, J., Shrestha, S.: Using system simulation to model the impact of human error in a maritime system. Safety Science 30(1-2), 235–247 (1998) Hauptmanns, U., Richei, A., Unger, H.: The human error rate assessment and optimizing system HEROS -— a new procedure for evaluating and optimizing the man–machine interface in PSA. Reliability Engineering & System Safety 72(2), 153–164 (2001) Higgins, J.J., Lee, K.W., Tillman, F.A.: A literature survey of the human reliability component in a man-machine system. IEEE Transactions Reliability 37(1), 24–34 (1998) Kardaras, D., Karakostas, B.: The use of cognitive maps to simulate the information systems strategic planning process. Information and Software Technology 41, 197–210 (1999) Kecklund, L.J., Svenson, O.: Human errors and work performance in a nuclear power plant control room: associations with work-related factors and behavioural coping. Reliability Engineering & System Safety 56(1), 5–15 (1997) Kirwan, B.: The development of a nuclear chemical plant human reliability management approach: HRMS and JHEDI. Reliability Engineering & System Safety 56(2), 107–133 (1997a) Kirwan, B.: The validation of three Human Reliability Quantification techniques – THERP, HEART and JHEDI: Part III – Practical aspects of the usage of the techniques. Applied Ergonomics 28(1), 27–39 (1997b) Kirwan, B., Kennedy, R., Adams, S.T., Lambert, B.: The validation of three human reliability quantification techniques – THERP, HEART and JHEDI: Part II – Results of validation exercise. Applied Ergonomics 28(1), 17–25 (1997) Kirwan, B., Robinson, L., Scannali, S.: A case study of a human reliability assessment for an existing nuclear power plant. Applied Ergonomics 27(5), 289–302 (1996) Kosko, B.: Fuzzy cognitive maps. International Journal of Man Machine Studies 24, 65–75 (1985) Kosko, B.: Neural networks and fuzzy systems. Prentice Hall, NJ (1992) Kosko, B.: Fuzzy thinking: the new science of fuzzy logic. Hyperion (1994) Latorella, K.A., Prabhu, P.V.: A review of human error in aviation maintenance and inspection. International Journal of Industrial Ergonomics 26(2), 133–161 (2000)

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Linstone, H.A., Turoff, M.: The Delphi method, techniques and applications. AddisonWesley, London (1975) Mahmood, M.A., Soon, S.K.: A comprehensive model for measuring the potential impact of information technology on organisational strategic variables. Decision Sciences 22 (1991) Margaritis, K.G., Tsadiras, A.K.: Cognitive mapping and certainty neuron fuzzy cognitive maps. Information Sciences 101, 109–130 (1997) Margaritis, K.G., Tsadiras, A.K.: An experimental study of the dynamics of the certainty neuron fuzzy cognitive maps. Neurocomputing 24, 95–116 (1999) Melchers, R.E., Steward, G.: Probabilistic risk assessment of engineering systems. Chapman & Hall, London (1997) Meredith, J.R., Raturi, A., Amoako-Gyampah, K., Kaplan, B.: Alternative research paradigms in operations. Journal of Operations Management 8(4), 297–326 (1989) Paradice, D.: SIMON: an object-oriented information system for coordinating strategies and operations. IEEE Transactions on Systems, Man and Cybernetics 22(3) (1992) Park, K.S., Jung, K.T.: Considering performance shaping factors in situation-specific human error probabilities. International Journal of Industrial Ergonomics 18(4), 325–331 (1996) Park, K.S., Kim, S.H.: Fuzzy cognitive maps considering time relationship. International Journal of Approximate Reasoning 2 (1995) Petersen, D.: Human error reduction and safety management. Van Nostrand Reinhold, New York (1996) Schneider, M., Shnaider, E., Kandel, A., Chew, G.: Automatic construction of FCMs. Fuzzy Sets and Systems 93 (1998) Shorrock, S.T., Kirwan, B.: Development and application of a human error identification tool for air traffic control. Applied Ergonomics 33(4), 319–336 (2002) Soresen, J.N.: Safety culture: a survey of the state-of-the-art. Reliability Engineering and System Safety 76, 189–204 (2002) Sträter, O., Bubb, H.: Assessment of human reliability based on evaluation of plant experience: requirements and implementation. Reliability Engineering and Systems Safety 63, 199–219 (1999) Taber, R.: Knowledge processing with fuzzy cognitive maps. Experts Systems with Applications 2, 83–87 (1991) Taber, R.: Fuzzy cognitive maps model social systems. AI Expert (1994) Taber, R., Siegel, M.: Estimation of expert weights with fuzzy cognitive maps. In: Proceeding 1st IEEE international conference on Neural Networks (ICNN 1987), pp. II:319–325 (1987) Toriizuka, T.: Application of performance shaping factor (PSF) for work improvement in industrial plant maintenance tasks. International Journal of Industrial Ergonomics 28(34), 225–236 (2001) Warren, K.: Exploring competitive futures using cognitive mapping. Long Range Planning 28(5) (1995) Zadeh, L.A.: Fuzzy sets. Information and Control, 338–365 (1965) Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transaction on Systems, Man and Cybernetics (1973) Zhang, W.R., Chen, S.S.: A generic system for cognitive map development and decision analysis. IEEE Transactions on Systems Man and Cybernetics 19(1) (1989) Zhang, W.R., Chen, S.S., Wang, W., King, R.: A cognitive-map-based approach to the coordination of distributed cooperative agents. IEEE Transactions on Systems Man and Cybernetics 22(1) (1992)

Chapter 17

Fuzzy Productivity Measurement in Production Systems Semra Birgün, Cengiz Kahraman, and Kemal Güven Gülen*

Abstract. Productivity is a measure relating a quantity or quality of output to the inputs required to produce it. Productivity measurement has an important role in production and service systems. In this chapter, productivity measurement is realized under vague and incomplete information. The fuzzy set theory is used for this purpose. Data envelopment analysis is also applied to a productivity optimization problem.

1 Introduction Productivity performance measurement has always been an important aspect in manufacturing. This is a long term or progressing measure. This measurement is essential to benchmark and improve a company performance. Measurement steers human and system behavior. The results can be used to monitor the changes in productivity levels and provide the direction for improvement. Hence, the appropriateness of and accuracy in productivity measurement is vital (Wazed and Ahmed 2008). Production is the process of creating, growing, manufacturing, or improving goods and services. In economics, productivity is used to measure the efficiency or rate of production. It is the amount of output (e.g. number of goods produced) per unit of input (e.g. labor, equipment, and capital). Productivity analysis refers to the process of differentiating the actual data over the estimated data of output and input measurement and presentation. In economics, productivity is the ratio of the output production per unit of input. It may also refer to the technical efficiency of production relative to the allocation of resources of enterprises. If the goal is to increase productivity, enterprises must produce more with the same level of input. The goal can also be done by maintaining the same level of output using fewer Semra Birgün and Kemal Güven Gülen Istanbul Commerce University, Department of Industrial Engineering, 34840 Küçükyalı, İstanbul Cengiz Kahraman Istanbul Technical University, Department of Industrial Engineering, 34367, Maçka İstanbul C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 417–430. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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inputs. The drive to increase productivity can be caused by various factors, but perhaps the most apparent is the aspiration of an enterprise to increase profitability. There are certain factors affecting the productivity of entities. General categories of the factors concerning productivity include the labor force, product, quality, process, capacity, and external influences. Resources are also important to consider in assessment of productivity of an entity. Measuring the production level of an entity may take certain processes that include data acquisition, data summary, and comparison. In obtaining data, documenting the activities of an entity helps in creating tangible reports of certain group transactions. Documents and files can be extremely valuable, particularly during the performance evaluation. Productivity analysis may be seen as an evaluative activity of the performance of an entity. The purpose of it being employed is to provide the appropriate solution to a problem that hinders the attainment of production goals in the present and future of the company. The findings from productivity analysis being undertaken are indeed of great help in providing an entity the necessary changes to be implemented for the realization of its production goals. An entity that is aiming for increased profitability should focus on the improvement of the aspect of productivity. Productivity analysis can be an important tool to employ to determine the things that need changes or improvement. Productivity analysis may be a part of performance evaluation exercise of an entity. It may be conducted after the production report is made and finalized. This activity may be undertaken by someone from the management level or an expert production analyst. A third party analyst may also be hired to conduct productivity analysis. Expert analysts independent from the entity could provide professional findings and effective recommendations using the proven formula. Some examples of productivity analysis objectives are to • Determine value added vs. non-value added work as well as any potential productivity improvements and/or cost reduction opportunities. • Provide recommendations for the most efficient way to perform the tasks. • Develop a valid assessment of the current operations. • Develop a valid assessment of the present workflow process. • Provide recommendations to improve utilization and productivity. When decision maker (DM)s’ judgments are not crisp and it is relatively difficult for them to provide exact numerical values, the evaluation data of inputs and outputs can be expressed in fuzzy numbers or linguistic terms. In this case, the fuzzy logic that provides a mathematical strength to capture the uncertainties associated with human cognitive process can be used. The aim of this chapter is to present classical productivity measurement techniques and then to extend them to be used under fuzziness. The rest of the chapter is organized as follows: Section 2 gives classical productivity measurement techniques in production systems. Section 3 presents the fuzzy versions of the classical techniques. Section 4 presents fuzzy data envelopment analysis as a productivity optimization tool for the fuzzy case. Finally, Section 5 includes conclusions.

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2 Literature Review: Productivity in Production Systems Freiheit et al. (2003) applied numerical models to predict the productivity of nontraditional manufacturing system configurations and to demonstrate their equivalency to buffered serial transfer lines. Rao et al. (2005) developed a knowledge-based prototype system for productivity analysis called PET (productivity evaluation technology). PET’s analysis is based on only total-factor productivity measurement models. In industrial engineering, productivity is generally defined as the relation of output (i.e. produced goods) to input (i.e. consumed resources) in the manufacturing transformation process. However, there are numerous variations on this basic ratio which is often too wide, a definition to be useful in practice. Examining the term from different perspectives, Tangen (2005) summarize a number of these variations found in different literatures as in Table 17.1. There are three broad categorizations of the term productivity: i) the technological concept: the relationship between ratios of output to the inputs used in its production; ii) the engineering concept: the relationship between the actual and the potential output of a process; and iii) the economist concept: the efficiency of Table 17.1 Examples of definitions of productivity Productivity - faculty to produce. Productivity is what man can accomplish with material, capital and technology. Productivity is mainly an issue of personal manner. It is an attitude that we must continuously improve ourselves and the things around us. Productivity - units of output/units of input. Productivity - actual output/expected resources used. Productivity - total income/(cost + goal profit). Productivity - value added/input of production factors. Productivity (output per hour of work) is the central long-run factor determining any population’s average of living. Productivity - the quality or state of bringing forth, of generating, of causing to exist, of yielding large result or yielding abundantly. Productivity means how much and how well we produce from the resources used. If we produce more or better goods from the same resources, we increase productivity. Or if we produce the same goods from lesser resources, we also increase productivity. By “resources”, we mean all human and physical resources, i.e. the people who produce the goods or provide the services, and the assets with which the people can produce the goods or provide the services. Productivity is a comparison of the physical inputs to a factory with the physical outputs from the factory. Productivity - efficiency * effectiveness = value adding time/total time Productivity - (output/input) * quality = efficiency * utilisation * quality Productivity is the ability to satisfy the market’s need for goods and services with a minimum of total resource consumption.

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resource allocation. Although the definition of productivity appears straightforward, for three major reasons it is difficult to deal with it (Wazed and Ahmed 2008). First, the outputs are usually expressed in different forms to the inputs. Outputs are often measured in physical terms such as units (e.g. cars produced), tonnes (of paper), kilowatts (of electricity), or value (euros) for example. However the inputs are usually physically different and include measures of people (numbers, skills, hours worked or costs) or materials (tonnes and costs) for example. Second, the ratio by itself tells us little about performance. A ratio of 0.75 is of little value unless it is compared with previous time periods, or a benchmark, or the potential productivity of the operation. Third, many different ratios can be used (both financial and nonfinancial, that can be used to create productivity ratios). Total productivity (TP) is the ratio of the total output of all products and services to the total resource inputs which can be disaggregated into separate product and service productivity, single factor productivity (SFP), e.g. the output of product X over the input resources for product X. TP is defined as in Eq.(17.1):

TP =

Total output O = Total input L+M +C + E +Q

(17.1)

where O: total output including sales, inventory, etc; L: Labour input factor; M: Material input factor; C: Capital input factor; E: Energy input factor; Q: other miscellaneous goods and services input factor. Sumanth (1985) considered the impact of all input factors on the output in a tangible sense:

TP =

Total tangible output Total tangible input

(17.2)

where Total tangible output = Value of finished units produced + Value of partial units produced +Dividends from securities + Interest from bonds + other income and Total tangible input = Value of (human +material + capital + energy + other expense) inputs used. For a better detail, types of tangible inputs are listed as follows: • Human - Workers, Managers, Professionals, Clerical staff. • Fixed capital - Land, Plant (buildings and structures), Machinery, Tools and equipment, and others • Working capital - Inventory, Cash, Accounts receivable, Notes receivable. • Materials - Raw materials, Purchased parts • Energy - Oil, Gas, Coal, Water, Electricity and etc. • Other expense -Travel, Taxes, Professional fees, Marketing, R&D, etc. Total productivity can be defined for the firm level and for a product as given in Eqs. (17.3) and (17.4), respectively:

TPF =

Total output of the firm Total input of the firm

(17.3)

Fuzzy Productivity Measurement in Production Systems

TPi =

Total output for product i Total input for product i

421

(17.4)

Multifactor productivity (MP) is calculated for the combination of two or more inputs as in Eq.(17.5) MP =

net output total output − materials and service purchased (17.5) = (labor + capital) input (labor + capital) input

Partial productivity (PP) is the ratio of output to one class of input. For example, labor productivity (the ratio of output to labor input) is a partial productivity measure. Similarly, capital productivity (the ratio of output to capital input) and material productivity (the ratio of output to materials input) are examples of partial productivities. For example, material productivity can be calculated by the ratio given in Eq.(17.7)

PPMaterial =

output material input

(17.6)

2.1 A Numerical Illustration Suppose that a company manufacturing electronic calculators produced 10,000 calculators by employing 50 people at 8 hours/day for 25 days. Then in this case • Production=10,000 calculators 10,000 calculators = 1calculator / man − hour 50*8*25 man − hours • Suppose this company increased its production to 12,000 calculators by hiring 10 additional workers at 8 hours/day for 25 days.

• Productivity (of labor)=

Productivity (of labor)=

12, 000 calculators = 1calculator / man − hour 60*8* 25 man − hours

• Clearly, the production of calculators has gone up 20 percent (from 10,000 to 12,000), but the labor productivity has not gone up at all. • We can easily show, by similar computations, that there could have been other extreme cases where in the labor productivity went down even though production went up; or, the labor productivity went up along with the production. • The point we are making is that an increased production does not necessarily mean increased productivity. • A decrease in “direct man-hours” is also often interpreted as an increase in labor productivity. This is another example of the confusion in interpreting the term productivity. Gold (1980) proposed a financial-ratio approach to productivity measurement. His measure focuses on the rate of return on investment, and it attributes profit sense

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to five specific elements of performances: product prices, unit costs, utilization of facilities, productivity of facilities, and allocation of capital resources between fixed and working capital. He combined these five elements as in Eq.(17.7) in order to measure the productivity of a company by the rate of return on its invested capital. ⎡ Product revenue

Profit Total investment

=⎢

⎣

output

−

Total costs ⎤

Output

Capacity

Fixed Investment

⎥ × Capacity × Fixed Investment × Total Investment ⎦

output

(17.7)

Multifactor Productivity/Performance Measurements Model (MFPMM) was developed by the American Productivity Center in 1977 for measuring productivity and price recovery, and for explicitly relating these results with profitability at the organizational/ functional levels (Sink 1985). Its primary focus is on a manufacturing/production unit with tangible outputs and inputs. It is suitable for a process that is stable, implying not-so-often changes in products being offered. Finally, the MFPMM can easily adapt the data from a typical accounting system. Multifactor productivity is the ratio of output to the sum of two or more inputs in the same breadth of time. It is given in Eq.(17.8): Output Output or or (Material + Labour) Inputs (Material + Labour + Machine) Inputs (17.8) Output Output or (Machine + Capital) Inputs (Machine + Labour + Capital) Inputs

MFPMM =

3 Fuzziness in Productivity Measurement The fuzzy set theory and fuzzy logic constitute the basis for linguistic approaches. Fayek et al. (2005) illustrated the application of fuzzy expert systems to effectively model predicting the labor productivity. Hougaard (2005) suggests a simple approximation procedure for the assessment of productivity scores with respect to fuzzy production plans. The procedure has a clear economic interpretation and all the necessary calculations can be performed in a spreadsheet making it highly operational. Cheng et al. (2007) aimed to develop a model for hierarchical structure of worker productivity improvement through interpretive structural modeling and to identify the priority weights of improvement for worker productivity through fuzzy analytic hierarchy process. They tested and validated their model in PCB manufacturing firms. When the elements of inputs and outputs are represented by triangular fuzzy ~

numbers, fuzzy total productivity ( TP ) is calculated as given in Eq.(17.9):

~ O TP = ~ ~ ~ ~ ~ L +M +C +E +Q ~

or

(17.9)

Fuzzy Productivity Measurement in Production Systems

(Ol , Om , Ou )

~

TP =

⎛ ⎜⎜ ⎝

M ,C , E , Q

M ,C , E ,Q

M ,C , E ,Q

⎞

i

i

⎠

∑ a , ∑ b , ∑ c ⎟⎟ i=L

i

i=L

i=L

423

⎛ ⎜ O O O = ⎜ M ,C , El ,Q , M ,C ,mE ,Q , M ,C , Eu ,Q ⎜ ⎜ ∑ ci ∑ bi ∑ ai i=L i=L ⎝ i=L

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

(17.10)

~ ~ O : fuzzy total output including sales, inventory, etc; L : fuzzy labor input ~ ~ ~ factor; M : fuzzy material input factor; C : fuzzy capital input factor; E : fuzzy ~ energy input factor; Q : other fuzzy miscellaneous goods and services input where

factor. ~ Fuzzy multifactor productivity ⎛⎜ MP ⎞⎟ is calculated for some combination of ⎝ ⎠ inputs. For example, Labor and Capital productivity is calculated using Eq.(17.11):

~ ~

MP L +C

NO Fuzzy Net output = = ~ ~ ( ) Fuzzy labor + capital input L + C

(17.11)

3.1 A Numerical Illustration A firm manufactures three products in a certain period by using fuzzy levels of inputs to produce a fuzzy output value. The data are given in Table 17.2. Calculate the total productivities of the three products. Calculate the multifactor productivity of energy and human inputs for the three products. Calculate the total productivity of the firm. Table 17.2 Fuzzy data for the numerical example Product I

Product II

Product III

Labor Input, $

(1450, 1500, 1550)

(2250, 2300, 2350)

(2800, 2880, 2960)

Material Input, $

(3400, 3500, 3600)

(2300, 2400, 2500)

(3110, 3210, 3310)

Capital Input, $

(2450, 2500, 2550)

(2750, 2800, 2850)

(1175, 1275, 1375)

Energy Input, $

(425, 450, 475)

(585, 615, 645)

(400, 435, 470)

Other Expense Input, $ (180, 200, 220)

(700, 750, 800)

(315, 335, 355)

Output, $

(18,000, 18,250, 18,500)

(15,000, 15,260, 15,520)

(14,000, 14,750, 15,500)

The total productivity of each product is calculated as follows: ~ 14,000 14,750 15,500 ⎛ ⎞ TPI = ⎜ , , ⎟ ⎝ 1550 + 3600 + 2550 + 475 + 220 1500 + 3500 + 2500 + 450 + 200 1450 + 3400 + 2450 + 425 + 180 ⎠ = (1.67, 1.81, 1.96 )

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~ 18,000 18, 250 18,500 ⎞ ⎛ TPII = ⎜ , , ⎟ ⎝ 2350 + 2500 + 2850 + 645 + 800 2300 + 2400 + 2800 + 615 + 750 2250 + 2300 + 2750 + 585 + 700 ⎠ = (1.97, 2.06, 2.15 )

~ 15,000 15, 260 15,520 ⎛ ⎞ TPIII = ⎜ , , ⎟ ⎝ 2960 + 3310 + 1375 + 470 + 355 2880 + 3210 + 1275 + 435 + 335 2800 + 3110 + 1175 + 400 + 315 ⎠ = (1.77 , 1.88, 1.99 )

With respect to the total productivities of the products, the rank is obtained as II > III > I. The multifactor productivities of the three products for Labor and Energy are: ~ 14,750 15,500 ⎞ = (6.91, 7.56, 8.27) ⎛ 14,000 , , MP I , L + E = ⎜ ⎟ + + + 425 ⎠ 1550 475 1500 450 1450 ⎝ ~ 18,250 18,500 ⎞ =(6.01, 6.26, 6.53) ⎛ 18,000 MP II , L+ E = ⎜ , , ⎟ + + + 585 ⎠ 2350 645 2300 615 2250 ⎝

~ 15,260 15,520 ⎞ =(4.39, 4.60, 4.85) ⎛ 15,000 MP III , L + E = ⎜ , , ⎟ 2960 470 2880 435 2800 + + + 400 ⎠ ⎝

With respect to the multifactor productivities, the rank is obtained as I > II > III. The total productivity of the firm (TPF) is calculated as in the following: ~ ⎛ 14,000 + 18,000 + 15,000 14,750 + 18,250 + 15,260 15,500 + 18,500 + 15,520 ⎞ , , TPF = ⎜ ⎟ 8150 + 8865 + 8135 7905 + 8585 + 7800 ⎠ ⎝ 8395 + 9145 + 8470

= (1.81,1.92, 2.04)

⎛ ⎝

⎞ ⎠

~

Fuzzy partial productivity for any input ⎜ PP X ⎟ can be measured using Eq.(17.12): ~

PP X =

(Ol , Om , Ou ) ⎛⎜ Ol Om Ou ⎞⎟ = , , , ( X l , X m , X u ) ⎜⎝ X u X m X l ⎟⎠

X = L, M , C , E , Q

(17.12)

3.2 A Numerical Illustration Suppose that a company manufacturing electronic calculators produced around 10,000 calculators by employing 50 people at about 8 hours/day for 25 days. Then in this case • Production = (9,500; 10,000; 10,500) calculators • Number of hours / day = (7, 8 ,9) man-hours

Fuzzy Productivity Measurement in Production Systems

• Productivity

(of

425

labor)= (9,500; 10,000; 10,500) calculators 50 * (7, 8, 9) * 25 man − hours

= (0.84, 1.0, 1.2) calculator/man − hour • Suppose this company increased its production to around 12,000 calculators by hiring 10 additional workers at about 8 hours/day for 25 days. (11,500; 12,000; 12,500) calculators • Productivity (of labor)= 60*(7, 8, 9)*25 man − hours = (0.85, 1.0, 1.19) calculator/man − hour • Clearly, the production of calculators has gone up 20 percent (from around 10,000 to around 12,000), but the labor productivity has not gone up at all.

4 Data Envelopment Analysis as a Productivity Optimization Tool Another technique used to empirically measure productive efficiency of decision making units (DMUs) is Data Envelopment Analysis (DEA). DEA is commonly used to evaluate the efficiency of a number of producers. Data envelopment analysis (DEA) is a linear programming methodology to measure the efficiency of multiple decision-making units (DMUs) when the production process presents a structure of multiple inputs and outputs. Some of the benefits of DEA are: • no need to explicitly specify a mathematical form for the production function • proven to be useful in uncovering relationships that remain hidden for other methodologies • capable of handling multiple inputs and outputs • capable of being used with any input-output measurement • the sources of inefficiency can be analysed and quantified for every evaluated unit In the DEA methodology, formally developed by Charnes, Cooper and Rhodes (1978), efficiency is defined as a weighted sum of outputs to a weighted sum of inputs, where the weights structure is calculated by means of mathematical programming and constant returns to scale (CRS) are assumed. In 1984, Banker, Charnes and Cooper developed a model with variable returns to scale (VRS).

4.1 A Numerical Illustration Assume that we have the following data (Wikipedia): • Unit 1 produces 100 pieces of items per day, and the inputs are 10 dollars of materials and 2 labour-hours • Unit 2 produces 80 pieces of items per day, and the inputs are 8 dollars of materials and 4 labour-hours

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• Unit 3 produces 120 pieces of items per day, and the inputs are 12 dollars of materials and 1.5 labour-hours To calculate the efficiency of unit 1, we define the objective function as • maximize efficiency = (u1 × 100) / (v1 × 10 + v2 × 2) which is subject to all efficiency of other units (efficiency cannot be larger than 1): • subject to the efficiency of unit 1: (u1 × 100) / (v1 × 10 + v2 × 2) ≤ 1 • subject to the efficiency of unit 2: (u1 * 80) / (v1 * 8 + v2 * 4) ≤ 1 • subject to the efficiency of unit 3: (u1 * 120) / (v1 * 12 + v2 * 1.5) ≤ 1 and non-negativity: • all u and v ≥ 0. But since linear programming cannot handle fraction, it is needed to transform the formulation, such that we limit the denominator of the objective function and only allow the linear programming to maximize the numerator. So the new formulation would be: • maximize efficiency = u1 * 100 • subject to the efficiency of unit 1: (u1 * 100) - (v1 * 10 + v2 * 2) ≤ 0 • subject to the efficiency of unit 2: (u1 * 80) - (v1 * 8 + v2 * 4) ≤ 0 • subject to the efficiency of unit 3: (u1 * 120) - (v1 * 12 + v2 * 1.5) ≤ 0 • all u and v ≥ 0. In the fuzzy case, the DEA model given above is fuzzified as follows: Fuzzy linear programming problem with fuzzy coefficients was formulated by Negoita (1970) and called robust programming. Dubois and Prade (1982) investigated linear fuzzy constraints. Tanaka and Asai (1984) also proposed a formulation of fuzzy linear programming with fuzzy constraints and gave a method for its solution which bases on inequality relations between fuzzy numbers. Shaocheng (1994) considered the fuzzy linear programming problem with fuzzy constraints and defuzzified it by first determining an upper bound for the objective function. Further he solved the so-obtained crisp problem by the fuzzy decisive set method introduced by Sakawa and Yana (1985). A linear programming problem with fuzzy technological coefficients can be given as in Eq.(17.13) (Gasimov and Yenilmez 2002): n

Max

∑c

j

xj

(17.13)

j =1

subject to n

∑ a~ x ij

j

≤ bi ,

1≤ i ≤ m

j =1

xj ≥ 0, where at least one x j f 0.

1≤ j ≤ n

Fuzzy Productivity Measurement in Production Systems

427

a~ij is a fuzzy number with the following linear membership function: ⎧1, if x p aij , ⎪ μ aij (x ) = ⎨(a ij + d ij − x ) d ij , if aij ≤ x p a ij + d ij ⎪ ⎩0, if x ≥ aij + d ij

(17.14)

where x ∈ R and d ij f 0 for all i = 1,..., m, j = 1,..., n. For defuzzification of this problem, we first fuzzify the objective function. This is done by calculating the lower and upper bounds of the optimal values first. The bounds of the optimal values, z l and z u are obtained by solving the standard linear programming problems. n

z1 = max ∑ c j x j j =1

Subject to n

∑a

ij

x j ≤ bi , i = 1,..., m

(17.15)

j =1

x j ≥ 0, j = 1,..., n and n

z 2 = max ∑ c j x j j =1

Subject to

∑ (a n

j =1

ij

+ d ij )x j ≤ bi

(17.16)

xj ≥ 0

z1 and z 2 while technological coefaij + d ij . Let z l = min( z1 , z 2 ) and

The objective function takes values between ficients

vary

between

a ij and

z u = max( z1 , z 2 ) . Then, z l and z u are called the lower and upper bounds of the optimal values, respectively.

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The problem (17.13) becomes to the following optimization problem

max λ μG(x) ≥ λ

μ C (x ) ≥ λ , 1 ≤ i ≤ m

(17.17)

i

x ≥ 0, 0 ≤ λ ≤ 1 where n ⎧ 0 , if c j x j p zl ∑ ⎪ j =1 ⎪ n ⎪⎪⎛ n ⎞ μ G ( x ) = ⎨⎜⎜ ∑ c j x j − z l ⎟⎟ ( z u − z l ), if z l ≤ ∑ c j x j p z u j =1 ⎠ ⎪⎝ j =1 ⎪ n ⎪1, if ∑ c j x j ≥ z u ⎪⎩ j =1

(17.18)

and n ⎧ 0 , b p aij x j ∑ i ⎪ j =1 ⎪ n ⎪⎪⎛ ⎞ n μ Ci (x ) = ⎨⎜⎜ bi − ∑ aij x j ⎟⎟ ∑ d ij x j , j =1 ⎠ j =1 ⎪⎝ ⎪ n ⎪1, bi ≥ ∑ (aij + d ij )x j ⎪⎩ j =1

n

∑a j =1

(17.19) ij x j ≤ bi p ∑ (a ij + d ij )x j n

j =1

Eq.(17.17) can be rewritten as in Eq.(17.20):

max λ n

λ ( z1 − z 2 ) − ∑ c j x j + z 2 ≤ 0, j =1

∑ (a n

j =1

ij

+ λd ij )x j − bi ≤ 0, 1 ≤ i ≤ m

(17.20)

x j ≥ 0, j = 1,..., n, 0 ≤ λ ≤ 1 The fuzzy DEA problem for the productivity problem considered above is then given by m

z1 = max ∑ ci u1 i =1

(17.21)

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429

Subject to n

ci u1 − ∑ a j v j ≤ 0, i = 1,..., m j =1

v j ≥ 0, j = 1,..., n u i ≥ 0, i = 1,..., m and m

z 2 = max ∑ ci u1 i =1

Subject to

ci u1 − ∑ (a j + d j )v j ≤ 0, i = 1,..., m n

(17.22)

j =1

v j ≥ 0, j = 1,..., n u i ≥ 0, i = 1,..., m And finally we have

max λ

(

)

n

λ z1 − z 2 − (c u − ∑ a j v ) + z 2 ≤ 0 i 1 j =1 j

(

)

n c u − ∑ a j + λd j v j ≤ 0, 1 ≤ i ≤ m i 1 j =1 v j ≥ 0, j = 1, 2,..., n, u = 1,..., m, 0 ≤ λ ≤ 1 i

(17.23)

5 Conclusions There are various productivity definitions in the literature. Its technological definition is the relationship between ratios of output to the inputs used in production. Its engineering definition is the relationship between the actual and the potential output of a process. Its economist definition is the efficiency of resource allocation. The theory of fuzzy logic provides a mathematical strength to capture the uncertainties associated with human cognitive processes, such as thinking and reasoning. The conventional approaches to knowledge representation lack the means for representating the meaning of fuzzy concepts. As a consequence, the approaches based on first order logic and classical probability theory do not provide an appropriate conceptual framework for dealing with the representation of commonsense

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knowledge, since such knowledge is by its nature both lexically imprecise and noncategorical. This chapter presented the fuzzy formulations for the technological definition of productivity. Data envelopment analysis is a good tool for the comparison of similar units for their productivity performances. A fuzzy DEA formulation was also given for the productivity maximization in a firm. For further research, we suggest the productivity performances of different firms to be compared using fuzzy DEA models.

References Banker, R.D., Charnes, R.F., Cooper, W.W.: Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30, 1078–1092 (1984) Charnes, A., Cooper, W., Rhodes, E.: Measuring the efficiency of decision-making units. European Journal of Operational Research 2, 429–444 (1978) Cheng, Y.L., Chiu, A.S.F., Tseng, M.L., Lin, Y.H.: Evaluation of worker productivity improvement using ISM and FAHP. In: Proceedings of the 2007 IEEE IEEM, pp. 109–113 (2007) Dubois, D., Prade, H.: System of linear fuzzy constraints. Fuzzy Sets and Systems 13, 1–10 (1982) Fayek, A.R., Asce, A.M., Oduba, A.: Predicting industrial construction labor productivity using fuzzy expert systems. Journal of Construction Engineering and Management, 938– 941 (2005) Freiheit, T., Shpitalni, M., Hu, S.J., Koren, Y.: Designing productive manufacturing systems without buffers. CIRP Annals - Manufacturing Technology 52(1), 105–108 (2003) Gasimov, R.N., Yenilmez, K.: Solving fuzzy linear programming problems with linear membership functions. Turk J. Math. 26, 375–396 (2002) Gold, B.: Practical productivity analysis for management accountants. Management Accounting, 31–44 (1980) Hougaard, J.L.: Asimple approximation of productivity scores of fuzzy production plans. Fuzzy Sets and Systems 152, 455–465 (2005) Negoita, C.V.: Fuzziness in management. OPSA/TIMS, Miami (1970) Rao, M.P., Miller, D.M., Lin, B.: PET: An expert system for productivity analysis. Expert Systems with Applications 29(2), 300–309 (2005) Sakawa, M., Yana, H.: Interactive decision making for multi-objective linear fractional programming problems with fuzzy parameters. Cybernetics Systems 16, 377–397 (1985) Shaocheng, T.: Interval number and fuzzy number linear programming. Fuzzy Sets and Systems 66, 301–306 (1994) Sink, D.S.: Productivity Management: Planning, Measurement, and Evaluation, Control, and Improvement. John Wiley & Sons, New York (1985) Sumanth, D.J.: Productivity engineering and management. McGraw-Hill, New York (1985) Tanaka, H., Asai, K.: Fuzzy linear programming problems with fuzzy numbers. Fuzzy Sets and Systems 13, 1–10 (1984) Tangen, S.: Demystifying productivity and performance. International Journal of Productivity and Performance Management 54(1), 34–46 (2005) Wazed, M.A., Ahmed, S.: Multifactor productivity measurements model (MFPMM) as effectual performance measures in manufacturing. Australian Journal of Basic and Applied Sciences 2(4), 987–996 (2008)

Chapter 18

Fuzzy Statistical Process Control Techniques in Production Systems Cengiz Kahraman, Murat Gülbay, Nihal Erginel, and Sevil Şentürk*

Abstract. Crisp Shewhart control charts monitor and evaluate a process as “in control” or “out of control” whereas the fuzzy control charts do it by using suitable linguistic or fuzzy numbers by offering flexibility for control limits. In this chapter, fuzzy attribute control charts and fuzzy variable control charts are developed and some numeric examples are given.

1 Introduction A variable that continues to be described by the same distribution when observed over time is said to be in statistical control, or simply in control. Control charts are statistical tools that monitor a process and alert us when the process has been disturbed so that it is now out of control. This is a signal to find and correct the cause of the disturbance. In the language of statistical quality control, a process that is in control has only common cause variation. Common cause variation is the inherent variability of the system, due to many small causes that are always present. When the normal functioning of the process is disturbed by some unpredictable event, special cause variation is added to the common cause variation. We hope to be able to discover what lies behind special cause variation and eliminate that cause to restore the stable functioning of the process. Many decision-making and problem-solving tasks are too complex to be understood quantitatively; however, people succeed by using knowledge that is imprecise rather than precise. The fuzzy set theory resembles human reasoning in its use Cengiz Kahraman Industrial Engineering Department, İstanbul Technical University, İstanbul, Turkey Murat Gülbay Industrial Engineering Department, Gaziantep University, Gaziantep Turkey Nihal Erginel Industrial Engineering Department, Anadolu University, Eskişehir, Turkey Sevil Şentürk Statistics Department, Anadolu University, Eskişehir, Turkey C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 431–456. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems. By contrast, traditional computing demands precision down to each bit. Since knowledge can be expressed in a more natural by using fuzzy sets, many engineering and decision problems can be greatly simplified. The fuzzy set theory implements classes or groupings of data with boundaries that are not sharply defined (i.e., fuzzy). Any methodology or theory implementing ‘‘crisp’’ definitions such as classical set theory, arithmetic, and programming, may be ‘‘fuzzified’’ by generalizing the concept of a crisp set to a fuzzy set with blurred boundaries. Statistical process control is one of the areas, to which the fuzzy set theory has been applied in the literature. In the remaining sections of this chapter, we will first develop fuzzy statistical process control techniques for attribute control charts and variable control charts. Later, in Section 3, some fuzzy unnatural pattern analyses are realized. Finally, conclusions are given Section 4.

2 Fuzzy Statistical Process Control Techniques ~

~

2.1 Fuzzy X and R Control Charts In the traditional approach, the control of process averages or mean quality levels is usually done by X charts. The process variability or dispersion can be controlled by either a control chart for the range, called R chart, or a control chart for ~ ~ the standard deviation, called S chart. In this section, fuzzy X - R control charts are introduced. The formulation of traditional X control chart based on sample ranges is given as follows (Montgomery 1991):

UCL X = X + A2 R

(18.1)

CL X = X

(18.2)

LCL X = X − A2 R

(18.3)

where, A2 is a control chart coefficient (Montgomery 1991) and R is the average of Ri’s that are the ranges of samples. A triangular fuzzy number is represented as ( X a , X b, X c ) for each fuzzy ob~ servation from a certain process. The center line CL is the arithmetic mean of

fuzzy sample means, which are represented by ( X a , X b , X c ) . Here, X a , X b , X c are called overall means and calculated as follows:

Fuzzy Statistical Process Control Techniques in Production Systems

433

n

∑X

a ji

i =1

X aj =

(18.4)

n n

∑X i =1

X bj =

b ji

(18.5)

n n

Xcj =

∑X

(18.6)

n

~ CL is calculated by

for i = 1, 2, L , n , j = 1, 2, L , m . And m

~ CL = ( X a , X b , X c ) = (

∑ j =1

c ji

i =1

m

∑

Xaj ,

m

j =1

m

∑X

Xbj

m

,

j =1

m

cj

)

(18.7)

~ where n is the size of fuzzy sample and m is the number of fuzzy samples and CL ~ is a center line for fuzzy X control chart. ~ 2.1.1 Fuzzy X Control Chart Based on Ranges ~ ~ The fuzzy X control chart based on ranges is calculated by using CL . Where ~ ~ ~ UCL and LCL are the upper and lower control limits of fuzzy X control chart with range. They are calculated as follows (Şentürk and Erginel 2008): % = CL% + A R = ( X , X , X ) + A ( R , R , R ) = ( X , + A R , X + A R , X + A R ) (18.8) UCL x 2 a b c 2 a b c a 2 a b 2 b c 2 c

~ CL X = ( X a , X b , X c ) % = CL% − A R = ( X , X , X ) − A ( R , R , R ) = ( X − A R , X − A R , X − A R ) LCL x 2 a b c 2 a b c a 2 c b 2 b c 2 a

(18.9) (18.10)

where Ra = Rb = Rc =

∑R

aj

m

∑R

bj

m

∑R

cj

m

(18.11)

(18.12)

(18.13)

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~ 2.1.2 α-Cut Fuzzy X Control Chart Based on Ranges An α-cut comprises of all elements whose membership degrees are greater than or equal to α . Applying α-cut of a fuzzy set, the values of X aα and X cα are determined as follows: X aα = X a + α ( X b − X a )

(18.14)

X cα = X c − α ( X c − X b )

(18.15)

~ Similarly, α-cut fuzzy X control chart limits based on ranges are stated as follows: % α = ( X α , X , X α ) + A ( Rα , R , Rα ) = ( X α + A Rα , X + A R , X α + A Rα ) (18.16) UCL x a b c 2 a b c a 2 a b 2 b c 2 c

~ CL x α = ( X aα , X b , X cα ) % α = ( X α , X , X α )) − A ( R α , R , R α ) = ( X α − A R α , X − A R , X α − A R α ) LCL x a b c 2 a b c a 2 c b 2 b c 2 a

(18.17) (18.18)

where

R aα = R a + α ( Rb − R a )

(18.19)

Rcα = Rc − α ( Rc − Rb )

(18.20)

~ 2.1.3 α-Level Fuzzy Midrange for α-Cut Fuzzy X Control Chart Based on Ranges α-level fuzzy midrange is one of four transformation techniques used to determine the fuzzy control limits. These control limits are used to give a decision such as “in-control” or “out-of-control” for a process. α-level fuzzy midrange is used as the fuzzy transformation technique while calculating the control limits:

UCLαmr − X = CLαmr − X + A2 (

Raα + Rcα ) 2

X aα + X cα ~ α CLαmr − X = f mr C L = ( ) −X 2 LCLαmr − X = CLαmr − X − A2 (

Raα + Rcα ) 2

(18.21)

(18.22)

(18.23)

~ The definition of α-level fuzzy midrange of sample j for fuzzy X control chart is,

Fuzzy Statistical Process Control Techniques in Production Systems

α S mr −x, j =

( X a j + X c j ) + α [( X b j − X a j ) − ( X c j − X b j )] 2

435

(18.24)

Then, the condition of process control for each sample is defined as: α α ⎫ ⎧⎪in − control , for LCLαmr − X ≤ S mr − X , j ≤ UCLmr − X ⎪ (18.25) Pr ocess control = ⎨ ⎬ ⎪⎭ ⎪⎩out − of control , for otherwise

~ 2.1.4 Fuzzy R Control Chart

Shewhart’s traditional R control chart is given by the following equations:

UCLR = D4 R

(18.26)

CL R = R

(18.27)

LCL R = D3 R

(18.28)

where D4 and D3 are control chart coefficients (Kolarik 1995):

~

Fuzzy R control chart limits are obtained in a similar way to traditional R control chart but they are represented by triangular fuzzy number as follows: ~ UC L R = D 4 ( R a , Rb , Rc ) (18.29) ~ CLR = ( Ra , Rb , Rc )

(18.30)

~ LCLR = D3 ( Ra , Rb , Rc )

(18.31)

~ 2.1.5 α-Cut Fuzzy R Control Chart

~

Control limits of α-cut fuzzy R control chart are stated by using: ~ UC LαR = D 4 ( R aα , Rb , Rcα )

(18.32)

~ CLαR = ( R aα , Rb , Rcα )

(18.33)

~ LCRαR = D3 ( Raα , Rb , Rcα )

(18.34)

~ 2.1.6 α-Level Fuzzy Midrange for α-Cut Fuzzy R Control Chart

~

Control limits of α-level fuzzy midrange for α-cut fuzzy R control chart are calculated as follows: ~ α UCLαmr − R = D 4 f mr (18.35) − R (CL )

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R aα + Rcα ~ α CLαmr − R = f mr − R (CL ) = 2 ~ α α LCL mr − R = D3 f mr − R (CL )

(18.36) (18.37)

~

The definition of α-level fuzzy midrange of sample j for fuzzy R control chart is: α Smr − R, j =

( Raj + Rcj ) + α [( Rbj − Raj ) − ( Rcj − Rbj )] 2

(18.38)

The condition of process control for each sample is defined as: α α ⎧⎪in − control ⎫ (18.39) , for LCLαmr − R ≤ S mr − R , j ≤ UCLmr − R ⎪ Process control = ⎨ ⎬ ⎩⎪out − of control , for otherwise ⎭⎪

~

~

2.2 Fuzzy X and S Control Charts The R chart is a very popular control chart used to monitor the dispersion associated with a quality characteristic. Its simplicity of construction and maintenance makes the R chart very commonly used and the range is a good measure of variation for small subgroup sizes. When the sample size increases (n>10), the utility of the range measure as a measure of dispersion falls off and the standard deviation measure is preferred (Montgomery 1991): The traditional X control chart based on standard deviation is given as follows: UCL X = X + A3 S

(18.40)

CL X = X

(18.41)

LCL X = X − A3 S

(18.42)

where, A3 is a control chart coefficient (Kolarik 1995):

S is calculated by the following equations: n

Sj =

∑(X i =1

ij

− X j )2 (18.43)

n −1 m

S=

∑S j =1

m

j

(18.44)

where S j is the standard deviation of sample j and S is the average of S j ’s.

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437

~ 2.2.1 Fuzzy X Control Chart Based on Standard Deviation ~ The theoretical structure of fuzzy X control chart and fuzzy been firstly presented by Şentürk and Erginel (2008).

~ S control chart has

~ S j is the standard deviation

of sample j and it is calculated as follows:

∑ [( X n

~ Sj =

a,

(

X b , X c )ij − X a , X b , X c

i =1

) j ]2 (18.45)

n −1

~

And the fuzzy average S is calculated by using standard deviation represented by the following triangular fuzzy number:

⎛ ⎜ ~ ⎜ S =⎜ ⎜ ⎜⎜ ⎝

m

∑

m

∑

S aj

j =1

,

m

m

∑S

Sbj

j =1

m

,

j =1

m

cj

⎞ ⎟ ⎟ ⎟ = S a , Sb , S c ⎟ ⎟⎟ ⎠

(

)

(18.46)

~ The control limits of fuzzy X control chart based on standard deviation are obtained as follows: ~ ~ UC L x = CL + A3 S = ( X a , X b , X c ) + A3 ( S a , S b , S c ) = ( X a + A3 S a , X b + A3 S b , X c + A3 S c )

~ CL X

= (X a , X b , X c )

~ ~ LCL x = CL − A3 S = ( X a , X b , X c ) − A3 ( S a , S b , S c ) = ( X a − A3 S c , X b − A3 S b , X c − A3 S a )

(18.47) (18.48) (18.49)

~ 2.2.2 α-Cut Fuzzy X Control Chart Based on Standard Deviation ~ α-cut fuzzy X control limits based on standard deviation are obtained as follows: ~ UCL x α = ( X aα , X b , X cα ) + A3 ( S aα , S b , S cα ) = ( X aα + A3 S aα , X b + A3 S b , X cα + A3 S cα )

~ CL x α = ( X aα , X b , X cα ) ~ LC L x α = ( X aα , X b , X cα ) − A3 ( S aα , S b , S cα ) = ( X aα − A3 S cα , X b − A3 S b , X cα − A3 S aα )

(18.50) (18.51) (18.52)

where S aα = S a + α ( S b − S a )

(18.53)

S cα = S c − α ( S c − S b )

(18.54)

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~ 2.2.3 α-Level Fuzzy Midrange for α-Cut Fuzzy X Control Chart Based on Standard Deviation ~ The control limits and center line for α-cut fuzzy X control chart based on standard deviation using α-level fuzzy midrange are: UCLαmr − X = CLα

~ mr − X

+ A3 (

S aα + S cα ) 2

(18.55)

X α + X cα ~ α (C L ) = a CLαmr − X = f mr −X 2 LCLαmr − X = CLαmr − X − A3 (

(18.56)

S aα + Scα ) 2

(18.57)

~ The definition of α-level fuzzy midrange of sample j for fuzzy X control chart is, α S mr −x, j =

( X a j + X c j ) + α [( X b j − X a j ) − ( X c j − X b j )]

(18.58)

2

The condition of process control for each sample is defined as: α ⎧⎪in − control , for LCLαmr − X ≤ S mr ≤ UCLαmr − X ⎫⎪ −X , j Process control = ⎨ ⎬ ⎪⎩out − of control , for otherwise ⎪⎭

(18.59)

~ 2.2.4 Fuzzy S Control Chart The traditional S control chart is given by UCLs = B4 S

(18.60)

CLs = S

(18.61)

LCLs = B3 S

(18.62)

~

where B3 and B 4 are control chart coefficients (Kolarik 1995). Fuzzy S control chart limits are obtained as follows: ~ UCLS = B4 ( Sa , Sb , Sc ) (18.63) ~ CLS = ( Sa , Sb , Sc )

(18.64)

~ LCLs = B3 ( Sa , Sb , Sc )

(18.65)

Fuzzy Statistical Process Control Techniques in Production Systems

439

~ 2.2.5 α-Cut Fuzzy S Control Chart

~

The control limits of α-cut fuzzy S control chart are obtained as follows: ~ UC LαS = B4 ( Saα , Sb , Scα ) (18.66) ~ CLαS = ( S aα , S b , S cα )

(18.67)

~ LCRSα = B3 ( Saα , Sb , Scα )

(18.68)

~ 2.2.6 α-Level Fuzzy Midrange for α-Cut Fuzzy S Control Chart

~

The control limits of α-level fuzzy midrange for α-cut fuzzy S control chart are

~

obtained in a similar way to α-cut fuzzy R control chart, like as: ~ α UCLαmr − s = B 4 f mr − S (CL ) S aα + S cα ~ α CLαmr − S = f mr − S (CL ) = 2 ~ α LCLαmr − S = B3 f mr − S (CL )

(18.69) (18.70) (18.71)

~

The definition of α-level fuzzy midrange of sample j for fuzzy S control chart is: α Smr −S , j =

( Saj + Scj ) + α [(Sbj − Saj ) − ( Scj − Sbj )] 2

(18.72)

The condition of process control for each sample is defined as: α α ⎫ , for LCLαmr − S ≤ S mr ⎪⎧in − control − S , j ≤ UCLmr − S ⎪ Process control = ⎨ ⎬ ⎩⎪out − of control , for otherwise ⎭⎪

(18.73)

2.3 Fuzzy ~p Control Chart 2.3.1 Fuzzy ~ p -Control Chart Based on Constant Sample Size

The fraction nonconforming is defined as the ratio of the number of nonconforming units in a population to the total number of units in that population. The units may have several quality characteristics that are examined simultaneously by the operator. If the unit does not conform to standard on one or more of these characteristics, the unit is classified as nonconforming (Montgomery 1991). The traditional p -control chart for known fraction nonconforming in the population would be as follows (Montgomery 1991):

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UCL p = p + 3

p (1 − p) n

CL p = p LCL p = p − 3

(18.74) (18.75)

p(1 − p) n

(18.76)

where; p is the fraction nonconforming in the population, n is the constant sample size. If the fraction nonconforming of population is unknown, sample fraction nonconforming is used instead of it. The sample fraction nonconforming is defined as the ratio of the number of nonconforming units, that is: pj =

dj

m

p=

∑

(18.77)

n m

dj

j =1

mn

=

∑p

j

j =1

m

(18.78)

where d j : the number of nonconforming units in the jth sample, p j :fraction nonconforming of jth sample, p : the average of sample fractions nonconforming, m : the number of sample, j = 1, 2, L, m The traditional p -control limits are computed from the average of sample fraction as (Montgomery 1991): UCL p = p + 3

p (1 − p ) n

(18.79)

UCL p = p + 3

p (1 − p ) n

(18.80)

LCL p = p − 3

p (1 − p ) n

(18.81)

(

)

In the fuzzy case, the number of nonconforming units is represented by the triangular fuzzy number d a j , d b j , d c j .

(p

The fraction nonconforming is expressed by a triangular fuzzy number such as . Here, ( p a , p b , p c ) are the fuzzy averages of the fraction noncona j , pb j , p c j

)

forming, where j = 1, 2, ..., m :

Fuzzy Statistical Process Control Techniques in Production Systems

pa j = pa =

da j

∑p

n aj

m

, pb j = , pb =

db j

, pc j =

n

∑p

bj

m

, pc =

441

dc j

(18.82)

n

∑p

cj

m

(18.83)

2.3.2 Fuzzy ~ p - Control Chart Based on Constant Sample Size

Fuzzy center line, fuzzy upper and fuzzy lower limits of fuzzy ~ p -control chart are obtained as follows:

⎛ p (1 − pa ) p (1 − pb ) p (1 − pc ) ⎞⎟ ~ UC L p = ⎜ pa + 3 a , pb + 3 b , pc + 3 c (18.84) ⎜ ⎟ n n n ⎝ ⎠ ~ CL p = ( p a , p b , p c ) (18.85) ⎛ p (1 − pc ) p (1 − pb ) p (1 − pa ) ⎞⎟ ~ , pb − 3 b , pc − 3 a LC L p = ⎜ p a − 3 c ⎜ ⎟ n n n ⎝ ⎠

(18.86)

2.3.3 α-Cut Fuzzy ~ p -Control Chart Based on Constant Sample Size

The mean of α-cut is a set which includes all elements whose membership degrees are greater than equal to α . With α-cuts, the values of p aα and p cα are determined as follows: paα = pa + α ( pb − pa ) and pcα = pc − α ( pc − pb )

(18.87)

α-cut fuzzy ~ p -control chart is obtained by the following equations:

⎛ p α (1 − paα ) p (1 − pb ) p α (1 − pcα ) ⎞⎟ ~ (18.88) UC Lαp = ⎜ paα + 3 a , pb + 3 b , pc + 3 c ⎜ ⎟ n n n ⎝ ⎠ ~α α α CL p = p a , p b , p c (18.89)

(

)

⎛ p α (1 − pcα ) p (1 − pb ) p α (1 − paα ) ⎞⎟ ~ (18.90) , pb − 3 b , pc − 3 a LCLαp = ⎜ paα − 3 c ⎜ ⎟ n n n ⎝ ⎠ 2.3.4 α-Level Fuzzy Median for α-Cut Fuzzy ~ p - Control Chart Based on Constant Sample Size

The fuzzy fraction nonconforming is transformed to the crisp numbers via the fuzzy transformation techniques. There are four transformation techniques that are

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α-level fuzzy midrange, fuzzy median, fuzzy average and fuzzy mode (Wang and Raz 1990). There is not any proposed way in literature to select transformation techniques. Here, fuzzy median transformation technique is used. The fuzzy median ( f med ) is expressed by the following equation:

∫

f med

aα

μ F ( x)dx =

∫

bα f med

μ F ( x)dx =

1 2

∫

bα

aα

μ F ( x)dx

(18.91)

where a ve b are the end points in the base variable of the fuzzy set F such α that a < b . For a sample j, α-level fuzzy median value ( S med − p , j ) is calculated as follows: α S med − p, j =

1 α pa , j + pb , j + pcα, j ) , j = 1, 2,.., m ( 3

(18.92)

By using these formulations, the fuzzy center line, fuzzy upper and fuzzy lower limits of α-level fuzzy median for α-cut fuzzy ~ p -control chart are calculated as below: CLαmed − p =

(

1 α pa + pb + pcα 3

)

CLαmed − p (1 − CLαmed − p )

UCLαmed − p = CLαmed − p + 3

n

LCLαmed − p = CLαmed − p − 3

CLαmed − p (1 − CLαmed − p ) n

(18.93)

(18.94)

(18.95)

The condition of process control for each sample is defined as: α α ⎫ , for LCLαmed − p ≤ Smed ⎪⎧in − control − p , j ≤ UCLmed − p ⎪ (18.96) Process control = ⎨ ⎬ ⎪⎩out − of control , for otherwise ⎪⎭

2.3.5 Fuzzy ~ p -Control Chart Based on Variable Sample Size

If the sample size is not constant, different number of units could be selected in each sample. In this case, the variable sample size should be used by calculating control limits in p-control chart. There are two approaches for variable sample size: calculating the control limits by using approximate sample size or calculating control limits for each sample size. Therefore, the approximate sample size ( n ) are defined as follows: m

∑n n=

j =1

m

j

(18.97)

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where n j express jth sample size, j = 1, 2, ..., m . n is used such as a constant sample size. If there is an unusually large variation in the size of a particular sample or if a point plots near the approximate control limits, then the exact control limits for that point should be determined and the point examined relative to that value (Montgomery 1991). If the sample size is not approximate to each other, it is required that the individual sample size for each of them should be used. The fuzzy fraction nonconforming for each sample and their fuzzy averages are calculated as follows;

pa j =

pa =

da j

, pb j =

nj

∑d ∑n

aj

, pb =

j

db j nj

, pc j =

∑d ∑n

bj

dc j

, pc =

j

(18.98)

nj

∑d ∑n

cj

(18.99)

j

where n j is the jth sample size and j = 1, 2, ..., m . 2.3.6 Fuzzy ~ p -Control Chart Based on Variable Sample Size

The control limits are calculated in fuzzy ~ p -control chart for each n j by using triangular membership functions and fuzzy averages of sample fraction nonconforming such as: ~ CL p , j = ( p a , p b , p c ) (18.100)

⎛ ~ p (1 − pa ) p (1 − pb ) p (1 − pc ) ⎞⎟ (18.101) UC L p , j = ⎜ pa + 3 a , pb + 3 b , pc + 3 c ⎟ ⎜ nj nj nj ⎠ ⎝ ⎛ ~ p (1 − pc ) p (1 − pb ) p (1 − pa ) ⎞⎟ (18.102) LC L p , j = ⎜ p a − 3 c , pb − 3 b , pc − 3 a ⎟ ⎜ nj nj nj ⎠ ⎝ 2.3.7 α-Cut Fuzzy ~ p -Control Chart Based on Variable Sample Size α-cut control limits for fuzzy ~ p -control chart based on variable sample size are given by: ⎛ ~ pα (1 − paα ) p (1 − pb ) pα (1 − pcα ) ⎞⎟ (18.103) UC Lαp , j = ⎜ paα + 3 a , pb + 3 b , pc + 3 c ⎜ ⎟ nj nj nj ⎝ ⎠

~ CLαp , j

(

= p aα , p b , p cα

)

(18.104)

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⎛ ~ pα (1 − pcα ) p (1 − pb ) pα (1 − paα ) ⎞⎟ (18.105) LCLαp , j = ⎜ paα − 3 c , pb − 3 b , pc − 3 a ⎜ ⎟ nj nj nj ⎝ ⎠

2.3.8 α-Level Fuzzy Median for α-Cut Fuzzy ~ p -Control Chart Based on Variable Sample Size

Control limits of α-level fuzzy median for α-cut fuzzy ~ p -control chart based on variable sample size are calculated by considering fuzzy median transformation technique as follows:

UCLαmed − p , j = CLαmed − p + 3 CLαmed − p

=

CLαmed − p (1 − CLαmed − p ) nj

(

1 α p a + p b + p cα 3

LCLαmed − p , j = CLαmed − p − 3

)

CLαmed − p (1 − CLαmed − p ) nj

(18.106)

(18.107)

(18.108)

α-level fuzzy median value for each sample is given as follows: α S med − p, j =

1 α pa , j + pb , j + pcα, j ) , j = 1, 2,..., m ( 3

(18.109)

The condition of process control for each sample is given by α α ⎫ , for LCLαmed − p , j ≤ Smed ⎪⎧in − control − p , j ≤ UCLmed − p , j ⎪ (18.110) Process control = ⎨ ⎬ ⎩⎪out − of control , for otherwise ⎭⎪

2.4 Fuzzy n~p Control Chart p -Control Chart Based on Constant Sample Size 2.4.1 Fuzzy n~ While p -control chart is related to the fraction of nonconforming, np -control chart is more convenient to deal with the number of nonconforming units. In many situations, observation of the number of nonconforming units is easier to interpret than the usual fraction nonconforming control chart (Montgomery 1991). In the conventional np -control chart for a known number of nonconforming units in the population is as follows (Montgomery 1991): UCLnp = np + 3 np(1 − p)

(18.111)

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CLnp = np

(18.112)

LCL np = np − 3 np(1 − p)

(18.113)

where; np is the number of nonconforming units in the population, n is a constant sample size. If the number of nonconforming units in the population is unknown, then the average of the sample number of nonconforming units, np , is used. The number of nonconforming units in the jth sample is expressed as d j , that is; m

np =

∑d

j

j =1

m

The limits of the traditional np -control chart are given as follows (Montgomery 1991): UCLnp = np + 3 np (1 − p )

(18.114)

CLnp = np

(18.115)

LCL np = np − 3 np (1 − p )

(18.116)

(

)

In the fuzzy case, the number of nonconforming units for each sample is stated by a triangular fuzzy number d a j , d b j , d c j . The average sample number of nonconforming units is expressed by a triangular fuzzy number (npa , npb , npc ) as follows: m

∑ np a =

j =1

m

∑

daj

m

, np b =

j =1

m

∑d

dbj

m

, np c =

j =1

cj

(18.117)

m

2.4.2 Fuzzy n~ p - Control Chart Based on Constant Sample Size

The limits of fuzzy n~ p -control chart are calculated with the following equations;

(

)

~ UC Lnp = npa + 3 npa (1 − pa ) , npb + 3 npb (1 − pb ) , npc + 3 npc (1 − pc ) (18.118)

~ CL np = (np a , np b , np c )

(

(18.119)

)

~ LCLnp = npa − 3 npc (1 − pc ) , npb − 3 npb (1 − pb ) , npc − 3 npa (1 − pa ) (18.120)

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2.4.3 α-Cut Fuzzy n~ p -Control Chart Based on Constant Sample Size

The limits of α-cut fuzzy n~ p -control chart are obtained as follows: ~ UC Lαnp = ⎛⎜ np aα + 3 np aα (1 − paα ) , npb + 3 npb (1 − pb ) , npcα + 3 npcα (1 − pcα ) ⎞⎟ (18.121) ⎝ ⎠

(

~ CLαnp = npaα , npb , npcα

)

(18.122)

~ LC Lαnp = ⎛⎜ npaα − 3 npcα (1 − pcα ) , npb + 3 npb (1 − pb ) , npaα − 3 npaα (1 − pαa ) ⎞⎟ (18.123) ⎝ ⎠

2.4.4 α-Level Fuzzy Median for α-Cut Fuzzy n~ p - Control Chart Based on Constant Sample Size

The following equations are expressed for α-cut fuzzy n~ p -control chart with αlevel fuzzy median: UCLαmed − np = CLαmed − np + 3 CLαmed − np (1 − CLαmed − np =

CLαmed − np

(

1 α npa + npb + npcα 3

LCLαmed − np = CLαmed − np − 3 CLαmed − np (1 −

n

)

(18.124)

)

(18.125)

CLαmed − np n

)

(18.126)

α α-level fuzzy median value S med − np , j for jth sample is given as follows;

α S med − np , j =

(

1 np αa , j + np b, j + np cα, j 3

)

(18.127)

The condition of process control for each sample is defined as; α α ⎫ (18.128) , for LCLαmed − np ≤ Smed ⎪⎧in − control − np , j ≤ UCLmed − np ⎪ Process control = ⎨ ⎬ out of control , for otherwise − ⎩⎪ ⎭⎪

2.5 Fuzzy ~c Control Chart In the crisp case, control limits for number of nonconformities are calculated by

CL = c

LCL = c − 3 c

(18.129) (18.130)

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447

UCL = c + 3 c

(18.131)

where c is the mean of the nonconformities. In the fuzzy case, each sample, or subgroup, can be represented by a trapezoidal fuzzy number (a, b, c, d) or a triangular fuzzy number (a, b, b, d). Note that a trapezoidal fuzzy number becomes triangular when b=c. For the ease of representation and calculation, a triangular fuzzy number is also represented as a trapezoidal fuzzy number by (a, b, b, d) or ~

(a, c, c, d). Center line,

( a , b , c , d ) where

CL is the mean of fuzzy samples, and it is represented by

a , b , c , and d are the arithmetic means of the values a, b, c,

and d, respectively. In the fuzzy case, it can be written as follows. n n n ⎛ n ⎜ ∑ a j ∑bj ∑c j ∑ d j ~ j =1 j =1 j =1 ⎜ j =1 CL = ⎜ , , , n n n n ⎜⎜ ⎝

⎞ ⎟ ⎟ ⎟ = (a , b , c , d ) ⎟⎟ ⎠

(18.132)

~

CL can be represented by a fuzzy number (Gülbay and Kahraman 2006, 2007) ~

whose fuzzy mode (multimodal) is the closed interval of

~

[b , c ] . CL , LCL , and

~

UCL are calculated by: CL = (a , b , c , d ) = (CL1 , CL2 , CL3 , CL4 )

(18.133)

(

(18.134)

~

)

LCL = CL − 3 CL = a − 3 d , b − 3 c , c − 3 b , d − 3 a = (LCL1 , LCL2 , LCL3 , LCL4 ) ~

~

~

(

) (

~ ~ ~ UCL = CL + 3 CL = a + 3 a , b + 3 b , c + 3 c , d + 3 d = UCL1, UCL2 , UCL3 , UCL4

) (18.135)

Using α-cut representations, fuzzy control limits can be rewritten as follows.

(

~

) (

CLα = a α , b , c , d α = CLα1 , CL2 , CL3 , CLα4

(

)

)

(18.136)

~ ~ ~ α α α α α α α α α =⎛ LCL = CL − 3 CL = a − 3 d , b − 3 c , c − 3 b , d − 3 a ⎜ LCL , LCL 2 , LCL 3 , LCL 4 ⎝ 1

(

~ ~ ~ α α α α α UCL = CL + 3 CL = a + 3 a , b + 3

b ,c + 3 c,dα

+3 d

α

)

⎞⎟ (18.137) ⎠

=⎛ ⎜ UCL , UCL 2 , UCL3 , UCL 4 ⎝ 1

α

α

⎞⎟ (18.138) ⎠

Results of these equations can be illustrated as in Figure 1. To retain the standard format of control charts and to facilitate the plotting of observations on the chart, it is necessary to convert the fuzzy sets associated with linguistic values into scalars referred to as representative values. This conversion may be performed in a number of ways as long as the result is intuitively representative of the range of the base variable included in the fuzzy set. Four ways, which are similar

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in principle to the measures of central tendency used in descriptive statistics, are fuzzy mode, α-level fuzzy midrange, fuzzy median, and fuzzy average. It should be pointed out that there is no theoretical basis supporting any one specifically and the selection between them should be mainly based on the ease of computation or preference of the user (Wang and Raz 1990). Conversion of fuzzy sets into crisp values results in loss of information in linguistic data. To retain information of linguistic data we prefer keeping fuzzy sets as themselves and comparing fuzzy samples with the fuzzy control limits. For this reason, a direct fuzzy approach (DFA) based on the area measurement is proposed for the fuzzy control charts. α~

~

~

α α α level fuzzy control limits, UCL , CL , and LCL , can be determined by fuzzy arithmetic as follows.

Fig. 18.1 Representation of fuzzy control limits

Decision about whether the process is in control can be made according to the ~

~

percentage area of the sample which remains inside the UCL and/or LCL defined as fuzzy numbers. When the fuzzy sample is completely involved by the fuzzy control limits, the process is said to be “in-control”. If a fuzzy sample is totally excluded by the fuzzy control limits, the process is said to be “out of control”. Otherwise, a sample is partially included by the fuzzy control limits. In this case, if the percentage area which remains inside the fuzzy control limits (βj) is equal or greater than a predefined acceptable percentage (β), then the process can be accepted as “rather in-control”; otherwise it can be stated as “rather out of control”. Possible decisions resulting from DFA are illustrated in Figure 2. Parameters for

Fuzzy Statistical Process Control Techniques in Production Systems

449

determination of the sample area outside the control limits for α-level fuzzy cut are LCL1, LCL2, UCL3, UCL4, a, b, c, d, and α. The shape of the control limits and fuzzy sample are formed by the lines of

LCL1 LCL2 , UCL3 UCL4 , a b , and

U c d . Sample area above the upper control limits, Aout , and sample area falling L

below the lower control limits, Aout , are calculated. Then, total sample area outside the fuzzy control limits,

Aout , is the sum of the areas below fuzzy lower con-

trol limit and above fuzzy upper control limit. Percentage sample area within the control limits is calculated as

βj = α

where

α S αj − Aout ,j

S αj

S αj is the sample area at α-level cut.

DFA provides the possibility of obtaining linguistic decisions like “rather in control” or “rather out of control”. Further intermediate levels of process control decisions are also possible by defining β in stages. For instance, it may be defined as given below which is more distinguished.

0.85 ≤ β j ≤ 1 ⎧in control , ⎪ ⎪ rather in control , 0.60 ≤ β j < 0.85 ⎪ Process Control= ⎨ ⎪ rather out of control , 0.10 ≤ β j < 0.60 ⎪ ⎪⎩out of control , 0 ≤ β j < 0.10

(18.139)

Fig. 18.2 Illustration of all possible sample areas outside the fuzzy control limits at α-level cut.

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2.6 Fuzzy u~ Control Chart If we are related to the number of nonconformities on one product, c-control chart is used. When the sample size is not be constant due to the process constraints, ucontrol chart is preferred to monitor and evaluation of process. The classical ucontrol chart limits proposed by Shewhart are given the following equations:

u u , CLu = u , LCLu = u − 3 nj nj

UCLu = u + 3

(18.140)

m

where u j =

cj

and u =

nj

∑u j =1

m

j

, j = 1, 2,..., m

where u j is the number of nonconformities per inspection unit and u is the average number of nonconformities per inspection unit, n j is the sample size, c j is total nonconformities in a sample of n j inspection units, and m is the number of sample. ~ -Control Chart 2.6.1 Fuzzy u

(

)

In this case, the number of nonconforming is expressed as a triangular fuzzy number ua j , ub j , uc j . The fuzzy averages of nonconforming values are calculated by

ua =

∑u m

aj

, ub =

∑u m

bj

, and u c =

∑u m

cj

.

The fuzzy u~ -control chart limits are given as follows:

⎛ u u ⎞ u ~ UC Lu = ⎜ u a + 3 a , ub + 3 b , u c + 3 c ⎟ ⎜ nj nj nj ⎟ ⎝ ⎠ ~ CLu = (u a , u b , u c ) ⎛ u u u ⎞ ~ LC Lu = ⎜ u a − 3 c , u b − 3 b , u c − 3 a ⎟ ⎜ nj nj nj ⎟ ⎠ ⎝

(18.141) (18.142) (18.143)

~ -Control Chart 2.6.2 α-Cut Fuzzy u When α-cut is adapted to the fuzzy sets, the values of u αa and u cα are determined as follows: uaα = ua + α (ub − ua ) and ucα = uc − α (uc − ub ) .

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α-cut fuzzy u~ -control chart is obtained by

⎛ uα u uα ~ UC Lαu = ⎜ u aα + 3 a , ub + 3 b , uc + 3 c ⎜ nj nj nj ⎝

(

~ CLαu = u aα , u b , u cα

⎞ ⎟ ⎟ ⎠

)

(18.144)

(18.145)

⎛ uα u uα ~ LCLαu = ⎜ u aα − 3 c , ub − 3 b , uc − 3 a ⎜ nj nj nj ⎝

⎞ ⎟ ⎟ ⎠

(18.146)

~ - Control Chart 2.6.3 α-Level Fuzzy Median for α-Cut Fuzzy u

~ -control chart is transformed to crisp numbers via the fuzzy transα-cut fuzzy u formation techniques. α-level fuzzy midrange, fuzzy median, fuzzy average and fuzzy mode (Wang and Raz 1990) are the transformation techniques. For a sample α j, α-level fuzzy median value ( S med − u , j ) is calculated as follows: α S med −u , j =

(

1 α u a , j + u b , j + u , αc , j 3

)

(18.147)

By using these formulations, the fuzzy center line, fuzzy upper and fuzzy lower ~ -control chart is obtained by: limits of α-level fuzzy median for α-cut fuzzy u UCLαmed −u = CLαmed −u + 3 CLαmed −u =

CLαmed −u nj

(

)

(18.149)

CLαmed −u nj

(18.150)

1 α u a + u b + u cα 3

LCLαmed −u = CLαmed −u − 3

(18.148)

The condition of process control for each sample is defined as: α α ⎧⎪in − control ⎫ , for LCLαmed −u ≤ Smed −u , j ≤ UCLmed −u ⎪ Process control = ⎨ ⎬ (18.151) ⎩⎪out − of control , for otherwise ⎭⎪

3 Fuzzy Unnatural Pattern Analysis Analysis of fuzzy unnatural patterns for fuzzy control charts can be performed via probability of fuzzy events. The formula for calculating the probability of a fuzzy event A is a generalization of the probability theory:

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⎧ μ A ( x)Px ( x)dx ⎪∫ P ( A) = ⎨ ⎪∑ μ A ( xi ) Px ( xi ) ⎩ i

, if X is continuous (18.152)

, if X is discrete

where PX denotes the probability distribution function of X. The membership degree of a fuzzy sample to belong to a region is directly related to its percentage area falling in that region, and therefore, it is continuous. For example, referring to Figure 3, a fuzzy sample may be in zone B with a membership degree of 0.4 and in zone C with a membership degree of 0.6. While counting fuzzy samples in zone B, that sample is counted as 0.4.

Fig. 18.3 The zones and probabilities of normal distribution

Numerous supplementary rules, like zone tests or run rules (Western Electric 1956, Nelson 1984 and 1985, Duncan 1986, Grant and Leavenworth 1988) have been developed to assist quality practitioners in detection of unnatural patterns for the crisp control charts. Run rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. If a run occurs, then this must mean that something has changed in the process to produce a nonrandom or unnatural pattern. Based on the expected percentages in each zone, sensitive run tests can be developed for analyzing the patterns of variation in the various zones. For fuzzy control charts, based on the Western Electric rules (1956), the following fuzzy unnatural pattern rules can be defined. Rule 1: Any fuzzy data falling outside the three-sigma control limits with a ratio of more than predefined percentage (β) of sample area at desired α-level. Membership function for this rule can subjectively be defined as below:

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, 0.85 ≤ x ≤ 1 , 0.60 ≤ x ≤ 0.85 , 0.10 ≤ x ≤ 0.60 , 0 ≤ x ≤ 0.10

⎧0 ⎪( x − 0.60) / 0.25 ⎪ μ1 = ⎨ ⎪( x − 0.10) / 0.50 ⎪⎩1

(18.153)

Rule 2: A total membership degree around 2 from 3 consecutive points in zone A or beyond. Probability of a sample being in zone A (0.0214) or beyond (0.00135) is 0.02275. Let membership function for this rule be defined as follows:

, 0 ≤ x ≤ 0.59

⎧0 ⎪ μ2 = ⎨( x − 0.59) /1.41 ⎪1 ⎩

, 0.59 ≤ x ≤ 2 ,2≤x≤3

(18.154)

Using the membership function above, fuzzy probability given in Eq. 18.152 can be determined by Eq. 18.155. x1

3

x2

3

∫ μ ( x)P ( x) = ∫ μ ( x)P ( x) + ∫ μ ( x)P ( x) + ∫ μ ( x)P ( x) 2

0

2

2

2

2

2

x1

0

x2

3

x1

x2

2

x2

2

(18.155)

= ∫ μ 2 ( x)P2 ( x) + ∫ μ2 ( x)P2 ( x)

where,

⎛ x − np ⎞ Px ( x) = Px ⎜ z ≥ ⎟⎟ . ⎜ npq ⎝ ⎠

To integrate the equation above, membership function is divided into sections each with a 0.05 width and μ2 ( x ) Px ( x ) values for each section are added. For

x1 = 0.59 and x2 = 2 , the probability of the fuzzy event, rule 2, is determined as 0.0015, which corresponds to the crisp case of this rule. In the following rules, the membership functions are set in the same way. Rule 3: A total membership degree around 4 from 5 consecutive points in zone C or beyond:

⎧0 ⎪ μ3 = ⎨( x − 2.42) /1.58 ⎪1 ⎩

, 0 ≤ x ≤ 2.42 , 2.42 ≤ x ≤ 4 ,4≤x≤5

Fuzzy probability for this rule is calculated as 0.0027.

(18.156)

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Rule 4: A total membership degree around 8 from 8 consecutive points on the same side of the centerline with the membership function below:

⎧0 ⎩( x − 2.54) / 5.46

μ4 = ⎨

, 0 ≤ x ≤ 2.54 , 2.54 ≤ x ≤ 8

(18.157)

The fuzzy probability for the rule above is then determined as 0.0039 Based on Grant and Leavenworth’s rules (1988), the following fuzzy unnatural pattern rules can be defined. Rule 1: A total membership degree around 7 from 7 consecutive points on the same side of the center line. Fuzzy probability of this rule is 0.0079 when membership function is defined as below:

⎧0 ⎩( x − 2.48) / 4.52

μ1 = ⎨

, 0 ≤ x ≤ 2.48 , 2.48 ≤ x ≤ 7

(18.158)

Rule 2: At least a total membership degree around 10 from 11 consecutive points on the same side of the center line. Fuzzy probability of this rule is 0.0058 when membership function is defined as below:

⎧0 ⎪ μ2 = ⎨( x − 9.33) / 0.77 ⎪1 ⎩

, 0 ≤ x ≤ 9.33 , 9.33 ≤ x ≤ 10 , 10 ≤ x ≤ 11

(18.159)

Rule 3: At least a total membership degree around 12 from 14 consecutive points on the same side of the center line. If membership function is set as given below, then fuzzy probability of the rule is equal to 0.0065.

⎧0 ⎪ μ3 = ⎨( x − 11.33) / 0.67 ⎪1 ⎩

, 0 ≤ x ≤ 11.33 , 11.33 ≤ x ≤ 12 , 12 ≤ x ≤ 14

(18.160)

Rule 4: At least a total membership degree around 14 from 17 consecutive points on the same side of the center line. Probability of this fuzzy event with the membership function below is 0.0062.

⎧0 ⎪ μ4 = ⎨( x − 13.34) / 0.66 ⎪1 ⎩

, 0 ≤ x ≤ 13.34 , 13.34 ≤ x ≤ 14 , 14 ≤ x ≤ 17

(18.161)

Fuzzy unnatural pattern rules based on Nelson’s Rules (1985) can be defined in the same way. Some of Nelson’s rules (Rules 3 and 4) are different from the Western Electric Rules and Grant and Leavenworth’s rules. In order to apply these

Fuzzy Statistical Process Control Techniques in Production Systems

455

rules to fuzzy control charts, fuzzy samples can be defuzzified using α-level fuzzy α midranges of the samples. The α-level fuzzy midrange, f mr , is defined as the midpoint of the ends of the α-cut. If aα and d α are the end points of α-cut, then, α f mr =

(

1 α a + dα 2

)

(18.162)

Then Nelson’s 3rd and 4th rules are fuzzified as follows: Rule 3: 6 points in a row steadily increasing or decreasing with respect to the desired α-level fuzzy midranges. Rule 4: 14 points in a row altering up and down with respect to the desired α-level fuzzy midranges. Probabilities of these fuzzy events are calculated using normal approach to binomial distribution. Probability of each fuzzy rule (event) above depends on the definition of the membership function which is subjectively defined provided that probability of each fuzzy rules are closest to that of classical rules for unnatural patterns.

4 Conclusions Statistical process control may be used when a large number of similar items are being produced. Every process is subject to variability. The variability present when a process is running well is called inherent variability. The purpose of statistical process control is to give a signal when the process mean has moved away from the target. A second purpose is to give a signal when item to item variability has increased. Sometimes variability cannot be measured with certainty. Instead, some measurements can be vague enough to handle them with the fuzzy sets. In this chapter, we developed the fuzzy control charts for variables and attributes. With fuzzy control charts, a more flexible and informative evaluation of the considered process can be made. For further research, fuzzy EWMA control charts can be developed.

References Duncan, A.J.: Quality control and industrial statistics. Irwin Book Company (1986) Erginel, N.: Fuzzy p̃ Control Chart. In: Proceedings of the 8th International FLINS Conference, Madrid, Spain, September 21-24 (2008) Grant, E.L., Leavenworth, R.S.: Statistical quality control. McGraw-Hill, New York (1988) Gülbay, M., Kahraman, C., Ruan, D.: α-cut fuzzy control charts for linguistic data. International Journal of Intelligent Systems 19, 1173–1196 (2004) Gülbay, M., Kahraman, C.: Development of fuzzy process control charts and fuzzy unnatural pattern analyses. Computational Statistics & Data Analysis 51, 434–451 (2006)

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Gülbay, M., Kahraman, C.: An alternative approach to fuzzy control charts: Direct fuzzy approach. Information Sciences 177, 1463–1480 (2007) Kolarik, W.J.: Creating quality- concepts, systems, strategies and tools. McGraw-Hill, New York (1995) Montgomery, D.C.: Introduction to statistical quality control. John Wiley & Sons, Chichester (1991) Nelson, L.S.: The Shewhart control chart-tests for special causes. Journal of Quality Technology 16, 237–239 (1984) Nelson, L.S.: Interpreting Shewhart x-bar control charts. Journal of Quality Technology 17, 114–116 (1985) Roberts, S.W.: Control chart tests based on geometric moving averages. Technometrics (1959) ~ ~ ~ ~ Şentürk, S., Erginel, N.: Development of fuzzy X − R and X − S control charts using α cuts. Information Sciences (2008) (Article in press) Wang, J.H., Raz, T.: On the construction of control charts using linguistic variables. Intelligent Journal of Production Research 28, 477–487 (1990) Western Electric: Statistical quality control handbook.Western Electric (1956)

Chapter 19

Fuzzy Acceptance Sampling Plans Cengiz Kahraman and İhsan Kaya*

Abstract. Acceptance sampling is one of the major components of the field of statistical quality control. It is primarily used for the inspection of incoming or outgoing lots. In recent years, it has become typical to work with suppliers to improve their process performance through the use of statistical process control (SPC). Acceptance sampling refers to the application of specific sampling plans to a designated lot or sequence of lots. Acceptance sampling procedures can, however, be used in a program of acceptance control to achieve better quality at lower cost, improved control, and increased productivity. In some cases, it may not be possible to define acceptance sampling parameters as crisp values. Especially in production environments, it may not be easy to define the parameters fraction of nonconforming, acceptance number, or sample size as crisp values. In these cases, these parameters can be expressed by linguistic variables. The fuzzy set theory can be used successfully to cope the vagueness in these linguistic expressions for acceptance sampling. In this paper, the two main distributions of acceptance sampling plans which are binomial and Poisson distributions are handled with fuzzy parameters and their acceptance probability functions are derived. Then fuzzy acceptance sampling plans are derived based on these distributions.

1 Introduction Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. It is widely used in industry for controlling the quality of shipments of components, supplies, raw materials, and final products. There are various methods of inspection in quality control for improving the quality of products. When inspection is for the purpose of acceptance or rejection of a product, based on adherence to a standard, the type of inspection Cengiz Kahraman Department of Industrial Engineering, İstanbul Technical University, 34367 Maçka İstanbul, Turkey İhsan Kaya Faculty of Engineering and Architecture, Department of Industrial Engineering, Selçuk University, 42075 Konya, Turkey C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 457–481. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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procedure employed is usually called acceptance sampling. Acceptance sampling plans are practical tools for quality assurance applications involving quality contract on product orders and it is an important aspect of statistical quality control. Acceptance sampling can be performed during inspection of incoming raw materials, components, and assemblies, in various phases of in-process operations, or during final product inspection. Acceptance samples of incoming materials may be required to verify conformity to their required specifications. In a welldeveloped quality system, suppliers’ measurements can be relied upon, which minimizes the amount of acceptance sampling required, thus reducing redundant costs in the value-adding chain from supplier to producer. In-process samples are needed for production process control. Finished product samples are needed for production process control, for product characterization, and for product release. The sampling plans provide the vendor and buyer with decision rules for product acceptance to meet the present product quality requirement. With the rapid advancement of manufacturing technology, suppliers require their products to be of high quality with very low fraction non-conformities often measured in parts per million. Acceptance sampling pertains to incoming batches of raw materials (or purchased parts) and to outgoing batches of finished goods. It is most useful when one or more of the following conditions is present: a large number of items must be processed in a short time; the costs of passing defective items is low; destructive testing is required; or the inspectors may experience boredom or fatigue in inspecting large numbers of items. Acceptance sampling is a compromise between not doing any inspection at all and 100% inspection. The scheme by which representative samples will be selected from a population and tested to determine whether the lot is acceptable or not is known as an acceptance plan or sampling plan. There are two major classifications of acceptance plans: based on attributes and based on variables. Sampling plans can be single, double, multiple, and sequential. In recent years, some studies have concentrated on acceptance sampling. Jamkhaneh et al. (2009) introduced average outgoing quality (AOQ) and average total inspection (ATI) for single sampling plans when proportion nonconforming was a triangular fuzzy number (TFN). They showed that AOQ and ATI curves of the plan were like a band having a high and low bound. Tsai et al. (2009) developed ordinary and approximate acceptance sampling procedures under progressive censoring with intermittent inspections for exponential lifetimes. The proposed approach allowed removing surviving items during the life test such that some extreme lifetimes could be sought, or the test facilities could be freed up for other tests. Pearn and Wub (2007) introduced an effective sampling plan based on process capability index, C pk , to deal with product acceptance determination for low fraction non-conforming products. The proposed new sampling plan was developed based on the exact sampling distribution rather than approximation. Practitioners could use this proposed method to determine the number of required inspection units and the critical acceptance value, and make reliable decisions in product acceptance. Kuo (2006) developed an optimal adaptive control policy for joint machine maintenance and product quality control. He included the interactions between the machine maintenance and the product sampling in the search for

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the best machine maintenance and quality control strategy for a Markovian deteriorating, state unobservable batch production system. He derived several properties of the optimal value function, which helped to find the optimal value function and identify the optimal policy more efficiently in the value iteration algorithm of the dynamic programming. The fuzzy set theory which was introduced by Zadeh (1965) provides a strict mathematical framework in which vague conceptual phenomena can be precisely and rigorously studied. It is an important method to provide measuring the ambiguity of concepts that are associated with human beings’ subjective judgments including linguistic terms, satisfaction degree and importance degree that are often vague. A linguistic variable is a variable whose values are not numbers but phrases in a natural language. The concept of a linguistic variable is very useful in dealing with situations, which are too complex or not well defined to be reasonably described in conventional quantitative expressions (Zimmermann 1991). In this chapter, acceptance sampling plans are analyzed when their main parameters are fuzzy. The rest of this chapter is organized as follows: The certain important terms relevant to acceptance sampling plans are discussed in Section 2. Discrete fuzzy probability distributions and acceptance probability functions of Binomial and Poisson distributions with fuzzy parameters are derived in Section 3. Acceptance probabilities of sampling plans, operating characteristic curve (OC), average outgoing quality (AOQ), average sample number (ASN), and average total inspection (ATI) are derived for single and double sampling plans under fuzzy environment in Section 4. Section 5 includes conclusions and future research directions.

2 Acceptance Sampling Plans An acceptance sampling plan tells you how many units to sample from a lot or shipment and how many defects you can allow in that sample. If you discover more than the allowed number of defects in the sample, you simply reject the entire lot. The principle of acceptance sampling to control quality is the fact that we do not check all units (N), but only selected part (n). Acceptance sampling plan is a specific plan that clearly states the rules for sampling and the associated criteria for acceptance or otherwise. Acceptance sampling plans can be applied for inspection of (i) end items, (ii) components, (iii) raw materials, (iv) operations, (v) materials in process, (v) supplies in storage, (vi) maintenance operations, (vii) data or records and (viii) administrative procedures. In the following, the main terms of acceptance sampling are briefly summarized (Schilling and Neubauer 2008, British Standard 2006, Montgomery 2005, Burr 2004, ISO 2859 1999, Juran and Godfrey 1998, Mitra 1998, John 1990, MIL STD 105E 1989, Duncan 1986, Schilling 1982): Item or Unit: an object or quantity of product or material on which observations (attribute or variable or both) are made. Lot or Batch: a defined quantity of product accumulated for sampling purposes. It is expected that the quality of the product within a lot is uniform.

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Attributes Method: where quality is measured by observing the presence or absence of some characteristic or attribute in each of the units in the sample or lot under consideration, and the number of items counted which do or do not possess the quality attribute, or how many events occur in the unit area, etc. Variables Method: where measurement of quality is by means of measuring and recording the numerical magnitude of a quality characteristic for each of the items. Nonconformity: the departure of a quality characteristic from its intended level, causing the product or service to fail to meet the specification requirement. If the product or service is also not meeting the usage requirements, it is called as a defect. Usually the terms "defect" and "nonconformity" are interchangeable but the word "defect" is more stringent. Nonconforming (Defective) Unit: a unit containing at least one nonconformity (defect). The terms "defective" and "nonconforming" are interchangeable but a defective unit will fail to satisfy the normally intended usage requirements. Proportion (Fraction) Defective or Proportion (Fraction) Nonconforming Units: This is the ratio of the number of nonconforming units (defectives) to the total number of (sampled) units. There are a number of different ways to classify acceptance-sampling plans. One major classification is by attributes and variables. Acceptance-sampling plans by attributes: (i) Single sampling plan, (ii) double sampling plan, (iii) multiplesampling plan, and (iv) sequential sampling plan.

2.1 Single Sampling Plans The single-sampling plan is a basic to all acceptance sampling. The simple acceptance sampling proceeds as follows: From the whole lot consisted from N units we choose a selection of n units. In the second step we must check these units, if they satisfy quality requirements. As a result, we get a number of spoiled units d. If this d is greater than the acceptance number c, then the lot will be rejected, otherwise the lot will be accepted. The operation of the plan is illustrated in Fig.19.1.

Fig. 19.1 Procedure for single sampling

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2.2 Double Sampling Plans Often a lot of items is so good or so bad that we can reach a conclusion about its quality by taking a smaller sample than would have been used in a single sampling plan. If the number of defects in this first sample (d1) is less than or equal to some lower limit (c1), the lot can be accepted. If the number of defects first and second sample (d2) exceeds an upper limit (c2), the whole lot can be rejected. But if the number of defects in the n1 sample is between c1 and c2, a second sample is drawn. The cumulative results determine whether to accept or reject the lot. The concept is called double sampling. The operation of the plan is shown in Fig.19.2.

Fig. 19.2 Procedure for double sampling

2.3 Multiple Sampling Plans Multiple sampling is an extension of double sampling, with smaller samples used sequentially until a clear decision can be made. In multiple sampling by attributes, more than two samples can be taken in order to reach a decision to accept or reject the lot. The chief advantage of multiple sampling plans is a reduction in sample size for the same protection.

2.4 Sequential Sampling Plans Single, double, and multiple plans assess one or more successive samples to determine lot acceptability. The most discriminating acceptance sampling procedure

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involves making a decision as to disposition of the lot or resample successively as each item of the sample is taken. Called sequential sampling, these methods may be regarded as multiple-sampling plans with sample size one and no upper limit on the number of samples to be taken. When units are randomly selected from a lot and tested one by one, with the cumulative number of inspected pieces and defects recorded, the process is called sequential sampling. Under sequential sampling, samples are taken, one at a time, until a decision is made on the lot or process sampled. After each item is taken a decision is made to (1) accept, (2) reject, or (3) continue sampling. Samples are taken until an accept or reject decision is made. Thus, the procedure is open ended, the sample size not being determined until the lot is accepted or rejected. Lot Tolerance Percent Defective The Lot Tolerance Percent Defective (LTPD) of a sampling plan is the level of quality routinely rejected by the sampling plan. It is generally defined as the percent defective (number of defectives per hundred units X 100%) that the sampling plan will reject 90% of the time. In other words, this is also the percent defective that will be accepted by the sampling plan at most 10% of the time. This means that lots at or worse than the LTPD are rejected at least 90% of the time and accepted at most 10% of the time. Acceptable Quality Level The Acceptable Quality Level (AQL) of a sampling plan is a level of quality routinely accepted by the sampling plan. It is generally defined as the percent defective (defectives per hundred units X 100%) that the sampling plan will accept 95% of the time. This means lots at or better than the AQL are accepted at least 95% of the time and rejected at most 5% of the time. AQL is the maximum percentage or proportion of nonconforming units in a lot that can be considered satisfactory as a process average for the purpose of acceptance sampling. Operating Characteristic Curve An important measure of the performance of an acceptance-sampling plan is the operating-characteristic (OC) curve. The operating characteristic (OC) curve describes how well an acceptance plan discriminates between good and bad lots. A curve pertains to a specific plan, that is, a combination of n (sample size) and c (acceptance number). It is intended to show the probability that the plan will accept lots of various quality levels. Due to sampling, one faces the risk of not accepting lots of AQL quality as well as the risk of accepting lots of poorer than AQL quality. One is therefore interested in knowing how an acceptance sampling plan will accept or not accept lots over various lot qualities. A curve showing the probability of acceptance over various lot or process qualities is called the operating characteristic (OC) curve and it is used to measure the performance of an acceptance or sampling plan. For the case of nonconformities per unit, the Poisson model is exact for OC function for single sampling as follows: Pa = P ( d ≤ c n, c, p ) c

λ d e−λ

d =0

d!

=∑

where λ = np

(19.1)

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463

The above OC function is also used as an approximation to binomial when n is large and p is small such that np < 5. Pa = P ( d ≤ c n, c, p ) c c ⎛ n⎞ n! n−d = ∑ ⎜ ⎟ pd qn− d = ∑ p d (1 − p ) d =0 ⎝ k ⎠ d = 0 d !( n − d ) !

(19.2)

The acceptance probability of double sampling can be calculated as follows: Pa = P ( d1 ≤ c1 ) + P ( c1 < d1 ≤ c2 ) P ( d1 + d 2 ≤ c2 ) c2 ⎧ c1 λ d1 e − n1 p ⎛ λ d1 e − n1 p c2 − d1 λ d2 e− n2 p ⎞ + ∑ ⎜⎜ ×∑ ⎪∑ ⎟ for Poisson d1 ! d 2 ! ⎟⎠ d1 > c1 ⎝ d2 = 0 ⎪d1 = 0 d1 ! =⎨ c c2 c2 − d1 ⎛ ⎛ n1 ⎞ ⎛ n2 ⎞ n1 − d1 n −d n −d ⎞ ⎪ 1 ⎛ n1 ⎞ d1 + ∑ ⎜⎜ ⎜ ⎟ p d1 (1 − p ) 1 1 × ∑ ⎜ ⎟ p d2 (1 − p ) 2 2 ⎟⎟ for Binomial ⎪ ∑ ⎜ d ⎟ p (1 − p ) d d 0 d = d > c d = 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 1 1 2 1 1 1 2 ⎝ ⎠ ⎩

(19.3)

The curve shows the ability of a sampling plan to discriminate between high quality and low quality lots. With acceptance sampling, two parties are usually involved: the producer of the product and the consumer of the product. When specifying a sampling plan, each party wants to avoid costly mistakes in accepting or rejecting a lot. The producer wants to avoid the mistake of having a good lot rejected (producer’s risk) because he or she usually must replace the rejected lot. Conversely, the customer or consumer wants to avoid the mistake of accepting a bad lot because defects found in a lot that has already been accepted are usually the responsibility of the customer (consumer’s risk). The producer's risk (α ) is the probability of not accepting a lot of AQL quality and the consumer's risk ( β ) is the probability of accepting a lot of LQL quality. Fig.19.3 shows the quality indices AQL, LQL and the associated risks α and β respectively on the OC curve.

Fig. 19.3 OC Curve and Producer’s and Consumer’s Risks

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Average Outgoing Quality (AOQ): The average outgoing quality (AOQ) can be defined as the expected quality of outgoing product following the use of an acceptance sampling plan for a given value of the incoming quality. For the lots accepted by the sampling plan, no screening will be done and the outgoing quality will be the same as that of the incoming quality p. For those lots screened, the outgoing quality will be zero, meaning that they contain no nonconforming items. Since the probability of accepting a lot is Pa, the outgoing lots will contain a proportion of p Pa defectives. If the nonconforming units found in the sample of size n are replaced by good ones, the average outgoing quality (AOQ) will be AOQ =

N −n Pa p N

(19.4)

For large N, AOQ ≅ Pa p

(19.5)

Fig.19.4 shows a typical AOQ curve as a function of the incoming quality for the plan (n=100, c=2). The maximum ordinate of the AOQ curve represents the worst possible average for the outgoing quality and is known as the average outgoing quality limit (AOQL).

Fig. 19.4 AOQ and AOQL

Average Sample Number (ASN): The average sample number (ASN) is defined as the average number of sample units per lot used for deciding acceptance or nonacceptance. For a single sampling plan, one takes only a single sample of size n and hence the ASN is simply the sample size n. In single sampling, the size of the sample inspected from the lot is always constant, whereas in double-sampling, the size of the sample selected depends on whether or not the second sample is necessary. Therefore, a general formula for the average sample number in double sampling is

Fuzzy Acceptance Sampling Plans ASN

465 = n1PI + ( n1 + n2 )(1 − PI ) = n1 + n2 (1 − PI )

(19.6)

Where PI is the probability of making a lot dispositioning decision on the first sample. This is PI = P {lot is accepted on the first sample} + P {lot is rejected on the first sample}

(19.7)

Fig.19.5 shows the average sample number curves for single and double sampling. Average total inspection (ATI): Another important measure relative to rectifying inspection is the total amount of inspection required by the sampling program. The average total inspection (ATI) can be defined as the average number of units inspected per lot based on the sample for accepted lots and all inspected units in lots not accepted. If the lots contain no defective items, no lots will be rejected, and the amount of inspection per lot will be the sample size n. If the items are all defective, every lot will be submitted to 100% inspection, and the amount of inspection per lot will be the lot size N. If the lot size is N and is of quality p, then the ATI for the single sampling plan is given by ATI = n + (1 − Pa )( N − n )

(19.8)

Fig. 19.5 ASN curve for single and double sampling

ATI curve for a single plan (N=3000, n=100, c=2) is illustrated in Fig.19.6. ATI for the double-sampling can be calculated from the followings: ATI = ASN + ( N − n1 ) P ( d1 > c2 ) + ( N − n1 − n2 ) P ( d1 + d 2 > c2 )

(19.9)

where P ( d1 > c2 ) = 1 − P ( d1 ≤ c2 )

P ( d1 + d 2 > c2 ) = 1 − Pa − P ( d1 > c2 )

(19.10)

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Fig. 19.6 ATI Curve for the plan (N=3000, n=100, c=2)

3 Discrete Fuzzy Distributions The two important distributions used in sampling plans to calculate the acceptance probability are binomial and poisson distributions. In this section these two distributions are analyzed for fuzzy parameters. Their procedure for calculating the acceptance probability is derived when the main parameters of them are fuzzy.

3.1 Fuzzy Binomial Distribution Assume that X = { x1 , x2 ,..., xn } and S is a nonempty subset of X . We have trials where the result is considered as ‘‘success’’ if the outcome xi is belonging to S . Otherwise, the result is considered as ‘‘failure’’. Let probability of success is P ( S ) = p and the probability of failure is P ( S ) = q = 1 − p . Also we know that 0 ≤ p ≤ 1 . Let n is the number of independent trials of this experiment. If Pk is the

probability of k successes in n trials, then ⎛n⎞ Pk = ⎜ ⎟ p k q n − k ⎝k ⎠

(19.11)

gives the binomial distribution. In the following subsection, the parameters of binomial distribution are analyzed under fuzzy environments.

3.1.1 Fuzzy Fraction of Defective Items A major assumption of sampling plan is that fraction of defective items ( p ) is crisp. However, sometimes we are not able to obtain exact numerical value for p . Many times this value is estimated or it is provided by experiment. Assume that in

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467

these trials P ( S ) is not known precisely and needs to be estimated or is obtained from expert opinions. This p value is uncertain and is denoted as p% . Therefore P%k represents the fuzzy probability of k successes in n independent trials and can be calculated as follows: ⎧⎪⎛ n ⎞ ⎪⎫ P%k (α ) = ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) ⎬ ⎪⎭ ⎪⎩⎝ k ⎠

,0 ≤ α ≤ 1

(19.12)

P%k (α ) = ⎡⎣ Pkl (α ) , Pkr (α ) ⎦⎤ ⎪⎧⎛ n ⎞ ⎪⎫ Pkl (α ) = min ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) ⎬ , k ⎪⎭ ⎩⎪⎝ ⎠ ⎧⎪⎛ n ⎞ ⎫⎪ Pkr (α ) = max ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) ⎬ k ⎪⎭ ⎩⎪⎝ ⎠

(19.13)

If p value is defined as triangular fuzzy numbers (TFNs) like p% = ( p1 , p2 , p3 ) , its α cuts can be derived as follows:

{

p (α ) = p1 + ( p2 − p1 )α , p3 + ( p2 − p3 ) α }

is defined as trapezoidal fuzzy numbers (TrFNs) p% = ( p1 , p2 , p3 , p4 ) , its α cuts can be derived as follows:

If

p

value

(19.14)

{

p (α ) = p1 + ( p2 − p1 )α , p4 + ( p3 − p4 )α }

like

(19.15)

Example-1. Let probability of success is defined as “Approximately 0.03” ( p% = TFN ( 0.02, 0.03, 0.04 ) ) and number of independent trials is 20 (n=20) and number of the success is determined as 4 (k=4). The fuzzy probability of 4 successes is calculated as follows: q% = 1 − p% = ( 0.96, 0.97, 0.98 )

p (α ) = {0.2 + 0.1α , 0.4 − 0.1α } q (α ) = {0.6 + 0.1α , 0.8 − 0.1α } ⎧⎪⎛ 20 ⎞ ⎪⎫ Pkl (α ) = min ⎨⎜ ⎟ p 4 q16 p ∈ p (α ) , q ∈ q (α ) ⎬ , ⎪⎭ ⎪⎩⎝ 4 ⎠ ⎧⎪⎛ 20 ⎞ ⎪⎫ Pkr (α ) = max ⎨⎜ ⎟ p 4 q16 p ∈ p (α ) , q ∈ q (α ) ⎬ ⎪⎭ ⎩⎪⎝ 4 ⎠

The α cuts of P%k are shown in Table 19.1 and its’ membership function is illustrated in Fig.19.7.

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468 Table 19.1 α cuts of probability of success trials

α

Pkl (α )

Pkr (α )

α

Pkl (α )

Pkr (α )

0.01

0.0006

0.0064

0.20

0.0008

0.0054

0.02

0.0006

0.0063

0.30

0.0009

0.0050

0.03

0.0006

0.0063

0.40

0.0011

0.0045

0.04

0.0006

0.0062

0.50

0.0013

0.0041

0.05

0.0006

0.0062

0.60

0.0015

0.0037

0.06

0.0006

0.0061

0.70

0.0017

0.0034

0.07

0.0006

0.0061

0.80

0.0019

0.0030

0.08

0.0006

0.0060

0.90

0.0021

0.0027

0.09

0.0007

0.0060

1.00

0.0024

0.0024

0.10

0.0007

0.0059

Fig. 19.7 Membership function of probability of success trials

The probability of success can also be defined as “Between 0.03 and 0.04”. TrFNs are more suitable to convert this definition into a fuzzy number. The definition can be converted as p% = TrFN ( 0.02, 0.03, 0.04, 0.05 ) and the α cuts of P%k are given in Table 19.2 when the probability of success is defined as TrFNs. 3.1.2 Fuzzy Number of Trials The number of trials can be defined by linguistic variables. TFNs or TrFNs can be used to define these linguistic variables. Assume that, number of trials is defined

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Table 19.2 α cuts of probability of success trials

α

Pkl (α )

Pkr (α )

α

Pkl (α )

Pkr (α )

0.01

0.0006

0.0132

0.20

0.0008

0.0117

0.02

0.0006

0.0132

0.30

0.0009

0.0109

0.03

0.0006

0.0131

0.40

0.0011

0.0102

0.04

0.0006

0.0130

0.50

0.0013

0.0095

0.05

0.0006

0.0129

0.60

0.0015

0.0088

0.06

0.0006

0.0128

0.70

0.0017

0.0082

0.07

0.0006

0.0127

0.80

0.0019

0.0076

0.08

0.0006

0.0127

0.90

0.0021

0.0070

0.09

0.0007

0.0126

1.00

0.0024

0.0065

0.10

0.0007

0.0125

by n% = TFN ( n1 , n2 , n3 ) or n% = TrFN ( n1 , n2 , n3 , n4 ) . Their alpha cuts can be derived from the followings equations, respectively: n (α ) = ( n1 + ( n2 − n1 ) α , n3 + ( n2 − n3 )α )

(19.16)

n (α ) = ( n1 + ( n2 − n1 ) α , n4 + ( n3 − n4 )α )

(19.17)

Then the fuzzy probability of k successes ( P%k ) can be calculated as follows: ⎛ n% ⎞ P%k = ⎜ ⎟ p% k q% n% − k ⎝k ⎠ ⎪⎧⎛ n ⎞ ⎪⎫ P%k (α ) = ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) ⎬ ,0 ≤ α ≤ 1 ⎪⎩⎝ k ⎠ ⎭⎪

(19.18)

(19.19)

or P%k (α ) = ⎣⎡ Pkl (α ) , Pkr (α ) ⎦⎤ ⎧⎪⎛ n ⎞ ⎪⎫ Pkl (α ) = min ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) ⎬ , ⎭⎪ ⎩⎪⎝ k ⎠ ⎧⎪⎛ n ⎞ ⎪⎫ Pkr (α ) = max ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) ⎬ k ⎪⎭ ⎩⎪⎝ ⎠

(19.20)

(19.21)

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470 Table 19.3 α cuts of probability of success n (α )

α

Pkl (α )

Pkr (α )

0.01

19.01

20.99

0.0005

0.0064

0.02

19.02

20.98

0.0005

0.0063

0.03

19.03

20.97

0.0005

0.0063

0.04

19.04

20.96

0.0005

0.0062

0.05

19.05

20.95

0.0005

0.0062

0.10

19.1

20.9

0.0005

0.0059

0.20

19.2

20.8

0.0007

0.0054

0.30

19.3

20.7

0.0008

0.0050

0.40

19.4

20.6

0.0009

0.0045

0.50

19.5

20.5

0.0010

0.0041

0.60

19.6

20.4

0.0012

0.0037

0.70

19.7

20.3

0.0014

0.0034

0.80

19.8

20.2

0.0016

0.0030

0.90

19.9

20.1

0.0018

0.0027

1.00

20

20

0.0024

0.0024

Example-2. Let us reconsider Example-1. Number of independent trials is defined as “Approximately 20” ( TFN (19, 20, 21) ) . P%k can be calculated by using Eqs. (19.6-19.11). α cuts of P%k are shown in Table 19.3. 3.1.3 Fuzzy Number of Success Another situation which should be taken into account is to define the number of success by linguistic variables. Fuzzy numbers can be used to represent this definition successfully. Assume that the number of success is defined as k% = TFN ( k1 , k2 , k3 ) or k% = TrFN ( k1 , k2 , k3 , k4 ) , then the P%k can be calculated as follows: ⎛ n% ⎞ % P%k = ⎜⎜ ⎟⎟ p% k q% n% − k % k ⎝ ⎠ ⎪⎧⎛ n ⎞ ⎪⎫ P%k (α ) = ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) , k ∈ k (α ) ⎬ , k ⎭⎪ ⎩⎪⎝ ⎠

or

(19.22)

0 ≤α ≤1

(19.23)

Fuzzy Acceptance Sampling Plans

471 P%k (α ) = ⎣⎡ Pkl (α ) , Pkr (α ) ⎦⎤

(19.24)

⎧⎪⎛ n ⎞ ⎪⎫ Pkl (α ) = min ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) , k ∈ k (α ) ⎬ , k ⎭⎪ ⎩⎪⎝ ⎠ ⎧⎪⎛ n ⎞ ⎪⎫ Pkr (α ) = max ⎨⎜ ⎟ p k q n − k p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) , k ∈ k (α ) ⎬ ⎪⎭ ⎩⎪⎝ k ⎠

(19.25)

Example-3. Let us reconsider Examples 1 and 2. Assume that the number of success is defined as “Approximately 4”. It should be converted to a fuzzy number as k% = TFN ( 3, 4, 5 ) . The α cuts of P%k are calculated by using Equations (19.1219.15) and are shown in Table 19.4. Table 19.4 α cuts of number of success and probability of success

α

p (α )

n (α )

k (α )

Pkl (α )

Pkr (α )

0.01

0.02

0.04

19.01

20.99

3.01

4.99

0.0005

0.0362

0.02

0.02

0.04

19.02

20.98

3.02

4.98

0.0005

0.0360

0.03

0.02

0.04

19.03

20.97

3.03

4.97

0.0005

0.0358

0.04

0.02

0.04

19.04

20.96

3.04

4.96

0.0005

0.0356

0.05

0.021

0.04

19.05

20.95

3.05

4.95

0.0005

0.0354

0.10

0.021

0.039

19.1

20.9

3.1

4.9

0.0005

0.0344

0.20

0.022

0.038

19.2

20.8

3.2

4.8

0.0007

0.0324

0.30

0.023

0.037

19.3

20.7

3.3

4.7

0.0008

0.0304

0.40

0.024

0.036

19.4

20.6

3.4

4.6

0.0009

0.0285

0.50

0.025

0.035

19.5

20.5

3.5

4.5

0.0010

0.0267

0.60

0.026

0.034

19.6

20.4

3.6

4.4

0.0012

0.0249

0.70

0.027

0.033

19.7

20.3

3.7

4.3

0.0014

0.0232

0.80

0.028

0.032

19.8

20.2

3.8

4.2

0.0016

0.0215

0.90

0.029

0.031

19.9

20.1

3.9

4.1

0.0018

0.0199

1.00

0.03

0.03

20

20

4

4

0.0024

0.0024

The n and k values should be integer numbers in the classical binomial distribution. Therefore the numbers with decimal points should be eliminated from α cuts. If we either eliminate or take into account as an integer the number which have decimal points for n and k values, the membership function of P%k for Example 3 can be obtained as shown in Fig.19.8.

C. Kahraman and İ. Kaya

472

Fig. 19.8 The membership function of P%k

3.2 Fuzzy Poisson Distribution The Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume. The Poisson distribution is used to model the number of events occurring within a given time interval. If the expected number of occurrences in this interval is λ, then the probability that there are exactly k occurrences is calculated as follows: f ( k; λ ) =

λ k e−λ k!

,

k = 0,1, 2,..., and λ > 0

(19.26)

3.2.1 Fuzzy Fraction of Defective Items Assume that the p value is uncertain and is denoted as p% . In this case, λ is also denoted as λ% .Therefore P%k represents the fuzzy probability of k events in n events and can be calculated as follows: λ% k e − λ

%

k = 0,1, 2,..., and λ% > 0

(19.27)

k − λ% ⎪⎧ λ% e ⎪⎫ f k , λ% (α ) = ⎨ λ ∈ λ (α ) ⎬ where λ% = np% ⎪⎩ k ! ⎭⎪

(19.28)

f k , λ% =

k!

,

( )

( )

P%k (α ) = ⎡ f l k ; λ% (α ) , f r k ; λ% (α ) ⎤ ⎣ ⎦

(19.29)

Fuzzy Acceptance Sampling Plans

473

k − λ% ⎫⎪ ⎪⎧ λ% e f l , k , λ% (α ) = min ⎨ λ ∈ λ (α ) ⎬ ⎪⎩ k ! ⎭⎪ % k −λ ⎪⎧ λ% e ⎪⎫ f r , k , λ% (α ) = max ⎨ λ ∈ λ (α ) ⎬ ⎪⎭ ⎩⎪ k !

(19.30)

Example-4. Assume that the probability of a single event is defined as “Approximately 0.03” ( p% = TFN ( 0.02, 0.03, 0.04 ) ) and there are n = 20 events, then the fuzzy value of the Poisson distribution function at k =4 is calculated as follows: λ% = np% = 20 × ( 0.02, 0.03, 0.04 ) = ( 0.4, 0.6, 0.8 ) λ (α ) = [ 0.4 + 0.2α , 0.8 − 0.2α ]

⎧⎪ λ% 4e − λ% f 4,λ% (α ) = ⎨ λ ∈ λ (α ) ⎩⎪ 4!

⎪⎫ ⎬ ⎪⎭

Table 19.5 α cuts of events

α

f l , k , λ% (α )

f r , k , λ% (α )

α

f l , k , λ% (α )

f r , k , λ% (α )

0.01

0.0007

0.0076

0.40

0.0014

0.0055

0.02

0.0007

0.0075

0.50

0.0016

0.0050

0.03

0.0008

0.0075

0.60

0.0018

0.0045

0.04

0.0008

0.0074

0.70

0.0021

0.0041

0.05

0.0008

0.0074

0.80

0.0023

0.0037

0.10

0.0009

0.0071

0.90

0.0026

0.0033

0.20

0.0010

0.0065

1.00

0.0030

0.0030

0.30

0.0012

0.0060

Fig. 19.9 Membership function of P%k

C. Kahraman and İ. Kaya

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The α cuts of P%k are calculated by using Eqs. (19.27-19.30) and they are shown in Table 19.5 and the membership function is illustrated in Fig.19.9. The probability of a single event can also be defined as “Between 0.03 and 0.04”. TrFNs are more suitable to convert this expression as a fuzzy number. The definition can be converted as p% = TrFN ( 0.02, 0.03, 0.04, 0.05 ) and the α cuts of P%k are shown in Table 19.6 when the probability of a single event is defined as a

TrFN. Table 19.6 α cuts of probability of events f l , k , λ% (α )

α

f r , k , λ% (α )

f l , k , λ% (α )

α

f r , k , λ% (α )

0.01

0.0007

0.0152

0.40

0.0014

0.0119

0.02

0.0007

0.0151

0.50

0.0016

0.0111

0.03

0.0008

0.0151

0.60

0.0018

0.0104

0.04

0.0008

0.0150

0.70

0.0021

0.0096

0.05

0.0008

0.0149

0.80

0.0023

0.0090

0.10

0.0009

0.0144

0.90

0.0026

0.0083

0.20

0.0010

0.0136

1.00

0.0030

0.0077

0.30

0.0012

0.0127

3.2.2 Fuzzy Number of Events The other two parameters of Poisson distribution n and k can be also evaluated as fuzzy numbers. Their α -cuts can be easily evaluated based on either TFNs or TrFNs. Then the fuzzy probability of k events can be derived as follows: f k% ,λ% =

%

λ% k e − λ k% !

%

,

k = 0,1, 2,..., and λ% > 0

⎧⎪ λ% k% e − λ% f k% ,λ% (α ) = ⎨ λ ∈ λ (α ) ,n ∈ n (α ) ,k ∈ k (α ) % ⎩⎪ k !

( )

⎪⎫ %% ⎬ where λ% = np ⎭⎪

( )

P%k (α ) = ⎡ f l k%; λ% (α ) , f r k%; λ% (α ) ⎤ ⎣ ⎦ ⎧⎪ λ% k% e − λ% ⎪⎫ f l , k% ,λ% (α ) = min ⎨ λ ∈ λ (α ) ,n ∈ n (α ) ,k ∈ k (α ) ⎬ % ⎪⎭ ⎪⎩ k ! % % ⎧⎪ λ% k e − λ ⎪⎫ f r , k% ,λ% (α ) = max ⎨ λ ∈ λ (α ) ,n ∈ n (α ) ,k ∈ k (α ) ⎬ % ⎪⎭ ⎩⎪ k !

(19.31)

(19.32) (19.33)

(19.34)

Fuzzy Acceptance Sampling Plans

475

Example-5. Let us reconsider Example 4 when there are “Approximately 20” ( n% = TFN (19, 20, 21) ) events and “Approximately 4” ( k% = TFN ( 3, 4, 5) ) successes. Then the fuzzy value of the Poisson distribution function at k% = TFN ( 3, 4, 5) is calculated by using Eqs. (19.27-19.34). The α cuts of P%k are shown in Table 19.7 as follows: Table 19.7 α cuts of probability of events

α

p (α )

n (α )

k (α )

Pkl (α )

Pkr (α )

0.00

0.02

0.04

19

21

3

5

0.00005

0.04265

0.01

0.02

0.04

19.01

20.99

3.01

4.99

0.00061

0.04237

0.02

0.02

0.04

19.02

20.98

3.02

4.98

0.00062

0.04210

0.03

0.02

0.04

19.03

20.97

3.03

4.97

0.00063

0.04183

0.04

0.02

0.04

19.04

20.96

3.04

4.96

0.00064

0.04156

0.05

0.021

0.04

19.05

20.95

3.05

4.95

0.00066

0.04129

0.10

0.021

0.039

19.1

20.9

3.1

4.9

0.00072

0.03995

0.20

0.022

0.038

19.2

20.8

3.2

4.8

0.00087

0.03734

0.30

0.023

0.037

19.3

20.7

3.3

4.7

0.00104

0.03481

0.40

0.024

0.036

19.4

20.6

3.4

4.6

0.00123

0.03238

0.50

0.025

0.035

19.5

20.5

3.5

4.5

0.00145

0.03004

0.60

0.026

0.034

19.6

20.4

3.6

4.4

0.00169

0.02779

0.70

0.027

0.033

19.7

20.3

3.7

4.3

0.00196

0.02564

0.80

0.028

0.032

19.8

20.2

3.8

4.2

0.00226

0.02358

0.90

0.029

0.031

19.9

20.1

3.9

4.1

0.00260

0.02162

1.00

0.03

0.03

20

20

4

4

0.00296

0.00296

The n and k values should be integer number in Poisson distribution. Therefore the number with decimal should be eliminated from α cuts. The membership function of P%k for Example 4 can be obtained as shown in Fig.19.10.

4 Fuzzy Acceptance Sampling Plans Sometimes the parameters of sampling plans cannot be expressed as crisp values. They can be stated as “approximately”, “around”, or “between”. Fuzzy set theory is a very usable tool to convert these expressions in to mathematical functions. In this case, acceptance probability of sampling plans should be calculated with

C. Kahraman and İ. Kaya

476

Fig. 19.10 Membership function of P%k

respect to fuzzy rules. In the previous section, binomial and Poisson distribution have been analyzed when their parameters are fuzzy. In this section single and double sampling plans are analyzed by taking into account these two fuzzy discrete distributions.

4.1 Fuzzy Single Sampling Assume that a sample whose size is a fuzzy number ( n% ) is taken and 100% inspected. The fraction nonconforming of the sample is also a fuzzy number ( p% ) . The acceptance number is determined as a fuzzy number ( c% ) . The acceptance probability for this single sampling plan can be calculated as follows: c% % d − λ% λ e P%a = P ( d ≤ c% n% , c%, p% ) = ∑ d! d =0 % % %. where λ = np

(19.35)

Pa (α ) = ⎡⎣ Pal , d , λ% (α ) , Par , d , λ% (α ) ⎤⎦

(19.36)

⎧⎪ c λ d e − λ Pal , d , λ% (α ) = min ⎨∑ λ ∈ λ ( α ) ,n ∈ n ( α ) ,c ∈ c ( α ) ⎩⎪d = 0 d ! d −λ ⎪⎧ c λ e Par , d , λ% (α ) = max ⎨∑ λ ∈ λ (α ) ,n ∈ n (α ) ,c ∈ c (α ) ⎪⎩ d = 0 d !

⎪⎫ ⎬ ⎪⎭ ⎪⎫ ⎬ ⎪⎭

(19.37)

If the binomial distribution is used, acceptance probability can be calculated as follows:

Fuzzy Acceptance Sampling Plans

477 c% ⎛ n% ⎞ P%a = ∑ ⎜ ⎟ p% d q% n% − d d =0 ⎝ d ⎠

(19.38)

c% ⎛ n% ⎞ ⎪⎧ c% ⎛ n% ⎞ ⎪⎫ P%a = ∑ ⎜ ⎟ p% d q% n% − d = ⎨∑ ⎜ ⎟ p% d q% n% − d p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) , c ∈ c (α ) ⎬ d d d =0 ⎝ ⎠ ⎪⎭ ⎩⎪d = 0 ⎝ ⎠

Pa (α ) = ⎡⎣ Pal (α ) , Par (α ) ⎦⎤

(19.39) (19.40)

⎪⎧ c ⎛ n ⎞ ⎪⎫ Pal (α ) = min ⎨∑ ⎜ ⎟ p d q n − d p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) , c ∈ c (α ) ⎬ , ⎪⎭ ⎪⎩ d = 0 ⎝ d ⎠ ⎪⎧ c ⎛ n ⎞ ⎪⎫ Par (α ) = max ⎨∑ ⎜ ⎟ p d q n − d p ∈ p (α ) , q ∈ q (α ) , n ∈ n (α ) , c ∈ c (α ) ⎬ ⎪⎭ ⎩⎪d = 0 ⎝ d ⎠

(19.41)

AOQ values for fuzzy single sampling can be calculated as follows: % ≅ P% p% AOQ a

(19.42)

AOQ (α ) = ⎡⎣ AOQl (α ) , AOQr (α ) ⎦⎤

(19.43)

{ AOQ (α ) = max { P p p ∈ p (α ) , P ∈ P (α )}

AOQl (α ) = min Pa p p ∈ p (α ) , Pa ∈ Pa (α )} , r

a

a

(19.44)

a

ATI curve can also be calculated as follows:

(

% = n% + 1 − P% ATI a

)( N% − n% )

ATI (α ) = ⎡⎣ ATI l (α ) , ATI r (α ) ⎦⎤

} { ATI (α ) = max {n + (1 − P )( N − n ) p ∈ p (α ) , P ∈ P (α ) , p ∈ N (α ) , N ∈ N (α )}

ATI l (α ) = min n + (1 − Pa )( N − n ) p ∈ p (α ) , Pa ∈ Pa (α ) , p ∈ N (α ) , N ∈ N (α ) , r

a

a

(19.45) (19.46) (19.47)

a

Example-6. Suppose that a product is shipped in lots of size “Approximately 5000”. The receiving inspection procedure used is a single sampling plan with a sample size of “Approximately 50” and an acceptance number of “Approximately 2”. If fraction of nonconforming for the incoming lots is “Approximately 0.05”, calculate the acceptance probability of the lot. Based on Eq. (19.37), the acceptance probability of the sampling plan is calculated as P%a = P ( d ≤ 2% ) = TFN ( 0.190, 0.544, 0.864 ) and its membership function is shown in Fig.19.11. % = TFN ( 0.008, 0.027, 0.052 ) by using Eq. (19.44). AOQ is calculated as AOQ ATI is also calculated as ATI = TFN ( 707.163, 2308.125, 4140.47 ) by using Eq. (19.47) and its membership function is illustrated in Fig.19.12.

C. Kahraman and İ. Kaya

478

Fig. 19.11 Membership function of acceptance probability

Fig. 19.12 Membership function of ATI for single sampling

4.2 Fuzzy Double Sampling Assume that we will use a double sampling plan with fuzzy parameters ( n%1 , c%1 , n%2 , c%2 ) . N% and p% are also fuzzy. If the Poisson distribution is used, the acceptance probability of double sampling can be calculated as follows: Pa = P ( d1 ≤ c%1 ) + P ( c%1 < d1 ≤ c%2 ) P ( d1 + d 2 ≤ c%2 ) P%a =

c%1

λ d e − n% p%

d1 = 0

d1 !

∑

1

1

+

c%2

⎛ λ d1 e − n%1 p%

∑ ⎜⎜

d1 > c%1

⎝

d1 !

×

c%2 − d1

∑

d2 = 0

λ d e − n% p% ⎞ 2

d2 !

2

⎟⎟ ⎠

(19.48) (19.49)

Fuzzy Acceptance Sampling Plans

479

Pa (α ) = ⎡⎣ Pal , d ; λ% (α ) , Par , d ; λ% (α ) ⎤⎦

(19.50)

c d −n p c2 ⎛ λ d1 e − n1 p c2 − d1 λ d 2 e − n2 p ⎞ ⎪⎧ 1 λ 1 e 1 + ∑ ⎜⎜ ×∑ Pal , d ;λ% (α ) = min ⎨ ∑ ⎟ d1 ! d 2 ! ⎟⎠ d1 > c1 ⎝ d2 = 0 ⎪⎩d1 = 0 d1 !

⎪⎫ ⎬ ⎭⎪

c c2 d −n p ⎛ λ d1 e − n1 p c2 − d1 λ d 2 e − n2 p ⎞ ⎪⎧ 1 λ 1 e 1 + ∑ ⎜⎜ ×∑ Par , d ; λ% (α ) = max ⎨ ∑ ⎟ d1 ! d 2 ! ⎟⎠ d1 > c1 ⎝ d2 = 0 ⎩⎪ d1 = 0 d1 !

⎪⎫ ⎬ ⎭⎪

(19.51)

where p ∈ p (α ) , n ∈ n (α ) , and c ∈ c (α ) . If the binomial distribution is used, acceptance probability can be calculated as follows: Pa =

⎛ ⎞ ∑ ⎜ d ⎟ p% (1 − p% )

n%1

c%1

d1 = 0

⎝

1

d1

n%1 − d1

⎠

+

⎛ n%1

⎛ ⎞ ∑ ⎜⎜ ⎜ d ⎟ p% (1 − p% ) c%2

d1 > c%1

⎝⎝

1

d1

n%1 − d1

⎠

×

⎛ n%2 ⎞ d 2 n%2 − d 2 ⎞ ⎟⎟ (19.52) ⎟ p% (1 − p% ) d2 = 0 ⎝ 2 ⎠ ⎠

c%2 − d1

∑ ⎜d

c2 c2 − d1 ⎧⎪ c1 ⎛ n1 ⎞ ⎛ ⎛ n1 ⎞ ⎛ n2 ⎞ n −d n −d n − d ⎞⎫ ⎪ Pal (α ) = min ⎨ ∑ ⎜ ⎟ p d1 (1 − p ) 1 1 + ∑ ⎜⎜ ⎜ ⎟ p d1 (1 − p ) 1 1 × ∑ ⎜ ⎟ p d2 (1 − p ) 2 2 ⎟⎟ ⎬ , d d d d1 > c1 ⎝ ⎝ 1 ⎠ d2 = 0 ⎝ 2 ⎠ ⎠ ⎭⎪ ⎩⎪ d1 = 0 ⎝ 1 ⎠

(19.53)

c2 c2 − d1 ⎧⎪ c1 ⎛ n ⎞ ⎛⎛ n ⎞ ⎛n ⎞ n −d n −d n −d ⎞⎫ ⎪ Par (α ) = max ⎨ ∑ ⎜ 1 ⎟ p d1 (1 − p ) 1 1 + ∑ ⎜⎜ ⎜ 1 ⎟ p d1 (1 − p ) 1 1 × ∑ ⎜ 2 ⎟ p d2 (1 − p ) 2 2 ⎟⎟ ⎬ d1 > c1 ⎝ ⎝ d1 ⎠ d2 = 0 ⎝ d 2 ⎠ ⎪⎩ d1 = 0 ⎝ d1 ⎠ ⎠ ⎭⎪

where p ∈ p (α ) , q ∈ q (α ) , n1 ∈ n1 (α ) , c1 ∈ c1 (α ) , n2 ∈ n2 (α ) , and c2 ∈ c2 (α ) . AOQ values for fuzzy double sampling can be calculated as in Section 4.1. ASN curve for double sampling can be calculated as follows:

(

)

(

= n%1P%I + ( n%1 + n%2 ) 1 − P%I = n%1 + n%2 1 − P%I

% ASN

)

(19.54)

ASN (α ) = ⎡⎣ ASN l (α ) , ASN r (α ) ⎦⎤

(19.55)

} { ASN (α ) = max {n + n (1 − P ) p ∈ p (α ) , n ∈ n (α ) , n ∈ n (α ) , P ∈ P (α )}

(19.56)

ASN l (α ) = min n1 + n2 (1 − PI ) p ∈ p (α ) , n1 ∈ n1 (α ) , n2 ∈ n2 (α ) , PI ∈ PI (α ) , r

1

2

1

I

1

2

2

I

I

ATI curve for fuzzy double sampling can also be calculated as follows:

(

)

(

)

% + N% − n% P ( d > c% ) + N% − n% − n% P ( d + d > c% ) % = ASN ATI 1 1 2 1 2 1 2 2 ATI (α ) = ⎡⎣ ATI l (α ) , ATI r (α ) ⎦⎤

{ (α ) = max { ASN + ( N − n ) P ( d

ATI l (α ) = min ASN + ( N − n1 ) P ( d1 > c2 ) + ( N − n1 − n2 ) P ( d1 + d 2 > c2 )} , ATI r

1

1

> c2 ) + ( N − n1 − n2 ) P ( d1 + d 2 > c2 )}

(19.57) (19.58) (19.59)

where p ∈ p (α ) , ASN ∈ ASN (α ) , n1 ∈ n1 (α ) , N ∈ N (α ) , n2 ∈ n2 (α ) , and c2 ∈ c2 (α ) . Example-7. Let us reconsider Example 6 for the case of fuzzy double sampling. The sample sizes are determined as “Approximately 75” and “Approximately

C. Kahraman and İ. Kaya

480

300” for the first and second samples, respectively. Also the acceptance numbers are determined as “Approximately 0” and “Approximately 3” for the first and second samples, respectively. Based on Eq. (19.51), acceptance probability of the double sampling plan is calculated as follows:

(

)

(

) (

)

(

) (

)

(

) (

)

P%a = P d1 ≤ 0% + ⎡ P d1 = 1% × P d 2 ≤ 2% ⎤ + ⎡ P d1 = 2% × P d 2 ≤ 1% ⎤ + ⎡ P d1 = 3% × P d 2 ≤ 0% ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = ( 0.0105, 0.0235, 0.2052 ) + ⎡⎣( 0.0105, 0.0882, 0.227 ) × ( 0, 0, 0.0024 ) ⎤⎦ + ⎡⎣( 0.0477, 0.1654, 0.224 ) × ( 0, 0, 0.0005 ) ⎤⎦ + ⎡⎣( 0.1088, 0.2067, 0.227 ) × ( 0, 0, 0.0001) ⎤⎦ = ( 0.0105, 0.0235, 0.2052 ) + ( 0, 0, 0.0005 ) + ( 0, 0, 0.0001) + ( 0, 0, 0 ) P%a = ( 0.0105, 0.0235, 0.2058) .

Its membership function is shown in Fig.19.13.

Fig. 19.13 Membership function of acceptance probability for double sampling

ASN is calculated as ASN = TFN ( 74.00, 213.08, 320.24 ) by using Eqs. (19.5456). Also AOQ is calculated as AOQ = TFN ( 0.00042, 0.001175, 0.01235 ) .

5 Conclusions Statistical process control (SPC) is an efficient method for improving a firm’s quality and productivity. The main objective of SPC is to detect quickly the occurrence of assignable causes or process shifts so that investigation of the process and corrective action may be undertaken before a large number of non-conforming units are manufactured. SPC has two main tools to control the process. One of them is ‘‘acceptance sampling” and the other one is ‘‘control charts”. Acceptance sampling is a practical, affordable alternative to costly 100 % inspection. It offers an efficient way to assess the quality of an entire lot of product and to decide whether to accept or reject it. The application of acceptance sampling allows

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481

industries to minimize product destruction during inspection and testing, and to increase the inspection quantity and effectiveness. Despite of the usefulness of acceptance sampling, it has a main difficulty in defining its parameters as crisp values. Sometimes it is easier to define these parameters by using linguistic variables. For these cases, the fuzzy set theory is the most suitable tool to analyze acceptance sampling plans. In this chapter, fuzzy binomial and fuzzy poisson distributions which are the two main distributions used in acceptance sampling are derived and sampling plans are analyzed based on them. The fuzzy set theory gives a flexible definition to sample size, acceptance number, and fraction of nonconforming. Acceptance probability, operating characteristic (OC) curve, average sample number (ASN), average outgoing quality limit (AOQL), and average total inspection number (ATI) are analyzed with fuzzy parameters. The obtained results show that the fuzzy definitions of parameters provide more flexibility and more usability. For future research, the effects of fuzzy parameters can be analyzed for variable sampling plans.

References British Standard: Acceptance sampling procedures by attributes — BS 6001 (2006) Burr, J.T.: Elementary statistical quality control. CRC Press, Boca Raton (2004) Duncan, A.J.: Quality control and industrial statistics. Irwin Book Company (1986) ISO 2859-1: Sampling procedures for inspection by attributes (1999) Jamkhaneh, E.B., Sadeghpour-Gildeh, B., Yari, G.: Preparation important criteria of rectifying inspection for single sampling plan with fuzzy parameter. Proceedings of World Academy of Science, Engineering and Technology 38, 956–960 (2009) John, P.W.M.: Statistical methods in engineering and quality assurance. John Wiley & Sons, Chichester (1990) Juran, J.M., Godfrey, A.B.: Juran’s quality handbook. McGraw-Hill, New York (1998) Kuo, Y.: Optimal adaptive control policy for joint machine maintenance and product quality control. European Journal of Operational Research 171, 586–597 (2006) MIL STD 105E: Military Standard-Sampling Procedures and Tables for Inspection by Attributes. Department of Defense, Washington, DC 20301 (1989) Mitra, A.: Fundamentals of quality control and improvement. Prentice-Hall, Englewood Cliffs (1998) Montgomery, D.C.: Introduction to statistical quality control. Wiley, Chichester (2005) Pearn, W.L., Chien-Wei, W.: An effective decision making method for product acceptance. Omega 35, 12–21 (2007) Schilling, E.G.: Acceptance sampling in quality control. CRC Press, Boca Raton (1982) Schilling, E.G., Neubaue, D.V.: Acceptance sampling quality in control. CRC Press, Boca Raton (2008) Tsai, T.R., Chiang, J.Y.: Acceptance sampling procedures with intermittent inspections under progressive censoring. ICIC Express Letters 3(2), 189–194 (2009) Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965) Zimmermann, H.J.: Fuzzy set theory and its applications. Kluwer Academic Publishers, Dordrecht (1991)

Chapter 20

Fuzzy Process Capability Analysis and Applications Cengiz Kahraman and İhsan Kaya1

Abstract. Process capability indices (PCIs) are very useful statistical analysis tools to summarize process’ dispersion and location through process capability analysis (PCA). PCIs are mainly used in industry to measure the capability of a process to produce products meeting specifications. Traditionally, the specifications are defined as crisp numbers. Sometimes, the specification limits (SLs) can be expressed in linguistic terms. Traditional PCIs cannot be applied for this kind of data. There are also some limitations which prevent a deep and flexible analysis because of the crisp definition of SLs. In this chapter, the fuzzy set theory is used to add more sensitiveness to PCA including more information and flexibility. The fuzzy PCA is developed when the specifications limits are represented by triangular or trapezoidal fuzzy numbers. Crisp SLs with fuzzy normal distribution are used to calculate the fuzzy percentages of conforming (FCIs) and nonconforming (FNCIs) items by taking into account fuzzy process mean, μ% and fuzzy variance, σ% 2 . Then fuzzy SLs are used together with μ% and σ% 2 to produce fuzzy PCIs (FPCIs). FPCIs are analyzed under the existence of correlation and thus fuzzy robust process capability indices are obtained. Then FPCIs are improved for six sigma approach. And additionally, process accuracy index is analyzed under fuzzy environment. The results show that fuzzy estimations of PCIs have much more treasure to evaluate the process when it is compared with the crisp case.

1 Introduction Process capability can be broadly defined as the ability of a process to meet customer expectations which are defined as specification limits (SLs). Some of the processes have a success for meeting SLs and therefore are classified as “capable process”, while others have not and are classified as “incapable process”. The measure of process capability summarizes some aspects of a process’s ability to meet SLs. The process capability analysis (PCA) is an approach to define a Cengiz Kahraman Department of Industrial Engineering, İstanbul Technical University, 34367 Maçka İstanbul, Turkey İhsan Kaya Faculty of Engineering and Architecture, Department of Industrial Engineering, Selçuk University, 42075 Konya, Turkey C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 483–513. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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relationship between the process’ ability and SLs. The process capability index is an approach for establishing the relationship between the actual process performance and the manufacturing specifications (Tsai and Chen, 2006). The process capability compares the output of a process to the specification limits by using capability indices. This comparison is made by forming the ratio of the width between the process specification limits to the width of the natural tolerance limits which is measured by 6 process standard deviation units. This method leads to make a statement about how well the process meets specifications (Montgomery, 2005). A process is said to be capable if with high the real valued quality characteristic of the produced items lies between a lower and upper specification limits (Kotz and Johnson, 2002). The PCA compares the output of a process to the SLs by using process capability indices (PCIs). In the literature some PCIs such as Cp, Cpk, Cpm, Cpkm, Cpc, Cpkc and Ca have been used to measure the ability of process to help us to decide how well the process meets the specification limits. Cp which is the first process capability index (PCI) to appear in the literature and called precision index (Kane, 1986) is defined as the ratio of specification width (USL − LSL ) over the process spread ( 6σ ) . The specification width represents customer and/or product requirements. The process variations are represented by the specification width. If the process variation is very large, the Cp value is small and it represents a low process capability. Cp indicates how well the process fits within the two specification limits. It never considers any process shift and it is calculated by using Eq. (20.1). Cp simply measures the spread of the specifications relative to the six-sigma spread in the process (Kotz and Johnson, 2002; Montgomery, 2005). Cp =

Allowable Process Spread USL − LSL = Actual Process Spread 6σ

(20.1)

where σ is the standard deviation of the process. USL and LSL are upper and lower specification limits, respectively. The process capability ratio Cp does not take into account where the process mean is located relative to specifications (Montgomery, 2005). Cp focuses the dispersion of the studied process and does not take into account the centering of the process and thus gives no indication of the actual process performance. Kane (1986) introduced index Cpk to overcome this problem. The index Cpk is used to provide an indication of the variability associated with a process. It shows how a process confirms to its specification. The index is usually used to relate the “natural tolerances ( 3σ ) ” to the specification limits. Cpk describes how well the process fits within the specification limits, taking into account the location of the process mean. Cpk should be calculated based on Eq. (20.2) (Kane, 1986; Kotz and Johnson, 2002; Montgomery, 2005).

{

C pk = min C pl , C pu } where C pu =

USL − μ μ − LSL and C pl = 3σ 3σ

(20.2)

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C p and C pk indices do not take into account the cost of failing to meet customer’s

requirements. To handle this situation C pm index shown in Eq. (20.3) can be used (Pearn, Kotz, 2006). C

pm

=

USL − LSL 6 σ + (μ − T ) 2

2

=

USL − LSL 2 6 E ⎡( X − T ) ⎤ ⎣ ⎦

(20.3)

In this chapter, process capability indices are analyzed under fuzzy parameters. The rest of this chapter is organized as follows: process capability indices are analyzed under fuzzy SLs and two ranking methods to compare fuzzy processes are presented in Section 2. Fuzzy normal distribution and its effects are investigated in Section 3. FPCIs are studied when the parameters have a correlation in Section 4. FPCIs are obtained based on α cuts and they are improved for six sigma approach in Section 5. Section 6 also includes a new capability index called accuracy index under fuzziness. Conclusions and future research directions are discussed in Section 6.

2 Fuzzy Process Capability Indices (FPCIS) Sometimes, SLs are not precise numbers and they are expressed in linguistic terms or fuzzy terms. The traditional PCIs are not suitable for these situations. Therefore, FPCIs should be applied. In this section, PCIs are analyzed using triangular and trapezoidal fuzzy numbers. The specifications are expressed in fuzzy numbers. After the inception of the notion of fuzzy sets by Zadeh (1965), many authors have applied this approach to very different areas such as statistics, quality control, and optimization techniques. These studies also affected process capability analyses. In recent years, some papers which have concentrated on different areas of PCIs using the fuzzy set theory have been published. They are summarized in the following briefly. Wu (2009) presented a set of confidence intervals that produces triangular fuzzy numbers for the estimation of Cpk index using Buckley’s approach with some modification. He also developed a three-decision testing rule and step-by-step procedure to assess process performance based on fuzzy critical values and fuzzy p-values. Kahraman and Kaya (2009) proposed fuzzy PCIs to control the pH value of dam's water for agriculture. They analyzed water stored in a dam to determine its suitability for irrigation. They illustrated an application which had been made for Kesikköprü Dam in Ankara, Turkey. Kaya and Kahraman (2008) applied the fuzzy process capability analyses when the specifications limits were triangular fuzzy numbers. They applied the proposed approach to teaching processes for some courses in a faculty. Kaya and Kahraman (2009a) analyzed the risk assessment of air pollution in Istanbul. The process capability indices (PCIs), which are very effective statistics to summarize the performance of process were used in that paper. Fuzzy PCIs were used to determine the levels of the air pollutants which were measured in different nine stations in Istanbul.

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Robust PCIs were used when air pollutants had a correlation. The fuzzy set theory was used for both PCIs and RPCIs to obtain more sensitive results. Kahraman and Kaya (2009b) used process capability indices to risk assessment of drought effects. The fullness rates of the dams in Istanbul were analyzed to avoid harmful effects of the drought by the help of PCIs. Additionally, process accuracy index (Ca), which measures the degree of the process centering and gives alerts when the process mean departures from the target value, was used for risk assessment. Its distinctive feature was used to determine the mean of the fullness rates that departure from the target value. The results were analyzed to improve precautions. They also analyzed the Ca index when the critical parameters were defined in linguistic terms. The specification limits and mean are defined by triangular or trapezoidal fuzzy numbers. Kaya and Kahraman (2009b) proposed a methodology based on PCIs to prevent air pollution. They used traditional and fuzzy process capability indices (FPCIs) for this aim. Also, they evaluated FPCIs in six-sigma approach and constructed the membership functions of PCIs based on the six-sigma approach. Kaya and Kahraman (2009c) used process accuracy index (Ca) which measures the degree of process centering and gives alerts when the process mean departures from the target value to solve a supplier selection problem. They modified the traditional process accuracy index to obtain a new tool under fuzziness. Chen and Chen (2008) presented a method to incorporate fuzzy inference with process capability. They proposed a fuzzy inference approach that employed the maximum-minimum product composition to operate fuzzy if-then rules to evaluate the multi-process capability based on distance values of a confidence box. They illustrated an example of color STN display demonstrated that the presented method was effective for assessment of multi-process capability. Hsu and Shu (2008) presented a method combining the vector of fuzzy numbers to produce the membership function of fuzzy estimator of Taguchi index, the loss-based process capability index Cpm, for further testing process capability. This approach allowed the consideration of imprecise output data resulting from the measurements of the products quality. They proposed two useful fuzzy inference criteria, the critical value and the fuzzy P-value, to assess the manufacturing process capability based on Cpm. Parchami and Mashinchi (2007) applied Buckley’s estimation approach to find fuzzy estimates of several process capability indices. They proposed an algorithm for fuzzy estimation of PCIs based on predefined α-cuts using Buckley’s approach. They created triangular fuzzy membership functions of PCIs using this approach. They also presented a method for comparing estimated PCIs. They illustrated some numerical examples to test the performance of the method. Parchami et al. (2006) obtained a (1 − α )100% fuzzy confidence interval for fuzzy process capability indices. They defined the specification limits as fuzzy numbers. They also presented some interpretations for the fuzzy confidence interval. Tsai and Chen (2006) extended the application of the process capability index Cp in the manufacturing industry to a fuzzy environment. They proposed a methodology for testing the Cp of fuzzy numbers. They formulated a pair of nonlinear functions to find the α-cuts of C% p . The membership functions of C% p are constructed from

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various values of α. They calculated the probability of rejecting the null hypothesis based on this membership function. Their methodology shows a grade of acceptability of the null hypothesis and the alternative hypothesis, respectively. Parchami et al. (2005) discussed the fuzzy quality. They analyzed fuzzy process capability indices. They introduced new process capability indices as triangular fuzzy numbers, where the engineering specification limits were also triangular fuzzy numbers. They determined the relations between the fuzzy process capabilities indices. They also presented a methodology based on a binary relation which was used for the comparison of fuzzy processes. They also applied two examples to clarify this methodology. Chen et al. (2003a) proposed a method to incorporate the fuzzy inference with the process capability index in the bigger-the-best type quality characteristics assessments. They used a concise score concept to represent the grade of the process capability. They also developed an evaluation procedure to use the method efficiently. Chen et al. (2003b) proposed a fuzzy inference method to select the best among the competing suppliers based on an estimated capability index of Cpm calculated from sampled data. Both input and output are described by linguistic variables to account for the uncertain information associated with them. Triangular and trapezoid membership functions are used to represent uncertain information about process variables. Gao and Huang (2003) emphasized that process tolerances had influences not only on manufacturing costs, but also on the achievement of the required specifications of a product. They dealt with the more complex nonlinear situations of manufacturing processes. They proposed a nonlinear optimal process tolerance allocation approach which was to optimize process tolerances based on manufacturing capability indices. The results of the comparison with the existing methods showed that the proposed approach was quite stable and was able to provide improvements in acceptable process probability, as the scrap rates were reduced. Lee (2001) proposed a model to calculate the fuzzy process capability index when observations were fuzzy numbers. This approach could mitigate the effect when the normal assumption was inappropriate. Lee (2001) focused on the construction of the membership function of the fuzzy process capability index. Lee et al. (1999) presented a model for designing process tolerances to maximize the process capability index. This model is consolidated into a single objective fuzzy programming. The proposed model simultaneously optimized the process capability of each operation. They determined the lower and upper bounds of the process capability index via a fuzzy membership function. They noted that low manufacturing cost resulted from wide process tolerances, whereas large process tolerances contributed to good process capability. Therefore, they transformed a multi-objective problem into a single objective formulation as a fuzzy model. Then they proposed a fuzzy approach to maximize the process capability index of each operation. Yongting (1996) defined a formula of process capability index Cpk to measure fuzzy quality. He determined the value of the fuzzy process capability index Cpk changing between 0 and 1, which was different from the classical range of [-∞, ∞].

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2.1 FPCIs when Specification Limits Are Triangular Fuzzy Numbers (TFNs) Suppose we have a fuzzy process with fixed σ, for which the upper and lower speci% = TFN ( a , a , a ) , LSL % = TFN ( b , b , b ) ∈ F (ℜ) . fication limits are the fuzzy as USL 1 2 3 1 2 3 T The width between fuzzy process specification limits is a triangular fuzzy number % ( a , a , a ) Ө LSL % ( b , b , b ) . The fuzzy process w% SL ∈ FT (ℜ) , defined by w% SL = USL 1 2 3 1 2 3 capability index is a triangular fuzzy number, C% p = w% SL Ø 6σ. Therefore, ⎛ a −b a −b a −b ⎞ C% p = TFN ⎜ 1 3 , 2 2 , 3 1 ⎟ 6σ 6σ ⎠ ⎝ 6σ

(20.4)

C% pk index can be calculated as follows: ⎛ a − μ a2 − μ a3 − μ ⎞ C% pu = TFN ⎜ 1 , , ⎟ 3σ 3σ ⎠ ⎝ 3σ

(20.5)

⎛ μ − b3 μ − b2 μ − b1 ⎞ C% pl = TFN ⎜ , , ⎟ 3σ 3σ ⎠ ⎝ 3σ

(20.6)

{

}

C% pk = min C% pl , C% pu , for TFNs.

(20.7)

2.2 FPCIs when Specification Limits Are Trapezoidal Fuzzy Numbers (TrFNs) Sometimes specification limits can be represented by TrFNs. Suppose that US̃L % = TrFN ( a , a , a , a ) and LSL % = TrFN ( b , b , b , b ) . and LS̃L are as follows: USL 1 2 3 4 1 2 3 4 The width between fuzzy process specification limits is a trapezoidal fuzzy number w% SL ∈ FT (ℜ) , defined by w% SL = TrFN ( a1 , a2 , a3 , a4 ) Ө TrFN ( b1 , b2 , b3 , b4 ) . The fuzzy process capability index is a trapezoidal fuzzy number, C% p = w% SL Ø 6σ. Therefore, ⎛a −b a −b a −b a −b ⎞ C% p = TrFN ⎜ 1 4 , 2 3 , 3 2 , 4 1 ⎟ 6σ 6σ 6σ ⎠ ⎝ 6σ

(20.8)

C% pk index can be calculated as follows: ⎛ a − μ a2 − μ a3 − μ a4 − μ ⎞ C% pu = TrFN ⎜ 1 , , , ⎟ 3σ 3σ 3σ ⎠ ⎝ 3σ

(20.9)

⎛ μ − b4 μ − b3 μ − b2 μ − b1 ⎞ C% pl = TrFN ⎜ , , , ⎟ 3σ 3σ 3σ ⎠ ⎝ 3σ

(20.10)

{

}

C% pk = min C% pl , C% pu , for TrFNs.

(20.11)

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2.3 Comparison of C% pu and C% pl The comparison of C% pl and C% pu is a necessity for using Eqs. (20.7) (20.10). In this chapter, C% pl and C% pu are compared with each other by using two ranking methods explained briefly as follows: 2.3.1 Yuan’s Ranking Method If two or more competitive fuzzy numbers exist, a criterion is needed to compare these processes. For this aim, the literature has different approaches. In this paper, firstly Yuan’s approach has been applied to compare fuzzy numbers. The details of this approach are as follows (Yuan, 1991): Let Ci and C j ∈ F(ℜ) be normal and convex. A fuzzy relation which compares the right spread of Ci with the left spread of Cj, is defined as Δij = ci+α

∫

(ci+α −c −jα )dα +

> c −jα

ci−α

μ (Ci , C j ) =

∫

(ci−α −c +jα )dα

(20.11)

> c +jα

Δij Δ ij + Δ ji

(20.12)

where μ (Ci , C j ) is the degree of largeness of Ci relative to Cj . Ci , C j ∈ F(ℜ) , then Ci is larger than Cj if and only if μ (Ci , C j ) > 0.5 . Ci and Cj are equal if and only if μ (Ci , C j ) = 0.5.

2.3.2 Tran and Duckstein’s Ranking Method The method proposed by Tran and Duckstein (2002) is based on the comparison of distances from fuzzy numbers (FNs) to some predetermined targets. These targets are called as the crisp maximum (Max) and the crisp minimum (Min), respectively. They are determined by the decision makers. The idea is that a FN is ranked first if its distance to the crisp maximum (Dmax) is the smallest but its distance to the crisp minimum (Dmin) is the greatest. If only one of these conditions is satisfied, a FN might be outranked by the others depending upon context of the problem (for example, the attitude of the decision-maker in a decision situation). The Max and Min are chosen based on Eq. (20.13): ⎛ I ⎞ Max ( I ) ≥ sup ⎜ U s A%i ⎟ ⎝ i =1 ⎠

( )

⎛ I ⎞ Min ( I ) ≤ inf ⎜ U s A%i ⎟ ⎝ i =1 ⎠

( )

(20.13)

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where s(Ãi) is the support of FNs Ai (i=1,…,I). Then Dmax and Dmin of FNs à can be computed as in Eq. (20.17). This equation is arranged for TFNs and TrFNs as follows (Tran and Duckstein, 2002):

D2

(

2 ⎧ ⎡⎛ a + a ⎤ ⎞ 1 ⎛ a + a3 ⎞ 3 −M⎟ + ⎜ 2 − M ⎟ × ⎣⎡( a4 − a3 ) − ( a2 − a1 ) ⎤⎦ + ⎥ ⎪ ⎢⎜ 2 ⎠ 2⎝ 2 ⎠ ⎪ ⎢⎝ 2 ⎥ ⎪⎢ 2 ⎥ ⎪ ⎢ 1 ⎛ a3 − a2 ⎞ + 1 ⎛ a3 − a2 ⎞ × ⎡( a − a ) + ( a − a ) ⎤ + ⎥ , if A% is TrFN 4 3 2 1 ⎦ ⎜ ⎟ ⎜ ⎟ ⎣ ⎪⎢ 3 ⎝ 2 ⎠ 6 ⎝ 2 ⎠ ⎥ ⎪⎢ ⎥ ⎪ 1 1 2 2 ⎥ A% , M = ⎨ ⎢ ⎡( a4 − a3 ) + ( a2 − a1 ) ⎤ − ⎣⎡( a2 − a1 ) × ( a4 − a3 ) ⎤⎦ ⎦ 9 ⎥ ⎪ ⎢⎣ 9 ⎣ ⎦ ⎪ ⎪⎡ 1 2 ⎤ ⎪ ⎢( a2 − M ) + 2 ( a2 − M ) ⎣⎡( a3 + a1 ) − 2 M ⎤⎦ + ⎥ ⎪⎢ ⎥ , if A% is TFN 1 2 2 ⎪⎢ 1 ⎡ ⎥ ⎤ − + − − ⎡ − − ⎤ a a a a a a a a ( ) ( ) ( )( ) 2 2 1 2 ⎦⎥ ⎪⎩ ⎢⎣ 9 ⎣ 3 ⎦ 9⎣ 2 1 3 ⎦

)

(20.17)

where M is either Max or Min, Dmax = D 2 ( A% , Max ) and Dmin = D 2 ( A% , Min ) .

2.4 Application Three machines produce aluminum rods. The diameter of rods is a critical quality characteristic. The following table shows the mean and standard deviations of rods’ diameters which are produced in different three machines. Table 20.1 Statistics of rods’ diameters Machine Mean (cm) Standard Deviation (cm) I

1.024

II

1.022

0.0025 0.0027

III

1.021

0.0024

2.4.1 Specification Limits Are TFN The lower and upper specifications for rod diameter have been determined as 1.016 and 1.032, respectively. In this paper, SLs have been represented by TFNs for all processes as follows: US̃L = Approximately 1.032 = TFN (1.030, 1.032, 1.034), and LS̃L= Approximately 1.016 = TFN(1.014, 1.016, 1.018). The fuzzy C% p s of these machines are derivated as follows: C% p1 = TFN ( 0.80, 1.07, 1.33 ) , C% p 2 = TFN ( 0.74, 0.99, 1.23) , C% p 3 = TFN ( 0.83, 1.11, 1.39 ) .

The membership functions of these FPCIs are presented in Fig. 20.1.

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Fig. 20.1 The Membership Functions of C% p s for alternative machines

The results of the pairwise comparisons for these PCI are presented in Table 20.2 with respect to Yuan’s approach. Table 20.2 The Results of the Comparisons for C% p s (Yuan’s approach) Processes

∆ij

∆ji

Decision

Degree

I -II

0.68

0.36

Machine I is better than Machine II

0.349

I − III

0.23

0.32

Machine III is better than Machine I

0.583

According to Table 20.2, Machine-III is better than Machine-I with a degree of 0.583. Machine-I is better than Machine-II with degree of 0.349. The machines are ranked as follow: {III,I,II} . The C% p s for alternative machines are also compared by Tran and Duckstein’s Method. The results are presented in Table 20.3. Table 20.3 The Results of the Comparisons for C% p s (Tran and Duckstein’s Method) Cp

a1

a2

a3

Dmax

Dmin

I

0.80

1.07

1.33

0.464

0.469

II

0.74

0.99

1.23

0.573

0.359

III

0.83

1.11

1.39

0.404

0.532

According to Table 20.3, Machine-III is the best alternative. Because it has the smallest Dmax and the largest Dmin value. Therefore the ranking is as follows: {III,I,II} . The ranking result is the same as Yuan’s method.

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Based on Eq. (20.6), the C% pl s are calculated as follows: C% pl −1 = TFN ( 0.80, 1.07, 1.33) , C% pl − 2 = TFN ( 0.49, 0.74, 0.99 ) , C% pl − 3 = TFN ( 0.42, 0.69, 0.97 ) .

Based on Eq. (20.5), the C% pu s are calculated as follows: C% pu −1 = TFN ( 0.80, 1.07, 1.33) , C% pu − 2 = TFN ( 0.99, 1.23, 1.48 ) , C% pu −3 = TFN (1.25, 1.53, 1.81) .

For a true ranking, we must also calculate C% pk values. C% pk s are calculated by using ranking methods and Eq. (20.7) as follows: C% pk −1 = TFN ( 0.80, 1.07, 1.33) , C% pk − 2 = TFN ( 0.49, 0.74, 0.99 ) , C% pk − 3 = TFN ( 0.42, 0.69, 0.97 ) .

It is clearly seen that the C% p1 and C% pk −1 are equal which means that the process is centered. For this point, Machine I is the best alternative. The machine ranking is as follows: {I,III,II} 2.4.2 Specification Limits Are TrFNs Using TrFNs is sometimes more suitable than using TFNs. In this section, the specification limits have been determined as TrFNs as follows: US̃L = Between 1.032 and 1.034 = TrFN(1.030, 1.032, 1.034, 1.036), and LS̃L= Between 1.016 and 1.018= TrFN(1.014, 1.016, 1.018, 1.020). The fuzzy C% p s of these processes are derived as follows by Eq. (20.8). C% p1 = TrFN ( 0.67, 0.93, 1.20, 1.47 ) , C% p 2 = TrFN ( 0.62, 0.86, 1.11, 1.36 ) , C% p 3 = TrFN ( 0.69, 0.97, 1.25, 1.53 ) .

The membership functions of C% p s for these processes are shown in Figure 20.2. The membership functions of the C% p s are compared with each other by Tran and Duckstein’s ranking method (2002). The results of the comparisons are summarized in Table 20.4. According to Table 20.4, Machine-III has the smallest Dmax and the largest Dmin value. Therefore it is the best process. Based on Eqs. (20.9-20.10), the C% pu s and C% pl s are derived as follows: C% pu −1 = TrFN ( 0.80, 1.07, 1.33, 1.60 ) , C% pu − 2 = TrFN ( 0.99, 1.23, 1.48, 1.73) , C% pu − 3 = TrFN (1.25, 1.53, 1.81, 2.08) C% pl −1 = TrFN ( 0.53, 0.80, 1.07, 1.33 ) , C% pl − 2 = TrFN ( 0.25, 0.49, 0.74, 0.99 ) , C% pl − I 3 = TrFN ( 0.14, 0.42, 0.69, 0.97 ) .

Based on Eq.(20.11), the C% pk s are derived as follows: C% pk −1 = TrFN ( 0.53, 0.80, 1.07, 1.33) , C% pk − 2 = TrFN ( 0.25, 0.49, 0.74, 0.99 ) , C% pk − I 3 = TrFN ( 0.14, 0.42, 0.69, 0.97 ) .

According to fuzzy process capability indices when the specifications are TrFN, the machines are ranked as follow: {III,I,II} .

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Fig. 20.2 Membership Functions for C% p s Table 20.4 The Results of the Comparisons for C% p s (Tran and Duckstein’s Method) Cp

a1

a2

a3

a4

Dmax

Dmin

I

0.67

0.93

1.20

1.47

0.488

0.477

II

0.62

0.86

1.11

1.36

0.560

0.399

III

0.69

0.97

1.25

1.53

0.449

0.521

3 Fuzzy Process Capability Analyses with Fuzzy Normal Distribution The process capability is defined as the percentage of the products which are within in the SLs. It is known that products or process’ outputs are inspected with respect to SLs and classified into two categories: accepted (conforming) and rejected (nonconforming). Consequently the percentages of accepted outputs (PCA) and rejected outputs (PRO) are two basic criteria for interpreting process’ ability or performance. PAO can be calculated by using Eq.(20.18): USL

PAO =

∫ P ( x ) dx = P (USL ) − P ( LSL )

(20.18)

LSL

where P ( x ) is the cumulative distribution function of the observed characteristics,

X . If the X fits normal distribution, N ( μ ,σ 2 ) , PRO can be calculated by using

Eq.(20.19): ⎛ LSL − μ ⎞ ⎡ ⎛ USL − μ ⎞ ⎤ PRO = Φ ⎜ ⎟ + ⎢1 − Φ ⎜ ⎟⎥ σ ⎝ ⎠ ⎣ ⎝ σ ⎠⎦

(20.19)

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where Φ (.) is the cumulative distribution function of the standard normal distribution and μ is the process mean. PAO can also be calculated by PAO = 1 − PRO . PRO helps us to produce the sigma quality level of the process and ppm values.

3.1 Fuzzy Normal Distribution Let N ( μ ,σ 2 ) denotes the crisp normal random variable with mean μ and variance σ 2 and f ( x; μ ,σ 2 ) , x ∈ ℜ be the density function of the crisp normal distri-

bution where 1 f ( x, μ , σ ) = e σ 2π

−( x − μ )

2

2

2σ 2

(20.20)

If the mean and variance are unknown, they estimated from a random sample and obtained fuzzy estimator μ% for μ and fuzzy estimator σ% 2 for σ 2 (Buckley and Eslami, 2004; Buckley 2004; 2005; 2006). So consider the fuzzy normal distribution denoted as N ( μ% ,σ% 2 ) for fuzzy mean μ% and variance σ% 2 . The fuzzy probability of obtaining a value in the interval [c, d ] will be denoted as P% [c, d ] . For α ∈ [0,1] , μ ∈ μ% [α ] and σ 2 ∈ σ% 2 [α ] , P% [c, d ] (α ) is given as in Eq. (20.21): 2 ⎧ d −( x − μ ) 1 ⎪ 1 2σ 2 P% [ c, d ] (α ) = ⎨ e dx μ ∈ μ% [α ] , σ 2 ∈ σ% 2 [α ] , ∫c πσ πσ 2 2 ⎪⎩

∞

∫e

−( x − μ ) 2σ 2

−∞

2

⎫ ⎪ dx = 1⎬ ⎪⎭

(20.21) ⎧⎪ 1 =⎨ ⎪⎩ 2π

z2

∫e

− z2 2

dz μ ∈ μ% [α ] , σ 2 ∈ σ% 2 [α ] ,

z1

where 0 ≤ α ≤ 1 , z1 =

1 2π

∞

∫e

− z2 2

−∞

⎪⎫ dz = 1⎬ ⎪⎭

c − μ% d − μ% and z2 = . σ% σ%

Eq. (20.21) gets the α -cuts of P% [c, d ] . Let P% [c, d ] (α ) = ⎡⎣ pL (α ) , pR (α ) ⎦⎤ . pL (α )

and pR (α ) are produced by the followings: ⎧⎪ pL (α ) = min ⎨ ⎪⎩ ⎪⎧ pR (α ) = max ⎨ ⎩⎪

1 2π 1 2π

z2

∫e

− z2 2

dz μ ∈ μ% [α ] , σ 2 ∈ σ% 2 [α ] ,

z1

z2

∫e z1

−z 2

2

1 2π

1 dz μ ∈ μ% [α ] ,σ ∈ σ% [α ] , 2π 2

2

∞

∫e

− z2 2

−∞

∞

∫e

−∞

⎪⎫ dz = 1⎬ ⎪⎭

− z2 2

⎪⎫ dz = 1⎬ ⎪⎭

(20.22)

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495

In this chapter fuzzy normal approximation is applied for PCIs. Fuzzy probabilities are calculated for the interval of [ LSL,USL ] . When the process mean is μ% and variation is σ 2 , the fuzzy probability of products which lie on between specification limits and called “conforming items”, can be calculated as follows: ⎧⎪ 1 P%z [ LSL,USL ] (α ) = ⎨ ⎩⎪ 2π

where z1 =

z2

∫e

− z2 2

dz μ ∈ μ% [α ] ,σ 2 ∈ σ% 2 [α ] ,

z1

1 2π

∞

∫e

− z2 2

−∞

⎪⎫ dz = 1⎬ = ⎡⎣ pl (α ) , pr (α ) ⎦⎤ ⎪⎭

(20.23)

LSL − μ% USL − μ% and z2 = . σ% σ%

Based on Eq. (20.23), the fuzzy probabilities of the products which fall out of the specification limits called “nonconforming items” are calculated as follows: P%z , d [ LSL,USL ] (α ) = ⎡⎣1 − pr (α ) ,1 − pl (α ) ⎤⎦ = ⎡⎣ pl , d (α ) , pr , d (α ) ⎤⎦

(20.24)

In Eqs. (20.23-20.24), fuzzy probabilities are calculated for crisp specification limits. This analysis should be extended to include SLs so that we can increase the sensitiveness of the results and have more information about the process. For this aim, SLs are defined as triangular and trapezoidal fuzzy numbers, respectively. When the fuzzy specification limits (FSLs) are defined, their α -cuts should be taken into account. In this case, the fuzzy probabilities of conforming and nonconforming items are calculated as follows: ⎧⎪ 1 z2 (α ) − z 2 1 2 2 P%z (α ) [ LSL,USL ] (α ) = ⎨ ∫ e 2 dz μ ∈ μ% [α ],σ ∈ σ% [α ], 2π ⎪⎩ 2π z1 (α ) P%z (α ), d [ LSL,USL ] (α ) = ⎡⎣1 − pr (α ) ,1 − pl (α )⎤⎦ = ⎡⎣ pl , d (α ) , pr , d (α )⎤⎦

where z1 (α ) =

LSL (α ) − μ (α )

σ (α )

and z2 =

∞

∫e

−∞

USL (α ) − μ (α )

σ (α )

− z2 2

⎪⎫ dz = 1⎬ = ⎡⎣ pl (α ) , pr (α ) ⎦⎤ ⎪⎭

(20.25)

.

3.2 Membership Functions of Process Mean and Variance Process mean and variance are two critical parameters in process capability analysis to determine the percentage of nonconforming items of a process. In Section 2, fuzzy normal approximation has been analyzed for fuzzy mean and variance. These parameters have been taken into account as TFNs or TrFNs. In this section, Buckley’s fuzzy estimation method is used to obtain the membership functions of the mean and variance (Buckley, 2004; 2005) 3.2.1 Fuzzy Membership Function of Process Mean Let x be a random variable which has a probability density function, N ( μ ,σ 2 ) , with unknown mean ( μ ) and known variance (σ 2 ) . A random sample x1 , x2 ,..., xn

C. Kahraman and İ. Kaya

496

from N ( μ ,σ 2 ) can be taken to estimate μ . The mean of this sample is a crisp number ( x ) . As it is known, ( x ) has a normal probability density function, ⎛ σ2 ⎞ x −μ N ⎜ μ , ⎟ , and has standard normal probability density function N ( 0,1) . σ n ⎝ ⎠ n

Therefore (Buckley, 2004; 2005): σ σ ⎞ ⎛ P ⎜ x − zβ ≤ μ ≤ x + zβ ⎟ =1− β 2 2 n n⎠ ⎝

(20.26)

where zβ is the z value of the probability of a N ( 0,1) random variable exceeding 2

β

2

. As a result, the (1 − β ) 100% confidence interval for μ can be obtained by

using Eq. (20.27) (Buckley, 2004; 2005): σ σ ⎤ ⎡ , x + zβ ⎡⎣θ1 ( β ) ,θ 2 ( β ) ⎤⎦ = ⎢ x − zβ ⎥ 2 2 n n⎦ ⎣

(20.27)

where zβ is defined as follows: 2

∫

zβ

−∞

2

N ( 0,1)dx = 1 −

β 2

(20.28)

If β values are taken into account as α − cuts , fuzzy estimator of μ , μ% , can be obtained by using Eq. (20.29): σ σ ⎤ ⎡ , x + zα ⎥ ⎣⎡ μl (α ) , μr (α ) ⎦⎤ z = ⎢ x − zα 2 2 n n⎦ ⎣

(20.29)

Additionally if x is a random variable which has a probability density function, N ( μ ,σ 2 ) , with unknown mean ( μ ) and variance (σ 2 ) , a random sample x1 , x2 ,..., xn from N ( μ ,σ 2 ) can be taken to estimate μ . The sample mean and vari-

ance are crisp numbers ( x and s 2 ) . μ% can be obtained by using Eq. (20.30) (Buckley, 2004; 2005): s s ⎤ ⎡ , x + tα ⎡⎣ μl (α ) , μ r (α ) ⎤⎦ t = ⎢ x − tα ⎥ 2 2 n n⎦ ⎣

(20.30)

In process capability analysis, the data have to be taken from a population which has the probability density function, N ( μ ,σ 2 ) . When the data are taken, the sample size should be large adequately. In this way, μ% can be obtained only by using Eq. (20.29) based on “z distribution” since “t distribution” is not suitable for process capability analysis.

Fuzzy Process Capability Analysis and Applications

497

3.2.2 Fuzzy Membership Function of Process Variance Let x be a random variable which has a probability density function, N ( μ ,σ 2 ) , with unknown mean ( μ ) and variance (σ 2 ) . A random sample x1 , x2 ,..., xn from N ( μ ,σ 2 ) can be taken to estimate σ 2 . Also it is known that

( n − 1) s 2 has a “chi2

σ square distribution” with n-1 degrees of freedom. Therefore the fuzzy estimator for σ 2 can be defined by the following confidence interval. ⎡ 2 2⎤ ⎢ ( n − 1) s , ( n − 1) s ⎥ ⎢ χ2 β χ L2, β ⎥ ⎢⎣ R , 2 ⎥⎦ 2

where χ R2 , β

2

and χ L2, β

(20.31)

are the points on the right and left sides of the χ 2 density 2

respectively, where the probability of exceeding (being less than) it is β 2 . This formula is a biased estimate for σ 2 . Buckley defined the following equations to obtain an unbiased fuzzy estimator ( 0.01 ≤ β ≤ 1.00 ) (Buckley, 2004; 2005): L ( λ ) = [1 − λ ] χ R2 ,0.005 + λ ( n − 1)

(20.32) R ( λ ) = [1 − λ ] χ

2 L ,0.005

+ λ ( n − 1)

Then the unbiased (1 − β )100% confidence interval for σ 2 should be calculated as: ⎡ ( n − 1) s 2 ( n − 1) s 2 ⎤ σˆ 2 = ⎢ , ⎥ , 0 ≤ λ ≤ 1. R ( λ ) ⎦⎥ ⎣⎢ L ( λ )

(20.33)

If β is taken into account as an α -cut level, the fuzzy triangular membership function for σ 2 is obtained from Eq. (20.33) as shown in Eq. (20.34). The triangular fuzzy membership functions can be built by placing these confidence intervals one on top of another. ⎤ ( n − 1) s 2 ( n − 1) s 2 , ⎥ 2 2 ⎣⎢ [1 − α ] χ R ,0.005 + ( n − 1) α [1 − α ] χ L ,0.005 + ( n − 1)α ⎦⎥ ⎡

σ 2 (α ) = ⎢

(20.34)

After σ 2 has been analyzed to obtain its fuzzy estimation, it is a necessity to reconsider the fuzzy estimation of μ . Eq. (20.30) can be reevaluated by taking into account σ% 2 as follows: μσ% (α ) = ⎡⎣ μl (α ) , μr (α ) ⎦⎤

(20.35)

where μl (α ) and μr (α ) represent the two sides of the fuzzy estimation of μ and they can be calculated by using Eqs. (20.36 -20.37).

C. Kahraman and İ. Kaya

498

σ i (α ) ⎞

⎛

μl (α ) = min ⎜ x − zα ⎝

i = 1,2

⎟ n ⎠

2

σ i (α ) ⎞

⎛

μr (α ) = max ⎜ x + zα ⎝

(20.36)

2

⎟ i = 1,2 n ⎠

(20.37)

where σ 1 (α ) and σ 2 (α ) represent the left and right sides of σ% 2 , respectively and can be calculated by using Eq. (20.37).

3.3 Fuzzy Process Capability Indices Process capability analysis is a statistical tool to evaluate process’ situation by taking into account process location and dispersion. The results of the analysis are summary statistics called PCIs which were explained in Section 1. It is known that the process mean and variance are two critical parameters to calculate PCIs. Their fuzzy estimations give us a chance to produce fuzzy PCIs (FPCIs). In this subsection, not only mean and variance but also SLs are considered as fuzzy numbers to increase sensitivity of PCIs. The fuzzy estimation of C p index can be obtained as follows: C p (α ) =

Suppose

that

the

upper

(USL − LSL )(α ) 6 σ 2 (α ) and

lower

(20.38) SLs

are

defined

as

% = TFN ( u , u , u ) and LSL % = TFN ( l , l , l ) . The α -cuts of index C can be obUSL 1 2 3 1 2 3 p

tained by using Eq. (20.39): ⎛ ⎜ ⎜ ( u − u + l − l )α + ( u1 − l3 ) ( u2 − u3 − l2 + l1 )α + u3 − l1 , C p (α ) = TFN ⎜ 2 1 3 2 n − 1) s 2 ( n − 1) s 2 ( ⎜ 6 ⎜ 6 [1 − α ] χ 2 [1 − α ] χ R2 ,0.005 + ( n − 1)α L ,0.005 + ( n − 1) α ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(20.39)

Also the fuzzy C pk can be derived by taking into account fuzzy SLs, variance and fuzzy estimation of μ all together. Firstly, α -cuts of C pu and C pl should be developed to obtain C% pk as follows: C pl (α ) =

( μ − LSL )(α ) = ⎛ μl (α ) − LSLr (α ) , μ r (α ) − LSLl (α ) ⎞ ⎜⎜ ⎟⎟ 3σ (α ) 3σ r (α ) 3σ l (α ) ⎝ ⎠

(20.40)

C pu (α ) =

(USL − μ )(α ) = ⎛ USLl (α ) − μr (α ) , USLr (α ) − μl (α ) ⎞ ⎜⎜ ⎟⎟ 3σ (α ) 3σ r (α ) 3σ l (α ) ⎝ ⎠

(20.41)

Fuzzy Process Capability Analysis and Applications

{

C pk (α ) = min C pl (α ) , C pu (α )}

499

(20.42)

% = TFN ( u , u , u ) and LSL % = TFN ( l , l , l ) ) , If the SLs are defined as TFNs ( USL 1 2 3 1 2 3

the α cuts of the indices C pu and C pl are obtained as follows: ⎛ ⎞ ⎛ ⎛ σ α ⎞ σ α ⎞ ⎜ min ⎜ x − zα i ( ) ⎟ − ( ( l2 − l3 ) α + l3 ) max ⎜ x + zα i ( ) ⎟ − ( ( l2 − l1 )α + l1 ) ⎟ 2 2 ⎜ ⎟ n n ⎝ ⎠ ⎝ ⎠ C pl (α ) = TFN ⎜ , ⎟ 2 2 − − n 1 s n 1 s ( ) ( ) ⎜ ⎟ 3 3 2 2 ⎜ ⎟ [1 − α ] χ L,0.005 + ( n − 1)α [1 − α ] χ R,0.005 + ( n − 1)α ⎝ ⎠ ⎛ σ α ⎞ σ α ⎞⎞ ⎛ ⎛ ⎜ ( ( u2 − u1 ) α + u1 ) − max ⎜ x + zα i ( ) ⎟ ( ( u2 − u3 ) α + u3 ) − min ⎜ x − zα i ( ) ⎟ ⎟ 2 2 ⎜ n ⎠ n ⎠⎟ ⎝ ⎝ C pu (α ) = TFN ⎜ , ⎟ n − 1) s 2 n − 1) s 2 ( ( ⎜ ⎟ 3 3 2 2 ⎜ ⎟ 1 n 1 1 n 1 α χ α α χ α − + − − + − ( ) ( ) [ ] [ ] L R ,0.005 ,0.005 ⎝ ⎠

(20.43)

(20.44)

The index C% pk can be derived by using Eq. (20.42) in the same way of calculating the minimum value of C% pl and C% pu for the two cases explained above. It is a necessity to use a ranking method since these indices are expressed as fuzzy numbers. In this chapter, the ranking methods explained in Section 2.3 are used.

3.4 An Application In this subsection, the pin diameter of pistons is analyzed by using FPCIs. The percentages of conforming and nonconforming items are also determined by using fuzzy normal approximation. For this aim, a Volvo Marine motor’s piston for diesel engine is considered and the measurements of pin diameter are saved. Sample size and sample number are 4 and 20, respectively. The first condition to apply PCA is that the process should be in statistical control and the data should come from a normal distribution. Generally control charts are used to satisfy the first condition. In the first stage, the data are controlled to determine whether or not they fit normal distribution. The probability plot of pin diameter measurements has been created by using MINITAB 14.0 and is illustrated in Fig. 20.3. According to Fig.20.3, the data fit the normal distribution since P value is determined as 0.608 and the significance level is 0.05. The next stage is to create control charts to check whether or not the process is in statistical control. x − R control charts which are illustrated in Fig. 20.4 are set up and pin diameter measurement are monitored. As it can be seen from Fig. 20.4, the process is in statistical control and the PCA can be applied. The analysis is executed by MINITAB 14.0. PCA of pin diameter is summarized in Fig. 20.5 The results for PCA are as follows: C p = 1.00 and C pk = 0.98 . Also the process mean and variance are estimated as 55.0006 and 0.00009926, respectively.

500

C. Kahraman and İ. Kaya

Fig. 20.3 Probability plot of Pin Diameter

Fig. 20.4 x − R control charts for pin diameter measurements

Before PCA has been executed, the fuzzy estimations of mean and variance are produced by using Eqs. (20.32-20.37). The fuzzy values of μ and σ 2 are compared with the corresponding crisp values and attached in Table 20.5. The fuzzy estimations of μ and σ 2 include more information than the crisp values. Notice that the crisp values belong to the fuzzy estimation with a membership value of 1.00. In the next phase, SLs are defined as fuzzy numbers to produce FPCIs and to determine the membership function of the percentages of conforming and nonconforming items. Assume that the USL and LSL are defined as approximately 55.03 and 54.97, respectively. The membership function of the index C p is obtained by using Eq. (20.37) as shown in Fig.20.6. The fuzzy estimation of C p includes 1.00

Fuzzy Process Capability Analysis and Applications

501

Fig. 20.5 PCA for pin diameter measurements Table 20.5 Fuzzy and Crisp Values for μ and σ 2 SLs

μ

σ2

Crisp

55.0006

0.0000992635

Fuzzy

(54.9958,55.0006,55.0054)

(0.00006814,0.000099263,0.00015558)

with a membership value of 1.00. The other values of C% p can be observed from Fig. 20.6 with different α -cut levels. Also the C% pu and C% pl indices can be derived by using Eqs. (20.40-20.41) and the index C% pk is obtained by taking into account the C% pu and C% pl indices and using Tran and Duckstein’s ranking method. The membership

functions

of

C% pu = TFN ( 0.63, 0.98, 1.42 ) ,

C% pu

and

C% pl

are

determined

C% pl = TFN ( 0.66, 1.02, 1.47 ) ,

as

follows: and

C% pk = TFN ( 0.63, 0.98, 1.42 ) .

The percentages of the fuzzy conforming and nonconforming items are analyzed for the considered process by taking into account Eq. (20.25) and the fuzzy estimations of μ and σ 2 . The membership function of the percentage of producing nonconforming items ( P%fd ) is determined as TFN ( 0.0006, 0.0026, 0.0308 ) . The crisp

value of to the percentage of producing nonconforming items is determined as 0.26 % as seen in Fig.20.5. This value belongs to the ( P%fd ) with a membership value of 1.00. Also the membership function of the percentage of producing conforming items P%z (α ) is determined as TFN ( 0.96917, 0.99735, 0.99939 ) .

(

)

502

C. Kahraman and İ. Kaya

Fig. 20.6 Membership function of the index C p

Fig. 20.7 Membership Function of P%d

The membership function of the percentages of FNCIs

( P% ( ) ) z α

is shown

in Fig.20.7.

4 Robust Process Capability Indices (RPCIS) Traditional PCIs are not suitable if the process observations have a correlation. In that case, they can cause misleading evaluations. They should be analyzed by taking into account the correlation. In statistics, correlation indicates the strength and direction of a linear relationship between two random variables. In general statistical usage, correlation refers to the departure of two variables from independence. Many industrial processes have been surrounded by correlation. In order to reduce the variability of the process its various components probably including correlations should be analyzed. The observed value for an arbitrary industrial process (Yt ) can be estimated as follows (Prasad and Bramorski, 1998): Yt = Zt + ε t . The standard deviation (σˆ c ) after accounting for the correlation can be estimated by the Eq. (20.45).

Fuzzy Process Capability Analysis and Applications

σˆ c =

∑ε

2 t

n −1

503

.

(20.45)

In statistics, regression analysis is a technique which examines the relation of a dependent variable to specified independent variables. Regression analysis can be used as a descriptive method of data analysis (such as curve fitting) without relying on any assumptions about underlying processes generating the data (Richard, 2004). The regression analysis can be applied as follows (Buckley, 2004): Assume that we have some data ( xi , yi ) , 1 ≤ i ≤ n, on two variables x and Y. The values of x are known in advance and Y is a random variable. We assume that there is no uncertainty in the x data. The future value of Y with certainty cannot be predicted so we focus on the mean of Y, E(Y). We have an assumption that E(Y) is a linear function of x. It can be formulated as follows: E (Y ) = a + b ( x − x ) . In this formula, x is the mean value. The regression model is as follows shown in Eq. (20.46): Yˆi = a + b ( xi − x ) + ε i

(20.46)

We wish to estimate values a and b. The crisp estimator of a is aˆ = y , the mean of the yi values. The crisp estimator of b is as follows: n

bˆ =

∑ y (x − x) i

i =1 n

i

∑ ( xi − x )

(20.47)

2

i =1

⎛1⎞ n ⎝ ⎠ i =1

Finally, σˆ c 2 = ⎜ ⎟ ∑ ⎡⎣ yi − aˆ − bˆ ( xi − x ) ⎤⎦ n

2

(20.48)

If another model is used to regression analysis such as non linear, cubic or logarithmic, the σˆ c 2 can be estimated as follows: ⎛1⎞

n

σˆ 2 = ⎜ ⎟ ∑ [ yi − yˆi ] ⎝ n ⎠ i =1

2

(20.49)

Traditional PCIs can be modified by substituting σˆ c (the standard deviation of the regression model, which is showed in Eqs. (20.48-20.49) for the standard deviation σ . The PCIs can be calculated by the following formulas when the process has correlation (Prasad and Bramorski, 1998): USL − LSL Cˆ pc = 6σˆ c

{

USL − μ μ − LSL Cˆ pkc = min Cˆ puc , Cˆ plc where Cˆ puc = and Cˆ plc = 3σˆ c 3σˆ c

}

(20.50) (20.51)

C. Kahraman and İ. Kaya

504

4.1 Fuzzy Robust Process Capability Indices (FRPCIs) Sometimes specification limits cannot be represented by crisp numbers. They are defined by fuzzy numbers. In this subsection, RPCIs are analyzed based on the fuzzy set theory. For this aim, the standard deviation of the regression model (σˆ c ) is handled based on Buckley’s approach (Buckley, 2004). (1 − β )100% confidence intervals for a and b given in Eq. (20.47) are as follows respectively: ⎡ ⎢ aˆ − t β ⎢⎣ 2 ⎡ ⎢ ⎢bˆ − t β ⎢ 2 ⎢ ⎣

σˆ 2

( n − 2)

nσˆ 2 n

( n − 2 ) ∑ ( xi − x )

2

, aˆ + t β 2

, bˆ + t β 2

i =1

⎤ ⎥ ( n − 2 ) ⎥⎦

(20.52)

⎤ ⎥ ⎥ n 2 ( n − 2 ) ∑ ( xi − x ) ⎥⎥ i =1 ⎦

(20.53).

σˆ 2

nσˆ 2

The fuzzy estimator for σ 2 can be defined by the following confidence interval ⎡

⎤

nσˆ 2 nσˆ 2 (Buckley, 2004): ⎢ 2 , 2 ⎥ where χ R2 , β and χ L2, β are the points on the right ⎢χ β χ β ⎥ 2 2 R, L,

⎢⎣

2

2

⎥⎦

and left sides of the χ density, respectively, where the probability of exceeding (being less than) it is β 2 . This formula is a biased estimate for σ 2 . To obtain an 2

unbiased fuzzy estimator, the following functions are defined (Buckley, 2004): ⎡

σˆ% c2 = ⎢

nσˆ 2

⎣⎢ [1 − λ ] χ

2 R ,0.005

⎤ ⎥, + λ n [1 − λ ] χ L2,0.005 + λ n ⎦⎥

,

nσˆ 2

0 ≤ λ ≤ 1.

(20.54)

If the λ is taken care into account as α , the fuzzy triangular membership function for σ c 2 is obtained from Eq. (20.54). The triangular fuzzy membership functions can be built by placing these confidence intervals one on top of another. Also we define specification limits as triangular fuzzy numbers (TFNs). Assume that upper specification limits (USL) and lower specification limits (LSL) are defined % = ( u , u , u ) , LSL % = (l , l ,l ) . as follows: USL 1 2 3 1 2 3 Then the FRPCIs can be derived by the following formulas:

(C%ˆ ) pc

α

⎛ ⎞ ⎜ ⎟ ⎜ ⎡( u2 − u1 ) + ( l3 − l2 ) ⎤⎦ × α + ( u1 − l3 ) ⎡⎣( u2 − u3 ) − ( l2 − l1 ) ⎤⎦ × α + ( u3 − l1 ) ⎟ =⎜⎣ , ⎟ (20.55) n × σˆ 2 n × σˆ 2 ⎜ 6× ⎟ 6× 2 2 ⎜ ⎟ − × + × − × + × α χ α n α χ α n 1 1 ( ) ( ) ( ) ( ) ,0.005 ,0.005 R L ⎝ ⎠

Fuzzy Process Capability Analysis and Applications

505

( C% )

⎛ ⎜ ⎡⎣( u2 − u1 ) × α + u1 ⎤⎦ − μ ⎡⎣( u2 − u3 ) × α + u3 ⎤⎦ − μ ⎜ =⎜ , 2 nσˆ nσˆ 2 ⎜ 3× 3× 2 ⎜ (1 − α ) × χ R ,0.005 + (α × n ) (1 − α ) × χ L2,0.005 + (α × n ) ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(20.56)

( C% )

⎛ ⎜ μ − ⎣⎡( l2 − l3 ) × α + l3 ⎤⎦ μ − ⎡⎣( l2 − l1 ) × α + l1 ⎤⎦ ⎜ =⎜ , 2 nσˆ nσˆ 2 ⎜ 3× 3× 2 ⎜ (1 − α ) × χ R ,0.005 + (α × n ) (1 − α ) × χ L2,0.005 + (α × n ) ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(20.57)

puc

plc

α

α

{

C% pkc = min C% pUc , C% plc

}

(20.58)

5 Fuzzy Process Capability Analysis Based on Alpha Cuts Estimation In some works, specification limits are crisp but a fuzzy estimation of PCI is made by defining the significance level as an α -cut level. In this section, this approach is used to obtain membership functions of PCIs. In the rest of this section, Buckley’s (2004) approach is applied to find fuzzy estimates of C p and C pk . Then fuzzy membership functions of C p and C pk are obtained based on six-sigma approach.

5.1 Buckley’s Approach In this chapter, Buckley’s approach (Buckley, 2004, 2005a, 2005b) for fuzzy estimation is used to produce triangular membership functions of PCIs. In this section, this approach is summarized briefly (Parchami and Mashinchi, 2007). Before the explanation of this approach, we should explain the notation. A triangular shaped fuzzy number “N” is a fuzzy subset of the real numbers “R” satisfying: • N ( x ) = 1 for exactly one x ∈ R. • For α ∈ ( 0,1] , the α-cut of N is a closed and bounded interval, which is denoted by Nα = ⎡⎣ n1 (α ) , n2 (α )⎦⎤ , where n1 (α ) is increasing, n2 (α ) is decreasing continuous functions. Let X is a random variable with probability density function (p.d.f.) f ( x;θ ) for a single parameter θ . Assume that θ is unknown and must be estimated from a random sample X1, . . . ,Xn. Let Y = u(X1, . . . ,Xn) is a statistic used to estimate θ . According to the values of these random variables, e.g. Xi = xi, 1 ≤ i ≤ n , we obtain a point estimate θˆ = y = u ( x1 ,..., xn ) for θ . There is no expectation that this

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point estimate is exactly equal to θ , so a (1 − β )100% confidence interval for θ is often computed. The (1 − β )100% confidence interval for θ

is denoted

as ⎡⎣θ1 ( β ) ,θ 2 ( β )⎤⎦ , for 0 < β < 1 . Thus the interval θ1 = ⎡⎣θˆ,θˆ ⎤⎦ is the 0% confidence interval for θ and θ0 = Θ is a 100% confidence interval for θ , where Θ is the whole parameter space. Consequently, it is obtained that a family of (1 − β )100% confidence intervals for θ , where 0 ≤ β ≤ 1 . β is used here since α , usually employed for confidence intervals, is reserved for α -cuts of fuzzy numbers. If these confidence intervals are placed one on top of the other, a triangular shaped fuzzy number θ whose α-cuts are the following confidence intervals is obtained (Buckley, 2004; 2005a; 2005b): θα = ⎣⎡θ1 (α ) ,θ 2 (α ) ⎦⎤

0 < α < 1, θ 0 = Θ and θ1 = ⎡⎣θˆ,θˆ,⎤⎦

for

(20.59)

5.1.1 Fuzzy Estimation of Cp The standard deviation, σ , of X in the traditional process capability formula which is given Eq. 20.1 can be estimated. We know that s is the natural estimator of σ . If X1, X2, . . . ,Xn are independent, and they are distributed as random variables with p.d.f. N ( μ ,σ 2 ) , then the sum of squared deviation from the mean is distributed by chi-square distribution ( χ 2 ) . Therefore, s 2 is distributed as σ2×

χ n2−1

( n − 1)

.

⎡ ⎤ 2 n − 1) s 2 ⎥ ( ⎢ ( n − 1) s 2 ≤σ ≤ =1− β . Pr ⎢ 2 χ χ 2 β ⎥⎥ ⎢ n −1,1− β n −1, 2 2 ⎣ ⎦

(20.60)

(20.61)

where Pr ⎡⎣ χ n2−1 ≤ χ n2−1,ε ⎤⎦ = ε . A random sample X1, X2, …, Xn from N ( μ ,σ 2 ) to estimate Cp is taken. Then (1β) 100 % confidence interval for σ 2 is (Buckley, 2004; 2005a; 2005b); ⎡ ⎤ 2 2 ⎢ ( n − 1) s ( n − 1) s ⎥ ⎡⎣σ ( β ) ,σ ( β ) ⎤⎦ = ⎢ 2 , 2 χ χ β ⎥⎥ ⎢ n −1,1− β n −1, 2 2 ⎣ ⎦ 2 1

2 2

(20.62)

where s is the natural estimator of σ , and (n-1) is the degree of freedom for chisquare distribution. According to Buckley’s approach;

Fuzzy Process Capability Analysis and Applications

⎡ ⎤ 2 2 ⎢ ( n − 1) s ( n − 1) s ⎥ , 2 =⎢ 2 χ χ α ⎥⎥ ⎢ n −1,1− α n −1, 2 2 ⎣ ⎦

(S ) 2

α

Let b ∈ ( S 2 )α ; C p (b ) =

507

(20.63)

α ∈ (0,1). Let us define

U −L then where (Parchami and Mashinchi, 2007): 6 b

(C )

p α

⎡ χ2 α χ2 α n −1, n −1,1− ⎢ˆ 2 2 , Cˆ p = ⎢C p 1 n n − −1 ⎢ ⎣⎢

⎤ ⎥ ⎥ ⎥ ⎦⎥

for

0 USL ) or ( μ < LSL ) ) . The complementary index Ca = 1 − k , referred to as the accuracy index, is defined to measure the degree of process centering relative to the manufacturing tolerance. This index can be defined as follows (Pearn et al. 1998; Pearn and Kotz, 2006): Ca = 1 −

μ −m d

(20.76)

The index Ca measures the degree of process centering, which alerts the user if the process mean deviates from its target value. Therefore, the index Ca only reflects the process accuracy.

6.1 New Insights to Index Ca Kaya and Kahraman (2009d) improve the PAI by removing the absolute operator from Eq. (20.76) to provide some new insights as presented in Table 20.6. Without the absolute operator, Eq. (20.76) becomes as in Eq. (20.76): Ca = 1 −

μ−m d

(20.77)

The suggested new formula for Ca successfully determines the location of the process mean, μ . For example, Ca > 2.00 indicates that μ is located out of the LSL. Ca = 2.00 indicates that μ is located on the LSL. Ca = 1.00 indicates that the process mean is located on the midpoint of specification limits. The possible situations for the location of the process mean are summarized in Table 20.6. Table 20.6 Ca values without absolute operator and the location of µ

C a values

Location of µ

Ca > 2.00

μ < LSL

Ca = 2.00

μ = LSL

1.00 < Ca < 2.00

LSL < μ < m

Ca = 1.00

μ=m

0.00 < Ca < 1.00

m < μ < USL

Ca = 0.00

μ = USL

Ca < 0.00

μ > USL

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~ 6.2 Fuzzy Process Accuracy Index ( C a ) In this section Ca is modified as a decision making tool. Unlike the studies given above, a completely different PCI is proposed under fuzziness. The process accuracy index is not related to process spread and dispersion. It just focuses on the location of process mean and the distance between mean and target value. Suppose we have a fuzzy process distribution for which the upper and lower speci% = TFN ( u , u , u ) , LSL % = TFN ( l , l , l ) ∈ F (ℜ) . fication limits are fuzzy as follows: USL 1 2 3 1 2 3 T The index Ca can be derived as follows: C% a = 1 −

μ% − m% d%

(20.78)

% + LSL % USL ⎛u +l u +l u +l ⎞ =⎜ 1 1, 2 2, 3 3⎟ 2 2 2 ⎠ ⎝ 2

(20.79)

% − LSL % USL ⎛u −l u −l u −l ⎞ d% = =⎜ 1 3, 2 2, 3 1⎟ 2 2 2 ⎠ ⎝ 2

(20.80)

m% =

Specification limits can also be represented by TrFNs. Let US̃L and LS̃L are de% = TrFN ( u , u , u , u ) , LSL % = TrFN ( l , l , l , l ) . fined as follows: USL 1 2 3 4 1 2 3 4 In this case, the process accuracy index can be derived as follows: C% a = 1 −

μ% − m%

(20.81)

d%

% + LSL % USL ⎛u +l u +l u +l u +l ⎞ =⎜ 1 1, 2 2, 3 3, 4 4⎟ 2 2 2 2 ⎠ ⎝ 2

(20.82)

% − LSL % USL ⎛u −l u −l u −l u −l ⎞ d% = =⎜ 1 4, 2 3, 3 2, 4 1⎟ 2 2 2 2 ⎠ ⎝ 2

(20.83)

m% =

Then the index C% a can be calculated based on Eq. (20.78). ⎧ μ% − m% ⎪1 − d% , ⎪ C% a = ⎨ ⎪ m% − μ% ⎪1 − , d% ⎩⎪

if μ% ≥ m%

(20.84) if μ% < m%

Eq. (20.84) requires a ranking method to be used since it includes a comparison between μ% and m% . The following formula does not need any comparison between μ% and m% and it has an advantage to clarify the exact location of μ% . μ% − m% C% a = 1 − d%

(20.85)

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7 Conclusions The tool called Process Capability Indices (PCIs) is a well known technique in Quality Control. Process capability analysis that produces some summary statistics called PCIs is a very useful to analyze to process’ performance. The process can be classified as capable if the PCIs are greater than predetermined critical values. Otherwise they can be labeled as incapable. Also the decision whether or not the process is centered can be made and the percentages of conforming and nonconforming items can be calculated by PCIs. PCIs are rapidly becoming a standard tool for quality reporting. Because of the importance of the PCIs, more flexibility and sensitiveness should be added to them for more information. Sometimes, spesification limits cannot be defined as crisp numbers. They can be defined by linguistic terms. Then the traditional capability indices are not suitable and have some problems for this situation. Therefore, the fuzzy set theory is applied to process capability indices. It provides an easy definition for specification limits. Crisp definitions of SLs, process mean and variance cause a limitation on PCIs. The fuzzy sets bring an advantage to the flexible definition and evaluation. The fuzzy values of process mean and variance have been produced by the estimation theory based on confidence intervals. For further research, the other fuzzy PCIs such as C% and C% can be reconpm

pmk

sidered by taking into account fuzzy SLs and the fuzzy estimations of μ and σ . The fuzzy percentages of conforming and nonconforming items can be calculated for six sigma approach and the α -cuts of parts per million (ppm) for nonconforming items can be evaluated. Also FPCIs can be improved for multi-criteria and multi-attribute decision making problems. 2

References Buckley, J.J.: Fuzzy statistics. Springer, Berlin (2004) Buckley, J.J.: Simulating fuzzy systems. Springer, Berlin (2005) Buckley, J.J.: Fuzzy probability and statistics. Springer, Berlin (2006) Buckley, J.J., Eslami, E.: Uncertain probabilities II: The continuous case. Soft Computing 8, 193–199 (2004) Chen, K.S., Chen, T.W.: Multi-process capability plot and fuzzy inference evaluation. International Journal of Production Economics 111(1), 70–79 (2008) Chen, T.W., Chen, K.S., Lin, J.Y.: Fuzzy evaluation of process capability for bigger-thebest type products. International Journal of Advanced Manufacturing Technology 21, 820–826 (2003a) Chen, T.W., Lin, J.Y., Chen, K.S.: Selecting a supplier by fuzzy evaluation of capability indices Cpm. International Journal of Advanced Manufacturing Technology 22, 534–540 (2003b) Gao, Y., Huang, M.: Optimal process tolerance balancing based on process capabilities. International Journal of Advanced Manufacturing Technology 21, 501–507 (2003) Hsu, B.M., Shu, M.H.: Fuzzy inference to assess manufacturing process capability with imprecise data. European Journal of Operational Research 186(2), 652–670 (2008) Kahraman, C., Kaya, İ.: Fuzzy process capability indices for quality control of irrigation water. Stochastic Environmental Research and Risk Assessment 23(4), 451–462 (2009a)

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Kahraman, C., Kaya, İ.: Fuzzy process accuracy index to evaluate risk assessment of drought effects in Turkey. Human and Ecological Risk Assessment: An International Journal 15(4), 789–810 (2009b) Kane, V.E.: Process capability indices. Journal of Quality Technology 18, 41–52 (1986) Kaya, İ., Kahraman, C.: Fuzzy process capability analyses: An application to teaching processes. Journal of Intelligent & Fuzzy Systems 19(4-5), 259–272 (2008) Kaya, İ., Kahraman, C.: Fuzzy robust process capability indices for risk assessment of air pollution. Stochastic Environmental Research and Risk Assessment 23(4), 529–541 (2009a) Kaya, İ., Kahraman, C.: Air pollution control using fuzzy process capability indices in sixsigma approach. Human and Ecological Risk Assessment: An International Journal 15(4), 689–713 (2009b) Kaya, İ., Kahraman, C.: Development of fuzzy process accuracy index for decision making problems. Information Sciences (2009c), doi:10.1016/j.ins.2009.05.019 Kotz, S., Johnson, N.: Process capability indices-a review 1992-2000. Journal of Quality Technology 34, 2–19 (2002) Lee, H.T.: Cpk index estimation using fuzzy numbers. European Journal of Operational Research 129, 683–688 (2001) Lee, Y.H., Wei, C.C., Chang, C.L.: Fuzzy design of process tolerances to maximise process capability. International Journal of Advanced Manufacturing Technology 15, 655–659 (1999) Lin, G.H.: A random interval estimation of the estimated process accuracy index. International Journal of Advanced Manufacturing Technology 27, 969–974 (2006) Montgomery, D.C.: Introduction to statistical quality control. John Wiley & Sons, Chichester (2005); Parchami, A., Mashinchi, M.: Fuzzy estimation for process capability indices. Information Sciences 177, 1452–1462 (2007) Parchami, A., Mashinchi, M., Maleki, H.R.: Fuzzy confidence interval for fuzzy process capability index. Journal of Intelligent & Fuzzy Systems 17, 287–295 (2006) Parchami, A., Mashinchi, M., Yavari, A.R., Maleki, H.R.: Process capability indices as fuzzy numbers. Austrian Journal of Statistics 34(4), 391–402 (2005) Pearn, W.L., Kotz, S.: Encyclopedia and handbook of process capability indices. Series on Quality, Reliability and Engineering Statistics, vol. 12. World Scientific, Singapore (2006) Pearn, W.L., Lin, G.H., Chen, K.S.: Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory and Methods 27(4), 985–1000 (1998) Prasad, S., Bramorski, T.: Robust process capability indices. Omega 26(3), 425–435 (1998) Richard, A.B.: Regression analysis: a constructive critique. Sage Publications, Thousand Oaks (2004) Tran, L., Duckstein, L.: Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy Sets and Systems 130, 331–341 (2002) Tsai, C.C., Chen, C.C.: Making decision to evaluate process capability index Cp with fuzzy numbers. International Journal of Advanced Manufacturing Technology 30, 334–339 (2006) Wu, C.W.: Decision-making in testing process performance with fuzzy data. European Journal of Operational Research 193(2), 499–509 (2009) Yongting, C.: Fuzzy quality and analysis on fuzzy probability. Fuzzy Sets and Systems 83, 283–290 (1996) Yuan, Y.: Criteria for evaluating fuzzy ranking methods. Fuzzy Sets and Systems 43, 139– 157 (1991) Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–359 (1965)

Chapter 21

Fuzzy Measurement in Quality Management Systems George T.S. Ho, Henry C.W. Lau, Nick S.H. Chung, and W.H. Ip*

Abstract. The fluctuating and competitive economy today is affecting the production industry worldwide. Under the stress of various forceful challenges, customer satisfaction and product loyalties form the necessary key for enterprises to survive or even thrive in this decade. The concept of Quality Management System (QMS) offers a chance for enterprises to win customer satisfaction by producing consistently high-quality products. With the use of Data Mining (DM) and Artificial Intelligence (AI) techniques, enterprises are able to discover previously hidden yet useful knowledge from large and related databases which assists to support the high-valued continuous quality improvement. Continuous quality improvement is of utmost importance to enterprises as it helps turn them potent to compete in today’s rivalrous global business market. In this chapter, Intelligent Quality Management System with the use of Fuzzy Association Rules is the main focus. Fuzzy Association Rule is a useful data mining technique which has received tremendous attention. Through integrating the fuzzy set concept, enterprises or users are able to decode the discovered rules and turn them into more meaningful and easily understandable knowledge, for instance, they can extract interesting and meaningful customer behavior pattern from a pile of retail data. In order to better illustrate how this technique is used to deal with the quantitative process data and relate process parameters with the quality of finished products, an example is provided as well to help explain the concept.

1 Introduction Product development and manufacturing process are the importance steps in producing high quality products and it is found that the quality performance can be improved in the manufacturing organizations by minimizing the production defects (Dhafr 2006). To better manage the production quality, the quality management system (QMS) is implemented to enable the enterprise to identify and control the manufacturing process by capturing the product production data so as to improve the production quality and performance. QMS is not an independent George T.S. Ho, Henry C.W. Lau, Nick S.H. Chung, and W.H. Ip Department of Industrial & Systems Engineering, The Hong Kong Polytechnic University C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 515–536. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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component of a work environment or project which they are part of the overall operation and product of a business unit, seamlessly integrated into the business processes and daily operations (Wiles et al. 2007). The purpose of having a quality management system in every functional process is to ensure that the products provided by each process meet the quality requirements resulted in continuous evaluation and improvement of the processes used to produce products. In collecting the production data throughout the manufacturing process, hidden relationships between all possible processes controls variables can be found out while the knowledge behind the production process regarding the product quality are discovered for further improvement. Through the use of data mining (DM) and artificial intelligent (AI) techniques, the relevant process data and experience can be captured in order to support the continuous improvement in quality management. This chapter is divided into four main sections focusing on quality measurement system with the use of fuzzy measurement. A literature review is conducted in Section 2 to review the existing measurement in quality management systems as well as the fuzzy measurement models in quality management adopted in the production industries. In Section 3, an intelligent quality measurement system is proposed using fuzzy association rules. This chapter is concluded in the final section by presenting the key findings and suggestions for further research.

2 Literature Review In manufacturing industries, due to globalization and mass customization, production process becomes more and more complicated. Mass customization is a dominant trend in modern manufacturing with the aim of producing a large variety of products that are customized for individual customers with the benefits that can only be achieved in high volume production (Pine, 1993; Eastwood, 1996). Trends of the industries would be characterized by globalization, parallelization, agility, virtual enterprise, customer's satisfaction and quality. Enterprises need to integrate and coordinate their business processes efficiently, in an inter-organizational environment and in continuous changing (in an inter-organizational and continuous changing environment), and that is necessary to model and plan the enterprise (Vemadat, 1996). In a virtual enterprise environment, a multi-functional or crossfunctional team from different departments establishes a network of linked decisions with interdependencies (Nahm et al., 2005). In order to survive and increase the market share in the fast growing and customer-oriented production environment, manufacturing companies aim to reduce production costs and improve product quality so as to meet customer requirements. This section aims to provide a comprehensive study on quality management regarding the academic publications and it is mainly focused on two areas including measurement in quality management system as well as the fuzzy measurement models in quality management. Through conducting the literature review, the existing quality measurement and improvement methods adopted are examined while the applications of different kinds of intelligent systems are reviewed to support the quality enhancement continuously.

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2.1 Measurement in Quality Management Systems Traditionally, a product is designed, manufactured and sold, while the product quality can only be improved based on customer feedback from the market. Any future problems that may occur in the market are not likely to be predicted or prevented during the product-design stage (Wu and Wu, 2000). Besides, major challenges faced by the production industries included a maturing workforce, excessive scrap, rework, lack of coordination, poor delivery performance, and a general lack of technical processes and product understanding (Kleiner et al. 1997). In order to produce products with constant quality, manufacturing systems need to be monitored for any unnatural deviations in the state of the process (Pacella et al. 2004). Qiao et al. (2008) stated that quality is the summation of the characters about the capabilities of product or service that satisfies the definite or hidden requirements.ïEach manufacturing stage is important and contributes to the final results and expected quality of the product, but it is very difficult to determine the overall effect of particular parameters of a certain manufacturing process on the final quality of the product and further complexities may exist due to any unknown (or unconfirmed) interrelationships between dimensions and parameters of the product (Shahbaz et al. 2006). Especially in manufacturing area, quantitative attributes such as state of process, condition of manufacturing, and measured quality of products, are necessary for quality control, manufacturing management, planning, and decision of strategy (Watanabe, 2004). The Total Quality management (TQM) has examined the positive relationships between the practices of quality management and various levels of organizational performance (Kaynak 2003) With the concept of total quality management (TQM)it is possible to develop an intelligent system to capture quality audit data from different processes during manufacturing so as to discover meaningful patterns and knowledge for future improvement (Ho et. al. 2006). Linderman et al. (2004) proposed an integrated view of quality and knowledge using Nonaka’s theory of knowledge creation. This integrated view helps illuminate how quality practices can lead to knowledge creation and retention. The knowledge perspective also provides insight into what it means to effectively deploy quality management practices. A self-organizing map plus a back-propagation neural network (SOM-BPNN) model is proposed for creating a dynamic quality predictor in a plastic injection molding process resulted in an accurate prediction of product quality (Chen et al. 2008).

2.2 Fuzzy Measurement Models in Quality Management Thanks to many researches concentrating on the use of AI, it is now made possible to structure a reliable, fast and practical procedure for executing quality evaluation (Paladini 2000). Most cases concerning the data mining and artificial intelligence technologies have been shown to provide various applications in supporting production quality control and improvement. Machine learning, data mining, and several related research areas are concerned with methods for the automated

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induction of models and the extraction of interesting patterns from empirical data depends on the fuzzy set theory (Hullermeier 2008). Ament and Goch (1999) introduced the concept of a fuzzy model based quality control, which allows an automated feedback to guarantee a constant quality of manufactured products. Based on a process model, the controller is able to interpret the measurement and adjust the process parameters. A quality intelligent prediction model Fuzzy Least Square Support Vector Machine (FLS-SVM) is proposed and taken as the intelligent kernel to set up quality prediction model for small-batch producing process which could avoid the disadvantages, such as over-training, weak normalization capability, etc., of artificial neural networks prediction (Dong et al. 2007). In particular, an application of quality control is suggested through developing an intelligent decision support system using a fuzzy MIN–MAX algorithm for heuristic knowledge and optimization for fundamental knowledge with the purpose of defect reduction (Lou and Huang 2003). Melin and Castillo (2007) demonstrated the use of an intelligent hybrid approach, combining type-2 fuzzy logic rule base containing the knowledge of human expects in quality control and neural networks, to the problem in automating of quality control during manufacturing. Li et al. (2002) proposed a genetic neural fuzzy system (GNFS) and a hybrid learning algorithm is used to train the GNFS for constructing quality prediction model in the injection process. A fuzzy model predictive control (FMPC) approach is introduced by Huang et al. (2000) to design a control system for a highly nonlinear process by a fuzzy convolution model that consists of a number of quasi-linear fuzzy implications (FIs) to minimize the prediction errors and control through a two-layered iterative optimization process. In the researches concerning the production quality improvement, association rule has been widely applied for data mining and knowledge discovery. By allowing for ‘‘soft’’ rather than crisp boundaries of intervals, fuzzy sets can avoid certain undesirable threshold effects (Sudkamp 2005). A mining-based knowledge support system for problem-solving on a production line is proposed by Liu and Ke (2007) in which association rule mining and sequential pattern mining are used to discover hidden decision-making and dependency knowledge patterns from historical problem-solving logs for quality improvement. Zeng (2008) investigated a data driven optimization model: Process Quality Optimization Model (PQOM) based on Association Rules Mining (ARM) and immune principle to support both static and dynamic optimization of process quality to find the implicit and hidden correlations in process systems. Chen et al. (2005) introduced the Root-cause Machine Identifier (RMI) method using the technique of association rule mining to solve the problem of analyzing correlations between combinations of machines and the defective products. Through reviewing of the application in AI and data mining (DM) techniques in manufacturing industries, it is found that there are many attempts to explore AI and DM techniques to optimize the processes with better finished quality. The use of knowledge-based techniques to assist quality engineers and solve a range of decision-making problems in manufacturing activities is also discussed. The quality engineers are required to select the most appropriate technique within different areas as more than one technique is available for use. To develop a sophisticated

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system in assisting the engineers to analyze the quality data both effectively and efficiently so that the manufacturing workflow within the manufacturing industry can be streamlined, an effective approach of Intelligent Quality Management System (IQMS) is developed based on the concept of fuzzy generalized mining algorithm (Hong et al. 2003). It supports the knowledge discovery and decision support from a mass of process data by discovering the appropriate parameter combinations to achieve the desired finished product quality in the complex manufacturing process.

3 Intelligent Quality Management System Using Fuzzy Association Rules The Intelligent Quality Management System (IQMS) is designed to capture the distributed data from different processes within the integrated workflow and convert the data into knowledge in terms of fuzzy association rules along the workflow which has positive or negative impacts on the quality of the finished products. In fact, it also allows process or quality engineers to access an objectoriented repository to retrieve the updated current inspection status of different processes in various departments. The mechanism of the proposed i-PM algorithm is briefly described below. The notations in this algorithm are tabulated as follows (Lau et al., 2009): Table 21.1 List of notation used

n

the number of integrated supply chain workflow records

N = {1,2, L , n}

the set of index of integrated supply chain workflow records

Wi

the i

s

the number of departments in an integrated workflow records

S = {1,2,L, s}

the set of indices of departments in an integrated workflow record

Da

the a department, ∀a ∈ S

m

the number of processes in an integrated workflow record

M = {1,2,L, m}

the set of index for processes in a supply chain workflow record

Paj

the j

k aj

the number of relevant process parameters in j

th

integrated supply chain workflow records, ∀i ∈ N

th

th

process in a department, ∀a ∈ S and ∀j ∈ M th

th

th

process of a de-

partment

K aj = {1,2, L , k aj } the set of index for relevant process parameters in j th process of

a th department Qajt

the t

th

relevant process parameter for Paj , ∀t ∈ K aj

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Table 21.1 (continued)

hajt

the number of fuzzy regions for Qajt

H ajt = {1,2, L , hajt } the set of index for fuzzy regions for Qajt th

fuzzy region of Qajt , ∀l ∈ H ajt

Rajtl

the l

PViajt

the quantitative value of Qajt of Paj of Da in Wi

f iajt

the fuzzy set converted from PViajt

f iajtl

the membership value of Qajt of Paj of Da of Wi in region Rajtl

countajtl

the summation of f iajtl values where ∀i ∈ N

max − countajt

the maximum count value among countajtl values, ∀l ∈ H ajt

max − Rajt

the fuzzy region of Qajt with max − countajt

α ajt

the predefined minimum support threshold of Qajt , ∀a ∈ S , ∀j ∈ M ,

∀t ∈ K aj

λ

the predefined minimum confidence threshold

Cr

the set of candidate itemsets with r items

Lr

the set of large itemsets with r items

Input: A set of n integrated workflow records. Each integrated workflow record consists of s departments and each department consists of m processes. Each process ( j th ) consists of different numbers of process parameters ( k aj ). A set of minimum support value α ajt , a predefined confidence value λ and a set of membership functions are predefined by process engineers. Output: A set of mined fuzzy association rules which identify the hidden relationships between the process parameters along the workflow and finished quality. Step 1: Transform the quantitative value PViajt of each relevant process content

Qajt of each process Paj of each department Da for each integrated workflow record Wi where ∀i ∈ N

appearing into a fuzzy set

f iajt represented as

( f iajt1 / Rajt1 + f iajt 2 / Rajt 2 + L + f iajth / Rajth ) using the given membership functions, where hajt is the number of fuzzy regions for Qajt , Rajtl is the l th fuzzy region of

Qajt , ∀l ∈ H ajt , and f iajtl is PViajt ’s fuzzy membership value in region Rajtl . Step 2: Calculate the count countajtl of each fuzzy region Rajtl in the workflow record.

Fuzzy Measurement in Quality Management Systems

count ajtl =

∑f

iajtl

521

(21.1)

∀i∈N

Step3: Find max − countajt = MAX (countajtl ) for ∀a ∈ S , ∀j ∈ M , ∀t ∈ K aj ∀l∈H ajt

where k aj is the number of relevant process parameters in j th process of Da . Step 4: Let max − Rajt be the region with max − countajt for relevant process parameter Qajt which will be used to represent the fuzzy characteristic of relevant process parameters in the later mining process. Step 5: Check whether the value max − countajt of a region max − Rajt , ∀a ∈ S , ∀j ∈ M , ∀t ∈ K aj , is larger than or equal to its predefined minimum support val-

ue α ajt . If a region max − Rajt is equal to or greater than its minimum support value, put it in the large 1- itemset (L1), that is L1 = { max − Rajt | max − countajt

≥ α ajt , ∀a ∈ S , ∀j ∈ M , ∀t ∈ K aj }

Step 6: Generate the candidate set C2 from L1 where the supports of all the large 1-itemset comprising each candidate 2-itemsets must be larger than or equal to the maximum ( ma ) of the minimum supports of items in the large 1-itemset. Step 7: For each newly formed candidate 2- itemsets with items (s1, s2) in C2 a. Calculate the fuzzy value of s in each workflow record Wi as f is = f is1 ∧ f is 2 where f is j is the membership value of Wi in region sj. If the min operator is used for the intersection, then f is = Min( f is1 , fis 2 ) . b. Calculate the count of s in the workflow record as counts =

∑f

is

(21.2)

∀i∈N

c. If counts is larger than or equal to the predefined minimum support value ma , put s in L2 . L2 = { s | counts

≥ ma , a = {1,2,L, C2 } }

Step 8: Check if L2 is null; then exit the algorithm; else do the next step. Step 9: Set r =2, where r is used to represent the number of items stored in the current large itemsets. Step 10: Generate the candidate set Cr +1 from Lr in a way similar to that in the Apriori algorithm. Store in Cr +1 itemsets having all their sub-r-itemsets in Lr except the supports of all the Lr comprising each candidate (r+1)-itemset s must be larger than or equal to the maximum ( ma ) of the minimum supports of items in the Lr .

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Step 11: For each newly formed (r+1)-itemset s with items ( s1 , s2 L sr +1 ) in Cr +1 : a. Calculate the fuzzy value of s in each workflow record Wi as f is = f is1 ∧ f is 2 ∧ L ∧ f is r +1 where f is j is the membership value of Wi in r +1

region sj. If the min operator is used for the intersection, then f is = Min f is j j +1

b. Calculate the count of s in the workflow record as

counts =

∑f

(21.3)

is

∀i∈N

c. If counts is equal to or larger than the predefined minimum support value

ma , put s in Lr +1 . Lr +1 = { s | counts

≥ ma , a = {1,2,L, Cr +1 } }

Step 12: Check if Lr +1 is null; If yes, then do the next step; else, set r=r+1 and repeat Steps 10-12. Step 13: Construct the association rules for all the large q-itemset s containing items ( s1 , s2 L sq ), q ≥ 2 , using the following substeps: a. Form all possible association s1 ∧ s2 ∧ L ∧ sk −1 ∧ sk +1 ∧ L ∧ sq → sw , w = {1,2, L , q}

rules

thus

b. Calculate the confidence values of all association rules using the formula:

∑f

is

∀i∈N

∑( f

∀i∈N

is1

(21.4)

∧ L ∧ f is k −1 , f is k +1 ∧ L ∧ f is q )

Step 14: Keep the rules with confidence values larger than or equal to the predefined confidence threshold λ . Step 15: Check whether the rule’s output satisfies the relevant constraint. The general form of the rule’s output is: IF a1 is b1 AND a 2 is b2 AND K AND a n is bn

c1 is d1 AND K AND c n is d n IF ( C ∈ Q AND NOT A ∈ Q ) THEN

THEN keep the rule ELSE discard the rule where A = {a1 , a2 L, an ) : the set of fuzzy variables in the condition part of a rule

B = {b1 , b2 L, bn ) : the set of linguistic terms of fuzzy variables in the condition part of a rule

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523

C = {c1 , c2 L, cn ) : the set of fuzzy variables in the consequent part of a rule

D = {d1 , d 2 L, d n ) : the set of linguistic terms of fuzzy variables in the consequent part of a rule Q = {q1 , q2 L, qn ) : the set of fuzzy variables related to finished quality characteristics Step 16: Output the rules to users as interesting rules. An example is given to illustrate how the fuzzy association rules are generalized from quantitative process data. Different symbols are used to represent various process parameters and finished quality features as shown in Table 21.2 and 21.3 respectively. The data set including the six workflow records is shown in Table 21.4. Each record contains the settings of process parameters in different processes from three functional departments and the inspected data of achieved Table 21.2 The symbols of process input features Process parameters

Range

Pressure of deposition (Pa)

0.12 – 0.36 A

Temperature setting of the machine (°C) 180 – 250

Symbol B

Fixture angle of the machine (°)

10 – 30

C

Wavelength of the machine (nm)

550 – 650

D

Compression stress (GPa)

0.4 – 1.2

E

Time of cleaning (min)

2–8

F

Table 21.3 The symbols of finished quality input features Finished quality features

Range Symbol

Hardness of the product (GPa) 25 – 42 G Thickness of the product (nm) 15 – 35 H

Table 21.4 The expanded production workflow Department 1 (D1)

Department 2 (D2)

Department 3 (D3)

Finished Quality (Q)

Process 1 Process 2 Process 1 Process 2 Process 1 Process 2 (P1) (P2) (P1) (P2) (P1) (P2) A

B

C

D

E

F

G

H

1

0.18

230

12

580

0.65

7

40

27

2

0.25

180

15

620

0.85

3

38.5

25

3

0.2

205

10

630

0.75

4

32

16

4

0.27

195

28

618

0.95

2.5

38

35

5

0.3

190

14

625

0.9

2.9

41

22

6

0.33

200

20

595

1.1

5.5

38

30

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Table 21.5 Minimum support of each process parameter Process parameter

A

B

C

D

E

F

G

H

Min. Support ( α ajt )

2.5

2.2

3.2

2.7

2

2.5

2

1.8

Membership value

Membership value Low

1

0 0.12

Medium

0.16

High

0.21

Pressure of deposition 0 (Pa)

0.3

0.36

Membership value Medium

High

13.5

18

0

30

24.5

Membership value Low

Medium

High

550

210

230

Medium

562

0.5

0.87

1.2

Low

1

Compression Stress (GPa)

0.4

Temperature (°C)

250

High

606

Wavelength (nm)

650

625

0

2

Medium

2.8

4.65

High

Very High

5.9

7.1

Time of cleaning (min)

8

Membership value

Membership value

0 25

High

Membership value

1

1

190

Low

1

Angle (°)

0 10

0

180

Medium

Membership value Low

1

Low

1

Medium

Low

High

1

Hardness (GPa)

28.5

33

39.7

42

0

Low

Medium

High

28

35

Thickness (nm)

15

17.5

Fig. 21.1 Membership functions used in the case example

finished quality. Fuzzy membership values are produced for all items in the workflow record based on the predefined membership functions as shown in Figure 21.1. The minimum support for each process parameter is predefined and shown in Table 21.5. Moreover the confidence value k is set at 0.9 and acts as the threshold for interesting association rules. In order to discover the hidden association between process parameters within the integrated workflow, steps 1 to 16 for extraction of fuzzy association rules are presented: Step 1: Transform the quantitative values of the process parameters and finished quality features in each workflow record into a fuzzy set using the

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525

membership functions given above. Take the first process parameter in the first workflow record as an example. As shown in Figure 21.2, the crisp value “0.18” of process parameter “A” is converted into the fuzzy set which is calculated as (0.4/ Medium+ 0.6/Low). This step is repeated for all items in six workflow records and the result is given in Table 21.6. The converted structure of process parameters with fuzzy regions is represented as “department.process.process_parameter.fuzzy_region”.

Fig. 21.2 Fuzzy set conversion of process parameter “A” Table 21.6 The fuzzy sets transformed from the process data WID 1

2

Quantitative values of process variables using fuzzy sets

(

0 .4 0 .6 1 1 )( )( ) + . . . . . D P B High D P D1.P1. A.Medium D1.P1. A.Low 1 2 2 1 C .Low

(

0.41 0.59 0.59 0.41 )( ) + + D2 .P2 .D.Low D2 .P2 .D.Medium D3 .P1.E.Medium D3 .P1.E.Low

(

1 0 .9 0 .1 0.92 0.08 )( )( + ) + D3 .P2 .F .VeryHigh D3 .P2 .F .High Q.G.High Q.H .Medium Q.H .Low

0.44 0.56 1 + )( ) D1 .P1 . A.High D1 .P1 . A.Medium D1 .P2 .B.Low 0.33 0.67 ( + ) D2 .P1 .C.Medium D2 .P1 .C .Low

(

(

0.74 0.26 0.95 0.05 + )( + ) D2 .P2 .D.High D2 .P2 .D.Medium D3 .P1 .E.Medium D3 .P1 .E.Low

(

0.82 0.18 0.11 0.89 + )( + ) D3 .P2 .F .Medium D3 .P2 .F .Low Q.G.High Q.G.Medium

(

0.71 0.29 + ) Q.H .Medium Q.H .Low

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G.T.S. Ho et al.

Table 21.6 (continued) 3

0.75 0.25 0 .8 0 .2 + )( + ) D1 .P1 . A.Medium D1 .P1 . A.Low D1.P2 .B.Medium D1.P2 .B.Low 1 ( ) D2 .P1 .C.Low (

1 0.68 0.32 )( + ) D2 .P2 .D.High D3 .P1.E.Medium D3 .P1.E.Low 0.65 0.35 ( + ) D3 .P2 .F .Medium D3 .P2 .F .Low (

( 4

0.78 0.22 1 + )( ) Q.G.Medium Q.G.Low Q.H .Low

0.67 0.33 0.25 0.75 )( ) + + D1.P1. A.High D1.P1. A.Medium D1 .P2 .B.Medium D1 .P2 .B.Low 1 ( ) D2 .P1.C.High

(

0.63 0.37 0.24 0.76 )( ) + + D2 .P2 .D.High D2 .P2 .D.Medium D3 .P1.E.High D3 .P1.E.Medium 1 ( ) D3 .P2 .F .Low

(

5

6

(

0.75 0.25 1 + )( ) Q.G.High Q.G.Medium Q.H .High

(

1 0.11 0.89 1 1 )( )( )( ) + D1 .P1 . A.High D1.P2 .B.Low D2 .P1.C.Medium D2 .P1.C.Low D2 .P2 .D.High

(

1 0.09 0.91 0.05 0.95 ) )( )( + + D3 .P1.E.High D3 .P1.E.Medium D3 .P2 .F .Medium D3.P2 .F .Low Q.G.High

(

0.43 0.57 + ) Q.H .Medium Q.H .Low

1 0 .5 0 .5 )( ) + D1 .P1 . A.High D1 .P2 .B.Medium D1 .P2 .B.Low 0.31 0.69 ( ) + D2 .P1.C.High D2 .P1 .C.Medium (

(

0.75 0.25 0.7 0 .3 )( ) + + D2 .P2 .D.Medium D2 .P2 .D.Low D3 .P1 .E.High D3 .P1 .E.Medium

(

0.75 0.25 0.68 0.32 + ) )( + D3 .P2 .F .High D3 .P2 .F .Medium Q.G.High Q.G.Medium

(

0.29 0.71 + ) Q.H .High Q.H .Medium

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527

Step 2: Calculate the count of each fuzzy region of each process parameter in the workflow record and put the process parameters in a set C1 . The process parameter in C1 is then called an itemset. Take process parameter A as an example. Since process parameter A has three fuzzy regions: “low”, “medium” and “high”, the “low” fuzzy region is chosen to demonstrate the calculation of the count. The count for the “low” fuzzy region of process parameter A is calculated from the 6 workflow records by adding the fuzzy count of D1.P1. A.Low for each record, and is calculated as (0.6+0+0.2+0+0+0) = 0.8. The counts for the fuzzy regions of other process parameters follow the same procedure and are shown in Table 21.7. Table 21.7 The fuzzy counts of the item sets in

C1

Process items

Count

Process items

Count

D1.P1.A.Low

0.8

D3.P1.E.Low

0.96

D1.P1.A.Medium

2.09

D3.P1.E.Medium

4.01

D1.P1.A.High

3.11

D3.P1.E. High

1.03

D1.P2.B.Low

3.50

D3.P2.F.Low

3.19

D1.P2.B. Medium

1.50

D3.P2.F. Medium

1.13

D1.P2.B.High

1.00

D3.P2.F. High

0.76

D2.P1.C.Low

3.56

D3.P2.F.VeryHigh

0.92

D2.P1.C. Medium

1.13

Q.G. Low

0.22

D2.P1.C.High

1.31

Q.G. Medium

1.46

D2.P2.D.Low

0.84

Q.G. High

4.32

D2.P2.D.Medium

1.79

Q.H. Low

1.96

D2.P2.D.High

3.37

Q.H. Medium

2.75

Q.H. High

1.29

Step 3 & 4: Find the maximum count, max − countajt , for each process parameter and the fuzzy region, max − Rajt , is the region with the maximum counts of a process parameter. For process parameter A, the counts for the 3 fuzzy regions, which are “low”, “medium” and “high”, are 0.8, 2.09 and 3.11 respectively. Since the “high” fuzzy region has the highest count, max − countajt is set as 3.11 and max − Rajt is set as HIGH for process parameter A. The maximum counts and the corresponding fuzzy regions for each process parameter are shown below in Table 21.8. Step 5: Check whether the value max − countajt of a region max − Rajt of a process parameter is equal to or larger than its predefined minimum support value α ajt . If the value of a region in Table 21.8 is equal to or greater than its minimum

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Table 21.8 The maximum fuzzy counts and corresponding fuzzy regions of the itemsets in C1 Process items

Count

Process items

Count

D1.P1.A.High D1.P2.B.Low

3.11

D3.P1.E.Medium

4.01

3.5

D3.P2.F.Low

3.19

D2.P1.C.Low

3.56

Q.G. High

4.32

D2.P2.D.High

3.37

Q.H. Medium

2.75

support value, it will be selected and put in the large 1-itemset ( L1 ). The itemsets

L1 found according to the minimum support in Table 21.5 are shown below. Since the counts of the fuzzy regions with maximum counts of all process parameters which are shown in Table 21.8 above are larger than its minimum support which is shown in Table 21.5, all the process parameters are then put in the large 1-itemset L1 which is shown in Table 21.9. Table 21.9 Items in large 1-itemset

L1

Process items

Count

D1.P1.A.High

3.11

D1.P2.B.Low

3.5

D2.P1.C.Low

3.56

D2.P2.D.High

3.37

D3.P1.E.Medium

4.01

D3.P2.F.Low

3.19

Q.G. High

4.32

Q.H. Medium

2.75

Step 6: The candidate set

C 2 is generated from L1 , and the supports of the two

items in each itemset in C 2 must be larger than or equal to the maximum of their predefined minimum support values. For example, the minimum support values for process parameter A and B are 2.5 and 2.2 respectively and the maximum of minimum support value of process parameter A and B is then taken as 2.5. Since the counts of D1.P1.A.High and D1.P2.B.Low are 3.11 and 3.5 respectively which is larger than their maximum of minimum support value (2.5), itemsets {D1.P1.A.High, D1.P2.B.Low} is put into the set of candidate 2-itemsets. On the other hand for the other possible 2-itemsets {D1.P1.A.High, D2.P1.C.Low}, since the count for D 1 . P1 . A . High is 3.11 which is smaller than their maximum of minimum support value (3.2). {D1.P1.A.High, D2.P1.C.Low} is not going to be put in C 2 . The other candidates in 2-itemsets are generated in the same way. The 2itemsets are shown below in Table 21.10.

Fuzzy Measurement in Quality Management Systems Table 21.10 Items in

529

C2

Process items

Process items

{ D1.P1.A.High, D1.P2.B.Low }

{ D2.P1.C.Low, D2.P2.D.High}

{ D1.P1.A.High, D2.P2.D.High}

{ D2.P1.C.Low, D3.P1.E.Medium }

{ D1.P1.A.High, D3.P1.E.Medium }

{ D2.P1.C.Low, Q.G. High }

{ D1.P1.A.High, D3.P2.F.Low }

{ D2.P1.C.Low, D3.P1.E.Medium }

{ D1.P1.A.High, Q.G. High }

{ D2.P2.D.High, D3.P2.F.Low }

{ D1.P1.A.High, Q.H. Medium }

{ D2.P2.D.High, Q.G. High }

{ D1.P2.B.Low, D2.P1.C.Low }

{ D2.P2.D.High, Q.H. Medium }

{ D1.P2.B.Low, D2.P2.D.High }

{ D3.P1.E.Medium, D3.P2.F.Low }

{ D1.P2.B.Low, D3.P1.E.Medium }

{ D3.P1.E.Medium, Q.G. High }

{ D1.P2.B.Low, D3.P2.F.Low }

{ D3.P1.E.Medium, Q.H. Medium }

{ D1.P2.B.Low, Q.G. High }

{ D3.P2.F.Low, Q.G. High }

{ D1.P2.B.Low, Q.H. Medium }

{ D3.P2.F.Low, Q.H. Medium }

Step 7: For each newly formed candidate 2- itemsets in C2 a. Calculate the fuzzy value and the count of each candidate 2-itemsets in the workflow record. The count for each 2-itemsets is calculated as the minimum of counts for each item in the workflow record. Take {D1.P1.A.High, D1.P2.B.Low} as an example. The count is calculated from each workflow record. Since {D1.P1.A.High, D1.P2.B.Low} does not occur in record 1 and 3 concurrently, the count is equal to 0 for record 1 and 3. While for the record 2, the value of D1.P1.A.High is 0.44 and D1.P2.B.Low is 1.0, the minimum of these two values is taken as the count for this record which is Table 21.11 The fuzzy counts of the itemsets in C2 Process items

Count

Process items

Count

{D1.P1.A.High, D1.P2.B.Low }

2.61

{ D2.P1.C.Low, D2.P2.D.High }

2.56 2.64

{ D1.P1.A.High, D2.P2.D.High }

2.07

{ D2.P1.C.Low, D3.P1.E.Medium }

{ D1.P1.A.High, D3.P1.E.Medium}

2.32

{ D2.P1.C.Low, Q.G. High }

2.56

{ D1.P1.A.High, D3.P2.F.Low}

2.06

{ D2.P2.D.High, D3.P1.E.Medium }

2.95

{ D1.P1.A.High, Q.G. High }

2.86

{ D2.P2.D.High, D3.P2.F.Low}

2.67

{ D1.P1.A.High, Q.H. Medium }

1.58

{ D2.P2.D.High, Q.G. High }

2.37

{ D1.P2.B.Low, D2.P1.C.Low }

1.81

{ D2.P2.D.High, Q.H. Medium}

1.14

{ D1.P2.B.Low, D2.P2.D.High}

2.62

{ D3.P1.E.Medium, D3.P2.F.Low}

2.91

{ D1.P2.B.Low, D3.P1.E.Medium}

3.16

{ D3.P1.E.Medium, Q.G. High }

3.18

{ D1.P2.B.Low, D3.P2.F.Low}

2.84

{ D3.P1.E.Medium, Q.H. Medium }

1.85

{ D1.P2.B.Low, Q.G. High }

3.07

{ D3.P2.F.Low, Q.G. High }

2.51

{ D1.P2.B.Low, Q.H. Medium }

1.64

{ D3.P2.F.Low, Q.H. Medium }

1.14

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G.T.S. Ho et al.

0.44 in this case. Summing up the counts for the 6 records is the count for the 2-itemsets {D1.P1.A.High, D1.P2.B.Low} which is calculated as (0+0.44+0+0.67+1+0.5) =2.61. Other 2-itemsets follows the same way to calculate the count. The result is shown in Table 21.11. b. The count of each 2-itemsets is then compared with its corresponding minimum support value. Take {D1.P1.A.High, D2.P2.D.High} as an example. The count for {D1.P1.A.High, D2.P2.D.High} is 2.07 and the minimum support for it is 2.5. Since the count of {D1.P1.A.High, D2.P2.D.High} is smaller than its minimum support, it is not going to be put in the large 2-itemsets L2 . While for {D1.P1.A.High, D1.P2.B.Low}, its count (2.61) is larger than its corresponding minimum support (2.5), so it is put into the large 2itemsets L2 . The rest of the 2-itemsets follows the same procedure. The result is shown in Table 21.12. Table 21.12 Items in large 2-itemsets

L2

Process items

Count

{D1.P1.A.High, D1.P2.B.Low }

2.61

{ D1.P1.A.High, Q.G. High }

2.86

{ D1.P2.B.Low, D3.P1.E.Medium}

3.16

{ D1.P2.B.Low, D3.P2.F.Low}

2.84

{ D1.P2.B.Low, Q.G. High }

3.07

{ D2.P2.D.High, D3.P1.E.Medium }

2.95

{ D3.P1.E.Medium, D3.P2.F.Low}

2.91

{ D3.P1.E.Medium, Q.G. High }

3.18

{ D3.P2.F.Low, Q.G. High }

2.51

Step 8: As L2 is not null, go on to the next step. Step 9: Set r = 2 ; Steps 6-9 are repeated and shown in Steps 10-12. Step 10: The candidate set C3 is generated from L2 , and the supports of the two items in each itemset in C3 must be larger than or equal to the maximum of their predefined minimum support values. Take {D1.P1.A.High, D1.P2.B.Low, Q.G. High} as an example. The minimum support value is taken as the maximum of the minimum support of {D1.P1.A.High, D1.P2.B.Low}, {D1.P1.A.High, Q.G. High} and {D1.P2.B.Low, Q.G. High} which are 2.5, 2.5 and 2.2 respectively and thus the maximum count, 2.5, is taken as the count for {D1.P1.A.High, D1.P2.B.Low, Q.G. High}. For {D1.P1.A.High, D1.P2.B.Low, Q.G. High} to be included in C3, the counts of all of its two itemsets including {D1.P1.A.High, D1.P2.B.Low}, {D1.P1.A.High, Q.G. High} and {D1.P2.B.Low, Q.G. High} which are 2.61, 2.86 and 3.07 respectively must be equal to or larger than the maximum of their minimum support which is 2.5 in this case. Since the counts for all two itemsets of {D1.P1.A.High, D1.P2.B.Low, Q.G. High} is larger than or equal to the minimum

Fuzzy Measurement in Quality Management Systems

531

support, {D1.P1.A.High, D1.P2.B.Low, Q.G. High} is put in C3. The candidate 3itemsets are then found as {D1.P1.A.High, D1.P2.B.Low, Q.G. High} , {D1.P2.B.Low, D3.P1.E.Medium, D3.P2.F.Low} , {D1.P2.B.Low, D3.P1.E.Medium, Q.G. High} , {D1.P2.B.Low, D3.P2.F.Low, Q.G. High} and {D3.P1.E.Medium, D3.P2.F.Low, Q.G. High }. Step 11: For each newly formed 3-itemsets C3: a. Calculate the fuzzy value and the count of each 3-itemsets in C3: Take {D1.P1.A.High, D1.P2.B.Low, Q.G. High} as an example. The count for record 1 and 3 is 0 since these three items do not occur concurrently. While for record 2 is 0.44 which is the minimum among D1.P1.A.High, D1.P2.B.Low and Q.G. High. Similarly, the count for record 4, 5 and 6 are equal to 0.67, 1.0 and 0.5 respectively. Thus the count of {D1.P1.A.High, D1.P2.B.Low, Q.G. High} is the summation of the counts for 6 records, (0+0.44 +0+0.67+1.0+0.5)=2.61. The counts of the other 3-itemsets are shown in Table 21.13. Table 21.13 The fuzzy counts of the itemsets in C3 Process items

Count

D1.P1.A.High, D1.P2.B.Low, Q.G. High

2.61

D1.P2.B.Low, D3.P1.E.Medium, D3.P2.F.Low 2.80 D1.P2.B.Low, D3.P1.E.Medium, Q.G. High

2.78

D1.P2.B.Low, D3.P2.F.Low, Q.G. High

2.51

D3.P1.E.Medium, D3.P2.F.Low, Q.G. High

2.48

b. Check the count of each itemset in C3 with its predefined minimum support value. Put it in L3 if its count is equal to or greater than its predefined minimum support. The support value is calculated as the maximum of their minimum support. For { D 1 . P1 . A . High , D 1 . P 2 . B . Low , Q .G . High }, the minimum support for process parameter A, B and G are 2.5, 2.2, 2 respectively. Thus the support for the 3-itemsets { D 1 . P1 . A . High , D 1 . P 2 . B . Low , Q .G . High } is 2.61. Comparing the count of 3-itemsets with its support value, those itemset with count equal to or larger than the support value is put in the large 3-itemsets L3. Since only {D1.P1.A.High, D1.P2.B.Low, Q.G. High}, {D1.P2.B.Low, D3.P1.E.Medium, D3.P2.F.Low}, {D1.P2.B.Low, D3.P1.E.Medium, Q.G. High} and {D1.P2.B.Low, D3.P2.F.Low, Q.G. High} have counts higher than its support value, they are put in the large 3-itemsets L3 and is shown in Table 21.14.

L3 is not null, the previous steps are repeated. No candidate 4itemset, C4, is generated and L4 is null. Step 12: Since

Step 13: The association rules for each large q-itemsets, q> = 2, are constructed by forming all possible association rules and by calculating the confidence values

532

G.T.S. Ho et al.

Table 21.14 Items in large 3-itemsets

L3

Process items

Count

D1.P1.A.High, D1.P2.B.Low, Q.G. High

2.61

D1.P2.B.Low, D3.P1.E.Medium, D3.P2.F.Low

2.80

D1.P2.B.Low, D3.P1.E.Medium, Q.G. High

2.78

D1.P2.B.Low, D3.P2.F.Low, Q.G. High

2.51

of all association rules. Take IF {D1.P1.A.High, D1.P2.B.Low}, THEN {Q.G. High} as an example. The confidence value of this rule is calculated as

( D1 .P1 . A.High ∩ D1 .P2 .B.Low ∩ Q.G.High) 2 . 61 = = 1 . 000 2 . 61 ( D1 .P1 . A.High ∩ DD1 .P2 .B.Low) Results for all association rules are shown below in Table 21.15 Table 21.15 All possible association rules and their corresponding confidence values Association rules

Confidence Value

If {D1.P1.A.High and D1.P2.B.Low}, then {Q.G. High}

2.61/2.61=

1.000

If {D1.P1.A.High and Q.G. High}, then { D1.P2.B.Low}

2.61/2.86=

0.914

If {D1.P2.B.Low and Q.G. High}, then { D1.P1.A.High}

2.61/3.07=

0.851

If {D1.P1.A.High }, then { D1.P2.B.Low and Q.G. High}

2.61/3.11=

0.839

If {D1.P2.B.Low }, then { D1.P1.A.High and Q.G. High}

2.61/3.5=

0.746

If {Q.G. High}, then { D1.P1.A.High and D1.P2.B.Low}

2.61/4.32=

0.604

If {D1.P2.B.Low and D3.P1.E.Medium}, then { D3.P2.F.Low}

2.80/3.16=

0.887

If {D1.P2.B.Low and D3.P2.F.Low}, then { D3.P1.E.Medium}

2.80/2.84=

0.987

If {D3.P1.E.Medium and D3.P2.F.Low}, then {D1.P2.B.Low}

2.80/2.91=

0.963

If {D1.P2.B.Low}, then {D3.P1.E.Medium and D3.P2.F.Low}

2.80/3.5=

0.800

If {D3.P1.E.Medium}, then {D1.P2.B.Low and D3.P2.F.Low}

2.80/4.01=

0.698

If {D3.P2.F.Low}, then {D1.P2.B.Low and D3.P1.E.Medium}

2.80/3.19=

0.878

If {D1.P2.B.Low and D3.P1.E.Medium}, then {Q.G. High}

2.78/3.16=

0.880

If {D1.P2.B.Low and Q.G. High}, then { D3.P1.E.Medium}

2.78/3.07=

0.906

If {D3.P1.E.Medium and Q.G. High}, then { D1.P2.B.Low }

2.78/3.18=

0.873

If {D1.P2.B.Low }, then {D3.P1.E.Medium and Q.G. High}

2.78/3.5=

0.794

If {D3.P1.E.Medium}, then { D1.P2.B.Low and Q.G. High}

2.78/4.01=

0.693

If {Q.G. High}, then {D1.P2.B.Low and D3.P1.E.Medium}

2.78/4.32=

0.643

If {D1.P2.B.Low and D3.P2.F.Low}, then {Q.G. High}

2.51/2.84=

0.886

If {D1.P2.B.Low and Q.G. High}, then { D3.P2.F.Low}

2.51/3.07=

0.819

If {D3.P2.F.Low and Q.G. High}, then { D1.P2.B.Low}

2.51/2.51=

1.000

If {D1.P2.B.Low}, then {D3.P2.F.Low and Q.G. High}

2.51/3.5=

0.718

Fuzzy Measurement in Quality Management Systems

533

Table 21.15 (continued) If {D3.P2.F.Low}, then {D1.P2.B.Low and Q.G. High}

2.51/3.19=

0.788

If {Q.G. High}, then {D1.P2.B.Low and D3.P2.F.Low}

2.51/4.32=

0.582

If {D1.P1.A.High}, then {D1.P2.B.Low}

2.61/3.11=

0.839

If {D1.P2.B.Low}, then {D1.P1.A.High }

2.61/3.5=

0.746

If {D1.P1.A.High}, then {Q.G. High}

2.86/3.11=

0.918

If {Q.G. High}, then {D1.P1.A.High }

2.86/4.32=

0.661

If {D1.P2.B.Low}, then {D3.P1.E.Medium}

3.16/3.5=

0.902

If {D3.P1.E.Medium}, then {D1.P2.B.Low}

3.16/4.01=

0.788

If {D1.P2.B.Low}, then {D3.P2.F.Low}

2.84/3.5=

0.811

If {D3.P2.F.Low}, then {D1.P2.B.Low}

2.84/3.19=

0.890

If {D1.P2.B.Low}, then {Q.G. High}

3.07/3.5=

0.876

If {Q.G. High}, then {D1.P2.B.Low}

3.07/4.32=

0.710

If {D2.P2.D.High}, then {D3.P1.E.Medium}

2.95/3.37=

0.876

If {D3.P1.E.Medium}, then {D2.P2.D.High}

2.95/4.01=

0.736

If {D3.P1.E.Medium}, then {D3.P2.F.Low}

2.91/4.01=

0.726

If {D3.P2.F.Low}, then {D3.P1.E.Medium}

2.91/3.19=

0.912

If {D3.P1.E.Medium}, then {Q.G. High}

3.18/4.01=

0.794

If {Q.G. High}, then {D3.P1.E.Medium}

3.18/4.32=

0.737

If {D3.P2.F.Low}, then {Q.G. High}

2.51/3.19=

0.788

If {Q.G. High}, then { D3.P2.F.Low}

2.51/4.32=

0.582

Step 14: Compare the confidence values of the rules with the predefined confidence threshold which is 0.9 in this case. Keep the rules with confidence values equal to or larger than the predefined confidence threshold (Table 21.16). Step 15: Keep the rules with finished quality variables in the consequent part only (Table 21.17). Step 16: Output the rules to users as interesting rules. Table 21.16 The association rules with confidence values >= 0.9 Association rules

Confidence Value

If {D1.P1.A.High and D1.P2.B.Low}, then {Q.G. High}

2.61/2.61=

1.000

If {D1.P1.A.High and Q.G. High}, then { D1.P2.B.Low}

2.61/2.86=

0.914

If {D1.P2.B.Low and D3.P2.F.Low}, then { D3.P1.E.Medium}

2.80/2.84=

0.987

If {D3.P1.E.Medium and D3.P2.F.Low}, then {D1.P2.B.Low}

2.80/2.91=

0.963

If {D1.P2.B.Low and Q.G. High}, then { D3.P1.E.Medium}

2.78/3.07=

0.906

If {D3.P2.F.Low and Q.G. High}, then { D1.P2.B.Low}

2.51/2.51=

1.000

If {D1.P1.A.High}, then {Q.G. High}

2.86/3.11=

0.918

If {D1.P2.B.Low}, then {D3.P1.E.Medium}

3.16/3.5=

0.902

If {D3.P2.F.Low}, then {D3.P1.E.Medium}

2.91/3.19=

0.912

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Table 21.17 The association rules satisfy the relevant constraints Association rules

Confidence Value

If {D1.P1.A.High and D1.P2.B.Low}, then {Q.G. High}

2.61/2.61=

1.000

If {D1.P1.A.High}, then {Q.G. High}

2.86/3.11=

0.918

The extracted fuzzy association rules contain high quality and actionable information but they should not be used directly for prediction without further analysis or domain knowledge (Han & Kamber, 2001). However, it is a helpful starting point for further exploration to produce quality rule sets. The patterns found from the rules can be interpreted and explained without spending too much effort. From the results obtained in step 15, two fuzzy association rules are generated to identify two different settings of process parameters which have a significant impact on the thickness of the finished product. In this example, it has been found that if a high setting is frequently used for the pressure of deposition and a low setting is frequently used for the temperature setting of the machine, then this will lead to a high level of the hardness of the product.

4 Conclusions and Suggestions for Further Research In recent years, attention has been focused on intelligent systems which have shown great promise in supporting TQM. However, only a small number of the currently used systems are reported to be operating satisfactorily because they are designed to maintain a quality level rather than to improve quality continuously. This research provided a generic methodology for the development of an intelligent quality management system with knowledge discovery and with the cooperative ability for monitoring the process effectively, efficiently and agilely. It will help to achieve dramatic improvements in critical contemporary measures, such as quality, cost, time of delivery, and utilization. Generally, process engineers need to assemble a group of experts to identify and evaluate improvement opportunities.

Acknowledgments The authors wish to thank the Research Committee of the Hong Kong Polytechnic University for their support of this project.

References Ament, C., Goch, G.: A learning fuzzy control approach to improve manufacturing quality. IEEE, 156–161 (September 1999) Chen, W.C., Tai, P.H., Wang, M.W., Deng, W.J., Chen, C.T.: A neural network-based approach for dynamic quality prediction in a plastic injection molding process. Expert Systems with Applications 35, 843–849 (2008)

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Chen, W.C., Tseng, S.S., Wang, C.Y.: A novel manufacturing defect detection method using association rule mining techniques. Expert Systems with Applications 29, 807–815 (2005) Dhafr, N., Ahmad, M., Burgess, B., Canagassababady, S.: Improvement of quality performance in manufacturing organizations by minimization of production defects. Robotics and Computer-Integrated Manufacturing 22, 536–542 (2006) Dong, H., Yang, S.Y., Wu, D.H.: Intelligent prediction method for small-batch producing quality based on fuzzy least square SVM. Systems Engineering - Theory & Practice 27(3), 98–104 (2007) Eastwood, M.A.: Implementing mass customization. Computers in Industry 30, 171–174 (1996) Ho, G.T.S., Lau, H.C.W., Lee, C.K.M., Ip, A.W.H., Pun, K.F.: An intelligent production workflow mining system for continual quality enhancement. International Journal of Advanced Manufacturing Technology 28, 792–809 (2006) Lau, H.C.W., Ho, G.T.S., Chu, K.F., Ho, W., Lee, C.K.M.: Development of an intelligent management system using fuzzy association rules. Expert Systems with Applications 36(2), 1801–1815 (2009) Han, J., Kamber, M.: Data mining: concepts and techniques. Morgan Kaufmann Publishers, America (2001) Hong, T.P., Lin, K.Y., Wang, S.L.: Fuzzy data mining for interesting generalized association rules. Journal of Fuzzy Sets and Systems 138, 225–269 (2003) Huang, Y.L., Lou, H.H., Gong, J.P., Edgar, T.F.: Fuzzy model predictive control. IEEE Transactions on Fuzzy Systems 8(6), 665–678 (2000) Hullermeier, E.: Fuzzy sets in machine learning and data mining. Applied Soft Computing, January 19 (2008) (in press, corrected proof) Kaynak, H.: The relationship between total quality management practices and their effects on firm performance. Journal of Operations Management 21, 405–435 (2003) Kleiner, B.M., Drury, C.G., Palepu, P.: A computer-based productivity and quality management system for cellular manufacturing. Computers & Industrial Engineering 34(1), 207–217 (1997) Li, E., Jia, L., Yu, J.: A genetic neural fuzzy system-based quality prediction model for injection process. Computers and Chemical Engineering 26, 1253–1263 (2002) Linderman, K., Schroeder, R.G., Zaheer, S., Liedtke, C., Choo, A.S.: Integrating quality management practices with knowledge creation process. Journal of Operations Management 22, 589–607 (2004) Liu, D.R., Ke, C.K.: Knowledge support for problem-solving in a production process: A hybrid of knowledge discovery and case-based reasoning. Expert Systems with Applications 33, 147–161 (2007) Lou, H.H., Huang, Y.L.: Hierarchical decision making for proactive quality control: system development for defect reduction in automotive coating operations. Engineering Applications of Artificial Intelligence 16, 237–250 (2003) Melin, P., Castillo, O.: An intelligent hybrid approach for industrial quality control combining neural networks, fuzzy logic and fractal theory. Information Sciences 177, 1543– 1557 (2007) Nahm, Y.E., Ishikawa, H.: A hybrid multi-agent system architecture for enterprise integration using computer networks. Robotics and Computer-Integrated Manufacturing 21, 217–234 (2005)

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Pacella, M., Semeraro, Q., Anglani, A.: Manufacturing quality control by means of a Fuzzy ART network trained on natural process data. Engineering Applications of Artificial Intelligence 17, 83–96 (2004) Paladini, E.P.: An expert system approach to quality control. Expert Systems with Applications 18, 133–151 (2000) Pine, B.J.: Mass customization: the new frontier in business competition. Mass Harvard Business School Press (1993) Qiao, C., Gou, B., Lu, C.: Research on key technology of quality management supporting digitized design and manufacturing. In: 9th International Conference on ComputerAided Industrial Design and Conceptual Design, CAID/CD 2008, pp. 1104–1107 (2008) Shahbaz, M., Srinivas, Harding, J.A., Turner, M.: Product design and manufacturing process improvement using association rules. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 220(2), 243–254 (2006) Sudkamp, T.: Examples, counterexamples, and measuring fuzzy associations. Fuzzy Sets and Systems 149(1), 57–71 (2005) Vemadat, F.B.: Enterprise modeling and integration: principles and applications. Chapman & Hall, Boca Raton (1996) Watanabe, T.: Mining fuzzy association rules of specified output field. In: 2004 IEEE International Conference on Systems, Man and Cybemetics, vol. 6, pp. 5754–5759 (2004) Wu, Y., Wu, A.: Taguchi method for robust design. American Society of Mechanical Engineers Press (2000) Zeng, H.F., Zhang, G.B., Huang, G.B., Wang, G.Q.: Process quality optimization model based on ARM and immune principle. In: International Symposium on Computational Intelligence and Design, vol. 1, pp. 389–393 (2008)

Chapter 22

Fuzzy Real Options Models for Closing/Not Closing a Production Plant Christer Carlsson, Markku Heikkil¨a, and Robert Full´er

Abstract. In traditional investment planning investment decisions are usually taken to be now-or-never, which the firm can either enter into right now or abandon forever. The decision on to close/not close a production plant has been understood to be a similar now-or-never decision for two reasons: (i) to close a plant is a hard decision and senior management can make it only when the facts are irrefutable; (ii) there is no future evaluation of what-if scenarios after the plant is closed. However, it is often possible to postpone, modify or split up a complex decision in strategic components, which can generate important learning effects and therefore essentially reduce uncertainty. If we close a plant we lose all alternative development paths which could be possible under changing conditions; on the other hand, senior management may have a difficult time with shareholders if they continue operating a production plant in conditions which cut into its profitability as their actions are evaluated and judged every quarter. In these cases we can utilize the idea of real options. The new rule, derived from option pricing theory, is that we should only close the plant now if the net present value of this action is high enough to compensate for giving up the value of the option to wait. Because the value of the option to wait vanishes right after we irreversibly decide to close the plant, this loss in value is actually the opportunity cost of our decision. In this work we will use fuzzy real option models for the problem of closing/not closing a production plant in the forest products industry sector. Christer Carlsson ˚ Akademi University, Joukahaisenkatu 3-5, FIN-20520 Abo ˚ IAMSR, Abo e-mail: [email protected] Markku Heikkil¨a ˚ Akademi University, Joukahaisenkatu 3-5, FIN-20520 Abo ˚ IAMSR, Abo e-mail: [email protected] Robert Full´er ˚ Akademi University, Joukahaisenkatu 3-5, FIN-20520 Abo ˚ IAMSR, Abo e-mail: [email protected] C. Kahraman & M. Yavuz (Eds.): Prod. Engr. & Manage., STUDFUZZ 252, pp. 537–560. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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1 The Probabilistic Model for Real Option Valuation Real options in option thinking are based on the same principals as financial options. In real options, the options involve ”real” assets as opposed to financial ones [1]. To have a ”real option” means to have the possibility for a certain period to either choose for or against something, without binding oneself up front. The value of a real option is computed by [9] ROV = S0 e−δ T N(d1 ) − Xe−rT N(d2 ) where d1 =

ln(S0 /X) + (r − δ + σ 2 /2)T √ , σ T

√ d2 = d1 − σ T ,

and where S0 is the present value of expected cash flows, N(d) denotes the probability that a random draw from a standard normal distribution will be less than d, X is the (nominal) value of fixed costs, r is the annualized continuously compounded rate on a safe asset, T is the time to maturity of option (in years), σ is the uncertainty of expected cash flows, and finally δ is the value lost over the duration of the option. Furthermore, the function N(d) gives the probability that a random draw from a standard normal distribution will be less than d, i.e. 1  d − x2 √ e 2 dx. N(d) = 2π −∞ Facing a deferrable decision, the main question that a company primarily needs to answer is the following: How long should we postpone the decision - up to T time periods - before (if at all) making it? From the idea of real option valuation we can develop the following natural decision rule for an optimal decision strategy [2]. Let us assume that we have a deferrable decision opportunity P of length L years with expected cash flows (cf0 , cf1 , . . . , cfL ), where cfi is the cash inflows that the plant is expected to generate at year i, i = 0, 1, . . . , L. Where the maximum deferral time is T , make the investment (exercise the option) at time t  , 0 ≤ t  ≤ T , for which the option, ROVt  , is positive and attends its maximum value, ROVt  = max ROVt = t=0,1,...,T

max {Vt e−δ t N(d1 ) − Xe−rt N(d2 )},

t=0,1,...,T

(22.1)

where Vt = PV(cf0 , . . . , cfL , βP ) − PV(cf0 , . . . , cft−1 , βP ) = PV(cft , . . . , cfL , βP ),

Fuzzy Real Options Models for Closing/Not Closing a Production Plant

539

is the present value of the aggregate cash flows generated by the decision, which we postpone t years before undertaking. Hence, L

t L cf j cf j cf j − cf0 − ∑ =∑ , j j j j=t (1 + βP) j=1 (1 + βP ) j=1 (1 + βP )

Vt = cf0 + ∑

where βP stands for the risk-adjusted discount rate of the decision (cf. [3] for details). Note 1. Obviously, in the case we obtain or learn some new information about the decision alternatives, their associated NPV table and the cash flows cfi which may change. Thus, we have to reapply this decision rule every time when new information arrives during the deferral period to see how the optimal decision strategy might change in light of the new information. Note 2. If we make the decision now without waiting, then V0 = PV(cf0 , . . . , cfL , βP ) =

L

cf j

∑ (1 + βP) j

j=0

and since we can formally write, lim d1 = lim d2 = +∞,

T →0

T →0

lim N(d1 ) = lim N(d2 ) = 1,

T →0

T →0

we obtain, ROV0 = V0 − X =

L

cf j

∑ (1 + βP) j − X,

j=0

That is, this decision rule also incorporates the net present valuation of the assumed cash flows Real options are used as strategic instruments, where the degrees of freedom of some actions are limited by the capabilities of the company. In particular, this is the case when the consequences of a decision are significant and will have an impact on the market and competitive positions of the company. In general, real options should preferably be viewed in a larger context of the company, where management does have the degree of freedom to modify (and even overrule) the pure stochastic real option evaluation of decisions and de-investment opportunities.

2 The Fuzzy Model for Real Option Valuation Let us now assume that the expected cash flows of the close/not close decision cannot be characterized with single numbers (which should be the case in serious decision making). With the help of possibility theory we can estimate the expected incoming cash flows at each year of the project by using a trapezoidal possibility distribution of the form

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cfi = (sLi , sRi , αi , βi ), i.e. the most possible values of the present value of expected cash flows lie in the interval [sLi , sRi ] (which is the core of the trapezoidal fuzzy number cfi ), and (sRi + βi ) is the upward potential and (sLi − αi ) is the downward potential for the present value of expected cash flows. In a similar manner one can estimate the expected costs by using a trapezoidal possibility distribution of the form X˜ = (xL , xR , α  , β  ), i.e. the most possible values of expected cost lie in the interval [xL , xR ] (which is the ˜ and (xR + β  ) is the upward potential and core of the trapezoidal fuzzy number X), L  (x − α ) is the downward potential for expected costs. Note 3. The possibility distribution of expected costs and the present value of expected cash flows could also be represented by nonlinear (e.g. Gaussian) membership functions. However, from a computational point of view it is easier to use linear membership functions and, more importantly, our experience shows that senior managers prefer trapezoidal fuzzy numbers to Gaussian ones when they estimate the uncertainties associated with future cash inflows and outflows. Let P be a deferrable decision opportunity with incoming cash flows and costs that are characterized by the trapezoidal possibility distributions given above. Furthermore, let us assume that the maximum deferral time of the decision is T , and the required rate of return on this project is βP . In these circumstances, we shall make the decision (exercise the real option) at time t, 0 < t < T , for which the value of the option, FROVt  is positive and reaches its maximum value [3]. That is, FROVt  =

max FROVt =

t=0,1,...,T

˜ −rt N(d2 (t))}, (22.2) max {V˜t e−δ t N(d1 (t)) − Xe

t=0,1,...,T

where, ˜ 0 , . . . , cf ˜ L , βP ) − PV(cf ˜ 0 , . . . , cf ˜ t−1 , βP ) Vt = PV(cf L ˜j cf ˜ t+1 , . . . , cf ˜ L , βP ) = ∑ = PV(cf j j=t (1 + βP ) and where, d1 (t) =

˜ + (r − δ + σ 2 /2)T ln(E(V˜t )/E(X)) √ , σ T

√ d2 (t) = d1 − σ T ,

˜ stands for the the and where, E denotes the possibilistic mean value operator, E(X) possibilistic mean value of expected costs and

σ :=

σ (V˜t ) E(V˜t )

Fuzzy Real Options Models for Closing/Not Closing a Production Plant

541

is the annualized possibilistic variance of the aggregate expected cash flows relative to its possi-bilistic mean (and therefore represented as a percentage value). However, to find a maximizing element from the set {FROV0 , FROV1 , . . . , FROVT }, is not an easy task because it involves ranking of trapezoidal fuzzy numbers. In our computerized implementation we have employed the following value function to order fuzzy real option values, FROVt = (ctL , ctR , αt , βt ), of trapezoidal form: v(FROVt ) =

ctL + ctR βt − αt + rA · , 2 6

where rA ≥ 0 denotes the degree of the investor’s risk aversion. If rA = 1 then the (risk neutral) manager compares trapezoidal fuzzy numbers by comparing their possibilistic expected values, i.e. he does not care about their downward and upward potentials. If rA > 1 then the manager is a risk-taker, and if rA < 1 then he is riskaverse.

3 The Binomial Model for Real Option Valuation For practical purposes and when working with senior management the binomial version of the real options model is easier to use and easier to explain in terms of the available data. For our case the basic binomial setting is presented as a setting of two lattices, the underlying asset lat-tice and the option valuation lattice; for adding insight we can also include a decision rule lattice. In Figure 22.1 the weights u and d describe the geometric movement (Brownian motion) of the cash flows V over time, q stands for a movement up and 1−q movement down, respectively. The value of the underlying asset develops in time according to probabilities attached to movements q and 1 − q, and weights u and d, as described in Figure 22.1. The input values for the lattice are approximated with the following set of formulae (see [7] for details), u = eσ d=e

√ Δt

(movement up)

√ −σ Δ t

1 1 α − 1/2 × σ q= + × 2 2 σ

(movement down) 2√

t

(probability of movement up)

The option valuation lattice is composed of the intrinsic values I of the latest time to invest retrieved as the maximum of present value and zero, the option values O generated as the maxi- mum of the intrinsic or option values of the next period (and their probabilities q and 1 − q) discounted, and the present value S − F of the period in question. This formulation describes two binomial lattices that capture the present values of movements up and down from the previous state of time PV and the incremental

542

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Asset value

q q

u 1-q

S 1-q

q

ud

d 1-q

0

u2

1

d2 2 Time

Fig. 22.1 Underlying asset lattice of two periods.

Fig. 22.2 Underlying asset lattice of two periods.

values I directly contributing to option value O. The relation of geometric movements up and down is captured by the ratio d = 1/u. The binomial model is a discrete time model and its accuracy improves as the number of time steps increases. Summary 1. In summary, the benefit of using fuzzy numbers and the fuzzy real options model - both in the Black-Scholes and in the binomial version of the real options model - is that we can represent genuine uncertainty in the estimates of future costs and cash flows and take these factors into consideration when we make

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the decision to either close the plant now or to postpone the decision by t years (or some other reasonable unit of time). The simpler, classical representation does not adequately show the uncertainty.

4 Closing Production Plants - When and Where The forest industry, and especially the paper making companies, has experienced a radical change of market since the change of the millennium. Especially in Europe the stagnating growth in paper sales and the resulting overcapacity have led to decreasing paper prices, which have been hard to raise even to compensate for increasing costs. Other drivers to contribute to the misery of European paper producers have been steadily growing energy costs, growing costs of raw material and the Euro/USD exchange rate which is unfavourable for an industry which is invoicing its customers in USD and pays its costs in Euro. The result has been a number of restructuring measures, such as closedowns of individual paper machines and production units. Additionally, a number of macroeconomic and other trends have changed the competitive and productive environment of paper making. The current industrial logic of reacting to the cyclical demand and price dynamics with operational flexibility is losing edge because of shrinking profit margins. Simultaneously, new growth potential is found in the emerging markets of Asia, especially in China, which more and more attracts the capital invested in paper production. This imbalance between the current production capacity in Europe and the better expected return on capital invested in the emerging markets has set the paper makers in front of new challenges and uncertainties that are different from the ones found in the traditional paper company management paradigm. In a global business environment both challenges and uncertainties vary from market to market, and it is important to find new ways of managing them in the current dynamic business environment. In Figure 22.3 is shown the population and the paper consumption in different parts of the world in 2005 (the numbers are indicative for our purposes, the trends have developed for the worse by 2008); the problem for the paper producing companies is that the production capacity is concentrated in countries where the paper consumption/capita has reached saturation levels, not where the potential growth of consumption is highest.

4.1 Production, Exports, Jobs and Investments The Finnish Forest Industries Federation continuously updates material on the forest industry on its website (cf. www.forestindustries.fi); from this material we can find the following key observations (cf. Figure 22.4). In 2006, the gross value of the forest industry’s production in Finland was about 21 billion, a third of which was accounted for by the wood products industry and two thirds by the pulp and paper industries. In 2006, the forest sector employed a total of 60,000, some 30,000 of whom worked for the paper industry and about 30,000 for

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Fig. 22.3 Population and paper consumption/capita worldwide [source: Finnish Forest Industries Federation].

Fig. 22.4 Value of the production, exports and imports of the forest industry in 2006.

the wood products industry. Finnish Forest Industries Federation member companies employed 50,000 people. In addition to their domestic functions, Finnish forest industry companies employed about 70,000 people abroad. The total investments of Finnish forest industry came to some e 2.2 billion in 2006, e 1.4 billion of which were invested abroad. World production of paper and paper board totals some 370 million tons. Growth is the most rapid in Asia, thanks mainly to the quick expansion of industry in China. Asia already accounts for well over a third of total world paper and paperboard production. In North America, by contrast, production is contracting; a number of Canadian mills have had to shut down because of weak competitiveness. Per capita consumption of paper and paperboard varies significantly from country to country and regionally. On average, one person uses about 55 kilos of paper a year; the extremes are 300 kilos for each US resident and some seven kilos for each African. Only around 35 kilos of paper per person is consumed in the populous area of Asia. This means that Asian consumption will continue to grow strongly in the

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coming years if developments there follow the precedent of the West. In Finland, per capita consumption of paper and paperboard is 205 kilos. Rapid growth in Asian paper production in recent years has increased the region’s self-sufficiency, narrowing the export opportunities available to both Europeans and Americans. Additionally, Asian paper has started to enter Western markets - from China in particular. Global competition has intensified noticeably as the new entrants’ cost level is significantly lower than in competing Western countries. The European industry has been dismantling overcapacity by shutting down unprofitable mills. In total, over five percent of the production volume in Europe has been closed down in the last couple of years. Globally speaking, the products of the forest industry are primarily consumed in their production country, so it can be considered a domestic-market industry. Globally speaking, the profitability of forest industry companies has been weak in recent years. Overcapacity has led to falling prices and this, coupled with rising production costs, is gnawing at the sector’s profitability. The Finnish forest industry has earlier enjoyed a productivity lead over its competitors. The lead is primarily based on a high rate of investment and the application of the most advanced technologies. Investments and growth are now curtailed by the long distance separating Finland from the large, growing markets as well as the availability and price of raw materials. Additionally, the competitiveness of Finnish companies has suffered because costs here have risen at a faster rate than in competing countries. Finnish energy policy has a major impact on the competitiveness of the forest industry. The availability and price of energy, emissions trading and whether wood raw material is produced for manufacturing or energy use will affect the future success of the forest industry. If sufficient energy is available, basic industry can invest in Finland. In decisions on how to use existing resources the challenges of changing markets become a reality when senior management has to decide how to allocate capital to production, logistics and marketing networks, and has to worry about the return on capital employed. The networks are interdependent as the demand for and the prices of fine paper products are defined by the efficiency of the customer production processes and how well suited they are to market demand; the production should be cost effective and adaptive to cyclic (and sometimes random) changes in market demand; the logistics and marketing networks should be able to react in a timely fashion to market fluctuations and to offer some buffers for the production processes. Closing or not closing a production plant is often regarded as an isolated decision, without working out the possibilities and requirements of the interdependent networks. Profitability analysis has usually had an important role as the threshold phase and the key process when a decision should be made on closing or not closing a production plant. Economic feasibility is of course an important consideration but - as pointed out - more issues are at stake. There is also the question of what kind of profitability analysis should be used and what results we can get by using different methods. Senior management worries - and should worry - about making the best possible decisions on the close/not close situations as their decisions will

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be scrutinized and questioned regardless of what that decision is going to be. The shareholders will react negatively if they find out that share value will decrease (closing a profitable plant, closing a plant which may turn profitable, or not closing a plant which is not profitable, or which may turn unprofitable) and the trade unions, local and regional politicians, the press etc. will always react negatively to a decision to close a plant almost regardless of the reasons. The idea of optimality of decisions comes from normative decision theory. The decisions made at various levels of uncertainty can be modelled so that the ranking of various alternatives can be readily achieved, either with certainty or with well-understood and non-conflicting measures of uncertainty. However, the real life complexity, both in a static and dynamic sense, makes the op-timal decisions hard to find many times. What is often helpful is to relax the decision model from the optimality criteria and to use sufficiency criteria instead. Modern profitability plans are usually built with methods that originate in neoclassical finance theory. These models are by nature normative and may support decisions that in the long run may be proved to be optimal but may not be too helpful for real life decisions in a real industry setting as conditions tend to be not so well structured as shown in theory and - above all - they are not repetitive (a production plant is closed and this cannot be repeated under new conditions to get experimental data). In practice and in general terms, for profitability planning a good enough solution is many times both efficient, in the sense of smooth management processes, and effective, in the sense of finding the best way to act, as compared to theoretically optimal outcomes. Moreover, the availability of precise data for a theoretically adequate profitability analysis is often limited and subject to individual preferences and expert opinions. Especially, when cash flow estimates are worked out with one number and a risk-adjusted discount factor, various uncertain and dynamic features may be lost. The case for good enough solutions is made in fuzzy set theory (cf. [4, 5]): at some point there will be a trade-off between precision and relevance, in the sense that increased precision can be gained only through loss of relevance and increased relevance only through the loss of precision. In a practical sense, many theoretically optimal profitability models are restricted to a set of assumptions that hinder their practical application in many real world situations. Let us consider the traditional Net Present Value (NPV) model - the assumption is that both the microeconomic productivity measures (cash flows) and the macroeconomic financial factors (discount factors) can be readily estimated several years ahead, and that the outcome of the project, such as a paper machine with an expected economic lifetime of 20-25 years, is tradable in the market of produc-tion assets without friction. In other words, the model has features that are unrealistic in a real world situation. The idea of the NPV is based on a fixed coupon bond that generates a fixed stream of cash flows during a pre-defined lifetime. For real investments with long economic life-times that are subject to intense competition, technological deterioration and radically changing context factors (currency exchange rates, energy costs, raw material costs, etc.) the NPV gives rather a simplistic picture of real life profitability. In reality, the decision makers have to face a complex set of

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interdependencies that change dynamically and are uncertain, and uncertain in their uncertainty. Having now set the scene, the problem we will address is the decision to close - or not to close - a production plant in the forest products industry sector. The plant we will use as a context is producing fine paper products, it is rather aged, the paper machines were built a while ago, the raw material is not available close by, energy costs are reasonable but are increasing in the near future, key markets are close by and other markets (with better sales prices) will require improvements in the logistics network. The intuitive conclusion is, of course, that we have a sunset case and senior management should make a simple, macho decision and close the plant. On the other hand we have the trade unions, which are strong, and we have pension funds commitments until 2013 which are very strict, and we have long-term energy contracts which are expensive to get out of. Finally, by closing the plant we will invite competitors to fight us in markets we have served for more than 50 years and which we cannot serve from other plants at any reasonable cost. We have shown in Subsection 4.1 that real options models will support decision making in which senior managers search for the best way to act and the best time to act. The key elements of the closing/not closing decision may be known only partially and/or only in imprecise terms, which are why we show that meaningful support, can be given with a fuzzy real options model. Following Heikkil¨a and Carlsson [8] the real world case will be introduced in Subsection 4.2 where we show the dilemma(s) senior management had to deal with and the (low) level of precision in the data to be used for making a decision. In Subsection 4.3 we will show the models we worked with and the results we were able to get with fuzzy real options models. Subsection 4.7, finally, summarizes some discussion points and offers some conclusions.

4.2 The Production Plant and Future Scenarios The production plant we are going to describe is a real case, the numbers we show are realistic (but modified for reasons of confidentiality) and the decision process is as close to the real process as we can make it. We worked the case with the fuzzy real options model in order to help senior management decide if the plant should (i) be closed as soon as possible, (ii) not closed, or (iii) closed at some later point of time (and then at what point of time). The background for the decisions can be found in the following general development of the profitability of the Finnish forest products companies (cf. Figure 22.5, the Finnish Forest Industries Confederation). The main reasons for the unsatisfactory development of the profitability are: (i) fine paper prices have been going down for six years, (ii) costs are going up (raw material, energy, chemicals), (iii) demand is growing slowly, (iv) production capacity cannot be used optimally, and (v) the e /USD exchange rate is unfavourable (sales invoiced in USD, costs paid in e ). The standard solution is to try to close the old, small and least cost-effective production plants.

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Fig. 22.5 Profit before taxes and ROCE, Finnish forest products companies.

Fig. 22.6 Production plant scenarios.

The analysis carried out for the production plant started from a comparison of the present production and production lines with four new production scenarios with different production line setups. In the analysis each production scenario is analyzed with respect to one sales scenario assuming a match between performed sales analysis and consequent resource allocation on production. Since there is considerable uncertainty involved in both sales quantities and sales prices the resource allocation decision is contingent to a number of production options that the man-agement has to consider, but which we have simplified here in order to get to the core of the case. There were a number of conditions which were more or less predefined. The first one was that no capital could/should be invested as the plant was regarded as a sunset plant. The second condition was that we should in fact consider five scenarios: the current production setup with only maintenance of current resources and four options to switch to setups that save costs and have an effect on production capacity used. The third condition is that the plant together with another unit has to carry considerable administrative costs of the sales organization in the country. The fourth condition is that there is a pension scheme that needs to be financed until 2013. The fifth condition is the power contract of the unit which is running until

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Fig. 22.7 Committing now vs. having options.

2013. These specific conditions have consequences on the cost structure and the risks that various scenarios involve. Each scenario assumes a match between sales and production, which is a simplification; in reality there are significant, stochastic variations in sales which cannot be matched by the production. Since no capital investment is assumed there will be no costs in switching between the scenarios (which is another simplification). The possibilities to switch in the future were worked out as (real) options for senior management; the opportunity to switch to another scenario is a call option. The option values are based on the estimates of future cash flows, which are the basis for the upward/downward potentials. In discussions with senior management they (reluctantly) adopted the view that options can exist and that there is a not-to-decide-today possibility for the close/not close decision. The motives to include options into the decision process were reasoned through with the following logic: • New information changes the decision situation (Good or Bad News in Figure 22.7) • Consequently, new information has a value and it increases the flexibility of the management decisions • The value of the new information can be analyzed to enable the management to make better informed decisions In the discussions we were able to show that companies fail to invest in valuable projects because the options embedded in a project are overlooked and left out of the profitability analysis. The real options approach shows the importance of timing as the real option value is the opportunity cost of the decision to wait in contrast with the decision to act immediately. We also worked out the use of decision trees as a way to work with the binomial form of the real options model (cf. Figure 22.7). We were then able to give the following practical description of how the option value is formed:

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Option value = Discounted cash flow ∗ Value of uncertainty (usually standard deviation) - Investment ∗ Risk free interest If we compare this sketch of the actual work with the decision to close/not close the production plant with the theoretical models we introduced in Subsection 4.1, we cannot avoid the conclusion that things are much simplified. There are two reasons for this: (i) the data available is scarce and imprecise as the scenarios are more or less ad hoc constructs; (ii) senior management will distrust results of an analysis they cannot evaluate and verify with numbers they recognize or can verify as ”about righ”.

4.3 Closing/Not Closing a Plant: A Case Study The capital investment in a new paper machine the type of project normally analysed with the real options models - is a project of several hundred million Euros. As a capital investment such a project is a long-time venture of 10-15 years of operational lifetime, which means that the productivity and profitability of the machine should be worked out over this period of time. Produc-tivity is largely defined by the technological deterioration rate. Generally, the longer the plant stays in the technological race for productivity, the longer it is able to compete profitably. The conventional wisdom in the paper making industry is to build a paper machine with the most advanced features of technology development, so that high profits can be retrieved during the early years to pay back the capital invested. The story is a bit different when we are nearing the end of the economic life time of a paper machine. Closing a paper mill is usually understood as a decision at the end of the operational life-time of the real asset. In the aging unit considered here the two paper machines were producing three paper qualities with different price and quality characteristics. The newer Machine 2 had a production capacity of 150 000 tons of paper per year; the older Machine 1 produced about 50 000 tons. The three products were, • Product 1, an old product with declining, shrinking prices, • Product 2, a product at the middle-cycle of its lifetime, • Product 3, a new innovative product with large valued added potential, As background information a scenario analysis had been made with market and price forecasts, competitor analyses and the assessment of paper machine efficiency. Our analysis was based on the assumptions of this analysis with five alternative scenarios to be used as a basis for the profitability analysis (cf. Figure 22.5). After a preliminary screening (a simplifying operation to save time) two of the scenarios, one requiring sales growth and another with unchanged sales volume were chosen for a closer profitability assessment. The first one, Scenario 1 (sales volume 200,000 tons) included two sub options, first 1A with the current production setup and 1B with a product specialization for the two paper machines. The 1B would offer possibilities for a close-down of a paper coating unit, which will result in savings of over e 700,000.

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Note 4. We have chosen Scenario 1A for analysis and for illustrations. Scenario 2 starts from an assumption of a smaller sales volume (150,000 tons) and allows a closedown of the smaller Machine 1, with savings of over e 3.5 M. In addition to operational costs a number of additional cost items needed to be considered by the management. There is a pension scheme agreement which would cause extra costs for the com-pany if Machine 1 is closed down. Additionally, the long term energy contracts would cause extra cost if the company wants to close them before the end term. The scenarios are summarised here as production and product setup options, and are modelled as options to switch a production setup. They differ from typical options - such as options to expand or postpone - in that they do not include major capital commitments; they differ from the option to abandon as the opportunity cost is not calculated to the abandonment, but to the continuation of the current operations.

4.4 Binomial Analysis Cash flow estimates for the binomial analysis were estimated for each of the scenarios from the sales scenarios of the three products and by considering changes in the fixed costs caused by the production scenarios. Each of the products had their own price forecast that was utilised as a trend factor. For the estimation of the cash flow volatility there were two alternative methods of analysis. Starting from the volatility of sales price estimates one can retrieve the volatility of cash flow estimates by simulation (the Monte Carlo method) or by applying expert opinions directly to the added value estimates. In order to illustrate the latter method the volatility is here calcu-lated from added value estimates (AVE) (with fuzzy estimates: a: AVE ∗ -10%, b: AVE ∗ 10%, α : AVE ∗ 10%, β : AVE ∗ 10%) (cf. Fig. 22.8). The annual cash flows in the option valuation were calculated as the cash flow of postponing the switch of production subtracted with the cash flows of switching now. The resulting cash flow statement of switching immediately is shown below (Figure 22.9). The cash flows were transformed from nominal to risk-adjusted in order to allow risk-neutral valuation. The switch immediately to Scenario 1A seems to be profitable. In the following option value calculation the binomial process results are applied in the row ”EBDIT, from binomial EBDIT lat-tice”. T he calculation shows that when given volatilities are applied to all the products and the retrieved Added Value lattices are applied to EBDIT, the resulting EBDIT lattice returns cash flow estimates for the option to switch, adding 24 million of managerial flexibility (cf. Fig. 22.10). The binomial process is applied to the Added Value Estimates (AVEs). The binomial process up and down parameters, u and d, are retrieved from the volatility (σ ) and time increment (dt). The binomial process is illustrated in Figure 22.11 - Figure 22.13.

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4.5 Fuzzy Interval Analysis The fuzzy interval analysis allows management to make scenario based estimates of upward potential and downward risk separately. The volatility of cash flows is defined from a possibility distribution and can readily be manipulated if the potential and risk profiles of the project change. Assuming that the volatilities of the

Fig. 22.8 Added value estimates, trapezoidal fuzzy interval estimates and retrieved volatilities (STDEV).

Fig. 22.9 Incremental cash flows and NPV with no delay in the switch to Scenario 1A.

Fig. 22.10 Incremental cash flows, the NPV and Option value assessment when the switch to Scenario 1A is delayed by 1 year.

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Fig. 22.11 Binomial value added process, and following steps.

Fig. 22.12 Binomial process, final node assessment.

three product-wise AVEs were different from the ones presented in Figure 22.8 to reflect a higher potential of Product 3 and a lower potential of Product 1, the following volatilities could be retrieved (Figure 22.14). Note that the expected value with products 1 and 3 now differs from the AVEs.

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Binomial process, example EBDIT 0 1 2 3 4

0 0 0 0 0 0

0 1 2 3 4

For NPV calculation with options cash flow of 1948444 is used instead of 1535949

1 0 0 0 0 0

2 1535949 811524 164086 -414548 -931692

3 0 2540158 1721123 989150 334984

4 0 0 4087253 3119647 2254411

5 0 0 0 4700974 3670258

6

7

8

9

0 0 0 5289020

0 0 0

0 0

0

1,948,444 1,035,607 2,345,683 346,506 1,335,483 2,785,291 0 497,294 1,700,238 3,257,473 0 0 713,701 2,129,549 3,748,311

497294 0

1700238 713701

497294 = 0* (1-0.696781795) + 713701 * 0.696781795

Risk-adjusted probability: 0.696781795 = (erf - down)/(up-down)

Fig. 22.13 Binomial process, node-wise comparison.

Fig. 22.14 Fuzzy Added Value intervals and volatilites.

The fuzzy cash flow based profitability assessment allows a more profound analysis of the sources of a scenario value. In real option analysis such an asymmetric risk/potential assessment is realised by the fuzzy ROV (cf. Section 2). Added values can now be presented as fuzzy added value intervals instead of single (crisp) numbers. The intervals are then run through the whole cash flow table with fuzzy

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Fig. 22.15 Fuzzy interval assessment, applying interval assumptions to Added Value.

Fig. 22.16 Fuzzy interval assessment, discounting a fuzzy number.

Fig. 22.17 Fuzzy interval assessment, NPV and Fuzzy Real Option Value (FROV).

arithmetic operators. The fuzzy intervals described in this way are called trapezoidal fuzzy numbers (cf. Fig. 22.15). In case of the risk-neutral valuation the discount factor is a single number. In our analysis the discounting is done with the fuzzy EBDIT based cash flow estimates by discounting each component of the fuzzy number separately. The expected value (EV) and the standard deviation (St.Dev) are shown in Figure 22.16 (see also Section 2). As a result from the analysis a NPV calculation now supplies results of the NPV and fuzzy ROV as fuzzy numbers. Also flexibility is shown as a fuzzy number (Figure 22.17).

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Fig. 22.18 Comparing the results from fuzzy interval method and binomial process graphically, the option to Switch to Scenario 1A at 2006.

Fig. 22.19 Results comparison.

4.6 A Comparative Analysis For illustrative reasons this comparative analysis is made by applying a standard volatility (10.3%) for each product, scenario and option valuation method. Figure 22.19 summarizes the results from binomial process and cash flow interval analysis (the analogous analysis of the switch to Scenario 2 has not been shown). The analysis shows that there are viable alternatives to the ones that result in closing the paper mill and that there are several options for continuing with the current operations. The uncertainties in the Added Value processes, which we have modeled in two different ways, show significantly different results when, on the one

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Fig. 22.20 Results from the binomial option valuation.

hand, both risk and potential are aggregated to one sin-gle (crisp) number in the binomial process and, on the other hand, there is a fuzzy number that allows the treatment of the downside and the upside differently. In this case study management is faced with poor profitability and needs to assess alternative routes for the final stages of the plant with almost no real residual value. The specific costs of closedown (the pension scheme and the energy contracts) are a large opportunity cost for an immediate closedown (the actual cost is still confidential). The developed model allows for screening alternative paths of action as options. The binomial assessment, based on the assumptions of the real asset tradability, overestimates the real option value, and gives the management flexibilities that actually are not there. On the other hand, the fuzzy cash flow interval approach allows an interactive treatment of the uncertainties on the (annual) cash flow level and in that sense gives the management powerful decision support. With the close/not close decision, the fuzzy cash flow interval method offers both rigor and relevance as we

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Fig. 22.21 Results from the Fuzzy Real Option Valuation.

get a normative profitability analysis with readily available uncertainty and sensitivity assessments. Here we showed one scenario analysis in detail and sketched a comparison with a second analysis. For the real case we worked out all scenario alternatives and found out that it makes sense to postpone closing the paper mill with several years. Note 5. The paper mill was closed on January 31, 2007 at a significant cost.

4.7 Discussion and Conclusions In decisions on how to use existing resources the challenges of changing markets become a reality when senior management has to decide how to allocate capital to production, logistics and marketing networks, and has to worry about the return on capital employed. The networks are interdependent as the demand for and the prices of forest industry products are defined by the efficiency of the customer

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production processes and how well suited they are to market demand; the production should be cost effective and adaptive to cyclic (and sometimes random) changes in market demand; the logistics and marketing networks should be able to react in a timely fashion to market fluctuations and to offer some buffers for the production processes. Closing or not closing a production plant is often regarded as an isolated decision, without working out the possibilities and requirements of the interdependent networks. The problem we have addressed is the decision to close or not to close a production plant in the forest products industry sector. The plant was producing fine paper products, it was rather aged, the paper machines were built a few decades ago, the raw material is not available close by, energy costs are reasonable but are increasing i n the near future, key markets are close by and other markets (with better sales prices) will require improvements in the logistics network. The intuitive conclusion was, of course, that we have a sunset case and senior management should make a simple, macho decision and close the plant. On the other hand we have the trade unions, which are strong, and we have pension funds commitments until 2013 which are very strict, and we have long-term energy c ontracts which are expensive to get out of. Finally, by closing the plant we will invite competitors to fight us in markets we have served for more than 50 years and which we cannot serve from other plants at any reasonable cost. We showed that real options models will support decision making in which senior managers search for the best way to act and the best time to act. The key elements of the closing/not closing decision may be known only partially and/or only in imprecise terms; then meaningful support can be given with a fuzzy real options model. We found the benefit of using fuzzy numbers and the fuzzy real options model both in the Black-Scholes and in the binomial version of the real options model - to be that we can represent genuine uncertainty in the estimates of future costs and cash flows and use these factors when we make the decision to either close the plant now or to postpone the decision by t years (or some other reasonable unit of time). We used a real world case to show the dilemma(s) senior management had to deal with and the (low) level of precision in the data to be used for making a decision. We worked the case with fuzzy real options models and were able to find ways to work out the consequences of closing or not closing the plant. Then, management made their own decision.

References 1. Alleman, J., Noam, E. (eds.): The New Investment Theory of Real Options and Its Implication for Telecommunications Economics. Kluwer Academic Publishers, Boston (1999) 2. Benaroch, M., Kauffman, R.J.: Justifying electronic banking network expansion using real options analysis. MIS Quarterly 24, 197–225 (2000) 3. Carlsson, C., Full´er, R.: A fuzzy approach to real option valuation. Fuzzy Sets and Systems 139, 297–312 (2003)

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4. Carlsson, C., Full´er, R., Majlender, P.: On possibilistic correlation. Fuzzy Sets and Systems 155, 425–445 (2005) 5. Carlsson, C., Full´er, R., Majlender, P.: A fuzzy real options model for R&D project evaluation. In: Liu, Y., Chen, G., Ying, M. (eds.) Proceedings of the Eleventh IFSA World Congress, Beijing, China, July 28-31, pp. 1650–1654. Tsinghua University Press and Springer (2005) 6. Carlsson, C., Full´er, R., Heikkil¨a, M., Majlender, P.: A fuzzy approach to R&D project portfolio selection. International Journal of Approximate Reasoning 44, 93–105 (2007) 7. Cox, J.C., Ross, S.A., Rubinstein, M.: Option Pricing: A Simplified Approach. Journal of Financial Economics 7, 229–263 (1979) 8. Heikkil¨a, M., Carlsson, C.: A Fuzzy Real Options Model for (Not) Closing a Production Plant: An Application to Forest Industry in Finland. In: Proceedings of the 12th Annual International Conference on Real Options, Rio de Janeiro, July 9-12 (2008) 9. Leslie, K.J., Michaels, M.P.: The real power of real options. The McKinsey Quarterly 3, 5–22 (1997)

Index

Acceptance Sampling Plans 459 Addition of two fuzzy numbers 78 Advanced Manufacturing Systems 125 Agile Manufacturing 59 Agreement index 90 Analytic Hierarchy Process 339 Analytic Network Process 339 Analytical Hierarchy Process 363 Arithmetic Operations on Fuzzy Numbers 216 Assignment problem 204 Assignment Problem 206, 229 Benefit-Cost Analysis 288 Boundedness 102 Budgeting 299 Bullwhip effect 201 Bullwhip Effect 238, 241 Capacity Requirement Planning 40 Center of gravity 27 Center of gravity 79 completion time 80 Conservativeness 103 Consumer Market Survey 8 Control chart 433 CONWIP 59 Data Envelopment Analysis 425 Delphi Method 8 Demand forecasts 239 Demand Signal Processing 242 Design of Plant Layout 364 deterministic model 52 Discrete Fuzzy Distributions 466 division of two fuzzy numbers 79 Double Sampling Plans 461 Due Date Bargaining 85 Earliness 71 Economic Order Quantity 28 Economic order quantity 30 Economic Production Quantity 31

Economical Analysis 282 Enterprise Resource Planning 40 Estimation in Simple Linear Regression 8 Exponential Smoothing 11 Extended Enterprise 201 Facility location 329 Facility location problem 329 Factor analysis 407 Financial Analysis 283 Flexible Linear Programming 223 Forecasting by Regression 8 Forecasting techniques 3 Forecasting Techniques 7 Forecasting Time Series 11 Forecasting 1 Fuzziness in Network Optimization 220 Fuzziness in Productivity Measurement 422 Fuzziness of Investment 283 Fuzzy Acceptance Sampling Plans 457, 476 Fuzzy AHP 343 Fuzzy arithmetic 250 Fuzzy arithmetic 27, 252 Fuzzy Association Rules 519 Fuzzy B/C Ratio Analysis 288 Fuzzy back propagation network 125 Fuzzy Binomial Distribution 467 Fuzzy budgets control 324 Fuzzy Cognitive Maps 381, 383 Fuzzy control 363 Fuzzy Cost Function 259 Fuzzy Double Sampling 479 Fuzzy Due Dates 69 Fuzzy Estimation 506 Fuzzy Forecasting 4 Fuzzy Fraction of Defective Items 467 Fuzzy Information Axiom 349 Fuzzy Inventory Management 25 Fuzzy Investment Analysis 285 Fuzzy linear programming 44

562 Fuzzy linguistic variables 363 Fuzzy Material Requirement Planning 39 Fuzzy Measurement 515 Fuzzy measurement 517 Fuzzy Membership Function of Process Mean 496 Fuzzy Membership Function of Process Variance 497 Fuzzy modeling 269 Fuzzy Normal Distribution 493, 494 Fuzzy Numbers 212 Fuzzy Payback Period Analysis 292 Fuzzy Petri Nets 99 Fuzzy Poisson Distribution 473 Fuzzy Present Worth 285 Fuzzy Process Accuracy Index 511 Fuzzy Process Capability Analysis 483 Fuzzy Process Capability Indices 485, 498 Fuzzy Production 299, 321 Fuzzy programming models 55 Fuzzy ranking 284 Fuzzy Rate of Return Analysis 287 Fuzzy Robust Process Capability Indices 504 Fuzzy Sets 209, 210 Fuzzy simple linear regression 1 Fuzzy Simple Linear Regression 10 Fuzzy Simulation 249 Fuzzy simulation of production 268 Fuzzy Single Sampling 477 Fuzzy Statistical Process Control Techniques 432 Fuzzy Time Series 12 Fuzzy TOPSIS 346 Fuzzy unnatural pattern analysis 452 Fuzzy Version of the Standard Bullwhip Model 244

Index Intelligent Quality Management System 515 Interval analysis 252 Interval Comparison 260 Inventory 25 Inventory management 25 Inventory models 32 Investment analysis 279 Investment Planning 279, 281 Job Cycle Time 126 Just-in-time (JIT) manufacturing 26 Just-in-time manufacturing 59 Kanban Cards 62 Kanban 59 Lead Time 77 Lead time management 77 Lean Management 59 Lean production 59 Lean Production Systems 59 Linear Equality Systems 220 Linear programming 205, 223 Linear programming 42 Liveness 103 Location Selection 329 Location Selection Techniques 336 Logistic system 269 Logistic Systems 249

Human factors 401 Human Reliability Analysis 381, 393 Human reliability assessment 402

Manufacturing System, 95 Materials Flow, 359, 362 Materials requirement planning, 26 Mathematical Programming, 337 Maximisation, 44 Measure of Possibility, 219 Minimisation, 44 Minimum cost flow problem 204 Minimum Cost Flow Problems 207, 232 Modeling of Manufacturing Systems 105 Moving Averages 11 MRP II 40 MRP 40 Multi-criteria Methods 338 Multiple Sampling Plans 462 Multiplication of a fuzzy number 78 Multiplication of two fuzzy numbers 79

Idle time 92 Information axiom 349

Network Optimization 203, 225 Normative Measures 217

General budgeting 300 General minimum cost flow problem 204 Grey forecasting model 13 Grey Forecasting Techniques 1 Grey Theory 12

563

Index One-Card System 62 Operations Budgeting 299, 321 Operations budgets control 309 Operations on Fuzzy Sets 215 Optimization of distributor’s decisions 273 Optimization of Production 249 Optimization 259 Optimum values 54 Order batching 239 Order Fulfillment 80 Order quantity 29 Periodic Review Models 34 Petri Nets 95 Plant Layout 359 Possibility Distributions 209, 217 Price fluctuations 240 probability distribution 252 Process Accuracy Index 509 Process capability 483 Processing time 80 Productivity in production systems 419 Productivity Measurement 417 Productivity Optimization 425 Pull 60 Push 60 Quality Management Systems 515 Quick Response Manufacturing 59 Ranking Method 489 Reachability 102 Reversibility 103 Robust Process Capability Indices 502

Safety Factor 67 Sales Force Composite 8 Scheduling 71 Scheduling 89 Scheduling jobs 77 Seasonal Time Series 12 Self-organization map 125 Sequential Sampling Plans 462 Setup time 92 Simple Additive Weighting 339 Simplex method 206 Single Sampling Plans 460 Single-Period Models 32 Six-sigma Approach 508 Specification Limits 488 Spoilage 204 Statistical Process Control 431 Stochastic Petri Nets 98 Stochastic PNs 107 Subtraction of two fuzzy numbers 79 Sup-min extension principle 216 Supply Chain Management 201, 202 Tardiness 71 Technical Analysis 282 Throughput Time 80 TOPSIS Method 341 Toyota 59 Transportation problem 204 Transportation Problem 205, 226 Two-Card System 64 Unnatural patterns 45