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Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Scheduling Problems and Solutions, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,

Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Scheduling Problems and Solutions, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,

COMPUTER SCIENCE, TECHNOLOGY AND APPLICATION

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SCHEDULING PROBLEMS AND SOLUTIONS

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COMPUTER SCIENCE, TECHNOLOGY AND APPLICATION

SCHEDULING PROBLEMS AND SOLUTIONS

HUSSEIN M. KHODR

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EDITOR

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Copyright © 2012 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‘ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Scheduling problems and solutions / editor, Hussein M. Khodr. p. cm. Includes index. ISBN:  (eBook) 1. Production scheduling. I. Khodr, Hussein M. TS157.5.S35 2011 658.5'3--dc23 2011025573

Published by Nova Science Publishers, Inc. † New York Scheduling Problems and Solutions, Nova Science Publishers, Incorporated, 2012. ProQuest Ebook Central,

CONTENTS Preface Chapter 1

Chapter 2

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Chapter 3

vii Integration of Operation Planning and Scheduling in Supply Chain Systems: A Review Amalia Nikolopoulou and Marianthi G. Ierapetritou

1

Apply Heuristics and Meta-Heuristics to Large-Scale Process Batch Scheduling Yaohua He and Chi-Wai Hui

21

Power Generation and Demand Scheduling Based on Stochastic Programming M. Baghdadi and S. S. Mortazavi

85

Chapter 4

Economic Load Scheduling of Thermal Power Generating Units Kamal K. Mandal and Niladri Chakraborty

125

Chapter 5

Concepts and Methods for Scheduling Field Machinery Operations D. D. Bochtis

179

Chapter 6

Single Machine Scheduling Problems under Learning Effect and Deteriorating Jobs M. Duran Toksarı and Ertan Güner

Chapter 7

Scheduling Problems and Solutions in Wimax Networks Jia-Ming Liang, You-Chiun Wang and Yu-Chee Tseng

Chapter 8

Scheduling at a Cross Dock Facility with Stochastic Truck Arrival Times Jeffery Karafa, Mihalis M. Golias, Stephanie Ivey, Georgios K. D. Saharidis and Nikolaos Leonardos

Chapter 9

Traffic Flow Scheduling Based on Local Regularity Prediction Flávio Henrique Teles Vieira and Scheila Guedes Garcez

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193 207

233

247

vi Chapter 10

Contents Load-Prediction Dynamic Scheduling and its Application to Biomedical Computer Simulations Wenfeng Shen, Daming Wei, Weimin Xu, Xin Zhu and Shizhong Yuan

Chapter 11

Scheduling Problems and Solution Approaches in Health Sector Şafak Kırış and Nihat Yüzügüllü

Chapter 12

Blood Transfusions on the Intensive Care Unit: The Role of Allogenous and Autologous Transfusions, and Pharmacological Agents Mukai Chimutengwende-Gordon and S. Wasim Khan

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Index

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267

283

311 321

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PREFACE In the last 50 years many papers and textbooks devoted to the scheduling problems and solutions of the production systems of many industries process have been written. In most part of them, the models and analysis techniques have been developed for the industries process of food, beverages, chemicals, and pharmaceuticals, brewing, communications sensors, electrical, mechanical and more. In recent years more attention has been devoted to the computer aided scheduling problems in industries process. Computer programs are now available so production engineer can develops a real feel for how a decision can be taken making the scheduling process easy to determine when, where and how to produce a set of products according to given requirements in a specific time period. Due to the discrete decisions involved, these scheduling problems have a combinatorial nature and therefore challenging from computational complexity standpoint. With the tools, scheduling studies can be run to simulate present industry conditions and to help with the long-range planning of new facilities for a given production plan. The tools also provide an opportunity for the production engineer to do such things as optimize the production process applying diverse mathematical optimization techniques. Optimization techniques are the systematized search for the best solution, the procedure used to make a scheduling problem as effective or functional as possible. In the last decades, optimization techniques have been extended in many industrial fields. The present book includes some of the most significant present and foreseen scheduling problems and their solution on the corresponding industries. This textbook consists of 11 Chapters that have been written by relevant academia and industry authors. Most of the chapters include algorithms, methods, know-how and examples of applications in real or academic industrial scheduling problem and its solution by the most adequate optimization technique. Researchers, engineers and students are also encouraged to write their own simple computer programs for many of the problems. A summary of the intent of each chapter follows. Chapter 1 introduces a literature review on the integration of operation, planning and scheduling in supply chain systems. It reports a mathematical programming and simulation approaches that consider the effects of uncertainty on solution optimality and feasibility. It gives a special reference to agent based simulation approaches taking into account the integration of activities, cooperation, coordination and information sharing throughout the

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Hussein M. Khodr

entire supply chain. The potential of integrated Supply Chain Management within a unified approach for future applications is discussed. Chapter 2 conducts an extensive review on the process scheduling, which includes the complexity of process scheduling and the available solution methods, and then the strategies to solve large-scale process scheduling problems are analyzed and summarized. In fact, it has dedicated quite some energy to the research on the solution of large-scale problems, focusing on heuristics and meta-heuristics. In this chapter, three types of process scheduling problems are solved using these methods. Chapter 3 elaborates the generation and demand scheduling problem in which both generation facilities and demand-side participations are to be scheduled. After a brief introduction of essential key concepts, the deregulation and electricity markets as well as the novel notion referred to as smart electric grids are reviewed. Next, the problem in question is formulated in the usual deterministic manner in which the power system random outages are either neglected or accommodated via ―N-1‖ security criterion. System loads are assumed to be constituted of various types which are gradually being more common with the advent of smart grids. Dispatchable, shiftable, and curtailable loads are considered in the modeling approach. Accordingly, the active contribution of demand-side entities is incorporated and the corresponding impacts on the system peak load, reserve requirement, operating cost, etc. are investigated. Chapter 4 addresses the problems and issues of on-line economic dispatch thermal power generating units. It uses the term ‗Economic Load Scheduling‘ in place of on-line economic scheduling to avoid confusions. The main objective of Economic Load Scheduling of electric power generating units is to schedule the committed generating units‘ outputs so as to meet the load demand at minimum operating cost while satisfying all unit and system equality and inequality constraints. Different classical, mathematical and more recently modern heuristic optimization techniques have been used to address the scheduling problems. This chapter also discusses the problems of economic load scheduling of thermal generating units and their solution techniques. Chapter 5 presents aspects related to two general types of scheduling problems that can be found in the bio-production systems domain. These types are the pure scheduling problems in the seasonal planning of field operations, and sequencing problems. For the sequencing scheduling case, the mixed integer programming formulation of the problem is presented and a study case is analyzed. Chapter 6 studies the scheduling problems which include a combination of linear/ nonlinear job deterioration and a position based learning effect. By the effects of learning and deterioration, the processing time of a job is defined by increasing function of its execution start time and position in the sequence. It considers objectives which are the single machine makespan, the total completion time, the sum of completion times (square) and the maximum lateness. For the single-machine case, it derives polynomial-time optimal solutions. Chapter 7 discusses the scheduling problems and their solutions under the three architectures of WiMAX networks, which covers the issues of how to improve network throughput, how to guarantee quality of service, and how to reduce energy consumption of devices. The comparison of these scheduling solutions is also given in the chapter. Chapter 8 proposes a game theoretic modeling framework based on the concept of Pareto optimality to schedule incoming trucks to the inbound doors of a freight distribution facility. It also attempts to address the issue of truck travel time variability from the place of origin to

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Preface

ix

the facility. The problem is formulated as a bi-objective mixed integer programming problem. A combination of an Evolutionary Algorithms-based heuristic and a simulation-based Pareto front algorithm is proposed to solve the resulting problem and produce an optimized truck-todoor assignment. In this chapter, numerical experiments indicate that the proposed truck scheduling approach provides improved schedules when compared to schedules where truck travel time variability is not considered. Chapter 9 introduces a new traffic flow scheduling scheme that makes use of the local regularity information of traffic processes. To this end, it firstly estimates the pointwise Hölder exponents in time windows based on the decay trends of wavelet coefficients. Next, it develops an adaptive algorithm for the prediction of pointwise Hölder exponents from the traffic flow data. The predicted Hölder exponents which measure the degree of local regularities of the traffic processes are intended to improve performance of the existing data flow scheduling strategies. Based on this traffic irregularity indicator, it proposes a new scheduling scheme for Internet data streams under the Generalized Processor Sharing scheduling discipline. The proposed data flow scheduling algorithm is extensively evaluated through simulations. Chapter 10 exposes various self-scheduling schemes such as pure self-scheduling (PSS), chunk self-scheduling, factoring self-scheduling, guided self-scheduling and trapezoid selfscheduling, which have successfully been used in shared memory multiprocessor systems. Of these, pure self-scheduling is the scheme that can achieve load-balancing in an extremely heterogeneous environment. However, pure self-scheduling cannot take full advantage of the General Purpose computation on Graphics Processing Units performance. In this chapter, load-prediction dynamic scheduling (LPDS) is introduced to solve this problem. To demonstrate the efficiency of LPDS for practical applications, LPDS was implemented to parallelize the computer simulation of electrocardiogram. Experimental results, presented in this chapter, show that LPDS is more efficient than PSS. Chapter 11 examines the need of scheduling structure of service sector. Health sector is one of the service sector analyzed in this chapter. The uncertain and varied structure of the system revealed the requirement of updates in scheduling process. In addition to this, human beings‘ lives can be affected with decisions on each step of the system. For this purpose firstly structure of the health care system and management problems are discussed. Then, the system design for health sector is investigated with an implementation for Emergency Departments. In this phase, assignment and scheduling problems are analyzed. Finally, solution approaches are proposed and tried to be generalized to other service units to raise customer/staff satisfaction. Chapter 12 - The decision to transfuse patients on the intensive care unit is made on an individual basis and should consider factors such as duration and severity of anaemia, symptoms, physiological parameters and comorbidities. Autologous blood transfusion has the benefit of avoiding some of the immunological and infective complications associated with allogenic blood transfusion. Pharmacological agents have a role in avoiding the need for blood transfusion.

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Chapter 1

INTEGRATION OF OPERATION PLANNING AND SCHEDULING IN SUPPLY CHAIN SYSTEMS: A REVIEW Amalia Nikolopoulou and Marianthi G. Ierapetritou Dept. of Chemical and Biochemical Engineering, Rutgers University, Piscataway, NJ, US

ABSTRACT

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In an increasingly competitive and complex business environment, Supply Chain Management (SCM) has emerged as a practice for improving the overall performance of companies. In recent years, strategies for SCM have been broadened to integrate operation planning and scheduling. Efficient implementation of such strategies for minimizing total cost and inventory, and increasing flexibility of processes, is the main challenge in SCM. Previous studies have shown that the lack of communication among different entities of the supply chain (plants, suppliers, warehouses, retailers), is a major obstacle in efficient decision making. Decisions on the operational and tactical level are often taken independently by the various players and thus, a global optimal solution cannot be guaranteed. Moreover, the influence of uncertainty inherent in production processes and demand is, traditionally, not taken into consideration. The integration of strategic level decisions with decisions on production planning and scheduling, is a promising approach for modeling the dynamic behavior of supply chains in order to meet such requirements. A literature review on the integration of operation planning and scheduling in supply chain systems is presented. We report mathematical programming and simulation approaches that consider the effects of uncertainty on solution optimality and feasibility. We give special reference to agent based simulation approaches taking into account the integration of activities, cooperation, coordination and information sharing throughout the entire supply chain. The potential of integrated SCM within a unified approach for future applications is discussed.



Author to whom correspondence should be addressed. Email: [email protected]; Tel: 732-445-2971; Fax: 732-445-2581.

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1. INTRODUCTION The coordination of activities along different stages of a supply chain has received much attention in production and operations management research. Supply chain is a network of facilities and business entities (i.e., plants, suppliers, warehouses, distributors and retailers), dispersed over a large geographic area and, in many cases, all over the globe. Typical functions in a supply chain involve procurement of raw materials, transformation of raw material into finished products, and distribution of finished products to customers. Various entities interact with each other in order to achieve high customer satisfaction level at lower cost and higher efficiency and productivity ([1], [2]). The motivation behind the supply chain integration is the growing broad based recognition that decisions for operation planning and scheduling are traditionally made in isolation, or dealt with on an individual basis rather than collectively. Moreover, the influence of uncertainty inherent in production processes and demand is, usually, not taken into consideration. Thus, there is a need for coordination and collaborative exchange of information within various levels of the supply chain. Several studies have shown that proper coordination of material and financial and information flow leads to improved efficiency and reduction of inventory levels ([3], [4]). The planning problems that have to be solved to achieve this coordination, cover a wide range of supply chain functions, from procurement and production to distribution and sales, and a wide range of time scales from short-term to long-term decisions. SCM strategies operate on three decision levels: strategic planning, tactical planning, and operational planning. Strategic (long term) planning determines the design and structure of the supply chain (i.e. facility location, plant or capital investments) and is typically a long range planning performed every few years when a supply chain needs to expand its activities. Tactical (medium term) planning is related to the optimization of flow of goods and services across the supply chain and involves decisions such as the assignment of production targets to facilities and the transportation from facilities to warehouses and to distribution centers. Typically, it is a medium range planning performed on a monthly basis. Operational (short term) planning is also referred to as scheduling and deals with the day-to-day production planning and inventory issues on the factory floor. Shapiro [5] explores major decision areas in SCM as well as the importance of integration of supply chain activities. A more recent work by Pinedo [6], examines decision making in integrated supply chains and discusses issues concerning uncertainty, robustness and reactive decision making. Due to interconnections among different levels of the supply chain, decisions across different time scales should be made simultaneously. In order to achieve that, Frayret et al. [7] suggest that entities must be able to: (i) exchange information promptly throughout the chain about products and services‘ availability and quality, production output, and demand, and (ii) react in a coordinated manner to correct any deviances or disturbances. The integration of strategic level decisions with decisions on production planning and scheduling, is a promising approach for modeling the dynamic behavior of supply chains. Many of the existing SCM approaches are based on optimization models, which often assume

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centralized management and are not suitable to efficiently undertake supply chain dynamics and uncertainty. On the contrary, simulation based approaches are able to deal with these two issues but they are not efficient to perform optimization of supply chain operations. Hybrid approaches proposed in the literature, offer the advantages of simulation based methods together with the optimization capabilities of mathematical programming models, for effective decision making. The objective of this paper is to review the literature on integrated planning and scheduling of operations and to identify areas where further research is needed. In section 2, challenges associated with effective decision making in SCM are discussed and the importance of integration is demonstrated. We provide an overview of traditional mathematical tools and simulation methods developed for supply chain integration. We then review the agent based modeling approach and present a number of alternative architectures existing in the literature. The main challenges in this area are outlined and promising research directions are explored. In section 3, we conclude with a summary and an overview of issues that have to be further investigated.

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2. INTEGRATION IN SUPPLY CHAIN DECISION-MAKING Challenges in effective decision-making involve the interoperability of dynamic components within a supply chain and the lack of cooperation among different decision makers. In a dynamic supply chain system, information is distributed across all levels and facilities. In addition to that, decision makers in a present day enterprise, may reside in different departments or even are spread in different places all over the world. The lack of communication is often a major bottleneck in efficient decision making and might be: (i) spatial, among different manufacturing plants, storage facilities and research and development centers, and (ii) temporal, among different decision making levels. As a result, decisions are optimized locally within the departments but do not assure a global optimum for the whole supply chain system. Numerous studies have emphasized the importance of information sharing within the supply chain ([8], [9], [10]). One of the main advantages of an integrated supply chain is the reduction of the bullwhip effect [11], where small changes on one level of the system may result in large fluctuations, large amount of stock, and increased lead times on other levels. Optimization in such a scenario, involves a variety of decision problems with different objective functions and time horizons, where decisions span the whole range of levels from the strategic to the operational level, passing through the tactical level. Most of the research work in this field, addresses SCM problems from either a strategic or a tactical point of view. As the need for integrating supply chain decision levels increases, the development of more sophisticated tools and decision information systems becomes a highly challenging problem.

2.1. Classification of Modeling Approaches Given the complex nature of supply chain systems it is difficult to establish a taxonomy of modeling approaches for integration of operation planning and scheduling. Several issues critical to SCM such as optimization, decision analysis, diagnostic evaluation, risk

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management, project planning as well as different areas of decision making [12], should be taken into consideration before moving to a systematic categorization. A review of the literature shows that many researchers support a primary categorization of modeling approaches as stochastic and deterministic. Stochastic models take into account random parameters and the inherent uncertainty within supply chains and can be further classified into optimal control and dynamic programming models [13]. On the contrary, deterministic models assume that all the model parameters are known. Considering this broad categorization, modeling approaches can be further classified into three specific types; process, statistical and mathematical models [14]. Mathematical modeling has been most commonly used to formulate operation planning and scheduling problems. The majority of relevant work in the literature ([15], [16], [17], [18], [19], [20]) distinguishes five primary categories: i) analytical, ii) simulation, iii) hybrid, iv) optimization models and v) Artificial Intelligence (AI). Analytical approaches find an optimal solution for problems which entails few decision variables and limiting assumptions to make it solvable. Conversely, simulation models do not aim to find an optimal solution, but to analyze a complex problem under different settings with relatively larger number of variables and parameters. An overview of modeling approaches for the integration of scheduling decisions and production planning decisions and the solution strategies is offered by Maravelias and Sung [21]. In addition, Kleijnen [22] gives a more specific overview of simulation based methods which are categorized into four approaches; i) spreadsheet simulation, ii) system dynamics, iii) discreteevent dynamic systems simulation, iv) and business games. A generic categorization scheme of modeling approaches for integration of operation planning and scheduling in supply chain systems as described above, is provided in Figure 1. Several attempts to classify different modeling approaches exist in the literature. According to Beamon [23], these methods can be grouped into four main categories: (i) deterministic models where all the parameters are known ([24], [25]), (ii) stochastic models, where at least one parameter is unknown but follows a probabilistic distribution ([26],[27]), (iii) economic models based on game-theory [28], and (iv) simulation models, which evaluate the performance of various supply chain strategies ([29], [30]). The work of Mula et al. [31], presents a classification of models for production planning under uncertainty, based on different approaches and distinguishes into conceptual, analytic, AI and simulation based methods. Sarmiento and Nagi [32] give an extensive analysis and categorization of integrated production and distribution planning problems. The authors attempt a categorization based on the type of decisions incorporated in the model, (i.e. production, distribution, inventory management) and on the number of locations per echelon. Thomas and Griffin [33] present a review of models addressing coordinated multi-stage supply chain planning and categorize them in two major categories; strategic and operational. As indicated in the thorough review of Sarimveis et al. [34], available tools for modeling supply chains may also vary from classical transfer function analysis to highly sophisticated control. Categorization of modeling approaches for integration of planning and scheduling according to several researchers is summarized in Table 1.

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Figure 1. Categorization of Modeling Approaches for the Integration of Planning and Scheduling Operations.

A large number of mathematical models use mixed integer programming (MIP) to solve the supply chain optimization problems. One of the first attempts for effective supply chain management is the work of Geoffrion and Graves [36], where an MIP model is formulated for the multicommodity location problem. The solution of the problem is based on Bender‘s Decomposition (BD) and involves the determination of Distribution Center (DC) locations, their capacities, customer zones and transportation flow patterns for all commodities. Timpe and Kallrath [37] describe a general mixed integer linear programming (MILP) model for managing a multi-plant production network with emphasis on chemical industries. The model combines aspects related to production, distribution and marketing. This work includes not only standard features of lot sizing problems (raw materials, production, inventories, demand) but also new conceptual aspects, such as how to define the capacity of a multi-site, multi-product production network, or how to approach complex planning problems. Arntzen et al. [38] develop an MIP model, called Global Supply Chain Model (GSCM), that can accommodate multiple products, facilities, stages (echelons), time periods, and transportation modes. Voudouris [39] develops a mathematical model designed to improve the schedule and enhance throughput in a supply chain. The model maximizes system flexibility and can be further utilized to anticipate the impact of introducing a new product line in the plant. Other mathematical formulations for operation management in multi-plant industrial networks are reported in the literature by Dondo et al. [40] and You and Grossmann [41]. The term Enterprise Wide Optimization (EWO) in multi-plant networks is used by Grossmann [42], in order to solve the combined production/distribution scheduling problem.

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Amalia Nikolopoulou and Marianthi G. Ierapetritou Table 1. Categorization of Modeling Approaches for Integration of Operation Planning and Scheduling appearing in the literature

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Categorization details Deterministic analytical, stochastic analytical, economic, simulation Optimization, meta-heuristic, IT-driven, hybrid Analytical, simulation Conceptual and non-quantitative, quantitative Conceptual, analytical, artificial intelligence (AI), simulation Analytical, mixed integer programming, system dynamics, agent-based modeling and simulation, discrete event simulation Analytical, simulation Deterministic, stochastic, hybrid, IT-driven Conceptual, analytic, artificial intelligence (AI) and simulation Stochastic programming, fuzzy mathematical programming, stochastic dynamic programming Strategic and operational

Authors Beamon [23] Bilgen and Ozkarahan [15] Chan and Chan [16] Ganeshan et al. [17] Giannocaro and Pontrandolfo [18]

Huang et al. [19] Hung et al. [35] Min and Zhou [13] Mula et al. [31]

Sahinidis [20] Thomas and Griffin [33]

Most of the optimization models are formulated with the single objective of minimizing the unit manufacturing cost of products. In some cases, performance measures (delivery reliability, quality, and responsiveness) are taken into account to complement cost analysis. Several other methods such as the formation of virtual enterprises to manage orders ([43], [44], [45]), project scheduling [46], extended trans-net [47], fuzzy set theory and fuzzy logics ([48], [49], [50]) are also considered for evaluating supply chain integration. Another major focus area of optimization models, is to determine the location of production, warehousing and sourcing facilities, and the paths the products take through them. Models in this area account mostly for strategic and strategic/tactical levels in order to achieve coordination of product and material flows between locations. Typical problems in this area are related to bringing products located at a central facility to geographically dispersed facilities at minimum cost. Kang and Kim [51] develop heuristic algorithms by simultaneously considering inventory and transportation decisions for a two-level supply chain. The objective in this problem is to determine the replenishment quantities and timing for the retailers as well as the amount of products delivered to the retailers by each vehicle for minimizing the total cost. Also, Bertazzi et al. [52] and Lee et al. [53] propose decomposition approaches to integrate inventory and transportation planning problems with deterministic demands. Çetinkaya et al. [54] present analytical results for the coordination of inventory and transportation decisions, considering stochastic demand. Karabuk [55], describes a complex transportation problem at a textile manufacturer and proposes a solution approach based on solving a series of mathematical programming models in interaction with schedulers, while supporting their workflow. Cohen and Lee [25], [25] consider global manufacturing and

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distribution networks and develop an MIP problem that supports resource deployment decisions. In order to overcome the limitations occurring when applying classical mathematical programming techniques to medium term production planning problems, AI was developed. AI is based on the theory of decision behavior in a fuzzy environment introduced by Bellman and Zadeh [56], where distinction is made between randomness and imprecision. Gen et al. [57] propose a fuzzy model with multiple objectives for aggregate planning, with objective function coefficients, technological coefficients, and resource right-hand side constraints represented by triangular fuzzy numbers. The limitations of applying classical mathematical programming techniques to medium-term production planning problems are reported by Wang and Fang [58]. The authors propose a fuzzy linear programming model for solving an aggregate planning problem with multiple objectives where the product price, subcontracting cost, manpower level, production capacity and market demand are considered fuzzy. Xing et al. [59] give an overview of recent publications related to the application of AI techniques to reverse supply chain with emphasis on certain types of product returns. The potential for addressing certain aspects of SCM with AI is also highlighted in this work. From the discussion above, it is evident that the complexity of the problems under consideration is a challenge to the Operations Research (OR) field, which are highly mathematical in nature. OR methods in supply chain planning and scheduling are usually algorithmic and apply the principles of linear programming, integer programming and dynamic programming as suggested by Simchi-Levi et al. [60]. Despite aforementioned limitations, remarkable advancements have been achieved in this field during the last decades. Studies by Chandra and Grabis [61], Dolgui and Proth [62], and Ivanov [63] provide a systematic summary of advanced OR based methods for SCM, especially for inventory management, tactical planning decisions, and supply contracts. Moreover, studies on modern Control Theory (CT) become of greater interest to researchers and practitioners as suggested by Lalwani et al. [64] and Hoberg et al. [65]. Although analytical techniques have been proven useful in many cases, they often rely on major simplifications to be of practical use for complex supply chain problems. Simulation based methodologies, inspired on system control concepts and on object-oriented architectures [35], have been proposed to address some of these challenges. Swisher et al. [66] describe key issues for applying parameter optimization within a simulation model and Fu [67] presents a survey on the available software solutions in this context. Bhaskaran and Leung [68] describe re-engineering of supply chains using queuing network models and simulation. Feigin et al. [69] look into enterprise modeling and simulation within an object oriented environment. Similar work has been done by Towill et al. [29] who use simulation techniques to evaluate the effects of various supply chain strategies on demand amplification. The objective of the simulation model is to determine which strategies are the most effective in smoothing the variations in the demand pattern. Wikner et al. [30] examine five supply chain improvement strategies, that they implement on a three-stage reference supply chain system. Recently, the powerful notion of evolutionary algorithms, among them Genetic Algorithms (GAs), has entered into the discussion of SCM. According to relevant work in the literature, this approach is mainly supported by the research in GAs combined with mathematical programming. Jeyanthi and Radhakrishnan [70] offer a review of methods for optimizing multi product inventory using GAs. Wang and Wang [71] propose a modeling tool

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which makes use of GAs for the optimization of the total cost of a multiple sourcing supply chain system. The system is exemplified by a multiple sourcing model with stochastic demand. A mathematical model is implemented to portray the stochastic inventory with demand and transportation parameters as well as price uncertainty factors. Mak and Wong [72] propose using GAs for determining the optimum stock level and quantities of production and transportation in an integrated production-inventory-distribution system. The proposed model consists of three echelons composed of several suppliers, one manufacturing plant and several retailers, respectively. In addition to the above methods, the use of modern Information Technology (IT) techniques has been applied for addressing the modeling lifecycle integration problem. Chandra and Grabis [73] discuss application of mainstream IT methods and tools in building integrated supply chain models. Knoblock et al. [74] develop a general IT system called Ariadne for rapidly building agents that integrate information from multiple heterogeneous sources including databases, web sites, and programs. Other researchers highlight the role of metaheuristics in SCM as sophisticated decision support systems based on powerful mathematical models and solution techniques. Xhafa and Abraham [75] survey computational models for scheduling problems and their resolution, using heuristic and meta-heuristic approaches. Siarry and Michalewicz [76] present advances in the domain of metaheuristics for solving complex supply chain related problems, derived by the importance of designing and managing the entire supply chain as a single entity. The main advantage of evolutionary approaches over other metaheuristics approaches is that they are capable of exploring a larger area of the solution space with a smaller number of objective function evaluations, meaning that high quality solutions can be obtained early in the simulation run. However, these approaches mostly deal with problems involving strategic decisions exclusively, such as the multi-stage facility location/allocation problem, rather than simultaneously integrate decisions belonging to the tactical or operational level. The application of IT for integrated SCM might also lead to improved efficiency, compared to existing logistics systems. Apart from the possible technological problems, that should not be neglected, a major challenge associated with this technology is the proper electronic data sharing among various entities. Another issue to be considered is that many decision modeling tools have limited support for advanced integration technologies and thus, development of wrapper applications is necessary in such cases. The usefulness of simulation tools when combined with mathematical models has been demonstrated through hybrid modeling approaches as suggested by Lee and Kim and Lim et al. [77]. Truong and Azadivar [78] develop an environment for solving supply chain design problems, where simulation is combined with GAs and MIP models. Strategic decisions regarding facility location and partner selection are primarily considered in this work. Moon et al. [79] propose an Integrated Process Planning and Scheduling (IPPS) model for the multiplant supply chain. The model is formulated as an integer programming problem, where the objective is to determine a global optimal schedule of machine assignments and operations sequences of all parts, so that the total tardiness is minimized. Similarly, Lee and Kim [80] propose a hybrid approach combining an analytic model with simulation for production and distribution planning in a supply chain, considering capacity constraints. The authors use simulation to check the capacity assumptions used for a simpler linear model in a more realistic environment with stochastic machine break-downs and to update these capacity parameters for the optimization. Byrne and Bakir [81] present a hybrid algorithm combining

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mathematical programming and simulation modeling of a manufacturing system for the multi-period and multi-product production planning problem. An emerging alternative of simulation methods is the new modeling approach using agents. In the next section we focus on the integration of supply chains from an agent based perspective and we give a detailed review of agent based models developed for this purpose.

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2.2. Agent-Based Models for SCM Alternative approaches to traditional OR-based and simulation methods were developed to address the planning and scheduling problem. The multi agent system (MAS) is a new modeling paradigm which combines object oriented modeling with distributed AI aspects as proposed by Galland et al. [82]. According to Jain et al. [83], multi agent based models offer a good approach for modeling supply chains with several autonomous firms that may operate with various levels of flexibility. The MAS architecture considers information exchange and individual relationship among the individual agents, which favors the cooperation between the agents and thus, improved solutions might be obtained. Through agents‘ learning capability, MAS can efficiently demonstrate the proactive and autonomous behavior of the participating agents in mitigating risks and rectifying supply chain disruptions in real time ([84], [85]). In this sense, high level of cross organizational collaboration is promoted in a computational and cost efficient manner. Industrial applications and case studies of agent based systems are presented by van Dyke Parunak [86], while Caridi and Cavalieri [87] survey different application domains of published multi-agent projects and examine their degree of maturity. Many research efforts focus on the development of frameworks for evaluating and improving the performance of supply chain systems as proposed by Valluri and Crosson [88] and Fereira and Borenstein [89]. Similarly, Swaminathan et al. [90] present a generic agent-based framework for developing customized decision support tools. Turoski [91] and Ghiassi and Spera [92] develop agent-based techniques for coordinating activities of e-commerce and internet-based supply chain for mass customization markets. Li and Feng [45] and Choy and Lee [44] propose an agent-based architecture to facilitate the formation and organization of virtual enterprises for order management. Bo and Zhiming [93] develop an agent based tool to evaluate various scheduling algorithms for orders allocated to different suppliers. Sindhu et al. [94] introduce a supply chain planning, execution and control approach with the help of multi-agent systems that perform both inter- and intra-organizational planning and execution tasks. Ouelhadj et al. [95] describe a new model for robust predictive/reactive scheduling of steel continuous casting, based on the use of multi-agent, tabu search and heuristic approaches. In another work, Ouelhadj et al. [96] describe a negotiation protocol proposed for inter-agent cooperation in a multi-agent system developed for optimization and dynamic integrated scheduling within steel production. A number of applications on agent-based modeling that have been published in the literature is related to operational planning and scheduling in chemical and petrochemical industries. Behdani et al. [97] develop an agent-based model of a lube additive manufacturing supply chain and examine the performance of the system under a significant range of behaviors, business policies, and environmental events. Garcia-Flores and Wang [98] present a multi-agent system for chemical supply chain simulation and management support. A

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platform for modeling and simulating chemical supply chains using a Grafcet-based agent description, is developed by Srinivasan et al. [99]. The authors discuss the applicability of this method for effective decision making and evaluating the effect of different business processes and configurations. An agent-based model for an oil refinery supply chain is described by van Dam et al. [100] and comparison is made with the equation-based model reported by Pitty et al. [101]. Supply chain optimization and management in chemical industries is also examined by Julka et al. [102]. The authors provide a focused review of the literature on the integration of refinery processes and present a framework which integrates the internal elements such as enterprises, their production processes, and associated business data and knowledge. The application of this framework to support the decision making process in a refinery, is examined in another work of Julka et al. [103]. A considerable number of studies have also focused on the integration of supply chain activities with manufacturing activities. The Integrated Supply Chain Management Systems (ISCM) project as described by Fox and Barbuceanu [104], is an extensive research program at the University of Toronto which considers the manufacturing enterprise as a network of operational nodes, and facilitates the decentralization of control using agent-based technology. ISCM provides an approach to the real time performance of supply chain functions. The CIIMPLEX framework for integrating manufacturing activities with customer services is presented by Peng et al. [105], [106]. CIIMPLEX uses functional agents with specialized expertise to integrate existing manufacturing scheduling, planning, analysis and execution systems. This integration facilitates information sharing amongst these systems hence, improves interactions amongst them. Shen and Norrie [107], propose the MetaMorph II agent-based architecture; a distributed intelligent environment that integrates manufacturing activities such as design, planning, scheduling, simulation execution, and product distribution, with the activities of suppliers and customers. Another complex manufacturing system and its supply network is integrated by Zhang et al. [108], using the concept of synergy of two emerging manufacturing concepts; agent-based agile manufacturing systems and e-manufacturing. This approach enables the dynamic generation of alternative scenarios of both the manufacturing system and its supply network, with respect to planning, scheduling, configuration and restructure. More recently, a number of alternative agent-based architectures for manufacturing enterprise integration and SCM have been proposed. GAs and multi-agent technology are combined to develop an integrated system, as proposed by Park et al. [109]. Compared with many other studies, this work has a great advantage in the sense that the integrated multiagent based system can reflect various changes in production, taking into account the supply network and outsources. The Agent Network for Task Scheduling (ANTS) and ways in which computer agent-based systems can assist human-based interaction and decision making in supply chains, are investigated by Sauter et al. [110]. The ISCM project previously discussed, has led to the development of a unified testbed used by the agents built, for supply chain functions; logistics, transportation management, order acquisition, resource management, scheduling and dispatching ([111], [112]). These agents share a common vocabulary for communication and use the services of Information Agents that automatically distribute information and manage information consistency and evolution. Another alternative approach is proposed by Madureira et al. [113] who describe a Multi-Agent Autonomic and BioInspired based framework with self-managing capabilities to solve complex scheduling problems using cooperative negotiation.

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2.3. Challenges in SCM From the above discussion, it is evident that despite considerable advances that have occurred throughout the years in SCM, an attempt to represent the entire supply chain comprising of all of its sub-processes is still missing. Challenges in this field are related to different levels of decision making, various activities within a company (purchasing, manufacturing, distribution, sales), and geographically distributed organizations (vendors, facilities and markets). Real-time coordination and integration of activities is crucial for ensuring a global solution, where constraints such as time precedence, duration, capacity and incompatibility need to be satisfied. The existing approaches reviewed, rely on complete information of services and resources and cannot adequately address the dynamics and uncertainties of the operating systems. While other approaches may induce some kind of synergy within the supply chain, this is limited to the coordination of only some of the functions of the whole system. The major weakness of the models introduced so far, lies in that they are based on assumptions which are not met in real world cases; therefore, theoretical optimum cannot be guaranteed. For instance, most models that account for manufacturing enterprise integration, assume infinite resource capacities on the shop floor, a common due date for all jobs, and idle plant when assigning resources to jobs. Their limited practical value is another drawback of existing models since they require a considerable amount of technical sophistication on behalf of the user, as well as a substantial amount of computation time. These practical considerations, as suggested by Blazewicz et al. [114], gave rise to interesting theoretical models covering challenging aspects of the scheduling problem while forming a good basis for realistic simulation. MAS, using the advantage of reactivity, anticipation and negotiation, are proved to be powerful tools for achieving greater efficiency when implemented in a distributed supply chain system. A comprehensive review of the literature though, reveals the low maturity level of the developed models. Although the simulation results presented in the literature have indicated the effectiveness of the agentbased approaches, there are several issues not yet considered, such as the ability of reusing and extending the model by modifying the behavioral rules of the agents. Another common drawback of a number of these models is the fact that the agents and their relationships cannot be redefined as the simulation runs. In addition, several models do not consider regulator agents for monitoring, structuring, and resolving conflicts, as in a real world supply chain. Improvement of MAS implies representing the entities of the supply chain as agents capable of recognizing and reconfiguring relationships with other agents during the supply chain simulation. Perspectives for further research should reflect to elaborating communication among agents and increasing information exchange between entities involved in the supply chain. Uncertainty consideration is crucial for effective decision making. Demand has been the most extensively studied source of uncertainty in the literature and is considered one of the most important factors that could affect the overall performance of the system. As Aytug et al. [115] report, the complexity nature of supply chains, has bearing on a number of other uncertain and complex factors as well. Variability in demand, development of products with short life cycles, raw material availability and competition in the global market are other sources of uncertainty that could affect supply chain performance. These factors increase the complexity within supply chain systems along with the computational complexity of the

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models developed to manage them. Mele et al. [116] develop an agent based model coupled with GAs, considering not only demand uncertainty but also transport and processing times. This work indicates the dependence of the computation cost of the proposed approach on the problem‘s dimension and its level of complexity. Consideration of uncertainty for the purpose of evaluating the associated risks within a supply chain system is also found in the literature ([117], [118], [119], [120], [121]). Hybrid methods for supply chain integration presented in this work, demonstrate a good potential for effectively representing real world applications. The compatibility of different modeling tools however, should be resolved. Data retrieval from appropriate data sources and positioning of the decision making components, are also critical to the development of more efficient and robust modeling tools. There is a need for developing more generic solutions which may serve as the basis to solve new problems instead of finding the solution of a particular supply chain under specific constraints. The development of integrated frameworks combining economic-based and environmental aspects should be examined as well.

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CONCLUSION A review of modeling approaches and solution strategies for the integration of operation planning and scheduling in supply chain systems is presented in this work. We focused on the key concepts of the various modeling and solution methods, and discussed some representative published work. Although the deterministic analysis requires less computational effort, the stochastic approach is generally proved more accurate when it comes to optimization, given that it provides a more realistic insight of the system under consideration. On the contrary, deterministic models cannot accurately represent the ever changing nature of supply chains and do not consider demand variability over time. In addition, most deterministic models focus mainly on profitability and ignore other important performance measures. Among the various methodologies that we have presented, the MAS framework seems to be particularly popular in the engineering community, as it proved successful in supporting decision making in supply chains. The autonomy of agents in the systems developed so far, gives this approach a competitive advantage compared to other simulation approaches. In the future, the proposed models could be further extended for incorporating agents with more complex decision making capabilities and exploring means to engender multi-actor collaboration across enterprises. Integration of agent based systems with optimization algorithms is also an appreciated contribution for decision making. Hybrid approaches combining advantages of different modeling approaches are a step forward in simulation based optimization methods that have been studied in the last several years. In this setting, a combined application of OR, CT and agent-based modeling may provide new insights into the supply chain integration domain. Current literature on supply chain integration reveals the lack of real world applications. A lot of approaches described in this work manage to capture a certain level of complexity inherent in supply chain systems however, there is still great potential for further improvement of the developed models.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from National Science Foundation under Grants CBET 0625515 and 0966861.

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[75] Xhafa, F. and A. Abraham, Metaheuristics for scheduling in industrial and manufacturing applications series: Studies in computational intelligence. 2008: Springer. [76] Siarry, P. and Z. Michalewicz, Advances in Metaheuristics for Hard Optimization. Springer ed. Natural Computing Series. 2008. [77] Lim, S.J., et al., A simulation approach for production-distribution planning with consideration given to replenishment policies. International Journal of Advanced Manufacturing Technology, 2006. 27: p. 593-603. [78] Truong, T.H. and F. Azadivar. Simulation based optimization for supply chain configuration design. in Proceedings of the 2003 Winter Simulation Conference. 2003. [79] Moon, C., J. Kim, and S. and Hur, Integrated process planning and scheduling with minimizing total tardiness in multi-plants supply chain. Computers and Industrial Engineering, 2002. 43(1-2): p. 331-349. [80] Lee, Y.H. and S.H. Kim, Optimal production-distribution planning in supply chain management using a hybrid simulation-analytic approach, in Proceedings of the 2000 Winter Simulation Conference 1 and 2. 2000. p. 1252-1259. [81] Byrne M.D. and M.A. Bakir, Production planning using a hybrid simulation-analytical approach. International Journal of Production Economics, 1999. 59: p. 305-311. [82] Galland, S., Grimaud, F., Beaune, P., and Campagne, J.P., MAMA-S: An introduction to a methodological approach for the simulation of distributed industrial systems. International Journal of Production Economics, 2003. 85: p. 11-31. [83] Jain, V., L. Benyoucef, and S.G. Deshmukh, A new approach for evaluating agility in supply chains using Fuzzy Association Rules Mining. Engineering Applications of Artificial Intelligence, 2008. 21(3): p. 367-385. [84] Kwon, O., G.P. Im, and K.C. Lee, MACE-SCM: A multi-agent and case-based reasoning collaboration mechanism for supply chain management under supply and demand uncertainties. Expert Systems with Applications, 2007. 33(3): p. 690-705. [85] Lu, L. and G. Wang, A study on multi-agent supply chain framework based on network economy. Computers and Industrial Engineering, 2008. 54(2): p. 288-300. [86] van Dyke Parunak, H., Characterizing the manufacturing scheduling problem. Journal of Manufacturing Systems, 1991. 10(3): p. 241-259. [87] Caridi, M. and S. Cavalieri, Multi-agent systems in production planning and control: an overview. Production Planning and Control. 15(2). [88] Valluri, A. and D.C. Croson, Agent learning in supplier selection models. Decision Support Systems, 2005. 39(2): p. 219-240. [89] Ferreira, L. and D. Borenstein, Normative agent-based simulation for supply chain planning. J. Oper. Res. Soc., 2010. [90] Swaminathan, J.M., S.F. Smith, and N.M. Sadeh, Modeling Supply Chain Dynamics: A Multiagent Approach. Decision Sciences, 1998. 29(3): p. 607-632. [91] Turoski, K., Agent-based e-commerce in case of mass customization. International Journal of Production Economics, 2002. 75(1-2): p. 69-81. [92] Ghiassi, M. and C. Spera, Defining the internet-based supply chain system for mass customized markets. Computers and Industrial Engineering, 2003. 45(1): p. 17-41. [93] Bo, X. and W. Zhiming, Modeling of supply chain: a multi-agent approach, in American control conference. 2003: Denver, CO, United States.

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[94] Sindhu, R., A. Wahid, and G.N. Purohit, Multi-Agent System Interaction in Integrated SCM. 2009, International Journal of Computer Science Issues, IJCSI. [95] Ouelhadj, D., et al., Inter-agent cooperation and communication for agent-based robust dynamic scheduling in steel production. Advanced Engineering Informatics, 2004. 18(3): p. 161-172. [96] Ouelhadj, D., P.I. Cowling, and S. Petrovic. Utility and stability measures for agentbased dynamic scheduling of steel continuous casting. in Robotics and Automation, 2003. Proceedings. ICRA '03. IEEE International Conference on. 2003. [97] Behdani, B., et al., Agent based model for performance analysis of a global chemical supply chain during normal and abnormal situations, in Computer Aided Chemical Engineering. 2009, Elsevier. p. 979-984. [98] Garcia-Flores, R. and X.Z. Wang, A multi-agent system for chemical supply chain simulation and management support. OR Spectrum, 2002. 24: p. 343. [99] Srinivasan, R., M. Bansal, and I.A. Karimi, A multi-agent approach to supply chain management in the chemical industry. Multiagent based supply chain management, in Studies in computational intelligence, J.P.M. B. Chaib-draa, Editor. 2006, Springer: Berlin. p. 419-448. [100] van Dam, K.H., et al. Benchmarking numerical and agent-based models of an oil refinery supply chain. in Proceedings of the European symposium on computer aided process engineering ESCAPE 18. 2008. Lyon, France. [101] Pitty, S., et al., Decision support for integrated refinery supply chains: 1. Dynamic simulation. Computers and Chemical Engineering Science, 2008. 32: p. 2767. [102] Julka, N., R. Srinivasan, and I. Karimi, Agent-based supply chain management--1: framework. Computers and Chemical Engineering, 2002. 26(12): p. 1755-1769. [103] Julka, N., R. Srinivasan, and I. Karimi, Agent-based supply chain management-2: A refinery application. Computers and Chemical Engineering,, 2002b. 26: p. 1771. [104] Fox, M.S. and M. Barbuceanu, Capturing And Modeling Coordination Knowledge For Multi-Agent Systems. International Journal of Cooperative Information Systems, 1996(5): p. 275-314. [105] Peng, Y., et al., A multi-agent system for enterprise integration. International Journal of Agile Manufacturing 1998 (2): p. 201-212. [106] Peng, Y., et al., Agent-based approach for manufacturing integration: The ciimplex experience. 1999, Taylor and Francis. p. 39 - 63. [107] Shen, W. and D.H. Norrie, An Agent-Based Approach for Manufacturing Enterprise Integration and Supply Chain Management, in Proceedings of the Tenth International IFIP WG5.2/WG5.3 Conference on Globalization of Manufacturing in the Digital Communications Era of the 21st Century: Innovation, Agility, and the Virtual Enterprise. 1998, Kluwer, B.V. p. 579-590. [108] Zhang, D.Z., et al., An agent-based approach for e-manufacturing and supply chain integration. Computers and Industrial Engineering, 2006. 51(2): p. 343-360. [109] Park, B.J., H. Choi, and M. Kang, Multi-agent Based Integration of Production and Distribution Planning Using Genetic Algorithm in the Supply Chain Management, in Analysis and Design of Intelligent Systems using Soft Computing Techniques, P. Melin, et al., Editors. 2007, Springer Berlin / Heidelberg. p. 696-706. [110] Sauter, J.A., H.V.D. Parunak, and J. Goic, ANTS in the supply chain. In Proceedings of workshop on agents for electronic commerce at agents, 1999.

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[111] Fox, M.S. and M. Gruninger, Enterprise modelling,, in AI Magazine. 1998. p. 109-121. [112] Laboratory, E.I. The integrated supply chain management project, http://www.eil.utoronto.ca/iscm-descr.html. 1996. [113] Madureira, A., F. Santos, and I. Pereira, Self-managing agents for dynamic scheduling in manufacturing, in Proceedings of the 2008 GECCO conference companion on Genetic and evolutionary computation. 2008, ACM: Atlanta, GA, USA. p. 2187-2192. [114] Blazewicz, J., K. Ecker, and D. Trystram, Recent advances in scheduling in computer and manufacturing systems. European Journal of Operational Research, 2005. 164(3): p. 573-574. [115] Aytug, H., et al., Executing production schedules in the face of uncertainties: A review and some future directions. European Journal of Operational Research, 2005. 161(1): p. 86-110. [116] Guillén, G., et al., An agent-based approach for supply chain retrofitting under uncertainty. Computers and Chemical Engineering, 2007. 31(5-6): p. 722-735. [117] Ahmed, S. and N.V. Sahinidis, An approximation scheme for stochastic integer programs arising in capacity expansion. Operations Research, 2003. 51: p. 461-474. [118] Ierapetritou, M.G. and E.N. Pistikopoulos, Batch plant design and operations under uncertainty. Industrial and Engineering Chemistry Research, 1996. 35: p. 772-787. [119] Levis, A.A. and L.G. Papageorgiou, A hierarchical solution approach for multi-site capacity planning under uncertainty in the pharmaceutical industry. Computers and Chemical Engineering, 2004. 28(707-725). [120] Guillen, G., et al., Addressing the design of chemical supply chains under demand uncertainty. Industrial and Engineering Chemistry Research, 2006. 45: p. 7566-7581. [121] Salema, M.I.G., A.P. Barbosa-Povoa, and A.Q. Novais, An optimization model for the design of a capacitated multi-product reverse logistics network with uncertainty. European Journal of Operational Research, 2007. 179(3): p. 1063-1077.

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In: Scheduling Problems and Solutions Editor: Hussein M. Khodr

ISBN 978-1-61470-689-2 © 2012 Nova Science Publishers, Inc.

Chapter 2

APPLY HEURISTICS AND META-HEURISTICS TO LARGE-SCALE PROCESS BATCH SCHEDULING Yaohua He a, b,  and Chi-Wai Hui b a

Dept. of Industrial and Systems Engineering / Office of the Provost (PVO), National University of Singapore, Singapore b Dept. of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Hong Kong, P. R. China

ABSTRACT

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Large-scale scheduling problems are still challenges to academic researchers and industrial practitioners. With the high complexity and multitudinous constraints existing in the process industry, process scheduling problems are more difficult to solve than the discrete machine scheduling problems. Traditionally, the process scheduling researchers usually formulate the problems into mathematical models in which so many constraints are expressed by equations or inequalities. The mathematical models can theoretically prove the optimality, and obtain optimal or acceptable solutions to small-sized problems. However, for the practical large-scale scheduling problems, it is still hard to achieve acceptable solutions within reasonable computational time by using such mathematical models. That is why scheduling by heuristics and experiences is still popular in real industrial world. In this chapter, heuristics and meta-heuristics have been developed to solve large-scale complex process batch scheduling problems, with the purpose to enhance the feasibility, solution quality and solution speed. In this chapter, an extensive review on the process scheduling is first conducted, which includes the complexity of process scheduling and the available solution methods, and then the strategies to solve large-scale process scheduling problems are analyzed and summarized. In fact, we have dedicated quite some energy to the research on the solution of large-scale problems, focusing on heuristics and meta-heuristics. In this chapter, three types of process scheduling problems are solved using these methods. 

Correspondence concerning this article should be addressed to Y. He at [email protected].

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Yaohua He and Chi-Wai Hui For the single-stage process scheduling in batch plants with parallel units, which is actually similar to the parallel machine scheduling, a comprehensive set of dispatching rules for a class of scheduling objectives were formed by using the impact factors analysis method. In selecting the suitable heuristic rule(s) for diverse scheduling problems, novel rule-evolutionary approaches were proposed to avoid tedious simulation experiments. These genetic-based approaches not only evolve the solutions of the problems, but also automatically select the effective heuristic knowledge to solve the problems. In solving the very difficult multi-stage process scheduling problems (similar to the hybrid flow shop), we have adopted forward/backward assignment strategies, active scheduling techniques and position selection rules in the proposed genetic algorithms. All these measures effectively enhance the solution quality and solution speed. Besides, based on the concept of evolutionary gradient, a global search framework was proposed to make full use of the search ability of the meta-heuristics for the globally optimal or near-optimal solutions. The scheduling of multi-purpose multi-product batch plants with network structures is more common and representative in the process industry, hence more widely studied by researchers, usually using mathematical models for small-size problems. Large-size problems are still great challenges for further study. In this chapter, a pattern matching method for the large-scale multi-purpose scheduling problems is introduced. Different from the conventional cyclic scheduling and decomposition methods, a significant feature of the pattern matching method is that the growth of problem size may not directly lead to the growth of computational time and complexity as usual. This is indeed practicable in solving large-scale real-world problems.

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1. INTRODUCTION 1.1. General Review on Process Scheduling The process industry covers areas such as food, beverages, specialty chemicals, pharmaceuticals and brewing. Scheduling plays a crucial role in raising the efficiency of any production system. In contrast to the long term concern with scheduling in the discrete parts manufacturing industry, the attention devoted to computer aided scheduling methodology in process industry is much more recent, beginning in the middle 1970‘s (Reklaitis, 1982; and Ku et al., 1987). The scheduling in process industry shows more complexity than that in discrete manufacture. The reasons why the research on batch and continuous process scheduling has received great attention from academia and industry in the past two decades are, on one hand, the increasing pressure to raise efficiency and reduce costs, and on the other hand, the significant progress in related modeling and solution techniques and rapidly developing computational power (Floudas and Lin, 2004). In general, process scheduling is also a decision making process to determine when, where and how to produce a set of products given requirements in a specific time horizon, a set of limited resources, and processing recipes. Due to the discrete decisions involved (e.g., equipment assignment, task allocation over time), these problems are combinatorial in nature, and hence very challenging from the computational complexity point of view (Pekny and Reklaitis, 1998). These problems are NP-complete (Garey and Johnson, 1979). As a result, all

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existing algorithms scale exponentially in the worst case. Therefore, a modest growth in problem size can lead to a significant increase in the computational requirements. This has important implications for the solution of scheduling problems. There have been significant research efforts over the past decades in the development of optimization approaches for process scheduling, and several excellent reviews can be found in Reklaitis (1992), Shah(1998), Pinto and Grossmann (1998), Pekny and Reklaitis (1998), Kallrath (2003), Floudas and Lin (2004), Mendez et al. (2006). Reklaitis (1992) reviewed the scheduling and planning of batch process operations, focusing on the basic elements of scheduling problems of chemical manufacturing systems and the available solution methods. Rippin (1993) summarized the development of batch process systems engineering with particular reference to the areas of design, planning, scheduling and uncertainty. Bassett, Dave, et al. (1996) presented an overview of existing strategies for implementing integrated applications based on mathematical programming models and examined four classes of integration including scheduling, control, planning and scheduling across single and multiple sites, and design under uncertainty. Applequist, Samikoglu, Pekny, and Reklaitis (1997) discussed the formulation and solution of process scheduling and planning problems, as well as issues associated with the development and use of scheduling software. Shah (1998) examined first different techniques for optimizing production schedules at individual sites, with an emphasis on formal mathematical methods, and then focused on progress in the overall planning of production and distribution in multisite flexible manufacturing systems. Pekny and Reklaitis (1998) discussed the nature and characteristics of the scheduling and planning problems and pointed out the key implications for the solution methodology for these problems. They reviewed the available scheduling technologies, including randomized search, rule-based methods, constraint guided search, simulation-based strategies, as well as mathematical programming formulation approaches using conventional and engineered solution algorithms. Pinto and Grossmann (1998) presented an overview of assignment and sequencing models used in process scheduling with mathematical programming techniques. They identified two major categories of scheduling models—one for single-unit assignment and the other for multiple-unit assignment—and discussed the critical issues of time representation and network structure. Balasubramanian and Grossmann (2002) reviewed the relevant literature on scheduling under uncertainty, and proposed a novel branch and bound algorithm for scheduling flow shop plants with uncertain processing times. Lin, Janak and Floudas (2004) presented an overview on the process scheduling under uncertainty, and proposed a novel robust optimization approach to address the problem of production scheduling with uncertain processing times, market demands, and/or prices of products and raw materials. Floudas and Lin (2004) presented an overview of developments in the scheduling of multi-product, multipurpose batch and continuous processes. They classified existing approaches for chemical process scheduling into discrete-time and continuous-time models based on the time representation. They pointed out that significant advances had been made in the area of scheduling of chemical processes in the past decade. Further research work is needed to address classes of important large-scale industrial applications. More specifically, future research efforts should aim at addressing (a) the development of mathematical models and algorithms that reduce and even close the integrality gap for medium and large scale

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short-term scheduling applications; (b) medium-term scheduling of batch and continuous processes; (c) the multi-site production and distribution scheduling; (d) the uncertainty in processing times, prices, changes in product demands, and equipment failure/breakdown; (e) the scheduling of manufacturing operations in the semiconductor industry in the presence of multiple reentrant flows; and (f) the integration of scheduling with design, synthesis, control and planning. Mendez et al. (2006) provided an up-to-date review of the state-of-the-art of batch process scheduling. Main features, strengths and limitations of existing modeling and optimization techniques as well as other available major solution methods were examined through this review. They first presented a general classification for scheduling problems of batch processes as well as for the corresponding optimization models. Subsequently, the models of representative optimization approaches for the different problem types were introduced in detail, focusing on both discrete and continuous time models. For the sake of completeness, other alternative solution methods applied in the field of process scheduling are also reviewed, followed by a discussion related to solving large-scale problems through rigorous optimization approaches. Finally, they listed available academic and commercial software for process scheduling. From the above reviews on process scheduling, the solution techniques mainly focus on mathematical programming, namely, exact methods. Actually, growing research and literature on meta-heuristic methods for machine scheduling have been seen in the area of industrial engineering and operational research. However, in the process industry, meta-heuristic methods are mainly applied to numerical function optimization. Kallrath (2003) came up with an overview on the solution techniques for process scheduling problems, mentioned and evaluated the application and importance of meta-heuristic methods for process scheduling. Mendez et al., (2006) have pointed out that for larger scheduling problems the use of metaheuristic methods such as simulated annealing, genetic algorithms, or tabu search may be preferable, since these algorithms can obtain good quality solutions within reasonable time.

1.2. Complexity of Process Scheduling Process industry shows considerable complexity compared to discrete parts manufacturing (Reklaitis et al., 1996). For instance, the complexity of scheduling batch processes is determined by such aspects as batch size constraints, shared intermediates, flexible proportions of input and output goods, carrying out processes without interruption, sequence and usage dependent cleaning operations, no-wait production for certain types of products, and so on. As a consequence, most of the scheduling approaches developed for discrete parts manufacturing are hardly applicable (Günther and van Beek, 2003). Hence, despitef the similarity of the fundamental planning and scheduling problems, there are a series of major issues that have to be reflected by the mathematical models and solution techniques employed. There are a number of factors concerned with most process scheduling problems, such as equipment-task assignment, sequencing and timing of activities. However, different problems may also change significantly in the following aspects, which present different requirements or degrees of difficulty for the modeling of these processes.

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1.2.1. Processing Sequences Based on the complexity of processing sequences employed to produce products, all the processes in multi-product multi-purpose plants can be classified into two different groups: 

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

Sequential processes: Different products follow the same processing sequence. It is usually possible to define processing stages, which can be a single stage or multiple stages. There can be only one unit per stage or parallel units at each stage. For this type of process, batches are used to represent production and it is thus not necessary to consider mass balances explicitly. Network-represented processes: When production recipes become more complex and/or different products have low recipe similarities, processing networks are used to represent the production sequences. This corresponds to the more general case in which batches can merge and/or split and material balances are required to be taken into account explicitly. Kondili, Pantelides, and Sargent (1993) proposed a general framework of state-task network (STN) for the ambiguity-free representation of such processes. The main advantages and drawbacks of STN were summarized by Kallrath (2003). Pantelides (1993) then proposed an alterative representation, the resource-task network (RTN), which describes processing equipment, storage, material transfer and utilities as resources in a unified way.

1.2.2. Intermediate Storage Policies There exist four major categories of treating intermediate storage. Unlimited intermediate storage (UIS): In this case, there is no need to model inventory levels. No intermediate storage (NIS): No storage tanks are available for intermediate materials. However, the materials can be held in the processing unit after the task is finished before they are transferred into the next unit. Zero-wait (ZW): Relevant intermediate materials are required to be consumed immediately after being produced. Special timing constraints are required to be incorporated. Finite intermediate storage (FIS): This corresponds to the most general case. 1.2.3. Changeovers There exist three main types of changeovers. Sequence dependent: When switched between tasks, a unit may require cleanup or setup for safety or quality reasons. The requirement depends on the unit and the tasks involved. Time or frequency dependent: A changeover may be needed after a certain amount of time or a certain number of batches (tasks). None: No changeover is needed between two tasks in a unit. 1.2.4. Operation Modes of Processing Tasks The processing tasks can be classified into batch and continuous tasks. Batch task: Materials are fed at the start of the task; after a certain period of time, products are produced at the end of the task. Continuous task: Materials are fed and/or products are produced continuously during the course of the task. The processing rate can either be fixed or within a certain range.

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1.2.5. Demand Patterns There exist two main classes of demand patterns. Demands due at the end of horizon: Demands for products are specified at the end of the horizon under consideration. Demands due at intermediate dates: Demands for products are specified at designated time instances within the time horizon.

Figure 1. Road-map for process scheduling problems.

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1.2.6. Resource Considerations There are two primary types of resource considerations. Renewable resources: The operations may require utilities, such as steam, cooling water, electricity, and/or manpower. These are regarded as renewable resources which are completely recovered at the time a task finishes. These resources can never exceed the maximum availability at any time during the production. None: No restrictions on resources are considered. 1.2.7. Scheduling Objectives Typical examples of overall objectives in process scheduling problems include: Minimize makespan: Given the production requirement, the objective is to find the optimal schedule with the shortest completion time of the whole process. Minimize earliness/tardiness/costs: Given the production requirement, the optimal schedule is considered to be the one with the lowest cost, which is measured by either simple deviations from specified due dates or total costs calculated in more sophisticated ways. Maximize profit: Given available equipment and other resources, the objective is to find the optimal schedule with the highest value of overall profit in a specified time horizon. Considering the above complexity of process scheduling, in order to provide a systematic characterization, Pinto and Grossmann (1998) presented a general road-map for classifying most relevant problem features. This roadmap was improved by Mendez et al. (2006) as shown in Figure 1, with the consideration of not only equipment and material issues, but also time and demand-related constraints. The classification in Figure 1 shows that there is a tremendous diversity of factors that must be accounted for in process scheduling, which makes the task of developing unified general methods quite difficult. The complexity of the scheduling problems makes it necessary to develop effective schemes for organizing the large amount of information required to describe most scheduling applications. For instance, Zentner, Elkamel, Pekny, and Reklaitis (1998) proposed a high level language as a compact and context independent means of expressing a wide variety of process scheduling problems. Honkomp, Lombardo, Rosen, and Pekny (2000) illustrated many of the features and complexity of the process scheduling problems described above. It is the features and complexity that make the scheduling problems difficult to be solved, and present challenges to the regular use of scheduling technologies.

1.3. Solution Methods for Process Scheduling The state-of-the-art technology based on mathematical, especially mixed-integer programming (MIP) for planning in process industry is quite advanced and appropriate for solving real-world planning problems (Kallrath, 2003). Commercial solvers, such as XPRESS-MP, CLPEX, are very efficient to solve mixed-integer linear programming (MILP) planning problems.

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Figure 2. Solution methods for process scheduling problems.

However, the complexity of scheduling problems can easily exceed today‘s hardware and algorithm capacities. Scheduling problems are usually NP-hard (Garey and Johnson, 1979), no standard solution techniques are available. Hence, in many cases, feasible solutions to the problems are more considered and practical, rather than optimal solutions (Kallrath, 2003). Mendez et al. (2006) summarized solution methods for process scheduling as shown in Figure 2. The main solution methods for process scheduling are: Heuristic methods: Much of the scheduling done in the process industry uses heuristic methods to quickly obtain a feasible, although not necessarily optimal, schedule. Dispatching rules (or scheduling rules) are considered as construction heuristics. These rules use certain empirical criteria to prioritize all the batches that are waiting for processing on a unit. For simple scheduling problems, they have demonstrated very good performance, although their efficiency is usually evaluated empirically. The usefulness of dispatching rules is still limited to quite a narrow variety of scheduling problems and optimality can be proved only in some special cases. Some relevant dispatching rules are: FCFS (first come first served), EDD (earliest due date), SPT (shortest processing time), LPT (longest processing time), ERD (earliest release date), WSPT (weighted shortest processing time). Often, composite dispatching rules involving a combination of basic rules can perform significantly better. Besides, dispatching rules can be easily embedded in exact models to generate more efficient hybrid approaches for large-scale scheduling problems. Despite the use of scheduling rules in industry, modern researchers in academia focus on more general mathematical programming techniques, which can provide optimal schedules for a range of problems, usually with small problem size. Exact methods: Exact methods are mathematical optimization techniques, including MILP and mixed-integer non-linear programming (MINLP). Mathematical programming methods formulate a scheduling problem as a set of variables representing the process state, a set of constraints on the values of those variables and an objective function that evaluates the quality of a schedule. Some of the models are in MILP form, others maybe in MINLP that is more difficult to be solved.

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MILP models are often solved by the branch and bound (B&B) method. In some cases, it is even not possible to find feasible integer solutions, because feasible integer solutions exist often very deep in the B&B tree. In many cases, even though the MILP models of the problems can be well described, but even using parallel algorithms and powerful hardware, scheduling problems might be too complex, and often cannot be solved with MILP methods, at least not yet. If we meet such cases, one choice is to simplify the MILP models, so that a solution can be found in a reasonable time. Another choice, it is worthwhile to apply constraint programming. Constraint programming (CP): CP is developed out of logic programming and constraint solving. CP originated from the work of Jaffar and Lassez (1987) and Jaffar and Maher (1994) and has been applied a large number of discrete optimization problems. CP is another solution approach for solving some classes of scheduling problems. CP is particularly effective for solving feasibility problems and seems to be better suited than traditional MILP approaches in special types of discrete optimization problems where finding a feasible solution is difficult. The lack of an obvious relaxation, however, makes CP worse for loosely constrained problems, where the focus is on finding the optimal solution among many feasible ones and proving optimality. Overall, CP and MILP have complementary strengths that can be combined into hybrid algorithms, yielding considerable computational improvements when compared to the standalone approaches. Examples of these are the work of Jain and Grossmann (2001) for single-stage process scheduling, Harjunkoski and Grossmann (2002) for the scheduling of multistage multiproduct plants, and Maravelias and Grossmann (2004) for the scheduling of multipurpose plants. Meta-heuristic methods: If CP also fails, a feasible resort might be to use meta-heuristic methods, including genetic algorithms (GA) (Holland, 1975 and Goldberg, 1989), simulated annealing (SA) (Kirkpatrick et al., 1983), tabu search (TS) (Glover, 1997), memetic algorithm (MA) (Moscato et al., 1989), particle swarm optimization (PSO) (Eberhart and Kennedy, 1995 and Eberhart and Shi, 2001), ant colony optimization (ACO) (Dorigo, 1991 and 1996), differential evolution (DE)( Storn and Price, 1997), and so on. For most scheduling problems, MILP solutions become prohibitive for large problems. Besides, the solution times for MILP algorithm are notoriously sensitive to problem data and unpredictable. Therefore, most of the literature on MILP for scheduling in the process industry focused on formulation of the problem, and resorted to commercial solver to solve the problem, more often only getting acceptable solutions to small-size problems. The mathematical methods are mainly of theoretical significance, but not of practical and applicable significance to large-scale industrial applications. Meta-heuristic methods have been widely reported in machine scheduling literature. Although meta-heuristic methods cannot guarantee optimal solutions, they take much less computational time and are insensitive to problem data, thus are more reliable and robust for practical applications. These factors make them very attractive for solving real-world, largesize scheduling problems. Although GA and SA have been widely reported in machine scheduling literature, they have been seldom applied to process scheduling before this research. Tandon et al. (1995) presented a solution methodology to obtain near-optimal solutions of the scheduling problems of multiple products on unrelated parallel units using simulated annealing algorithm. Nearoptimal solutions were obtained for problems with 20~100 products and 3~11 units.

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Chen et al. (1996) developed an approach for applying genetic algorithms to the continuous flow shop scheduling problems. Azzaro-Pantel et al. (1998) adopted a GA for solving an industrial-size short-term batch process scheduling problem, their numerical experiments indicate that GA represents a good alternative for the solution of large combinatorial problems. Wang et al. (2000) proposed a genetic algorithm for onlinescheduling of a multi-product polymer batch plant. It was seen that the genetic algorithm outperformed mathematical programming. Moreover, the modeling effort for GA was considerably lower than that for mathematical programming. Tabu search was invented by Glover (1997) in the 1970‘s and has been used to solve a wide range of hard optimization problems, such as the job shop scheduling, the traveling salesman problems (TSP), graph coloring and continuous function optimization. Together with SA and GA, TS has been singled out by the committee on Next Decade of Operation Research as ―extremely promising‖ for the future treatment of practical applications (Wang, 1999). Practically, long-time tabu search is often used to search the best solution to large-size combinatorial problems. Nowicki and Smutnicki (1996) proposed an algorithm based on a tabu search technique with a specific neighborhood definition that employed a block of jobs notion, computational experiments to the permutation flow-shop problem (up to 500 jobs and 20 machines) showed its excellent numerical properties. Grabowsk and Wodecki (2004) proposed a new very fast local search procedure based on a tabu search approach for the permutation flow shop problem with the makespan criterion. They solved flow-shop instances up to 500 jobs and 20 machines with high accuracy in very short time. TS is just beginning to see application to process industry (Wang, 1999 and Lin, 2004), and mainly used for continuous parameter optimization. Artificial intelligence (AI): With the main goal of making a more efficient use of the process information as well as the essential knowledge provided by human schedulers, AI techniques have also been widely applied to scheduling problems. AI is the mimicking of human thought and cognitive processes to solve complex problems automatically. It uses techniques for writing computer code to represent and manipulate knowledge. Different techniques mimic the different ways that people think and reason. Some interesting applications based on AI technologies for addressing real-world scheduling problems have been reported in Henning and Cerda (2000), Sauer and Bruns (1997) and Zweben and Fox (1994). Hybrid methods: A new research tendency is to develop hybrid methods by combining optimization techniques, both exact and heuristic. For example, GA is often combined with SA or TS for various problems. GA is well known for its global search ability, but its local optimization ability for complex problem is limited. On the other hand, some meta-heuristic search methods, such as SA and TS, are famous for local search ability, which overcome the shortcomings of GA. The following publications are some studies on hybrid methods: Timple (2002) reported a successful application of a combined MIP-CP approach to a scheduling problem in the process industry. Maravelias and Grossmann (2004) presented a hybrid MILP/CP framework for STN scheduling problems. Roe et al. (2005) presented a novel hybrid CLP/MILP algorithm for scheduling complex multipurpose batch processes. The scheduling problem was decomposed into two sub-problems: first an aggregate planning problem was solved using an MILP model, and then a sequencing problem was solved using CP techniques.

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Spina et al. (2003) presented a hybrid approach for solving manufacturing scheduling problems, based on the integration between CP and GA. Wang and Zheng (2001) proposed a hybrid heuristic method for flow shop scheduling. The method incorporated GA and SA together. Nearchou (2004) presented a new hybrid simulated annealing algorithm (hybrid SAA) for solving the flow-shop scheduling problem (FSSP). The hybrid SAA integrated the basic structure of a SAA together with features borrowed from the fields of genetic algorithms and local search techniques. Xia et al. (2005) proposed approach that makes use of particle swarm optimization (PSO) to assign operations on machines and simulated annealing (SA) algorithm to schedule operations on each machine. To sum up, although significant advances have been made in the area of process scheduling in the past decades, there are still a number of major challenges and questions that remain unresolved. One of the challenges is the large-scale industrial applications (Floudas and Lin, 2004). From a mathematical perspective, most scheduling problems found in industrial environments can be regarded as very large-scale combinatorial and complex optimization problems, which rarely can be solved to optimality within a reasonable amount of computational time. Such a combinatorial explosiveness has to do with the increased number of products to be processed, the long sequence of processing stages, the multiple units available for each task and the length of the scheduling horizon to be considered. The complexity arises from a wide range of operational constraints that often need to be taken into account in real world problems. Fortunately, much better scheduling techniques knock at our doors and might, within 5 to 10 years, play a similar role as does MIP in planning nowadays (Kallrath, 2003). In this chapter, we try to study and develop meta-heuristic methods and their hybrids for large-scale complex process scheduling problems.

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1.4. Strategies for Large-Scale Process Scheduling Usually, a complex process scheduling problem can be changed or simplified into one of the typical process scheduling problems with solution methods available. However, a largesize problem can not be reduced to a level at which optimality is possible. For a large-scale problem, on one hand, we will consider how to decompose the problem into sub-problems; on the other hand, we should seek new effective and efficient solution techniques. Therefore, for the large-scale process scheduling, the following strategies and methods are often applied: Decomposition/aggregation techniques are the first choice of mathematical programming researchers (Grossmann and Biegler, 2004). Decomposition has been long recognized as a fundamental technique in large-scale optimization. Realff, Shah and Pantelides (1996) proposed a rigorous decomposition procedure for their MILP problem. Harjunkoski and Grossmann (2001) used a decomposition approach to solve the challenging scheduling problem of the steelmaking-continuous casting plant. Harjunkoski and Grossmann (2002) presented decomposition techniques for multi-stage scheduling problems using MILP and CP methods. When Wu and Ierapetritou (2003) found that the MILP model for the scheduling of multi-purpose multi-product batch plants proposed by Ierapetritou and Floudas (1998) was difficult to solve the problems within long time horizons, they proposed several decomposition strategies. Apart from decomposition, cyclic (or periodic) scheduling is widely used for the problems within long time horizons.

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Cyclic scheduling is developed to make the operation decisions easier and profitable. It constructs an operation schedule and makes it executed repeatedly. In addition to the advantage of easy management and control of the plant, mathematically the problem is limited to a smaller time horizon and can be thus solved more efficiently. Shah, Pantelides, and Sargent (1993) modified the formulation of Kondili et al. (1993) and extended it to the periodic scheduling of batch plants using a discrete time representation. Schilling and Pantelides (1999) presented a periodic scheduling formulation which is based on their earlier work on continuous-time representation. Because of the difficulty in linearizing the nonlinear function, a branch-and-bound algorithm that branches on both discrete and continuous variables was proposed. Castro et al. (2003) modified their early RTN-based formulation to fit periodic scheduling requirement for an industrial application. Wu and Ierapetritou (2004) extended the work by Ierapetritou and Floudas (1998) based on the STN representation, and developed the cyclic scheduling formulation with the inherited advantage of using few binary variables. Although the idea of cyclic scheduling is to overlook the start-up and finishing phases, in order to obtain a feasible solution for short-term scheduling problem, especially when long time horizons are considered where such a solution is a challenge, a detailed consideration of start-up and finishing phases was proposed in the work by Wu and Ierapetritou (2004). In fact, cyclic scheduling is still a decomposition technique. What should be noticed is that in cyclic scheduling the objective function may turn into a non-linear equation which results in an MINLP model and increases the difficulty to solve the problem. Decomposition techniques are problem dependent and no underlying general theory has evolved (Hoffman, 2000). The drawback of decomposition is that the global optimum may never be found despite the possible near-optimal solutions. So decomposition is a compromise under the current technology condition. As stated above, meta-heuristic methods, such as simulated annealing, tabu search and the genetic algorithm, are the second choices (Mendez et al., 2006). In recent years, there is vast literature on meta-heuristic methods, which are currently the best choice for ―good enough‖ solutions within reasonable time. Although some process scheduling researchers do not think much of these methods, we strongly believe that these methods have great potential to be developed for process scheduling. Although meta-heuristic methods do not guarantee the global optimum, they often achieve optimal or near-optimal solutions, depending on the problem sizes. The third alternative is the reliance on parallel computing. Due to the demand for higher performance, lower cost and sustainable productivity, the past several years have witnessed an ever-increasing acceptance and adoption of parallel computing. With the parallel implementation of the solution approaches, even the users do not have high performance supercomputers, they still can realize high performance computing in a cluster of homogenous or heterogeneous computers, even in a cluster of personal computers (PC). With the fast development of grid computing technology, the middleware for grid computing theoretically enable the users to take advantage of the computational resource all over the world. In process engineering area, the early parallel computing was used for sparse matrices processing and matrix multiplication. In recent years, researchers have begun to apply parallel computing to optimization at modeling development level or at solution technique level.

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Currently, many engineering modeling platforms are being redesigned to exploit advances in parallel computing architecture (Grossmann and Biegler, 2004).

1.5. Summary of the Research Background

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From the overview of process scheduling, the research background can be summarized as follows: (1) Process scheduling shows more complexity than machine scheduling. From a mathematical perspective, most scheduling problems in industrial environments can be regarded as very large-scale complex combinatorial optimization problems, which rarely can be solved to optimality within a reasonable time. (2) Researchers in this area pay more attention to exact methods and often use small-size sample problems with customized data to illustrate their models. For large-size problems, decomposition or cyclic scheduling techniques are often used for nearoptimal solutions. (3) Meta-heuristic methods have been widely studied in the area of machine scheduling and regarded as appropriate tools to solve the large-scale combinatorial problems including scheduling problems. (4) However, in the process scheduling area, some researcher argued that meta-heuristic methods are not suitable for process scheduling problems. Due to the high complexity and constraints of process scheduling, the initial feasible solution(s) for coming evolution is or are not easy to obtain. There are relatively fewer publications on meta-heuristic methods for process scheduling. During recent years, metaheuristic methods have been gradually accepted by some scholars and regarded to be techniques to obtain acceptable solutions for large-scale problems within reasonable time. (5) Hybrid methods are natural choices in order to take advantage of the complementary strengths of several techniques. (6) Parallel computing is regarded as a practical and applicable technique to realize high performance computing in the situation without high performance supercomputers. The parallel implementation of the solution approaches enables high performance computing to be carried out in a cluster of homogenous or heterogeneous computers, even in a cluster of PCs.

1.6. Problems to be investigated In terms of plant configurations, batch processes can be classified, in increasing order of complexity, as single-stage (i.e. multiple parallel units), multi-stage and multi-purpose processes. This chapter focuses on three typical classes of process scheduling problems (most examples are benchmark problems widely studied): (1) Single-stage multi-product scheduling problems (SMSP) in batch plants with parallel units. These problems have the same features as the parallel machine scheduling

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problem (PMSP), but with high constraints, such as order or unit release times, forbidden changeovers or forbidden processes. Considering all these constraints in large-size SMSP, even feasibility is a challenge for the existing MILP models, optimality is almost impossible. Hence, it is practical to develop a method for nearoptimal or acceptable solutions. (2) Multi-stage multi-product scheduling problems (MMSP) in batch plants. These problems have the same features as the flexible flow shop scheduling (FFSSP), but with high constraints similar to those in SMSP. In each stage, there are several parallel units available to process the products. This makes it much more complex than the flow shop scheduling (FSSP). MMSP are regarded as the most difficult process scheduling problems. Existing MILP models are often formulated with simplified constraints and illustrated by small-size examples with customized or tailored data. In this study, the proposed methods for SMSP are adjusted for largesize random MMSP. (3) Multi-purpose multi-product scheduling problems (MPSP) in batch plants with network structures. Multi-purpose multi-product batch plants can be usually expressed by using STN or RTN networks, thus called network processes, and the corresponding scheduling problems are referred to as network process scheduling problems. Compared with the above two types of sequential scheduling, MPSP are more common and representative in the process industry, hence more widely studied by researchers often using MP models for small-size problems. Large-size MPSPs are still great challenges for further study. In this chapter, we will try to develop novel methods beyond MP models for large-size MPSP. The algorithms in this book chapter were implemented in C language. The computation tests of both the MILP models and the meta-heuristic algorithms were run on a PC with an Intel Pentium M 1500MHz CPU and 768MB of memory. GA is just for illustration, not the only choice. Other meta-heuristics, such as tabu search, and hybrid algorithms, may be more effective in solving large-size problems.

2. RULE-EVOLUTIONARY APPROACHES FOR SMSP 2.1. Problem Description In the single-stage multi-product scheduling problem (SMSP), a fixed number (denoted as M) of production units (forming a set of units U) are available to process a number (denoted as N) of all customer orders (forming a set of orders O). Each order involves a single product, requiring a single processing stage, has a predetermined due date, and can only be processed in a subset of the units available. The production units have different processing capacities. Hence, the process time of the same order is fixed and production unit dependent. A production unit processes only one order at a time. When one order changes over to another order, time is required for the preparation of the unit for the changeover. The changeover times are sequence-dependent or both sequence- and unit-dependant. Forbidden changeovers and processes, called CP constraints, may exist in the problem.

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Time-based scheduling objectives are commonly considered in the literature because cost-based objectives usually can be surrogated by time-based objectives (Pinedo, 1995). Common time-based objectives are described by Pinedo (1995). Makespan, total tardiness and total earliness are three typical scheduling objectives to be minimized. (1) Makespan (Cmax): The makespan, defined as:

Cmax  max{ C1, C2 , ..., C j , ..., CN } j = 1, 2, …, N

(1)

is the completion time of the last order to leave the system, where Cj is the completion time of order j. A minimum makespan usually implies a high utilization of the units. (2) Total tardiness (T): The total tardiness is the sum of the tardiness of all orders, defined as: N

T  T j

(2)

j 1

where Tj is the tardiness of order j: Tj  max{ C j  d j , 0} , dj is the due date of order j. (3) Total earliness (E): The total earliness is the sum of the earliness of all orders, defined as: N

E  Ej

(3)

j 1

where Ej is the earliness of order j, E j  max{ d j  C j , 0} .

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(4) Total flow time (F): The total flow time is the sum of completion time of all orders: N

F  C j

(4)

j 1

The scheduling objective in this work is to minimize the makespan Cmax or total tardiness T, depending on the due dates of the orders. If it is obvious that all the orders can be completed before their due dates, it makes no sense to minimize the total tardiness. Just as done in the literature (Cerda et al., 1997; Hui and Gupta, 2001), when minimizing the makespan of Example 2-1, the due dates of the orders are not considered as constrains. If due dates are considered as constraints when minimizing makespan, the complexity of MILP model increases. However, in our work, it is just an option, not an actual limitation, that the due dates are not considered as constraints when minimizing makespan. Actually, adding a penalty term in the objective function for the orders violating due dates enable the algorithm to find a good schedule without tardy orders, assuming that there are no tardy orders at optimum. In case of tardy orders at optimum, soft due dates (He and Hui, 2007a) should be used, which is determined by minimizing the tardiness, hence increasing computational effort. Therefore, we do not consider due dates as constrains when minimizing makespan in Example 2-1.

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Yaohua He and Chi-Wai Hui

(5) The minimization of the makespan and the minimization of the total tardiness can be conflicting objectives. If the due dates of the orders are loose, it is easy to find a schedule with zero total tardiness; but the makespan of the schedule may be still very long. To consider both makespan and total tardiness at the same time, we propose a new compound objective TC:

TC  T  Cmax

(5)

where T is the total tardiness, and Cmax is the makespan, and α and β are weight coefficients. We may let α = β =1. This compound objective is used in order to satisfy the order due dates. (6) Similarly, to consider both total flow time and total tardiness simultaneously, we put forward a new compound objective TF: N

TF   (T j  C j )

(6)

j 1

In this chapter, Example 2-1 is to be used for illustration of the proposed approaches. Example 2-1 is a random SMSP in which 10 orders (N = 10) are to be assigned to 3 units (M = 3). Each order involves only one batch, required to be completed before the due date di. The due date di is produced randomly, di∈(20, 60). All the order release times and unit release times are null. Changeover time cij is produced randomly, cij∈(0.50, 2.00); and process time piu is also produced randomly, piu∈(5.00, 20.00). Neither forbidden changeover nor forbidden process exists in this example. The data for Example 2-1 is presented in Table 1.

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Table 1. Changeover times, due dates and process times of Example 2-1 Changeover times

i1 i2 i3 i4 i5 i6 i7 i8 i9 i10

Due date

j1

j2

j3

j4

j5

j6

j7

j8

j9

j10

— 0.78 1.66 0.95 0.72 1.24 1.96 0.97 1.07 1.54

1.02 — 1.87 1.41 1.44 1.60 1.99 1.78 1.81 1.76

0.89 1.86 — 1.06 1.68 1.68 1.85 1.32 0.71 1.27

0.80 1.96 1.69 — 1.81 1.26 0.81 0.67 1.57 1.46

0.98 1.69 1.88 1.30 — 1.37 0.70 1.22 0.78 1.51

1.24 1.97 0.81 1.02 1.26 — 1.13 1.89 1.83 0.83

1.83 1.37 1.43 0.71 1.17 1.11 — 0.99 1.34 1.19

0.84 1.03 0.78 1.45 1.74 1.39 1.83 — 1.43 0.51

1.58 0.96 1.49 1.26 0.77 0.96 1.34 0.66 — 1.14

1.02 1.66 0.63 0.71 0.74 0.65 0.75 1.77 0.54 —

31 39 22 34 55 28 55 29 26 42

Process times u1

u2

u3

14.07 15.64 16.49 10.70 15.53 8.20 14.41 6.83 5.08 8.60

12.20 8.95 14.29 16.18 5.13 10.68 17.24 9.70 6.43 13.11

5.40 19.41 8.56 5.97 10.68 7.19 5.73 8.02 10.41 14.12

2.2. MILP Model for SMSP For SMSP, Cerda et al. (1997) did the original work with MILP model, later Karimi and McDonald (1997), Hui and Gupta (2001) and Chen et al. (2002) presented their modified MILP models. All these works focused small- size problems, and the later improvements

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were marginal, not substantial, especially for large-size problems. So the bi-index MILP model based on continuous-time representation (Hui and Gupta，2001) is representative, maybe not very efficient.

2.2.1. Notations (A) Indices i, j= different customer orders u, v= different processing units

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(B) Sets I= a set of orders to be processed (with N orders), or set O U= a set of units (with M units) Ui= units available to process order i in U Iu= orders that can be processed in unit u PRi= feasible predecessors of order i SUi= feasible successors of order i PSi= orders processed either just before or immediately after order i (C) Parameters N= the number of orders M= the number of units Piu= process time of order i on unit u cij= changeover time when order i changeover to order j (not unit dependant) ori= order release time: the earliest time at which order i can start its processing uru= unit release time: the earliest time at which unit u can get ready di= due date: the committed shipping or completion date of order i (the date the order is promised to the customer) BIGM= a big number (D) Variables Positive Variables: STi= start time of order i Ti= tardiness of order i Cmax= makespan T= total tardiness Binary Variables: Xij= assignment of order j after order i Wiu= assignment of order i to unit u Siu= order i is first assigned to unit u, i.e. order i is the starting order on unit u

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2.2.2. Milp Model (A) Problem Constraints (1) Assignment of consecutive orders in a unit:

W

Wiu 

jv

vU j ,v u

 Xij  Xji  2  0, i  I, j  PSi , u U i

(7)

(2) Each order has at the most one unique successor:

X

jSUi

ij

 1, i  I

(8)

(3) Each order has at the most one unique predecessor:

X

iPRi

ji

  Siu  1, i  I

(9)

uU i

(4) Each unit has a unique starting order:

S

iI u

 1, u U

iu

(10)

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(5) Each order is processed:

W

uU i

iu

 1, i  I

(11)

(6) Relation between the starting times of the consecutive orders in a unit: (1  X ij )  BIGM  ST j  STi   (Wiu  piu )  cij , i  I , j  SU i

(12)

uU i

(7) Starting time of the first order in the unit:

STi  Wiu  Max(uru , ori ), i  I

(13)

uUi

(8) Relation between variables Wiu and Siu:

Wiu  Siu , i  I , uUi

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(14)

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(9) Tardiness at the completion of order:

Ti  ( STi  Wiu  piu )  di , i  I

(15)

uUi

(10) Makespan: Cmax  ( STi 

W

uU i

iu

 piu )  Max (ori , Min(uru )), i  I

(16)

uU iu

(B) Objective Functions

min Cmax

(17) N

min T   Ti

(18)

i 1

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2.2.3. Solutions for Example 2-1 Actually, the single-stage problem can be regarded as an irregular assignment problem in which the orders to be processed by each unit can be limited under a certain number n. Furthermore, the value of Big M also influences the performance of the model. If the Big M is selected a suitable value, much computational effort can be saved in obtaining a better solution. In the bi-index MILP model (Hui and Gupta, 2001) and the tri-index model (Cerda et al., 1997), these two factors were not considered. Therefore, besides the constraints in the bi-index model, another constraint is given as follows: N

W i 1

iu

n

(n  N )

(19)

The bi-index model is applied to Example 2-1, being formulated in GAMS (General Algebraic Modeling System) (Brooke et al., 1992) and solved by the OSL solver, and the results are presented in Table 2 under the original model. The results of Example 2-1 by the improved model are also presented in Table 2. Table 2. Results of Example 2-1 by MILP

Iterations 592 1000 2000 20000 200000 2000000 3000000

Original model (Big M = 1000) Nodes CPU time(s) No integer solution yet 46 0.27 81 0.50 769 4.53 6896 45.64 77642 768.05 118249 1176.95

Cmax 36.63 36.63 31.74 31.53 28.79 28.79

Improved model (n = 4, Big M = 150) Iterations Nodes CPU time(s) 844 No integer solution yet 1000 27 0.22 2000 56 0.42 20000 596 4.01 200000 5042 39.46 2000000 56843 542.78 3000000 88667 1044.68

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Cmax 47.29 37.53 37.53 28.31 28.31 28.31

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Yaohua He and Chi-Wai Hui

In (He and Hui, 2007b), we have analyzed the difficulties for the MILP model to solve the SMSP. In short, MILP cannot solve a large-size instance to optimality or even to feasibility in acceptable time.

2.3. Heuristic Rules and Random Search To solve large-size scheduling problems, due to the difficulties of the MP, the preferred method in industry is to use scheduling rules, such as the shortest processing time first (SPT) rule and the earliest due date first (EDD) rule. According to a scheduling rule, jobs are sequenced in decreasing priority order and then one by one assigned to machines or processing units. During the last 30 years, the performance of a large number of scheduling rules has been studied extensively by using simulation techniques. Research on the scheduling rules has shown that there is no single universal rule, and the effectiveness of a scheduling rule depends on the scheduling objective and the prevailing shop or plant conditions. Scheduling objectives include time-based and cost-based objectives. Most of the scheduling rules are proposed and investigated for time-based scheduling objectives which can be classified into two categories: makespan related objectives (such as makespan Cmax, total tardiness T, total flow time F, and compound objectives TC and TF defined above) and earliness related objectives (see Section 3). For the former, we try to let the products be completed as early as possible. For the later, we try to let the product complete as near as possible to the due date, but not after the due date.

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Table 3. Seven rules for minimization of the makespan related objectives in SMSP Rule Rule 1

Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 Rule 7

Detailed description Assign the order on the unit that makes the order‘s possible start time be as early as possible, that is, assign the order on the first available unit (FAU) Assign the order on the unit that makes the order‘s changeover time on the unit be the shortest Assign the order on the unit that makes the order‘s process time on the unit be the shortest Assign the order on the unit that makes the order‘s start time be as early as possible Assign the order on the unit that makes the sum of the order‘s possible start time and process time on the unit be the shortest Assign the order on the unit that makes the sum of the order‘s changeover time and process time on the unit be the shortest Assign the order on the unit that makes the order‘s completion time be as early as possible

Shortened form earliest possible start time, FAU

shortest changeover time, SCT shortest process time, SPT earliest start time, EST shortest possible start time + process time, SPsPT shortest changeover time + process time, SCPT earliest completion time, ECT

In random search, the tasks or products are randomly sequenced, and then one by one assigned to machines or processing units by using a certain scheduling rule. Through exploring a set of random solutions, feasible solutions better than the simple rule-based method can be attained. In (He and Hui, 2008), we have proposed a random search based on heuristic rules, which outperforms MILP in solving large-size scheduling problems in single-

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stage batch plants with parallel units; Furthermore, GA has used to evolve a set of random feasible solutions and obtains much better solutions than the random search. In both random search and GA, heuristic rules play a very important role in reducing search space. Now that the heuristic rules are so important to algorithms, we should construct the rule base scrupulously, not letting some possible useful rules lie out of our consideration. In practice, the rule base can be enlarged if necessary. On the other hand, each heuristic rule needs a subroutine inlaid into the host algorithm. If the rule base is large, the workload to write code will be huge. Moreover, managing a large rule base is also a tough task. Therefore, our work (He and Hui, 2007b) first proposes a rule construction method with the purpose to construct a comprehensive, but not very large set of rules according to the analysis of impact factors with respect to makespan related objectives; and then presents three approaches for the host algorithm to automatically select suitable rules from the rule set for solving a specific problem (see sub-section 2.4).

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2.3.1. Seven Rules for the Minimization of Makespan Related Objectives In (He and Hui, 2007b), seven rules have been summarized based on the analysis of impact factors with respect to makespan related objectives in Table 3, where Rules 1-3 are simple rules, and Rules 4-7 are compound rules. 2.3.2. Performance of Different Rules Natural numbers 1, 2, 3, …, N are used to denote N orders in SMSP. An order sequence  = (1, 2, 3, …, N) is produced randomly, i ∈{1, 2, 3, …, N}, i=1, 2, 3, …, N. And then, from 1 to N, one by one, each order will be assigned to the units according to a certain heuristic rule above. As a result, a schedule is formed with an objective value, f(). f() can be calculated by the functions from (1) to (6). Figure 3 shows the procedure to synthesize an order sequence into a schedule according to one selected rule. Assume that a random order sequence is  = (3, 2, 7, 6, 4, 5, 9, 10, 1, 8) in Example 2-1. The above seven rules are used respectively to assign all the orders in the order sequence to the units, and different schedules with various objective values are formed, see Table 4. For Cmax, T and TC, Rule 7 is the first best rule, Rule 6 the second. But for F and TF, Rule 6 is the first best rule, Rule 5 the second. For the same order sequence, when different rules are used for order assignment, different quality schedules are obtained.

Figure 3. Schedule synthesis by a heuristic rule.

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Yaohua He and Chi-Wai Hui Table 4. A random order sequence scheduled by different rules

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Order sequence  Rule used Makespan Cmax Total tardiness T Total flow time F Compound objective TC Compound objective TF

Rule 1 38.24 10.46 224.13 48.70 234.59

Rule 2 58.86 47.86 260.46 106.72 308.32

3 2 7 6 4 5 9 10 1 8 Rule 3 Rule 4 Rule 5 37.62 38.24 31.94 6.62 10.46 2.94 182.79 224.13 184.82 44.24 48.70 34.88 189.41 234.59 187.76

Rule 6 30.72 1.72 176.89 32.44 178.61

Rule 7 30.14 0.09 197.72 30.23 197.81

2.3.3. Procedure of the Genetic Algorithm GA is used to evolve the chromosomes. Each chromosome, which is a sequence of orders, is synthesized into a schedule by using a specific rule. Each rule is a method to change a sequence of orders into a schedule, and such a method is realized by a subroutine (or function in C language). When GA is applied to solve an SMSP, solutions to the problem are represented by chromosomes. Permutation-based representation is adopted in our work, which is a kind of integer coding. As stated previously, natural numbers 1, 2, 3, …, N are used to denote N orders. A random order sequence  = (1, 2, 3, …, N) is produced, i ∈{1, 2, 3, …, N }.  is called a chromosome in GA. For example,  = (6, 3, 5, 4, 8, 2, 7, 1, 9, 10) is a sample chromosome of the 10-order problem in Example 2-1. The evaluation of a chromosome is a procedure to synthesize the chromosome (according to a pre-selected heuristic rule) into a schedule with an objective value. The procedure is shown in Figure 3. Traditionally, only one rule is applied for the evaluation of all the chromosomes in the GA process. At the beginning of GA, an initial generation of chromosomes is produced randomly. Assume the number of chromosomes in the initial generation is popsize, which depends on the problem size and is an important parameter in GA for controlling the solution quality. In all the genetic algorithms in this section, popsize=200. At every iteration of GA, a new generation will be produced through crossover, mutation and selection, and popsize will remain constant. Throughout genetic evolution, because of the mechanism of selection, crossover and mutation, good-quality offspring are born from the previous generation (parents). Generation by generation, stronger chromosomes are the survivors in a competitive environment. At the end of GA, optimal or near-optimal solutions can be achieved. The detailed components of GA were described in the paper (He and Hui, 2006). The procedure of GA is shown in Figure 4. Assume the number of the chromosomes that are selected from a generation to crossover is xsize; the number of the chromosomes that are selected from the generation to mutate is msize. We can let msize + xsize = popsize. The ratio Cr=xsize/popsize is called crossover rate, often Cr∈[0.5,0.9]; and the ratio Mr= msize/ popsize called mutation rate, often Mr∈[0.1,0.3]. So Cr + Mr =1. Cr and Mr are two important parameters that influence the convergent performance of GA. In general, if Mr increases, GA converges slowly on a final solution, thus, GA has more chances to arrive at better solutions. But if Mr is too large, GA tends to be like random search. In all the genetic algorithms in this paper, we let Cr=0.8 and Mr=0.2. Due to popsize=200, we get xsize=160 and msize=40. The termination criteria for GA is that the algorithm stops when the objective value difference between the worst chromosome and the best one in the current generation is equal to or less than 0.001.

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Figure 4. Flow chart of GA.

2.3.4. Simulation Experiments of GA Combined with Different Rules GA was combined with the seven rules in Table 3 respectively for solving Example 2-1. Table 5 presents the results of Example 2-1 by GA combined with different rules. For each method, 10 tests of computation were performed. The makespan of the optimal schedules is 28.31. It can be seen that from Table 5: (1) Rule 5 and Rule 7 are the best rules that enables the GA to obtain the optimal makespan in every test. (2) Rule 1, Rule 2 and Rule 4 are also good rules, but do not enable GA to obtain the optimal makespan in every test. (3) The other two rules are poor-quality rules that never enable the GA to obtain the optimal makespan. Rule 3 is the worst. An optimal schedule obtained by GA combined with Rule 7 is shown in Figure 5, where  = (8, 7, 4, 2, 1, 6, 9, 3, 10, 5), f() = Cmax = 28.31. Table 5. Results of Example 2-1 by GA combined with different rules (min Cmax)

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Method GA_R1 (FAU)* GA_R2 (SCT) GA_R3 (SPT) GA_R4 (EST) GA_R5 (SPsPT) GA_R6 (SCPT) GA_R7 (ECT) *

Best 28.31 28.31 36.31 28.31 28.31 30.53 28.31

Mean Cmax 28.36 29.85 36.31 28.36 28.31 30.53 28.31

Mean Dev. from opt. 0.17 5.44 28.26 0.17 0.00 7.84 0.00

Mean CPU time(s) 0.0715 0.066 1 (for infeasible schedules), or w = 1 (for feasible schedules). In our computation, w is randomly given as w  [1.01, 1.20] for infeasible schedules. We note that such penalty method makes the search time longer in solving large-size instances. One may argue why not use direct repair of solutions. Penalty methods to handle the prefixed constraints and the infeasibility arising in the course of solution search demonstrate complete different performance in meta-heuristics. From our experience, the adoption of penalty methods in the MILP model increases the solution time; but the penalty techniques used meta-heuristic methods increase solution speed. Furthermore, in meta-heuristics, sometimes the infeasible solutions encountered during the search process are bridges to find the optimal or near-optimal solutions. We have tested directly repairing solutions and found that repair wastes plenty of CPU time. Repair has no function to lead the search direction, but penalty has.

3.3.4. Comparison of GA and MILP In this sub-section, first, IGA is applied to Examples 3-1 and 3-2 for minimizing the total process time PT. In Examples 3-1 and 3-2, all orders can be completed before their due dates, so that only one IGA procedure to minimize PT is required. In the IGA procedure, every chromosome is synthesized into a schedule according to the following requirements for minimization of the total process time: (1) Backward Assignment; (2) Technique (similar to active scheduling) to reduce the idle times of the units and intermediate storage time of the semi-finished products; (3) For each stage, choose a proper position to assign an order according to Rule 8. The parameters used in IGA are Cr = 0.7, Mr = 0.3. The termination condition for IGA is that the algorithm stops when the objective value difference between the worst chromosome and the best one in the current generation is equal to or less than small value, like 0.001. Next, IGA is applied to Examples 3-1 and 3-2 for minimizing the total flow time F (He and Hui, 2010a). In the IGA procedure, every chromosome is synthesized into a schedule according to the requirements for minimization of the total flow time: the forward assignment strategy, the active scheduling technique and the position selection rule - Rule 7. The parameters and the termination condition are the same as previous. And then, IGA is applied to Example 3-3 for minimizing PT (He and Hui, 2010a). Example 3-3 is an example with tardy orders at the optimum, hence first minimize T and set soft due dates of the orders, then minimize PT. Table 7 presented the best results of Example 3-1 (with problem size from 5 to 24 orders) solved by IGA, OGA and MILP. Table 7 shows that both IGA and OGA perform much better than MILP in terms of solution quality and search time. Compared to OGA, IGA increases solution quality, though it requires a little more search time due to the position selection. With the increasing of the problem size, the difference increases. For the large-size 24-order instance, IGA performs much better. Figure 13 shows a schedule obtained by IGA, where no idle times are seen.

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Table 7. Best results of Example 3-1 by IGA, OGA and MILP (min PT) IGA/OGA

NM popsize

Iterations

CPU (s)

MILP Best PT

PT

Iterations

Diff1*

Diff2**

CPU (s)

55

200/200

5/10

0.055/