Studies in Quantitative Decision Making: Selected Papers from XXIII Annual International Conference of the Society of Operations Management (Asset Analytics) 9811658196, 9789811658198

Citation preview

Asset Analytics Performance and Safety Management Series Editors: Ajit Kumar Verma · P. K. Kapur · Uday Kumar

Diptesh Ghosh · Avijit Khanra · S. V. Vanamalla · Faiz Hamid · Raghu Nandan Sengupta   Editors

Studies in Quantitative Decision Making Selected Papers from XXIII Annual International Conference of the Society of Operations Management

Asset Analytics Performance and Safety Management

Series Editors Ajit Kumar Verma, Western Norway University of Applied Sciences, Haugesund, Rogaland Fylke, Norway P. K. Kapur, Centre for Interdisciplinary Research, Amity University, Noida, India Uday Kumar, Division of Operation and Maintenance Engineering, Luleå University of Technology, Luleå, Sweden

The main aim of this book series is to provide a floor for researchers, industries, asset managers, government policy makers and infrastructure operators to cooperate and collaborate among themselves to improve the performance and safety of the assets with maximum return on assets and improved utilization for the benefit of society and the environment. Assets can be defined as any resource that will create value to the business. Assets include physical (railway, road, buildings, industrial etc.), human, and intangible assets (software, data etc.). The scope of the book series will be but not limited to: • • • • • • • • • • • • •

Optimization, modelling and analysis of assets Application of RAMS to the system of systems Interdisciplinary and multidisciplinary research to deal with sustainability issues Application of advanced analytics for improvement of systems Application of computational intelligence, IT and software systems for decisions Interdisciplinary approach to performance management Integrated approach to system efficiency and effectiveness Life cycle management of the assets Integrated risk, hazard, vulnerability analysis and assurance management Adaptability of the systems to the usage and environment Integration of data-information-knowledge for decision support Production rate enhancement with best practices Optimization of renewable and non-renewable energy resources

Review Process The proposal for each volume follows multi-pronged review process. The first level of review (single blind) is conducted by the series editors who may or may not decide to enlist the help of editorial board members for a second level of review. Proposals may also undergo a third level of peer review (double blind) if recommended by the Series Editors. The series follows Ethics Statement found in the Springer standard guidelines here. https://www.springer.com/us/authors-editors/journal-author/journal-aut hor-helpdesk/before-you-start/before-you-start/1330#c14214

More information about this series at https://link.springer.com/bookseries/15776

Diptesh Ghosh · Avijit Khanra · S. V. Vanamalla · Faiz Hamid · Raghu Nandan Sengupta Editors

Studies in Quantitative Decision Making Selected Papers from XXIII Annual International Conference of the Society of Operations Management

Editors Diptesh Ghosh Production and Quantitative Methods Area Indian Institute of Management Ahmedabad Ahmedabad, Gujarat, India

Avijit Khanra Industrial and Management Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India

S. V. Vanamalla Industrial and Management Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India

Faiz Hamid Industrial and Management Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India

Raghu Nandan Sengupta Industrial and Management Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India

ISSN 2522-5162 ISSN 2522-5170 (electronic) Asset Analytics ISBN 978-981-16-5819-8 ISBN 978-981-16-5820-4 (eBook) https://doi.org/10.1007/978-981-16-5820-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To Manjari, Anusha, Amrita, baba, and ma —Diptesh Ghosh Dedicated to my parents —S. V. Vanamalla To all my Ph.D., M.Tech., and MBA students of IIT Kanpur —Faiz Hamid To all my Ph.D., M.Tech., and MBA students of IIT Kanpur —Raghu Nandan Sengupta

Preface and Introduction

In light of ever-increasing competition and environmental uncertainties, the management of resources has become a matter of critical importance for every organization. In this context, different areas of quantitative techniques, operations research (OR), decision sciences, etc., along with their applications in supply chain management (SCM), play important roles in the efficient and effective conversion of resources into desired products and services. To meet this demand, innovative mathematical tools are required and this is where this edited volume titled Studies in Quantitative Decision Making comes into relevance. This volume discusses some interesting papers from the XXIII Annual International Conference of Society of Operations Management, IIT Kanpur, India, 2019, in topics related to linear, nonlinear, pure integer, and mixed-integer programming. The volume consists of ten papers ranging from theoretical ones to interesting application work. Chapter “Inventory Optimization Under Lost Sales and Fractional Lead Times” analyzes a single item, single location, and periodic review inventory procurement problem where the demand is modeled as a continuous-time stochastic process, while lead time is also non-deterministic. Apart from considering order-up-to-S policies, the paper finds the ordering policy’s optimal properties also. The next chapter (“A New Insight on Order Quantity Allocation for Coordinated Multi-tier Multiple Supplier Networks Under Demand Uncertainty”) throws light on new insights on optimal order quality allocation decisions based on the problem which considers coordinated multi-tier multiple supplier’s network, operating under uncertain demand environment. An MINLP model is formulated considering multiple conflicting objectives. Two-stage simulation-based designed experimentation is also taken into consideration to identify important variables. Two sets of data based on automobile component are analyzed. This work proposes to help practitioners design sourcing strategies for a centralized multi-tier multiple supplier’s network. Chapter “Designing Distribution Network for Indian Agri-fresh Food Supply Chain” and “A Study on Inventory Models for Perishable Items in a Serial Supply Chain Operating with Price Markdowns” are related to agribusiness SCM. Chapter “Designing Distribution Network for Indian Agri-fresh Food Supply Chain” proposes vii

viii

Preface and Introduction

aggregate product movement in the context of Indian agri-fresh food supply chain to overcome high transportation cost and low profitability. To minimize total distribution cost, multi-period MINLP and MILP models for multi-product four-echelon supply chain system are formulated and solved. A real case study is considered, and sensitivity analysis is performed. Challenge for perishable goods to sell them before their shelf life expires is the theme of chapter “A Study on Inventory Models for Perishable Items in a Serial Supply Chain Operating with Price Markdowns,” whereby mathematical models to study pricing and inventory control of perishable items in a multistage supply chain with price-dependent demand are analyzed. Ordering policies for a multistage supply chain are analyzed, and a fresh produce supplier in India is studied. Two different cases of modes of inventory position calculations are performed. A capacitated vehicle routing problem (CVRP) is used to model supply chain delivery and pickup of goods in chapter “Material Flow Optimisation in a Manufacturing Plant by Real-Coded Genetic Algorithm (RCGA).” The authors solve both capacity and time using various heuristics and meta-heuristics methods which are developed and solved using GA. The multiple response optimization (MRO) problem is the idea in chapter “A Multi-objective Particle Swarm Optimisation-Based Solution Approach for Multiple Mean-Responses Optimisation Considering Empirical Model Uncertainties.” Responses are often correlated, and hence, trade-off solutions are inevitable. To derive improved and efficient solutions in the context of MRO, the authors propose the multi-objective particle swarm optimization-based solution approach. Two MCDM techniques are proposed to aid the decision maker. The motivation of chapter “A Mathematical Model for Scheduling and Rostering of Staff with Real-Life Considerations: The Case of Indian Call Centres” is to analyze the deterministic multi-skill employee call center problem that considers staff scheduling and rostering. Ideas related to the allocation of flexible shifts with the flexibility of starting the shift are considered, whereby the authors incorporate ideas related to days off scheduling and flexible break assignment. Real-life considerations demand the problem of shift scheduling and rostering to be solved in an integrated manner, and for this, an MILP formulation is proposed and solved. Chapter “Product Rate Variation (PRV) Problem and Its Variants” studies an interesting product rate variation (PRV) problem and some of its variants. Branch and bound algorithm is developed, and apart from that, a simple heuristic is designed and solved. A variety of problems related to scheduling round-robin tournaments and scheduling interviews in a business school context is discussed. The chapter also shows an application with numerical examples. The theme of bidirectional courier distribution network design problem for the multimodal load movements is at the heart of chapter “Optimized Multimodal Transportation for Efficient Parcel Movement in Courier Industry.” In order to deliver products at specified locations, given a specific turnaround time (TAT) the authors develop a hub-spoke model, whereby the movement of items across multiple hubs, depending

Preface and Introduction

ix

on available connectivity options, is formulated as a network flow problem. Important costs and time factors are considered for a better analysis of the multimodal transportation model. In order to minimize loss due to fire in public transport system or luxury bus, chapter “Enhancement for Easy Egress Through Emergency Gate of Bus” proposes the concept of improved type emergency egress doors, windows, and hatches and validates the same through a prototype design in a luxury bus. Ahmedabad, India Kanpur, India Kanpur, India Kanpur, India Kanpur, India

Diptesh Ghosh Avijit Khanra S. V. Vanamalla Faiz Hamid Raghu Nandan Sengupta

Acknowledgments

This edited volume “Studies in Quantitative Decision Making” is a part of a collection of selected papers from the XXIII Annual International Conference of the Society of Operations Management (SOM2019) held in December 2019 at IIT Kanpur, INDIA. First and foremost the editors would like to duely acknowledgement the unflinching support of Society of Operations Management and Indian Institute of Technology Kanpur, INDIA in facilitating the overall conduct of this international conference with a phenomenal success which was evident in its overwhelming participation response. The editors would also like to thank the department of Industrial & Management Engineering at Indian Institute of Technology Kanpur, INDIA for its abutment during each and every step of conducting SOM2019. The editors would also recognize the dedication, enthusiasm and detailed work of each and every reviewer who gave useful comments to make each and every paper apt, keeping in mind the general theme of SOM2019 as well as for this volume. Last but not the least appreciation is also due to Springer for making this volume possible.

xi

About this book

This edited volume is an in-depth collation of the usage of different quantitative decision making techniques in practical areas such as lean & green supply chain, reverse logistics, perishable logistics, closed loop supply chain, sustainable project management, retail management, block chain applications, optimal supplier selection problem, demand/supply modelling, forecasting under uncertainties, scheduling & sequencing, resource constraint logistics, dynamic network supply chain, risk evaluation, and so on. Additionally, the book also solves these issues in theoretical and practical context using innovative mathematical tools. Consisting of selected papers from the 23rd Annual International Conference of the Society of Operations Management, this book’s highlight is not only the coverage of interesting topics, but also how these topics are dealt with, such that post-graduate students as well as researchers and industry personnel working in areas like engineering, economics, social sciences, management, mathematics, etc., can derive the maximum benefit by reading or referring to this book. Apart from the emphasis on new mathematical, operations research, operations management, and statistical techniques, the authors also ensure that all the concepts are made clear by highlighting their practical significance in different areas of applications of operations management. By using novel presentation methods, the book offers a good practical flavor of all the different topics relevant to operations management in the coming decades.

Aim and Scope With a growing demand and urgent need for different mathematical, statistical, and quantitative tools for solving a variety of theoretical as well as practical SCM problems, there is an immediate and urgent need to come up with an interesting, in-depth, and updated review book that caters to the growing demand of academicians and practitioners alike. The innovative idea of this edited volume (which consists of 10 different interesting papers) is limited not only to the coverage of relevant topics xiii

xiv

About this book

but also to the fact how these topics are dealt with, such that postgraduate students as well as researchers from diverse fields working in SCM and decision making can derive the maximum benefit by reading/referring to this edited volume. Apart from the emphasis on techniques, the authors make all the relevant concepts clear by considering real-life data which brings an added flavor to this book. Thus, this edited volume is a sincere attempt to consolidate a whole gamut of tools, techniques, and methodologies developed and used in different areas of applications of SCM, logistics, design, etc. This edited volume, Studies in Quantitative Decision Making, will definitely do justice to the research in applied OM by covering different practical applications from a mathematical viewpoint along with different interesting examples and thus aid theoreticians and practitioners alike.

Key Features (1)

(2)

(3)

The edited volume covers a variety of relevant topics which are interesting to motivate interesting ideas of future research in the field of SCM and quantitative decision making. Rather than being the sole authority, this book attempts to complement as well as supplement the existing diverse application areas in SCM and quantitative decision making. The edited work deals with the theoretical concepts and considers nice data from the field of SCM which is in itself quite attractive.

Contents

Inventory Optimization Under Lost Sales and Fractional Lead Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ganesh Janakiraman and John A. Muckstadt

1

A New Insight on Order Quantity Allocation for Coordinated Multi-tier Multiple Supplier Networks Under Demand Uncertainty . . . . Avinash Bagul and Indrajit Mukherjee

23

Designing Distribution Network for Indian Agri-fresh Food Supply Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rakesh Patidar and Sunil Agrawal

49

A Study on Inventory Models for Perishable Items in a Serial Supply Chain Operating with Price Markdowns . . . . . . . . . . . . . . . . . . . . . . Saravanya Sankarakumaraswamy, Arshinder Kaur, Chandrasekharan Rajendran, and Hans Ziegler Material Flow Optimisation in a Manufacturing Plant by Real-Coded Genetic Algorithm (RCGA) . . . . . . . . . . . . . . . . . . . . . . . . . . K. C. Bhosale and P. J. Pawar

75

99

A Multi-objective Particle Swarm Optimisation-Based Solution Approach for Multiple Mean-Responses Optimisation Considering Empirical Model Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Abhinav Kumar Sharma and Indrajit Mukherjee A Mathematical Model for Scheduling and Rostering of Staff with Real-Life Considerations: The Case of Indian Call Centres . . . . . . . 143 Sweety Hansuwa and Chandrasekharan Rajendran Product Rate Variation (PRV) Problem and Its Variants . . . . . . . . . . . . . . 165 S. Rahul and G. Srinivasan

xv

xvi

Contents

Optimized Multimodal Transportation for Efficient Parcel Movement in Courier Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Mahidhar Surapaneni, Abhaya Kumar Sahoo, Sarang Jagdale, and Sonia Kushwaha Enhancement for Easy Egress Through Emergency Gate of Bus . . . . . . . 199 Vinay Kumar Singh, Rahul Kumar, and Rahul Varshney

About the Editors

Dr. Diptesh Ghosh is a professor at IIM Ahmedabad. He is Fellow from IIM Calcutta and B.Tech., Hons. From IIT Kharagpur. Prior to joining IIM Ahmedabad he was a postdoctoral researcher at the University of Groningen, and a faculty member at IIM Lucknow. His research focuses on the use of metaheuristics, specially applied to location, layout, and routing problems. He has published in several national and international journals such as Computers & Operations Research, Discrete Optimization, European Journal of Operational Research. He serves on the editorial board of OPSEARCH, the official publication of the Operational Research Society of India (ORSI). Dr. Avijit Khanra is a faculty member in the Department of Industrial and Management Engineering of Indian Institute of Technology, Kanpur. He obtained his PhD from Indian Institute of Management, Ahmedabad. His research interests are in the areas of Inventory Control, Supply Chain Management, Scheduling, and Mathematical Modelling and Optimization. His research work has been published in European Journal of Operational Research, Operations Research Letters, etc. He has also been involved in industrial consultancy.

xvii

xviii

About the Editors

Dr. S. V. Vanamalla is currently an Associate Professor in the Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur. She obtained her PhD from the Department of Management Studies, Indian Institute of Science Bangalore. Her research interests include Operations Research and Game Theory. Some of her research works have been published in Journal of the Operational Research Society and Annals of Operations Research.

Dr. Faiz Hamid is Assistant Professor in the Department of Industrial & Management Engineering at Indian Institute of Technology Kanpur. He obtained his PhD from Indian Institute of Management Lucknow and PDF from Telecom SudParis, France. His research work has been published in international journals of repute such as Networks, Annals of Operations Research, IEEE Transactions on Signal Processing, etc. His areas of research interest are Combinatorial Optimization applied to Network Optimization and Transportation, and Applications of Data Mining.

Dr. Raghu Nandan Sengupta is Professor, Department of Industrial & Management Engineering, Indian Institute of Technology Kanpur. He obtained his PhD from Indian Institute of Management Calcutta and PDF from Princeton University, USA. His research interests are in areas of Sequential Estimation, Statistical and Mathematical Reliability Theory, Risk Analysis, Optimization Techniques in Finance, Meta Heuristic Techniques, Reliability based Optimization, Robust Optimization. His research work has been published in EJOR, QF, CSDA, Communication in Statistics, Journal of Applied Statistics, Metrika, Statistics, etc. He has also edited the book titled Decision Sciences: Theory and Practice, CRC Press, 2016.

Inventory Optimization Under Lost Sales and Fractional Lead Times Ganesh Janakiraman and John A. Muckstadt

1 Introduction We study a single item, single location and periodic review inventory procurement system which operates as follows. At the beginning of each period, a procurement order is placed on a supplier. The lead time for an order is a random variable, which is strictly positive but always assumes a value less than the length of a period. Demand occurs throughout the period. Demand is assumed to be described by a continuous random variable. All demand that arises before the receipt of this order in excess of the inventory on hand at the time the order is placed is assumed to be lost. Next, the inventory that was ordered at the beginning of that period arrives, which increases the quantity of stock on hand. Following the receipt of this order, random demand continues to occur. Demand occurring from the time just after the receipt of this period’s order until the placement of the next order in excess of supply is also lost. We also assume demands are independent and identically distributed from period to period. However, non-stationarity of demand throughout a period and correlation of demand within a period are permitted. Slightly more restrictive assumptions about the demand process and the lead times will be specified in the statement of two propositions we make in Sect. 4. We assume linear purchase costs and linear holding costs. Holding costs are charged at the beginning of every period based on inventory on hand. At the end of a period, linear lost sales costs are charged proportional to the number of units of sales lost during that period. The discounted sum of these expected period costs over a finite horizon represents the problem’s objective function.

G. Janakiraman (B) Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, USA e-mail: [email protected] J. A. Muckstadt Operations Research and Information Engineering, Cornell University, Ithaca, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_1

1

2

G. Janakiraman and J. A. Muckstadt

Our goals in this paper are to (1) show the objective function is a convex function, (2) establish properties of the optimal ordering policy, (3) determine bounds on the optimal order quantities, (4) derive monotonicity properties for the probability of not stocking out and on-hand stock at the end of a period as a function of lead time length and (5) prove a convexity property of the objective function when the commonly used order-up-to or base stock policy is followed. To place our problem in context, consider a retailer who places replenishment orders at the beginning of every week and receives delivery within a week—in other words, the length of a period in this example is a week and the lead time is between zero and one week. In retail, any demand that occurs when the product is out of stock is typically lost—thus, the assumption of lost sales of unmet demand is appropriate. Another application context is that of service parts management. Consider a firm which uses a high value resource to generate revenues—this resource could be an airplane in the airline industry or a fab in the semiconductor industry. Let us say that components or service parts required to keep the resource productive are replenished daily and that the deliveries happen within 24 h. However, if a component required to fix a resource that has broken down is out of stock, then an emergency supply source is used to procure a unit of that component as opposed to wait for the scheduled replenishment. This emergency procurement is considerably more expensive than regular replenishment but provides the advantage of immediate delivery. Here, every unit of demand served using the emergency source can be viewed as a “lost sale,” and the premium paid for emergency procurement over regular replenishment per unit can be viewed as the “lost sales penalty cost.” The problem we study in this paper applies to both the retail and service parts examples discussed above. The study of optimal policies in inventory theory is largely based on the assumption that excess demand is backordered. However, the lost sales inventory problem with lead times has been recognized to be an important problem in terms of practical importance and as a mathematically challenging problem to solve/analyze. The first attempt at studying this problem was the seminal paper by Karlin and Scarf (1958) where the problem is studied when the lead time is exactly one period in length. Some properties on the structure of the optimal property—monotonicity of the order quantity with respect to the inventory on hand, for instance—are derived. Later, Morton (1969) generalized these basic results to the periodic review lost sales problem with an arbitrary, but fixed lead time that is an integer number of periods. In addition, he developed easily computable bounds on the optimal order quantity for a given inventory vector (the vector of the quantities of inventory in different stages of the pipeline). The next steps of progress in analyzing the lost sales inventory problem occurred much later but, relatively speaking, in quick succession. The reader is referred to the survey paper by Bijvank and Vis (2011) for a detailed discussion of this literature. Some notable results include the asymptotic optimality of base stock/order-up-to policies when the shortage cost approaches ∞ (Huh et al., 2009) and the asymptotic optimality of constant order policies when the lead time approaches ∞ (Goldberg et al., 2016; Xin & Goldberg, 2016). Another recent paper (Xin, 2021) proposes a policy that combines the structures of order-up-to and constant order policies and establishes its effectiveness.

Inventory Optimization Under Lost Sales …

3

It is important to note that almost all papers that deal with periodic review inventory procurement problems assume that lead times are integral multiples of the period lengths. Our problem differs from the existing literature on this count. Our assumption that lead times are between zero and the length of a period may appear to be a minor alteration of the ones made by others; doing so, however, surprisingly complicates the analysis significantly. Also, by permitting non-stationarity of demand throughout a period and correlation of demand within a period, we also extend previous results. The remainder of this paper is organized as follows. In Sect. 2, we describe the notation and assumptions used throughout the paper. In Sect. 3, we develop key analytical results about the optimal ordering quantities. In Sect. 4, we consider the case when the lead times are deterministic and present two results on the effect of increasing the lead times. In Sect. 5, we restrict attention to the class of order-up-to policies and prove a useful convexity result. Finally, we conclude the paper in Sect. 6. The results developed in Sect. 3 are extensions of those found in Karlin and Scarf (1958) and Morton (1969). Using methods similar to theirs, we are able to prove the convexity of the cost function, monotonicity of order quantities as a function of on-hand stock at the beginning of a period and some bounds on the order quantity. In Sect. 5, we consider the class of order-up-to policies. While these policies are known to be suboptimal for lost sales problems, they are commonly used in practice. See Karlin and Scarf (1958) and Nahmias (1979), for example.

2 Notation and Assumptions 2.1 Notation Assume that there are N periods in the planning horizon which we index in a backward fashion,1 i.e., period N − 1 occurs after period N , and period 1 is the last period in the planning horizon. As we mentioned earlier, a linear purchase cost is allowed; but it is easy to show that this problem is equivalent to a problem where the purchase cost is zero. This transformation is useful in simplifying the analysis. To make this simplification possible, we assume that the on-hand inventory at the end of period 1 can be salvaged at the purchase price.

1

The optimal ordering decision in a period depends on the current inventory and the parameters describing the problem in the remaining periods and not on the parameters of the problem in the past. Thus, the optimal decision in a period depends on the number of periods remaining in the horizon, hence, the convention of numbering periods backwards.

4

G. Janakiraman and J. A. Muckstadt

Let us now state the notation that is used in this paper. α = discount factor. xn = on-hand inventory at the beginning of period n. x0 = on-hand inventory at the end of period 1. qn = quantity ordered in period n. Dn1 = random variable representing the demand that occurs between the start of period n and the time the order of size qn is received. Dn = random variable representing the total demand that occurs in period n. Dn2 = Dn − Dn1 = the random variable representing the demand that occurs between the time that we receive the order of size qn and the end of period n. D n = the random vector (Dn1 , Dn2 ) . yn = on-hand inventory just after receiving qn . = (xn − Dn1 )+ + qn . h = holding cost incurred per unit of inventory (charged at the beginning of a period). b = lost sales cost incurred for every unit of sales lost during a period (charged at the end of a period). c = purchase cost per unit (charged at the beginning of a period). The dynamics of the system’s operation as discussed in Sect. 1 are depicted in Fig. 1. The timing of events and the manner in which costs are incurred are critical to our analysis. Each period is linked to the following one through the evolution of

qn An Demand = Dn1

Beginning of period n On Hand Inventory = xn Place order for qn units

Shipment qn qn-1 arrives. Inventory = yn Bn A(n-1) Demand =Dn2

Demand =D(n-1)1

Shipment qn-1 arrives. Inventory = yn-1 B(n-1) Demand =D(n-1)2

Beginning of period n-1 On Hand Inventory = xn-1 Place order for qn-1 units

Fig. 1 Dynamics of the order/demand process

qn-2

Beginning of period n-2 On Hand Inventory = xn-2 Place order for qn-2 units

Inventory Optimization Under Lost Sales …

5

the on-hand inventory at the beginning of a period. The equation that describes this relationship from period n to n − 1 is given by xn−1 = ((xn − Dn1 )+ + qn − Dn2 )+ .

(1)

Next, we define the one-period cost and the expected discounted cost. Note that all the expectation operators in this paper will be subscripted with the set of random vectors or variables over which the expectation is taken. Vn (xn , qn ) = expected period n purchase, holding and lost sales costs if we start period n with xn units of inventory on hand and we order qn units   = E D n c · qn + h · xn + b · (Dn1 − xn )+ + b · (Dn2 − yn )+ .

(2)

f n (xn ) = minimum expected sum of all discounted future costs over the planning horizon if we start period n with xn units of inventory on hand   = min Vn (xn , qn ) + α · E D n [ f n−1 (xn−1 )] . qn ≥0

(3)

f n (xn , qn ) = minimum expected sum of all discounted future costs if we start period n with xn units of inventory on hand and if we order qn units in period n qn∗ (x)

= Vn (xn , qn ) + α · E D n [ f n−1 (xn−1 )].

(4)

= arg min ( f n (x, q)) , and therefore q≥0

f n (x) = f n (x, qn∗ (x)).

In addition to these costs incurred in periods N , N − 1, . . ., 1 there is an end of horizon cost f 0 (x0 ) which depends on x0 , the inventory on hand at the end of period 1. f 0 (x0 ) = cost incurred at the end of the horizon, i.e., the end of period 1 = h · x0 − c · x0 .

(5)

Observe that at the end of the horizon, we are charged a holding cost for the inventory on hand less the salvage value associated with that inventory. The finite horizon problem is to determine the optimal ordering policy, that is, the optimal order function qn∗ (x), for all n ∈ {N , N − 1, . . . , 1} and for all x ≥ 0.

6

G. Janakiraman and J. A. Muckstadt

2.2 Assumptions The following conditions are assumed throughout the paper unless specifically stated to be otherwise. 1. 0 ≤ α ≤ 1. 2. The cost parameters satisfy the relation α(b − h) ≥ c (such a set of costs will be called “valid"). This implies that the present value of losing a sale in the period is greater than the purchase price of the unit plus the present value of holding that unit in inventory throughout the period. Thus, if the sale of a unit of stock is certain to occur some time during the period, it will be best to purchase the unit of stock. 3. {D n } = {(Dn1 , Dn2 )} is a sequence of i.i.d. random vectors. 4. All the random variables representing demand possess continuous density functions. This assumption implies the existence of all the derivatives that appear in Sect. 3. Let us now define the necessary probability density functions and cumulative distribution functions.

φ1 (u 1 ) = the probability density of Dn1 at u 1 . 1 (u 1 ) = the cumulative distribution function of Dn1 at u 1 = P( Dn1 ≤ u 1 ). φ2 (u 2 ) = the probability density of Dn2 at u 2 . 2 (u 2 ) = the cumulative distribution function of Dn2 at u 2 = P( Dn2 ≤ u 2 ). φ(u) = the probability density of Dn at u. (u) = the cumulative distribution function of Dn at u = P( Dn ≤ u ).  ∗ 1 (u) = P( Dn + Dn−1,1 ≤ u ). 1 (u 1 ) = the complementary cumulative distribution function of Dn1 at u 1 = 1 − 1 (u 1 ). 2 (u 2 ) = the complementary cumulative distribution function of Dn2 at u 2 = 1 − 2 (u 2 ). ˜ (u) = P(Dn2 + D(n−1)1 ≤ u). φ12 (u 1 , u 2 ) = probability density function of D n at (u 1 , u 2 ). The randomness in the lead time is captured in the probability density function of D n = (Dn1 , Dn2 ).

Inventory Optimization Under Lost Sales …

7

3 Properties of the Optimal Ordering Function Our first lemma states that the finite horizon problem with a positive unit purchase cost of c can be transformed into a finite horizon problem with zero purchase cost per unit. This transformation is useful in simplifying the analysis. In the following lemma, the superscript denotes the purchase, holding and lost sales cost parameters in that order. ˜ b), ˜ ∃ another set of cost paramLemma 1 For all valid sets of cost parameters (c, ˜ h, eters (0, h, b) such that ˜ ˜

f n(c,˜ h,b) (xn , qn ) = f n(0,h,b) (xn , qn ) + a ter m independent o f the policy. and b ≥ h. Proof This result is a straightforward consequence of the analysis in Janakiraman and Muckstadt (2004).  As a consequence of this lemma, we know that the optimal ordering policy for a ˜ b) ˜ is the same as the optimal policy for an equivalent problem with parameters (c, ˜ h, transformed problem with parameters (0, h, b). From now on, we will be working only with the transformed problem. In other words, we will assume that the purchase cost c is zero in the remainder of the paper. We now derive basic properties of the function f n and the optimal ordering function, qn∗ . The key facts that we present in the following theorem are (i) the function f n (x) is convex, (ii) there exists a limiting quantity x¯n on the on-hand inventory beyond which the optimal order quantity is zero, and (iii) the optimal ordering function qn∗ (x) is a non-increasing function with slope greater than -1, that is qn∗ (x + 1) − qn∗ (x) ≤ 1∀x ≥ 0. Theorem 1 For any period 1 ≤ n ≤ N ,

(a) f n (x) ≥ 0 ∀ x ≥ 0,

(b) f n (∞) ≥ 0, (c) Fn (x, q) =

(6) (7)

∂ f n (x, q) increases with q ∀ x ≥ 0, ∂q

(d) f n (x) ≥ h − b ∀ x ≥ 0, (e) ∃ x¯n such that qn∗ (x) > 0 if and only if x < x¯n , dq ∗ (x) ≤ 0 ∀ x ≥ 0. (f) −1 ≤ n dx

(8) (9) (10) (11)

Proof The proof is by induction. Recall that f 0 (x) = h · x. Thus, statements (a), (b) and (d) are trivially true for f 0 . Now, we assume that the statements (a), (b) and (d) in the theorem are true for functions f 0 , f 1 , . . . , f n−1 . Now, let us show that all the statements (a)–(e) are true for period n as well.

8

G. Janakiraman and J. A. Muckstadt

Next, to simplify notation, let θ = ((x − u 1 )+ + qn∗ (x) − u 2 ), θ1 = x − u 1 + qn∗ (x) − u 2 , θ2 = qn∗ (x) − u 2 where u 1 represents a realization of Dn1 and u 2 represents a realization of Dn2 . From (2) and (4), we get f n (x, q) = h · x + b · E Dn [(Dn1 − x)+ ] + b · E Dn [(Dn2 − ((x − Dn1 )+ + q)+ ] +α · E Dn [ f n−1 (((x − Dn1 )+ + q − Dn2 )+ )]. (12) Differentiating this expression with respect to q, we get Fn (x, q) = −b · P((x − Dn1 )+ + q < Dn2 )

+α · E D n [ f n−1 ((x − Dn1 )+ + q − Dn2 )1((x − Dn1 )+ + q − Dn2 > 0)] (13)  x  ∞  ∞  ∞ = −b φ12 (u 1 , u 2 ) du 2 du 1 − b φ12 (u 1 , u 2 ) du 2 du 1 u 1 =0 u 2 =x−u 1 +q  x−u 1 +q x

u 1 =x

 +α

 +α

u 1 =0 u 2 =0 ∞  q u 1 =x

u 2 =0

u 2 =q

f n−1 (x − u 1 + q − u 2 )φ12 (u 1 , u 2 ) du 2 du 1



f n−1 (q − u 2 ) φ12 (u 1 , u 2 ) du 2 du 1 .

(14)

But Fn (x, q) is increasing in q and is non-negative for sufficiently high values of q. (x,q) equals Also, Fn (x, q) increases in x. Note that qn∗ (x) is the value at which ∂ Fn∂q zero. This implies that ∃x¯n such that qn∗ (x) > 0 if and only if x < x¯n .

(15)

Thus, we have shown statements (c) and (e).

d f n (x, qn∗ (x)) dx = h − b · P(x < Dn1 ) − b · P(x > Dn1 , θ1 < 0) +α · E Dn [ f n−1 (θ + )1(x > Dn1 , θ > 0)]  x  ∞ = h − b · 1 (x) − b · φ12 (u 1 , u 2 ) du 2 du 1

f n (x) =

 +α ·

x

u 1 =0



u 2 =x−u 1 +qn∗ (x)

u 1 =0

x−u 1 +qn∗ (x) u 2 =0

×(u 1 , u 2 ) du 2 du 1 .

(16)



f n−1 (x + qn∗ (x) − u 1 − u 2 ) φ12 (17)

Inventory Optimization Under Lost Sales …

9





But, f n (0) = h − b and, by assumption statement (d) holds for n − 1. Thus, f n (x) ≥ h - b. Also f n (∞) ≥ 0, using (16), and statement (b) which holds for n − 1. Thus, we have shown statements (b) and (d). Differentiating both sides of Eq. (17), we get

f n (x) = b · φ1 (x) + b · −b · +α ·

 ∞ u 2 =qn∗ (x)

 x

  dq ∗ (x) φ12 (u 1 , x − u 1 + qn∗ (x)) du 1 · 1 + n dx u 1 =0

 x

φ12 (x, u 2 ) du 2

 x−u +q ∗ (x) 1 n

u 1 =0 u 2 =0



 dq ∗ (x) f n−1 (x + qn∗ (x) − u 1 − u 2 ) · 1 + n dx

×φ12 (u 1 , u 2 ) du 2 du 1

 x dq ∗ (x) f n−1 (0)φ12 (u 1 , x − u 1 + qn∗ (x)) · (1 + n ) du 1 +α · dx u 1 =0  q ∗ (x) n +α · f n−1 (qn∗ (x) − u 2 ) φ12 (x, u 2 ) du 2 . u 2 =0

(18)

Along the curve y = qn∗ (x), Fn (x, qn∗ (x)) = 0∀x ∈ (0, x¯n ). Let H (x) = Fn (x, qn∗ (x)). Thus dH = 0, 0 < x < x¯n and dx  ∗      dH ∂qn (x) ∂ Fn (x, q) ∂ Fn (x, q) , when 0 < x < x¯n . + = dx ∂x ∂q ∂x q=qn∗ (x) q=qn∗ (x)

Observe that

 x  dH ∂q ∗ (x) φ12 (u 1 , x − u 1 + qn∗ (x)) du 1 = (b + α · f n−1 (0)) 1 + n dx ∂x u 1 =0  ∗  ∞ ∂qn (x) +(b + α · f n−1 (0)) φ12 (u 1 , qn∗ (x)) du 1 ∂x u 1 =x    x  x−u 1 +qn∗ (x) ∂q ∗ (x) +α f n−1 (x + qn∗ (x) − u 1 − u 2 )φ12 (u 1 , u 2 ) du 2 du 1 · 1 + n ∂x u 1 =0 u 2 =0   ∗  ∞  qn∗ (x) (x) ∂q n +α φ12 (u 1 , u 2 )du 2 du 1 in the region 0 Dn1 , θ1 < 0)



+α E Dn [ f n−1 (θ + )1(θ > 0)] − α E Dn [ f n−1 (θ + )1(x < Dn1 , θ > 0)] (from (16)) = h − b P(x < Dn1 ) − b P(x > Dn1 , θ1 < 0) +[Fn (x, qn∗ (x)) + b P(θ < 0)] − α E Dn [ f n−1 (θ + )1(x < Dn1 , θ > 0)]

Inventory Optimization Under Lost Sales …

11

(from Eq. (13)) = h − b P(x < Dn1 ) − b P(x > Dn1 , x + qn∗ (x) < Dn1 + Dn2 )

+b P((x − Dn1 )+ + qn∗ (x) < Dn2 ) − α E Dn [ f n−1 (θ + )1(x < Dn1 , θ > 0)] (since Fn (x, qn∗ (x)) = 0) = h − b P(x < Dn1 ) + b P(x < Dn1 , qn∗ (x) < Dn2 )

−α E Dn [ f n−1 (qn∗ (x) − Dn2 )1(x < Dn1 , qn∗ (x) > Dn2 )] = h − b P(x < Dn1 , qn∗ (x) > Dn2 )

−α E Dn [ f n−1 (qn∗ (x) − Dn2 )1(x < Dn1 , qn∗ (x) > Dn2 )].

Evaluating the above expression at x = x¯n , we see that f n (x¯n ) = h by using the fact that qn∗ (x¯n ) = 0 ∀ n.



Since f n−1 (x¯n ) = h and f n (·) is convex,

f n (x) ≥ h − (b + αh)P(x < Dn1 , qn∗ (x) > Dn2 ) ≥ h − 1 (x)(b + α h) = −(b + α h − h) + 1 (x)(b + α h) which establishes one side of the inequality stated in the theorem. Now, we prove the other side of the inequality. Rewriting (16), we get

f n (x) = h − b P(x < Dn1 ) − b P(x > Dn1 , x + qn∗ (x) < Dn1 + Dn2 )

+α E D n [ f n−1 ((x − Dn1 )+ + qn∗ (x) − Dn2 )1(x > Dn1 , x + qn∗ (x) > Dn1 + Dn2 )]

= h − b P(x < Dn1 ) − b P(x > Dn1 , x + qn∗ (x) < Dn1 + Dn2 )

+α E D n [ f n−1 (x − Dn1 + qn∗ (x) − Dn2 )1(x > Dn1 , x + qn∗ (x) > Dn1 + Dn2 )] ≤ h − b1 (x) + α E[h1(x > Dn1 , x + qn∗ (x) > Dn1 + Dn2 )]

(since f n−1 (x − Dn1 + qn∗ (x) − Dn2 ) ≤ f n−1 (x¯n ) = h) ≤ h − b1 (x) + αh1 (x) ≤ (h − b) + (b + αh)1 (x),

which proves the other side of the inequality stated in the theorem.



Next, we construct bounds on the probability of not running out of stock in the interval of time between the receipt of two successive orders. Let πn (x, q) = P((x − Dn1 )+ + q ≥ Dn2 + D(n−1)1 ) = P(x > Dn1 , x + q − Dn1 ≥ Dn2 + D(n−1)1 ) πn∗ (x)

+ P(x < Dn1 , q ≥ Dn2 + D(n−1)1 ), and = πn (x, qn∗ (x)).

12

G. Janakiraman and J. A. Muckstadt

πn (x, q) is the probability that we will not run out of stock in the time between the receipt of the order for q units, given x are on hand when the order is placed, and the receipt of the subsequent order. But   πn (x, q) = E Dn E D(n−1)1 1((x − Dn1 )+ + q − Dn2 ≥ D(n−1)1 )

 = E Dn 1 ((x − Dn1 )+ + q − Dn2 ) . The following theorem gives bounds on the probability of not stocking out while b−h b and U = b+αh . using the optimal policy. Let L = b+αh Theorem 3 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N , L ≤ πn∗ (x) ≤ U. Proof 0 = Fn (x, qn∗ (x))

= −b P((x − Dn1 )+ + qn∗ (x) < Dn2 ) + α E D n [ f n−1 (θ + )1(θ > 0)]

= −b P(θ < 0) + α E D n [ f n−1 (θ)1(θ > 0)]  = −b + E D n (b + α · f n−1 (θ))1(θ > 0) ≥ −b + E D n [(b + α · [(h − b − α · h) + 1 (θ)(b + α · h)])1(θ > 0)] (using Theorem 2)   = −b + E D n E D n−1 (b + α · (h − b − α · h) + α(b + α · h)1(D(n−1)1 < θ))1(θ > 0) (because 1 (θ) = P(D(n−1)1 < θ) = E D n−1 [1(D(n−1)1 < θ)])   ≥ −b + E D n E D n−1 (b + α · (h − b − α · h) + α(b + α · h)1(D(n−1)1 < θ))1(θ > D(n−1)1 ) (because 1(θ > 0) ≥ 1(θ > D(n−1)1 )) = −b + (b + α · h)P(θ > D(n−1)1 )

= −b + (b + α · h)P((x − Dn1 )+ + qn∗ (x) − Dn2 > D(n−1)1 ) = −b + (b + α · h)πn∗ (x)

(from the definition of πn∗ (x)).

b Therefore, we have πn∗ (x) ≤ b+αh . To prove the other inequality, we again start with the expression Fn (x, qn∗ (x)) = 0.

0 = Fn (x, qn∗ (x))

(19)

= −b + b P((x − Dn1 )

+

+ qn∗ (x)



+

> Dn2 ) + α E D n [ f n−1 (θ )1(θ > 0)]



= −b + E D n [(b + α f n−1 ((x − Dn1 )+ + qn∗ (x) − Dn2 ))1((x − Dn1 )+ + qn∗ (x) > Dn2 )] ≤ −b + E D n [(b + α(h − b) + α(b + αh)1 ((x − Dn1 )+ + qn∗ (x) − Dn2 )) · 1((x − Dn1 )+ + qn∗ (x) > Dn2 )] (using Theorem 2) ≤ −b + E D n [b + α(h − b) + α(b + αh) · 1 ((x − Dn1 )+ + qn∗ (x) − Dn2 )]

Inventory Optimization Under Lost Sales …

13

(since 1((x − Dn1 )+ + qn∗ (x) > Dn2 ) ≤ 1

and (b + α(h − b) + α(b + αh) · 1 ((x − Dn1 )+ + qn∗ (x) − Dn2 )) ≥ 0)

= α(h − b) + α(b + αh)πn∗ (x) (using Eq. (19)).

Therefore, πn∗ (x) ≥

b−h b + αh



Next, we state easily computable bounds on the optimal order quantity qn∗ (x). Lemma 2 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N , ˜ −1 (U ). qn∗ (x) ≤  Proof ˜ n∗ (x)) = P(D(n−1)1 + Dn2 ≤ qn∗ (x)) (q ≤ P(D(n−1)1 + Dn2 ≤ (x − Dn1 )+ + qn∗ (x)) = πn∗ (x) ≤ U.



Another upper bound on qn∗ (x) is established in Lemma 3 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N , qn∗ (x) ≤ ( ∗ 1 )−1 (U ) − x, or x + qn∗ (x) ≤ ( ∗ 1 )−1 (U ). Proof  ∗ 1 (x + qn∗ (x)) = P(Dn1 + Dn2 + D(n−1)1 ≤ x + qn∗ (x)) = P(Dn2 + D(n−1)1 ≤ x − Dn1 + qn∗ (x)) ≤ P(Dn2 + D(n−1)1 ≤ (x − Dn1 )+ + qn∗ (x)) = πn∗ (x) ≤ U. We now establish a lower bound on qn∗ (x), as shown in Lemma 4 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N , ˜ −1 (L) − x. qn∗ (x) ≥ 



14

G. Janakiraman and J. A. Muckstadt

Proof L ≤ πn∗ (x) = P((x − Dn1 )+ + qn∗ (x) ≥ Dn2 + D(n−1)1 ) ˜ + qn∗ (x)). ≤ P(x + qn∗ (x) ≥ Dn2 + D(n−1)1 ) = (x When x = 0, qn∗ (0) = q¯n ; consequently,



˜ −1 (L). q¯n ≥ 

(20)

A second lower bound on qn∗ (x) is stated in Lemma 5 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N ,  ˜ −1 (U ))1 (x) . x + qn∗ (x) ≥ ( ∗ 1 )−1 L − 1 ( Proof L ≤ πn∗ = P(x ≥ Dn1 , x + qn∗ (x) ≥ Dn1 + Dn2 + D(n−1)1 ) + P(x ≤ Dn1 , qn∗ (x) ≥ Dn2 + D(n−1)1 ) ≤  ∗ 1 (x + qn∗ (x)) + P(x ≤ Dn1 , qn∗ (x) ≥ D(n−1)1 ) =  ∗ 1 (x + qn∗ (x)) + 1 (x)1 (qn∗ (x)) ˜ −1 (U )), ≤  ∗ 1 (x + qn∗ (x)) + 1 (x)1 ( ˜ n∗ (x)) ≤ U from Lemma 2. since (q



A third lower bound on qn∗ (x) is given by Lemma 6 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N , qn∗ (x) ≥ −1 1



L − U 1 (x) 1 (x)

.

Proof Observe from Theorem 2 that

f n (qn∗ (x)) ≤ (h − b) + (b + αh)1 (qn∗ (x)).

Note also that Eq. (19), along with the fact that f n−1 (·) is an increasing function, implies that

f n (x) ≥ h − b P(x < Dn1 , qn∗ (x) > Dn2 )

−α E Dn [ f n−1 (qn∗ (x))1(x < Dn1 , qn∗ (x) > Dn2 )]

≥ h − b P(x < Dn1 ) − α E Dn [ f n−1 (qn∗ (x))1(x < Dn1 )]. Combining (21) and (21), we get

(21)

Inventory Optimization Under Lost Sales …

  f n (x) ≥ h − b + α(h − b) + α(b + α · h)1 (qn∗ (x)) 1 (x).

15

(22)

Finally, using (22) and Theorem 2, we see that (h − b) + (b + α · h)1 (x) ≥ h−[b + α(h − b) + α(b + α · h)1 (qn∗ (x))]1 (x). Simplifying this equation and using the definitions of L and U yield the result.



We can also state a lower bound on x¯n as follows. Lemma 7 x¯n ≥ ( ∗ 1 )−1 (L). Proof Recall that by definition qn∗ (x¯n ) = 0 and πn∗ (x¯n ) = πn (x¯n , 0). But πn∗ (x¯n ) = P(x¯n ≥ Dn1 + Dn2 + D(n−1)1 ) = ( ∗ 1 )(x¯n ) ≥ L .



Thus, we have developed a series of bounds on the optimal order quantity qn∗ (x). These bounds can potentially be used to establish heuristics for computing order quantities.

4 Monotonicity Properties with Respect to the Lead Times In this section, we assume that the lead time for all the orders is a known constant between 0 and the length of one period. For simplicity, we assume that time is scaled so that the length of a period is one time unit. Thus, the order lead time, L, is less than 1. We examine the effect of increasing the lead time on the probability of not stocking out and on the on-hand inventory at the end of a period and at the time of receipt of the next period’s order. We will show these results by examining two situations, one when the lead time is L and the other, when the lead time is L + , where 0 < L +  ≤ 1. Also, to prove these lemmas, we make the following two assumptions about the demand process. 1. For any (t1 , t2 ) such that 0 ≤ t1 ≤ t2 ≤ 1, Dn (t1 , t2 ) is i.i.d., from period to period where Dn (t1 , t2 ) is the random variable for the demand that occurs in the interval (t1 , t2 ) of period n. 2. The stochastic process Dn (0, t) has independent increments. That is, Dn (t1 , t2 ) and Dn (t3 , t4 ) are probabilistically independent if (t1 , t2 ) and (t3 , t4 ) are disjoint subintervals of the interval (0, 1). The first lemma in this section says that the probability of not stocking out is a decreasing function of the lead time for a given pair (x, q), that is, when we start

16

G. Janakiraman and J. A. Muckstadt

period n with x units on hand and we place an order for q units. Before stating the lemma, let us define πnL (x, q) and πn(L+) (x, q) to be the same as πn (x, q) as defined in Sect. 3 when the lead time is L and L + , respectively. Lemma 8 For all x ∈ [0, x¯n ] and any 1 ≤ n ≤ N , πnL (x, q) ≥ πn(L+) (x, q). Proof Let the superscript L (and L + ) represent the order lead time in the situation L L + D(n−1)1 has the same distribution considered. As a result of our assumptions, Dn2 L+ L+ as Dn2 + D(n−1)1 . Let us define D as a random variable with this distribution. Since L+ L ≥ Dn1 for every sample path, Dn1 (L+)

πn

L+ + L )+ + q ≥ D) = π L (x, q). (x, q) = P((x − Dn1 ) + q ≥ D) ≤ P((x − Dn1 n

 In other words, Lemma 8 says that for the same on-hand inventory x, the order size q that is required to maintain a certain probability of not stocking out increases with the lead time. Next, we compare the on-hand inventory random variable at the end of the current period and at the time of receiving an order in the next period in the two situations. Denote the on-hand inventory at the end of period n by I AL and at the time of receiving the order qn−1 (in period n − 1) by I BL when the lead time is L. The corresponding quantities when the lead time is L +  are I AL+ and I BL+ . (The subscripts A and B refer to the time points An−1 and Bn−1 of Fig. 1m, respectively.) Lemma 9 For a given pair (xnL , qnL ) = (xnL+ , qnL+ ) = (x, q),

∀s ≥ 0, P(I BL+

I AL+ ≥ I AL , ≥ s) ≤ P(I BL ≥ s).

L+ L − Dn1 . Clearly, ≥ 0 for any Proof Define the random variable to be Dn1 sample path of the stochastic process Dn (0, t). In addition, for every sample path of L+ L+ L+ L L L + Dn2 = Dn1 + Dn2 by definition. Then, we have Dn2 = Dn2 − Dn (0, t), Dn1 . Let us compare I AL and I AL+ for any such sample path of demands. L+ + L+ + ) + q − Dn2 ) I AL+ = ((x − Dn1 L + L = ((x − Dn1 − ) + q − Dn2 + )+ L + L ≥ ((x − Dn1 ) − + q − Dn2 + )+ L + L + = ((x − Dn1 ) + q − Dn2 ) = I AL

Now let us prove the second part.

Inventory Optimization Under Lost Sales …

17

L L Our assumptions about the demand process imply that (Dn2 + D(n−1)1 ) has the same distribution as D, the random variable representing total demand in any period. L+ L+ + D(n−1)1 ) has the same distribution as D. We say X ∼d Y In the same way, (Dn2 if the two random variables X and Y have the same distribution function.

Since I BL I BL Since I BL+

=

L + L L ((x − Dn1 ) + q − Dn2 − D(n−1)1 )+ ,

L )+ + q − D)+ . ∼d ((x − Dn1

=

L+ + L+ L+ ((x − Dn1 ) + q − Dn2 − D(n−1)1 )+ ,

L − )+ + q − D)+ . I BL+ ∼d ((x − Dn1

Comparing the expressions for the distributions of I BL+ and I BL , we can see that the second inequality stated in the lemma is just a consequence of the fact that ≥ 0  for every sample path of Dn (0, t).

5 Base Stock Policies: A Convexity Result We now consider a particular class of policies, the “base stock" or “order-up-to S" policies. Clearly, such policies are not necessarily optimal, but are often used in practice. When following such policies, an order is placed in each period to raise the inventory position to S. One interesting research question is “How well do such policies perform?". Another equally important and interesting question is “How do we determine the value of S that minimizes the cost among this class of policies?". The latter is the issue that we address in this section. In the absence of simple analytic methods to determine the optimal value of S, it is common to use simple search techniques in combination with simulation to decide on a value of S. For example, “infinitesimal perturbation analysis" (Glasserman & Tayur 1995) is an effective and efficient technique for computing stock levels. However, these techniques can be guaranteed to yield optimal solutions only if the cost function is known to be convex. With this as the motivation, we next prove the convexity of the discounted cost function for our finite horizon problem in the parameter S. To obtain the result, we show that it is true for almost every sample path of the demand and lead time processes. It is important to note that the result is valid under much weaker assumptions about the demand distributions than we have been making. Convexity results for other lost sales inventory problems using base stock policies are available in Downs et al. (2001) and Janakiraman and Roundy (2004). First, we develop the notation required in the analysis. Let xn = on-hand inventory at the beginning of period n, that is, the point in time at which the order is placed in period n, qn = order placed in period n which equals S − xn , and ln = the amount of lost sales in period n. Assume x N = S, that is, we start the system with S units of inventory on hand. Now we write recursive equations describing the evolution of the on-hand inventory through time. Subsequently, we show that the on-hand inventory at the beginning of

18

G. Janakiraman and J. A. Muckstadt

any period can be written as a piecewise linear function of S. We also characterize the set of values of S at which this function could potentially have a derivative discontinuity. Then, we show that the one-period cost function is linear in S and the on-hand inventory at the end of a period or, equivalently, at the beginning of the next period. This, along with the piecewise linearity of xn (S), implies that the discounted cost function is a piecewise linear function in S and therefore trivially convex at all these values of S. However, to prove the convexity of the discounted cost function over the entire domain of S, we need to verify that the slope, or derivative, of this cost function increases at every derivative discontinuity, that is, the left-hand derivative is smaller than the right-hand derivative. Suppose we have a given sample path of demands throughout the horizon, {(D N 1 , D N 2 ), . . . , (D11 , D12 )}. We know that

Furthermore,

x N = S, and xn + qn = S. +  xn−1 = (xn − Dn1 )+ + qn − Dn2 = [xn − min(xn , Dn1 ) + qn − Dn2 ]+

(23) (24) (25) (26)

= [S − Dn2 − min(xn , Dn1 )]+ , and (27) + + + ln = (Dn1 − xn ) + [Dn2 − (xn − Dn1 ) − qn ] . (28) At the beginning of a period, just prior to the time an order is placed, all the inventory in the system is on hand. As soon as the order is placed, the inventory position takes the value S. Hence, xn−1 can be computed as the inventory position at the beginning of the previous period (S) less the total depletion in inventory in the previous period (Dn − ln ). Thus, xn−1 = S − (Dn − ln ), or ln − Dn + xn−1 − S.

(29) (30)

Define the one-period cost function in period n to be vn (S) = h · xn−1 + b · ln , or vn (S) = (h + b)xn−1 − b · S + b · Dn .

(31) (32)

Compare this definition with definition (2) in Sect. 2. The definition of the one-period cost function here (31) is the same as (2) without the expected value operator except for the following difference. The holding cost for the on-hand inventory at the end of period n (beginning of period n − 1) is charged in period n in (31), whereas it was charged proportional to the inventory on hand at the beginning of period n in for algebraic simplicity. We are (2). We define the cost function vn this way purely N α N −m vn (S). interested in proving the convexity of the function m=1

Inventory Optimization Under Lost Sales … Table 1 Computing {δn , γn } Period n 5 Dn1 Dn2 δn−1 γn−1

7 2 9 7

S=1

40

19

4

3

2

1

12 9 21 21

2 14 14 23

12 1 1 28

1 10 11 2

S=9 S=11 S=14

S=21 S=23

S=28

x5(S)

35

x4(S)

x 0(S)

x1(S) x2(S)

30 25

x3(S)

20 15 10 5 0

0

5

10

15

20

25

30

35

40

S

Fig. 2 Plots of xn versus S

For the given sample path, define δn−1 = inf{S : xn−1 > 0}, and

(33)

γn−1 = inf{S : xn > Dn1 }.

(34)

By examining Eq. (27), it is apparent that δn−1 and γn−1 are two obvious values of S where xn−1 could be potentially discontinuous. We illustrate the dynamics of the sequence {xn } and also record the values of δn and γn in the following example (see Table 1). In the following graph (see Fig. 2), Eqs. (23) and (27) are used to generate the plots for xn (S) for this example. These equations also determine the values of δn and γn which are given in table 1. It is important to note from the graph (see Fig. 2) that all the values of S where some xn (S) has a derivative discontinuity belong to the set {δn } {γn }. Since we assume that the random variables Dn1 and Dn2 have continuous density functions, we can see from Eq. (27) that for almost every sample path, if j = k, then

20

G. Janakiraman and J. A. Muckstadt

δk = δ j , γk = γ j and δk = γ j . Though we do not state it explicitly, we consider only such sample paths in all the results in this section. In the following lemma, we show that xn (S) is piecewise linear in S with derivative discontinuities only at values of S that are equal to δk or γk for some k. In addition, we also show that xn (S) has a slope of 0 or 1 in all its linear segments and also that it always lies between 0 and S. We use xnR (S) and xnL (S) to denote the left-hand and right-hand derivatives of xn (S) with respect to S. xn (S) will denote the derivative of xn (S) when it exists. Lemma 10 The following statements hold for every 1 ≤ n ≤ N . (a) xn (S) is continuous in S. (b) ∀S = δk or γk , k ≥ n, xn (S) is linear at S and xn (S) ∈ {0, 1}. (c) 0 ≤ xn ≤ S. Proof The proof is by induction. Observe that the three statements are trivially true when n = N since x N (S) = S. Now, assume that the statement is true for all j ≥ n (n ≥ 1). We will now verify that the statements are true for n − 1. Recall that xn−1 (S) = [S − (Dn2 + min(xn (S), Dn1 ))]+ .

(35)

We use this fact throughout the proof. First, observe that by assumption, xn (S) is a continuous function, and therefore xn−1 (S) is a continuous function of S as well. Second, since 0 ≤ xn (S) ≤ S, 0 ≤ xn−1 (S) ≤ S, too. To prove statement (b) for n − 1, assume that S = δk or γk , k ≥ n − 1. Suppose S is in the region {S > δn−1 , S > γn−1 }. Then we see that xn−1 (S) = S − Dn2 − Dn1 , and therefore xn−1 is linear at S with slope 1 in this region. Next, suppose that S is in the set {S > δn−1 , S < γn−1 }. For all S in this region, xn−1 (S) = S − xn − Dn1 . Since xn (S) is linear and xn (S) ∈ {0, 1} at all these values of S, xn−1 (S) is linear in S and xn (S) ∈ {0, 1}. Finally, suppose that S is in the set {S < δn−1 }. This condition implies that xn−1 (S) = 0 in this region. Consequently, xn−1 is linear with slope 0 at these values of S. Thus, we have shown  statement (b) to be true for xn−1 (S). This lemma implies is a piecewise linear function of S. Consequently,  Nthat xNn (S) α −m vn (S) is a piecewise linear function of S. Therefore, Eq. (32) implies that m=1 N to prove the convexity of m=1 α N −m vn (S), we need to make sure that the slopes of N N −m the linear segments of m=1 α xn−1 (S) increase as S increases. This is proved as a corollary to the following theorem. N R L Theorem 4 For every 1 ≤ n ≤ N , Wn (S) = m=n am · (xm−1 (S) − xm−1 (S)) ≥ 0∀S and any non-negative and non-decreasing sequence {am }. Proof As was the case for the preceding theorem, we prove this one by induction. R L First observe that (xm−1 (S) − xm−1 (S)) = 0 if xm−1 (S) is differentiable, that is, the left-hand derivative equals the right-hand derivative at S. When n = N , W N (S) = a N · (x NR −1 (S) − x NL −1 (S)). Also x N −1 = [(S − D N 1 )+ − D N 2 ]+ , which is the same

Inventory Optimization Under Lost Sales …

21

as (S − D N )+ . This is clearly differentiable ∀ S = D N and the slope increases from 0 to 1 at S = D N . Hence, the statement of the theorem is true for n = N . Assume the conjecture is true for all j ≥ n for some n > 1, and let us now try to prove the statement for n − 1. Recall that Wn =

N 

R L R L am · (xm−1 (S) − xm−1 (S)) + an · (xn−1 (S) − xn−1 (S)).

m=n+1

Case I S < δn−1 and xn−1 = 0. In this region xn−1 (S) is differentiable everywhere R L (S) − xn−1 (S)) = 0 everywhere. and hence (xn−1 Case IIa S ≥ δn−1 , Sγn−1 , xn−1 = S − xn − Dn2 . Case IIb S ≥ δn−1 , S ≥ γn−1 , xn−1 = S − Dn1 − Dn2 . In this region, xn−1 (S) is difR L (S) − xn−1 (S)) = 0 everywhere. ferentiable everywhere and hence (xn−1 R L Since we have just seen that (xn−1 (S) − xn−1 (S)) = 0 in the regions represented by Case I and Case IIb, the theorem is clearly true for n − 1 if S lies in one of these regions. To complete the proof, we need to consider Case IIa and the points of intersection between the three cases. In region I, xn−1 (S) is zero and in region IIa, xn−1 (S) ∈ {0, 1}. So at the boundary of these two regions, the change in the derivative of xn−1 (S) is non-negative. Using almost identical reasoning, we see that the change in the derivative of xn−1 (S) is non-negative at the point of intersection between regions IIa and IIb and between I and IIb. Now, let us examine Case IIa. In this region, xn−1 = S − xn − Dn2 . ThereR L (S) − xn−1 (S)) = −(xnR (S) − xnL (S)). Substituting this in the expresfore, (xn−1 N R L am · (xm−1 (S)−xm−1 (S))+(an+1 −an ) · sion for Wn (S), we get Wn (S) = m=n+2 R L (xn−1 (S)−xn−1 (S)). This is nothing but Wn+1 (S) for the sequence of coefficients {0, . . . , 0, an+1 − an ,n+2 , . . . , a N −1 , a N } which is clearly non-negative and nondecreasing. But we have assumed the statement of the theorem to be true for n + 1. Therefore, Wn (S) ≥ 0 in region IIa. We have thus shown the theorem to be true for all values of S.  N N −m Corollary 1 vm (S) is convex in S. m=1 α

Proof Application of the theorem with am = α N −m , which is non-negative and  N above R L N −m (xm−1 (S) − xm−1 (S)) ≥ 0. Consequently, non-decreasing, shows that m=1 α N N −m R L (vm (S) − vm (S)) ≥ 0∀S. Therefore, the slope of Eq. (32) implies that m=1 α N N −m α v (S) increases at every value of S where the derivative is not defined. m m=1 N At all the other values of S, this function is linear. Therefore, m=1 α N −m vm (S) is convex in S.  Thus, we have shown that the discounted cost function is convex in S for almost every realization of the sequence of random variables {(Dn1 , Dn2 )}, which implies the convexity of the expected discounted cost function.

22

G. Janakiraman and J. A. Muckstadt

6 Conclusions We have considered a periodic review, lost sales inventory procurement problem which differs from existing problems in the literature. Significant differences are the incorporation of lead times that are random and between zero and the length of a period, and non-stationary and correlated demand within a period. An expected discounted cost function was considered as the objective function to be minimized. We proved that the cost that results from the use of the optimal ordering policy is a convex function of the inventory on hand at the beginning of a period. The optimal ordering quantity was shown to be a monotonic function of the beginning inventory. Easily computable bounds were derived for the optimal order size. These bounds can easily be used to develop a variety of heuristics to decide the order quantity. We also considered the class of order-up-to policies and proved a convexity result that justifies the use of simple search methods to determine the optimal value of the order-up-to parameter. Acknowledgements This research was partially funded by the National Science Foundation (grant DMI 0075627) and Aspen Technology.

References Bijvank, M., & Vis, I. F. (2011). Lost-sales inventory theory: A review. European Journal of Operational Research, 215(1), 1–13. Downs, B., Metters, R., & Semple, J. (2001). Managing inventory with multiple products, lags in delivery, resource constraints and lost sales: A mathematical programming approach. Management Science, 47(3), 464–479. Glasserman, P., & Tayur, S. (1995). Sensitivity analysis for base-stock levels in multiechelon production-inventory systems. Management Science, 41(2), 263–281. Goldberg, D. A., Katz-Rogozhnikov, D. A., Lu, Y., Sharma, M., & Squillante, M. S. (2016). Asymptotic optimality of constant-order policies for lost sales inventory models with large lead times. Mathematics of Operations Research, 41(3), 898–913. Huh, W. T., Janakiraman, G., Muckstadt, J. A., & Rusmevichientong, P. (2009). Asymptotic optimality of order-up-to policies in lost sales inventory systems. Management Science, 55(3), 404–420. Janakiraman, G., & Muckstadt, J. A. (2004). Inventory control in directed networks: A note on linear costs. Operations Research, 52(3), 491–495. Janakiraman, G., & Roundy, R. O. (2004). Lost-sales problems with stochastic lead times: Convexity results for base-stock policies. Operations Research, 52(5), 795–803. Karlin, S., & Scarf, H. (1958). Inventory models of the Arrow-Harris-Marschak type with time lag. Studies in the Mathematical Theory of Inventory and Production (pp. 155–178). Morton, T. E. (1969). Bounds on the solution of the lagged optimal inventory equation with no demand backlogging and proportional costs. SIAM Review, 11(4), 572–596. Nahmias, S. (1979). Simple approximations for a variety of dynamic leadtime lost-sales inventory models. Operations Research, 27(5), 904–924. Xin, L. (2021). Understanding the performance of capped base-stock policies in lost-sales inventory models. Operations Research, 69(1), 61–70. Xin, L., & Goldberg, D. A. (2016). Optimality gap of constant-order policies decays exponentially in the lead time for lost sales models. Operations Research, 64(6), 1556–1565.

A New Insight on Order Quantity Allocation for Coordinated Multi-tier Multiple Supplier Networks Under Demand Uncertainty Avinash Bagul and Indrajit Mukherjee

1 Introduction To gain a competitive advantage, organizations focus on outsourcing non-core activities (Aissaoui et al., 2007). An obvious outcome of outsourcing the non-core activities is an increase in the number of suppliers, generally referred to as extended enterprises. The organization’s success depends on how well it can leverage the supplier’s capabilities to gain a competitive advantage. As the cost of purchased materials constitutes a significant part of the total costs, there exist many opportunities for cost reduction through close integration of supply network operations (Christopher, 2016). In this context, supplier selection and optimal order quantity allocation (SSOQA) decisions can help an organization optimize its supply network cost performance. However, optimal order quantity decision also depends on the degree of coordination among the supply chain members. The decision-making becomes more complex in case of uncertain demand scenarios with multiple objectives or goals. Recently, researchers (Jia et al., 2020) also highlighted the need to consider sustainability issues for SSOQA decisions. In this context, the cost of CO2 emissions in terms of carbon pricing is considered an additional objective for the SSOQA decision-making problem (Kellner & Utz, 2019). Thus, SSOQA can be formulated as a multi-objective optimization problem considering supply network and environmental sustainability cost. CO2 emissions are primarily due to logistics activities for the transfer of materials within the supply network.

A. Bagul School of General Management, National Institute of Construction Management and Research, Pune, India e-mail: [email protected] I. Mukherjee (B) Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, Mumbai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_2

23

24

A. Bagul and I. Mukherjee

This paper provides a new insight for SSOQA decision-making in a centralized multi-tier multiple supplier network based on earlier work (Bagul & Mukherjee, 2019). The optimal model formulation considers customer demand uncertainties and sustainability cost aspects to determine the best solution. Depending on its manufacturing capacity, an OEM can opt for a single or multiple sourcing strategy for a coordinated multi-tier supply network. Figure 1a shows a typical two-tier multiple

a

Tier -II Suppliers S 211

( d211,,σ211 )

Tier -I Suppliers S11

S 212

S 221

( d212 ,σ212 )

( d 11 ,σ11 ) Manufacturer

( d221 ,σ221 ) S12

S 222

S 231

( d 12 ,σ12 ) OEM

( d ,σ d )

( d 222 ,σ2222 )

( d231 ,σ231 )

( d 13 ,σ13 ) S13

S232

( d232 ,σ232 )

b Tier -II Suppliers S 211

( d211,,σ211 )

Tier -I Suppliers S11

S 212

S 221

( d212 ,σ212 )

( d 11 ,σ11 ) Manufacturer

( d221 ,σ221 ) S12

S 222

S 231

( d 12 ,σ12 ) OEM

( d ,σ d )

( d 222 ,σ2222 ) ( d231 ,σ231 )

( d 13 ,σ13 ) S13

S232

( d232 ,σ232 )

Fig. 1 a A typical two-tier supply network with a multi-sourcing situation. b A typical two-tier supply network with a single-sourcing scenario

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

25

supplier network multi-sourcing situation, whereas Fig. 1b shows a supply network with a single-sourcing scenario. In Fig. 1a, multiple suppliers are selected from Tier-I and Tier-II (shown as gray) to fulfill the overall demand of OEM. The primary objective of a multi-sourcing strategy is to reduce the risk associated with the suppliers. In Fig. 1b, out of many available suppliers in Tier-I and Tier-II, OEM only selects a single supplier from each tier (shown as gray) of the network. Treleven and Schweikhart (1988) defined a singlesourcing strategy as ‘fulfillment of all of the organization’s needs for a particular item from a specific vendor by choice.’ A prerequisite for any single-sourcing strategy is mutual trust in the integrated scheduling of production activities and coordination of inventory information between the buyer and the supplier. Thus, coordination between different players in the supply network is critical to reducing the cost associated with the procurement process. Some of the advantages of coordination are (i) shared authentic information, (ii) reduced overall cost of purchasing, and (c) reduction in transaction costs (Schotanus, 2007). A centralized coordination model based on multi-echelon inventory can provide optimal overall supply chain cost (Mendoza & Ventura, 2010). In this work, an MINLP model formulation is used to derive an optimal solution with multiple objectives (e.g., supply network cost and sustainability cost). The model is based on a multi-echelon inventory concept. The model also assumes an unrestricted manufacturing capacity for each selected supplier. This chapter is organized as follows. Section 2 provides a review of relevant literature on SSOQA and highlights the research gaps. Section 3 provides the multiple objective SSOQA model formulation for multi-tier multiple supplier sourcing network under demand uncertainty. Section 4 demonstrates the applicability of the proposed strategy for SSOQA decision-making based on a real-life case study on two different automotive components. Section 5 concludes with key points from the case study and future research directions.

2 Literature Review Aitken (1998) defined supply chain as ‘A network of connected and interdependent organizations which are mutually and cooperatively working together, to control, manage, and improve the flow of materials and information from suppliers to end users.’ With the increasing globalization of supply chains, the complexity of the supply chain is increasing as organizations try to source materials from diverse locations or regions. Thus, close coordination among the different entities in the global supply chains can reduce supply chain costs and help improve product service levels. Simatupang et al. (2002) defined supply chain coordination as a means by which two or more organizations work jointly to plan and execute supply chain operations yielding better results than when they act in isolation. Arshinder et al. (2011) defined supply chain coordination as ‘identifying interdependent supply chain activities between supply chain members and devising a mechanism to manage these

26

A. Bagul and I. Mukherjee

interdependencies.’ As supply chain members are distinct and interdependent entities with conflicting objectives, lack of coordination may lead to uncertainties in the supply chain (Arshinder et al., 2008). For a three-level supply chain, Jaber and Goyal (2008) advocated coordination and stated that coordination leads to a decrease in supply chain cost. Viswanathan and Piplani (2001) studied the benefits of inventory coordination for one vendor and multiple buyer scenarios following standard replenishment times. Treleven (1987) and Maloni and Benton (1997) stated few advantages of coordination for suppliers as (i) improved product quality, (ii) reduced transaction cost due to economies of scale, (iii) decreased administrative and switching costs, (iv) process integration, and (v) ability to provide quantity discounts. Sarmah et al. (2006) argued that buyer–vendor coordination is an essential aspect of achieving competitive advantage and lowering supply chain cost. A buyer–vendor coordination can be achieved through an optimal SSOQA decision. Various researchers (Ahmad & Mondal, 2016; Bagul & Mukherjee, 2018; Mendoza & Ventura, 2010, 2012) advocated coordination for SSOQA decisionmaking to reduce the overall supply network cost. Zhang and Zhang (2011) proposed a methodology for SSOQA with suppliers offering different prices and restrictions on minimum and maximum order quantities. Mendoza and Ventura (2010) developed a model for SSOQA using a multi-echelon inventory concept for a multistage supply chain consisting of suppliers, manufacturers, warehouses, distribution centers, and retailers. They considered a deterministic demand scenario for the model development. Choudhary and Shankar (2014) proposed a multiple objective integer linear programming model for a single-stage, multi-period supply network for SSOQA. Grimm et al. (2014) addressed the management of Tier-II suppliers concerning transparency in contracts and the direct involvement of these suppliers with the OEM. They argued that these issues are not discussed in earlier literature from the perspective of sustainability. Guo and Li (2014) proposed a single-objective SSOQA model, based on the multi-echelon inventory concept, for a multistage supply chain consisting of suppliers, warehouses, and retailers with uncertain demand scenarios. Jadidi et al. (2014) suggested a multi-objective optimization model for SSOQA to minimize cost, rejects, and lead time for a single-stage supply network. Jain et al. (2015) proposed an SSOQA model, based on the coopetition concept and discounts offered by the suppliers, for a single-stage supply network. They developed an MINLP model and solved it using a genetic algorithm, artificial bee colony, and chaotic bee colony optimization strategy. Lee et al. (2015) developed a solution methodology for SSOQA, considering agility criteria. They recommended fuzzy AHP and fuzzy TOPSIS to determine weights for the criteria. They studied OQA decisions under agile and nonagile environments. Pazhani et al. (2016) developed a composite single-objective SSOQA optimization model for a multi-tier supply network. They considered overall inventory and transportation costs as multiple objectives with a deterministic demand scenario. Scott et al. (2015) developed an integrated approach for SSOQA for a singlestage supply network, considering multiple criteria and multi-stakeholder requirements. They used analytic hierarchy process (AHP), quality function deployment (QFD), and a chance-constrained optimization algorithm to resolve the problem.

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

27

Cárdenas-Barrón et al. (2015) developed a SSOQA model for multi-product, multi-period, and deterministic demand scenario. Ahmad and Mondal (2016) proposed an MINLP model for a two-echelon supply network for SSOQA decision. They assumed a deterministic demand scenario. Bagul and Mukherjee (2019) analyzed a multi-tier supply network optimization model considering demand uncertainties. They demonstrated the cost advantage of a centralized supply network over a decentralized supply network OQA strategy. In the context of SSOQA model formulation, researchers have started emphasizing sustainability. Mohammed et al. (2018) proposed an integrated four-phase approach to solve a single-period SSOQA problem, considering economic, environmental, and social criteria. They developed a multi-objective optimization model for OQA, considering demand uncertainties. Arabsheybani et al. (2018) used a fuzzy multiobjective optimization model for SSOQA decisions. They assumed multi-period, multi-product deterministic demand scenarios. Supplier risk was also assessed using failure mode and effects analysis. Park et al. (2018) developed a two-phase strategy for SSOQA in the context of global business and logistics management. In the first phase, sustainable supplier regions were identified based on multi-attribute utility theory. They considered regional sustainability indices for economic and social factors. In the second phase, a multi-product, multi-objective integer linear programming model was analyzed to determine an optimal number of supplies and assign order quantities in a deterministic demand scenario. Cheraghalipour and Farsad (2018) developed an SSOQA model for multi-period, multi-item, and multi-supplier environments. They considered quantity discounts and disruption risks. Weights were assigned to the suppliers based on the multi-criteria best–worst method. The biobjective mixed-integer linear programming model was resolved by a revised multichoice goal programming approach. However, the model was confined to a single-tier supply network with a deterministic demand environment. Vahidi et al. (2018) proposed a bi-objective two-stage mixed stochastic programming model for SSOQA, considering operational and disruption risks. They considered a single-period single-tier supply network with a deterministic demand environment. Torres-Ruiz and Ravindran (2018) resolved the multi-objective OQA model to minimize procurement cost, transportation costs, lead time, sustainability risks, and greenhouse gas (GHG) emissions for a single-tier network. They also suggested the selection of primary and backup suppliers. Khoshfetrat et al. (2019) developed a single-stage, fuzzy multi-objective, multiproduct, and multi-period model for SSOQA. They considered inflation, risk, and fuzzy uncertainties. Arabsheybani et al. (2019) developed a multi-objective model for SSOQA considering multi-supplier, multi-product, multi-item, and multi-period scenarios. They resolved the model using NSGA II and multiple objective particle swarm optimization-based search strategy. However, they considered a deterministic demand scenario. Mohammed et al. (2019) proposed an SSOQA strategy, considering economic, environmental, and social sustainability. They used multi-criteria decision-making and fuzzy multi-objective optimization techniques. Their study was limited to a single-tier supply network with deterministic demand. Kellner and Utz (2019) suggested a multi-objective SSOQA model using Markowitz portfolio theory.

28

A. Bagul and I. Mukherjee

They considered sustainability, purchasing cost, and supply risk to determine optimal solutions. Almasi et al. (2019) formulated a multi-period, multi-objective SSOQA model, considering economic, environmental, and social criteria. They also considered the risk and inflation factors. The model was resolved using the weighted sum and augmented ε-constraint method. Li et al. (2021) suggested a two-stage approach for the SSOQA problem. They considered the risk and environmental issues for a multi-period, multi-product supply network scenario. For screening the suppliers, they adopted a best–worst method. A multi-objective model was resolved for OQA in a new energy vehicle industry. You et al. (2020) solved a multi-objective linear programming SSOQA model, considering quantity discount and uncertain information environment. Jia et al. (2020) proposed a robust goal programming strategy for single-tier SSOQA decisions, considering demand uncertainties and CO2 emissions. As per the critical review of the literature, the following research gaps are identified: (i) (ii)

Research works related to SSOQA are primarily restricted to a single-tier supply network with deterministic demand scenarios. There is no evidence of open literature that provides new insight on SSOQA strategy for centralized, coordinated multi-tier multiple supplier sourcing network, subject to demand uncertainties and sustainability cost.

This work attempts to extend the work of Bagul and Mukherjee (2019) and address the gaps mentioned above. In the model suggested by Bagul and Mukherjee (2019), environmental sustainability cost (e.g., CO2 emission cost) is added to this research’s objective function. In addition, unrestricted capacities of the suppliers are considered for the coordinated multi-tier sourcing network. Real-life case data sets of two different automotive components are used to analyze the optimal solutions. Subsequently, a two-stage design of experiment strategy is used to understand the influence of significant factors on each cost objective.

3 Model Formulation for SSOQA Under Demand Uncertainties Bagul and Mukherjee (2019) proposed a solution framework and model formulation for an SSOQA problem. The problem considered sourcing strategy for a centralized multi-tier multiple supplier networks with demand uncertainties. In the model formulation, multi-echelon inventory and transportation costs were considered to form a single composite objective function. The model was subjected to supplier’s manufacturing capacity and purchasing organizations’ storage space constraints. The maximum amount to be spent on transportation was also added as a constraint condition. In this work, the above formulation is relaxed by considering all the suppliers’ unlimited manufacturing capacity. The revised model also considers the availability

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

29

of ample storage space for the purchasing organizations. A sustainability cost objective is also added to the model objectives. Model assumptions and notations are given in Annexure A. The first part of the objective function represents multi-echelon inventory cost and transportation cost (Bagul & Mukherjee, 2019). The second part of the model represents the sustainability cost incurred for the transportation of materials within the supply network. Detailed explanation on sustainability cost formulation is provided below. Sustainability cost is considered as the cost to the society. The cost occurs due to CO2 equivalent gas emissions from the vehicles used to transport materials within the supply chain. To calculate sustainability cost, one needs to calculate CO2 eq. emission (per order cycle time for each supplier in the network) as: CO2 eq. emission per order cycle time for Tier-I supplier (S ij ) = Fuel consumption per liter × FEF × Number of trips.

(1)

Assuming that the materials are transported using diesel trucks, the amount of diesel consumed (in liters per trip) can be calculated by multiplying distance (distij ) with the average mileage of the trucks in terms of km/l. In Eq. (1), FEF represents the amount of CO2 eq. emission per liter of fuel consumption. The number of trips depends on the number of orders (J ij ) given to the supplier (S ij ). Thus, CO2 eq. emission per order cycle time for Tier-I supplier (S ij ) can be expressed as CO2 eq. emission per order cycle time for Tier-I supplier (S ij ): = FCi j × FEF × Ji j , ∀i = 1, j

(2)

Using Eq. (2), the total CO2 eq. emission per order cycle time for all the Tier-I suppliers is expressed as =

n 

FCi j × FEF × Ji j , ∀i = 1, j

(3)

j=1

The total CO2 eq. emission per unit time for all the Tier-I suppliers can be calculated by dividing Eq. (3) with the order cycle time. Thus, total CO2 eq. emission per unit time for all the Tier-I suppliers is: d × FEF × = Q

 u

 FCi j × Ji j n , ∀i = 1, j j=1 Ji j

j=1

(4)

Similarly, total CO2 eq. emission per unit time for Tier-II suppliers is di j × FEF = × Q



v k=1 FCi+1 jk × Ji+1 jk  n j=1 n i+1 jk Ji+1 jk

 , ∀i = 1, j, k

(5)

30

A. Bagul and I. Mukherjee

According to World Bank (2021), carbon pricing is an instrument that captures the external costs of greenhouse gas (GHG) emissions, usually in the form of the price of carbon dioxide (CO2 ) emissions. Based on Eqs. (4) and (5), sustainability cost per unit time is calculated by multiplying total CO2 eq. emission per unit time with carbon pricing. Thus, sustainability cost per unit time for Tier-I suppliers is expressed as d × FEF × CP × = Q

 u

 FCi j × Ji j n , ∀i = 1, j j=1 Ji j

j=1

(6)

Similarly, sustainability cost per unit time for Tier-II suppliers is di j × FEF × CP = × Q



v k=1 FCi+1 jk × Ji+1 jk  n j=1 n i+1 jk Ji+1 jk

 , ∀i = 1, j, k

(7)

In this study, both supply network cost and sustainability cost are considered as objective functions in the final model formulation. A supply network cost (f 1 ) consists of seven different costs for Tier-I and Tier-II suppliers in the network. These costs include setup cost (f 11 ), inventory holding cost (f 12 ), safety stock holding cost (f 13 ), stock-out cost (f 14 ), materials procurement cost (f 15 ), materials procurement cost for safety stock (f 16 ), and transportation cost (f 17 ), per unit time. Thus, supply network cost (f 1 ) can be represented as: f 1 = f 11 + f 12 + f 13 + f 14 + f 15 + f 16 + f 17

(8)

where  v  di j Ji+1 jk (K i+1 jk + δi+1 jk ) k=1 × + ∀i = 1, j (9) v Q k=1 n i+1 jk Ji+1 jk ⎡ ⎛ ⎞⎤  u  v v a×Q bQ ⎣ k=1 Ji+1 jk Pi+1 jk ⎝  j=1 Ji j Pi j v u × × f 12 = × n i+1 jk − v ⎠⎦∀i = 1, j + 2 2 k=1 Ji+1 jk j=1 Ji j d × f 11 = Q

 u

j=1 Ji j (K i j + δi j ) u j=1 Ji j



k=1

 u

j=1 Ji j Z i j σi j Pi j u j=1 Ji j

f 13 = a ×

d × f 14 = Q

 +b×

 u

j=1 Bi j Ji j pi j ≥ Z i j u j=1 Ji j

 u f 15 = d ×

j=1

u

Ji j Pi j

j=1

Ji j





 v

jk Ji+1 jk Z i+1 jk σi+1 jk Pi+1 jk k=1 n i+1 v k=1 n i+1 jk Ji+1 jk

di j × + Q

 v

(10)



jk Ji+1 jk pi+1 jk ≥ Z i+1 jk k=1 Bi+1 v k=1 n i+1 jk Ji+1 jk

∀i = 1, j 

(11) ∀i = 1, j

(12)   v k=1 n i+1 jk Ji+1 jk Pi+1 jk v + di j × k=1 n i+1 jk Ji+1 jk

∀i = 1, j (13)

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

 u f 16 =

j=1 Ji j Z i j σi j Pi j u j=1 Ji j

 +

 u f 17 = d × w ×

j=1 Ji j T ci j u j=1 Ji j

 v

31



jk Ji+1 jk Z i+1 jk σi+1 jk Pi+1 jk k=1 n i+1 ∀i = 1, j v k=1 n i+1 jk Ji+1 jk

 + di j × w ×

 v



(14)

n i+1 jk Ji+1 jk T ci+1 jk k=1 v ∀i = 1, j k=1 n i+1 jk Ji+1 jk

(15) The decision variables considered in Eqs. (8)–(15) are J ij , J ijk , nijk , Q, Z ij , and Z ijk . Sustainability cost (f 2 ) per unit time is the sum of the sustainability costs for Tier-I and Tier-II suppliers and is expressed as:  f2 =

  u v FCi+1 jk × Ji+1 jk FEF × CP j=1 FCi j × Ji j k=1  u × d× + di j × ∀i = 1, j v Q k=1 n i+1 jk Ji+1 jk j=1 Ji j

(16)

The decision variables considered in Eq. (16) are J ij , J ijk , nijk , and Q. As objectives, f 1 (supply network cost) and f 2 (sustainability cost) may be correlated, and there is a possibility of a significant correlation between these objectives (Derringer & Suich, 1980; Kim & Lin, 2000). Thus, a min–max approach is used to define a single composite objective function. The objective function and the constraint conditions for the model (Model-I) are given below Minimize[Maximum( f 1 , f 2 )]

(17)

Subject to: w×

u 

Ji j T ci j ≤ ri ×

j=1

w×

v 

u 

Ji j Pi j , ∀i = 1, j

(18)

j=1

n i jk Ji jk T ci jk ≤ ri j ×

k=1

v 

n i jk Ji jk Pi jk , ∀ i = 2, j, k

(19)

k=1 v 

n i+1 jk Ji+1 jk =Ji j , ∀i = 1, j

(20)

Ji j (qi j − q j ) ≥ 0, ∀ i = 1, j

(21)

Ji jk (qi jk − qi ) ≥ 0, ∀ i = 2, j, k

(22)

δi j = g, ∀ qi j < qi , i = 1, j

(23)

k=1 u  j=1 v  k=1

32

A. Bagul and I. Mukherjee

δi j = 0, ∀ qi j ≥ qi , i = 1, j

(24)

δi jk = g  , ∀ qi jk < qi , i = 2, j, k

(25)

δi jk = 0, ∀ qi jk ≥ qi , i = 2, j, k

(26)

Ji j ≥ 1, and integer, ∀i = 1, j

(27)

Ji jk ≥ 1 and integer, ∀i = 2, j, k

(28)

n i+1 jk ≥ 1 and integer, ∀i = 1, j , k

(29)

Q ≥ 1,

(30)

Ji j d i j = d × u j=1

Ji j

, ∀i = 1, j

Ji+1 jk d i+1 jk = d i j × v , ∀i = 1, j, k j=1 Ji+1 jk

(31) (32)

In the above formulation, Eqs. (18) and (19) represent constraints on transportation cost for Tier-I and Tier-II suppliers. Equation (20) represents the equality of the number of units ordered for both the OEM and the Tier-I suppliers in an order cycle. Quality constraints are represented in Eqs. (21) and (22) for Tier-I and TierII suppliers, respectively. Equations (23)–(26) provide a setup cost penalty. Equation (27) represents the number of allocations to Tier-I suppliers, and Eq. (28) represents the number of allocations to Tier-II suppliers. Order quantity multipliers for Tier-II suppliers are given in Eq. (29). Equation (30) represents order quantity to be allocated to Tier-I and Tier-II suppliers. The proportion allocation for Tier-I and Tier-II suppliers is determined by using Eqs. (22) and (23), respectively. Model-I is an MINLP problem, and a suitable nonlinear optimization technique can be used to derive the best solution.

3.1 Influence of Variables on Supply Network Cost and Sustainability Cost To identify the influential factors (or variables) that affect the objectives, initial potential variables are selected based on the literature review. Subsequently, Taguchi’s

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

33

orthogonal array (OA) design (Ross, 1996; Taguchi, 1990) is used to screen significant variables for the experimental simulation runs. There are two conflicting objectives (e.g., supply network cost and sustainability cost). The problem may be considered a multiple response optimization (MRO) problem. Tong et al. (1997) suggested Taguchi’s quality loss function-based multiple response S/N ratio (MRSN) technique to determine significant variables. In this study, each potential variable is varied at three different levels for the simulation trials. In each trial (e.g., a specific combination of levels of selected variables), best supply network cost (SNC) and sustainability cost (SC) are derived by solving Model-I. MRSN values for each trial are calculated as per Tong et al. (1997). Based on all the trial-run MRSN values, significant factors are screened for the second stage analysis. In the second stage of analysis, full factorial experimentation (e.g., factors identified after MRSN analysis) and ANOVA results are used to determine the critical factor that impacts supply network cost and/or sustainability cost. The complete analysis is based on real-life case data sets collected from the automobile industry in western India. Full details of cases and solution results are discussed in the following section.

4 Real-Life Case Data Analysis and New Insights In this study, two different automotive component (i.e., ‘rocker housing’ and ‘push rod’ components) data sets are used to analyze the optimization models. Both the components have two-tier, multiple supplier networks. The OEM is located in the western part of India and produces diesel engines. The organization emphasizes environmental sustainability issues. The supply network of ‘rocker housing’ and ‘push rod’ components is shown in Fig. 2a and b, respectively. Table 1 provides the basic data sets collected from OEM and suppliers for the ‘rocker housing’ component. This data is used to derive the best-expected supply network cost and sustainability cost, using Model-I formulation. A suitable MINLP solution search technique was adopted to solve Model-I in LINGO 16 software interface. OEM had set the upper limit of transportation cost for the ‘rocker housing’ component as 10% of materials cost per year. The upper limit on the total number of orders to be allocated to all the Tier-I suppliers was set to 100 to control the order cycle’s length. To calculate the sustainability cost, truck mileage on average was assumed as 4 km/liter of diesel consumption. The amount of CO2 emitted per liter of diesel consumption was taken as 3.24 kg/l. Carbon pricing was assumed as Rs. 654.9 [US$8.85] (as per Torres-Ruiz & Ravindran, 2018). Table 2 compares the proportion allocation, considering the restricted capacity model (Bagul & Mukherjee, 2019) and unrestricted capacity model (Model-I) for the ‘rocker housing’ component. Table 3 provides the actual data set collected for the ‘push rod’ component. Based on this data, expected supply network cost and sustainability cost are calculated by using Model-I. OEM had set the upper limit on transportation cost for the ‘push rod’

34

A. Bagul and I. Mukherjee

a

b

Fig. 2 a Supply network of ‘rocker housing’ component. b Supply network of ‘push rod’ component

component as 15% of materials cost per year. The upper limit on the total number of orders allocated to all Tier-I suppliers was set to 10 to control the order cycle’s length. Table 4 compares the proportion allocation, considering the restricted capacity model (Bagul & Mukherjee, 2019) and unrestricted capacity model (Model-I) for the ‘push rod’ component.

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

35

Table 1 Actual data set of the ‘rocker housing’ component a OEM

and supplier data

d

σd

q1

q2

a

b

w

36,520

390.27

0.95

0.95

15%

10%

1.1

Tier-I suppliers S 11

S 12

S 13

qij

0.96

0.99

0.94

K ij

1830

2460

1075

Pij

1450

1660

1590

σ ij

50.38

39.03

55.19

Bij

60,000

100,000

70,000

Tcij

30

110

95

distij

360

1320

1140

Tier-II suppliers S 211

S 212

S 221

S 222

S 231

S 232

qijk

0.97

0.99

0.94

0.97

0.93

0.99

K ijk

2020

2270

2740

3380

3420

3800

Pijk

1050

1190

1155

1400

1275

1395

σ ijk

29.77

35.22

12.61

15.93

19.22

19.22

Bijk

50,000

70,000

40,000

60,000

100,000

70,000

Tcijk

40

52

80

73

95

60

distijk

480

624

960

876

1080

720

d , σd , σi j , σi jk values are based on the last five years’ data. qij , qijk are the average values of eight quarters

a

Based on the actual case data sets of the components (given in Tables 1 and 3), expected supply network and sustainability cost (in Rs.) are calculated, considering restricted (Bagul & Mukherjee, 2019) and unrestricted capacity model (ModelI). Each MINLP model was solved in LINGO 16 interface. Table 5 provides the percentage cost advantage derived for the two components.

4.1 Significant Variables and Their Impact on Supply Network and Sustainability Costs Based on the earlier research works, potential variables that may impact the responses are identified. Levels are selected based on earlier studies (Al-Othman et al., 2008; Berger et al., 2004; Ekici, 2013; Ghodsypour & O’Brien, 2001; Glasserman & Tayur, 1995; Mateen et al., 2015; Meena & Sarmah, 2016; Pazhani et al., 2016; Rameshan et al., 1991; Ruiz-Torres & Mahmoodi, 2007). Each variable is varied at three different

36

A. Bagul and I. Mukherjee

Table 2 Proportion allocation of the ‘rocker housing’ component

Restricted capacity

Tier-I suppliers

Demand (d 1j )

No. of orders (J 1j )

Service level (%)

Tier- II suppliers

Demand (d 1j )

No. of orders (J 2ij )

S 11

17,987.5

33

99.7

S 211

7195

2

92.5

S 212

10,792

3

99.7

S 221

6904

2

96

S 222

3452

1

99.7

S 221

5451

2

97.6

S 222

2725

1

99.6

S 12

10,356.4

S 13 Unrestricted capacity

8176.1

19

99.8

15

99.7

Service level (%)

S 11

36,520

1

97.8

S 211

36,520

1

99

S 212

0

0

–

S 12

0

0

–

S 221

0

0

–

S 222

0

0

–

S 221

0

0

–

S 222

0

0

–

S 13

0

0

–

Table 3 Actual data set of the ‘push rod’ component a OEM

and supplier data d

σd

a

b

q1

q2

w

24,140

793.63

15%

10%

0.95

0.95

0.25

Tier-I suppliers

Tier-II suppliers

S 11

S 211

S 212

S 221

S 12

S 222

qij

0.98

0.96

qijk

0.99

0.97

0.98

0.94

K ij

2565

3830

K ijk

3200

2260

1980

2730

Pij

230

193

Pijk

152

128

120

156

σ ij

102.45

121.22

σ ijk

61.47

77.75

36.65

44.89

Bij

70,000

70,000

Bijk

50,000

50,000

45,000

45,000

Tcij

72

59

Tcijk

60

72

36

55

distij

864

708

distijk

720

864

432

660

d , σd , σi j , σi jk values are based on the last five years’ data. qij , qijk are the average values of eight quarters

a

levels (low, current, and high level) for all the Tier-I and Tier-II suppliers. Table 6 summarizes the selected levels for the Tier-I supplier (S 11 ). Component-wise data (i.e., ‘push rod’ and ‘rocker housing’), as provided by the OEM, is reported in Table 6. The last column of Table 6 shows variations (high and low levels) considered for each variable for the experimental trials. Similarly, variable levels are also selected for all the remaining suppliers in the supply network (e.g., Tier-I and Tier-II).

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

37

Table 4 Proportion allocation of the ‘push rod’ component Tier-I Demand suppliers (d 1j ) Restricted capacity

S 11

No. of Service Tier-II Demand orders level suppliers (d 2j ) (J 1j ) (%)

13,794.3 4

S 12

99.34

10,345.7 3

Unrestricted S 11 capacity

0

S 12

24,140

99.34

0

–

10

99.96

No. of Service orders level (J 2ij ) (%)

S 211

6897.1

1

99.4

S 212

6897.2

1

93.8

S 221

5173

1

97.4

S 222

5173.7

1

99.3

S 211

0

0

–

S 212

0

0

–

S 221

8046.7

1

97.9

S 222

16,093.3 2

99.8

Table 5 Cost advantage for an unrestricted capacity model Restricted capacity cost (Rs./year)

Unrestricted capacity cost (Rs./year)

Cost advantage (%)

Rocker housing

104,493,600 ($1,412,076)

94,661,370 ($1,279,208)

9.41

Push rod

9,371,919 ($126,648)

8,484,409 ($114,694)

9.43

a

$1.00 = Rs. 74.00

Table 6 Variables and levels selected for the screening experimentations Variables (or factors)

Component: push rod (current level)

Component: rocker housing (current level)

High and low levels

d

24,140

36,520

± 10%

K 11

2565

1830

± 10%

P11

230

1450

± 10%

σd

793.63

390.27

Upper and lower limit of d [confidence interval]

B11

70,000

60,000

± 10%

Tc11

72

30

± 10%

a

0.15

0.15

0.2 & 0.1

CP

654.9

654.9

± 10%

For the simulation experimentation study, it was assumed that if the level of a particular variable (for any specific supplier in Tier-I or Tier-II) changes from current to the high level, levels of all other suppliers in the network will also change from current to the high level. In other words, if the price increases for one of the suppliers in the network from current to a higher level, there would be an increase in price for

38

A. Bagul and I. Mukherjee

all other suppliers accordingly. Table 6 indicates that full factorial experimentation will require 38 trials (e.g., eight factors at three levels). Thus, to reduce the number of trials and to screen prominent factors, Taguchi’s OA design (L27 ) was selected for the study. Different factor combinations are considered in each trial (based on L27 ), and corresponding optimal costs (supply network and sustainability cost) are calculated by solving Model-I. Tong et al.’s (1997) concept of MRSN is used to determine significant factors. ANOVA of the S/N ratio (with criteria p-value < 0.05) helped identify critical factors for further analysis. Common significant factors for ‘rocker housing’ and ‘push rod,’ as determined from the MRSN study are demand rate (d), unit item price (Pij ), the standard deviation of demand (σ d ), inventory holding rate (a), and carbon pricing (CP). Subsequently, full factorial three-level experiments (based on Model-I) helped to identify the specific influence of factors on supply network and sustainability costs. The main effect plot of experimental results for supply network cost for both the components is shown in Fig. 3a and b, whereas impact on sustainability cost for both the components is shown in Fig. 4a and b. In all the four figures (Figs. 3a, b and 4a, b), the horizontal axis represents three different levels of the variables [i.e., low level (L1), current level (L2), and high level Main Effects Plot of Supply Network Cost for Rocker Housing Mean Supply Network Cost

Unit Price

Demand Rate

Standard Deviation of Demand

Carbon Pricing

Inv Holding Rate

115000000 110000000 105000000 100000000 95000000

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

Mean Supply Network Cost

Main Effects Plot of Supply Network Cost for Push Rod Standard Deviation of Demand

Unit Price

10000000

Carbon Pricing

Inv Holding Rate

Demand Rate

9500000 9000000 8500000 8000000

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

Fig. 3 a Main effect plot of ‘supply network cost’ for ‘rocker housing.’ b Main effect plot of ‘supply network cost’ for ‘push rod’

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

Main Effects Plot of Sustainability Cost for Rocker Housing

a

Unit Price

Mean Sustainability Cost

39

Standard Deviation of Demand

Demand Rate

Carbon Pricing

Inv Holding Rate

14000 13000 12000 11000 10000 9000 8000 7000 6000 L1

L2

L3

L1

L2

L3

L1

L2

L3

L2

L1

L3

L1

L2

L3

Main Effects Plot of Sustainability Cost for Push Rod

b Unit Price

Standard Deviation of Demand

Demand Rate

Carbon Pricing

Inv Holding Rate

Mean Sustainability Cost

10000

9000

8000

7000

6000 L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

L1

L2

L3

Fig. 4 a Main effect plot of ‘sustainability cost’ for ‘rocker housing.’ b Main effect plot of ‘sustainability cost’ for ‘push rod’

(L3)]. The vertical axis in Fig. 3a and b represents the mean supply network cost, whereas the vertical axis in Fig. 4a and b illustrates the mean sustainability cost. Tables 7 and 8 summarize the impact of change in levels of variables on supply network cost and sustainability cost for ‘rocker housing’ and ‘push rod,’ respectively. Table 9 provides a cross-case summary of main effect plots and generalized insights, considering both the components. Table 7 Summary of ANOVA for the ‘rocker housing’ component Experimental variable

Variable notation

Change in variable values

Impact on Supply network cost

Sustainability cost

Unit price

Pij

Increase

Increase

No impacta

Std. deviation of demand

σd

Increase

No impacta

Decrease

Demand rate

d

Increase

Increase

No impacta

impacta

Increase Increase

Inventory holding rate

a

Increase

No

Carbon pricing

CP

Increase

No impacta

a

Variable is not statistically significant (p > 0.05)

40

A. Bagul and I. Mukherjee

Table 8 Summary of ANOVA for the ‘push rod’ component Experimental variable

Variable notation

Change in variable values

Impact on Supply network cost

Sustainability cost

Unit price

Pij

Increase

Increase

No impacta

Std. deviation of demand

σd

Increase

No impacta

Decrease

Demand rate

d

Increase

Increase

No impacta

Inventory holding rate

a

Increase

No impacta

Increase

Carbon pricing

CP

Increase

No impacta

Increase

a

Variable is not statistically significant (p > 0.05)

Table 9 Cross-case generalized insight of main effects for both the components Experimental variable

Variable notation

Unit price

Pij

Std. deviation of demand

σd

Increase

No

Demand rate

d

Increase

Increase

No impacta

Inventory holding rate

a

Increase

No impacta

Increase

Carbon pricing

CP

Increase

No impacta

Increase

a

Change in variable values

Impact on Supply network cost

Sustainability cost

Increase

Increase

No impacta

impacta

Decrease

Variable is not statistically significant (p > 0.05)

The above result provides details on influential variables and their impact on relevant costs or objectives. These results are expected to help practitioners develop a sound strategy for coordinated sourcing in a multi-tier supply network.

5 Conclusions This paper provides valuable insights on optimal supply network and sustainability costs for a coordinated multi-tier sourcing scenario. The model considers the unrestricted capacity of suppliers and uncertain demand environment. Real-life automotive case data of two different components helped to study the influence of variables on the selected cost objectives. Key highlights of this research are:

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

(i) (ii) (iii)

(iv)

41

The proposed model (Model-I) can help practitioners derive optimal OQA for a coordinated multi-tier multi-supplier sourcing environment. The proposed two-stage design of the experiment can help identify the influence of variables on supply network and sustainability cost. The case study reveals that demand rate and unit item price influence supply network cost. In comparison, the standard deviation of the demand, inventory holding rate, and carbon pricing are the influential factors affecting sustainability cost. The case study also indicates that organizations may adopt a single-sourcing policy if suppliers have unrestricted capacity.

There is also a scope to consider more scientific sampling-based simulation runs to enhance the quality of analysis. The suggested model may be further refined to study social impact cost. Supplier risk may be considered for further research insights.

Annexure 1 All notations and symbols used to formulate a centralized multi-tier multiple supplier sustainable supply network under demand uncertainty are given in Sect. A.1.

A.1 Notations and Symbols Notations and symbols used to formulate the model are given in Tables 10, 11, and 12. Table 10 Sets i = 1, 2

i

The stage under consideration for the supply network

j

The serial number for the selected Tier-I supplier

j = 1, …, u

k

Serial number for selected Tier-II supplier associated with Tier-I supplier (j)

k = 1, …, v

42

A. Bagul and I. Mukherjee

Table 11 Notations Bij

Cost per stock-out occasion at stage (i) with Tier-I supplier (j)

Bijk

Cost per stock-out occurrence at stage (i) with Tier-II supplier (k) associated with Tier-I supplier (j)

CP

Carbon pricing in Rs. per ton of CO2 emission

d

Average demand, per unit time, of manufacturer

dLij

Average demand during lead time (L ij ) at stage (i) from Tier-I supplier (j)

dij

Average demand per unit time at stage (i) from Tier-I supplier (j)

dijk

Average demand per unit time at stage (i) from Tier-II supplier (k) associated with Tier-I supplier (j)

Distij

The distance at stage (i), between the OEM and Tier-I supplier (j)

Distijk

The distance at stage (i), between Tier-I supplier (j) and Tier-II supplier (k) associated with it

FCij

Fuel consumption per trip (in liters) for the supply of materials at stage (i) from Tier-I supplier (j)

FCijk

Fuel consumption per trip (in liters) for the supply of materials at stage (i) from Tier-II supplier (k) to Tier-I supplier (j)

FEF

Fuel emission factor (amount of CO2 equivalent emitted per liter of fuel consumption)

J ij

The number of orders allocated at stage (i) to Tier-I supplier (j) per order cycle

J ijk

The number of orders allocated at stage (i) to Tier-II supplier (k) associated with Tier-I supplier (j) per order cycle

K ij

Setup cost per order at stage (i) for Tier-I supplier (j)

K ijk

Setup cost per order at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

L ij

Lead time at stage (i) for Tier-I supplier (j)

L ijk

Lead time at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

nijk

Order quantity multiplier at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

Pij

Unit item price at stage (i) for Tier-I supplier (j)

Pijk

Unit item price at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

pij ≥ (Z ij )

Probability that the unit normal variable takes a value Z ij or larger

pijk ≥ (Z ijk ) Probability that the unit normal variable takes a value Z ijk or larger qi

Minimum product acceptance rate for all suppliers at stage (i)

qij

Acceptance rate at stage (i) for Tier-I supplier (j)

qijk

Acceptance rate at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

ri

Upper limit on % of materials cost per unit time at the manufacturer, which can be spent on transportation at stage (i) by all Tier-I suppliers (continued)

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

43

Table 11 (continued) Bij

Cost per stock-out occasion at stage (i) with Tier-I supplier (j)

r ij

Upper limit on % of materials cost per unit time at Tier-I supplier (j), which can be spent transportation at stage (i) by all Tier-II suppliers associated with Tier-I supplier (j)

Tcij

Transportation cost per Kg at stage (i) for Tier-I supplier (j) to transport materials to manufacturer

Tcijk

Transportation cost per Kg at stage (i) for Tier-II supplier (k) to transfer materials to Tier-I supplier (j)

σ

Demand standard deviation of manufacturer

σ ij

Standard deviation of demand at stage (i) for Tier-I supplier (j)

σ ijk

Standard deviation of demand at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

W

Unit weight of item in (Kg)

δ ij

Per order setup cost penalty at stage (i) for Tier-I supplier (j) not adhering to minimum acceptance rate (qi ) set for Tier-I suppliers

δ ijk

Per order setup cost penalty at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j) and not adhering to minimum acceptance rate (qij ) set for Tier-II suppliers

Table 12 Decision variables J ij

The number of orders allocated at stage (i) to Tier-I supplier (j) per order cycle

J ijk

The number of orders allocated at stage (i) to Tier-II supplier (k) associated with Tier-I supplier (j) per order cycle

nijk

Order quantity multiplier at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

Q

Optimum order quantity at stage 1

Sr ij

The service level at stage (i) of Tier-I supplier (j)

Sr ijk

The service level at stage (i) of Tier-II supplier (k) associated with Tier-I supplier (j)

Z ij

Safety factor for inventory at stage (i) for Tier-I supplier (j)

Z ijk

Safety factor for inventory at stage (i) for Tier-II supplier (k) associated with Tier-I supplier (j)

A.2 Model Assumptions Under Sustainability and Demand Uncertainties The following assumptions are considered for model formulation and determine supply network cost and sustainability cost for a centralized coordinated multi-tier multiple supplier network: (i) (ii)

An OEM faces uncertain demand requirements with an average demand of d with standard deviation of demand (σ) per unit time. Demand is normally distributed for all the suppliers and the manufacturer.

44

(iii)

(iv) (v) (vi)

(vii)

(viii) (ix) (x) (xi) (xii)

A. Bagul and I. Mukherjee

Each Tier-I supplier (S ij ) has deterministic lead time (L ij ) with a standard deviation of demand during the lead time (σ ij ). Suppliers supply the order quantity of (Q) units. Holding cost per unit, per unit time for Tier-I supplier (S ij ) during the lead time is a product of inventory holding rate % per unit time (a) for Tier-I suppliers (Nahmis & Oslen, 2015), unit item price (Pij ), and inventory safety factor (Simchi-Levi et al., 2004) for the supplier (Z ij ). Tier-I suppliers supply (Q) quantity to the manufacturer at the end of the lead time, after receipt of an order. The manufacturer maintains a safety stock, and the stock-out cost per stockout occurrence is (Bij ) for Tier-I supplier (S ij ). Tier-II supplier (S ijk ) also has deterministic lead time (L ijk ) with a standard deviation of demand during lead time as (σ ijk ). The inventory safety factor defined for each Tier-II supplier is (Z ijk ), and the stock-out cost per stock-out occurrence is (Bijk ). For replenishment of inventory, at each time point, an order of ‘nQ’ (n is an integer) quantity is placed with Tier-II suppliers. Holding cost per unit, per unit time, for Tier-II suppliers is the product of inventory holding rate % per unit time (b) for Tier-II suppliers and unit item price (Pijk ). All the Tier-I and Tier-II suppliers have infinite manufacturing capacity. Material holding cost for the manufacturer is considered to be higher than Tier-I suppliers. Material weights are considered the same as they flow from Tier-II suppliers to the OEM. Coordination cost for a centralized multi-tier supply network is considered negligible for the centralized sourcing strategy. Each time an order is placed with the supplier, materials are transferred to the manufacturer using a single truck.

References Ahmad, M. T., & Mondal, S. (2016). Dynamic supplier selection model under two-echelon supply network. Expert Systems with Applications, 65, 255–270. Aissaoui, N., Haouari, M., & Hassini, E. (2007). Supplier selection and order lot sizing modeling: A review. Computers & Operations Research, 34(12), 3516–3540. Aitken, J. M. (1998). Supply chain integration within the context of a supplier (Doctoral dissertation). Al-Othman, W. B. E., Lababidi, H. M. S., Alatiqi, I. M., & Al-Shayji, K. (2008). Supply chain optimization of petroleum organization under uncertainty in market demands and prices. European Journal of Operational Research, 189, 822–840. Almasi, M., Khoshfetrat, S., Rahiminezhad Galankashi, M. (2019). Sustainable supplier selection and order allocation under risk and inflation condition. IEEE Transactions on Engineering Management, 68, 823–837. Arabsheybani, A., Mohammad Mahdi, P., & Abdul Sattar, S. (2019). Sustainable supplier selection and order allocation applying metaheuristic algorithms. International Journal of Supply and Operations Management, 7(2), 164–177.

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

45

Arabsheybani, A., Paydar, M. M., & Safaei, A. S. (2018). An integrated fuzzy MOORA method and FMEA technique for sustainable supplier selection considering quantity discounts and supplier’s risk. Journal of Cleaner Production, 190, 577–591. Arshinder, K., Kanda, A., & Deshmukh, S. G. (2011). A review on supply chain coordination: Coordination mechanisms, managing uncertainty and research directions. Springer. Arshinder, K. A., & Deshmukh, S. G. (2008). Development of a decision support tool for supply chain coordination using contracts. Journal of Advances in Management Research, 5(2), 20–41. Bagul, A., & Mukherjee, I. (2018). An insight into centralised sourcing strategy for enhancing performance and sustainability of multi-tier supply network. International Journal of Intelligent Enterprise, 5(1–2), 50–69. Bagul, A. D., & Mukherjee, I. (2019). Centralized vs decentralized sourcing strategy for multi-tier automotive supply network. International Journal of Project Management, 68(3), 587–607. Berger, P. D., Gerstenfeld, A., & Zeng, A. Z. (2004). How many suppliers are best? A decisionanalysis approach. Omega, 32, 9–15. Cárdenas-Barrón, L. E., González-Velarde, J. L., & Treviño-Garza, G. (2015). A new approach to solve the multi-product multi-period inventory lot sizing with supplier selection problem. Computers & Operations Research, 64, 225–232. Cheraghalipour, A., & Farsad, S. (2018). A bi-objective sustainable supplier selection and order allocation considering quantity discounts under disruption risks: A case study in plastic industry. Computers & Industrial Engineering, 118, 237–250. Choudhary, D., & Shankar, R. (2014). A goal programming model for joint decision making of inventory lot-size, supplier selection and carrier selection. Computers & Industrial Engineering, 71, 1–9. Christopher, M. (2016). Logistics and supply chain management (5th edn.). Pearson Pub. Derringer, G., & Suich, R. (1980). Simultaneous optimization of several response variables. Journal of Quality Technology, 12, 214–219. Ekici, A. (2013). An improved model for supplier selection under capacity constraint and multiple criteria. International Journal of Production Economics, 141(2), 574–581. Ghodsypour, S. H., Brien, C. O. (2001). The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint. International Journal of Production Economics, 73(1), 15–27. Glasserman, P., Tayur, S., Glasserman, P., & Tayur, S. (2017). Sensitivity analysis for base-stock levels in multiechelon production-inventory systems. Management Science, 41(2), 263–281. Grimm, J. H., Hofstetter, J. S., & Sarkis, J. (2014). Critical factors for sub-supplier management: A sustainable food supply chains perspective. International Journal of Production Economics, 152, 159–173. Guo, C., & Li, X. (2014). A multi-echelon inventory system with supplier selection and order allocation under stochastic demand. International Journal of Production Economics, 151, 37–47. Jaber, M. Y., & Goyal, S. K. (2008). Coordinating a three-level supply chain with multiple suppliers, a vendor and multiple buyers. International Journal of Production Economics, 116, 95–103. Jadidi, O., Zolfaghari, S., & Cavalieri, S. (2014). A new normalized goal programming model for multi-objective problems: A case of supplier selection and order allocation. International Journal of Production Economics, 148, 158–165. Jain, V., Kundu, A., Chan, F. T. S., & Patel, M. (2015). A Chaotic Bee Colony approach for supplier selection-order allocation with different discounting policies in a coopetitive multi-echelon supply chain. Journal of Intelligent Manufacturing, 26, 1131–1144. Jia, R., Liu, Y., & Bai, X. (2020). Sustainable supplier selection and order allocation: Distributionally robust goal programming model and tractable approximation. Computers & Industrial Engineering, 140, 106267. Kellner, F., & Utz, S. (2019). Sustainability in supplier selection and order allocation: Combining integer variables with Markowitz portfolio theory. Journal of Cleaner Production, 214, 462–474. Khoshfetrat, S., Rahiminezhad Galankashi, M., & Almasi, M. (2019). Sustainable supplier selection and order allocation: A fuzzy approach. Engineering Optimization, 52(9), 1494–1507.

46

A. Bagul and I. Mukherjee

Kim, K. J., & Lin, D. (2000). Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions. Journal of the Royal Statistical Society: Series C: Applied Statistics, 49, 311–325. Lee, J., Cho, H., & Kim, Y. S. (2015). Assessing business impacts of agility criterion and order allocation strategy in multi-criteria supplier selection. Expert Systems with Applications, 42(3), 1136–1148. Li, F., Wu, C. H., Zhou, L., et al. (2021). A model integrating environmental concerns and supply risks for dynamic sustainable supplier selection and order allocation. Soft Computing, 25(1), 535–549. Maloni, M. J., & Benton, W. C. (1997). Supply chain partnerships: Opportunities for operations research. European Journal of Operational Research, 101(3), 419–429. Mateen, A., Chatterjee, A. K., & Mitra, S. (2015). VMI for single-vendor multi-retailer supply chains under stochastic. Computers & Industrial Engineering, 79, 95–102. Meena, P. L., & Sarmah, S. P. (2016). Supplier selection and demand allocation under supply disruption risks. International Journal of Advanced Manufacturing Technology, 83, 265–274. https://doi.org/10.1007/s00170-015-7520-5 Mendoza, A., & Ventura, J. A. (2010). A serial inventory system with supplier selection and order quantity allocation. European Journal of Operational Research, 207(3), 1304–1315. Mendoza, A., & Ventura, J. A. (2012). Analytical models for supplier selection and order quantity allocation. Applied Mathematical Modelling, 36(8), 3826–3835. Mohammed, A., Harris, I., & Govindan, K. (2019). A hybrid MCDM-FMOO approach for sustainable supplier selection and order allocation. International Journal of Production Economics, 217, 171–184. Mohammed, A., Setchi, R., Filip, M., et al. (2018). An integrated methodology for a sustainable two-stage supplier selection and order allocation problem. Journal of Cleaner Production, 192, 99–114. Nahmis, S., & Oslen, T. L. (2015). Production and operations analysis (7th ed.). Waveland Press Inc. Park, K., Okudan Kremer, G. E., & Ma, J. (2018). A regional information-based multi-attribute and multi-objective decision-making approach for sustainable supplier selection and order allocation. Journal of Cleaner Production, 187, 590–604. Pazhani, S., Ventura, J. A., & Mendoza, A. (2016). A serial inventory system with supplier selection and order quantity allocation considering transportation costs. Applied Mathematical Modelling, 40, 612–634. Rameshan, R. V., Ord, J. K., Hayya, J. C., & Pan, A. (1991). Sole versus dual sourcing in stochastic lead time (s, Q) inventory models. Management Science, 37(4), 428–444. Ross, P. J. (1996). Taguchi techniques for quality engineering (2nd ed.). McGraw-Hill. Ruiz-Torres, A. J., & Mahmoodi, F. (2007). The optimal number of suppliers considering the costs of individual supplier failures. Omega, 35, 104–115. Sarmah, S. P., Acharya, D., & Goyal, S. K. (2006). Buyer vendor coordination models in supply chain management. European Journal of Operational Research, 175, 1–15. Schotanus, F. (2007). Horizontal cooperative purchasing. University of Twente, Beta Research School for Operations Management and Logistics. Printpartners Ipskamp B.V. Scott, J., Ho, W., Dey, P. K., & Talluri, S. (2015). A decision support system for supplier selection and order allocation in stochastic, multi-stakeholder and multi-criteria environments. International Journal of Production Economics, 166, 226–237. Simatupang T. M., & Sridharan, R. (2002). The collaborative supply chain. International Journal of Logistics Management, 13(1), 15–30. Simchi-Levi, D., Philip, K., & Edith, S. (2004). Managing the supply chain: The definitive guide for the business professional (3rd edn.). McGraw-Hill Companies. Taguchi, G. (1990). Introduction to quality engineering. Asian Productivity Organization, Tokyo, Japan.

A New Insight on Order Quantity Allocation for Coordinated Multi-tier …

47

Tong, L. I., Su, C. T., & Wang, C. H. (1997). The optimization of multi-response problems in the Taguchi method. International Journal of Quality & Reliability Management, 14, 367–380. Torres-Ruiz, A., & Ravindran, A. (2018). Multi-objective optimisation model for the selection of critical suppliers integrating sustainability criteria. Journal of Operations Management, 33(2), 208–238. Treleven, M. (1987). Single sourcing: A management tool for the quality supplier. Journal of Purchasing and Materials Management, 23(1), 19–24. Treleven, M., & Bergman Schweikhart, S. (1988). A risk/benefit analysis of sourcing strategies: Single vs. multiple sourcing. Journal of Operations Management, 7(4), 93–114. Vahidi, F., Torabi, S. A., & Ramezankhani, M. J. (2018). Sustainable supplier selection and order allocation under operational and disruption risks. Journal of Cleaner Production, 174, 1351–1365. Viswanathan, S., & Piplani, R. (2001). Coordinating supply chain inventories through common replenishment epochs. European Journal of Operational Research, 129, 277–286. World Bank. (2021). World Bank. https://carbonpricingdashboard.worldbank.org/what-carbon-pri cing. Accessed April 24, 2021 You, S., Zhang, L., Xu, X., & Liu, H. (2020). A new integrated multi-criteria decision making and multi-objective programming model for supplier selection and order allocation. Symmetry, 12(2), 302–320. Zhang, J.-l., & Zhang, M.-y. (2011). Supplier selection and purchase problem with fixed cost and constrained order quantities under stochastic demand. International Journal of Production Economics, 129(1), 1–7.

Designing Distribution Network for Indian Agri-fresh Food Supply Chain Rakesh Patidar and Sunil Agrawal

1 Introduction India is the world’s largest producer of several agricultural products and has 11.3% of the global arable land (MoAFW, 2017). Agriculture sector provides a livelihood to 70% of India’s population and contributes nearly 16% of the nation’s GDP (MoAFW, 2018). Thereby, the agriculture sector is the lifeline of the Indian economy. Agricultural productivity and farmers’ profitability are the key drivers of all agriculturalrelated activities. The farmers are not only the producers of crops, but also a major consumer community for most of the commodities. Therefore, the farmers’ profitability is the prime mover for agriculture as well as the economic growth of the nation. On the other side, the cost of input resources—like fuel, fertilizers, pesticide— required for agricultural production is continuously increasing, whereas the rise in the average selling price of products is lower than the cost of input resources. Therefore, the farmers’ earning in the farming profession is unprofitable. The farmers are looking for other job opportunities, and 40% of the farmers want to quit farming (NSSO, 2005). Many times, due to the low profitability, farmers are drowned by high debt. Unfortunately, this high debt enforces farmers to end their lives. According to the NCRB report, 8007 farmers committed suicide in the year 2015 (NCRB, 2016). Agri-fresh food supply chain (AFSC) includes all the processes starting from growing of crops to delivering short shelf-life products (Shukla & Jharkharia, 2013; Tsolakis et al., 2014). Figure 1 shows the working and role of each entity of traditional Indian AFSC. In this chain, a farmer grows crops and independently transports his products into the agricultural market for selling where a commission agent does an R. Patidar (B) · S. Agrawal Department of Mechanical Engineering, Indian Institute of Information Technology, Design & Manufacturing, Jabalpur, India S. Agrawal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_3

49

50

R. Patidar and S. Agrawal

Fig. 1 Working of traditional Indian AFSC

open auctioning, and a wholesaler buys these products through bidding the highest price in the auctioning. Afterward, the wholesaler supplies these products to a retailer, and finally, the retailer sells the products to customers by visiting their proximity regions (Patidar et al., 2018). The players between farmer and customer, i.e., agent, wholesaler, and retailer, are known as intermediaries in traditional Indian AFSC. These large numbers of unorganized intermediaries—govern and control the chain— add service charges and profit margin into the product price. As a result, the customers pay the higher price, and the farmers receive only one-third of the price paid by customers (Hegde & Madhuri, 2013; Kundu, 2013; Panda & Sreekumar, 2012). Another major issue of Indian AFSC is high wastage of agricultural production as identified in a set of articles (Balaji & Arshinder, 2016; Dandage et al., 2017; Gardas et al., 2017; Kashyap & Agarwal, 2019; Negi & Anand, 2017; Rais & Sheoran, 2015; Shukla & Jharkharia, 2013). According to these papers, 15–25% of the production is lost due to poor distribution network, a large number of intermediaries, and lack of demand–supply integration. Further, other operational issues like lack of loading–unloading facilities, lack of standard quality control and harvesting methods, lack of tracking, and traceability facilities in the chain are also identified as sources of food wastages. The literature review papers (Dandage et al., 2017; Ganeshkumar et al., 2017; Gardas et al., 2017; Negi & Anand, 2017; Rajurkar & Jain, 2011; Samuel et al., 2012; Siddh et al., 2017) revealed that researchers have mainly analyzed the existing chain rather than redesigning distribution network. However, few researchers (Anjaly & Bhamoriya, 2011; Sihariya et al., 2013; Sohoni & Joshi, 2015) developed the case study-based conceptual models for the distribution of fresh products. Recently, Patidar et al. (2018) reported that the poor distribution network is responsible for the low profitability of farmers, high wastage of production, and lack of demand–supply integration. The authors developed strategies for designing a sustainable distribution network, and one of the developed strategies is recommended to aggregate product collection from villages (farmers). There are also opportunities to reform the agriculture sector by designing a suitable distribution network of Indian AFSC (Parwez, 2016; Patidar et al., 2018).

Designing Distribution Network for Indian Agri-fresh Food …

51

The major challenge in designing an AFSC is the small farm holding farmers who are abundant in numbers (Mahapatra & Mahanty, 2018). The average agricultural land possessed per family has been reduced from 2.28 ha in 1970–71 to 1.16 ha in 2010–11 due to continuous population growth and other infrastructural developments. Hence, the small farmers having farming land less than 2 ha have increased from 70 to 85% in the last four decades (NABARD, 2014). The set of studies (Gandhi & Namboodiri, 2004; Hegde & Madhuri, 2013; Kundu, 2013; Panda & Sreekumar, 2012; Raghunath & Ashok, 2009) examined the traditional Indian AFSC and reported that during the product movement from production site to market premises, farmers manage and pay the cost of transportation services. Farmers find it difficult to visit every time to the market by traveling long distances to sell their produce. Rather than selling in the market, they sell their products to local intermediaries. The intermediaries take advantage of this situation; farmers are deprived of a considerable profit margin, which is instead taken by the intermediaries, and results in the customers are paying a higher price (Gardas et al., 2019). On the other side, it is impractical for retailers or wholesalers to procure products directly from farmers because 85% of the total farmers belong to the small farming community and are geographically dispersed which makes the movement of items cost-ineffective. From the above discussion, it can be concluded that the traditional Indian AFSC is unsustainable for the farmers due to low profitability, high post-harvest losses, and higher transportation cost. It is observed that aggregate product movement (APM) can mitigate the effects of small farm holdings and geographically dispersed farmers as well as it can reduce transportation cost in the Indian AFSC. The concept of APM can be implemented using clustering of farmers for aggregation of products at different central locations. Further, the transportation of aggregate products from cluster centers to the market enables the movement of products from less than truckload to full truckload movement; hence, it will reduce transportation cost by sharing economics (Simchi-Levi et al., 2016). An attempt has been made in this work to redesign the distribution network by considering clustering of farmers. It is observed that the APM by clustering of farmers is a genuine requirement of the system as well as it has the potential to increase their profit and minimize food wastage by reducing the transportation cost. The remnant of this research paper is organized as follows: Sect. 2 presents a literature review on designing of the distribution network and identification of research gaps. Section 3 describes and formulates the proposed model. Section 4 discusses data preparation, computational results, and sensitivity analysis for the case study problem. Finally, Sect. 5 presents a summary, managerial implications, and future research.

52

R. Patidar and S. Agrawal

2 Literature Review Distribution network design is a strategic decision in which researchers plan supply chain structure and configuration over the next several years (Chopra, 2003). The distribution network design problem determines the location and allocation of facilities to serve customer demand; therefore, this problem is also known as the location–allocation problem (Amiri, 2006). These decisions are very important as well as critical in efficient and effective flow of products, and the effect of these decisions can be seen for a long time (Daskin et al., 2005). The location–allocation problem can be formulated as discrete or continuous, single objective or multi-objective, single period or multi-period with or without the restriction of capacitated facilities (Melo et al., 2009). Literature review on the modeling of the distribution network is divided into four sections. The first section discusses the papers related to the designing of a distribution network. The second section describes the multi-period distribution network models. The third section explains some relevant papers on the clustering of suppliers in designing of a supply chain. Finally, the fourth section reports a research gap in the literature.

2.1 Distribution Network Design Jayaraman and Ross (2003) developed a classical distribution network model of the four-echelon supply chain for multiple products. The authors formulated the model in two stages; the first stage identifies the location and allocation of various entities and accordingly determines the flow of products into the second stage. OlivaresBenitez et al. (2012) developed a bi-objective location–allocation model to optimize transportation cost and time for the distribution of product from plants to distribution centers and from distribution centers to customers. Hiremath et al. (2013) designed a hybrid and flexible multi-objective outbound logistics network for different variants of items (i.e., fast, slower, and very slow moving items) of the four-echelon supply chain. Chandrasekaran and Ranganathan (2017) proposed a supply chain planning model for traditional Indian agriculture supply chain to reduce the post-harvest loss occurring through respiration and CO2 emission produced by the selected produces during logistics. In recent years, researchers have started modeling the agriculture supply chain in order to solve real-life problems. The integrated supply chain of production and distribution for fresh items was formulated as mixed-integer linear programming (MILP) (Ahumada & Villalobos, 2011; Brulard et al., 2018; Farahani et al., 2012). Etemadnia et al. (2015) designed a three-echelon distribution network model for fruit and vegetable supply chain in the USA. The authors formulated a model for a single product considering lower and upper bound on hub capacity. Yu et al. (2015) developed a multi-objective dual-channel (traditional and e-commerce) supply chain

Designing Distribution Network for Indian Agri-fresh Food …

53

network model for agri-fresh product to enhance supply performance. The authors validated the model by considering a real case problem of Chinese vegetable company. Nourbakhsh et al. (2016) developed a mathematical model for designing the logistics network by considering quantitative and qualitative post-harvest losses in the distribution. The model determines an optimal plan for transportation and infrastructure investment by identifying roadway/railway capacity expansion and locations of preprocessing facilities, respectively.

2.2 Multi-period Distribution Network Design The multi-period model helps in determining the impact of supply and demand variability on the distribution network structure. It ascertains whether it is worth to use a more complex multi-period model for distribution network design (Melo et al., 2009). Khamjan et al. (2013) developed a multi-period mathematical model to determine the location–allocation and capacity of the existing and new sugarcane loading stations for Thailand. Gelareh et al. (2015) formulated a hub location model for hub-and-spoke network structure, including installation, maintenance, and closing costs associated with each hub. Zhu et al. (2018) developed a multi-period supply chain model to integrate production, inventory, and distribution decisions for perishable products. Gholamian and Taghanzadeh (2017) designed a five-echelon wheat supply chain network for Iran with considering the blending of wheat, quality variant, and multi-mode transportation along with location and allocation decisions into the formulation. Similarly, a mixed-integer nonlinear programming (MINLP) model for the food grain supply chain of the public distribution system in India was developed to minimize the total cost including bulk food grain shipment, storage, and operational cost (Mogale et al., 2017).

2.3 Clustering in Distribution Network Design Clustering of supply chain players helps in collaborative transportation by aggregating products at a common point in order to take the benefits of sharing economies. Bosona and Gebresenbet (2011) built a cluster of farmers and found the optimal location of collection centers for the integration of logistics activities in the local food supply chain. Boudahri et al. (2011) developed a distribution network model of the three-echelon chicken meat supply chain with considering clustering of customers. The complete model was decomposed into two sub-models; the first model determines the centroid of customers’ clusters to establish retailer outlets, and accordingly, the second model determines the location–allocation of facilities. In the literature, researchers pay little attention to aggregate product collection and transportation. However, in recent years some authors used covering radius for cluster formation.

54

R. Patidar and S. Agrawal

Rancourt et al. (2015) determined a set of food distribution centers out of potential locations using a coverage radius for the region of Garissa in Kenya. Similarly, Khalilpourazari and Khamseh (2017) used the covering radius to establish temporary and permanent blood collection facilities in the disaster relief supply chain.

2.4 Research Gap The introduction section points out the following shortcomings: low profitability of farmers, high post-harvest losses, and a large number of small farm landholding farmers. The small farmers are individually selling their products in which they have to bear higher transportation costs in the range of 30–40% of total transaction cost (Gandhi & Namboodiri, 2004; Panda & Sreekumar, 2012). Govindan (2018) identified poor transportation structure, inappropriate infrastructure, and lack of demand– supply integration as critical barriers in designing a food supply chain. Also, the author recommended incorporating a suitable transportation strategy in order to cooperate and collaborate among the supply chain partners. Recent review paper (Zhu et al., 2018) reported that the involvement and integration of small farmers were neglected in existing supply chain models. It is observed that APM has the potential to solve the above-mentioned problems of Indian AFSC and it has been ignored in designing a supply chain in the literature. It enables the farmers to participate in a supply chain in a consolidated form and has capabilities to mitigate the identified shortcomings. The APM using the clustering of farmers is capable of reducing transportation cost. The reduced transportation cost can be realized by farmers, which increases farmers’ profitability. It also helps in easy handling of various operational activities and demand–supply integration, which can reduce post-harvest losses. Therefore, the main aim of this paper is to propose a mathematical model for designing a distribution network with incorporating clustering of farmers into the formulation.

3 Model Description and Formulation A location–allocation problem is considered to design a distribution network for Indian AFSC. The problem optimizes location and allocation of facilities among the set of available potential sites and movement of product between these facilities to minimize total distribution cost (TDC). The introduction section reports that the farmers individually transport their product to markets (hubs) for selling in traditional AFSC. This approach is an inefficient way of product movement from farmers to hubs. Aggregate transportation supports the idea of sharing economy since the products of nearby farmers are aggregated at central locations, and the movement of these aggregated products to hubs will result in the transport economy. We modify the existing AFSC by aggregating products at farmers’ cluster centers (FCCs) and

Designing Distribution Network for Indian Agri-fresh Food …

55

transporting these aggregated products to the hubs. The proposed approach helps in hassle-free supply of product to hubs and reduces the transportation cost in the distribution of fresh items. The four-echelon location–allocation model integrates the supply nodes (i.e., farmers) and demand nodes (i.e., customer zones (CZs)) via FCCs and hubs, as shown in Fig. 2. The FCCs and hubs manage the transshipment of the product to satisfy demand economically. A multi-period mathematical model for multiple products is developed in this section. The supply availability of agricultural products changes from day to day, and the proposed multi-period model incorporates these changes into location–allocation of facilities. In the model, we first define farmers’ clusters (FCs) and the locations of the respective FCCs. An FCC is defined at any available farmer’s location of an FC. The assignment of farmers to the respective FCCs is based on the maximum distance traveled by a farmer, who brings their products to the common points known as FCCs for aggregation of products. In the next stage, the aggregated products are transported from FCCs to the hubs. Further, these products are delivered from hubs to CZs to fulfill the demand of the products. Figure 2 depicts the structure of the proposed distribution network model, and symbols used in this figure are described in Sect. 3.2.

Fig. 2 Proposed distribution network for Indian AFSC

56

R. Patidar and S. Agrawal

3.1 Assumptions The model is formulated under the following assumptions: • The values of customer demand, farmer supply, and other model parameters are fixed, and hence, the model is deterministic. • The model is multi-period; therefore, excess items are used in future periods. • Each farmer can be assigned to only one FCC in each period. • An FCC can supply partial amounts to any hub. Similarly, a CZ can receive partial supply from any hub. • Inter-hub product movement is not considered in the model.

3.2 The Model Proposed MINLP model determines the following decisions for each period: • The optimal number of FCCs and their respective locations for aggregation of products. • The optimal locations of hub from the available potential locations to match demand and supply of products. • The quantities of each product type flow between any two nodes at different echelons. • The inventory level of products at each hub. Sets and indices f

Index of farmers (suppliers), f ∈ F

i

Index of FCCs, i ∈ I ∈ F

j

Index of hubs, j ∈ J

k

Index of CZs (retailers), k ∈ K

p

Index of product types, p ∈ P

t

Index of periods (days), t ∈ T

Parameters H tf p

Harvest quantity available to supply of product type p from farmer f (kg)

t Dkp

Quantity demanded by CZ k for product type p (kg)

D1 f i

Distance from farmer f to FCC i (km)

D1c

Maximum distance traveled by a farmer belonging to the any FCC (km)

D2i j

Distance from FCC i to hub j (km)

D3 jk

Distance from hub j to CZ k (km) (continued)

Designing Distribution Network for Indian Agri-fresh Food …

57

(continued) T C1

Unit transportation cost from a farmer to an FCC (INR/km/kg)

TC2

Unit transportation cost from an FCC to a hub (INR/km/kg)

TC3

Unit transportation cost from a hub to a CZ (INR/km/kg)

W jp

Inventory holding cost of product type p for a per period at hub j (INR/kg/period)

LBi p

Lower bound on supply from FCC i to hub(s) for product type p (kg)

UBi p

Upper bound on FCC capacity i to handle product type p (kg)

UB j p

Upper bound on hub capacity j to handle product type p (kg)

NH

Number of hubs to be opened

FC1i

Fixed cost for forming FCC i (INR)

FC2 j

Fixed cost for opening hub j (INR)

Decision variables  Fit  Z tj

1 if hub j is opened in period t; 0 otherwise.

 Y tf i

1 if FCC i is formed in period t; 0 otherwise.

1 if farmer f is assigned to FCC i in period t; 0 otherwise.

Sitp

Quantity aggregated at FCC i in period t for product type p (kg)

Q it j p

Quantity transported from FCC i to hub j in period t for product type p (kg)

t

Q jkp

Quantity transported from hub j to CZ k in period t for product type p (kg)

B tj p

Inventory level of product type p at hub j in period t (kg)

Objective function Minimize Z = C1 + C2 + C3 + C4 + C5 + C6 C1 =

T  I 

(1)

FC1i Fit

(1a)

D1 f i TC1 H tf p Y tf i

(1b)

FC2 j Z tj

(1c)

t=1 i=1

C2 =

T  I  F  P  t=1 i=1 f =1 p=1

C3 =

T  J  t=1 j=1

58

R. Patidar and S. Agrawal T  I  J  P 

C4 =

D2i j TC2 Q it j p

(1d)

t=1 i=1 j=1 p=1

C5 =

P T  J  

W j p B tj p

(1e)

t=1 j=1 p=1

C6 =

T  J  K  P 

D3 jk TC3 Q tjkp

(1f)

t=1 j=1 k=1 p=1

Subject to: D1 f i Y tf i ≤ D1c , ∀ f, ∀i, ∀t I 

(2)

Y tf i Fit = 1, ∀ f, ∀t

(3)

H tf p Y tf i = Sitp , ∀i, ∀ p, ∀t

(4)

i=1 F  f =1 J 

Q it j p ≤ Sitp , ∀i, ∀ p, ∀t

(5)

H tf p Y tf i ≤ UBi p Fit , ∀i, ∀ p, ∀t

(6)

j=1 F  f =1 J 

Q it j p ≥ LBi p Fit , ∀i, ∀ p, ∀t

(7)

j=1 J 



t t Q jkp ≥ Dkp , ∀k, ∀ p, ∀t

(8)

j=1

B t−1 jp +

I 

Q it j p −

i=1

B t−1 jp +

K 



t Q jkp = B tj p , ∀ j, ∀ p, ∀t

(9)

k=1 I  i=1

Q it j p ≤ UB j p Z tj , ∀ j, ∀ p, ∀t

(10)

Designing Distribution Network for Indian Agri-fresh Food … K 



t Q jkp ≤ UB j p Z tj , ∀ j, ∀ p, ∀t

59

(11)

k=1 J 

Z tj = NH, ∀t

(12)

j=1

Y tf i , Fit , Z tj = {0, 1}, ∀ f, ∀i, ∀ j, ∀t

(13)

The objective function (1) minimizes the TDC incurred for the distribution of products at various stages. TDC includes three types of costs, i.e., fixed costs, transportation costs, and inventory holding cost at hubs, and their explanations are as follows: The costs C1 and C3 denote fixed costs of forming FCCs and opening hubs, respectively. The costs C2, C4, and C6 describe the transportation costs from farmers to FCCs, FCCs to hubs, and hubs to CZs, respectively. The cost C5 represents total inventory holding cost at all hubs. Following constraints represent various conditions for each period. Constraints (2) and (3) define the FCs and identify the associated farmers to these FCCs based on maximum distance traveled by a farmer for the aggregation of product. Constraint (4) determines the aggregated product quantity at each FCC for each product type. Constraint (5) ensures that the quantity of product shipped from each FCC to all hubs is less than or equal to its availability. Constraint (6) ensures the upper bound on FCC capacity and inward product movement to the formed FCC. The lower bound on supply from FCC and outward product movement from the formed FCC are warranted by constraint (7). Constraint (8) guarantees that the quantity delivered to each CZ is greater than or equal to the demand of each product type. Inventory flow balance is described by constraint (9) for each product type at each hub. The upper bound on hub capacity and inward/outward product movement to/from the open hub are warranted by constraint (10)/(11). Constraint (12) ensures the total number of hubs to be opened. The binary integer variables are ensured by constraint (13).

3.3 Linearization of MINLP An MINLP model is complex in execution and unable to solve the large-size problem in a reasonable time (Kaur & Singh, 2017). Therefore, we transform the proposed MINLP model (Eqs. 1–13) into MILP in this section. The MINLP model is converted into a MILP model by replacing nonlinear constraint (3) with a set of linear constraints from (14) to (18). To obtain equivalent linear constraints, a new auxiliary binary variable (Uit ) is introduced for each FCC as defined below.  Uit

=

1 if Fit ≥ 0 can hold (can form FCC); 0 if Fit = 0 must hold (will not form FCC).

60

R. Patidar and S. Agrawal

With the help of the above-defined auxiliary binary variable, nonlinear constraint (3) in the original MINLP model can now be replaced by the following set of linear constraints. Uit ≤ Fit , ∀i, ∀t

(14)

Y tf i ≤ Uit , ∀ f, ∀i, ∀t

(15)

Y tf i + Fit − 1 ≤ Uit , ∀ f, ∀i, ∀t

(16)

F 

Y tf i = 1, ∀i, ∀t

(17)

Uit = {0, 1}

(18)

f =1

Constraints (14)–(16) ensure assignment of any farmer to the formed FCC only. The assignment of each farmer to only one FCC is ensured by constraint (17). The binary restriction on the auxiliary variable is ensured by constraint (18). Hence, the proposed MILP consists of Eqs. (1), (2) and (4)–(18).

4 Model Implementation and Numerical Results We consider a case study problem to check the validity and viability of the proposed MINLP/MILP model for a real scenario. To conduct various experiments, the problem is divided into small (A), medium (B), and large (C) categories on the basis of its size (Table 1). Section 4.1 explains the description of data preparation for the problem. Section 4.2 compares the objective function values and computational time of both models for the problems A and B. Further, Sects. 4.3 and 4.4 discuss the results and sensitivity analysis of the proposed MILP model for the case study problem C, respectively. Table 1 Size of the problem

Parameter

Problem size Small: A Medium: B Large: C

Number of farmers (F)

8

14

94

Number of hubs (J)

2

3

10

Number of customer zones 3 (K)

6

40

Types of products (P)

5

5

5

Number of periods (T)

3

3

10

Designing Distribution Network for Indian Agri-fresh Food …

61

4.1 Data Preparation Madhya Pradesh is the central province and presents the variability of different parts of India in terms of cultural, geographical, climatic, and agricultural diversity. In Madhya Pradesh region, the common vegetables are generally consumed by customers (NSSO, 2014) and produced by farmers (MoAFW, 2017) and are identified. These vegetables are classified into five groups, as suggested by Dhaliwal (2017) and Pennington (2003). These groups of vegetables are mentioned as product types in the model and are described in Table 2. To reduce the cost of data collection, authors prepare data for Mandsaur district of Madhya Pradesh province. In Mandsaur district, there are 19 towns and 446 villages (Census of India, 2011). The demand comes from highly populated towns, and vegetables are supplied by farmers from the villages to these towns. In this work, we assume each village as a farmer unit and its location as a farmer’s location. Mandsaur city is assumed to be a demand area (town) which consists of several wards, and each ward is considered here as CZ. For simplicity, we consider 94 farmers and 40 CZs. Among these CZs, 10 locations also act as potential hub locations to serve all the CZs based on their geographical locations. The population of CZs and consumption of vegetables (per person/day) are referred to estimate demand from the Census of India (Census of India, 2011) and NSSO (2014) reports, respectively. Based on these data, the average demand for vegetables is calculated for all CZs for a single period. Similarly, authors estimate the average supply of vegetables to Mandsaur city based on farmers’ landholding (Census of India, 2011) and crop yield per hectare (MoAFW, 2017) for a single period. These single-period estimates of demand and supply are used to generate multi-period demand and supply with the help of uniform distribution. We conduct several experiments for each product type to identify the values of lower bound on supply from FCCs, upper bound on FCC capacity, and upper bound on hub capacity for the estimated data. These values for different product types are presented in Table 3. Distance between farmers, distance from FCC to the hub, and distance from the hub to CZ are determined using Google Maps (2018). Fixed cost of opening FCC and hub, transportation cost from farmer to FCC, FCC to hub, and hub to CZ as well as holding cost at hub are assumed based on our experience and prevailing conditions of Mandsaur district as shown in Table 4. Table 2 Group of vegetables Product type

Name of vegetables

p1

Coriander and spinach

p2

Brinjal, chili, bell paper (capsicum), bitter, cauliflower, beans

p3

Radish, carrot, bottle gourd, cucumber, lady’s finger, pointed gourd

p4

Tomato and cabbage

p5

Potato, onion, pumpkin

62

R. Patidar and S. Agrawal

Table 3 Estimated value of parameters in kg Product type

Farmer supply capacities

CZ demand quantities

Lower bound on supply from FCC

Upper bound on FCC capacity

Upper bound on hub capacity

p1

5–23

15–30

8

100

1100

p2

50–86

100–160

45

500

2640

p3

50–83

105–150

45

500

2640

p4

34–64

82–102

30

400

1980

p5

130–182

238–338

120

1000

6600

Table 4 Value of various cost parameters

Parameter

Value (INR)

(1) Fixed cost of each facility for each period FCC

500

Hub

5000

(2) Holding cost for each hub (per kg/period) p1

5

p2

3

p3

3

p4

2

p5

2

(3) Transportation cost (per kg/km) From farmer to FCC

1

From FCC to hub

1

From hub to CZ

1

4.2 Comparison Between MINLP and MILP Outcomes To compare the efficiency of nonlinear and linear models in terms of time complexity and solution accuracy, we run both models for each problem A and B (Table 1). The MINLP and MILP models are coded and solved in LINGO 12.0 software using branch and bound technique. The experiments are conducted on the PC having Windows 10 operating system, Intel Core i5 processor, and 8 GB of RAM configuration. The MINLP solution gives local optima because of nonlinear constraint (3), which makes the model a non-convex model, whereas the results of the MILP model show global optima in the solution report due to the linear nature of the problem (Table 5). Further, the feasibility of results obtained by MILP is checked for the nonlinear constraint of MINLP. It is observed that the results of MILP are feasible and better in terms of solution accuracy. Table 5 compares the numerical values of the objective function, elapsed computational time, and other parameters of nonlinear and linear models for both the problems. From the results, it is noted that the elapsed computational time

Designing Distribution Network for Indian Agri-fresh Food …

63

Table 5 Comparison between MINLP and MILP solutions S. No.

Parameter

Values for problem A (NH = 1, D1c = 16.5) MINLP (x1 )

MILP (x2 )

Percentage reduction

(x 1 −x 2 )×100 x1

Values for problem B Percentage (NH = 2, D1c = reduction (x 1 −x 2 )×100 16.5) x1

MINLP (x1 )

MILP (x2 )

1

C1

7000

7500

− 7.14

11,500

13,500

− 17.39

2

C2

27,573

22,660

17.81

52,033

32,825

36.91

3

C3

15,000

15,000

0

30,000

30,000

0

4

C4

91,126

95,322

− 4.6

207,568

216,982

− 4.53

5

C5

142

114

19.71

248

325

− 31.04

6

C6

3641

3641

0

13,441

13,441

0

7

TDC

144,483

144,238

0.16

314,791

307,074

2.45

8

Time (mm:ss)

00:20

00:01

95

44:58

00:02

99.92

9

No. of FCCs

4–5

4–6

7–8

8–10

10

Variables Total

709

733

1801

1843

Nonlinear

216

0

630

0

Integer

222

246

639

681

11

Constraints Total

842

1250

1706

2924

Nonlinear

24

0

42

0

12

State

Local opt Global opt

Local opt Global opt

13

Solver

B and B

B and B

B and B

B and B

of the nonlinear model is high and rapidly increasing in comparison with the linear model as increasing the size of the problem. The nonlinear model is run for the largesize problem C, and it does not report the solution after 80 h of computational time. In a similar kind of work, Kaur and Singh (2017) also reported that for the large-size problem, the MINLP model takes very high computational time in comparison with the MILP. Therefore, the MILP model can be used to solve the large-size problem.

4.3 Results of a Large-Size Problem The proposed MILP model is used to solve problem C (Table 1) by referring to data from Tables 3 and 4. The model has 81,277 variables and 144,643 constraints. The total number of FCs to be formed depends on lower bound on supply from

64

R. Patidar and S. Agrawal

Table 6 Numerical results for large-size problem Number of hubs to be opened (NH)

Maximum distance traveled by a farmer (D1c ) 9.5

10.5

3

No feasible solution found

High computational time

13.5

16.5

19.5

2,616,356 (8:11a )

2,601,141 (7:10)

2,592,008 (9:27)

2,586,079b (10:30)

6

2,649,760 (5:26)

2,634,545 (12:02)

2,625,598 (5:47)

2,619,849 (7:32)

8

2,694,881 (4:01)

2,679,682 (6:17)

2,670,579 (3:23)

2,665,211 (4:38)

10

2,744,416 (2:35)

2,728,984 (4:07)

2,719,729 (5:38)

2,714,373 (5:57)

4

a b

Elapsed computational time in mm:ss. Best result.

FCCs, upper bound on FCC capacity as well as maximum distance traveled by a farmer to reach an FCC. Similarly, which hubs should be opened depends on demand and supply, hub capacity as well as the distance between two nodes of consecutive echelons. To find the best result, we conduct 16 experiments by changing the distance traveled by a farmer and the number of hubs to be opened each at four levels. The results of these experiments are shown in Table 6. It is reported that the experiment which has 4 numbers of hubs and 19.5 km distances traveled by a farmer reports the best result (minimum TDC) out of these 16 experiments. The solver reports an optimal cost of INR 2.586 × 106 by solving the proposed MILP model in 10 min and 30 s. The best result reports that in order to meet the demand and supply of the defined region, FCCs in the range between 27 and 34 are formed in different periods for the collection of the products from 94 farmers. All costs incurred in the proposed distribution network model are analyzed using a Pareto chart and shown in Fig. 3. Figure 4a and b shows the locations of farmers ( ), FCCs ( ), hubs ( ), and CZs ( ) for Mandsaur region for period 1. In order to understand the dynamic nature of our model, some group of farmers is selected. The selected farmers are shown in Fig. 4a in a dashed rectangle; within this rectangle, the FCs and respective FCCs are presented in dashed loops for period 1. Similarly, the grouping of these farmers to form FCs and the corresponding position of FCCs for period 2 is shown in Fig. 4c in loops. The selected farmers, their supply availability, and distance between these farmers are shown in Tables 7 and 8, respectively. The dynamics of the supply chain in terms of formation of an FC and FCC are explained as follows. • The FCC at farmer 18 and the associated farmers (12, 18, 25, 67, 69, 86) satisfy upper bound on FCC capacity for period 1. At this FCC, the supply availability of these farmers increases in period 2. Therefore, the upper bound on the FCC capacity exceeds. Hence, it can be seen that the FCC at 18 has only farmers (12, 18, 69, 86) for this period. It is to be noted that farmer at 67 has joined in another

Designing Distribution Network for Indian Agri-fresh Food …

65

Fig. 3 Pareto analysis of cost incurred in the proposed distribution network

FCC at 15 and farmer at 25 is grouped into a new FCC at 62 as shown in Fig. 4c for period 2. • In our model, FCC formation depends on the location of farmers (distance between farmers) and their supply. For example, we can see that as the supply by a farmer at 67 increases, the FCC is shifted from farmer 10 in periods 1 to 15 in period 2. • The third dynamics is due to the setting of lower bound on supply from FCC to hubs in our model. For example, in period 1 two FCCs are formed at farmers’ location 87 and 5 for the collection and transfer of products, whereas in period 2 FCC is formed only at location 5 because FCC at 87 is not able to supply its minimum required lower bound; therefore, FCC at 87 in period 2 is merged with FCC at 5.

4.4 Sensitivity Analysis This section examines the impact of changing the parameter value of maximum distance traveled by a farmer to reach an FCC (D1c ) and the number of hubs to be opened (NH) on the TDC of the supply chain. Two sets of experiments (each set consists of 4 experiments) are discussed for the sensitivity analysis in the following paragraphs. The first set of experiments examines the effect of changing D1c with a fixed value of NH as 6. Figure 5 shows the variations in costs C1, C2, C4, and TDC with increasing D1c . It is reported that increasing D1c increases the size of an FC (greater number of farmers can access an FCC) and reduces the number of FCCs which in turn minimizes cost C1. Further, reducing the number of FCCs results in decreasing transportation cost from the FCCs to the hub (C4) since the total distance traveled from all FCCs to the hub is decreased. On the other hand, transportation cost

Fig. 4 a Locations of farmers, FCCs, hubs, and CZs of Mandsaur region based on the proposed distribution network model for t = 1; b locations of hubs and CZs in zoomed figure for t = 1; and c locations of farmers and FCCs of dotted rectangle for t = 2

66 R. Patidar and S. Agrawal

Designing Distribution Network for Indian Agri-fresh Food …

67

Table 7 Selected farmer’s product supply availability Farmer number

For t = 1 period p1

Total

p2

p3

p4

p5

For t = 2 period p1

p2

Total

p3

p4

p5

5

10

51

71

40

159

331

10

68

55

50

141

324

10

10

55

64

64

140

333

18

63

71

62

141

355

12

13

84

59

61

176

393

12

71

53

56

178

370

15

7

79

77

34

164

361

15

61

53

54

145

328

18

6

76

51

38

131

302

7

78

82

53

134

354

23

23

55

80

49

157

364

7

57

69

37

169

339

25

23

80

68

56

178

405

5

59

67

46

180

357

27

20

67

54

58

170

369

22

81

51

53

148

355

34

13

67

57

59

154

350

19

66

60

48

161

354

50

14

71

71

41

169

366

21

85

61

48

142

357

62

13

57

77

50

180

377

21

81

64

54

140

360

67

7

63

52

45

174

341

15

77

59

59

176

386

69

20

80

79

49

150

408

18

54

61

56

167

356

86

6

71

59

37

154

327

7

50

83

48

181

369

87

21

56

77

44

179

377

7

59

63

46

136

314

from farmers to FCCs (C2) increases due to increasing distance traveled by farmers. Therefore, the TDC is almost constant, as shown in Fig. 5. The second set of experiments examine the impact of changing NH by keeping a constant value of D1c as 16.5 km. The variations in costs C3, C4, C6, and TDC with increasing NH are shown in Fig. 6. By increasing NH, the cost required to open hub (C3) increases because each hub is associated with fixed opening cost. On the other side, transportation costs from FCCs to hubs (C4) and hubs to CZs (C6) are almost constant because the potential locations of all hubs are very close in the defined region, i.e., Mandsaur city. Therefore, the TDC increases with increase in NH.

5 Conclusions and Future Research In traditional Indian AFSC, farmers independently bring their products to the market (hubs) for selling, which leads to the inefficient flow of products. The paper proposes an MINLP model for designing a distribution network to resolve the shortcomings of Indian AFSC. The model aims to minimize the TDC through APM by considering the clustering of farmers. The APM can reduce transportation cost and post-harvest losses, which will enhance the farmers’ profitability. The MINLP model determines the following decisions: the optimal number of FCCs with their corresponding locations for aggregation of products, locations of hub, allocation of facilities (FCC and hub), product flow between facilities, and inventory level of products at each hub. We

0

21.27

14.8

26.5

11

2.24

12.5

19.3

17.33

27.59

14.13

24.52

8.21

12.7

4.89

5

10

12

15

18

23

25

27

34

50

62

67

69

86

87

5

Farmer number

15.38

6.18

11.51

13.38

7.14

6.33

3.94

6.28

9.3

16.35

11.84

5.23

10.17

0

21.27

10

10.46

4.36

6.65

9.74

8.72

15.34

11.68

4.52

6.32

12.82

3.78

8.21

0

10.17

14.8

12

20.61

11.41

16.74

9.46

12.37

7.6

9.17

3.69

14.53

21.57

11.98

0

8.21

5.23

26.5

15

6.68

1.58

2.87

13.52

4.94

18.16

7.9

8.33

2.54

9.04

0

11.98

3.78

11.84

11

18

Table 8 Distance between selected farmers

5.61

10.62

9.48

25.8

9.19

22.65

12.39

20.57

11.59

0

9.04

21.57

12.82

16.35

2.24

23

8.24

4.12

4.43

16.06

2.4

15.62

5.36

10.83

0

11.59

2.54

14.53

6.32

9.3

12.5

25

14.97

8.88

11.16

10.16

13.42

10.82

10.22

0

10.83

20.57

8.33

3.69

4.52

6.28

19.3

27

11.44

4.97

7.58

17.32

3.2

10.27

0

10.22

5.36

12.39

7.9

9.17

11.68

3.94

17.33

34

4.06

9.83

4.1

21.77

8.47

21.93

11.67

16.55

9.81

10.03

8.26

20.24

12.03

15.61

8.7

41

21.7

12.51

17.84

14.32

13.46

0

10.27

10.82

15.62

22.65

18.16

7.6

15.34

6.33

27.59

50

5.3

12.81

9.17

24.75

13.54

26.98

16.72

19.52

12.79

7.06

11.23

25.88

15.01

20.66

5.8

60

8.25

8.17

4.38

18.46

0

13.46

3.2

13.42

2.4

9.19

4.94

12.37

8.72

7.14

14.13

62

20.2

14.1

16.39

0

18.46

14.32

17.32

10.16

16.06

25.8

13.52

9.46

9.74

13.38

24.52

67

3.87

4.45

0

16.39

4.38

17.84

7.58

11.16

4.43

9.48

2.87

16.74

6.65

11.51

8.21

69

7.72

8.51

4.07

20.45

7.11

20.56

10.3

15.23

9.5

2.11

6.94

19.47

10.71

14.24

3.54

83

8.26

0

4.45

14.1

8.17

12.51

4.97

8.88

4.12

10.62

1.58

11.41

4.36

6.18

12.7

86

0

8.26

3.87

20.2

8.25

21.7

11.44

14.97

8.24

5.61

6.68

20.61

10.46

15.38

4.89

87

68 R. Patidar and S. Agrawal

Designing Distribution Network for Indian Agri-fresh Food …

Millions

C1

C2

69 TDC

C4

3 2.5

Cost

2 1.5 1 0.5 0

10.5

13.5

16.5

19.5

Maximum distance travelled by a farmer

Millions

Fig. 5 Cost against change in maximum distance traveled by a farmer C3

3

C4

TDC

C6

2.5

Cost

2 1.5 1 0.5 0

4

6

8

10

Number of hubs to be opened

Fig. 6 Cost against change in the number of hubs to be opened

transform the original model into the linear model to reduce the complexity of the developed MINLP model. Both MINLP and MILP models are coded and solved in LINGO 12.0 software. A case study of Mandsaur district (India) is considered in this paper with small-, medium-, and large-size problems. Firstly, the small- and mediumsize problems are considered to compare the results obtained from both models. It is reported that the MILP model is better over the MINLP in terms of objective value and computational time. Further, implementation and sensitivity analysis of the MILP model are reported for the large-size problem.

70

R. Patidar and S. Agrawal

The proposed model addresses three dimensions of sustainability: It addresses the economical dimension directly by minimizing the TDC incurred in the chain. On the other hand, other two dimensions, i.e., social and environmental, are addressed indirectly. The reduction in transportation cost from farmers to hub can be benefited to the farmers (social dimensions). The aggregate product transportation supports easy product handling and reduces carbon emissions and traffic congestion; hence, it is an eco-friendly strategy for product transportation (environment dimensions). Hence, this novel model may reform traditional Indian AFSC by distributing fresh items in sustainable ways as well as reducing the actual shortcomings of the system. Finally, the paper not only suggests a novel transportation strategy, but is also a model of a new way of doing agribusiness. In real life, this model can be easily implemented with the help of suitable information sharing applications to integrate and collaborate with supply chain partners and their activities.

5.1 Managerial Implications The results allow us to draw a few implications by implementing the proposed model on the real scenario of Indian AFSC as follows. • The model suggests finding the optimal location of FCCs as well as hubs from available potential locations in each period. Therefore, this model guides managers and policymakers to identify optimal locations for facilities where suitable infrastructure can be developed. • The results indicate that the developed model is robust concerning the changes in D1c and NH. • Manager or policymaker can decide the value of D1c for a particular distribution network as per the policy of the organization and geographical situation for the concerned region. • The APM will reduce carbon emissions and traffic congestion; hence, it is an eco-friendly strategy for product transportation.

5.2 Future Research Based on our study, the following recommendations are suggested for future research work: • From the Pareto chart (Fig. 3), it is observed that 90% of the TDC in the proposed distribution network is due to transportation cost from farmers to FCCs (C2) and further FCCs to hubs (C4). These costs C2 and C4 can be reduced by opening more numbers of FCCs with considering vehicle routing for picking up the products from FCCs to hubs in future research work.

Designing Distribution Network for Indian Agri-fresh Food …

71

• The criteria for cluster formation can be refined by considering other parameters like the accessibility of vehicles of farmers’ locations (road connectivity). • The perishability of products can be incorporated into designing the distribution network model. • Researchers can develop conceptual framework followed by policies and information technology model (app/Web site) for the proposed distribution network. The model may be helpful to the existing retail chain network like Reliance Fresh, Big Bazaar, and Bigbasket for the collection of fruits and vegetables from farmers. Entrepreneurs can use the proposed model to develop a fresh food retail chain for Indian scenarios under the Startup India and Smart City schemes (Vibrant Gujarat Global Summit, 2017).

References Ahumada, O., & Villalobos, J. R. (2011). A tactical model for planning the production and distribution of fresh produce. Annals of Operations Research, 190(1), 339–358. Amiri, A. (2006). Designing a distribution network in a supply chain system: Formulation and efficient solution procedure. European Journal of Operational Research, 171(2), 567–576. Anjaly, B., & Bhamoriya, V. (2011). Samriddhii: Redesigning the vegetable supply chain in Bihar. Indore Management Journal, 2(4), 40–52. Balaji, M., & Arshinder, K. (2016). Modeling the causes of food wastage in Indian perishable food supply chain. Resources, Conservation & Recycling, 114, 153–167. Bosona, T. G., & Gebresenbet, G. (2011). Cluster building and logistics network integration of local food supply chain. Biosystems Engineering, IAgrE, 108(4), 293–302. Boudahri, F., Sari, Z., Maliki, F., & Bennekrouf, M. (2011). Design and optimization of the supply chain of agri-foods: Application distribution network of chicken meat. In 2011 International Conference on Communications, Computing and Control Applications, CCCA 2011. Available from https://doi.org/10.1109/CCCA.2011.6031424 Brulard, N., Cung, V., Catusse, N., & Dutrieux, C. (2018). An integrated sizing and planning problem in designing diverse vegetable farming systems. International Journal of Production Research, 1–19. Census of India. (2011). District Census Handbook Mandsaur (24th ed.). Directorate of Census Operations Madhya Pradesh. Available from http://censusindia.gov.in/2011census/dchb/2316_P ART_B_DCHB_MANDSAUR.pdf Chandrasekaran, M., & Ranganathan, R. (2017). Modelling and optimisation of Indian traditional agriculture supply chain to reduce post-harvest loss and CO2 emission. Industrial Management & Data Systems, 117(9), 1817–1841. Chopra, S. (2003). Designing the distribution network in a supply chain. Transportation Research Part e: Logistics and Transportation Review, 39(2), 123–140. Dandage, K., Badia-Melis, R., & Ruiz-García, L. (2017). Indian perspective in food traceability: A review. Food Control, 71, 217–227. Daskin, M. S., Snyder, L. V., & Berger, R. T. (2005). Facility location in supply chain design, logistics systems: Design and optimization. Available from https://doi.org/10.1007/0-387-249 77-X_2 Dhaliwal, M. S. (2017). Classification of vegetable crops. Handbook of vegetable crops (third ed., pp. 11–17). Kalyani Publishers.

72

R. Patidar and S. Agrawal

Etemadnia, H., Goetz, S. J., Canning, P., & Tavallali, M. S. (2015). Optimal wholesale facilities location within the fruit and vegetables supply chain with bimodal transportation options: An LP-MIP heuristic approach. European Journal of Operational Research, 244(2), 648–661. Farahani, P., Grunow, M., & Günther, H.-O. (2012). Integrated production and distribution planning for perishable food products. Flexible Services and Manufacturing Journal, 24(1), 28–51. Gandhi, V. P., & Namboodiri, N. V. (2004). Marketing of fruits and vegetables in India: A study covering the Ahmedabad, Chennai and Kolkata Markets. Available from https://web.iima.ac.in/ assets/snippets/workingpaperpdf/2004-06-09vpgandhi.pdf Ganeshkumar, C., Pachayappan, M., & Madanmohan, G. (2017). Agri-food supply chain management: Literature review. Intelligent Information Management, 9(2), 68–96. Gardas, B. B., Raut, R. D., Cheikhrouhou, N., & Narkhede, B. E. (2019). A hybrid decision support system for analyzing challenges of the agricultural supply chain. Sustainable Production and Consumption, 18, 19–32. Gardas, B. B., Raut, R. D., & Narkhede, B. (2017). Modeling causal factors of post-harvesting losses in vegetable and fruit supply chain: An Indian perspective. Renewable and Sustainable Energy Reviews, 80, 1355–1371. Gelareh, S., Neamatian Monemi, R., & Nickel, S. (2015). Multi-period hub location problems in transportation. Transportation Research Part e: Logistics and Transportation Review, 75, 67–94. Gholamian, M. R., & Taghanzadeh, A. H. (2017). Integrated network design of wheat supply chain: A real case of Iran. Computers and Electronics in Agriculture, 140, 139–147. Google Maps. (2018). Mandsaur District. Available from https://www.google.com/maps/place/ Mandsaur,+Madhya+Pradesh/@24.0609876,74.9963644,11z/data=!4m5!3m4!1s0x39642ea9 db15e09f:0x89a3e5ea4399695b!8m2!3d24.076836!4d75.0692952 Govindan, K. (2018). Sustainable consumption and production in the food supply chain: A conceptual framework. International Journal of Production Economics, 195, 419–431. Hegde, R. N., & Madhuri, N. V. (2013). A study on marketing infrastructure for fruits and vegetables in India. National Institute of Rural Development. Available from http://www.nird.org.in/ NIRD_Docs/rs2013/RS91.pdf Hiremath, N. C., Sahu, S., & Tiwari, M. K. (2013). Multi objective outbound logistics network design for a manufacturing supply chain. Journal of Intelligent Manufacturing, 24(6), 1071–1084. Jayaraman, V., & Ross, A. (2003). A simulated annealing methodology to distribution network design and management. European Journal of Operational Research, 144(3), 629–645. Kashyap, D., & Agarwal, T. (2019). Food loss in India: Water footprint, land footprint and GHG emissions. Environment, Development and Sustainability, 2–14. Kaur, H., & Singh, S. P. (2017). Heuristic modeling for sustainable procurement and logistics in a supply chain using big data. Computers and Operations Research, 98, 1–21. Khalilpourazari, S., & Khamseh, A. A. (2017). Bi-objective emergency blood supply chain network design in earthquake considering earthquake magnitude: A comprehensive study with real world application. Annals of Operations Research, 1–39. Khamjan, W., Khamjan, S., & Pathumnakul, S. (2013). Determination of the locations and capacities of sugar cane loading stations in Thailand. Computers and Industrial Engineering, 66(4), 663– 674. Kundu, T. (2013). Design of a sustainable supply chain model for the Indian agri-food sector: An interdisciplinary approach (M.Tech Thesis). Jadavpur University. Mahapatra, M. S., & Mahanty, B. (2018). India’s national food security programme: A strategic insight. S¯adhan¯a, 43. Melo, M. T., Nickel, S., & Saldanha-da-gama, F. (2009). Facility location and supply chain management—A review. European Journal of Operational Research, 196(2), 401–412. Ministry of Agriculture & Farmers Welfare (MoAFW) Government of India. (2017). Horticultural statistics at a glance 2017. Oxford University Press. Available from https://doi.org/10.1787/hea lth_glance-2017-en Ministry of Agriculture & Farmers Welfare (MoAFW) Government of India. (2018). Agricultural statistics-2018. Available from http://agricoop.gov.in/sites/default/files/agristatglance2018.pdf

Designing Distribution Network for Indian Agri-fresh Food …

73

Mogale, D. G., Dolgui, A., Kandhway, R., Kumar, S. K., & Tiwari, M. K. (2017). A multi-period inventory transportation model for tactical planning of food grain supply chain. Computers and Industrial Engineering, 110, 379–394. NABARD. (2014). Agricultural land holdings pattern in India. NABARD Rural Pulse, I, 1–4. National Crime Records Bureau (NCRB). (2016). Accidental deaths & suicides in India 2015, New Delhi. Available from http://ncrb.nic.in/StatPublications/ADSI/ADSI2015/adsi-2015-full-report. pdf. Accessed November 16, 2017 National Sample Survey Office (NSSO). (2014). Household consumption of various goods and services in India 2011–12 (Vol. 558). New Delhi. Available from http://mospi.nic.in/sites/def ault/files/publication_reports/Report_no558_rou68_30june14.pdf National Sample Survey Organisation (NSSO). (2005). Situation assessment survey of farmers: Some aspects of farming (Vol. 496). Available from http://planningcommission.nic.in/sec tors/agri_html/some%20aspects%20of%20farming%2059%20tround%202003.pdf. Accessed February 7, 2017 Negi, S., & Anand, N. (2017). Post-harvest losses and wastage in Indian fresh agro supply chain industry: A challenge. IUP Journal of Supply Chain Management, 14(2), 7–24. Nourbakhsh, S. M., Bai, Y., Maia, G. D. N., Ouyang, Y., & Rodriguez, L. (2016). Grain supply chain network design and logistics planning for reducing post-harvest loss. Biosystems Engineering, 151, 105–115. Olivares-Benitez, E., González-Velarde, J. L., & Ríos-Mercado, R. Z. (2012). A supply chain design problem with facility location and bi-objective transportation choices. TOP, 20(3), 729–753. Panda, R. K., & Sreekumar. (2012). Marketing channel choice and marketing efficiency assessment in agribusiness. Journal of International Food & Agribusiness Marketing, 24(3), 213–230. Parwez, S. (2016). A conceptual model for integration of Indian food supply chains. Global Business Review, 17(4), 1–17. Patidar, R., Agrawal, S., & Pratap, S. (2018). Development of novel strategies for designing sustainable Indian agri-fresh food supply chain. S¯adhan¯a, 43(10), 1–16. Pennington, J. A. (2003). Definitions and classifications for fruit and vegetables. IARC Handbooks of Cancer Prevention (pp. 1–21). Raghunath, S., & Ashok, D. (2009). Delivering simultaneous benefits to the farmers and the common man: Time to unshackle the agricultural produce distribution system. In S. Chopra, P. Meindl, & D. V. Kalra (Eds.), Supply chain management: Strategic, planning and operation, third (pp. 133– 139). Pearson Prentice Hall. Rais, M., & Sheoran, A. (2015). Scope of supply chain management in fruits and vegetables in India. Journal of Food Processing & Technology, 6(3), 1–7. Rajurkar, S. W., & Jain, R. (2011). Food supply chain management: Review, classification and analysis of literature. International Journal Integrated Supply Management, 6(1), 33–72. Rancourt, M. È., Cordeau, J. F., Laporte, G., & Watkins, B. (2015). Tactical network planning for food aid distribution in Kenya. Computers and Operations Research, 56, 68–83. Samuel, M. V., Shah, M., & Sahay, B. S. (2012). An insight into agri-food supply chains: A review. International Journal of Value Chain Management, 6(2), 115–143. Shukla, M., & Jharkharia, S. (2013). Agri-fresh produce supply chain management: A state of the art literature review. International Journal of Operations & Production Management, 33(2), 114–158. Siddh, M. M., Soni, G., Jain, R., Sharma, M. K., & Yadav, V. (2017). Agri-fresh food supply chain quality (AFSCQ): A literature review. Industrial Management & Data Systems, 117(9), 2015–2044. Sihariya, G., Hatmode, V. B., & Nagadevara, V. (2013). Supply chain management of fruits and vegetables in India. International Journal of Operations and Quantitative Management, 19(2), 113–122. Sohoni, V., & Joshi, A. (2015). Nisarg Nirman: The social farming venture from India. Emerald Emerging Markets Case Studies, 5(8), 1–16.

74

R. Patidar and S. Agrawal

Tsolakis, N. K., Keramydas, C. A., Toka, A. K., Aidonis, D. A., & Iakovou, E. T. (2014). Agrifood supply chain management: A comprehensive hierarchical decision-making framework and a critical taxonomy. Biosystems Engineering, 120, 47–64. Vibrant Gujarat Global Summit. (2017). Government of India initiatives. Vibrant Gujarat Global Summit 2017. Available from https://vibrantgujarat.com/writereaddata/images/pdf/PresentationGovernment-of-India-Initiatives.pdf. Accessed February 10, 2017 Yu, J., Gan, M., Ni, S., & Chen, D. (2015). Multi-objective models and real case study for dual-channel FAP supply chain network design with fuzzy information. Journal of Intelligent Manufacturing, 29(2), 389–403. Zhu, Z., Chu, F., Dolgui, A., Chu, C., & Zhou, W. (2018). Recent advances and opportunities in sustainable food supply chain: A model-oriented review. International Journal of Production Research, 1–23.

A Study on Inventory Models for Perishable Items in a Serial Supply Chain Operating with Price Markdowns Saravanya Sankarakumaraswamy, Arshinder Kaur, Chandrasekharan Rajendran, and Hans Ziegler

1 Introduction The fruits and vegetables vendor often reduce the selling price of old items in order to sell them before the end of their shelf-life. They reduce the per unit price or offer more quantity for the same price. The unsold and old produce will have to be discarded due to perishing. Hence the fresh produce retailers face the problem of deciding how much to order each time and what should be the selling price for older items in order to increase sales and reduce discarding perished items. The same set of decisions is needed for upstream members in the fresh produce supply chain as well. With this motivation, we have performed this study on inventory models for perishable items in a serial multi-stage supply chain. A case scenario is considered for this study inspired from the fruits and vegetables supply chain in India (see Fig. 1). Fruits and vegetables vendors selling fresh produce to consumers procure from wholesale markets who in turn procure their produce from suppliers who are generally the farmers. The data

Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/978-981-16-5820-4_4) contains supplementary material, which is available to authorized users. S. Sankarakumaraswamy (B) · A. Kaur · C. Rajendran Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India A. Kaur e-mail: [email protected] C. Rajendran e-mail: [email protected] H. Ziegler School of Business, Economics and Information Systems, University of Passau, 94032 Passau, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_4

75

76

S. Sankarakumaraswamy et al.

Fig. 1 Illustration of fruits and vegetables supply chain in India

used for this study was inferred from vegetable vendors in the Indian state of Tamil Nadu. Supply chains dealing with food and related raw materials such as fresh fruits and vegetables are very different from other types of products with respect to their perishable nature and limited shelf-life. Perishable inventory management involve decisions on (Shen et al., 2011) Ordering policy (when and how much to order), Issuing policy (sequence in which items are removed from the stockpile), Disposal policy (when and how much to dispose under stochastic demand when perishing occurs), and Pricing policy (dealing with ordering policy separately or simultaneously with markdown price as a decision variable). Perishable items are constrained by the limited selling period due to shelf-life constraints or new product arrival in the market. This forces the firm selling the perishable item to dispose off the units as it approaches the end of selling season. Since, the perceived value of the items by the consumers will be less as the end of the selling season nears, the goods are required to be marked down for price. Hence, pricing for the perishable items become an important decision to be made by the firms. This makes the demand for these items to be price-sensitive. Decisions like when to markdown the price and how much to markdown are critical for the firms so that their profit is also maximized and loss due to perishing or outdating is reduced. This leads to the question of how much to order in the next period and how much of old items to carry to the next period, if leftover occurs. Hence the need for joint decision making on pricing and stocking of perishable items arises.

2 Literature Review and Background In the existing literature many researchers have studied ordering policies for perishable inventory management. Abdel-Malek and Ziegler (1988), Kanchanasuntorn and Techanitisawad (2006), Yong-bo et al. (2008) and Jia and Hu (2011) presented optimal ordering policies for determination of order quantities and order frequencies for multi-stage perishable item supply chain. Some authors analyzed the problem of price markdown for perishable items. Urban and Baker (1997) studied a single-period inventory model with a deterministic and a multivariate demand function of price, time and inventory level. Two models were presented, one where price remained static and the other in which price was marked down. Mostly perishable inventory is studied at a single stage or for a single period. The inventory ordering policies for such items may change for multi-stage and multi-period.

A Study on Inventory Models for Perishable Items …

77

Berk and Gurler (2008) studied the performance of (r, Q) inventory policy (similar to the (s, Q) continuous review policy discussed in this paper, where s is the reorder level) for a perishable inventory system with fixed shelf life, constant lead times and lost sales. They considered the concept of effective self-life of products by which the remaining shelf life of products are considered at the instances when the inventory level hits Q. Sethupathi and Rajendran (2010) and Sethupathi et al. (2014) developed mathematical models and heuristic algorithms to determine optimal and heuristic base-stock levels and review periods that minimize the total system cost for a serial supply chain dealing with non-perishable products. Haijema (2013) developed a new class of stock-level dependent ordering policy (s, S, q, Q), which is an order-up-to level policy with order quantity restricted to a minimum and maximum value for perishable products with a short maximum shelf-life. Lu et al. (2016) studied and compared inventory system for perishable items with limited replenishment capacity where the demand rate depended on the stock quantity displayed and the sales price fixed by the retailer. The paper studied and compared the optimization problem of joint static pricing and replenishment policy with the joint dynamic policy concluding that the latter is a better policy. However, the work considered only a single stage in the supply chain. Dobson et al. (2017) included a linear age-dependent demand rate to analyze an EOQ approximation model for perishable items sold by retailers. Their work considered a deterministic setting where the demand rate is same between order cycles with zero lead time and without any shortage to the retailer. This work provides a basis for considering perishable items with age-dependent demand in a deterministic setting and a single stage supply chain. Chen et al. (2006), Chew et al. (2009), Chang et al. (2010), Rong et al. (2011), and Chung and Erhun (2012) studied the integrated decision-making problem of order quantity and price for perishable items for a single-stage supply chain setting. From the literature review, it is seen that little work has been done with respect to pricing of products based on the age of inventory when demand is dependent on price and age of the product. In addition, no work seems to exist which simultaneously considers markdowns and ordering policy of perishable items. Hence there is a need to address these gaps with respect to ordering policies for perishable inventory and especially in a multi-stage supply chain setting. There should be a consideration to study various ordering policies and select the best policy for each scenario. With this understanding, we propose the following approach and technique to solve the ordering policy issues for perishable items in a multi-stage supply chain. In this paper, we develop a mathematical formulation for ordering policies that determine the order parameters and markdown prices for different ages of inventory. This approach can be considered to be a first step in addressing ordering policy decision making problems for multi-stage perishable supply chains. For similar large-scale problems, heuristics approach can be considered. Dolgui et al. (2018) used evolutionary techniques such as genetic algorithm to analyse a three-stage production-inventory-distribution problem for limited shelf life perishable items

78

S. Sankarakumaraswamy et al.

including cosmetics, fast moving consumer goods and pharmaceutical industry products. Their work considered a constant and known demand for each period with inventory control at retailer’s end only. In this work we consider two different cases for the computation of inventory position. The formulation is developed for six different order policies such as (R, S), (s, Q), (s, S), (R, s, Q), (R, s, S), and (s, S, q, Q) (see Haijema, 2013) and their performance is compared through computational experimentation of the models developed in this paper. In these models, s refers to the reorder level, Q refers to the order quantity, S refers to the order-up-to level and R refers to the review period in the inventory models. Detailed list of notations is presented in Sect. 3. Finally, a sensitivity analysis is also performed with respect to parameters of price-dependent demand function and shelf life.

3 Description of the Models for Different Order Policies The supply chain system considered in this study consists of three members, a supplier, a wholesaler and a retailer. The wholesaler procures items from a supplier who has infinite capacity of the items and then ships them to a retailer. The product is assumed to have a limited and fixed shelf-life after entering the retailer end. The retailer faces a price-dependent and deterministic demand (i.e. we sample the daily demand from a uniform distribution, and this sampled demand stream is deterministic and known a priori) from the customer for different ages of inventory. The retailer places the replenishment order to the wholesaler based on his/her ordering policy. While ordering, the retailer considers two cases for determining on-hand inventory. The first case (Case A) considers the entire on-hand inventory for all different ages to compute the inventory position of the retailer at the end of the current time period. The second case (Case B) considers the on-hand inventory of items surviving on the day of receipt of order replenishment to compute the inventory position for retailer at the end of the current time period. The wholesaler ships the ordered quantity to the retailer after receiving the order request, either fully or with the available on-hand inventory, and this inventory shipment reaches the retailer after the transportation lead time. The retailer chooses one of these cases for determining his/her inventory position. The wholesaler also follows the same ordering policy as that of retailer, and based on it, places the replenishment order to the supplier. The supplier ships the order quantity to the wholesaler, and it reaches the wholesaler after the corresponding transportation lead time. The decision variables in the proposed model are the price markdowns (for retailer alone) and ordering policy parameters such as order quantity, order-up-to level, reorder level and review period depending upon the order policy for every member in the supply chain. The objective function is to maximize the total supply chain profit. The total supply chain profit is determined by computing the difference between the revenue and total costs which includes purchase cost, holding cost, ordering cost and disposal cost. It is to be noted that we

A Study on Inventory Models for Perishable Items …

79

sample the customer demand from the given demand distribution with the consideration of markdown price and shelf-life. These sampled customer demands (price and product age dependent) are known over the finite time horizon. This facilitates the development of mathematical models for different inventory policies.

3.1 Assumptions 1. 2. 3. 4.

5.

6. 7.

8. 9. 10. 11. 12. 13. 14. 15.

A single type of perishable product is considered to be shipped from the supplier to the retailer via the wholesaler. Time is assumed to be discrete and the unit of time is assumed to be a day. The planning horizon is finite. The product considered has a fixed and limited shelf-life, τ . At the end of its shelf-life, the units are discarded at a cost (i.e., outdating cost or cost of disposal). Perishability of the product is assumed to happen only at the retailer and not at other members. The age of unit inventory is assumed to be 0 when it reaches the retailer at the beginning of day t (index for time unit); at the beginning of day (t + τ ), the age of the corresponding inventory is (τ − 1), and hence it becomes obsolete or outdated at the end of day (t + τ ). Retailer faces a price-dependent discrete demand from the customers, which is deterministic and known a priori. Retailer is assumed to have a set of discrete markdown prices from which the best price is to be selected for each age of inventory units. The price markdown is assumed to be in discrete intervals, since in real-life price is marked down in discrete fashion and not in continuum. The price-sensitivity of demand varies for units of different ages of inventory. Every member other than the retailer faces a discrete demand from their respective downstream members. The supplier has an infinite capacity. Lead-time for transportation exists between every member in the serial supply chain. Information lead-time is assumed to be negligible or zero. Every member in the serial supply chain operates with the same ordering policy. Unmet demand is not backlogged and is considered as lost sales. Each member in the supply chain incurs a fixed ordering cost per order and a carrying cost.

80

S. Sankarakumaraswamy et al.

3.2 Notations 3.2.1

General Notations

t

Index for time unit

i

Index for product’s age at the retailer; 0 ≤ i ≤ (τ − 1), i = 0 for fresh products with τ = 1

j

Index for members in the supply chain; 1 ≤ j ≤ N

k

Index for markdowns

3.2.2

T

Input Parameters

Total number of days over which the supply chain is deterministically simulated

τ

Shelf life of the product

N

Total number of members in the supply chain

K

Total number of markdowns; where k = 1, . . . , K

v

Cost of disposal per unit for retailer at the end of shelf-life

LT j

Replenishment lead time with respect to installation j

hj

Holding cost rate for installation j

Oj

Ordering cost for installation j

M

A large positive number

RCP

Retailer’s purchasing cost per unit

CP j

Purchasing cost per unit for installation j, 2 ≤ j ≤ N

SP j

Selling price per unit of the quantity shipped by installation j to j − 1,2 ≤ j ≤ N

3.2.3

Notations with Respect to System Variables

OIbj,t

On-order inventory refers to the order quantity placed for installation j at the beginning of time t, j ≥ 1

OIej,t

On-order inventory refers to the order quantity placed for installation j at the end of time t, j ≥ 1

b RIi,t

Retailer’s on-hand inventory of units with age i at the beginning of time t

e RIi,t

Retailer’s on-hand inventory of units with age i at the end of time t (continued)

A Study on Inventory Models for Perishable Items …

81

(continued) RSIet

Retailer’s sum of inventory of units at the end of time t over age 0 to τ* (note: RSIet =

τ∗  i=0

e ; note: (i) τ ∗ = RIi,t (τ − 1) − 1 for Case A when the on-hand inventory is

computed as the sum of on-hand inventory of the units with age 0 to age (τ − 1) − 1 at the end of time t; (ii) τ ∗ = (τ − 1) − LT1 for Case B when on-hand inventory is computed as the sum of on-hand inventory of the units with age 0 to age (τ − 1) − LT1 at the end of time t so that the inventory that is not aged at the beginning of (τ − 1) − LT1 is reckoned with) I bj,t

On-hand inventory at the beginning of time t for installation j, j > 1

I ej,t

On-hand inventory at the end of time t for installation j ( j > 1) (note: for j = 1 we e = τ −2 RIe ) have I1,t i=0 i,t

RIPt

Inventory position for the retailer at the end of time t (note: this is similar to x(t) used to represent inventory policy w.r.t time t; see Silver et al., 1998)

IP j,t

Inventory position for installation j at the end of time t, j > 1

RLi,t

Number of units of lost sales with age i at retailer with respect to time t

L j,t

Lost sales for installation j at time t, j > 1



RDi,k,t

Customer demand faced by the retailer for units with age i and with selling price with respect to markdown k at time t

RDi,t

Customer demand faced by retailer for units with age i at time t

RSDt

Sum of retailer’s demand for units of all ages at time t

D j+1,t

Demand for installation j + 1 from downstream installation j in time t

RQSi,t

Number of units of demand for the product with age i met by the retailer

QS j,t

Quantity shipped by installation j to installation j − 1 at time t



pi,k

Retailer’s selling price per unit with respect to markdown k for units with age i

REVi,k,t Revenue for retailer at time t with respect to the number of demand met by the retailer to the customer, corresponding to the product of age i and markdown k RSPi

3.2.4



Retailer’s selling price per unit of the product with age i

Notations with Respect to Decision Variables in the Respective Inventory Policies

Reorder level for installation j, j ≥ 1

sj 

Qj 

Order quantity for installation j, j ≥ 1

Sj

Order-up-to level for installation j, j ≥ 1

R

Review period for installation j, R  ∈ {1, 2, . . . , τ } (see Silver et al. (1998) for representation of notations)

82

3.2.5

S. Sankarakumaraswamy et al.

Binary Variables

δ ∗j,t

Binary variable that takes the value 1 if on day t an order is placed by installation j to upstream member j + 1

δi,k

Binary variable that indicates the selection of markdown k for units with age i

The first binary variable, δ ∗j,t uses a ‘*’ superscript to differentiate from the second binary variable, δi,k when the two variables take same subscripts in instances of the programming iteration.

3.3 Sequence of Events The sequence of events that take place with respect to each supply chain member j (index for members in the supply chain) in every time t is as follows. The common steps are presented here, whereas the steps specific to inventory ordering policy are discussed in the respective sections for the models. 1. 2.

3.

4.

5.

6. 7. 8.

Any replenishment order quantity that is due to arrive at an installation j will take place in the beginning of time t. The retailer decides on the selling prices to be fixed for inventory of different ages at the beginning of time t, which have an impact on the corresponding customer demand. For other upstream members in the supply chain, the demand is realized at the beginning of time t from the downstream member j − 1. For the retailer, the beginning inventory of age i (i = 0, 1, . . . , τ − 1) is equal to the end inventory of age i + 1 of previous time t − 1, provided that the items have remaining shelf-life. Units whose shelf-life is reached at the end of time t are discarded. Demand is satisfied from the available on-hand inventory. If sufficient on-hand inventory is not there to meet the demand, then the unmet demand is considered as lost sales. The downstream member j − 1 receives the order quantity after the corresponding replenishment lead-time with respect to installation j. If the downstream member j − 1 is the customer, then he/she receives it on the same day t. The order is placed to upstream member j + 1, with respect to the order policy considered. On-hand inventory and on-order inventory for installation j are updated at the end of time t. Quantity shipped by installation j to j −1 is computed to calculate the revenue for installation j.

A Study on Inventory Models for Perishable Items …

9. 10.

83

Costs such as fixed ordering cost, purchase cost and holding cost for each installation and outdating cost for retailer are computed in every time t. Similar sequence of events gets repeated for all installations j for every time t by incrementing the value of t by one time unit (day) until the end of planning horizon. From the total revenue earned and cost realized, the total supply chain profit is computed over the planning horizon.

3.4 Mathematical Model Formulation for Perishable Items for Various Ordering Policies In this study, we develop mathematical formulations of the models Hybrid-Haijema, Periodic review, Continuous review and Hybrid review policies. This paper presents the complete formulation for the Hybrid-Haijema (s, S, q, Q) policy as given below. Hybrid-Haijema (s, S, q, Q) policy (see Haijema, 2013): this is basically an order-up-to level (s, S) policy with the order quantity restricted by a minimum q and a maximum Q. When the inventory position falls to or below the reorder level, an order equal to max{q, min{Q, S − RIPt }} by the retailer and  quantity  max q, min Q, S − IP j,t by the installation j is placed to the upstream installation. Notations specific for (s, S, q, Q) policy: 

Minimum order quantity that can be placed by installation j to ( j + 1)

qj 

Qj 

γ j,t ∗ γ j,t

Maximum order quantity that can be placed by installation j to ( j + 1) Binary variable to determine the minimum value of order quantity for installation j at time t Binary variable to determine the maximum value of order quantity for installation j at time t

minval j,t

Minimum value of order quantity determined for installation j at time t

maxval j,t

Maximum value of order quantity determined for installation j at time t ( it is the actual order quantity placed by installation j to ( j + 1))

For Hybrid-Haijema (s, S, q, Q) policy, the sequence of events presented in Sect. 3.3 holds good except for Step 6 which is replaced as follows: The order is placed to upstream member j + 1, if the inventory position is equal to or below the re-order point and order quantity of the retailer and at installation   j at the end  of current time t equals max{q, min{Q, S − RIPt }} and max q, min Q, S − IP j,t respectively. The objective function of the problem (in terms of total supply chain cost) described for Hybrid-Haijema (s, S, q, Q) policy is as given below: Maximize

84

S. Sankarakumaraswamy et al.

⎞⎞ ⎡ ⎛ ⎛ τ −1 K T τ −1  N    

⎣ SP j − CP j − RCP⎠⎠ Z= REVi,k,t + ⎝ RQSi,t × ⎝ t=1

⎛

−⎝

i=0 k=1

N 

⎞

i=0



O j × δ ∗j,t ⎠ − h 1

τ −2 

j=1

j=1

e RIi,t

i=0

⎤ ⎞ ⎛ N 

e ⎠ h j × I j,t − v × RIeτ −1,t ⎦ −⎝ j=2

(1) The selling price per unit of the jth entity is same as purchasing cost per unit of ( j + 1)th entity, hence will get cancelled for intermediate members. So, the order cost incurred will be only with respect to cost price of installation N , as seen in the second term of Eq. 2. Hence Eq. 1 can be represented by Eq. 2. The model formulation is described below. Maximize Z=

 τ −1 K T   t=1

 − h1

REVi,k,t

i=0 k=1 τ −2 

−

 τ −1 

CP N × RQSi,t

⎞ ⎛ N  O j × δ ∗j,t ⎠ −⎝

i=0

e RIi,t

⎤ ⎞ ⎛ N 

e e h j × I j,t ⎠ − v × RIτ −1,t ⎦ −⎝

i=0

j=1

(2)

j=2

subject to K 



pi,k δi,k = RSPi for i = 0, 1, . . . , (τ − 1)

(3)

k=1 K 

δi,k = 1 for i = 0, 1, . . . , (τ − 1)

(4)

k=1

RSPi ≥ RSPi+1 for i = 0, 1, . . . , (τ − 2)

(5)

{(do the following for t = 1, . . . , T ) K 



RDi,k,t δi,k = RDi,t for i = 0, 1, . . . , (τ − 1)

k=1 

(note: assumed that RDi,k,t = 0 for i = 1, 2, . . . , (τ − 1) with t = 1, .., i and ∀k),

(6)

A Study on Inventory Models for Perishable Items …

85

OIb1,t = OIe1,t−1 − QS2,t−LT1 (note: QS2,t−LT1 = 0 for t ≤ LT1 )

(7)

{ RIb0,t = QS2,t−L T1 for t ≥ (LT1 + 1)

(note : QS2,t−L T1 = 0 for t ≤ L T1 

and RIb0,t = S1 )

(8)

} b e RIi,t = RIi−1,t−1 for i = 1, 2, . . . , (τ − 1)

(9)

e b RIi,t − RLi,t = RIi,t − RDi,t for i = 0, 1, . . . , (τ − 1)

(note : RDi,t = 0 for i = 1, 2, . . . , (τ − 1) t = 1, 2, .., i)

(10)

b e RQSi,t = RIi,t − RIi,t

(11)

 

 REVi,k,t ≤ RQSi,t × pi,k + M 1 − δi,k

(12)

 

 REVi,k,t ≥ RQSi,t × pi,k − M 1 − δi,k

(13)

REVi,k,t ≤ Mδi,k

(14)

∗

RSIet

=

τ 

e RIi,t

i=0

(note: τ ∗ = ((τ − 1) − 1) for Case A; τ ∗ = ((τ − 1) − L T1 ) for Case B;

(15)

RIPt = OIb1,t + RSIet

(16)

 ∗ RIPt ≤ s1 + M 1 − δ1,t

(17)

86

S. Sankarakumaraswamy et al. 

∗ RIPt ≥ s1 + 1 − Mδ1,t

(18)

∗ minval1,t ≤ Mδ1,t

(19)

  

∗ minval1,t ≤ S1 − RIPt + M 1 − δ1,t

(20)

 ∗ minval1,t ≤ Q 1 + M 1 − δ1,t

(21)

     

∗ minval1,t ≥ S1 − RIPt − M γ1,t − M 1 − δ1,t

(22)

 

  ∗ minval1,t ≥ Q 1 − M 1 − γ1,t − M 1 − δ1,t

(23)

∗ maxval1,t ≤ Mδ1,t

(24)



∗ maxval1,t ≥ minval1,t − M 1 − δ1,t

(25)

 ∗ maxval1,t ≥ q1 − M 1 − δ1,t

(26)

∗ ∗ maxval1,t ≤ minval1,t + Mγ1,t + M 1 − δ1,t

(27)



 ∗ ∗ maxval1,t ≤ q1 + M 1 − γ1,t + M 1 − δ1,t

(28)



∗ γ1,t ≤ δ1,t

(29)

∗ ∗ γ1,t ≤ δ1,t

(30)

∗ D2,t ≤ maxval1,t + M 1 − δ1,t

(31)

∗ D2,t ≥ maxval1,t − M 1 − δ1,t

(32)

∗ D2,t ≤ Mδ1,t

(33)

OIe1,t = OIb1,t + D2,t

(34)

} (for retailer, i.e., j = 1).

A Study on Inventory Models for Perishable Items …

87

{ OIbj,t = OIej,t−1 − QS j+1,t−L T j

(35)

b e I j,t = I j,t−1 + QS j+1,t−LT j

(36)

e b I j,t − L j,t = I j,t − D j,t

(37)

e IP j,t = OIbj,t + I j,t

(38)

 IP j,t ≤ s j + M 1 − δ ∗j,t

(39)



IP j,t ≥ s j + 1 − Mδ ∗j,t

(40)

minval j,t ≤ Mδ ∗j,t

(41)

  

minval j,t ≤ S j − I P j,t + M 1 − δ ∗j,t

(42)

 minval j,t ≤ Q j + M 1 − δ ∗j,t

(43)

     

minval j,t ≥ S j − IP j,t − M γ j,t − M 1 − δ ∗j,t

(44)

 

  minval j,t ≥ Q j − M 1 − γ j,t − M 1 − δ ∗j,t

(45)

maxval j,t ≤ M δ ∗j,t

(46)



maxval j,t ≥ minval j,t − M 1 − δ ∗j,t

(47)

 maxval j,t ≥ q j − M 1 − δ ∗j,t

(48)

∗ maxval j,t ≤ minval j,t + Mγ j,t + M 1 − δ ∗j,t

(49)



 ∗ maxval j,t ≤ q j + M 1 − γ j,t + M 1 − δ ∗j,t

(50)



γ j,t ≤ δ ∗j,t

(51)

88

S. Sankarakumaraswamy et al. ∗ γ j,t ≤ δ ∗j,t

(52)



D j+1,t ≤ maxval j,t + M 1 − δ ∗j,t

(53)

D j+1,t ≥ maxval j,t − M 1 − δ ∗j,t

(54)

D j+1,t ≤ Mδ ∗j,t

(55)

b e QS j,t = I j,t − I j,t

(56)

OIej,t = OIbj,t + D j+1,t

(57)

for j = 2, 3, . . . , N ; QS N +1,t = D N +1,t

(58)

for t = 1, 2, .., T ; with initial conditions given as follows: 

e I j,0 = S j for j > 1 

(59)

RIb0,t = S1 for t = 1

(60)

e RIi−1,0 = 0 for i = 1, 2, . . . , (τ − 1)

(61)

OIej,0 = 0 ∀ j

(62)

QS j+1,t−L T j = 0 ∀t ≤ LT j and for j = 1, 2, . . . , N

(63)





sj ≤ Sj ∀ j 







qj ≤ Q j ∀ j sj ≤ Q j ∀ j

(64) (65) (66)

A Study on Inventory Models for Perishable Items … 



89

qj ≤ Sj ∀ j

(67)

γ j,t ∈ {0, 1} ∀ j and ∀t

(68)

∗ γ j,t ∈ {0, 1} ∀ j and ∀t

(69)

δ ∗j,t ∈ {0, 1} ∀ j and ∀t

(70)

δi,k ∈ {0, 1} for i = 0, 1, .., (τ − 1) and for k = 1, 2, .., K

(71)



and all other variables ≥ 0, for all j ≤ N and for all t ≤ T . Equation 2 represents the objective function to maximize the profit function of the proposed (s, S, q, Q) policy, which takes into account the revenues, the purchase price, ordering costs holding costs at all installations in the supply chain and outdating costs for the retailer. Equations 3–5 are the price constraints for the retailer representing the selection of the actual selling price from the available set of markdown prices and ensures the price of age i is greater than age i + 1. Equation 6 represents the demand constraint for the retailer. Equation 7 updates the on-order inventory for the retailer at the beginning of time t. Equations 8 and 9 represent the beginning inventory for retailer for each age of inventory i at time t. Equation 10 shows the inventory and lost sales balance equation for the retailer. Equation 11 represents the quantity shipped by retailer to the customer for each age of inventory i. Equations 12– 14 represent the constraints used to linearize the retailer’s revenue component in the objective function since the terms RQSi,t and REVi,k,t are variables in the model. Equation 15 represents the sum of end inventory computed with respect to the three different scenarios proposed for analysis. Equation 16 updates the inventory position for retailer at time t. Equation 17 is used to ensure that the retailer’s inventory position is equal or below the reorder point. Equation 18 is used to ensure that the retailer’s inventory position is above the reorder point. Equation 19 ensures the computation ∗ ∗ = 1. When δ1,t = 0 no of minimum value among Q and (S − RIPt ) only when δ1,t need to explicitly compute the minimum value so that execution time can be saved. In fact, our computational experiments have shown that such a scheme of mathematical modeling has helped us to cut down the execution time. Equations 20–23 ∗ = 1. Equation 24 ensures are used to determine the minimum value only when δ1,t the computation of maximum value among the minimum value (found above) and ∗ = 1. Equations 25–29 are used to determine the maximum value q, only when δ1,t ∗ = 1. Equations 30 and 31 are also used to ensure the computation only when the δ1,t ∗ = 1. Equations 32–34 represent of minimum and maximum value only when δ1,t the order placement by the retailer to the upstream member j. Equations 35 and 36 updates the on-order and on-hand inventory of installation j. Equation 37 represents the balance equation for on-hand inventory at the end of time t and lost sales for

90

S. Sankarakumaraswamy et al.

installation j. Equation 38 represents the inventory position for installation j. Equation 39 is used to ensure that the inventory position of installation j is equal to or below the reorder point. Equation 40 is used to ensure that the inventory position of installation j is above the reorder  Equation 41 ensures the computation of   point. minimum value among Q and S j − IPt only when δ ∗j,t = 1. When δ ∗j,t = 0, no need to explicitly compute the minimum value, so that execution time can be saved. In fact, our computational experiments have shown that such a scheme of mathematical modelling has helped us to drastically cut down the execution time. Equations 42–45 are used to determine the minimum value only when the δ ∗j,t = 1. Equation 46 ensures the computation of maximum value among the minimum value (found above) and q only when δ ∗j,t = 1. Equations 47–50 are used to determine the maximum value only when the δ ∗j,t = 1. Equations 51 and 52 are also used to ensure the computation of minimum and maximum value only when δ ∗j,t = 1. Equations 53–55 represent the order placement by the retailer to the upstream member j. Equation 56 updates the quantity shipped by installation j to j − 1 at time t. Equation 57 updates the on-order inventory for installation j at the end of time t. Equation 58 represents the quantity shipped by the supplier to the most upstream member N . Equations 59–67 represent the initial conditions for the proposed mathematical formulation. Equations 68–71 represent the binary variables used in the mathematical formulation. The model formulations for the other five policies developed as part of this study— (R, S), (s, Q), (s, S), (R, s, Q), and (R, s, S) policies are presented in a separate link and described briefly as follows. Periodic review—(R, S) Policy: In periodic review policy, inventory is reviewed at periodic intervals of time and an order (whose quantity equal to the order-up-to level minus the inventory position) is placed if the current time equals the review period. Continuous review—(s, Q) and (s, S) Policies: In this continuous review system inventory is reviewed continuously and when the inventory position drops to or below a particular level called re-order level, an order is placed. Here the decision variables are the re-order level and order quantity for (s, Q) policy, and re-order level and order-up-to level for (s, S) policy. Hybrid review—(R, s, Q) and (R, s, S) policies: In this type of review system, inventory is reviewed periodically at fixed intervals and order is placed only if the inventory position is equal to or below the re-order level. Here the decision variables are the review period, re-order level and order quantity for (R, s, Q) policy and review period, re-order level and order-up-to level for (R, s, S) policy.

4 Computational Experimentation The supply chain considered for computational evaluation is a serial system, which includes a retailer, wholesaler and a supplier with infinite capacity. The input parameter values are presented in Table 1.

A Study on Inventory Models for Perishable Items …

91

Table 1 Experimental setting Installation parameters

Supplier ( j = 3)

Wholesaler ( j = 2)

Retailer ( j = 1)

Holding cost per unit per day (h j )

–

0.75

1

Ordering cost per order (O j )

–

250

500

Outdating cost per unit (v)

–

–

6

Purchase price (CP j )

–

6

8

Selling price (SP j )

6

8

{20, 18, 16}a

Lead time (LT j )

–

2

2

a This pricing applicable only to retailer

The input parameter values used for this study are inferred from real life fruits and vegetables supply chain entities in India. The price and cost data are derived from such supply chains handling perishable items. The retailer is assumed to face a price-dependent demand function corresponding to a given age of the product. A set of selling prices are available for the retailer to choose from for each age of inventory items as a means of markdown policy since items with less remaining shelf-life needs to be priced less than the fresher items with more remaining shelf-life (see Table 1, where {20, 18, 16} denote the set of markdown prices with respect to age 0, 1 and 2 respectively). The price-dependent demand functions used to derive forecasted demand are given in Appendix 1. Hence, the mathematical model is solved for two sets of demand streams, i.e., first run and the antithetic run (explained in Sect. 1 of the appendix). The run length considered for analysis is 9 days in view of the associated large computational effort for solving the mathematical models; note: for T = 12 days, the computational time exceeded 25 h with number of iterations by solver exceeding 109,355,157. Hence, we have restricted T to be 9 days for computational evaluation of the developed models. Each run is solved for the two different cases of the modes of inventory position. The shelf-life of the perishable item is assumed to be 3 days for the purpose of computational evaluation. The beginning inventory for each installation is fixed as 100 units, which is derived as approximately twice the average demand for two days. The total supply chain profit (TSCP) is computed as the measure of performance. The results obtained for this experimental setting (known as Base Case in this paper) is presented in the following section.

5 Results of Computational Experimentation The mathematical models developed are coded in LINGO 11 optimization software and solved for the experimental setting presented in the previous section with a computer with Intel (R) Core i5 processor with 4 GB RAM and 3.10 GHz. The results obtained for (s, S, q, Q) policy with τ = 3 and bi ∈ {1, 3, 4}, termed as base case results, are presented in Table 2.

92

S. Sankarakumaraswamy et al.

Table 2 Average (of first run and antithetic run) value of performance indicators for (s, S, q, Q) policy with τ = 3 and, bi ∈ {1, 3, 4} Performance indicator

Case A

Total supply chain profit Revenue

Case B

5095.10

4977.70

12,880.00

11,435.00

Purchase cost

3888.00

3492.00

Retailer’s total ordering cost

3000.00

2000.00

500.00

375.00

Wholesaler’s total ordering cost Retailer’s total holding cost

239.00

249.00

Wholesaler’s total holding cost

154.80

341.20

3.00

0.00

Outdating cost

In Table 2, Case A considers the sum of on-hand inventory with age 0 to age ((τ − 1) − 1). Case B considers the sum of on-hand inventory of the units with age 0 to age ((τ − 1) − L T1 ). From Table 2 we can observe that Case A gives higher profit since the holding costs are less for Case A. This is due to the nature of the inventory position computation which considers fewer amounts of units in on-hand inventory. Case A has an average outdating cost of 3, apart from ordering costs and holding costs. The total supply chain profit with respect to Z (see Eq. 1) for all other policies and including (s, S, q, Q) policy for the base case of with τ = 3 and bi ∈ {1, 3, 4}, are presented in Table 3. The profit reported in the table is average of the first run and antithetic run for each case of retailer’s on-hand inventory calculation. From Table 3, we see that (s, S, q, Q) policy performs better than all other policies for the two cases of inventory position computation. We can also infer that continuous review policies yield higher profits owing to the tight monitoring of inventory positions than periodic review policies. We have also solved the mathematical models for the experimental setting presented in the previous section except for the outdating cost. It is assumed to be negligible due to the very less outdating cost associated with disposing of the spoiled produce (such as fruits and vegetables). These costs could be about 10–20% (at most) of the purchase price of the item. The results obtained for (s, S, q, Q) Table 3 Total supply chain profits for τ = 3 and, bi ∈ {1, 3, 4} Ordering policy

Case A

Case B

(R, S)

3255.25

3255.25

(s, Q)

4632.25

4972.25

(s, S)

4367.25

4367.25

(R, s, Q)

3858.00

3858.00

(R, s, S)

3257.25

3257.25

(s, S, q, Q)

5095.13

4977.25

A Study on Inventory Models for Perishable Items …

93

Table 4 Average (of first run and antithetic run) values of performance indicators for (s, S, q, Q) policy with τ = 3 and, bi ∈ {1, 3, 4} Performance indicator

Case A

Total supply chain profit

5183.75

Revenue

14,420.0

Case B 4975.25 11,435.0

Purchase cost

4368.0

3492.0

Retailer’s total ordering cost

3500.0

2000.0

Wholesaler’s total ordering cost

250.0

375.0

Retailer’s total holding cost

339.0

251.5

Wholesaler’s total holding cost

341.25

779.25

Table 5 Average (of first run and antithetic run) value for total supply chain profits for τ = 3 and, bi ∈ {1, 3, 4} Ordering policy

Case A

Case B

(R, S)

3255.25

3255.25

(s, Q)

4972.25

4972.25

(s, S)

4367.25

4367.25

(R, s, Q)

3859.75

3859.75

(R, s, S)

3257.25

3257.25

(s, S, q, Q)

5183.75

4975.25

policy without the outdating cost are presented in Table 4. The profits for all other policies and including (s, S, q, Q) policy for the base case is presented in Table 5. To see the impact of shelf-life and price-sensitivity parameter, sensitivity analysis is performed for the same experimental setting described in Sect. 4 of the paper except for the change in price-sensitivity and shelf-life in two different scenarios. The computation setting and the results are presented in Appendix 2.

6 Implications and Concluding Remarks The proposed models in this paper will be helpful for inventory managers in any stage of a supply chain dealing with perishable inventory such as fresh produce (fruits and vegetables) and other grocery items with limited shelf life. Specifically, these models can be handy to the retailers in order to estimate order quantities and frequencies to provide cost savings and increase their profits. The retailers or wholesalers can test these models for their specific product category and identify the best policy that suits their needs. Our experimental results for the multi-stage serial supply chain setting suggest the use of continuous review ordering policies for higher profit yield.

94

S. Sankarakumaraswamy et al.

This paper has addressed the problem of joint pricing and ordering decisions for a perishable inventory supply chain by considering different ages of inventory of items with price-dependent demand and multi-period setting. Most of the literature existing in this area has studied the two period perishable inventory problems with the consideration of either pricing or order policies, but not both simultaneously. In our work, we have addressed the issue of a multi-period case of perishable inventory management with the simultaneous consideration of the determination of markdown price and inventory policy parameters. We have studied the performance of different inventory control policies and compared them to find the best policy for the problem setting. It is found that (s, S, q, Q) policy is the best policy, with (s, Q) performing the next best. The main contribution of this work can be stated as being the first attempt to study the most widely used inventory control policies for a multi-stage serial supply chain problem dealing with perishable items by providing mathematical formulation and simultaneous determination of selling price and markdown for perishable items with remaining shelf-life. This work has also considered two different cases of modes of inventory position computation for order placement in a multi-period problem setting. Our attempt to present these models can be considered as a first step to address ordering policy decisions for multi-stage supply chains of perishable commodities. Further for solving such large-size problems we suggest a heuristics-based approach. Acknowledgements The first two authors of this paper are grateful to Department of Science and Technology, India for the grants, which helped significantly in carrying out this research work. The first author is thankful to Indian Institute of Technology Madras, University of Passau and DAAD for the grant provided to carry out a part of this work at University of Passau.

Appendix 1: Demand Forecast The forecasted demand for the given age i and markdown k is, , dˆi,k = a − bi × pi,k

(72)

In Eq. 72, a refers to the scale parameter of demand function (a = 100), bi is , refers to the price-sensitivity parameter of demand function (bi ∈ {1, 3, 4}), pi,k , selling price set by retailer for units ( pi,k ∈ {20, 18, 16}) corresponding to age of inventory, i (i ∈ {0, 1, 2}) and with k being the index for markdown (k ∈ {1, 2, 3}). The demand functions corresponding to the first (k = 1), second (k = 2) and third (k = 3) markdown are given below by Eq. 73. , k = 1, 2, and, 3 dˆi,1 = a − bi × pi,k

(73)

Based on the above data and the demand functions, the forecasted demands obtained are: [80, 40, 20], [82, 46, 28] and [84, 52, 36] using Eq. 73 with respect

A Study on Inventory Models for Perishable Items …

95

to age of inventory [0, 1, 2]. It is assumed that the price-sensitivity increases for units with higher age of inventory, in relation to that of fresh units with zero age of inventory. The retailer’s daily demand (known a priori) is sampled using a uniform distribution with mean demand computed from the demand function presented in Eq. 74. Daily demand di,k is given as follows di,k = dˆi,k ±

   10% of dˆi,k × (a uniform random no.)

(74)

In Eq. 74, a uniform random number (u) ranges from 0 to 1 is considered; it is assumed without loss of generality that the actual demand deviates by 10% from the price-dependent demand. This expression is represented as Lower limit and Upper limit terms in Eqs. 75 and 76 respectively.

Lower limit LLi,k = 0.9dˆi,k

(75)



Upper limit ULi,k = 1.1dˆi,k

(76)

Range = ULi,k − LLi,k

(77)

Two sets of demand streams are sampled a priori as in Eqs. 78 and 79 known as first run and antithetic run respectively and provided as input to the mathematical programming model.   Demand 1 = 0.9dˆi,k + 0.2dˆi,k × u

(78)

  Demand 2 = 0.9dˆi,k + 0.2dˆi,k × (1 − u)

(79)

Appendix 2: Sensitivity Analysis Two different scenarios considered for sensitivity analysis are explained below. Scenario 1: Run-length, T = 9 days, Shelf-life, τ = 3 days, bi ∈ {1, 3.5, 4.5}; Scenario 2: Run-length, T = 9 days, Shelf-life, τ = 4 days, bi ∈ {1, 3.5, 4.5, 4.7}. The results obtained for (s, S, q, Q) policy are presented in Table 6. Increasing the shelf-life (as in Scenario 2), the two cases differ significantly by giving different average profits and higher profits. Scenario 1 profits for all six policies are presented in Figs. 2 and 3 for Case A and Case B respectively. It can be inferred that consistently (s, S, q, Q) policy performs

96

S. Sankarakumaraswamy et al.

Table 6 Average (of the first run and the antithetic run) values of performance indicators for (s, S, q, Q) policy for Scenario 1 and Scenario 2 Performance indicator

Scenario 1 Case A

Total supply chain profit Revenue

Scenario 2 Case A

Case B

4905.80

Case B 4606.50

5712.3

5760.3 14,270

13,700.00

10,830.00

13,920

Purchase cost

4141.20

3315.00

3542.6

4281.1

Retailer’s total ordering cost

3500.00

2000.00

2500.0

2750.0

Wholesaler’s total ordering cost

250.00

250.00

250.0

375.0

Retailer’s total holding cost

183.00

210.00

886.0

816.0

Wholesaler’s total holding cost

705.00

448.50

399.1

287.6

15.00

0.00

0.00

0.00

Outdating cost

Fig. 2 Profits for Scenario 1 (Case A vs. Base)

Fig. 3 Profits for Scenario 1 (Case B vs. Base)

A Study on Inventory Models for Perishable Items …

97

Table 7 Total supply chain profits for Scenario 2 with τ = 4 days, bi ∈ {1, 3.5, 4.5, 4.7} (R, S)

(s, Q)

(s, S)

(R, s, Q)

(R, s, S)

(s, S, q, Q)

Case A

3181.6

4855.2

4547.5

3579.0

3183.2

5712.3

Case B

3389.3

5058.5

4623.2

3739.8

3443.5

5760.3

better than all other policies for the two cases of Scenario 1. The same trend is observed for Scenario 2 as shown in Table 7.

References Abdel-Malek, L. L., & Ziegler, H. (1988). Age dependent perishability in two-echelon serial inventory systems. Computers & Operations Research, 15(3), 227–238. Berk, E., & Gürler, Ü. (2008). Analysis of the (Q, r) inventory model for perishables with positive lead times and lost sales. Operations Research, 56(5), 1238–1246. Chang, C. T., Chen, Y. J., Tsai, T. R., & Wu, S. J. (2010). Inventory models with stock- and price dependent demand for deteriorating items based on limited shelf space. Yugoslav Journal of Operations Research, 20(1), 55–69. Chen, Y. F., Ray, S., & Song, Y. (2006). Optimal pricing and inventory control policy in periodicreview systems with fixed ordering cost and lost sales. Naval Research Logistics, 53(2), 117–136. Chew, E. P., Lee, C., & Liu, R. (2009). Joint inventory allocation and pricing decisions for perishable products. International Journal of Production Economics, 120(1), 139–150. Chung, Y. T., & Erhun, F. (2012). Designing supply contracts for perishable goods with two periods of shelf life. IIE Transactions, 45(1), 53–67. Dobson, G., Pinker, E. J., & Yildiz, O. (2017). An EOQ model for perishable goods with agedependent demand rate. European Journal of Operational Research, 257(1), 84–88. Dolgui, A., Tiwari, M. K., Sinjana, Y., Kumar, S. K., & Son, Y. J. (2018). Optimising integrated inventory policy for perishable items in a multi-stage supply chain. International Journal of Production Research, 56(1–2), 902–925. Haijema, R. (2013). A new class of stock-level dependent ordering policies for perishables with a short maximum shelf life. International Journal of Production Economics, 143(2), 434–439. Jia, J., & Hu, Q. (2011). Dynamic ordering and pricing for perishable goods supply chain. Computers and Industrial Engineering, 60(2), 302–309. Kanchanasuntorn, K., & Techanitisawad, A. (2006). An approximate periodic model for fixed-life perishable products in a two-echelon inventory–distribution system. International Journal of Production Economics, 100(1), 101–115. Lu, L., Zhang, J., & Tang, W. (2016). Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand. International Journal of Systems Science, 47(6), 1480–1494. Rong, A., Akkerman, R., & Grunow, M. (2011). An optimization approach for managing fresh food quality throughout the supply chain. International Journal of Production Economics, 131(1), 421–429. Sethupathi, P. V. R., & Rajendran, C. (2010). Optimal and heuristic base-stock levels and review periods in a serial supply chain. International Journal of Logistics Systems and Management, 7(2), 133–164. Sethupathi, P. V. R., Rajendran, C., & Ziegler, H. (2014). A comparative study of periodic-review order-up-to (T, S) policy and continuous-review (s, S) policy in a serial supply chain over a finite

98

S. Sankarakumaraswamy et al.

planning horizon. In U. Ramanathan & R. Ramanathan (Eds.), Supply chain strategies, issues and models (pp. 113–152). Springer. Shen, Z., Dessouky, A., & Ordonez, F. (2011). Perishable inventory management system with a minimum volume constraint. Journal of the Operational Research Society, 62(12), 2063–2082. Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling (3rd ed.). Wiley. Urban, T. L., & Baker, R. C. (1997). Optimal ordering and pricing policies in a single-period environment with multivariate demand and markdowns. European Journal of Operational Research, 103(3), 573–583. Yong-bo, X., Jian, C., & Xiao-lin, X. (2008). Fresh product supply chain coordination under CIF business model with long distance transportation. Systems Engineering—Theory & Practice, 28(2), 19–34.

Material Flow Optimisation in a Manufacturing Plant by Real-Coded Genetic Algorithm (RCGA) K. C. Bhosale and P. J. Pawar

1 Introduction In the basic vehicle routing problem (VRP), a number of vehicles are used to supply the material to various locations from a central depot. In a single trip, the vehicles visit to all the locations only once. The vehicle routing problem (VRP) is the most important problem for the supply chain management. This problem is faced by all the industries in a day-to-day life in the form of transportation of material, pickup and delivery of goods and materials, etc. The number of possible solutions is large in numbers. As the numbers of customers are increased, the number of solutions is increased. This makes the VRP Np-hard. The capacitated vehicle routing problem (CVRP) is one of the versions of the VRP. In CVRP, the demands of each customer are known in advance and deterministic and cannot be split up. The vehicles have equal and limited material carrying capacity, and they all start from a single central depot. In CVRP, the vehicle visits each location exactly once. The start and end point of the vehicle gives the route of the vehicle. The objective is considered to minimise the route length by considering the constraint of vehicle capacity. The real-life conditions of all the problems vary from instance to instance. So, the objectives and constraints vary from problem to problem. Many software and algorithms are developed for solving the VRPs. Various meta-heuristics and heuristics are developed for VRP.

K. C. Bhosale (B) Department of Mechanical Engineering, Sanjivani College of Engineering, Kopargaon, India Savitribai Phule Pune University, Pune, Maharashtra, India P. J. Pawar Department of Production Engineering, K. K. Wagh Institute of Engineering Education and Research, Nashik, India Savitribai Phule Pune University, Pune, Maharashtra, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_5

99

100

K. C. Bhosale and P. J. Pawar

For the optimisation of CVRPs, researches have proposed meta-heuristics such as artificial bee colony (ABC) algorithm, ant colony optimisation (ACO), genetic algorithm (GA), simulated annealing, Tabu search and neural network. Bhagade and Puranik (2012) used the travelling salesman problem for VRP optimisation by using nearest neighbour method. Evaluation results are then compared with artificial bee colony algorithm. A tour length and route were optimised. Alvarado-Iniesta et al. (2013) have considered CVRP for minimising the distance travelled by a worker from central warehouse to production lines. Alvarado-Iniesta et al. (2013) have considered one operator and 10 production lines and vehicle capacity constraint as 15. For optimisation, Alvarado-Iniesta et al. (2013) proposed artificial bee colony algorithm and achieved a standard time of 7 min. Ng et al. (2017) have considered a vehicle scheduling problem by considering the traffic in the area. According to the density of the traffic, the vehicles are rescheduled to avoid the late delivery. They have proposed multiple colonies, artificial bee colony algorithm. Wang and Lin (2013) have considered a capacitated vehicle routing problem with stochastic travel times. In this problem, the vehicle travel time is considered lower than the actual maximum travel time. This approach is used in the congestions such as accidents, traffic conditions and weather conditions. For the optimisation, simulation-based algorithm is used. The objectives considered are to minimise the total cost consisting of travel cost of vehicles and overtime wages. Particle swarm optimisation (PSO) is a population-based search method that mimics the behaviour of group organism as a searching method. Venkatesan, Logendran and Chandramohan (2011) have considered a CVRP to optimise total cost. They used sweep, Clark and Wright algorithm for forming the cluster. For the optimisation, they proposed PSO. Akhand, Peya, and Murase(2017) have considered benchmark problems of CVRP and proposed adaptive sweep clustering method. For generating the routes, they proposed velocity tentative particle swarm optimisation. In the genetic algorithm (GA), a random population is generated. From this random population, some solutions are selected randomly for the crossover process. After the crossover operation, mutation is carried out to reduce the chance of entrapping the solution in the local optima. Because of the usefulness of the GA, researchers have used GA in the basic as well as in the different forms. The CVRPs are solved with cluster-first, route-second approach. In this method, first a group of customers is formed and then from this group routes are generated. Korayem et al. (2015) have considered problem instances of bench problems. They proposed grey wolf optimisation algorithm for the optimisation. They used K-means clustering method. Yakub (2017) have proposed nearest neighbour algorithm and particle swarm optimisation (PSO) algorithm for optimisation. PSO was found better as compared to nearest neighbour algorithm. Cassettari et al. (2018) have considered a CVRP with time and distance constraint. They used simulated annealing for optimisation. Peya, Akhand, and Murase (2018) have considered sweep clustering for the grouping of the routes. They considered benchmark problems of CVRP and optimised the routes by velocity tentative PSO. Feld et al. (2019) have considered benchmark problems of CVRP. For the optimisation, quantum annealing was proposed.

Material Flow Optimisation in a Manufacturing Plant by Real-Coded …

101

In this paper, two case studies are considered which are dependent on CVRP. The first case study based on Alvarado-Iniesta et al. (2013) is considered to optimise the material flow. The second case study based on Venkatesan et al. (2011) is considered to optimise the total cost. To solve both the case studies, real-coded genetic algorithm (RCGA) is proposed due to its usefulness to solve the problems of VRP. The data of both case studies is of discrete type. Hence, to solve the problem related to discrete type, an algorithm must be used to tackle the issue. In this paper, an attempt is made to use RCGA on discrete type of data. In the next section, case study of Alvarado-Iniesta et al. (2013) is presented.

2 Case Study #1 In this section, a case study based on Alvarado-Iniesta et al. (2013) is considered. The problem is related to capacitated vehicle routing problem (CVRP). In this problem, a central depot is considered from which the material is to be supplied to 10 lines. The capacity of the operator is 15. The objective is considered to optimise the material flow. For delivery and pickup, single operator is considered, i.e. M = 1, and 10 production lines, i.e. N = 10. The demands are known in advance and given in Table 1. In this case, the material is considered as homogeneous for each of the lines, and capacity of operator is considered as q = 15. The distances between warehouse and lines are known in advance and given in Table 2. Table 1 Demand of lines (Alvarado-Iniesta et al., 2013) Line

1

2

3

4

5

6

7

8

9

10

Demand

7

5

4

6

7

6

5

4

3

3

Table 2 Distance between the lines 0

1

2

3

4

5

6

7

8

9

10

0

0

1033

1502

1690

1596

939

1877

1033

1830

1971

1502

1

1033

0

1924

1877

1267

1033

2816

2112

2816

2534

2347

2

1502

1924

0

469

1173

2394

1408

1690

1643

3379

2675

3

1690

1877

469

0

845

2534

1877

2112

2112

3661

3004

4

1596

1267

1173

845

0

2159

2534

2347

2675

3520

3051

5

939

1033

2394

2534

2159

0

2769

1690

2628

1502

1502

6

1877

2816

1408

1877

2534

2769

0

1173

329

3098

2159

7

1033

2112

1690

2112

2347

1690

1173

0

986

1924

1033

8

1830

2816

1643

2112

2675

2628

329

986

0

2816

1877

9

1971

2534

3379

3661

3520

1502

3098

1924

2816

0

939

10

1502

2347

2675

3004

3051

1502

2159

1033

1877

939

0

102

K. C. Bhosale and P. J. Pawar

The objective function, f (x), for a given solution x, is given in Eq. (1), f (x) = c(x)+ ∝ q(x)

(1)

where c(x) =



ci j xi j

(2)

i∈D j∈D

where cij = the travelled distance from point j to i and x ij = if the route from point j to i is chosen. For a given solution, q(x) = the total violation of the capacity constraint (penalty factor). The capacity constraint is 

di

i∈N



xi j ≤ q

(3)

i∈D

where d i = the demand of each point q = the current capacity of the vehicle α = a self-adjusting parameter that is modified in each iteration. If the constraint of vehicle capacity is violated, then the penalty for violation is assigned. The distance between production lines is shown in Fig. 1. However, the distance between the lines was not provided by Alvarado-Iniesta et al. (2013). So, it is calculated and presented in Table 2. For the optimisation of the problem, RCGA is implemented. In the next section, working of RCGA is presented.

3 Real-Coded Genetic Algorithm (RCGA) Genetic algorithm (GA) is based on the concept of nature’s law of survival of fittest. The evolution of nature is based on the principle of survival of fittest which states that the stronger ones live for many years, they produce many offsprings and pass on many attributes to their children, while the weaker ones die out early without any reproduction. Real-coded genetic algorithm (RCGA) is a part of GA in which coding is represented in the form of real numbers. The following steps are followed in RCGA:

Material Flow Optimisation in a Manufacturing Plant by Real-Coded …

103

80 9 70 5 60 1

10

50 0 40 7 30

4

20 3

8

2

10

6

0 0

10

20

30

40

50

60

70

80

Fig. 1 Graphical representation of warehouse and production lines (Alvarado-Iniesta et al., 2013)

• • • • •

Initialisation of population Selection of solutions Crossover operation Mutation operation Termination of the algorithm.

Step 1: Initialisation of population In this stage, a random sequence of the routes of the vehicle is determined. For this random sequence, an objective function is obtained. Thus, this solution is called as a chromosome. Similarly, many chromosomes are generated randomly. This will be called as population. Step 2: Selection of solutions RCGA is based on the principle of survival of fittest. So, from the population generated above, two solutions are selected randomly. For the selection process, a Roulette wheel selection method is used. Roulette wheel selection determines the fitness function of each solution. If the fitness function of the solution is high, then the chance of the solution to get selected is higher. In this selection method, two random solutions are selected by generating random numbers which are called as parent solutions. Step 3: Crossover operation After selecting two parent solutions, two child solutions are obtained by performing crossover operation. The crossover location can be selected as two-point crossover or multi-point crossover.

104

K. C. Bhosale and P. J. Pawar

Step 4: Mutation operation After the crossover operation, it may be possible that the solution may get entrapped into the local optimum solution. So, to avoid this, mutation operation is performed. This mutation operation can perform swapping method, insert method, etc. Step 5: Termination criteria: This complete cycle of selection–crossover–mutation completes one iteration. Similarly, many iterations are carried out till the termination criteria are reached. There is usefulness of RCGA over GA. There is no coding–decoding required in RCGA. So, it is simple and less time consuming. In RCGA, real numbers are used. So, the length of chromosomes is short.

3.1 Working of RCGA As discussed in Sect. 3, the working of RCGA is explained below: Step 1: Parameters of RCGA The parameters of real-coded genetic algorithm used are: Population size: 10 Number of generations = 50 Crossover type: single point Crossover probability: 0.8 Mutation probability: 0.2. The size of population is determined by conducting various experiments. Different sizes of population ranging from 10 to 20 are checked; it is observed that after population size 10, there is no significant change in solutions, and it is also observed that after 50 generations, there is no significant change in results. Step 2: Selection of solutions In this step, 10 initial solutions are generating for random sequence and presented in Table 3. Using Roulette wheel selection method, some solutions are selected to perform crossover. Roulette wheel selection method is explained with a simple example in Fig. 2. Step 3: Crossover operation In the crossover operation, two parent solutions are selected at random and then crossover is performed at single point as shown in Fig. 3. As shown in Fig. 3, the crossover site is selected at 5th location. After completing the crossover operation, some numbers are repeated and some numbers are omitted. So, in the next step, duplicate numbers are removed and shown in Fig. 4.

Material Flow Optimisation in a Manufacturing Plant by Real-Coded …

105

Table 3 Fitness function calculated according to Eq. 1 S. no.

Initial random solutions

Objective value

1

1–10-5–2-3–4-9–7-6–8

23,480.75

2

9–10-7–2-4–5-6–8-3–1

19,820

3

9–5-4–6-8–10-7–1-2–3

21,087.18

4

8–4-3–7-10–6-2–5-1–9

23,513.27

5

9–2-10–6-7–8-4–3-5–1

24,372.47

6

10–7-5–2-3–4-9–6-8–1

18,866.93

7

2–5-7–9-1–3-6–8-4–10

23,231.67

8

9–8-6–1-4–10-2–3-5–7

22,354.36

9

1–3-6–8-10–2-7–9-5–4

22,448.23

10

2–1-5–9-8–10-3–6-4–7

23,856.21

ROULETTE WHEEL SELECTION

Fig. 2 Example of Roulette wheel selection method

10 9%

1 9%

9 10%

2 11%

8 10%

3 11%

7 10%

4 9% 6 12%

5 9%

Step 4: Mutation operation Mutation operation is carried out by considering mutation probability of 0.1, i.e. single-point mutation. A random site is selected for mutation. Then from the available set of machine numbers, only a random machine number is selected and mutation is carried out. The mutation operation is explained in Fig. 5. Step 5: Termination criteria After reaching the termination criteria, the iterations are stopped. The best solution is recorded.

106

K. C. Bhosale and P. J. Pawar

1

10

5

2

3

4

9

7

6

8

7

2

4

5

6

8

3

1

Parent solution 1 9

10

Parent Solution 2 a) Before crossover operation 1

10

5

2

3

5

6

8

3

1

7

2

4

4

9

7

6

8

Child solution 1 9

10

Child Solution 2 b) After crossover operation Fig. 3 Crossover operation

1

10

5

2

3

5

6

8

3

1

7

5

6

8

3

9

2

7

5

6

8

3

9

2

4

5

6

8

3

9

a) Duplicate numbers after crossover 1

10

4

2

b) Duplicate numbers are removed Fig. 4 Duplicate number removal 1

10

4

a) Before mutation 1

10

7

b) After mutation

Fig. 5 Mutation procedure

4 Case Study #2 The second case study is based on the capacitated vehicle routing problem (CVRP) of Venkatesan et al. (2011). This work considers clustering the several customer points from the warehouse and finds out optimal route of each cluster. Here, the problem involves both the cases, i.e. allocation of vehicle for warehouse to customer and to find out shortest route between them. The mathematical model for the above problem is as follows: Equation 4 represents the number of customers with respect to total capacity of vehicle. Equations 5–7 represent the constraint, i.e. how to visit customer i and customer j. Here, the travelling salesman problem (TSP) is used to form a cluster. In

Material Flow Optimisation in a Manufacturing Plant by Real-Coded …

107

the problem, there are N cities that should be visited exactly once by the travelling salesperson, with the minimum cost of course. Once the cluster is formed for each vehicle, then to find out shortest route between them will be the solution to the problem, which is computationally complex. If the number of customers is more, it makes the problem still difficult to solve. Each cluster is representing a warehouse and associated customer. The objective function is Minimise

n N n   

Ci j X i jk

(4)

i=0 j=0, j=i k=1

Subject to n 

X i jk =

i =0 i = j

n 

X jik = 1,

(5)

j =0 j = i N n   k=1

X i jk = 1,

j = 1, 2, . . . , n

(6)

i =0 i = j n n   i=0



j = 1, . . . , n, k = 1, . . . .N

d j X i jk ≤ C

(7)

j =1 j = i

X i jk ≥ 1, ∀s ⊆ {1, 2, . . . , n}, s = , k = 1, . . . , N

(8)

i⊆s j∈s

X i jk ∈ {0, 1}, i = 0, 1, . . . , n, j = 0, 1, . . . , n, i = j, k = 1, . . . , N

(9)

Table 4 shows the X and Y coordinates and demand of each customer and warehouse (Venkatesan et al., 2011).

5 Result and Discussion For the case study#1 of Alvarado-Iniesta et al. (2013), RCGA is implemented as discussed in the previous section. The objective function value obtained by AlvaradoIniesta et al. (2013) by using ABC is 18,820 in., whereas in this work RCGA is used for the same case study and has obtained the objective function value as 15,722 in.

108

K. C. Bhosale and P. J. Pawar

Table 4 X and Y coordinates and demand of each customer and warehouse (Venkatesan et al., 2011) S. no.

X coordinate value

Y coordinate value

Demand

0

0

0

Warehouse

1

35

29

5

2

−14

−4

23

3

35

−29

14

4

−10

−35

13

5

−10

−60

8

6

0

24

18

7

−34

−65

19

8

−35

−54

10

9

40

−51

18

10

6

−55

20

11

11

14

5

12

−3

−41

9

13

−35

−44

23

14

25

30

8

15

12

−16

18

16

30

−25

10

17

31

−65

24

18

17

9

13

19

16

29

14

20

−19

−25

8

21

26

30

10

22

5

−6

19

23

10

4

14

24

−10

20

13

25

−3

−61

14

26

−25

−60

2

27

−4

−61

23

28

16

6

15

29

40

−1

8

30

0

−61

20

31

26

14

24

32

−35

−20

3

Material Flow Optimisation in a Manufacturing Plant by Real-Coded …

109

Table 5 Optimum objective function Optimisation method

Optimal route

Objective function value (in.)

% saving in objective function

ABC

0–5-0–6-2–3-0–8-10–9-7–0-4–1-0

18,820

–

RCGA

0–8-6–7-0–2-3–4-0–10-9–5-0–1-0

15,722

16.45

Thus, there is a 16.45% saving in the objective function value. This is shown in Table 5. Alvarado-Iniesta et al. (2013) have used ABC to optimise the objective function value. In this paper, RCGA is proposed to optimise the objective function value. RCGA has given 16.45% improvement in objective function value. The RCGA is used to optimise the CVRP. Much more efficient optimisation algorithms are developed during the last decade. However, most of the algorithms are used for solving continuous type of data. RCGA is population-based evolutionary algorithm and developed to solve the discrete data problems in its original form. The results are shown in Table 5. The optimum route for the solution obtained by RCGA is presented in Fig. 6. The convergence graph for Alvarado-Iniesta et al. (2013) is shown in Fig. 7. The optimum solution of case study #2 obtained by sweep algorithm is 664, whereas the solution obtained by RCGA is 658. The optimum solutions obtained by RCGA are shown in Table 6. 80 9 70 5 60

1

10

50

0

40 7 30

4

20 3

8

2

10

6

0 0

10

20

30

40

50

Fig. 6 Best route for Alvarado-Iniesta et al. (2013), RCGA

60

70

80

110

K. C. Bhosale and P. J. Pawar

Convergence Graph of RCGA 25000

Material flow

23000 21000 19000

96

86

91

81

76

71

66

56

61

46

51

41

36

31

26

21

16

6

11

15000

1

17000

Number of iterations

Fig. 7 Convergence graph of RCGA for Alvarado-Iniesta et al. (2013)

Table 6 Optimum solution obtained by RCGA Sr. Vehicle route no.

Vehicle Total Vehicle route load distance

Sweep algorithm

Vehicle Total load distance

RCGA

1

0,24,6,19,14,21,1,31,11,0 95

116

0,24,6,19,14,21,1,31,29,0 100

144

2

0,20,26,7,8,13,32,2,0

94

161

0,20,26,7,8,13,32,2,0

88

161

3

0,12,30,25,27,5,4,0

92

132

0,10,30,25,27,5,4,0

98

134

4

0,10,17,9,3,16,29,0

94

193

0,12,17,9,3,16,15,0

93

166

5

0,22,15,18,28,23,0

71

62

0,22,11,18,28,23,0

66

53

Total

446

664

445

658

6 Conclusion In this paper, two case studies which are based on capacitated vehicle routing problem (CVRP) are considered. Alvarado-Iniesta et al. (2013) have considered the objective to minimise the total distance travelled by the worker, and the capacity of the vehicle is considered as constraint for this problem. Alvarado-Iniesta et al. (2013) have considered ABC to optimise the problem. For this problem, RCGA is proposed and has obtained 16.45% saving in the objective function value. In the second case study, Venkatesan et al. (2011) have considered a problem instance of CVRP. They proposed sweep algorithm to optimise the objective function value. The objective function value obtained by sweep algorithm is 664, whereas the objective function value obtained by RCGA is 658. The results show that the results obtained by RCGA are dominating to previously used algorithm by solution quality.

Material Flow Optimisation in a Manufacturing Plant by Real-Coded …

111

References Akhand, M. A. H., Peya, Z. J., & Murase, K. (2017). Capacitated Vehicle routing problem solving using adaptive sweep and velocity tentative PSO. International Journal of Advanced Computer Science and Applications (IJACSA), 8(12). Alvarado-Iniesta, A., Garcia-Alcaraz, J. L., Rodriguez-Borbon, M.I., & Maldonado, A.: Optimization of the material flow in a manufacturing plant by use of artificial bee colony algorithm. Expert Systems with Applications, 40(12), 4785−4790 (2013) Bhagade, A. S., & Puranik, P. V. (2012). Artificial bee colony (ABC)algorithm for vehicle routing optimization problem. International Journal of Soft Computing and Engineering, 2, 29–333. Cassettari, L., Demartini, M., Mosca, R., Revetria, R., & Tonelli, F. (2018). A multi-stage algorithm for a capacitated vehicle routing problem with time constraints. Algorithms, 11, 69. Bhosale, K. C., & Pawar, P. J. (2019). Material flow optimisation of production planning and scheduling problem in flexible manufacturing system by real coded genetic algorithm (RCGA). Flexible Services and Manufacturing Journal, 31, 381–423. Ng, K. K. H., Lee, C. K. M., Zhang, S. Z., Wu, K., & Ho, W. (2017). A multiple colonies artificial bee colony algorithm for a capacitated vehicle routing problem and re-routing strategies under time-dependent traffic congestion. Computers & Industrial Engineering, 109, 151–168 Korayem, L., Khorsid, M., & Kassem, S. S. (2015). Using Grey Wolf Algorithm to solve the capacitated vehicle routing problem. IOP Conference Series: Materials Science and Engineering, 83. Peya, Z. J, Akhand, M. A. H., & Murase, K. (2018). Capacitated vehicle routing problem solving through adaptive sweep based clustering plus swarm intelligence based route optimization. Oriental Journal of Computer Science and Technology,11(2). Feld, S., Roch, C., Gabor, T., Seidel, C., Neukart, F., Galter, I., Mauerer, W., & Linnhoff-Popien, C. (2019). a hybrid solution method for the capacitated vehicle routing problem using a quantum annealer. Frontiers in ICT, 6, 13. Venkatesan, S. R., Logendran, D., & Chandramohan, D. (2011). Optimization of capacitated vehicle routing problem using PSO. International Journal of Engineering Science and Technology, 3, 7469–7477. Wang, Z., & Lin, L. (2013). A simulation-based algorithm for the capacitated vehicle routing problem with stochastic travel times. Journal of Applied Mathematics, 2013, Article ID 127156, 10p

A Multi-objective Particle Swarm Optimisation-Based Solution Approach for Multiple Mean-Responses Optimisation Considering Empirical Model Uncertainties Abhinav Kumar Sharma and Indrajit Mukherjee

1 Introduction In a manufacturing process, the overall quality of a product is usually judged based on multiple critical characteristics. These characteristics are so-called critical-toquality (CTQ) output characteristics or responses. To achieve minimum acceptable overall quality, all responses must adhere to customer specification limits (internal or external). However, it is desired that all responses strike the optimal target values (specified by customers). Problems related to simultaneous optimisation of multiple responses are known as ‘multiple response optimisation (MRO)’ problems. As the responses are often correlated and conflicting in nature, a change in the existing process setting conditions to improve the quality of one response may deteriorate the quality of another response(s). Thus, a ‘compromised’ or ‘trade-off’ setting condition is sought to achieve a balanced quality. In multi-objective optimisation (MOO) strategy, efficient compromised solutions are so-called ‘nondominated’ or ‘Pareto’ solutions (Coello et al., 2007). As per Park and Kim (2005), multi-objective optimisation-based solution search strategy can efficiently resolve varied multiple response optimisation problems. The mapping function related to independent predictor variables with a dependent response variable is the so-called response surface (RS) model (Montgomery, 2005). This response surface is generally developed based on specific Design of Experiment (DoE) trials or ‘as is’ production data (Bera & Mukherjee, 2018a, 2018b; Mukherjee & Ray, 2006).

A. K. Sharma School of Business Management, Narsee Monjee Institute of Management Studies, Mumbai, India e-mail: [email protected] I. Mukherjee (B) Shailesh J Mehta School of Management, Indian Institute of Technology Bombay, Mumbai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_6

113

114

A. K. Sharma and I. Mukherjee

Bera and Mukherjee (2018a, 2018b) provide the basic steps for response surfacebased multiple response optimisation solution approaches. They recommend six critical steps for response surface-based solution approach, (i) identify process and response variables, (ii) collect relevant data based on appropriate data collection scheme, (iii) develop appropriate data-driven empirical response surface(s), (iv) formulate the overall multiple response optimisation problem, (v) determine tradeoff solutions using appropriate optimal iterative search strategy, and (vi) evaluate final solution(s) for implementation. A critical review of the literature reveals that very few researchers had considered response surface model uncertainties and their influence on the quality of multi-objective optimisation-based multiple response optimisation solutions. While deriving efficient trade-off solutions, ignoring response surface model uncertainties may result in considerable deviations in the expected and true values of the responses for a given process setting condition. In this context, response surface model coefficients (or parameters) and response predictive uncertainties play an essential role in determining implementable process setting conditions. Unbiased point estimates of the parameters and predicted mean responses might suggest setting condition(s) that provide an actual response(s) outside the customer specification(s). As per the review of open literature (discussed in Sect. 2), there are only a few multi-objective optimisation-based solution approaches for multiple response optimisation problems that simultaneously consider model parameters uncertainties and response predictive uncertainties. The suggested approach, proposed in this work, determines plausible process setting conditions that appear most frequently on the nondominated fronts of multiple alternate response surface models (considering model parameter estimation uncertainties). The solutions also attempt to ensure that all responses’ simultaneous prediction interval (PI) falls within the customer specifications. This chapter is organised as follows. Section 2 reviews the relevant open literature to identify the research gaps. Subsequently, the set of objectives and scope of research is defined for the study. Section 3 elaborates on the proposed multi-objective optimisation-based solution approach for mean multiple response optimisation problems. Section 4 illustrates the step-wise implementation of the proposed solution approach on varied mean multiple response optimisation cases. Comparison of the solution results with different solution approaches is also discussed in this section. Finally, Sect. 5 concludes with research highlights, limitations, and scope for future research.

2 Literature Review Bera and Mukherjee (2018b) provide a critical review on response surface-based solution approaches for multiple response optimisation problems. They classify these approaches into two broad categories: (i) single composite objective optimisation (SCOO) and (ii) multi-objective optimisation (MOO). In single composite

A Multi-objective Particle Swarm Optimisation-Based …

115

objective optimisation, multiple response functions are transformed into single composite objective function using an appropriate mathematical transformation. Single composite objective optimisation provides a single best solution in single finite computational run. However, there is little guarantee that the best solution will lie on a Pareto front. Whereas the multi-objective optimisation-based solution approach simultaneously optimises multiple responses and can provide multiple Pareto solutions in a single finite computational run. In the single composite objective optimisation category, few popular response surface-based solution approaches (Bera & Mukherjee, 2018b) are desirability function (Kim & Lin, 2000), generalised distance (Khuri & Conlon, 1981), loss function (Vining, 1998), dual response surface (DRS) optimisation (Del Castillo et al., 1999), and goal attainment approach (Xu et al., 2004). Most response surface-based single composite objective optimisation approaches derive the best solutions based on point estimates of responses and ignored model uncertainties. However, few studies on single composite objective optimisation-based multiple response optimisation solution approaches also considered model uncertainties. He et al. (2012) recommended a simultaneous confidence interval (CI) and desirability function-based solution approach to resolve multiple response optimisation problems. Ouyang et al. (2015) proposed a weighted loss function and individual prediction interval-based solution approach for multiple response optimisation problems. Ouyang et al. (2016) demonstrated a prediction region-based loss function approach to resolve multiple response optimisation problem, considering model parameter uncertainties. Ouyang et al. (2017) propose a loss function-based approach considering model parameters and process setting implementation errors. Costa and Lourenço (2017) used two different metrics [viz. prediction standard error (se) and quality of prediction (QoP)] to assist the decision-maker (DM) in selecting the best process setting condition, considering response uncertainties. In the second category (multi-objective optimisation-based solution approach), Costa et al. (2012) demonstrated Pareto frontiers’ role and their specific advantages to resolve multiple response optimisation problems. Ishibuchi et al. (2011) suggest a population-based metaheuristic to derive optimal fronts of optimisation problems with correlated objectives. Chapman et al. (2014a) recommend the Pareto front approach to resolve multiple response optimisation problems. They use grid point samples in the search space to generate nondominated solutions. Chapman et al. (2014b) further show the impact of model parameter estimation uncertainties on Pareto front quality. They develop multiple sets of alternate response surface models based on experimental data. Process setting conditions that appear most frequently on the nondominated front of multiple response surface models are suggested for implementation. Lu et al. (2017) extend the work of Chapman et al. (2014b) to address higher-dimensional multiple response optimisation problems. They suggest graphical summaries (e.g. trade-off and desirability weight volume plot) and numerical summaries (e.g. composite desirability index and synthesised efficiency) to assist decision-makers to select the best solution. Sharma and Mukherjee (2020) propose a multi-objective optimisation-based solution approach for multiple response optimisation problems considering only response predictive uncertainties. They recommend

116

A. K. Sharma and I. Mukherjee

using simultaneous prediction interval for responses to determine efficient solutions for a multiple response optimisation problem iteratively. Wang et al. (2019) propose a multi-objective solution approach, considering response surface model parameters uncertainties. They optimised total rejection cost and expected quality loss to determine the best process setting for a laser micro-drilling process. A summary of relevant multiple response optimisation approaches, considering response surface model uncertainties in the two categories, is provided in Table 14 of Appendix 1. Based on the critical review of the literature and analysis of the summary provided in Table 14 of Appendix 1, the following specific research gaps and lacunas in earlier relevant research are observed: (i)

(ii)

(iii)

(iv)

(v)

There seems an absence of multiple response optimisation literature that simultaneously deals with response predictive uncertainties and empirical model parameter uncertainties. Single composite objective optimisation approach proposed by He et al. (2012) considers only response predictive uncertainties. They recommend simultaneous confidence intervals to address response predictive uncertainties. However, researchers suggest simultaneous prediction interval, as more precise interval estimations to address responses uncertainties for unknown predictive conditions (Meeker et al., 2017; Montgomery & Peck, 1992). Chapman et al. (2014a, 2014b) multi-objective optimisation-based approaches recommend individual prediction interval concept to assess the impact of response predictive uncertainties on the final solution. However, authors did not incorporate the idea of uncertainties in the overall multiple response optimisation problem formulation and iterative search strategy. Wang et al. (2019) consider model parameter uncertainties. However, the approach does not confirm that the best solutions fall on the optimal nondominated fronts of all alternate response surface models. Sharma and Mukherjee (2020) use simultaneous prediction intervals to determine feasible nondominated solutions for correlated multiple response optimisation problems. However, they had not considered model parameter uncertainties to derive final implementable solutions.

Considering the research gaps, lacunas, and summary of the earlier research (Table 14 of Appendix 1), there seems a scope to develop a multi-objective optimisationbased solution approach that can synergistically identify efficient solutions for a multiple response optimisation problem considering model uncertainties. Thus, the specific objectives of this research are set to (i)

(ii)

(iii)

develop a guided multi-objective optimisation-based solution approach for mean multiple response optimisation problems, which can identify best implementable solutions, considering response surface model uncertainties; suggest appropriate multi-criteria decision-making ranking scheme, which can help decision-maker(s) or practitioner(s) to select the best process operating conditions for real-life implementation; and to demonstrate the suitability and superiority of the solution approach in varied mean multiple response optimisation problems.

A Multi-objective Particle Swarm Optimisation-Based …

117

The key differences between this research and earlier relevant single composite objective optimisation or multi-objective optimisation-based multiple response optimisation studies are: (i)

(ii)

The majority of the reported multiple response optimisation studies investigate the effect of response uncertainties on the solution quality. Only few studies consider the impact of both model parameters and response uncertainties. No previous mean multiple response optimisation research has considered both these uncertainties into the overall multiple response optimisation formulation to determine efficient solutions. In this research, promising robust solutions are derived based on alternate response surface models (e.g. considering parameter uncertainties of the fitted response surface models). The proposed approach is significantly different from the grid point sampling approach to identify promising solutions (Chapman et al., 2014b). The efficiency of the grid point sampling approach can decrease with an increase in problem complexity (e.g. curse of dimensionality).

Specific new contributions to the existing body of knowledge are given below: (i)

(ii)

The proposed multi-objective optimisation-based multiple response optimisation solution approach simultaneously considers response surface model parameters and response predictive uncertainties. Thus, the recommended process settings are expected to minimise product rejection (e.g. adhering to specifications) and enhance process capability (e.g. closeness of outputs to target specifications) The suitability of the proposed solution approach is verified based using varied mean multiple response optimisation cases. The derived case solutions are compared with reported best solution(s). Reported solution(s), for a specific case, are derived based on various single composite objective optimisation and multi-objective optimisation solution approaches (e.g. DRS, composite desirability, goal programming, generalised distance, and loss function). The superiorities of a solution are claimed based on two different recommended performance metrics [e.g. nondominated frequency (NDF), and closeness to target responses (e.g. Mahalanobis distance)] and statistical multiple comparison testing.

The following section elaborates on the proposed multi-objective optimisationbased solution approach.

3 Proposed Multi-objective Optimisation-Based Solution Approach for Multiple Response Optimisation Problems The proposed multi-objective optimisation-based solution approach is illustrated in Fig. 1. Intrinsic details on each step are elaborated below.

118

A. K. Sharma and I. Mukherjee Step 1: Identify controllable factors and critical-to-quality response variables of the process. Operating range of all control factors, target and specifications of all responses are also defined at this step.

Yes

Step 2: Select appropriate design for Design of Experiment(DOE) and run all experimental trials

Step 3: Pre-process the experimental data. Generate multiple best fit response surface (RS) models using appropriate regression technique. Check all model assumptions and adequacy.

Is there a significant deviation between predicted and actual response values?

Step 4 : Generate alternate RS models considering parameter uncertainties of the best fit RS model

Step 5 : Formulate MRO problem, using each RS model, as a MOO problem, considering response predictive uncertainties

No

Process Control & Feedback loop

Monitor the actual responses after setting the implementable process condition

Step 6: Evaluate trial runs on fitted response models to select appropriate simultaneous PI method (e.g. Bonferroni or Scheffe's PI) for a specific MRO problem

Step 7:Determine promising robust solutions that appear on overlapping regions of non-dominated fronts of alternate RS model sets. Use suitable metaheuristic search strategy to determine non-dominated fronts (e.g. MOPSO)

Step 8: Rank and select best implementable solutions using appropriate decision making technique (e.g. TOPSIS and VIKOR) and ranking criteria.

Fig. 1 Proposed eight-step multi-objective optimisation approach

Step 1: Identify the controllable factors and critical-to-quality response variables of the process. In this step, the controllable independent factors (X) and dependent response variables (R) are identified from the process. The operating range of all factors is also defined. The targets and customer specifications of all the responses are also defined in this step. Step 2: Select an appropriate design for Design of Experiment (DOE) and run all experimental trials. The process data needs to be collected based on an appropriate statistical design to develop the empirical response functions. A specific statistical design (Montgomery, 2005) is selected based on the number of predictors, selected levels, number of replicates, and needed polynomial order of the response surface. The experimenter

A Multi-objective Particle Swarm Optimisation-Based …

119

must ensure that the process is statistically stable and corner points cover a reasonable feasible operating space of predictors (X). Step 3: Pre-process the experimental data and develop the best-fit response surface models. Once the experimental data is collected, pre-processing (e.g. standardisation and data cleaning) of data is necessary to develop an appropriate response surface model to map the relationships between the control factors (X) and responses (R). Regression analysis is generally recommended for creating the best-fit response surface model(s) (Montgomery & Peck, 1992). For a given rth response (yr ), the functional relationship between the control factors (X) and response is expressed as yr = β0 +

P  i=1

βi xi +

P  i=1

βii xi2 +

P P−1  

βi j xi x j + r

(1)

i=1 j=i+1

whereβ0 , βi , βii , and βi j are unknown regression model parameters and r is the error in prediction. The matrix notation of Eq. (1) is Yr = X r βr + er

(2)

where Yr is the n × 1 vector of the observations of the rth response; X r is the n × br design matrix; br is the total number of model parameters in the rth response model; βr is the b × 1 matrix of the unknown regression model parameters; and er is the n × 1 vector of the residual (error). The unknown regression model parameters βr can be estimated using the ordinary least squares (OLS) method. The estimate of βr using the OLS is expressed as   βˆr = X r X r X r Yr

(3)

where βˆr is the unbiased estimator of βr . It is recommended to check the goodness of fit for a regression model. The model diagnostic checks on residuals are performed to prove the generalisation ability of the model (Montgomery & Peck, 1992). Step 4: Generate alternate response surface models considering parameter uncertainties. In this step, the response surface model parameter uncertainties are considered to develop multiple response surface models (Chapman et al., 2014b). The point estimates of multiple response surfaces for a given process setting condition show the variability in the predicted responses. The process setting conditions that frequently appear in the nondominated fronts of the multiple response surfaces are preferred for future implementation. In this context, the multiple response surfaces, considering parameter uncertainties, can be generated based on the following expression:

120

A. K. Sharma and I. Mukherjee

  −1  βˆrs ∼ M V N βˆr , σr2 X r X r , ∀s = 1, 2, . . . , S ∀r = 1, 2, . . . , R

(4)

where βˆrs is the set of the rth response model parameters in the sth response surface. Here σr2 is the fitted response model error variance for the rth best-fit response and can be estimated based on the following equation: σˆ r2 = SSRes,r /(n − br )

(5)

In Eq. (5), SSRes,r is the residual sum of squares for the rth fitted response surface; n is the number of experimental data points, and br is the number of regression model parameters in the rth fitted response surface. An illustration of the variation in response value due to response surface model parameter uncertainty at a given process setting conditions is shown in Fig. 2. Figure 2 shows the surface plot of three alternate response surfaces generated from Eq. (4) and how the response values (represented in the Y-axis) change for the combination of factors (x1 , x2 ) at different levels. Thus, an optimal process setting condition derived from a specific response surface may yield different predicted response values for alternate response surface models. In other words, the optimal process settings for a particular response surface may not be optimal for an alternate response surface model.

Fig. 2 Surface plot of three different alternate response surfaces and change in predicted response values at two different parameter combinations

A Multi-objective Particle Swarm Optimisation-Based …

121

Step 5: Formulate multiple response optimisation problem using each response surface as a multi-objective optimisation problem considering predictive uncertainty. In this step, the multiple response optimisation problem is formulated using each response surface (generated considering parameter uncertainties) as a constrained multi-objective optimisation problem. A general expression of constrained multiobjective optimisation formulation is given below: Minimise f 1 (C1 ), f 2 (C2 ), . . . , f M (C M )

(6)

Subject to gi (C gi ) ≥ bi , ∀i = 1, 2, . . . , I

(7)

h j (C h j ) = 0, ∀ j = 1, 2, . . . , J

(8)

Cm , C gi , C h j ∈ C, ∀i = 1, 2, . . . , I, ∀ j = 1, 2, . . . , J, ∀m = 1, 2, . . . , M cnL ≤ cn ≤ cnU , ∀n = 1, 2, . . . , N

(9) (10)

Here, f 1 (C1 ), f 2 (C2 ), . . . , f M (C M ) are the M objective functions considered for the multi-objective optimisation formulation. gi (C gi ) and h j (C h j ) are the inequality and equality constraints, respectively. Cm , C gi , and C h j are the subset of decision variables for the objective functions, equality constraints, and inequality constraints, respectively; and Cm , C gi , C h j ∈ C. cnL and cnU are the lower and upper bounds of the nth decision variable. If the objective function is to be maximised, it can be rewritten as a minimisation problem by multiplication of −1. For a multiple response optimisation problem, the multi-objective optimisation objective functions are developed based on a suitable transformation of the responses, which depends on the type of response [smaller-the-better (STB), larger-the-better (LTB), or nominal-the-best (NTB)]. For STB and LTB types of responses, one can simply define the objective as a minimisation and maximisation function, respectively. For NTB type, the objective is generally expressed as a minimisation of the absolute deviation from a target value. Using each alternate response surface model, the following discussion elaborates on the intrinsic details of the formulation of the multiple response optimisation problem to a multi-objective optimisation problem by considering the response predictive uncertainty. Add simultaneous prediction intervals in the overall multi-objective optimisation formulation to address response uncertainties. For a given process setting condition (Xe ), an empirical response function can only provide a point estimate ( yˆr ) or predicted response for rth response model. However, the actual value of the response may fall anywhere within the range of its prediction interval [say ( yˆr,L , yˆr,U ) for an Xe setting condition at the α0 level of significance]. Thus, any point Xe is considered feasible if and only if the prediction intervals of all the responses fall within their respective customer specification limits.

122

A. K. Sharma and I. Mukherjee

Fig. 3 Feasible zone for nominal-the-best type of response considering prediction interval

yˆ rU

yˆ rL

LSL

yˆ r ( X e )

USL

As an example, Fig. 3 illustrates the zones of the feasible solutions for an NTB type of response by considering the prediction interval. The simultaneous prediction interval (Montgomery & Peck, 1992) for the rth estimated response at a given overall level of significance (αo ) is expressed as [ yˆr L , yˆrU ] = [ yˆr (Xe ) ±  ∗ se( yˆr (Xe ))], ∀r = 1, 2, . . . R

(11)

where yˆr L and yˆrU are the lower and upper predicted limits of the rth response. In Eq. (11),  is a statistic that depends on the selection of the prediction interval calculation method and se ( yˆr (Xe )) is the standard error associated with a point estimate of the r th response (at Xe ); it is expressed as se[ yˆr (Xe )] =

    (X  X )−1 x ) σˆ r2 1 + xe,r e,r r r

(12)

  xe,r is expressed as xe,r = 1,x1 , . . . , x p , x12 , . . . , x 2p , x1 x2 , . . . , x p−1 x p for a response surface model of the form given in Eq. (1). Overall multi-objective optimisation problem formulation of the multiple response optimisation problem considering response uncertainties, process setting robustness, and correlation. The multiple response optimisation problem can be formulated as a multiobjective optimisation problem by considering the response predictive uncertainties. As an example, let us consider a mean multiple response optimisation problem with an only nominal-the-best type of responses. The overall multi-objective optimisation problem is expressed as Minimise abs|[ yˆ1 − T1 ]|, abs|[ yˆ2 − T2 ]|, . . . , abs|[ yˆ R − TR ]|

(13)

Subject to yˆr L ≥ L S L r , ∀r = 1, 2, . . . , R;

(14)

yˆrU ≤ U S L r , ∀r = 1, 2, . . . , R;

(15)

x pL ≤ x p ≤ x Up ∀ p = 1, 2, . . . , P.

(16)

A Multi-objective Particle Swarm Optimisation-Based …

123

Here yˆ1 , yˆ2 , . . . , yˆ R ∀ r ∈ R are response surface models of the responses considered for the multiple response optimisation study; T1 , T2 , ..., TR ∀ r ∈ R are the response target values. The constraints in Eqs. (14) and (15) ensure that the simultaneous prediction interval limits of the responses are within specification limits. Step 6: Evaluate trial runs to select an appropriate simultaneous prediction interval method for a specific multiple response optimisation problem. Montgomery and Peck (1992) suggest the Bonferroni and Scheffe methods to calculate the simultaneous prediction intervals. For a specific regression model, researchers (Montgomery & Peck, 1992) recommended a prediction interval method with the lowest average prediction interval range for all responses. An overall level of significance (α0 ) is assumed to calculate the simultaneous prediction intervals. Using Bonferroni’s method,  in Eq. (11) is replaced by a statistic t(αo /R)/2,n−br , where αo /R is the individual level of significance for a specific response. R is the number of response functions, n is the total number of experimental design points, and br is the number of regression coefficients in the rth response. For Scheffe’s method,    is replaced by R × Fαo ,R,n−br . This statistic controls an individual level of significance to account for an overall level of significance (α0 ) (Miller, 1981). Trial runs, based on the fitted response surface models and a given αo value, provide the best  statistic for generating simultaneous prediction interval for a point estimate of the response. Step 7: Determine promising solutions using multi-objective optimisation search strategy. In this step, promising robust solutions for multiple response optimisation problem are identified. In this study, promising solutions are determined based on the approach suggested by (Villa et al., 2013) for a multi-objective optimisation problem under model uncertainties. As compared to the grid point approach (Chapman et al., 2014b), their approach determined solutions from nondominated fronts of multiple alternative objective functions. Metaheuristic search strategy is generally recommended to generate nondominated fronts for multi-objective optimisation problems (Coello et al., 2007). Thus, to solve an overall multi-objective multiple response optimisation formulation, multi-objective particle swarm optimisation (MOPSO) (Coello et al., 2004) is used in this study. Specific pseudo-code of MOPSO is provided in Fig. 6 (Appendix 2). As an illustration, promising solutions obtained from Villa et al. (2013) and Chapman et al.’s (2014b) approach for a three-response optimisation problem with two controllable process variables are shown in Fig. 4. In Fig. 4, 50 different alternative response surface models are generated for a specific problem. Using Chapman et al. (2014b) approach, 6917 grid points were sampled from the search space, considering a spacing of 0.03 for each factor dimension. The spacing of 0.03 was selected to balance between computational time and adequate representation of the search space.

124

A. K. Sharma and I. Mukherjee

Fig. 4 Comparison of promising general solutions identified by Villa et al. (2013) approach and Chapman et al. (2014b) approach

Whereas promising solutions from the Villa et al. (2013) approach was obtained by solving each response surface model using MOPSO (Coello et al., 2004). As promising solutions are determined, using both the approaches, the percentage of times (frequency) they appeared on the nondominated fronts of multiple response surface models is recorded. In Fig. 4, only a few solutions (e.g. sampled from the complete solution pool) are shown for both approaches for clarity. The percentage of promising setting conditions, appearing at different nondominated frequency ranges from both the approaches, is given in Table 1. From Table 1, it is evident that 45.11% of promising settings, identified from the Villa et al. (2013) approach, has a higher frequency (81–100%) on multiple nondominated response surfaces as compared to Chapman et al. (2014b) grid point approach (e.g. 10.3%). Thus, based on the comparison of two approaches, it is evident that nondominated solution points identified by Villa et al. (2013) approach has higher likelihood to appear in nondominated fronts. Table 1 Percentage of solutions appeared in NDF using two different approaches Frequency of nondomination (in %)

Percentage of solutions by Chapman et al. (2014b) approach

Percentage of solutions by Villa et al. (2013) approach

0–20

65.98

6.4

21–40

8.28

8.7

41–60

7.7

13.4

61–80

7.74

26.6

81–100

10.3

45.11

A Multi-objective Particle Swarm Optimisation-Based … Table 2 Selected parameters for MOPSO algorithm

125

Parameter

Value

Population size ( pop)

100

Repository size (r ep)

200

Number of divisions (ndiv)

30

Mutation rate (mutrate)

0.5

Inertia weight (w)

0.4

In this study, for a specific multiple response optimisation problem, promising solutions are identified by solving Eqs. (13)–(16) using varied alternate response surface models. Among the promising solutions, solutions corresponding to nondominated overlapping regions are retained. These solutions are expected to have a higher chance to appear in nondominated fronts and also satisfy response specifications. Parameter Selection of MOPSO algorithm The performance of an evolutionary algorithm depends on the selection of its intrinsic parameter values (Coello et al., 2007). Coello et al. (2004) provided an exhaustive experimental analysis of MOPSO for varied multi-objective test problems. This study adopts the same MOPSO strategy to resolve a multiple response optimisation problem, and the recommended optimal parameter values are given in Table 2. Step 8: Rank and select the best implementable process settings condition using MCDM technique. The best implementable process setting conditions can be selected using an appropriate technique. In this context, Chapman et al. (2014b) suggested to predefine a ‘preference point’ and then select the best process setting condition in the vicinity. They recommended a desirability index to identify the best vicinity point. In this study, the implementable process settings are selected based on two different MCDM techniques [e.g. VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) (Tzeng & Huang, 2011)]. The proposed eight-step solution approach also recommends confirmatory trial runs before selecting a best implementable process setting.

4 Multiple Response Optimisation Case Studies This section discusses and reports solutions derived from the proposed multiobjective optimisation solution approach on varied mean multiple response optimisation problems. The solutions are also compared with other reported solutions (e.g. derived from various approaches). Summary details of each case study are provided in Table 3. In this study, the source data and response surface models (Step 1–Step

Case source

Montgomery (2005)

Khuri and Conlon (1981)

Khuri and Conlon (1981)

Vining (1998)

Wang et al. (2019)

Derringer and Suich (1980)

Case

Case-1

Case-2

Case-3

Case-4

Case- 5

Case-6

Dual RS (Del Castillo et al. 1999)

Multi-objective optimisation (Wang et al., 2019)

Composite loss function (Ouyang et al., 2017)

Goal attainment (Xu et al., 2004)

Generalised distance function (Khuri & Conlon, 1981)

Composite desirability function (He et al., 2012) Multi-objective optimisation (Chapman et al., 2014b)

Proposed solution approach compared with

Table 3 Summary of multiple response optimisation cases

This case investigates the effect of hydrated silica level, silane coupling agent level, and sulphur level on quality characteristics of tyre tread compound (PICO abrasion index and elongation at break)

This case investigates the effect of the average power, pulse frequency and cutting speed on geometrical property, and machining precision of Nd:YLF laser micro-drilling of silicon slices

This case investigates the effect of the reaction time, reaction temperature, and amount of catalyst on the conversion of a polymer and thermal activity

This case investigates the effect of the heating temperature, PH level, redox potential, sodium oxalate and sodium sulphate on foaming properties of whey protein concentrates (maximum overrun, time at first drop, undenatured protein, and soluble protein)

This case investigates the effect of cysteine and calcium chloride on the quality characteristics of whey protein concentrate gel systems (hardness, cohesiveness, springiness, and compressible water)

This case investigates the effect of the reaction time and temperature on the yield, viscosity, and molecular weight of a chemical process

Brief description of the cases

126 A. K. Sharma and I. Mukherjee

A Multi-objective Particle Swarm Optimisation-Based …

127

Table 4 Least feasible αo and best  method for each case Case

Lowest feasible α0

Best  method

Case -1

0.85

Scheffe

Case-2

0.05

Bonferroni

Case-3

0.05

Bonferroni

Case-4

0.5

Scheffe

Case-5

0.80

Scheffe

Case-6

0.10

Bonferroni

3 in Fig. 1) are taken from open literature and assumed to be satisfactory for a fair comparison of the proposed approach with other solution approaches. For each multiple response optimisation case, a total of 500 response surface model sets are generated [499 alternate response surface model sets are generated from Eq. (4) and one set of best-fit response surface models as discussed in Step 3]. The selection of 500 alternative response surface models is based on balancing between an adequate number of alternate response surface models and computational time (Chapman et al., 2014b). Subsequently, based on all relevant information provided for a specific case, the multi-objective optimisation problem is formulated in this study (Step 5). For each response surface model, the mean effective response function is also derived in this step. Initial trial runs on fitted response surface models provide the necessary support to select the appropriate  method (e.g. Bonferroni or Scheffe) for calculating simultaneous prediction intervals for each case (Step 6). The lowest possible αo (e.g. the value αo that provides feasible solutions), based on average prediction interval width for the selected cases, are given in Table 4.

4.1 Select Best Implementable Solutions To identify promising solutions, the multi-objective optimisation formulation of each response surface model set (e.g. 500 in number) is solved using MOPSO (Step 7) search strategy. MOPSO program code was developed in MATLAB 2019b software interface. All computational runs are executed on a specific personal laptop with an Intel core i7-4700HQ processor, 8 GB RAM, 512 GB SSD, and a Windows 10 operating system. All solution points and their corresponding responses are rounded to the nearest decimal place (as per the least count reported in the experimental data).

4.1.1

Rank the Solutions Using MCDM Techniques

From the promising solutions, the best robust solutions are determined using two different MCDM techniques. Tzeng and Huang (2011) describe the MCDM as a series of sequential steps that involve (i) defining the problems, (ii) evaluating the

128

A. K. Sharma and I. Mukherjee

alternative(s), and (iii) determining the best alternative(s). They provide intrinsic details of different MCDM techniques. In this study, the best alternatives from the robust solutions are derived based on TOPSIS and VIKOR techniques (Tzeng & Huang, 2011). The ranking of alternatives, determined by TOPSIS, is based on the shortest Euclidean distance from a ‘positive ideal solution’ and the farthest Euclidean distance from a ‘negative ideal solution’. The positive and negative ideal solutions are determined from the weighted normalised decision matrix. A decision matrix comprises criteria arranged as a column vector, with the alternatives arranged as a rows vector. The weighted normalised decision matrix is obtained by multiplying the elements of the normalised decision matrix with the corresponding criterion weight. The vectors of the individual best and worst values of the criteria are termed the positive ideal solution and negative ideal solution, respectively. In VIKOR, the best alternatives are ranked according to their closeness to the ideal solution. The promising robust solutions for the mean multiple response optimisation problems are compared based on the (i) nondominated frequency values and (ii) average worst-case Mahalanobis distance (WMD) value. For a specific solution vector (Xe ), the average WMD is obtained by taking the average of WMDs, which is calculated from the various alternate response surface models and target values of responses. The worst-case response values are derived from the simultaneous prediction intervals limits of the responses. The WMD of a specific solution vector (Xe ) from a particular target vector (T ) is expressed as

MDX0 =

(Yˆkw (Xe ) − T )

−1 y

(Yˆkw (Xe ) − T )

(17)

 In Eq. (17), Y is the variance–covariance matrix of the responses derived from the experimental data and Yˆkw (Xe ) is the predicted worst-case vector of responses at Xe . The NDF of a specific solution setting (Xe ) is calculated based on the following equation: S

NDs

× 100, S ⎧ ⎨ 1 if solution i appear on non-dominated where NDs = surface of sth RS mode ⎩ 0 if otherwise

NDFXe =

s=1

(18)

Here S is the total number of response surface models considered for the study. Table 5 summarises the performance comparison of the proposed approach with existing solution approaches, suggested by researchers, for varied mean multiple response optimisation cases, respectively. ANOVA analysis was used to confirm significant statistical differences in the WMD values, and multiple comparison procedure provides grouping information (at a 5% level of significance). In each case,

A Multi-objective Particle Swarm Optimisation-Based …

129

Table 5 Comparison of the proposed approach with different solution approaches considering different performance metrics Existing solution approach

Authors

Performance metrics NDF

Composite desirability function

He et al. (2012)

✓

Generalised distance function

Khuri and Conlon (1981)

✓

Goal attainment

Xu et al. (2004)

✓

Composite loss function minimisation

Ouyang et al. (2017)

✓

Multi-objective optimisation Chapman et al. (2014b) Wang et al. (2019) Dual RS optimisation

Average WMD

✓ ✓

Del Castillo et al. (1999)

‘✓’ indicates that the specific performance metric value, based on best-ranked solution(s) [VIKOR and/or TOPSIS], derived from the proposed approach is significantly better than the metrics values derived from the existing solution approach

the assumption of the equality of sample variances is checked using Levene’s test. Tukey’s multiple comparison test is used if sample variances are equal and Games Howell’s multiple comparison test is suggested if sample variances are unequal. In the Montgomery (2005) case, Tukey’s multiple comparison test was used due to equal sample variances of WMD values. In the rest of the cases, sample variances of WMD values were significantly different, and Games Howell’s multiple comparison test was used. Summary results in Table 5 clearly indicate that the proposed solution approach provides significantly superior nondominated frequency values compared to most existing approaches. However, the average WMD values were found to be significantly better only in comparison to Wang et al. (2019) approach. This suggests further investigation and scope for improvement. Specific values of performance metrics for each solution approach are provided in Tables 6, 7, 8, 9, 10, 11, and 12. Specific values of point estimate and corresponding simultaneous prediction interval of the best-ranked solutions by TOPSIS and VIKOR in each case are provided in Table 13. Figure 5 demonstrates the promising solutions and the nondominated fronts for the alternate response surface models in Case 6. The grey-coloured locations in Fig. 5 are nondominated solutions obtained by solving each response surface model. The locations marked with black-coloured ‘diamond’ symbol in the controllable variable space (Fig. 5a) correspond to the nondominated overlapping regions in the objective function space (Fig. 5b).

130

A. K. Sharma and I. Mukherjee

Table 6 Values of performance metrics for the solution by proposed approach and composite desirability function approach (He et al., 2012) in Case-1 Solution approach

Process setting

Proposed approach

Best solution ranked by TOPSIS and VIKOR

He et al. (2012)

Performance metrics NDF

Average WMD*

(-0.145, 0.953)

100

3.367a

(0.172, -0.852)

26.6

2.997b

*

Values with different alphabets (a and b) indicate a significant difference between the means, based on Tukey’s multiple comparison test

Table 7 Values of performance metrics for the solution by proposed approach and generalised distance function approach (Khuri & Conlon, 1981) in Case-2 Solution approach Proposed approach

Process setting

Performance metrics NDF

Average WMD*

Best solution ranked (0.013, 0.769) by TOPSIS

99.8

21.898a

Best solution ranked (0.268, −0.924) by VIKOR

94.8

21.26b

(−0.46, −0.138)

98.2

21.817a

Khuri and Conlon (1981)

* Values with different alphabets (a and b) indicate a significant difference between the means, based on Games Howell’s multiple comparison test

Table 8 Values of performance metrics for solution by proposed approach and goal attainment approach (Xu et al., 2004) in Case-3 Solution approach Proposed approach

Process setting

Performance metrics NDF

Average WMD*

Best solution ranked by TOPSIS

(0.868, 0.835, 0.38, 0.032, −1.064)

100

7.388a

Best solution ranked by VIKOR

(−1.142, −0.693, − 0.618, 1.507, 0.647)

75.2

6.739b

(−1.457, 0.037, 0.768, 0.624, 1.377)

26

6.944c

Xu et al. (2004) *

Values with different alphabets (a, b, and c) indicate significant difference between the means, based on Games Howell’s multiple comparison test

5 Conclusions In this paper, a synergistic response surface-based multi-objective optimisation solution approach was proposed for varied mean multiple response optimisation problems. The approach considers empirical model parameters and response uncertainties

A Multi-objective Particle Swarm Optimisation-Based …

131

Table 9 Values of performance metrics for solution by proposed approach and composite loss function optimisation approach (Ouyang et al., 2004) in Case-4 Solution approach

Process setting

Proposed approach

Performance metrics NDF

Average WMD*

Best solution ranked by TOPSIS

(−0.643, 1.22, − 0.494)

92

1.875a

Best solution ranked by VIKOR

(−0.602, 1.313, − 0.4)

49.4

1.72b

(−0.415, 1.682, − 0.473)

3.8

1.28c

Ouyang et al. (2017) *

Values with different alphabets (a, b, and c) indicate significant difference between the means, based on Games Howell’s multiple comparison test

Table 10 Values of performance metrics for the solution by proposed approach and multi-objective optimisation approach (Chapman et al., 2014b) in Case-1 Solution approach Proposed approach

Best solution ranked by TOPSIS & VIKOR

Chapman et al. (2014b)

Process setting

Performance metrics NDF

Average WMD*

(−0.145, 0.953)

100

3.367a

(−0.1, −0.8)

18.4

2.997b

* Values with different alphabets (a and b) indicate a significant difference between the means, based on Tukey’s multiple comparison test

Table 11 Values of performance metrics for solution by proposed approach and multi-objective optimisation approach (Wang et al, 2019) in Case-5 Solution approach Proposed approach

Wang et al. (2019)

Process setting Best solution ranked by TOPSIS & VIKOR

Performance metrics NDF

Average WMD*

(−1.248, −1.682, − 0.38)

87.5

3.889a

(0.967, −0.615, 1.282)

100

5.262b

* Values with different alphabets (a and b) indicate a significant difference between the means, based on Games Howell’s multiple comparison test

to iteratively determine efficient Pareto solutions. The superiority of the proposed approach is verified using varied mean multiple response optimisation cases.

132

A. K. Sharma and I. Mukherjee

Table 12 Values of performance metrics for the solution by proposed approach and dual response optimisation approach (Del Castillo et al., 1999) in Case-6 Solution approach

Process setting

Performance metrics NDF

Average WMD*

Proposed approach Best solution ranked (−0.324, 1.408, − by TOPSIS & 0.296) VIKOR

99.6

2.391a

(−0.563, 0.445, − 0.697)

100

2.112b

Del Castillo et al. (1999)

* Values with different alphabets (a and b) indicate a significant difference between the means, based on Games Howell’s multiple comparison test

Table 13 Point estimates and simultaneous prediction interval of responses for the best-ranked solutions by TOPSIS and VIKOR in each case Case

Best-ranked Process solution by setting

  ˆyµ1 ˆyµ1U, ˆyµ1L

  ˆyµ2 ˆyµ2U, ˆyµ2L

  ˆyµ3 ˆyµ3U, ˆyµ3L

  ˆyµ4 ˆyµ4U, ˆyµ4L

Case-1 TOPSIS & VIKOR

(−0.145, 78.4 (78.7, 78.1) 0.953)

64.6 (67, 62)

3187 (3348, 3026)

Case-2 TOPSIS

(0.013, 0.769)

1.06 (1.79, 0.32)

0.62 (0.69, 0.54)

1.67 (1.85, 1.48)

0.54 (0.68, 0.39)

VIKOR

(0.268, −0.924)

1.68 (2.42, 0.99)

0.61 (0.68, 0.53)

1.7 (1.8, 1.51)

0.48 (0.63, 0.32)

Case-3 TOPSIS

(0.868, 0.835, 0.38, 0.032, − 1.064)

923 (1201, 645)

23.1 (30, 16.2)

47.3 (54.7, 39.9)

91 (101, 82)

(−1.142, 1189 (1514, 864) −0.693, −0.618, 1.507, 0.647)

17.1 (25.1, 9.1)

77 (86.6, 69.2)

85 (96, 74)

Case-4 TOPSIS

(−0.643, 90 (97, 83) 1.22, − 0.494)

56.4 (58.5, 54.2)

VIKOR

(−0.602, 91 (97, 84) 1.313, − 0.4)

56.8 (58.9, 54.6)

VIKOR

Case-5 TOPSIS & VIKOR

(0.967, −0.615, 1.282)

17.13 (18.5, 15.76) 0.985 (0.994, 0.976)

Case-6 TOPSIS & VIKOR

(−0.324, 143 (158, 128) 1.408, − 0.296)

442 (496, 387)

A Multi-objective Particle Swarm Optimisation-Based …

133

Fig. 5 Nondominated solutions (a) and nondominated fronts (b) for alternate response surface models in Case-6

Essential insights from this research are, (i)

(ii)

(iii)

The proposed solution approach can provide better solutions in terms of nondominated frequency, as compared to the majority of the existing solution approaches. The solution approach can determine promising robust settings considering the simultaneous prediction intervals of the responses. Thus, there exists a lower chance probability of the actual responses falling outside the customer specifications. In other words, the number of defects or defective items is expected to be less. Superior response surface prediction models (with lower σˆ r2 [as per Eq. (5)]) can result in a lower value of the overall level of significance (α0 ). α0 is considered for calculating simultaneous prediction intervals [Eqs. (11) and (12)]. A Pareto solution (or robust process setting), derived from a lower overall α0 (or type-I error), is more likely to result in actual responses that fall within the customer specifications. The reverse will be valid for higher σˆ r2 valued response surface models. In the Montgomery (2005) case, the lowest possible α0 to derive efficient solutions was recorded as 0.85. This may be attributed to the high MSE values for the ‘viscosity’ (y2 ) response, which has tighter specification limits (62 ≤ y2 ≤ 68).

134

A. K. Sharma and I. Mukherjee

(iv)

With superior response surface prediction models, the expected range of the model parameter coefficients can be lower [as given in Eq. (4)]. Thus, fewer alternate response surface models can be used to estimate the impact of the response surface model parameter uncertainties. With inferior fitted response surface models (with higher σˆ r2 ), the intervals of the model parameters are farther spread out, and the likelihood of the predicted prediction interval limits exceeding close tolerance response specifications can increases. In most of the selected cases, the earlier reported solutions have lower nondominated frequency than the MCDM-based best-ranked solutions derived from the proposed approach. Chapman et al. (2014b) also identify promising solutions based on the nondominated frequency metrics. However, they ignore the response predictive uncertainties and explore a limited promising region in the search space.

(v)

Future research may be directed to process sensitivity and study the influence of noises on the quality of solutions.

Appendix 1 Summary of relevant multiple response optimisation research in different solution categories (SCOO and MOO). A summary of different multiple response optimisation solution categories (SCOO and MOO), proposed by researchers, and lacunas is provided in Table 14.

Appendix 2: Pseudo-code of MOPSO See Fig. 6.

Solution category (SCOO/MOO)

SCOO

SCOO

SCOO

Authors (year)

Khuri and Conlon (1981)

Del Castillo et al. (1999)

Xu et al. (2004)

Goal attainment

Dual RS optimisation

Generalised distance minimisation

Solution approach

None

None

None

Type(s) of uncertainty considered for the study

Table 14 A summary of multiple response optimisation research in different solution categories

NA

NA

NA

Predicted response interval estimations

(continued)

The proposed approach requires response functions to be continuous and twice differentiable. The approach minimises the deviation of the predicted response vector from the target response vector. In some cases, this may result in misleading decisions, as two or more solutions with different response values can have equal deviations from the target vector

In their dual RS optimisation approach, only one of the responses is optimised, and the other response is treated as constrained. This may result in misleading optimal results

This approach requires all the response models to be fitted with a common design matrix. This may lead to the overfitting of some response variables

Lacuna of research

A Multi-objective Particle Swarm Optimisation-Based … 135

Solution category (SCOO/MOO)

SCOO

MOO

Authors (year)

He et al. (2012)

Chapman et al. (2014b)

Table 14 (continued)

MOO using grid point sampling

Composite desirability function

Solution approach

Model parameter and response uncertainties

Response uncertainties

Type(s) of uncertainty considered for the study

Individual PI

Simultaneous CI

Predicted response interval estimations

(continued)

The solutions are derived from grid points identified in the search space, rather than a guided search. The approach seems computationally expensive and challenging to implement for higher-dimensional problems. Besides, the response predictive uncertainties are not integrated with the overall MOO formulation. Thus, all Pareto solutions may not be feasible for implementation owing to the response uncertainties

Simultaneous confidence intervals are constructed by controlling the family error rate. However, as per researchers, the use of confidence interval seems inadequate (Meeker et al., 2017; Montgomery & Peck, 1992)

Lacuna of research

136 A. K. Sharma and I. Mukherjee

Solution category (SCOO/MOO)

SCOO

SCOO

SCOO

Authors (year)

Ouyang et al. (2015)

Ouyang et al. (2016)

Ouyang et al. (2017)

Table 14 (continued)

Composite loss function

Composite loss function

Composite loss function

Solution approach

Model parameter uncertainties

Model parameter uncertainties

Response uncertainties

Type(s) of uncertainty considered for the study

NA

NA

Individual PI

Predicted response interval estimations

(continued)

Uncertainties in model parameters are represented by the estimation of the covariance matrix of model parameters. Simultaneous PIs are not used to determine the optimal solutions

The suggested approach requires all models to be fitted with the same degree of the polynomial. This may result in the overfitting of the RS models

In this work, PI is used to estimate the impact of response uncertainties. Use of the individual PI of responses seems inadequate, as the responses are correlated (Montgomery & Peck, 1992)

Lacuna of research

A Multi-objective Particle Swarm Optimisation-Based … 137

Solution category (SCOO/MOO)

SCOO

MOO

Authors (year)

Costa and Lourenço (2017)

Lu et al. (2017)

Table 14 (continued)

MOO using grid point sampling

Optimisation of global criterion

Solution approach

Model parameter and response uncertainties

Response uncertainties

Type(s) of uncertainty considered for the study

Individual PI

Individual PI

Predicted response interval estimations

(continued)

Response uncertainties are not considered to identify Pareto solutions. The impact of uncertainties on the responses is only considered for the evaluation of implementable solutions

Pareto solutions are generated based on point estimates of responses. Response uncertainties are only considered to examine the maximum variability associated with the point estimates after deriving the final solution. Thus, all Pareto solutions may not be feasible for implementation owing to the response uncertainties

Lacuna of research

138 A. K. Sharma and I. Mukherjee

MOO

MOO

Sharma and Mukherjee (2020)

Wang et al. (2019)

NA Not applicable

Solution category (SCOO/MOO)

Authors (year)

Table 14 (continued)

MOO using NSGA-II

MOO using NSGA-II

Solution approach

Model parameter uncertainties

Response uncertainties

Type(s) of uncertainty considered for the study

NA

Simultaneous PI

Predicted response interval estimations

The optimal process settings are derived based on fitted RS models. However, the approach does not confirm that the best solutions fall on the optimal or near-optimal nondominated fronts of alternate RS models

The work does not consider model parameter uncertainties to derive implementable solutions. Also, the correlation information among the responses is not used to derive the Pareto solutions

Lacuna of research

A Multi-objective Particle Swarm Optimisation-Based … 139

140

Fig. 6 Pseudo-code of MOPSO

A. K. Sharma and I. Mukherjee

A Multi-objective Particle Swarm Optimisation-Based …

141

References Bera, S., & Mukherjee, I. (2018a). A solution framework for response surface-based multiple quality characteristics optimization. International Journal of Reliability, Qualitive and Safety Engineering, 25(05), 1850025-1–1850025-27 Bera, S., & Mukherjee, I. (2018b). Advances in solution methods for optimisation of multiple quality characteristics in manufacturing processes. International Journal of Productve and Qualitive Management, 24(4), 475–494. Chapman, J. L., Lu, L., & Anderson-Cook, C. M. (2014a). Process optimization for multiple responses utilizing the Pareto front approach. Quality Engineering, 26(3), 253–268. Chapman, J. L., Lu, L., & Anderson-Cook, C. M. (2014b). Incorporating response variability and estimation uncertainty into Pareto front optimization. Computers & Industrial Engineering, 76, 253–267. Coello, C. A. C., Lamont, G. B., & Van Veldhuizen, D. A. (2007). Evolutionary algorithms for solving multi-objective problems, 2nd edn. Springer Science & Business Media Coello, C. A. C., Pulido, G. T., & Lechuga, M. S. (2004). Handling multiple objectives with particle swarm optimization. IEEE Transactions on Evolutionary Computation, 8(3), 256–279. Costa, N. R., Lourenço, J., & Pereira, Z. L. (2012). Multiresponse optimization and pareto frontiers. Quality and Reliability Engineering International, 28(7), 701–712. Costa, N., & Lourenço, J. (2017). Reproducibility of nondominated solutions. Chemometrics and Intelligent Laboratory Systems, 168, 1–9. Del Castillo, E., Fan, S. K., & Semple, J. (1999). Optimization of dual response systems: A comprehensive procedure for degenerate and nondegenerate problems. European Journal of Operational Research, 112(1), 174–186. Derringer, G., & Suich, R. (1980). Simultaneous optimization of several response variables. Journal of Quality Technology, 12(4), 214–219. He, Z., Zhu, P. F., & Park, S. H. (2012). A robust desirability function method for multi-response surface optimization considering model uncertainty. European Journal of Operational Research, 221(1), 241–247. Ishibuchi, H., Akedo, N., Ohyanagi, H., & Nojima, Y. (2011). Behavior of EMO algorithms on many-objective optimization problems with correlated objectives. In: 2011 IEEE Congress of Evolutionary Computation. IEEE, pp 1465–1472 Khuri, A. I., & Conlon, M. (1981). Simultaneous optimization of multiple responses represented by polynomial regression functions. Technometrics, 23(4), 363–375. Kim, K. J., & Lin, D. K. J. (2000). Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions. Journal of Royal Statistical Society: Series C-Applied Statistics, 49(3), 311–325. Lu, L., Chapman, J. L., & Anderson-Cook, C. M. (2017). Multiple response optimization for higher dimensions in factors and responses. Quality and Reliability Engineering International, 33(4), 727–744. Meeker, W. Q., Hahn, G. J., & Escobar, L. A. (2017). Statistical intervals : A guide for practitioners and researchers. Wiley. Miller, R. G. J. (1981). Simultaneous statistical inference (2nd ed.). Springer. Montgomery, D. C. (2005). Design and analysis of experiments. Wiley. Montgomery, D. C., & Peck, E. A. (1992). Introduction to linear regression analysis. Wiley. Mukherjee, I., & Ray, P. K. (2006). A review of optimization techniques in metal cutting processes. Computers & Industrial Engineering, 50(1–2), 15–34. Ouyang, L., Ma, Y., & Byun, J. H. (2015). An integrative loss function approach to multi-response optimization. Quality and Reliability Engineering International, 31(2), 193–204. Ouyang, L., Ma, Y., Byun, J. H., et al. (2016). A prediction region-based approach to model uncertainty for multi-response optimization. Quality and Reliability Engineering International, 32(3), 783–794.

142

A. K. Sharma and I. Mukherjee

Ouyang, L., Ma, Y., Wang, J., & Tu, Y. (2017). A new loss function for multi-response optimization with model parameter uncertainty and implementation errors. European Journal of Operational Research, 258(2), 552–563. Park, K. S., & Kim, K. J. (2005). Optimizing multi-response surface problems: How to use multiobjective optimization techniques. IIE Transactions, 37(6), 523–532. Sharma, A. K., & Mukherjee, I. (2020). Enhanced multi-objective solution approach for multiple quality characteristics optimisation problems considering predictive uncertainties. International Journal of Quality & Reliability Management, 37(1), 112–144. Tzeng, G. H., & Huang, J. J. (2011). Multi-attribute Decision Making. CRC Press. Villa, C., Lozinguez, E., & Raphaël, L. (2013). Multi-objective optimization under uncertain objectives: Application to engineering design problem. International Conference on Evolutionary Multi-Criterion Optimization (pp. 796–810). Springer. Vining, G. G. (1998). A compromise approach to multi-response optimization. Journal of Quality Technology, 30(4), 309–313. Wang, J., Ma, Y., Tsung, F., et al. (2019). Economic parameter design for ultra-fast laser microdrilling process. International Journal of Production Research, 57(20), 6292–6314. Xu, K., Lin, D. K. J., Tang, L. C., & Xie, M. (2004). Multiresponse systems optimization using a goal attainment approach. IIE Transactions, 36(5), 433–445.

A Mathematical Model for Scheduling and Rostering of Staff with Real-Life Considerations: The Case of Indian Call Centres Sweety Hansuwa and Chandrasekharan Rajendran

1 Introduction Call centre services have been growing since the 1970s in the world. The increased trend of call centre services in India began in 1998 after the liberalization of the Indian telecom sector in 1994. In the last two decades, the Indian industry has been growing significantly and providing services for many organizations across the world. India’s call centre market size is anticipated to reach USD 54.42 billion by 2025, according to Adroit Market Research (2018). The growth in market size is driven by the major factors that include cost-effective employees, technology advancement and skilled professionals. The call centre’s service market is a labour-intensive operation. Call centres hire employees (both full-time and part-time) to satisfy the varying demands in various service categories. Staffing accounts for a major share of the total call centre operational costs (Cordone et al., 2011; Taskiran & Zhang, 2017). Hence, staffing scheduling is a crucial issue for the call centre workforce. In India, the demographics of the country favour call centre services being outsourced by other countries, mainly the USA and the UK. However, this service demands the flexibility of working hours at the workplace and the necessity of availability of staff. The staff scheduling in India’s call centre is a challenging issue, given the mode and flexibility of Indian staff scheduling and rostering operations. A staff scheduling problem is typically solved in multiple steps or phases, e.g. (1) the number of staff or workforce (of varying skills) required in each shift by taking into account the number and type of calls (over the planning horizon) that require the service of staff with varying skill levels; (2) scheduling the shift, i.e. determining the start time of the shift (mostly fixed shifts are considered in the literature), followed by the duration of work to every worker; (3) assigning the break to every staff or S. Hansuwa · C. Rajendran (B) Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_7

143

144

S. Hansuwa and C. Rajendran

the employee during the respective shift (mostly accounting for the meal break); (4) assigning days off to the staff by taking into account their preferences or as per the nature of employment, i.e. full-time or part-time; and (5) assigning the tasks or calls to the staff of matching skill (i.e. sometimes referred to as rostering that involves the construction of line of work during the respective shift period). The challenge in the call centre’s staff scheduling problem is associated with scheduling and rostering with respect to multi-skilled employees in the workforce (also see Taskiran & Zhang, 2017). Our paper addresses the staff scheduling problem in call centres. We consider the start of a shift to be flexible in terms of starting at the beginning of any hour (prevalent in India) instead of the fixed-shift daily schedule (mostly assumed so in the literature), coupled with the flexible break assignment during a shift, and all these pose challenges for the optimization of staff schedules at call centres. In addition, an employee is required to work for forty hours per week, and hence she/he works for eight hours per day (excluding the break for one hour) for five days over the sevenday period (or a week). A full-time employee gets two consecutive days off, while a part-type employee works for no more than five days over the seven-day period. Our paper work aims to provide a detailed schedule (including day off scheduling, flexible break assignment and overtime assignment) for every employee. This can be helpful for a manager for scheduling and rostering of staff based on the demand for call services. The contributions of this paper are as follows. Many mathematical models address the problem of shift scheduling, task or call assignment and staff assignment at the macrolevel, i.e. they do not consider the specific employee and hence derive her/ his specific schedule of work every day and the associated break assignment, and also the matching of her/ his skill set with the calls that can be attended to. Many solution approaches involve the solution to the problem of shift scheduling and rostering in a sequential manner, with the output of a stage of the procedure becoming the input to the next stage of the procedure to arrive at the final schedules and rosters. Such approaches are not globally optimal. Moreover, such approaches are not capable of incorporating the flexible shift schedule, i.e. the start of a shift to be flexible in terms of starting at the beginning of any hour of the day instead of the fixed-shift daily schedule that allows for mostly three fixed shifts in a day. It also means that in case of flexible shift schedules, we need to allow the shift to begin in the evening and get carried over to the early morning of the following day. In addition, we need to ensure that the start of the next day’s work can commence with a gap of at least twenty four hours with respect to the start of the earlier day’s shift. As for the flexible break assignment, an employee is allowed for a meal break either after working for four hours or five hours, and this break lasts for one hour in most call centres. An employee works for eight hours excluding this break period of one hour, and she/ he can be assigned with one hour of overtime. As for days off, a full-time employee is given two days off over the seven-day period. The off-days are considered to be consecutive. However, part-time employees work for no more than five days over the seven-day period and take days of break that need not be consecutive. Such considerations necessitate the development of an integrated model of flexible shift scheduling and rostering with respect to every employee (and employees have different skill sets),

A Mathematical Model for Scheduling and Rostering of Staff …

145

and our model is possibly the first of its kind to address this complex problem in a single model and in an integrated manner, and that too, with respect to flexible shift scheduling and rostering of every full-time employee and every part-type employee. The paper is organized as follows. Section 2 presents a brief literature review (related to personnel scheduling in several systems, and also specifically to call centres), and Sect. 3 presents the problem description. Section 4 presents the MILP model with the description of parameters, variables, constraints and the objective function. Section 5 discusses the computational experiments, and Sect. 6 presents the conclusion and the direction for future research.

2 A Brief Literature Review Ernst et al. (2004) reviewed the literature on personnel (staff) scheduling or rostering problems (in the context of transportation systems such as railways and airlines, healthcare systems, emergency services, civil services such as postal and university staffing services, financial services, hospitality and tourism services, retail business and manufacturing systems). They treated personnel scheduling and rostering as synonymous (sometimes they are not treated the same by other researchers) and classified rostering problems into sub-problems or sub-modules in terms of temporal staff requirements, work schedules, break assignment and days off schedules. Typically, the rostering process begins with the determination of requirements of staff and concluding with the work to be performed, by each employee or an individual over the rostering (planning) horizon. It is possible that these sub-modules may be combined into one single procedure. This procedure involves the construction of a line of work, called work schedules (also called roster lines) over the planning horizon for every employee. This calls for task assignment and staff assignment. It is common to decompose the rostering problem into separable modules or sub-modules depending upon the business scenario and computational restrictions. The call centre is one of such application areas. Typically, mathematical programming models, and constraint programming and heuristic approaches are used (also see Aggarwal, 1982; Tien & Kamiyama, 1982). Bhulai, Koole, and Pot (2008) proposed a two-step approach in which the first step determined the staffing of each agent type for each period, and the second step computed the schedule by solving an integer programming model. Pot et al. (2008) discussed the algorithm for staffing in a multi-skill call centre to meet the service-level constraints. Avramidis et al. (2010) proposed simulation-based algorithms for solving the agent scheduling problem in a multi-skill call centre with the constraints on the expected service level. Robbins and Harrison (2010) discussed the problem for call centre staffing which considered the uncertainty in call volumes with the consideration of service-level constraints. They formulated a mixed-integer stochastic programming formulation that combined the server sizing and staff scheduling. Cordone et al. (2011) discussed the multi-skill call centre staff scheduling and staff contracting as two sub-problems. The contracts depended on the

146

S. Hansuwa and C. Rajendran

assignment of work shifts with days off and lunch break periods. They formulated the integer linear programming models and mathematical-programming-based heuristic algorithms. Örmeci, Salman, and Yücel (2014) studied multi-skill rostering problems with agent pick-and-drop services, and proposed mixed-integer programming by considering a multi-objective function that aimed to minimize the overtime cost, transportation costs, and unsatisfied demand penalty. Taskiran and Zhang (2017) studied the cross-training staff-scheduling problem and solved the problem using integer programming. They used an objective function to minimize the labour cost of employees and penalty costs for uncovered demand. Mattia et al. (2017) studied the shift-scheduling problem in a multi-shift, flexible call centre. They proposed a two-stage robust integer program formulation in the first stage, a daily shift schedule generated for a week and then the personnel reallocation according to the actual period-by-period need. Türker and Demiriz (2018) studied shift scheduling and rostering problems for inbound call centres where overlapping shift systems are used. An integer programming the model that determined which shifts would be opened and how many employees would be assigned to these shifts was proposed for the shift-scheduling problem. For the rostering problem, integer programming and constraint programming models were developed to determine assignments of employees to all shifts, weekly days off and meal and relief break times of the employees. Ta et al. (2021) proposed simulation-based decomposition approaches for a staffing problem for multi-skill call centres under arrival uncertainty. They formulated a two-stage stochastic programming model, with chance constraints at the second stage. They proposed decomposition techniques using sample average approximation and discussed the numerical illustrations. In comparison with existing approaches, unique characteristics of our work are that our model captures daily flexible shift scheduling of employees on an hourly basis (that can be granulated for any time window), flexible scheduling of shifts with days off in a week for full-time and part-time staff, and the flexible break assignment within a given break-time window.

3 Problem Description This section discusses the staff scheduling and rostering problem. We first discuss the problem statement, and thereafter its four modules.

3.1 Problem Statement Our study focuses on the staff scheduling and rostering problem to find the optimal allocation of employees so as to assign calls to multi-skilled employees on the given granulated time period, for the given employees’ shift assignment on a daily basis associated with the break assignment and days off in a week, and with the consideration of a call centre that operates twenty four hours in a day. We can configure our

A Mathematical Model for Scheduling and Rostering of Staff …

147

model to take into account the appropriate time granularity in a day, for example, comprising of 48 half-an-hour time periods in a day or 96 quarter-hour time periods in a day. The problem is sub-divided into four modules: (a) shift scheduling, (b) call assignment (also called rostering with the associated construction of line of work), (c) break assignment and (d) days off scheduling associated with full-time employees and part-time employees.

3.2 Shift Scheduling A shift defines a specific period during which an employee is available to handle calls. It is described by its start time, its end time (after 9 h including the break time) and the possible overtime of one hour to handle any demand variation. The shift generally starts at the beginning of an hour (even though our model is flexible to handle the commencement of a shift from, say, 8:30 h or 8:15 h). In addition, our model allows for the shift to start within a given day and to end on the following day without any interruption other than the allowed break (i.e. spill-over of a shift to the following day). It means that an employee can start to work at the beginning of hour 18 and the last working hour can be at the beginning of hour 27 (i.e. hour 3 on the following day), with one hour of break after working for 4 h or 5 h (flexible break time), and the last hour of work on the following day corresponding to the overtime work, starting from hour 27. Moreover, given that an employee has started her/ his shift on a given day, the employee can start the shift on the following day with a gap of at least 24 h between two consecutive shifts’ start times (i.e. in this case, any time after the beginning of hour 18 on the next day). It is possible that we have operational constraints with respect to working hours of full-time and part-time employees; for example, the total number of hours that all full-time employees work per week or per day is more than that of all part-time employees (e.g. the total number of hours worked by part-time employees is not more than 50% of the total number of hours spent by full-time employees), and similarly, the total number of hours spent by the part-time employees on overtime is not more than 10% of that of the full-time employees.

3.3 Task Assignment A call can be attended by an employee if she/he has the skill that is required to handle that call. The reason is that different employees (including full-time and part-type employees) have different skill sets, and hence, all cannot handle or address all types of calls. The problem of rostering therefore arises. For example, let us assume that full-time employee i has the capability to attend to calls of types 1, 2 and 3 with the respective expected call-service durations (in minutes) by this full-time employee being 5, 6 and 7, respectively. Suppose that the number of calls between 8:00 h and

148

S. Hansuwa and C. Rajendran

9:00 h corresponds to 3 calls of type 1, 3 calls of type 2 and 1 call of type 3. Fulltime employee i can handle all these calls during this one-hour period. However, part-time employee i has the capability or proficiency to attend calls of types 1 and 2 only, with expected call-service duration (in minutes) 10 and 12, respectively (possibly different from the expected call-service durations with respect to a full-time employee). Therefore, part-time employee i cannot attend to 3 calls of types 1 and 3 calls of type 2 in this one-hour period.

3.4 Break Assignment Employees normally take a break for one hour during their shift spanning over nine hours, and spanning over ten hours if they work overtime for one hour. This break can be flexible in the sense that an employee can take the break after working for four hours or five hours. In our model, this stipulation is made flexible in that we can schedule the commencement of the break after four hours of work or five hours of work for every employee so as to adjust to the demand of calls.

3.5 Days Off Scheduling A full-time employee is given two days off over the seven-day period. The offdays are considered to be consecutive. However, part-time employees work for no more than 5 days over the seven-day period and take days of break that need not be consecutive. For example, a full-time employee i can have possible shift calls or roll calls over a seven-day period such as (1111100) or (1111001) or (1110011) with one indicating a working day and zero indicating a day off. A part-time employee j can have possible shift calls such as (1010111) or (1001011) or (1110011).

4 Model Formulation We now describe the mathematical model in detail.

4.1 Sets N1 N2 N i

number of full-time employees number of part-time employees total number of employees (inclusive of full-time and part-time) index for an employee

A Mathematical Model for Scheduling and Rostering of Staff …

J j T

t β

R r

Ci

Oj M

149

the maximum number of levels of skills required by the staff to attend to calls index for levels of skill the set of time periods in the planning horizon (equal to number of days × 24 × β + 9 × β, or number of days × 24 × β, given that the maximum shift duration is 10 h, including the break time and the possible overtime of one hour) a time period (with the granularity corresponding to 60 min or 30 min or 15 min, with the corresponding β being 1 or 2 or 4, and the multiples of such a time period lead to one hour), with t = 1, 2, …, T number of periods in an hour for different granularity with respect to one hour (e.g. β = 1 when we refer to 60 min as one time period, β = 2 when we refer to 30 min as one time period, and β = 4 when we refer to 15 min as one time period) number of shift calls or roll calls in the planning horizon index for the shift call or roll call for a given employee i, with the start of the corresponding shift assumed to be at the beginning of an hour in the interval [1 + (r − 1) × 24, 24 × r] (note that shift call or roll call r corresponds to the commencement of the shift by considering the 24-h period), thereby allowing for the possible spill-over of this shift call to be the following day the set of skills possessed (i.e. types of calls that can be attended) by employee i / ∗ e.g.: C i ∈ {1, 2, 3} means that employee i can attend to calls of types 1, 2 and 3 ∗ / the set of employees who can attend to calls of type j a big number.

4.2 Parameters τ j ,i N j ,t

the expected service duration (in minutes) of call j when attended by employee i number of calls requiring the service by skill level j of an employee during time period t.

4.3 Decision Variables Binary Variables: x i,r ,t i,r

is equal to 1 if employee i is assigned for work at the beginning of time period t in roll call (i.e. shift call) r; 0 otherwise is equal to 1 if employee i works in roll call r (note that this roll call r can start at the beginning of any hour in day r only, and not in any other day); 0 otherwise

150 

 i,r

bi,r ,t zi,r ,t 

z i,r ,t

S. Hansuwa and C. Rajendran

equals 1 indicating that the status of roll call of employee i is the same in roll call r (given by i,r ) and also in the previous roll call (given by i,r −1 ), and 0 if the status of employee i is the same in roll call r (given by i,r ) and also in the next roll call (given by i,r +1 ); this binary variable ensures the constraint on (consecutive) days off during the seven-day period is equal to 1 if employee i is assigned for a break at the beginning of time period t in roll call r; 0 otherwise is equal to 1 if employee i is assigned for work commencing at the beginning of time period t, in roll call r; 0 otherwise is equal to 1 if employee i is assigned for a break commencing at the beginning of time period t in roll call r; 0 otherwise.

Integer Variables: OT i,r S i,r  S i,r F i,r nj ,i,t

overtime work of employee i in roll call r the shift start time for employee i in roll call r the break start time for employee i in roll call r the shift end time for employee i in roll call r number of calls of type j handled by employee i during time period t.

4.4 The Proposed MILP Model Objective function Z maximizes the total number of calls attended: Maximize Z =

T J  

n j,i,t

(1)

j=1 i∈O j t=1

subject to the following set of constraints. Constraint (2) ensures that each full-time employee works for five shift calls over a period of seven consecutive days: r +6 

i,r  = 5, r = 1, . . . , (R − 6), i = 1, . . . , N1

(2)

r  =r

Constraint (3) stipulates that each part-time employee works for at most five shift calls over a period of seven consecutive days: r +6 

i,r  ≤ 5 r = 1, . . . , (R − 6), i = N1 + 1, . . . , N

(3)

r  =r

Constraints (4)–(7) ensure that the two days off correspond to two consecutive days over seven consecutive days for full-time employees. For example, i,r values for seven roll calls can have possible values such as (0011111), (1001111) and

A Mathematical Model for Scheduling and Rostering of Staff …

151

(1110011), but cannot take values as (0101111), (1011011) and (1111010). This  is accomplished by using  i,r that tracks the status of the given i,r and the status of the preceding or succeeding i,r . Consider a sample of three sets of feasible roster calls for full-time employees: (0011111), (1001111) and (1100111) (i.e. giving two consecutive rosters free). In all these sets, Δi,r (with r = 2, 3, …, 6) is definitely either equal to i,r −1 or equal to Δi,r+1 , and hence four constraints pertaining to Δ’ i,r ,  Δi,r , Δi,r −1 , and Δi,r+1 will hold (including redundancy). It means that Δ i,r can take either take 0 or 1. It is to be noted that one of the two binary values is feasible. It is also to be noted that the sum of Δi,r over 7 given consecutive rosters equals 5, thereby (0111110) is also considered feasible because in a rolling window of 7 rosters, two consecutive rosters will be off for a full-time employee. We now consider a sample set of infeasible roster calls for full-time employees: (0101111), (1011011) and (1111010). By invoking the above arguments, it can be seen that these sets of roster calls are indeed infeasible indicated by the non-satisfaction of four constraints  involving Δ i,r , Δi,r , Δi,r −1 , and Δi,r+1 . In other words, the set of constraints forces the model yield a feasible set of rosters, wherein two consecutive rosters are necessarily free for a full-time employee out of 7 consecutive rosters.  r = 2, . . . , (R − 1), i = 1, . . . , N1 i,r ≤ i,r +1 + 2 × i,r

(4)

 i,r ≥ i,r +1 − 2 × i,r r = 2, . . . , (R − 1), i = 1, . . . , N1

(5)

 i,r ≤ i,r −1 + 2 × (1 − i,r ) r = 2, . . . , (R − 1), i = 1, . . . , N1

(6)

 i,r ≥ i,r −1 − 2 × (1 − i,r ) r = 2, . . . , (R − 1), i = 1, . . . , N1

(7)

Constraint (8) indicates that the total hours of work include the minimum number of hours (i.e. 9 h) and the possible overtime of one hour. Further explanation is the following: we include 9β in the expression for T because there is a possibility that an employee starts to work at 23:00 h on the last roster call (i.e. at the beginning of 23:00 h on the last roster call) and continues to work for a maximum 9 h (including the possible overtime) on the following day (even though the full-time employee’s last roster R has commenced on the day previous to the current day of check-out time). It means a roster can go beyond a given number of days to ensure that an employee who has checked into the last roster call works for the period of maximum of 10 h (including the possible overtime). Alternatively, the problem statement may stipulate that no employee will work beyond R days, i.e. with no possibility of the last roster call’s work extending (beyond R days) to the following day. Accordingly, this leads to two possibilities of giving input with respect to defining the calls and the planning horizon corresponding to the two cases discussed—it means that either the calls are given as input over every given period of time (defined by β) and over the planning horizon defined by the number of days R (i.e. T equals R × β) or over the planning horizon defined by the number of days R plus 9β periods (T equals R × β

152

S. Hansuwa and C. Rajendran

+ 9β), thereby the maximum number of days being R plus 1 (effectively), whereas the maximum number of rosters is defined by R. In one case, due to the feasibility condition that an employee has to work for at least 9 h (and with the possibility of one hour of overtime), an employee can start the work on the last day (i.e. on the last shift call) such that she/ he will not overshoot the work beyond R × β periods. In the other case, an employee can start to work at 23:00 h on the last roster call (i.e. at the beginning of 23:00 h on the last roster call) and continue to work for a maximum 9 h (including the possible overtime) on the following day. We define our model in a generic manner so as to address these two cases. 24×β×r +9×β



xi,r,t = 9 × β × i,r + O Ti,r ∀r, ∀i

(8)

t=1+(r −1)×24×β

Constraints (9)–(11) refer to the overtime work of an employee in the given shift call. If the given shift call is not there, then the overtime corresponding to that shift call is zero. However, if the given shift call is there and overtime work is there, then xi,r,t beyond the ninth hour of work will exist (or vice versa). Note that this overtime can spill over to the following day of the roll call depending on when the roll call has started: 24×β×r +9×β

O Ti,r ≥



xi,r,t − 9 × β − 10 × β × (1 − i,r ) ∀r, ∀i

(9)

t=1+(r −1)×24×β 24×β×r +9×β

O Ti,r ≤



xi,r,t − 9 × β + 10 × β × (1 − i,r ) ∀r, ∀i

(10)

O Ti,r ≤ β × i,r ∀r, ∀i

(11)

t=1+(r −1)×24×β

Constraints (13)–(15) are related to the status (i.e. tracking) of the start time of a given roll call with respect to t, given by the binary variable zi,r,t . Constraint (12) ensures that the start time of an employee in a roll call is computed only when the corresponding roll call is assigned to the employee. If an employee commences her/ his work for a given roster r at the beginning of the period corresponding to the beginning of a day, then corresponding x i,r,t = 1, and hence zi,r,t = 1; 0 otherwise. In the latter case, when the roster call commences at the beginning of any other time period, the binary variable zi,r,t is used to identify the corresponding commencement time period—it is to be noted that the set of constraints (13) and (14), coupled with the summation of zi,r,t being equal to Δi,r , helps us to identify the time period of commencement of the work in a given roster. In addition, this variable zi,r,t also helps us to maintain the continuity of x i,r,t (with respect to work, if present in that roster) without interruption. We have

A Mathematical Model for Scheduling and Rostering of Staff …

153

24×β×r +9×β



z i,r,t = i,r ∀r, ∀i

(12)

t=1+(r −1)×24×β

z i,r,t + xi,r,t−1 ≥ xi,r,t t = 2 + (r − 1) × 24 × β, . . . , 24 × β × r + 9 × β ∀r, ∀i

(13)

z i,r,t ≤ xi,r,t t = 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 9 × β ∀r, ∀i z i,r,t = xi,r,t t = 1 + (r − 1) × 24 × β ∀r, ∀i

(14) (15)

Constraint (16) ensures that employee i, if assigned to shift call r, can work at the beginning of the time period t + 1 (and possibly up to ninth hour of the following day, thereby accounting for any overtime) of following day of that shift call r, only when the employee has been working during the previous time period t of the corresponding roll call r: xi,r,t ≥ xi,r,t+1 t = 23 × β × r + 1, . . . , 24 × β × r + 9 × β − 1, ∀r, ∀i

(16)

Constraint (17) ensures that at most one shift call is operational for each employee in a given time period: r +1 

xi,r  ,t ≤ 1 t = 24 × β × r + 1, . . . , 24

r  =r

× β × r + 9 × β, r = 1, . . . , (R − 1), ∀i

(17)

Constraint (18) ensures that at least one break should be allocated to an employee if she/he is allocated to the corresponding roll call: 24×β×r +5×β+1



bi,r,t = β × i,r ∀r, ∀i

(18)

t=4×β+1+(r −1)×24×β

Constraint (19) ensures that the break is assigned in the roll call if she/he is working during the corresponding roll call, but having the break in time period t during that roll call: bi,r,t ≤ xi,r,t t = 4 × β + 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 5 × β + 1, ∀r, ∀i

(19)

154

S. Hansuwa and C. Rajendran

Constraint (20) ensures that the start time of the break in a roll call for an employee is computed only when the roll call is assigned to the employee: 24×β×r +5×β+1



 z i,r,t = i,r ∀r, ∀i

(20)

t=4×β+1+(r −1)×24×β

Constraints (21)–(23) are related to the status (i.e. tracking) of the start time of  the break in a given roll call with respect to t, given by the binary variable z i,r,t . In addition, this variable ensures the continuity of the break without interruption: 

z i,r,t + bi,r,t−1 ≥ bi,r,t t = 4 × β + 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 5 × β + 1, ∀r, ∀i

(21)



z i,r,t ≤ bi,r,t t = 4 × β + 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 5 × β + 1, ∀r, ∀i bi,r,t = 0 t = 1 + (r − 1) × 24 × β, . . . , 4 × β + (r − 1) × 24 × β, ∀r, ∀i

(22)

(23)

Constraints (24)–(25) pertain to the shift start time for each employee in roll call r: Si,r ≥ 24 × β × (r − 1) − T × (1 − i,r ) r = 2, . . . , R, ∀i

(24)

Si,r ≤ (24 × β × (r − 1) + 23 × β) × i,r ∀r, ∀i

(25)

Constraint (26) ensures that the shift start time for each employee in her/his shift call should be greater than equal to the gap of 24-hour period after the start time of the previous shift call: Si,r +1 ≥ Si,r + 24 × β − T × (2 − i,r − i,r +1 ) r = 1, . . . , R − 1, ∀i

(26)

Constraint (27) computes the shift end time for each employee in her/his roll call by considering the corresponding start time and the shift duration time: 24×β×r +9×β

Fi,r = Si,r +



t=1+(r −1)×24×β

xi,r,t ∀r, ∀i

(27)

A Mathematical Model for Scheduling and Rostering of Staff …

155

Constraint (28) computes the shift end time including the number of hours of work and the possible overtime, if an employee is working in the roll call: Fi,r − Si,r = 9 × β × i,r + O Ti,r ∀r, ∀i

(28)

Constraints (29)–(30) assign the break for each shift call of an employee, and this break can take place after four hours or five hours of work: (t − 1) × bi,r,t ≥ (Si,r + 4 × β)−(T + 1) × (1 − bi,r,t ) t = 4 × β + 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 5 × β + 1, ∀r, ∀i

(29)

(t − 1) × bi,r,t ≤ (Si,r + 5 × β) + (T + 1) × (1 − bi,r,t ) t = 4 × β + 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 5 × β + 1, ∀r, ∀i

(30)

Constraint (31) pertains to the determination of the shift start time for each employee in a given shift call, based on the binary variable z i,r,t : 

24×β×r

Si,r =

(t − 1) × z i,r,t ∀r, ∀i

(31)

t=1+(r −1)×24×β

Constraints (32)–(34) pertain to the determination of the start time of break for each employee in a given shift call (after four hours of work or five hours of work),  : based on the binary variable z i,r,t  Si,r =

24×β×r +5×β+1



 ∀r, ∀i (t − 1) × z i,r,t

(32)

t=4×β+1+(r −1)×24×β  Si,r ≥ Si,r + 4 × β × i,r ∀r, ∀i

(33)

 Si,r ≤ Si,r + 5 × β × i,r ∀r, ∀i

(34)

Constraint (35) is related to the number of calls handled by an employee during the shift, and the allocation of a call is based on its type and the corresponding skill set of the employee:  j∈Ci

n j,i,t × τ j,i ≤

60 × (xi,r,t − bi,r,t ) β

t = 1 + (r − 1) × 24 × β, . . . , 24 × β × r + 9 × β, ∀r, ∀i

(35)

156

S. Hansuwa and C. Rajendran

It is to be noted that from the left-hand side of the expression in (35), it is possible thereafter to determine the utilization and the number of calls attended by employee i in every roster r. Constraint (36) indicates that the number of calls of a given call type allocated to employees cannot exceed the demand (i.e. the number of calls to be handled during a time period): 

n j,i,t ≤ N j,t ∀ j, ∀t

(36)

i∈O j

Constraint (37) imposes the business condition that the total overtime to be no more than 10% of total regular time hours: N  R 

O Ti,r ≤ 0.1 ×

i=1 r =1

N  R 

24×β×r +9×β



xi,r,t

(37)

i=1 r =1 t=1+(r −1)×24×β

Constraint (38) pertains to the business constraint (though not restrictive in our model) that the beginning of the shift time always starts from the beginning of an hour, irrespective of the granularity of time, e.g. when β = 4 corresponding to 15 minutes’ time period, the shift may start at 08:00 am (i.e. hour 8), but not at 8:15 am or 8:30 am or 8:45 am: z i,r,t = 0 if ((t mod β) + (β − 1)) mod β = 0, t = 1 + (r − 1) × 24 × β, . . . , 24 × β × r, ∀r, ∀i

(38)

4.5 Special Case 1 When the time period corresponds to one hour (i.e. interval of 60 min). Put β = 1 in Constraints (2)–(28), (31)–(37) (and remove (38)), and modify Constraints (29)–(30) as follows, in order to account for the break assignment after 4 h or 5 h of work commencement: 24×r +5+1 

(t − 1) × bi,r,t ≥ (Si,r + 4) − (T + 1) × (1 − i,r ) ∀r, ∀i

(39)

(t − 1) × bi,r,t ≤ (Si,r + 5) + (T + 1) × (1 − i,r ) ∀r, ∀i

(40)

t=4+1+(r −1)×24 24×r +5+1  t=4+1+(r −1)×24

A Mathematical Model for Scheduling and Rostering of Staff …

157

4.6 Special Case 2 When the time period β corresponds to one hour and a shift lasts for nine hours, including the break time, and the roll call (or the shift call) starts only at the beginning of either hour 1 or hour 9 or hour 17 of a given day (so that a full-time employee works for 40 h per week), and T = 24 × R—the case of shift calls having the fixed time of commencement, as against Special Case 1 having flexible shift calls: The same set of constraints holds as in Special Case 1 (of Sect. 4.5) and with the additional condition: z i,r,t = 0 if ((t mod 8) + 7) mod 8 = 0, t = 1 + (r − 1) × 24, . . . , 24 × r ∀r, ∀i (41)

5 Computational Experimentation In this section, we first present numerical examples for illustrating the proposed mathematical model and its computational evaluation thereafter. We present the numerical illustrations with specific problem instances in order to bring out the mechanism of the formulation and also to highlight the flexible shift in relation to the fixed shift.

5.1 Numerical Illustrations We present three numerical illustrations: the first one pertains to Special Case 1 (in order to simplify the Gantt chart presentation), the second one pertains to Special Case 1, but with no overtime being allowed in this numerical illustration, and the third one pertains to Special Case 2 with the fixed-shift calls, with no overtime being allowed in this numerical illustration. The objective is to show how we can exploit the overtime and flexible shift calls to maximize the number of calls attended by the employees. We have taken a specific problem instance that considers three types of calls, two full-time employees and one part-time employee. We consider the fulltime employees to handle all types of calls, and the part-time employee to handle only type-one and type-two calls. The calls’ duration for the three types (required by full-time employees) is assumed to be 2, 3 and 3, respectively, while those for types 1 and 2 for part-time employees are assumed to be 3 and 4, respectively. We assume that the demand for calls of three types is assumed to be [0, 10], [0, 5] and [0, 4], respectively. We execute the proposed model and generate the shift schedule and rostering for a week. In the first example, we execute Special Case 2 with the value of β =1 and with the consideration of flexible shift schedule, flexible break assignment, days off scheduling and possible overtime. We obtain the total number of

158

S. Hansuwa and C. Rajendran

calls attended to be 1390 calls attended by employees, and the schedule is presented in the Gantt chart in Fig. 1. In the second example, we execute again the same model with no overtime allowed for employees. The total number of calls attended is 1278, and the schedule is presented in the Gantt chart in Fig. 2. In the third example, we execute Special Case 2 with the fixed shifts in the shift calls (i.e. roll calls), and no overtime is allowed for employees. The three shifts are considered to start at the beginning of 0, 9 and 17 h in a given day, but the first (fixed) shift employee will work up to the end of 9 hours, while the second (fixed) shift employee will work up to the end of 17 h and the third (fixed) shift employee will work up to the end of 1 h on the following day. We obtain the total number of calls to be 1231, and the Gantt chart is presented in Fig. 3.

5.2 Computational Results of the Proposed MILP Model We generate random problem instances considering the number of full-time employees varying from 18 to 30, in steps of 6, and the part-type employees varying from 1 to 6. We consider three types of calls and assume that full-time employees are capable of handling all three types of calls that belong to {1,2,3}, and the part-time employee can handle calls of types 1 and 2 only. To generate the demand (in terms of the number of calls per given time period), we sample the number of calls per time period with equal probability in the interval [20,40]. For a given call, we assign its type with equal probability among {1,2,3}. The deterministic durations of calls of types 1, 2 and 3 (taken by a full-time employee) are assumed to be 6, 9 and 12 min, respectively, while the durations of calls of types 1 and 2 are assumed to be 9 and 12 min, respectively. The time period is assumed to correspond to 30 min (i.e. β = 2), and the number of days (i.e. roster calls, R) equal to 7. The MILP model has been implemented in OPL CPLEX 12.8, and the models have been executed (up to 7200 s.) using a computing processor with Intel Xeon 2.4 GHz (dual-core) and 64 GB RAM. Table 1 presents the computational details concerning problem instances with 18 full-time employees and part-time employees varying from 1 to 6. Table 2 presents the computational details concerning problem instances with 24 full-time employees and part-time employees varying from 1 to 6, and Table 3 presents the computational details concerning problem instances with 30 full-time employees and part-time employees varying from 1 to 6.

6 Summary and Scope for Future Research We have studied the call centre problem that considers shift scheduling and rostering, and with the consideration of flexible shift scheduling, flexible break assignment, days off and task allocation with the consideration of full-time and part-type employees. We proposed a mixed-integer linear programming model in this study.

A Mathematical Model for Scheduling and Rostering of Staff …

Fig. 1 Special Case 1 with the value of β = 1

159

160

S. Hansuwa and C. Rajendran

Fig. 2 Special Case 1 with the value of β = 1, but with no overtime

A Mathematical Model for Scheduling and Rostering of Staff …

Fig. 3 Special Case 2 with the fixed-shift calls

161

162

S. Hansuwa and C. Rajendran

Table 1 Computational results with the consideration of 18 full-time employees Employees

MILP model

Full-time

Part-time

Objective-function value

Upper bound

Optimal gap (%)

Run time (s)

18

1

2837

5974.25

110.58

7200

18

2

5480

6199.25

13.12

7200

18

3

3740

6424.25

71.77

7200

18

4

1692

6572.90

288.47

7200

18

5

4083

6631.42

62.42

7200

18

6

3985

6727

68.81

7200

Table 2 Computational results with the consideration of 24 full-time employees Employees

MILP model

Full-time

Part-time

Objective-function value

Upper bound

Optimal Ggap (%)

Run time (s)

24

1

4373

6712.69

53.50

7200

24

2

4196

6727

60.32

7200

24

3

3739

6727

79.91

7200

24

4

4580

6727

46.88

7200

24

5

5525

6727

21.76

7200

24

6

5960

6727

12.87

7200

Table 3 Computational results with the consideration of 30 full-time employees Employees

MILP model

Full-time

Part-time

Objective-function value

Upper bound

Optimal gap (%)

Run time (s)

30

1

5584

6727

20.47

7200

30

2

4356

6727

54.43

7200

30

3

6120

6727

9.92

7200

30

4

3586

6727

87.59

7200

30

5

6279

6727

7.13

7200

30

6

8150

8862

8.74

7200

The proposed model is tested with many problem instances. Future research on this problem can consider the extension in terms of proposing a heuristic approach to solve the problem, and with call durations being stochastic. Another extension may include additional real-life consideration of short breaks (tea breaks) and preference of employees for shift timing.

A Mathematical Model for Scheduling and Rostering of Staff …

163

Acknowledgements We thank the reviewer and the Editor for their thorough review, and we highly appreciate the comments and suggestions, which have significantly contributed to improving the quality of the chapter.

References Adroit Market Research. (2018). The transition of India call center market (BPO) to business process management. https://www.openpr.com/news/1466995/The-Transition-of-India-Call-Center-Mar ket-BPO-to-Business-ProcessManagement.html. Accessed on April 20, 2020. Aggarwal, S. C. (1982). A focussed review of scheduling in services. European Journal of Operational Research, 9(2), 114–121. Avramidis, A. N., Chan, W., Gendreau, M., L’ecuyer, P., & Pisacane, O. (2010). Optimizing daily agent scheduling in a multiskill call center. European Journal of Operational Research, 200(3), 822–832. Bhulai, S., Koole, G., & Pot, A. (2008). Simple methods for shift scheduling in multiskill call centers. Manufacturing & Service Operations Management, 10(3), 411–420. Cordone, R., Piselli, A., Ravizza, P., & Righini, G. (2011). Optimization of multi-skill call centers contracts and work-shifts. Service Science, 3(1), 67–81. Ernst, A. T., Jiang, H., Krishnamoorthy, M., & Sier, D. (2004). Staff scheduling and rostering: A review of applications, methods and models. European Journal of Operational Research, 153(1), 3–27. Mattia, S., Rossi, F., Servilio, M., & Smriglio, S. (2017). Staffing and scheduling flexible call centers by two-stage robust optimization. Omega, 72, 25–37. Örmeci, E. L., Salman, F. S., & Yücel, E. (2014). Staff rostering in call centers providing employee transportation. Omega, 43, 41–53. Pot, A., Bhulai, S., & Koole, G. (2008). A simple staffing method for multiskill call centers. Manufacturing & Service Operations Management, 10(3), 421–428. Robbins, T. R., & Harrison, T. P. (2010). A stochastic programming model for scheduling call centers with global service level agreements. European Journal of Operational Research, 207(3), 1608–1619. Ta, T. A., Chan, W., Bastin, F., & L’Ecuyer, P. (2021). A simulation-based decomposition approach for two-stage staffing optimization in call centers under arrival rate uncertainty. European Journal of Operational Research. https://doi.org/10.1016/j.ejor.2020.12.049 Taskiran, G. K., & Zhang, X. (2017). Mathematical models and solution approach for cross-training staff scheduling at call centers. Computers & Operations Research, 87, 258–269. Tien, J. M., & Kamiyama, A. (1982). On manpower scheduling algorithms. SIAM Review, 24(3), 275–287. Türker, T., & Demiriz, A. (2018). An integrated approach for shift scheduling and rostering problems with break times for inbound call centers. Mathematical Problems in Engineering. https://doi. org/10.1155/2018/7870849

Product Rate Variation (PRV) Problem and Its Variants S. Rahul and G. Srinivasan

1 Introduction Consider n unique products each with demand di . It is desirable to have a smooth flow of production where at any instance we produce proportionate amount of each product. It is assumed that the demands are integers. One way to measure smoothness or proportionate production is to minimize the actual production to the proportions, and this led to the Product Rate Variation problem (PRV problem). The PRV is given by U=

DT  n   k=1 i=1

n di DT x ik yik U

di xik − k DT

2

number of unique products. demand for product i. total n number of units for all products in a sequence or total demand; DT = i=1 di . total number of units of product i produced over stages 1 to k; k = 1…DT . 1 if job i is produced at stage k of the sequence. PRV of the sequence.

The objective is to find the sequence that minimizes PRV. Additional constraints are that we produce only one job in each stage and that the demands have to be met. These are given by S. Rahul Indian Institute of Technology Madras, Chennai 600036, India G. Srinivasan (B) Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_8

165

166

S. Rahul and G. Srinivasan n 

yik=1

∀k

i=1 DT 

yik = di ∀i

k=1

xik =

k 

yil

l=1

yik = 0, 1; xik ≥ 0 It is observed that this optimization problem has a quadratic objective function with linear constraints and binary variables and becomes a binary quadratic programming problem. The most important application stated in the literature is in just-in-time manufacturing where it was formally observed that an effective JIT system has a smooth flow that can be measured as a flow with minimum PRV. Minimizing PRV as a measure of smoothness was proposed by Miltenburg (1989) and Kubiak (1993). This problem, called the ORV (Output rate variation) problem, originated from just in time manufacturing where we minimize the variance of the consumption rate of components. This problem primarily concerned components that were required to make the products, and each product required a different number of one or more components. The problem was to minimize the deviations in the rate of procurement of the components and two different objectives—minimizing the squared difference from the ideal proportions and minimizing absolute deviation from the ideal proportions were considered. Subsequent researchers concentrated more on the PRV objective and studied it extensively. It was also shown that under some assumptions, the PRV problem can be solved as an assignment problem. Kubiak (1993) presented a survey on models related to minimizing variation production rates and also established that the PRV problem is NP-Hard. Bautista et al. (1996) studied the similarity between the PRV problem and the apportionment problem, where a given integer is divided into integral parts proportional to requirement. Balinski and Shahidi (1998) observed that while researchers are clear about minimizing some measure of variation of production quantities, they are not unanimous regarding the choice of the objective function. They also feel that PRV is a complicated way of measuring a simple term because of which the complexity of computations increases. They also propose alternative measures based on minimizing total deviation and minimizing maximum deviation. They, however, restrict themselves only to positions in the sequence of production. Boysen et al. (2009a) provide a review on sequencing mixed-model assembly lines. They describe three research problems—mixed-model sequencing, car sequencing and level scheduling. In the level scheduling problem, sequences that are in line with the JIT philosophy are determined. Here, ideal production rates are

Product Rate Variation (PRV) Problem and Its Variants

167

defined, and deviations from the ideal rates are minimized. They also observe that the PRV problem that minimizes the product rate was carved out of the original problem where parts and not products were considered. The problem also became much simpler when products were considered instead of parts. Boysen et al. (2009b) mention the importance of the mixed-model assembly line problem from a practical perspective and state that several industries face this problem. They observe that minimizing PRV has been studied in the literature as an approximate model under certain assumptions. They perform computational experiments and discuss the relevance of the assumptions in a practical setting and conclude that PRV may not be very appropriate for use in today’s real-world mixed-model assembly systems. Nazar and Madhusudanan Pillai (2015a) presented a bit-wise mutation algorithm to solve the PRV problem. Their genetic algorithm solves problems with up to 100 jobs within reasonable time. They also provide a two-objective extension to the problem considering PRV and makespan and use the algorithm to generate nondominated solutions. Nazar and Madhusudanan Pillai (2015b) extend the PRV problem by including setup times and restrictions on available times. The time taken for a sequence includes both internal and external set-up times, and therefore, some sequences become infeasible if there is a total time restriction. Their modified bit-wise mutation algorithm solves the constrained PRV problem and provides very good results. Khadka and Dhamala (2013) studied the bottleneck PRV problem with batching. They formulate separate problems for batching and sequencing and present an algorithm to solve the problems and provide relationships between the problems. Razali, Rashid, and Abdullah (2016) have addressed the mixed-model assembly line balancing problem with resource constraints. They presented a multi-objective mathematical model considering cycle times, resources and PRV as three objective functions. Corominas, Kubiak, and Pastor(2010) provided a mathematical formulation of the response time variability problem. They mention the application of the problem to manufacturing and services. The manufacturing application is very close to the PRV problem, while in service, this could be applied to waste collection, scheduling of advertisements, etc. This paper presents an efficient mathematical programming formulation for the problem that can solve up to 40 jobs optimally. Khadka and Becker (2017) provide an improved largest function value of a feasible solution for the problem when the mth power of the maximum deviation between the actual and the ideal cumulative productions has to be minimized. Zhu and Zhang (2011) develop an ant colony optimization algorithm with elitist ant for mixed-model assembly line sequencing problem. They compare the PRV value of their solutions with three other methods including a genetic algorithm and show superior results. Mohammadi and Mohammadi (2016) propose an optimization model to minimize the total deviation of actual production rates from the desired production rates. They also propose an algorithm and show that it is efficient and gives better results for large problem sizes.

168

S. Rahul and G. Srinivasan

Thapa and Silvestrov (2016) present mathematical models of the ORV problem. They review the heuristic approaches that are available to solve the problem. They also discuss a couple of the open problems regarding ORV. In the rest of the paper, we discuss few properties related to the PRV problem. We develop a branch and bound algorithm based on the properties and present computational experiences with the algorithm. We also propose a simple heuristic to solve this problem and illustrate its performance. We discuss variants of this problem from scheduling round robin tournaments and well as from scheduling interviews in a business school context.

2 Solving the PRV Problem In this section, we develop a branch and bound algorithm and a heuristic algorithm to solve the PRV problem. We discuss few properties related to the problem to develop a lower bound. Notations dik = k DdiT [dik ] dik  dik  xik ∗ xik Uik Uik∗   U ∗ = i k Uik∗

represents the ideal number of batches of product i at stage k rounded integer value of d ik lower integer value of d ik upper integer value of d ik number of batches of product i at stage k in a given sequence number of batches of product i at stage k in the optimum sequence minimum contribution of job i at stage k to PRV contribution of job i at stage k to PRV in the optimum sequence sum of the contributions Uik∗

Property 1 Given job i at stage k, the value of x ik that minimizes (xik − dik )2 is given by xik = [dik ] where [dik ] is the rounded integer value of d ik . Explanation: Since dik = k DdiT . Since we are minimizing (xik − dik )2 where x ik is an integer, the minimum value happens at the rounded value of d ik . Property 2 Uik ≤ Uik∗ Explanation: Let Uik = ([dik ] − dik )2 . Let Uik∗ be the contribution of job i at  ∗ 2 − dik will be greater than or equal to stage k to the optimum PRV U*. Uik∗ = xik ([dik ] − dik )2 . Therefore, Uik ≤ Uik∗ .   Property 3 L B1 = k L k is a lower bound to U* where L k = i ([dik ] − dik )2 2  ∗ Explanation: From Property 2, L k ≤ xik − dik at stage k and therefore is the sum of  the lower estimates of the contributions of the jobs to U*. Also L k = i Uik . Hence k L k is a lower bound to U*.

Product Rate Variation (PRV) Problem and Its Variants

169

Property 4 If at stage k.  of stage k to the lower bound to U* k, then L 1k is the contribution [d ] i ik > n m where L 1k = i=1 (dik − dik )2 + i=m+1 (dik − [dik ])2 . Let J be the set of jobs where [d ik ] ≥ dik and jobs are arranged in increasing  order k +ma. At stage k, we have i dik = k of the fractional values of d ik . Let i [dik ] =  and we should also have k xik = k. Since i [dik ] > k, all jobs where [dik ] = dik cannot have the minimum contribution to the lower bound through dik . Few jobs will have to contribute through their lower integer value. The first m jobs in the list J will now contribute through the lower integer value and the rest through the rounded value. Property 5 If at stage k,  contribution of stage k to the lower bound to U* where ] ik i [d m < k, L 1k is 2the  n L 1k = i=1 (dik − dik ) + i=m+1 (dik − [dik ])2 . Let J be the set of jobs where [dik ] ≤  dik and jobs are arranged in decreasing . Let − m. At stage k, we have order of the fractional values of d ik i [dik ] = k   i dik = k and we should also have k x ik = k. Since i [dik ] < k, all jobs where [dik ] = dik cannot have the minimum contribution to the lower bound through dik . Few jobs will have to contribute through their upper integer value. The first m jobs in the list J will now contribute through the lower integer value and the rest through the rounded value. Property 6 If at stage k  [d ] ik i  = k, L 1k is the contribution of stage k to the lower bound to U* where L 1k = in (dik − [dik ])2 . Explanation: From Property 2.   Property 7: L B2 = k L 1k is a tighter lower bound to U* than L B1 = k L k . > k and i dik < Explanation: Properties  5 and 6 show that L 1k ≥ L k when i dik  k. Property 3 shows that k L k is a lower bound to U*. Therefore, k L 1k is a tighter lower bound to U* than k L k . Property 8  Given a feasible partial sequence  DT up to stage t, lower bound to U* is given n t 2 by L B3 = i=1 − d + (x ) ik k=t+1 L 1k , where L 1k values are computed k=1 ik using the expressions in Properties 4, 5 and 6. Explanation: The objective function is the sum of the contributions at various stages. The contribution of feasible partial sequence is fixed. Property 3 ensures that the expression is a lower bound to the contribution to the rest of the sequence. Property 9 Given a feasible partial sequence up to stage t, the contribution of stage t + 1 to the lower bound is given by L 2t+1 =

m n   2   dit+1 − yit + 1  , (dit+1 − yit + 1 )2 + i=1

i=m+1

170

S. Rahul and G. Srinivasan

if

 dit+1 > t + 1 i

L 2t+1

m n    2  dit+1 − yit+1 , = (dit+1 − yit+1 )2 + i=1

 if dit+1 < t + 1.

i=m+1

i

L 2t+1 =

n   2  dit+1 − yit+1 i

 dit+1 = t + 1. if i



where yi,t+1 = maximum xit, dit+1 . Explanation: Since a feasible solution should satisfy xi,t+1 ≥ xit , from Property 1, we are minimizing (xit+1 − dit+1 )2 subject to xit+1 ≥ xit ; xit ≥ 0 and integer, the minimum value happens at maximum of xit, dit+1 . This is included in the results of Properties 3, 5 and 6. Property 10 Given partial sequence up to stage t, lower bound to U* is t na feasible DT 2 given by L B4 = i=1 − d + L + (x ) ik ik 2t+1 k=1 k=t+2 L 1k , where L 1k values are computed using the expressions in Properties 4 and 6, 5 and 6. It is easy to prove that L 2t+1 ≥ L 1t+1 and is a lower bound to the contribution of stage k. Therefore, LB4 is a lower bound to the contribution of stages t + 1 to DT to U*. Property 11 Given a feasible partial sequence up to stage k, LB4 is a tighter lower bound to U* than LB3 . Explanation: From Property 10. Using Properties 1–11 described earlier, we propose a heuristic algorithm and a branch and bound algorithm for the PRV problem. The steps are given below: Heuristic Algorithm The heuristic algorithm uses Properties 1 and 2. We add job i to the sequence at stage k if this job results in minimum increase in PRV value. Ties are broken arbitrarily. The steps are given below: 1. 2. 3. 4. 5.

Initialize X i0 = 0 for all i At stage k, X ik = X i,k −1 . 2  Find job j such that x jk − d jk is minimum. Break ties arbitrarily. X jk = X jk + 1 Repeat steps 2–4 till all stages are completed.

Branch and Bound algorithm (BB1)

Product Rate Variation (PRV) Problem and Its Variants

171

The branch and bound algorithm uses Properties 3–11. Initial lower bound is found by setting each d ik to its lower integer value and finding the contribution to PRV. It is further improved by adding a component that ensures that at stage k, the total batches produced is k. Lower bound for a partial sequence is given in Properties 8–11. The steps in the branch and bound algorithm are given below: 1. 2. 3. 4.

5. 6. 7.

Compute LB2 . Node 0 has all X ik = 0 Compute an upper bound UB using the heuristic. Choose an active node with the lowest value of LB4 . Let k jobs be fixed in this solution. Create a n new nodes where in new node j, X j,k+1 = X jk + 1 and X j,k+1 is between dk+1  and dk+1 . Compute LB4 for the newly created nodes. If LB4 > UB, fathom the node by bound. If the node has a feasible solution, fathom it by feasibility compute the PRV. Update the solution and the value of UB if the new solution has lower PRV. Repeat steps 3–6 till no node is active. The best solution saved is optimum.

3 Numerical Illustration and Computational Testing of the Proposed Algorithms Consider four jobs A, B, C and D with d i = 5, 3, 1, 1, respectively. For k = 1, we compute d A1 = 0.5, d B1 = 0.3, d C1 = 0.1, d D1 = 0.1, L 1 = 0.36. We compute L 2 = 0.24, L 3 = 0.44, L 4 = 0.36, L 5 = 1, L 6 = 0.36, L 7 = 0.44, L 8 = 0.24, L 9 = 0.36, L 10 = 0. Using Property 3, we compute LB1 = 3.8. The values of i [dik ] for stages 1–10 are 1, 2, 3, 3, 7, 7, 8, 8, 10, 10. While computing these values, we have rounded to the higher integer when the fractional value of d ik is0.5. 4 At k = 4, i=1 [di4 ] = 3. By Property 5, m = 1. Jobs C, D and B have fractional values of 0.4, 0.4 and 0.2. L 4 = 0 + 0.04 + 0.36 + 0.16 = 0.56. 6 At k = 6, i=1 [di6 ] = 7. By Property 4, m = 1. Jobs B, C and D have fractional values of 0.8, 0.6 and 0.6. L 4 = 0 + 0.04 + 0.36 + 0.16 = 0.56. We observe this at k = 5, 7 and 9, but Property 4does not increase the contribution to the lower bound. LB2 = 4.2 which is tighter than LB1 = 3.8 Consider a feasible partial sequence ABB with three jobs in the first three stages. 3 4  The fixed portion of the schedule contributes to i=1 (xik − dik )2 = 2.64. We k=1 4 calculate the contribution of stages 4–10 to the lower bound. Stage 4 has i=1 [di4 ] = 4 3. L 14 = 0.56 (Properties 5and 9). Stages 5, 6, 7 and 9 have i=1 [dik ] > k. L 15 = 1, L 16 = 0.56, L 17 = 0.59, L 19 = 0.36 (Properties 4 and 9). Stages 8 and 10 have  4 i=1 [dik ] = k. L 18 = 0.24 and L 1,10 = 0 (Properties 6 and 9). LB3 = 2.64 + 0.56 + 1 + 0.56 + 0.59 + 0.36 + 0.24 = 5.95. Using Property 10, L 24 = 0 + 0.64 + 0.16 + 0.16 = 0.96 and LB4 = 6.35.

172

S. Rahul and G. Srinivasan

Table 1 Performance of the heuristic algorithm for the PRV problem Jobs (DT )

(H – LB2 )/LB2 (Average)

(H – LB2 )/LB2 (Maximum)

50

0.220969

0.450623

75

0.239178

0.435821

100

0.223012

0.402364

200

0.209256

0.355454

300

0.226097

0.348144

400

0.208048

0.35052

500

0.205038

0.339558

1000

0.205077

0.324407

2000

0.200614

0.33224

3000

0.203826

0.336182

4000

0.205636

0.328925

5000

0.193986

0.322174

We apply the heuristic to the illustration. At stage 1, the solution is S 1 = [1 0 0 0] which represents X A1 = 1, X B1 = X C1 = X D1 = 0. We have S 2 = [1 1 0 0] where S 2 is the solution at stage 2. S 3 = [2 1 0 0]. S 4 = [2 1 1 0]. Here, the tie was arbitrarily broken, and X34 = 1. S5 = [3 1 1 0]. Tie was broken arbitrarily, and X 15 = 3. S 6 = [3 2 1 0], S 7 = [3 2 1 1], S 8 = [4 2 1 1], S 9 = [4 3 1 1], S 10 = [5 3 1 1] and PRV = 4.2 At stage 5, if the tie was broken to yield S 5 = [2 1 1 1], the rest of the solution would have been S 6 = [3 1 1 1], S 7 = [3 2 1 1], S 8 = [4 2 1 1], S 9 = [4 3 1 1], S 10 = [5 3 1 1] and PRV = 4.6 Table 1 shows the performance of the heuristic, and Table 2 shows the performance of the branch and bound algorithm. The heuristic has been tested on randomly generated problem sets with DT values ranging from 50 to 5000. It is observed that the heuristic is able to provide solutions with up to 20% deviations on an average and up to 45% maximum. This is primarily due to two reasons. When the tie breaking rule is applied in Step 3, which happens many times during the implementation, we choose the first among the tied values. This can result in inferior solutions. The other issue is when the value di is very small when compared to DT , the choice made by the heuristic does not seem to be the best. Based on the data given in Table 1, we propose that we require better heuristics, particularly tie breaking rules based on a look-ahead heuristic where the next stage could be considered when tie breaking decisions are made. Table 2 shows the performance of the branch and bound algorithm. The program was coded and run on an I7 processor 8 GB RAM machine. A time limit of 5 s and node limit of 150,000 was set for these problems. Ten instances were run for each size. We were able to get the optimum solution in several cases and were able to solve up to DT = 40.

Product Rate Variation (PRV) Problem and Its Variants

173

Table 2 Performance of the branch and bound algorithm for the PRV problem n=5

n=6

DT Average nodes

Maximum nodes

Average TLE Average CPU instances nodes time

Maximum nodes

Average TLE CPU instances time

10

0

0

< 0.01

0

0

0

< 0.01

0

12

37.1

101

< 0.01

0

167

1034

< 0.01

0

14

408.5

1497

< 0.01

0

3864.7

20,243

< 0.01

0

16

296.8

964

< 0.01

0

4122.9

24,151

< 0.01

0

18

878

3497

< 0.01

0

12,539.9

34,232

0.07

0

20

4844

18,512

< 0.01

0

2974.8

9160

< 0.01

0

22

30,579.2

98,218

0.049 0

325,916.5 1,603,894

0.371 0

24

68,349.5

351,189

0.11

0

76,722.8

> 1,500,000

0.142 1

26

53,871.7

> 1,500,000

0.073 1

31,027.8

> 1,500,000

0.062 1

28

46,062.5

42,637

0.123 0

68,253

> 1,500,000

0.149 2

30

16,142.1

> 1,500,000

0.025 1

65,103.9

> 1,500,000

0.138 3

32

76,104

> 1,500,000

0.15

150,779

> 1,500,000

0.373 1

34

223,204.4 1,059,271

0.531 0

78,949

> 1,500,000

0.253 2

36

65,523.7

> 1,500,000

0.153 2

74,079.8

> 1,500,000

0.255 3

38

38,402.4

> 1,500,000

0.138 2

68,818.9

> 1,500,000

0.309 3

40

80,700.5

> 1,500,000

0.223 1

52,738

> 1,500,000

0.261 2

1

(TLE = Time limit of 5 s CPU time exceeded)

4 Variant of the PRV Problem—Application from Round Robin Tournament Scheduling We now describe the application of the PRV concept by considering the scheduling problem in round robin tournaments. We consider the schedule of the cricket world cup 2019 as well as the schedule of IPL 2019 and show the relevance of solutions that minimize PRV. The 2019 cricket world cup was played in England and Wales and involved ten teams. Each time played against all the teams resulting in 45 round robin matches. The top four played the semifinals, and the winners played the finals. We consider only the round robin stage of the tournament involving 45 matches. The ideal situation has nine rounds; each of five matches where in every round each team plays one match with a different opponent. A feasible solution exists for the ideal situation and happens to be the minimum PRV solution. There are 45 stages, each representing a match and the ideal proportion for each team at each stage is 0.2. Since each match is played between two teams, each stage results in two teams increasing their x ik by 1. Considering nine rounds, the feasible solution for the ideal situation has the minimum PRV of 72.

174

S. Rahul and G. Srinivasan

The schedule for the 2019 cricket world cup is available in https://www.cri cbuzz.com/cricket-series/2697/icc-cricket-world-cup-2019/matches. The PRV for the given schedule is calculated to be 122. It is also observed that all the teams had played equal number of matches at the end of the eighth round (40th match). This has resulted in a high PRV for the implemented schedule. It is also observed that when one of the teams was playing its first match, the opponent was playing its third match and this resulted in an unbalanced schedule if we consider the proportions and the PRV. A slight rescheduling between matches 8–10 could bring down the PRV considerably and would have resulted in a schedule where each team had played two matches at the end of round 2. It is also to be noted that there are other considerations that can act as side constraints to the round robin scheduling problem that can increase the PRV. One of them is to ensure that there is a reasonable gap between the matches played by each team. Assuming that the 45 matches are played on 45 consecutive days, the minimum gap of 2 days is possible in the ideal minimum PRV schedule. However, if we have a schedule where Saturdays and Sundays have two matches each (in two different places), the PRV will increase if we impose a gap of 2 days between matches played by a team. The present schedule had two matches in a day on very few days of the week and the 45 matches were played over 38 consecutive days with 7 days having two matches each. These seven days included six Saturdays and a Wednesday. It is also required that matches involving the home team and matches involving traditional opponents are to be scheduled on holidays and weekends. This soft condition along with the constraint on the minimum gap as well as continuous days and multiple matches in a day prevents the optimum PRV solution from being implemented. However, scheduling these tournaments to get the minimum PRV solution with additional constraints given the number of days for the round robin stage of the tournament is an interesting optimization problem to work on. If we implement the ideal solution with minimum PRV where the teams play an equal number of matches at the end of each round, the points earned would provide an immediate comparison of the relative performance of the teams. If we implement any other solution, teams would have played different number of matches at the end of each round. Would such a scenario result in higher expectation from teams having fewer points brings a behavioural angle to the problem and can be addressed separately. We now address the problem of round robin scheduling in the Indian Premier League (IPL) cricket tournament. We use the data from the 2019 tournament (https:// www.iplt20.com/results/men) for our analysis and consider only the round robin stage involving 56 matches. Eight teams play each other twice (one home match and one away match). There were played over 44 consecutive days with 12 days hosting two matches. The schedule included seven weekends and on two Saturdays there was only 1 match, and in the remaining Saturdays and Sundays, there were two matches on each day. There were 5 weekends with two matches on Saturdays and Sundays, and all the eight teams played one match in one of the two days. The minimum gap between matches for any team was 2 days.

Product Rate Variation (PRV) Problem and Its Variants

175

The ideal situation would be to have seven rounds of four matches each where at the end of each round all the teams would have played equal number of matches. This is repeated for the reverse rounds where each team plays every opponent second time resulting in 56 matches. Assuming that DT = 56 and d i = 14 and each match increases the x ik by 1 for two teams, the minimum PRV possible for the feasible schedule for the ideal situation is 70. It is also possible to ensure a minimum gap of 2 days if the matches are played over 56 consecutive days. The present schedule involves several additional constraints as mentioned earlier and has a PRV score of 92. At the half way stage at the end of match #28, each team had not played seven matches; one team had played its 8th match. The teams had played equal number of matches at the end of the first three rounds only. It is also not possible to implement the ideal situation with minimum PRV due to the additional constraints of two matches on many weekend days and the constraint that a team does not play matches on two consecutive days. We present the constraints that are to be incorporated into the IPL scheduling formulation and provide expressions for these constraints in terms of the decision variables. The constraints are as follows. 1. 2. 3. 4. 5.

Each team plays the other once at home (since the home match for one team is an away match for the other, this ensures that 56 matches are scheduled). Teams i and j do not have home and away matches (between them) on the same day. Each day has a fixed number of matches. A team plays utmost one match on a day. A team does not play matches on consecutive days.

We define a decision variable x ijk = 1 if team i plays a home match with team j on day k. The variables X iik are excluded because no team plays against itself. Throughout the formulation, X ijk are considered where i = j. Constraints 1 to 5 are given below: D 

xi jk = 1; ∀i, j

k

xi jk + x jik ≤ 1 ∀i, j, k n  n 

xi jk = Nk ∀k = 1 to D

i=1 j=1 n  j=1 n  j=1

xi jk +

n  j=1

xi jk +

n 

x jik ≤ 1 for i, k

j=1

xi j,k+1 +

n  j=1

x jik +

n  j=1

x ji,k+1 ≤ 1∀i, k

176

S. Rahul and G. Srinivasan

Table 3 PRV of few feasible solutions to the IPL scheduling problem PRV

Maximum gap

Total trips

Given solution

92

6

84

Solution 1

76

7

85

Solution 2

108

8

87

Solution 3

124

9

91

The formulation with n = 8 teams and D = 44 days has 2464 binary variables and 2028 constraints. Constraints 3 and 5 with N k = 2 on the two days in the weekend ensure that the ideal solution with minimum PRV cannot be implemented. We have not included the following soft constraints in our formulation. 1. 2. 3.

4.

Number of consecutive home and away matches Maximum and minimum gap between matches. Minimizing the maximum gap Minimizing the total number of trips. Two consecutive away matches or one home match followed by an away match or one away match followed by a home match would involve a trip Balancing the trips of the teams (ensuring that the teams have equal number of trips).

We solved some instances optimally with suitable linear objective functions representing number of trips and gaps between matches. We also applied the proposed heuristic by modifying it to suit the constraints. Table 3 presents a summary of few feasible heuristic solutions to the IPL tournament scheduling problem, considering 2019 data. It is observed that proposed Solution 1 has lesser PRV than the present solution. It also performs well in terms of minimizing the total number of trips of all eight teams. The maximum gap between matches is 7 indicating a good spread of matches. The number of trips varies from 9 to 12 for the teams. Incorporating the PRV concept gives an alternative that is expectedly better in terms of proportion of matches played with acceptable values of other parameters such as number of trips and gap between matches.

5 Variant of the PRV Problem—Application from Scheduling Campus Interviews We now address the problem of scheduling campus interviews in the context of a business school. Here candidates (students) give their choice of domain and the companies indicate the domains that they specialize in. Here, we assume that the student gives only one choice of domain while the companies can give multiple domains. We explain this problem using a sample case with DT = 49 students; n =

Product Rate Variation (PRV) Problem and Its Variants

177

7 types of domains (called A to G) with d i = 4, 12, 16, 3, 5, 8 and 1. In this case, the interviews were scheduled over 11 days with different number of students getting their jobs each day. If we consider the each of the 49 students as a stage, the existing solution (or order of getting the jobs) is AABBBCCCCCCBBBBADCCCCCEEEEEDDCCCCFFFFFFFBCFGBBBBA with PRV = 924.12. We applied the heuristic proposed in Sect. 3. The solution obtained has PRV value of 32.4898 and is given by CBFECABCDFBCCBEFCBACFBCDEGCBFCBACFEBCCBFDCBACEFBC. This is under the assumption that the student will get his/her job irrespective of the order of interviews. The proposed solution also assumes that it is possible to schedule the interviews in the order given which may be difficult to implement. We therefore formulate the problem to include practical constraints that can affect the actual scheduling of interviews. We use the following notation: n di DT M bk x ik yik U

number of interviews demand for domain i n total number of students placed or total demand; DT = i=1 di . number of interviews number of students placed (size) in stage (interview) k total number of students in domain i placed over interviews 1 to k; k = 1…M. 1 if domain i is interviewed at stage k of the sequence. PRV of the sequence.

The objective is to find the sequence that minimizes PRV. Additional constraints are that we produce only one job in each stage and that the demands have to be met. These are given by n 

yik = 1 ∀k

i=1 M 

yik bk = di ∀i

k=1

xik =

k 

bl yil

l=1

yik = 0, 1; xik ≥ 0 The objective is to minimize PRV and is given by U=

M  n   k=1 i=1

di xik − k DT

2

178

S. Rahul and G. Srinivasan

The formulation assumes that at stage k, we place certain number of students of domain I, and each stage addresses only one domain. In our example, there were 22 interviews and the order of the interviews was A2, B1, B2, C2, C3, C1, B4, A1, D1, C5, E3, E2, D2, D2, C1, C3, F7, B1, C1, F1, G1, B1, A1 and B3. Only at one stage an interview resulted in A1, D1 which means that 1 student each in domains A and D picked up jobs. The existing solution with 22 stages and DT = 49 has a PRV of 411.6. The LB1 for PRV using the algorithm was found to be 12.78, and the improved LB2 was 36.34. We modified the proposed heuristic to suit this application, and this resulted in a solution B1, C1, C2, B4, C5, A2, E3, B1, D2, C3, B3, F7, G1, C3, A1, D1, B2, E2, C2, A1, F1 with PRV = 230.89. The proposed heuristic was modified to include A1 and D1 together since there was an instance of an interview (stage) resulting in two domains. The heuristic solution shows less PRV and more spread among the interviews in terms of meeting the demands of the domains. It is true that in this case, the l k associated with stage k would not be known apriori while d i , M and DT would be known. It is possible to use previous year data as an indication to schedule the interviews with minimum PRV. The problem can be solved by scheduling the M interviews in T days and by keeping days as stage. In this case, we have to define another parameter that defines the number of students expected to be placed on day k.

6 Conclusions In this article, we addressed the Product Rate Variation (PRV) problem and some of its variants. We proposed bounds using which we developed a branch and bound algorithm and a heuristic algorithm to solve this problem. We discussed variants of this problem from scheduling round robin tournaments as well as from scheduling interviews in a business school context. The PRV problem that originated from JIT manufacturing has applications in a variety of contexts. The problem has to be modified to include side constraints, which are difficult to solve considering the nonlinear nature of the PRV objective. Heuristics incorporating the side constraints have to be developed for each application because the side constraints are different for each application. This gives us enough scope and opportunity to study the PRV problem and its variations further considering various applications and associated side constraints.

Product Rate Variation (PRV) Problem and Its Variants

179

References Balinski, M., & Shahidi, N. (1998). A simple approach to the product rate variation problem via axiomatics. Operations Research Letters, 22, 129–135. Bautista, J., Companys, R., & Corominas, A. (1996). A note on the relation between the product rate variation (PRV) problem and the apportionment problem. Journal of the Operational Research Society, 47, 1410–1414. Boysen, N., Fliedner, M., & Scholl, A. (2009). Sequencing mixed-model assembly lines: Survey, classification and model critique. European Journal of Operational Research, 192, 349–373. Boysen, N., Fliedner, M., & Scholl, A. (2009). The product rate variation problem and its relevance in real world Mixed-model assembly lines. European Journal of Operational Research, 197, 818–824. Corominas, A., Kubiak, W., Pastor, R. (2010). Mathematical programming modeling of the response time variability problem. European Journal of Operational Research, 200(2), 347–357 De, P, Ghosh, J. B., & Wells, C. E. (1992). On the minimization of completion time variance with a Bicriteria Extension. Operations Research, 40 [online] https://doi.org/10.1287/opre.40.6.1148 https://www.cricbuzz.com/cricket-series/2697/icc-cricket-world-cup-2019/matches. https://www.iplt20.com/results/men Khadka, S. R., & Becker, T. (2017). On upper bound for the bottleneck product rate variation problem dynamics in logistics (pp. 391–399). Springer Publication Khadka, S. R., & Dhamala, T. N. (2013). On Bottleneck product rate variation problem with batching. Iranian Journal of Optimization, 5, 412–425 Kubiak, W. (1993). ‘Minimizing variation or production rates in just in-time systems: A survey’, European Journal of Operational. Research, 66, 259–271. Miltenburg, J. (1989). Level schedules for mixed-model assembly lines in just-in time production systems. Management Science, 35, 192–207. Merten, A. G., & Muller, M. E. (1972). Variance minimization in single machine sequencing problems. Management Science, 18(9), 518–528. Mohammadi, G., & Mohammadi, D. (2016). Minimizing products rates variation in just-in-time mixed-model manufacturing. American Journal of Operations Research, 6, 147–152. Nazar, A. K P., & Madhusudanan Pillai, V. (2015a). A bit-wise mutation algorithm for mixedmodel sequencing in JIT production systems. International Journal of Production Research, 53(19), 5931–5947. https://doi.org/10.1080/00207543.2015.1032438 Nazar, A. K. P., & Madhusudanan Pillai, V. (2015b). Mixed-model sequencing problem under capacity and machine idle time constraints in JIT production systems. Computers and Industrial Engineering, 118, 226–236. Razali, M M, Ab.Rashid, M F. F., & Abdullah, M. R. (2016) Mathematical Modelling of MixedModel Assembly Line Balancing Problem with Resources Constraints International Engineering Research and Innovation Symposium (IRIS) IOP Publishing IOP Conf. Series: Materials Science and Engineering 160 012002 doi:https://doi.org/10.1088/1757-899X/160/1/012002 Thapa, G. B., & Silvestrov, S. (2016). Output Rate Variation Problem: Some Heuristic Paradigms and Dynamic Programming, Engineering Mathematics II. Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, vol. 179, pp. 313–325) Zhu., Q., & Zhang, J. (2011). Ant colony optimisation with elitist ant for sequencing problem in a mixed model assembly line, International Journal of Production Research, 49, 4605–4626 (published online: 21 Sep 2010)

Optimized Multimodal Transportation for Efficient Parcel Movement in Courier Industry Mahidhar Surapaneni, Abhaya Kumar Sahoo, Sarang Jagdale, and Sonia Kushwaha

1 Introduction The problem statement is to identify the most optimal route for moving a parcel between an origin and a destination along with the mode of transport for a given turnaround time (TAT) requirement. The product types in courier industry are linked with turnaround time (TAT). Between an origin–destination pair, the fastest product might have a requirement of a next day delivery, while the slowest delivery can have a service-level/TAT requirement of 5 days (hypothetical example). The differences in the TAT requirement have different implications in the routes and the mode of transport to be selected. This paper presents an algorithm to arrive at the same. The parcel pickup–delivery system is a part of logistics courier industry. Logistics represent a collection of activities that ensures the availability of the right shipments to the right customers at or before the actual promised delivery time. These activities serve as the link between source and destination locations separated by distance and time window of delivery. This requires focus on transfer including pickup and delivery of the parcels, shipments, loads or physical good, transfer of information

M. Surapaneni (B) · A. K. Sahoo IBM India Pvt.Ltd, Varanasi, India e-mail: [email protected] A. K. Sahoo e-mail: [email protected] S. Jagdale Opex Analytics, Pune, India e-mail: [email protected] S. Kushwaha DTDC, Navi Mumbai, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_9

181

182

M. Surapaneni et al.

through information technology (IT) system and transferring goods from one business to another or from business to the consumer people. The changing requirements and environmental factors have created the need for logistics as a competitive tool to improve customer service and reduce the total cost of providing customer service. The changes also have created the need constantly to review and redesign the various logistics operations as deployed by companies. Transportation management deals with vendor selection, transportation mode, fleet size, route selection, vehicle scheduling, freight consolidation and distribution network. According to Logistics Skill Council (2017), Indian logistics industry is valued at USD $150 billion, contributing 14.4% of country’s GDP. In terms of movement of parcel, there are different modes which include air, road, rail and ship. Statista Research Department (May 2020) states that road is the dominant mode of transport which accounts for 67% of freight movement in India. Trucks are the most widely used mode of transportation in India. In our model, we have considered only road and air movement and omitted the train movement as there is dependency on the its timings and schedule creating hard constraint. The growth of logistics outsourcing in India is attributable to better transportation solutions; greater focus on core businesses; impact on cost reduction; improvements in services; development of necessary technological expertise; availability of computerized systems; and the need for more professional and better prepared logistics services. The loads are categorized as documents or non-documents. Within the nondocuments, load can be bulky or small parcels have different service levels to be achieved. Depending on these attributes, the parcel must be picked up and delivered in a different fashion. Parcel pickup–delivery system has developed an effective solution approach for the strategic network design using level of load consolidations. There are five different delivery services provided, each with a different service level defined in terms of turnaround time (TAT) taken to deliver the product. Depending on the product types, these parcels are managed in separate facilities, have different processes to be performed and require separate transportation movement. This initiative is proposed to maintain integrated operations across these products and leads to enhanced customer services aligning with market realities. This study develops a model that would identify the most optimum network structure and an efficient route planning for the unified operating model given the future demand estimate. Ben-Ayed (2013) discussed the parcel distribution network design problem for ground shipments. He considered a modified version of the hub location problem (Campbell, 1996) and suggested an optimization model based on actual economies of scale assuming a complete hub-to-hub network. The parcel pickup–delivery system runs on a hub-spoke model having facilities at the origin and destination. In this study, the smaller centers which are present near the customer are called branches and larger centers which are mostly away from the customer are called major hubs, thus creating a hub-spoke model. Also, there medium space consolidation points called smaller hubs are present in between the branches and major hubs. Each branch is connected to either major hubs or smaller hubs. At each of the facilities, there are different processes including consolidation and deconsolidation. The optimal set of routes are determined for each of this facility as per network alignments.

Optimized Multimodal Transportation for Efficient Parcel …

183

Our strategy consists of identifying the fundamental elements of a transportation system, such as existing and potential facility locations for load consolidation and deconsolidation, facility capacity planning to perform the different processes and how to design the physical network for line haul and long operations as per parcel booking point aligned to the respective consolidation–deconsolidation centers. Multiple levels of numerical experiments were conducted taking into consideration the different potential locations and their load handling capacity that operates on a hub-and-spoke network. The accuracy of the mathematical model and the behavior of the system under different load situations were examined. The study suggests that significant reductions in operational costs are expected as the result of using the proposed optimization model. The different inputs, parameters and assumption to capture the realities used in the model are given in the tables. Client is a prominent player in the domestic courier industry. The organization has been a pioneer in express courier services in India and is one of the few pan-India e-commerce logistics service providers. The client has more than 40 hubs across the entire country and operates across multiple movement types: road and air movement. Given the complexity of the operations, client wanted to evaluate potential scope of optimization in their regular operations of parcel movement between origin and destination branches. The paper is organized as follows: The introduction includes problem definition, understanding and components pertaining to the logistic network. The literature review section discussed the different studies done till date in the field of logistic courier industry and the different methodologies used to solve the similar issues of this industry. The next section describes the theory behind the methodology adopted. The following section to this gives the solution procedure and reports the result, for example, network. Finally, the conclusion of the work and future scope of this study is described.

1.1 Key Components of Logistics Network The entire logistics network is arranged as a hub-n-spoke model. The movement of load in logistic network consists of three legs: collection (branch–hub movement), hub–hub movement and distribution (hub–branch movement). The overview of the network is detailed in Fig. 1. Collection: The load into the logistics network originates at the branches. The branches act as the collection center for parcel/loads from multiple customers in the catchment. The branches are optimally located to ensure maximum catchment area with respect to the customer. Each of the branches has a collection time window. All the parcels from the customers are collected within the collection time window. Every branch is connected to a hub. At the end of the day, all the load collected at the branch is deposited at the hub.

184

M. Surapaneni et al.

Fig. 1 Nodes of the logistics company

Hub-to-hub movement: Hub-to-hub movement forms the key aspect of the overall load movement. The load collected from the branch is deposited at the source hub. Depending the parcel destination, the load is forwarded from the source hub to the destination hub to which the delivery branch is attached to. Hub acts as a consolidation–deconsolidation node for all loads generated at the branches. This consolidation of load helps improve the transit time and the cost. The hub-to-hub movement is critical to maintaining the right turnaround time (TAT) level while ensuring the cost efficiency is maintained. The hub-to-hub movement consists of multiple modes of transport: road and air. The road movement between the hubs consists of wholly owned truck movement. Road movement: Road movement consists of load movement in full trucks from source hub to the destination hub. Each of the routes has a connection starting and ending time at the source and destination hub, respectively. This is one of the most cost-effective options. However, the transit time is very high. Air movement: The air movement consists of sending load via the domestic air carriers. All the domestic air carriers offer space in the passenger flights for cargo movement as well. There are different parameters associated with the air movement. There are multiple connections between the same origin and destination hub. And the cost of moving the load varies by the connection timing. Hence, selection of the right connection time for each of the product types is imperative to ensure service levels (TAT) are maintained at optimal costs. The source hub to the destination hub might consist of multiple intermediary hubs with a combination of different transportation modes (Fig. 2). The key requirement is to identify the optimal route and the transportation type for the load movement.

Optimized Multimodal Transportation for Efficient Parcel …

185

Intermediary Hub

Source Hub

Destination Hub

Fig. 2 Movement via intermediary hub while moving from source to destination hub

Distribution: Once the parcel/load reaches the destination hub, it is forwarded to the respective destination branch depending on the customer catchment area. The parcel is then delivered from the destination branch to the end customer. Turnaround time (TAT): The other key aspect while deciding the nature of the connection is the service level of the product. The service level is defined as the turnaround time taken for the parcel to reach the end customer. There are different services provided to the customer depending on the service levels offered starting from within a day delivery to within a week. The turnaround time calculation comprises two key components: • Physical transit time: The time it takes for physically moving the parcel from one location to the other. • Cooling time: Time spent by the parcel at the hub/branch from the arrival time of incoming connection to the departure time of the outgoing connection.

2 Literature Review In this chapter, we mainly focus on bidirectional strategic service network design for a courier service provider company. As per Martin et al. (2015), the network design optimization problems consist of two interrelated problems: to determine the number and locations of hubs and depots, and to allocate geographic areas to these facilities. Both the types of problems belong either to the class of hub location or to the group of p-hub median problems. Campbell (1994) presents mathematical programming formulations for several discrete hub location problems, which are analogous to four fundamental types of discrete facility location problems: the p-median problem, the un-capacitated facility location problem, the p-center problem and covering problems. In our approach, the demand generation points are considered as the origin location and the consumption point of the services are defined as the destination locations. The hubs are defined as discrete input points for optimal network model design. Hakimi demonstrates that the p-median problem has been a fundamental problem in locational research since its inception. Campbell et al. (2005) defined the efficient single allocation p-hub median analogous to a p-median problem. The p-hub median problem has straightforward applications to transportation networks in which the objective is to minimize the total cost of movement. Martin et al. (2015)

186

M. Surapaneni et al.

introduced a distribution network structure for letter and parcel mail services based on location–allocation model using weighted directed graph. Ishfaq (2010) applied the multiple allocation p-hub median mathematical model approach considering the dynamics of individual modes of transportation considering the transportation cost parameters under service time requirements using tabu search meta-heuristic method. According to Alumur and Kara (2008) and Alumur et al. (2009), hubs are the special facilities that serve as switching, transshipment and sorting points in manyto-many distribution systems. The hub location problem is concerned with locating hub facilities and allocating demand nodes to hubs in order to route the traffic between origin–destination pairs. He proposed some recent trends on hub location over the synthesis of various literature reviews. Network design problems have become relevant and critical across all the industries considering multiple transport models—air, rail, sea and road network. On a generic level, all the industries have fixed their logistic network considering strategic, tactical and operational. Previously, Karimi et al. (2014) considers the modeling of the capacitated single allocation hub location problem with a hierarchical approach where different levels of services have been considered for transportation network and solved using a mixedinteger programming formulation for the problem with three-index variables using discount factors. We are trying to distinguish our mathematical model and incorporate actual economies of scale instead of the application of theoretical discount factor which are mostly available in the literatures. Crainic (2000) proposed a service network design formulation considering the explicit decision variables for a multimode, multicommodity model that integrates the service selection and traffic distribution problems. They considered the service capacity restrictions for the targeted utilization and over-assignment of traffic expense with allowing the additional costs and delays. Crainic presents a state-of-the-art review of service network design modeling efforts and mathematical programming developments for single allocation network design. As per Crainic (2000), service network design formulations are typically developed to assist the tactical planning of operations, although this planning level may be referred to as strategic/tactical or tactical/operational according to the planning traditions and horizons of the firm. The goal of such formulations is to plan services and operations to answer demand and ensure the profitability of the firm. Tactical planning of operations is comprised of a set of interrelated decisions that aim to ensure an optimal allocation and utilization of resources to achieve the economic and customer service goals of the company. There are many types of distribution transport topology optimization methods. Hub-and-spoke distribution paradigm is one of the forms in which there are many origins, many destinations and several hubs. Bryan and O’Kelly (1999) states that all the transport flow must travel from an origin to a destination via either one, two or many hubs. There are certain intermediate hubs which act as load transshipment facilities for a respective origin–destination load pair. This chapter deals with the multiple allocation hub-and-spoke networks considering to be un-capacitated and with given number of hubs. The given number of hubs is the total sum of existing hubs in the current network and the potential candidates which can be derived based on the

Optimized Multimodal Transportation for Efficient Parcel …

187

future load projection. The multiple allocation network model allows each origin– destination pair of nodes to select the optimized network alignment with minimizing the overall cost considering the required service-level time window which is called as turnaround time (TAT) through the set of hubs. The modeling approach used in this chapter is to formulate the intermodal hub network design as an interacting hub location–allocation problem. In general, a network model is represented by an undirected graph G(N, A), where N is a collection of nodes connected through a set of arcs A. According to Ishfaq and Sox (2011), the arcs have weights which may represent distance between nodes, unit cost per kg to travel between nodes or time of travel between nodes. The network is composed of a few central locations (relative to the total number of locations) which act as central hub or act as a transshipment point. Each non-hub location is assigned to one or more hub(s) based on the geographical direction of load movement. Such network designs are graphically represented as a wheel with the hub at the center connected to the non-hub locations through the spokes of the wheel. For this reason, hub networks are also called hub-and-spoke networks.

3 Mathematical Model Formulation The modeling framework discussed in this section falls within the context of a road– air intermodal network. However, our approached framework can also be applicable to other intermodal networks, such as road–rail and road–barge. In this network, the load collected from different originating locations (cities) delivered through trucks at the hub is called collection hub. The packages are sorted and grouped with other packages destined for a geographical region. The consolidated shipments are loaded into truck based on the types of load and delivery time window. For shorter time window of committed delivery time window, premium products are handed over to the co-leaders to dispatch the load through air cargos based on the cargo connection time. The shipments sent by air can be directly dispatched at the destination hub or enrooted to the destination hub based on the next leg connectivity. The shipments sent by road are loaded onto a long-haul truck. A mathematical formulation for a multimodal transportation network has been developed. The service time restrictions are defined for each origin–destination pair. A location–allocation decision impacts the shipment transit times which are composed of the travel time during pickup, interhub travel and drop-off. Furthermore, shipments spend additional time at the hubs, waiting for consolidation (at originating hub), transshipment hub and destination hub (at dispatching hub) called cooling time. The objective of our model is to identify optimal locations which are both economical and feasible with respect to the service time requirements.

188

M. Surapaneni et al.

3.1 The Problem is Modeled Using the Following Notation and Mathematical Formulation 3.1.1

Assumptions for Formulation of Planning Solution

• • • • •

There is no delay on flight time. Every flight is assumed to operate on every day of the week. Co-loader offload from the assigned flight is not considered for the modeling. Load for any O–D pair reaches airport on time. Load of any product category is dispatched based on the mapping mode-type movement. • There is no maximum weight discrepancy at airport. • Maximum number of hub transmissions are limited to 4: Simulation was done to understand that maximum of 4 nodes were enough to connect the farthest origin and destination points in the country based on the expected service level. This will also reduce the unnecessary handling and misplacement of the parcels/load in the hubs. 3.1.2 N H P MT M NT DH HC 3.1.3 Wo,d, p To,d, p Co,h1 CTo,h1 CEo,h1

Notations The set of all existing location demand nodes: origin o ∈ N and destination d ∈ N. The set of all hub locations including all the potential candidate hubs. The set of product types to be delivered for each O–D pair. The set of all available transportation mode types, i.e., road or airways. The set of all transportation modes based on available mode types MT. Number of planning time period in days. Day bucket planning time window is 24 h. The set of load handling capacities at each hub. Parameters Total amount of the load originates from origin location o ∈ N to destination location d ∈ N of product type p ∈ P. Turnaround time for the shipment originates from origin location o ∈ N to destination location d ∈ N of product type p ∈ P. Collection cost which is the unit cost of transportation from the origin location o ∈ N to origin hub location h1 ∈ H . Collection time which is the travel time from the origin location o ∈ N to origin hub location h1 ∈ H . Collection end time at origin hub for the day which is shipped from the origin location o ∈ N to origin hub location h1 ∈ H .

Optimized Multimodal Transportation for Efficient Parcel …

Dh2,d

DTh2,d DEh2,d DFh2,d

TCh1,h2,m

TSh1,h2,m

TTh1,h2,m

THh1,h2,m

TEh1,h2,m

189

Delivery cost which is the unit cost of transportation for distributing the load from the destination hub location h2 ∈ H to end destination demand node location d ∈ N . Delivery time which is the travel time transportation from the destination hub location h2 ∈ H to destination demand node location d ∈ N . Distribution start time from the destination hub locations h2 ∈ H to the destination demand node locations d ∈ N . Delivery frequency time for major destination demand node locations d ∈ N having multiple times of deliveries once the load has arrived at destination hub locations h2 ∈ H . Transfer cost which is the unit cost of transportation from the hub locations h1 ∈ H to the hub locations h2 ∈ H using transportation mode m ∈ M. Transshipment cost which is the unit cost of transshipment of the shipment from the hub locations h1 ∈ H to the hub locations h2 ∈ H using transportation mode m ∈ M. Transfer time which is the travel time including the load processing at origin hub locations h1 ∈ H and the sorting time of load arrived at destination hub locations h2 ∈ H using transportation mode m ∈ M. Transfer start time from the hub locations h1 ∈ H to the hub locations h2 ∈ H using transportation mode m ∈ M based on day’s flight itinerary. Transfer end time at the hub locations h2 ∈ H for the load start from the hub locations h1 ∈ H using transportation mode m ∈ M based on day’s flight itinerary.

3.2 Construction Algorithm for Intermediate Model Parameters A heuristic algorithm is designed to solve a problem in a faster and more efficient way compared to the traditional methods by sacrificing optimality, accuracy and precision. Heuristic algorithms that build a set of routes are known as construction algorithms and are used to build an initial feasible solution for the problem. We have used the construction algorithms to generate all the set of feasible routes for all the valid origin–destination (O–D) pair. All the routes are generated based on the timeoriented neighborhood search method within the allowable service time window for each O–D pair. Baker and Schaffer (1986) proposed the first sequential construction algorithm. The extension of this algorithm can be interpreted as the savings heuristic of Clarke and Wright (1964). Clarke and Wright algorithm works based on the maximum saving between two routes on each iteration. Here, we have created all the feasible routes considering the elapsed time factors at each operation for all the O–D pairs as a saving opportunity.

190

M. Surapaneni et al.

TC3h1,m1,h2,m2,h3 : 3-Hub link transfer cost is the unit cost of transportation from the hub location h1 ∈ H to intermediate hub location h2 ∈ H to destination hub h3 ∈ H using transportation modes m1, m2 ∈ M. TS3h1,m1,h2,m2,h3 : 3-Hub link transshipment cost is the unit cost of transshipment of the shipment from the hub location h1 ∈ H to intermediate hub location h2 ∈ H to destination hub h3 ∈ H using transportation modes m1, m2 ∈ M TT3h1,m1,h2,m2,h3 : 3-Hub link transfer time to travel the 3-hub links {h1, h2, h3} ∈ H from the origin hub locations h1 to destination hubs h3 ∈ H through intermediate hub locations h2 ∈ H using transportation modes m1, m2 ∈ M. TL3h1,m1,h2,m2,h3 : 3-Hub link cooling time is the calculated waited time at node to capture the next available connections using the 3-hub links {h1, h2, h3} ∈ H from the origin hub locations h1 to destination hubs h3 through intermediate hub locations h2 using transportation modes m1, m2 ∈ M. TC4h1,m1,h2,m2,h3,m3,h4 : 4-Hub link is the unit cost of transportation from the hub locations h1 ∈ H to intermediate hub links {h2, h3} ∈ H to destination hubs h4 ∈ H using transportation modes m1, m2, m3 ∈ M. TS4h1,m1,h2,m2,h3,m3,h4 : 4-Hub link is the unit cost of transshipment of the shipment from the hub locations h1 ∈ H to intermediate hub links {h2, h3} ∈ H to destination hub h4 ∈ H using transportation modes m1, m2, m3 ∈ M. TT4h1,m1,h2,m2,h3,m3,h4 : 4-Hub link is the transfer time to travel the 4-hub links {h1, h2, h3, h4} ∈ H from the origin hub locations h1 to destination hub h4 through intermediate hub links {h2, h3} using transportation modes m1, m2, m3 ∈ M. TL4h1,m1,h2,m2,h3,m3,h4 : 4-Hub link is the cooling time to travel the 4-hub links {h1, h2, h3, h4} ∈ H from the origin hub locations h1 to destination hub h4 through intermediate hub links {h2, h3} using transportation modes m1, m2, m3 ∈ M. AWh1,h2,m : Airway baggage lot weight for a hub link pair {h1, h2} ∈ H using the transportation mode m ∈ M through the mode type t ∈ M T : Air . FW p,m : Unit airway baggage lot cost for the product type p ∈ P shipped using the transportation mode m ∈ M through the mode type t ∈ M T : Air . L ∈ o, d, p denote the flow mapping combination set of origin city o ∈ N , destination city d ∈ N and the product type p ∈ P. L1 ∈ o, d, p denote the flow mapping combination set of origin city o ∈ N , destination city d ∈ N and the product type p ∈ P where “o and d” are not connected to the same hub.

3.3 Decision Variable Used for the Mathematical Model ⎧ ⎨ 1 if origin city location o ∈ N is allocated X l,o,h1 to origin hub h1 ∈ H for all l ∈ L; ⎩ 0 Otherwise

Optimized Multimodal Transportation for Efficient Parcel …

191

⎧ ⎨ 1 if destination city location d ∈ N is allocated Yl,h2,d destination hub h2 ∈ H for all l ∈ L; ⎩ 0 Otherwise ⎧ ⎨ 1 if the flow l ∈ L uses the hub links {h1, h2} from h1 ∈ H Z 2l,h1,m,h2 to h2 ∈ H with transportation mode m ∈ M; ⎩ 0 Otherwise ⎧ 1 if the flow l ∈ L uses the hub links {h1, h2, h3} ⎪ ⎪ ⎪ ⎪ ⎪ from h1 ∈ H to intermediate ⎪ ⎪ ⎪ ⎨ hub h2 ∈ H with itermediate transport mode m1 ∈ M Z 3l,h1,m1,h2,m2,h3 ⎪ and to end hub h3 ∈ H with transportation mode ⎪ ⎪ ⎪ ⎪ ⎪ m2 ∈ M; ⎪ ⎪ ⎩ 0 Otherwise ⎧ 1 if the flow l ∈ L uses the hub links {h1, h2, h3, h4} ⎪ ⎪ ⎪ ⎪ from h1 ∈ H to first intermediate hub h2 ∈ H using ⎪ ⎪ ⎪ ↔ ⎪ ⎪ ⎨ transport mode m 1 ∈ M of mode type road to Z 4l,h1,m1,h2,m2,h3,m3,h4 second intermediate hub h3 ∈ H using transport mode ⎪ ⎪ ⎪ m2 ∈ M and to end destination hub h4 ∈ H using ⎪ ⎪ ⎪ ⎪ ⎪ transporation mode m3 ∈ M of mode type road; ⎪ ⎩ 0 Otherwise W Nh1,h2,m : Positive integer variable for the multiple shipment lot hub links {h1, h2} from h1 ∈ H to h2 ∈ H using the transportation mode m ∈ M. W Tl : Positive variable calculates the total service time for serving the flow L ∈ o, d, p.

3.4 Mathematical Model Formulation Minimize Total Cost: Collection Cost + (Hub Transfer Cost2 + Hub Transfer Cost3 + Hub Transfer Cost4) + Distribution Cost + (Hub Transshipment Cost3 + Hub Transshipment Cost4) + Total Baggage Cost where Collection Cost = l∈o,d, p h1 X l,o,h1 ∗ Co,h1 ∗ Wo,d, p

192

M. Surapaneni et al.

Hub Transfer Cost2 = l∈o,d, p h1,m,h2 Z 2l,h1,m,h2 ∗ TCh1,h2,m ∗ Wo,d, p Hub Transfer Cost3 = l∈o,d, p h1,m1,h2,m2,h3 Z 3l,h1,m1,h2,m2,h3 ∗ TC3h1,m1,h2,m2,h3 ∗ Wo,d, p Hub Transfer Cost4 = l∈o,d, p h1,m1,h2,m2,h3,m3,h4 Z 4l,h1,m1,h2,m2,h3,m3,h4 ∗ TC4h1,m1,h2,m2,h3,m3,h4 ∗ Wo,d, p Hub Transshipment Cost3 = l∈o,d, p h1,m1,h2,m2,h3 Z 3l,h1,m1,h2,m2,h3 ∗ T S3h1,m1,h2,m2,h3 ∗ Wo,d, p Hub Transshipment Cost4 = l∈o,d, p h1,m1,h2,m2,h3,m3,h4 Z 4l,h1,m1,h2,m2,h3,m3,h4 ∗ TS4h1,m1,h2,m2,h3,m3,h4 ∗ Wo,d, p Distribution Cost = l∈o,d, p h2,d Yl,h2,d ∗ Dh2,d ∗ Wo,d, p Total Baggage Cost =  p∈P,h1∈H,h2∈H,m∈M WNh1,h2,m ∗ FW p,m ∗ NT The objective function is to minimize the sum of total transportation cost for all the origin–destination pair product shipments such as collection cost which signifies the load shipment to origin hubs, hub-to-hub load shipment cost (Hub Transfer Cost2 + Hub Transfer Cost3 + Hub Transfer Cost4), distribution cost which signifies the load delivery cost at the destination from the destination hub, the transshipment cost for load loading and unloading at intermediate hubs (Hub Transshipment Cost3 + Hub Transshipment Cost4), and total multiple lot baggage cost. There is a fixed baggage cost associated for each air mode shipment between two hubs. So, loads need to be sorted and packed based on the baggage weight size. The total baggage cost signifies the total load shipped between each hub pair through each air mode. Subject to: h1∈H X l,o,h1 ≥ 1 ∀l ∈ L : o, d ∈ N , p ∈ P

(1)

h2∈H Yl,h2,d ≥ 1 ∀l ∈ L : o, d ∈ N , p ∈ P

(2)

X l,o,h1 = Yl,h2,d ∀l ∈ L : o, d ∈ N , p ∈ P, h1, h2 ∈ H : h1 = h2

(3)

Optimized Multimodal Transportation for Efficient Parcel …



X l,o,h1 ≥

Z 3l,h1,m1,h2,m2,h3

h2,h3∈H,m1,m2∈M



+



Z 2l,h1,m,h2 +

h2∈H,m∈M

193

Z 4l,h1,m1,h2,m2,h3,m3,h4

h2,h3,h4∈H,m1,m2,m3∈M

∀l ∈ L1 : o ∈ N , p ∈ P, h1, h2 ∈ H : h1! = h2 

Yl,h2,d ≥

h1∈H,m∈M

Z 3l,h1,m1,h3,m2,h2

h1,h3∈H,m1,m2∈M



+



Z 2l,h1,m,h2 +

(4)

Z 4l,h1,m1,h3,m2,h4,m3,h2

h1,h3,h4∈H,m1,m2,m3∈M

∀l ∈ L1, h2 ∈ H  h1,h2∈H,m∈M

+

(5) 

Z 2l,h1,m,h2 +

Z 3l,h1,m1,h2,m2,h3

h1,h2,h3∈H,m1,m2∈M



Z 4l,h1,m1,h2,m2,h3,m3,h4 = 1

h1,h2,h3,h4∈H,m1,m2,m3∈M

∀l ∈ L1 : o, d ∈ N : o = d, p ∈ P

(6)

The constraint (1) ensures that every load pair l ∈ L is assigned to more than one origin hub based on the directional movement of destination load. Constraint (2) ensured that every destination node d ∈ N is assigned to more than one hub as per the directional alignment to serve the destination demand. Constraint (3) is for flow conversion for delivering the load at destination city using a single hub node allocation for each product. Constraint (4) implies that each load l ∈ L1 which is not aligned with single hub is aligned to multiple hub pairs based on the directed connection linkages to send shipment from the origin hub. Similarly, constraint (5) ensures that each load l ∈ L1 which is not aligned with single hub is aligned to multiple hub pairs based on the nearest destination hub to deliver the load at the destination. Constraint (6) implies the load l ∈ L1 which is not assigned through any single hubs is to be delivered at destination using any one of the multiple hub pair movements. It establishes a unique transportation mode for a unique hub-to-hub pair to align with the destination demand using more than one hub node.  l∈L:o,d∈N , p∈P



Wo,d, p Z 2l,h1,m,h2 ∗ ceil NT



 Wo,d, p NT l,∈L:o,d∈N , p∈P,h3∈H,m2∈M    Wo,d, p Z 3l,h3,m2,h1,m1,h2 ∗ ceil + NT l∈L:o,d∈N , p∈P,h3∈H,m2∈M +





Z 3l,h1,m1,h2,m2,h3 ∗ ceil

194

M. Surapaneni et al.

 Wo,d, p NT l∈L:o,d∈N , p∈P,h3,h4∈H,m2,m3∈M    Wo,d, p Z 4l,h3,m2,h1,m1,h2,m3,h4 ∗ ceil + NT l∈L:o,d,∈N , p∈P,h3,h4∈H,m2,m3∈M    Wo,d, p Z 4l,h3,m2,h4,m3,h1,m1,h2 ∗ ceil + NT l∈L:o,d∈N , p∈P,h3,h4∈H,m2,m3∈M 



+

Z 4l,h1,m1,h2,m2,h3,m3,h4 ∗ ceil

≤ AWh1,h2,m ∗ WNh1,h2,m ∀h1, h2 ∈ H : h1 = h2, m, m1 ∈ M : m = m1 (7) The airway loads need to be sorted and packaged based on the maximum baggage weight. Each air carrier will collect the load based on the multiple bags. The load l ∈ L in weights is considered at annual level and is averaged to a per day basis by dividing with number of working days (NT). Based on the daily planning bucket, the constraint (7) calculates the minimum number of baggage to be shipped through air for each hub pair link. 



X l,o,h1 ∗ CTo,h1 +

h1∈H

h2∈H, CEo,h1 ≤DEh2,d ,DFh2,d =0



+

Yl,h2,d ∗ DEh2,d − CEo,h1



Yl,h2,d ∗ DEh2,d − CEo,h1 + DH

h2∈H,CEo,h1 >DEh2,d ,DFh2,d =0



+

Yl,h2,d ∗ DFh2,d +

h2∈H,CEo,h1 ≤DEh2,d ,DFh2,d >0



Yl,h2,d ∗ DTh2,d ≤ To,d, p

h2∈H

∀l ∈ L : o, d ∈ N , p ∈ P

(8)

Constraint (8) ensures that the total transit time from the origin node to the destination node assigned through a single hub is within the service time window for each load l ∈ L. 



X l,o,h1 ∗ CTo,h1 +

h1∈H



+

h1,h2∈H,m∈M,CEo,h1 ≤THh1,h2,m



Z 2l,h1,m,h2 ∗ THh1,h2,m − CEo,h1 + DH

h1,h2∈H,m∈M,CEo,h1 >THh1,h2,m



+

h1,h2∈H,m∈M

+

Z 2l,h1,m,h2 ∗ TTh1,h2,m + 

h1,h2∈H,m∈M,DEh2,d ≥TEh1,h2,m ,DFh2,d =0



+

Z 2l,h1,m,h2 ∗ THh1,h2,m − CEo,h1



Z 2l,h1,m,h2 ∗ DFh2,d

h1,h2∈H,m∈M,DFh2,d >0

Z 2l,h1,m,h2 ∗ DEh2,d − TEh1,h2,m

Z 2l,h1,m,h2 ∗ DEh2,d − TEh1,h2,m + DH

h1,h2∈H,m∈M,DEh2,d ≤TEh1,h2,m ,DFh2,d =0

+



Yl,h2,d ∗ DTh2,d ≤ To,d, p

h2∈H

∀l ∈ L : o, d ∈ N , p ∈ P

(9)

Optimized Multimodal Transportation for Efficient Parcel …

195

Constraint (9) ensures the total transit time from the origin node to the destination node assigned through a hub pair {h1, h2} ∈ H using the transportation mode m ∈ M is within the service time window for each load l ∈ L. 

X l,o,h1 ∗ CTo,h1

h1∈H



+



Z 3l,h1,m,h2,m2,h3 ∗ THh1,h2,m − CEo,h1

h1,h2,h3∈H, m1,m2∈M,CEo,h1 ≤THh1,h2,m



+



Z 3l,h1,m1,h2,m2,h3 ∗ THh1,h2,m − CEo,h1 + DH

h1,h2,h3∈H,m1,m2∈M, CEo,h1 >THh1,h2,m



+

Z 3l,h1,m1,h2,m2,h3 ∗ TT3h1,m1,h2,m2,h3 + TL3h1,m1,h2,m2,h3

h1,h2,h3∈H,m1,m2∈M



+

Z 3l,h1,m1,h2,m2,h3 ∗ DFh2,d

h1,h2,h3∈H, m1,m2∈M,DFh2,d >0



+

Z 3l,h1,m1,h2,m2,h3 ∗ DEh2,d − TEh1,h2,m

h1,h2,h3∈H,m1,m2∈M, DEh2,d ≥TEh1,h2,m ,DFh2,d =0



+

Z 3l,h1,m1,h2,m2,h3 ∗ (DEh2,d − TEh1,h2,m

h1,h2,h3∈H,m1,m2∈M, DEh2,d ≤TEh1,h2,m ,DFh2,d =0

+ DH) +



Yl,h2,d ∗ DTh2,d ≤ To,d, p

h2∈H

∀l ∈ L : o, d ∈ N , p ∈ P

(10)

Constraint (10) is for the service time window constraint for shipment flow using 3 hub links {h1, h2, h3} ∈ H from h1 to intermediate hub h2 to destination hub node h3 using the transportation modes m1, m2 ∈ M.  h1∈H

+

X l,o,h1 ∗ CTo,h1 

Z 4l,h1,m,h2,m2,h3,m3,h4 ∗ THh1,h2,m − CEo,h1

h1,h2,h3∈H,m1,m2∈M,CEo,h1 ≤THh1,h2,m

+



Z 4l,h1,m,h2,m2,h3,m3,h4

h1,h2,h3,h4∈H,m1,m2,m3∈M,CEo,h1 >THh1,h2,m



∗ THh1,h2,m − CEo,h1 + DH  Z 4l,h1,m,h2,m2,h3,m3,h4 ∗ (TT4h1,m1,h2,m2,h3 + h1,h2,h3,h4∈H,m1,m2,m3∈M

+ TL4h1,m1,h2,m2,h3 ) +



Z 4l,h1,m,h2,m2,h3,m3,h4 ∗ DFh2,d

h1,h2,h3,h4∈H,m1,m2,m3∈M,DFh2,d >0



+

Z 4l,h1,m,h2,m2,h3,m3,h4

h1,h2,h3,h4∈H,m1,m2,m3∈M,DEh2,d ≥TEh1,h2,m ,DFh2,d =0



∗ DEh2,d − TEh1,h2,m +



h1,h2,h3,h4∈H,m1,m2,m3∈M,DEh2,d ≤TEh1,h2,m ,DFh2,d =0

Z 4l,h1,m,h2,m2,h3,m3,h4

196

M. Surapaneni et al.



∗ DEh2,d − TEh1,h2,m + DH + Yl,h2,d ∗ DTh2,d ≤ To,d, p h2∈H

∀l ∈ L : o, d ∈ N , p ∈ P

(11)

Constraint (11) ensures the service time window constraint for shipment flow using 4 hub links {h1, h2, h3, h4} ∈ H from h1 to intermediate hub links {h2, h3} to destination hub node h4 using the transportation modes m1, m2, m3 ∈ M.

4 Computational Study The purpose of the study is to evaluate the performance of large-scale mixed-integer programming (MIP) solution approach (using branch and cut algorithm) which is evaluated based on the optimization gap (0.01), the distance from lower bound. CPLEX MIP optimizer utilizes the state-of-the-art algorithms and techniques to solve complex models having mixed-integer variables. The prominent algorithm used in CPLEX is branch and cut algorithm. The solver creates subproblems by systematically relaxing LP constraints via an efficient branching methodology. The CPLEX solver builds a tree with a linear relaxation of the original MIP at root. Then, it creates various subproblems across tree nodes to arrive at practical solution. CPLEX also contains multiple inbuilt sophisticated heuristics to prune non-basic variables for finding initial feasible solution space, thereby improving model run efficiency. The scenarios were run on an i7 CPU octa-core processor with 4.2 GHz clock and 64 GB RAM. The size of a problem is represented by the number of nodes, n = 372 with 42 number of hubs as a load consolidation center. The problem scenarios were solved to optimality with CPLEX solver 12.6 using IBM ILOG Decision Optimization Center (DOC) Version 3.8. The script language of IBM OPL was used to run the problem instances in an efficient manner. IBM DOC is comprising of eclipse IDE with an inbuilt functionality of application data model (ADM). ADM is a relational data model that defines and holds the data structures used by the Decision Optimization Center, and it provides several extensions that helps perform the task of importing schemas from external databases using SQL queries.

4.1 Scenario Evaluation and Results In the current logistics model, we considered 15,000+ origin destination product combinations for identifying the optimal movement. This led to more than 2 million decision variables being generated for evaluation. For baseline scenario evaluation, the existing connections were enforced to generate the baseline cost. To-be scenario was evaluated by evaluating all possible network combinations. Instead of a single branch to hub connection, we explored the possibility of multiple branches to hub

Optimized Multimodal Transportation for Efficient Parcel …

197

movement depending on the parcel destination. This led to better load alignment with respect to the demand data. When compared with the baseline scenario, the to-be scenario resulted in a potential cost saving of close to 14%. The benefit realized was primarily due to identification of optimal mode of transport between the source and destination hub considering the cost and the provided service levels. Instead of costlier air connection, given the TAT requirements, possibilities of load movement via road were identified resulting in huge cost-benefits on the route. Also, potential benefits were realized due to reduction in multiple flight connections required for load movement between source and the destination hubs. The final set of benefits were realized from optimal alignment of branch movement to hub depending on the destination branch. The other key benefit involves identification of origin destination and product combinations where the service levels (TAT) can be improved with the same operation costs. This helps improve customer delight, thereby having a potential positive impact on the revenue.

5 Conclusion The current optimization model shows scope for potential operation cost reduction. However, the current model considers data parameters as static: demand, transit times, connection times, etc. Also, currently there is no air carrier offloading considered. In the future, we plan to enable uncertainty across the data parameters as discussed above and enable the probability of offloading in the load sent via air mode. These considerations will be the subject to the future work.

References Ahuja, R. K., Magnanti, T. L., & Orlin, J.B. (1993). Network flows: Theory, algorithms, and applications. Pearson Publication. Alumur, S., & Kara, B. Y. (2008). Network hub location problems: The state of the art. European Journal of Operational Research, 190, 1–21. Alumur, S., Kara, B. Y., & Karasan, O. (2009). The design of single allocation incomplete hub networks. Transportation Research Part B, 43, 936–951. Alumur, S., Kara, B. Y., & Karasan, O. (2012). Multi-modal hub location and hub network design. The International Journal of Management Science, 40, 927–939. Baker, E. K., & Schaffer, J. R. (1986). Solution improvement heuristics for the vehicle routing and scheduling problem with time window constraints. American Journal of Mathematical and Management Sciences, 6, 261–300. Ben-Ayed, O. (2013). Parcel distribution network design problem. Operational Research International Journal, 13, 211–232. Bryan, D. L., & O’Kelly, M. E. (1999). Hub-and-spoke networks in air transportation: An analytical review. Journal of Regional Science, 39, 275–295. Campbell, J. F. (1994). Integer programming formulations of discrete hub location problems. European Journal of Operations Research, 72, 387–405.

198

M. Surapaneni et al.

Campbell, J. F. (1996). Hub location and the p-hub median problem. Operations Research, 72, 387–405. Campbell, J. F., Ernst, A. T., & Krishnamoorthy, M. (2005). Hub arc location problems: Part II—Formulations and optimal algorithms. Management Science, 51, 1556–1571. Clarke, G., & Wright, W. (1964). Scheduling of vehicles from a central depot to a number of delivery points. Operations Research (12), 568581. Crainic, T. G. (2000). Service network design in freight transportation. European Journal of Operations Research, 122, 272–288. Dutta, B., & Jackson, M. O. (2003). Networks and groups: Models of strategic formation. Springer. Freight movement by road transport India FY 2001–2017. Statista Research Department, May 8, 2020. https://www.statista.com/statistics. Ishfaq, R. (2010). Hub network design models for intermodal logistics. A Dissertation, The University of Alabama. Ishfaq, R., & Sox, C. R. (2011). Hub location–allocation in intermodal logistic networks. European Journal of Operational Research, 210, 213–230. Karimi, M., Eydi, A. R., & Korani, E. (2014). Modeling of the capacitated single allocation hub location problem with a hierarchical approach. International Journal of Engineering, 27(4), 573–586. Logistics Skill Council. Overview on logistics industry. https://lsc-india.com. Martin, N. B., Halil, I. G., Thomas, M., & Sebastian, H. (2015). Strategic planning of optimal networks for parcel and letter mail, quantitative approaches in logistics and supply chain management. Lecture notes in logistics. Springer.

Enhancement for Easy Egress Through Emergency Gate of Bus Vinay Kumar Singh, Rahul Kumar, and Rahul Varshney

1 Introduction In case of an accidental situation, the occupants of a bus have to come out of the vehicle as quickly, and most of the times the situation becomes very tight and creates panic among the passengers as they do not find a quick escape. In recent years, the rising volume of traffic and the growing congestion on roads and building of expressways with higher speed limits have led to a marked increase in the number of accidents involving commercial and passenger vehicles resulting in severe injuries and death cases. The vehicle manufacturing industry follows many rules and specifications for the designing which reduces the accident rate and increases protection or safety of the passengers. But most of the Indian commercial vehicle manufactures prefer selling their drive away chassis due to business reasons, the fleet owner also prefers the same due to financial reasons, and they prefer the body design and building by local manufactures who do not care much about safety and regulations resulting in large amounts of fatal accidents (Mohan et al., 2009). In recent past, many such cases have been recorded, in one of the cases recently happened (26 March 2019) in Agra–Lucknow Expressway, SCANIA metro link multi-axle met with fire accident at Yamuna Expressway resulting in death of five persons, and twenty passengers were grievously injured. A passenger and her child were burnt alive due to the lack of good emergency exit provision. V. K. Singh (B) Digital Product Development System, Tata Technologies Ltd., Lucknow, India e-mail: [email protected] R. Kumar Digital Product Development System, Tata Technologies Ltd., Jamshedpur, India e-mail: [email protected] R. Varshney ERC, Tata Motors Ltd., Lucknow, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 D. Ghosh et al. (eds.), Studies in Quantitative Decision Making, Asset Analytics, https://doi.org/10.1007/978-981-16-5820-4_10

199

200

V. K. Singh et al.

Fig. 1 Road fatalities in India. Source Ministry of Road Transport and Highway, News Reports

Proposed modification in existing emergency exit will enable easy and safe egress of passengers from a bus wreckage considering various age groups and their physical abilities. Figure 1 shows the variation of death rate year wise which shows continuous increase due to increase speed limit, building of expressway and deteriorating bus body quality due to the lack of suitable clauses in certification and regularity documents like AIS052 and central motor vehicles rules (CMVR) for bus body interiors like seating, flooring, curtains, etc.

2 Methodology AIS052 is a code of practice for bus body design and approval govern by Ministry of Road Transport and Highways. Emergency door/window is designed in accordance with AIS052. Proposed change is meeting all requirements of AIS052 except 3.4.5.13 for emergency door for which consultation is to be done with Automotive Research Association of India (ARAI). Hinge provision is to be added on bottom side with proper demonstration to ARAI. Bus body means the portion of a bus that encloses the bus’s occupant space, exclusive of the bumpers, the chassis frame and any structure forward and rear word of the chassis. Type I: Medium to high capacity vehicles, designed and constructed for urban and sub urban transport. Type II: Used for inter-urban transport with space for standing passengers. Type III: Designed for long-distance, interstate passenger transport, not designed for standing passenger (Figs. 2 and 3). Figure 4 shows the schematic representation of various kinds of buses manufactured across in medium heavy commercial vehicle (MHCV) buses in automotive industry. These are classified based on wheel base, engine position, door options, floor height and passenger capacity, etc.

Enhancement for Easy Egress Through Emergency Gate of Bus Fig. 2 Minimum dimensions of the emergency exits. Source AIS052

Fig. 3 Minimum requirements of the emergency windows. Source AIS052

201

Category

Height (mm)

Width (mm)

Type I

1250

550

Type II

1250

550

Type III

1250

550

Category

Area (cm2)

Type I

4000

Type II

4000

Type III

4000

Remark It shall be possible to inscribe in this area a 50 cm high and 70 cm wide rectangle.

Fig. 4 Schematic bus classification

Emergency Door means a door intended for use as an exit by passengers in an emergency only. Emergency Window means a window intended for use as an exit by passengers in an emergency only. Escape Hatch means a roof opening intended for use as an exit by passengers in an emergency. It is additional to the emergency doors and windows and may be fitted in the roof for all category of vehicles.

202

V. K. Singh et al.

Proposed change validity. • Emergency door/window is designed in accordance with AIS052. • In accordance with AIS052, emergency door/window is hinged similar to regular (vertically hinged) one like passenger and driver doors. Proposed change is meeting all requirements of AIS052 except 3.4.5.1.3 for emergency door where it differs in terms of hinge position, and it is to be added on bottom side with proper demonstration to ARAI.

3 Literature Review In an attempt to understand the issues and challenges faced during the emergency situation in passenger vehicles, the passengers were not able to find a quick escape from the bus body, and a literature review was carried out on three major disastrous events that happened across the globe. These include fire in buses, buses drowned in water and structural damages. India, unfortunately, ranks at the top with the highest number of fatalities with about 11% share in the world. The total number of accident-related deaths in 2018 stood at 151,417 indicating an increase of 2.3% over the figures for 2017 (Bentama et al., 2017). Research on bus safety has been carried out mainly by institutions on contract with regulatory bodies in different countries on the dynamics of passenger’s safety and how to better protect passengers (Maputi, 2010). Rear exits are the most preferred in buses as they are accessible in most accident situations. The analysis however does not consider exit for disabled or aged persons (Fig. 5). The usability of the individual emergency exits is different in different bus categories Fig. 4 (e.g. low floor city bus, high-decker tourist coach, etc.) and also in different accident situations (e.g. frontal collision, rollover, fire, side impact, rear impact, bus is in shallow water, etc.). Fires in buses are a relatively common incident, and this indicates that the fire safety level in buses is not good enough. Hence, the material used and the construction and design of interior materials must be reviewed for fire safety by considering reaction to fire, ignitability, fire propagation, smoke production and toxicity (Johansson & Axelsson, 2006).

Fig. 5 Bus categorisation of buses

Enhancement for Easy Egress Through Emergency Gate of Bus

203

A fuel tank of a Uttar Pradesh state road transport corporation bus has burst after a collision with a lorry on 05 June 2017, which is travelling from New Delhi to Gonda District travelling with two drivers and thirty-eight passengers, and fire engines reached the spot after 90 min. Twenty-two people were buried in this fire accident. On 31 October 2017, a private bus caught fire near Dhaula Kuan in South Delhi which is carrying thirty-three students of Kendriya Vidyalaya, Naraina branch, and it was a narrow escape. The central locking system of the luxury bus jammed preventing people from getting out of the emergency door and windows. On 20 January 2015, a Volvo bus travelling with forty-five passengers from Mumbai–Hyderabad caught fire in the Telangana state Medak District. The driver stopped the bus and alerted the passengers immediately after noticing the rear engine side fire, and no one was injured but there was a property loss. A private bus caught fire in Maharashtra, five passengers died and ten injured near Talegaon Wardha District, and the bus was travelling from Jalgon to Nagpur on 29 May 2014. A bus travelling from Pune to Pandharpur on 5 May 2014 caught fire due to short circuit, and the driver and conductor off loaded all the forty-six passengers in Maharashtra. The vehicle has buried totally which belongs to Maharashtra State Transport. On November 2013, a KSRTC bus going from Hassan to Bangalore caught fire; the driver and conductor immediately alerted the passengers inside the bus at Gorgunteplaya. On 30 October 2013, a private Volvo bus caught the fire near Mahbubnagar on the way from Bangalore to Hyderabad which killed forty-five people and seven were injured. In the same way, many fire accidents are caused not only in India but also in many countries (Chafekar et al., 2013). In highway accidents where passengers got trapped inside a burning bus by not opening of the regular and emergency doors, many fatalities have occurred. This happened due to the possibility of their pivots or sliding surfaces locking by reason of rust or dirt and by friction around pulleys or pivots (Robert, 1951). Bus fires have caused a large number of fatalities due to unavailability of fire extinguishers cylinders, emergency doors not got open, use of non-standard vehicle accessories, electricalrelated causes and absence of flame proof upholstery and floorboards (Hofmann & Dülsen, 2012). From April 2020, all buses with seating capacity of more than twentythree passengers besides driver will require to have an inbuilt safety feature to detect engine fire and suppress it automatically while creating additional alarm. Fire detection and separation system (FDSS) will be able to douse a fire in 5–6 s caused by any mishappening (Sapper & Reep, 2006). The new bus body code includes features like more cabin space for drivers, fire extinguishers, emergency exits, proper illumination and ventilation, more space between two rows of seats and insulation of doors and windows to avoid rain water, dust and pollution (Young et al., 1997). The primary focus of the cause of accident has been narrowing down on unsafe driver behaviour, speeding, fatigue and alcohol use. Section 135 of the Motor Vehicles Act highlights the need to frame a mechanism for investigation of accident cases, and most states have no such system in place (Young et al., 1997). Man behind the wheel plays an important role in most of the fatal accidents. There are three types of accidental deaths. These are instantaneous where death happens within seconds, death happens in early minutes or early hours, and death happens days after the impact. Out of this,

204

V. K. Singh et al.

Fig. 6 Buses caught fire and totally buried

the second category deaths can be saved if victims get medical assistance on time (Singh & Misra, 2004) (Fig. 6). Accidents Scenario and Crashworthiness Based on the studies conducted, we found that following are the possible major scenarios: • • • • •

Frontal impact Side impact Rear impact Rollover Fire.

In case of frontal impact, there is possibility of driver and main door getting damaged to the point of not opening and thereby becoming an obstruction. Secondary factors like the destruction of a fuel tank and electrical short circuit can affect survivability when a fire breaks out. In case of fire, there is a time limit, in which the bus must be evacuated (smoke, poisoning gases, high temperature). Rollover is the most complex accident; after the accident, the bus could land in different positions. It covers all other accident situations when the bus is standing on its wheels.

4 Concept This study is based on finding major differences in the conventional emergency exit vs the newly introduced concept with bottom hinged position during emergency passenger egress out of bus. A comparative study was carried out after the proto vehicle updation with new concept to find out its impact on the time period required by the passenger to safely exit out of the bus. Currently, emergency door is located at the height from the ground, and pocket steps are given for egress of passengers.

Enhancement for Easy Egress Through Emergency Gate of Bus

205

Fig. 7 Existing arrangement of emergency exit

Problem with Existing arrangement: • In case of emergency, passengers are in panic situation, and locating pocket steps increases the time for egress. • In most of the cases, passenger tends to jump from height which can also cause injury (Fig. 7). What can be alternate solution? We proposed the revision of design of existing emergency gate with minor modification which will enable in quick egress of passengers from a bus wreckage considering various age groups and their physical ability in case of accident or fire. Changes are as follows. • Instead of vertical mounting/hinge of emergency door/window on bus body, horizontal mounting points are to be given. In revised emergency gate, it will enable emergency door/window like a slider, and quicker and easier egress of passengers would be possible. Door will be locked from inside through locked latch which can be opened easily by passengers from both inside and outside. In case of emergency, gate will be opened through latch which will be convenient and easy for all age groups to timely evacuate from bus (Figs. 8 and 9). In case of frontal crash, fire door will open slopping towards ground, which will work like a slider for easy egress of all age groups of people. Below listed are the preventive measures that can be implemented for better passenger safety:

206

V. K. Singh et al.

Fig. 8 Emergency door opening mechanism

Fig. 9 Emergency door opening mechanism

• Enhancement of emergency door and windows • Develop efficient fire detection and suppression system • Install fire and some detection, suppression system in engine and luggage compartments • Conduct periodic vehicle inspection and audit for all safety requirements • Improved gangway design to allow free movement of passengers • Passenger should not be allowed to travel with inflammable and explosive material.

Enhancement for Easy Egress Through Emergency Gate of Bus

207

5 Results and Analysis Emergency evacuation tests were carried out in proto vehicles by a set of passengers based on some independent variable (age) in which three groups were selected and divided as follows.

We have considered 30 members in each group of people based on their age and physical ability. • Group 1: 7–15 years old • Group 2: 20–45 years old • Group 3: 60–75 years old. Emergency egress activities break-up for different age group of people is given in Table 1. For identifying the significant difference between the mean time of three independent groups, we used one-way analysis of variance (ANOVA) technique (Ardelean, 2017). The survey was analysed digitally using SPSS20.0 software for the interpretation and comparison of results. The ANOVA test results shown in Table 2 indicated that there was a statistically significant difference for door with slider between groups as determined by oneway ANOVA (F(2,87) = 7.812E3, p = 0.00). The value of F is 7.8, which reaches significance with a p-value of 0.00 (which is less than the 0.05 alpha level). There was a statistically significant difference for door without slider between groups as determined by one-way ANOVA (F(2,87) = 448.687, p = 0.00). The value of F is 448.6, which reaches significance with a p-value of 0.00 (which is less than the 0.05 alpha level). There was a statistically significant difference for window with slider between groups as determined by one-way ANOVA (F(2,87) = 1.273E4, p = 0.00). The

208

V. K. Singh et al.

Table 1 Time study for easy egress S. No.

Evacuation method

Group 1 time (s)

Group 2 time (s)

Group 3 time (s)

Remarks (s)

1

Emergency door with slider (600 mm) high

125

145

235

Each time when

2

Emergency door without slider

215

205

a

3

Emergency window with slider (600 mm) high

280

320

559

4

Emergency window without slider

530

480

a

Average time saving

480

527.5

514.5

a

Participant could not complete exercise

Table 2 ANOVA Door with slider

Door without slider

Window with slider

Window without slider

Sum of squares

df

Mean square

F

Sis

Between groups

216,960.622

2

108,480.311

7.812E3

0.000

Within groups

1208.100

87

13.886 448.687

0.000

1.273E4

0.000

1.116E3

0.000

Total

218,168.722

89

Between groups

82,047.489

2

41,023.744

Within groups

7954.467

87

91.431

Total

90,001.956

89

Between groups

1,395,763.889

2

698,381.944

Within groups

4771.267

87

54.842

Total

1,401,535.156

89

Between groups

267,849.622

2

133,924.811

Within groups

10,436.333

87

119.958

Total

278,285.956

89

value of F is 1.273, which reaches significance with a p-value of 0.00 (which is less than the 0.05 alpha level). There was a statistically significant difference for window without slider between groups as determined by one-way ANOVA (F(2,87) = 1.116E3, p = 0.00). The value of F is 1.116, which reaches significance with a p-value of 0.00 (which is less than the 0.05 alpha level). Tables 2 and 3 shows summary of survey results. The ANOVA test multiple comparison results in Table 3 indicated that for egress in case of ‘door with slider’,

Enhancement for Easy Egress Through Emergency Gate of Bus

209

Table 3 Multiple comparisons Dependent variable

(I) Group

(J) Group

Mean difference (I-J)

Door with slider

1

2

−15.06667*

0.96216

3

−110.86667*

0.96216

1

15.06667*

3

−95.80000*

1

2 3 Door without slider

1 2 3

Window with slider

1 2 3

Window without slider

1 2 3

*

Std. error

Sig

95% Confidence interval Lower bound

Upper bound

0.000

−17.3609

−12.7724

0.000

−113 1609

−108.5724

0.96216

0.000

12.7724

17.3609

0.96216

0.000

−98 0942

−93.5058

110.86667*

0.96216

0.000

108.5724

113.1609

2

95.80000*

0.96216

0.000

93.5058

98.0942

2

7.90000*

2.46888

0.005

2.0130

13.7870

3

−59.73333*

2.46888

0.000

−65 6203

−53.8453

1

−7.90000*

2.46888

0.005

−13.7870

−2.0130

3

−57.63333*

2.46888

0.000

−73.5203

−61.7453

1

59.73333*

2.46888

0.000

53.8463

65.6203

2

57.63333*

2.46888

0.000

61.7463

73.5203

2

−39.16667*

1.91210

0.000

−43.7260

−34.6073

3

−281.66667*

1.91210

0.000

−2S6.2260

−277.1073

1

39.16667*

1.91210

0.000

34 6073

43.7250

3

−242.50000*

1.91210

0.000

−247.0594

−237.9405

1

281.66667*

1.91210

0.000

277.1073

286.2250

2

242.50000*

1.91210

0.000

237.9406

247.0594

2

21.03333*

2.82793

0.000

14.2902

27.7755

3

−103.76667*

2.82793

0.000

−110.5098

−97.0235

1

−21.03333*

2.82793

0.000

−27.7765

−14.2902

3

−124.80000*

2.82793

0.000

−131 5431

−118.0559

1

103.76667*

2.82793

0.000

97 0235

110.5098

2

124.80000*

2.82793

0.000

118 0569

131.5431

The mean difference is significant at the 0.05 level

mean time taken by age group (7–15 years) was − 15.067 s which is less than the time taken by age group (20–45 years) and significant at p-value = (0.00). This is probably due to the fact that one sample of 20–45 years group is predominated by people of age group 15–20 years. In 15–20 age group, all were athletic and agile which made easy and faster egress. For egress, in case of ‘door with slider’, mean time taken by age group (60– 75 years) was − 110.86 s greater than the time taken by age group (7–15 years) and significant at p-value = (0.00). This happens due to emergency door being located at a height from the ground which increases the time for egress. For egress, in case of ‘door with slider’, the mean timings are similar to the above two cases.

210

V. K. Singh et al.

For egress, in case of ‘door without slider’, mean time taken by age group (7– 15 years) was 7.9 s greater than the time taken by age group (20–45 years) and significant at p-value = (0.00). This is probably due to the emergency door pockets not being provided at appropriate height on the bus body which ultimately increases the time for egress. For egress, in case of ‘door without slider’, mean time taken by age group (60– 75 years) was − 59.7 s lesser than the time taken by age group (20–15 years) and significant at p-value = (0.00). This might be due to the fact that one sample of 60–75 years group is predominated by the people of age group 7–15 years, and in this age group, people were sporty and swift which made faster egress. For egress, in case of ‘door without slider’, the mean timings are similar to the above two cases. For egress, in case of ‘window with slider’, mean time taken by age group (7– 15 years) was − 39.16 s lesser than the time taken by age group (20–45 years) and significant at p-value = (0.00). 7–15 years group people can escape easily from window compared to 20–45 years group people due to the physique of people in this group. For egress, in case of ‘window with slider’, mean time taken by age group (7– 15 years) was − 281.67 s greater than the time taken by age group (60–75 years) and significant at p-value = (0.00). This happens due to the greater height of windows from the ground surface which creates panic among the passengers, and they do not find a quick escape. For egress, in case of ‘window with slider’, the mean timings are similar to the above two cases. For egress, in case of ‘window without slider’, mean time taken by age group (7–15 years) was 21.03 s greater than the time taken by age group (20–45 years) and significant at p-value = (0.00). This is possible due to the poor design of emergency window, its height and operating mechanism lead to increase in the time for egress, and in this case passenger tends to jump from height which can also cause the injury and slow rate of egress. For egress, in case of ‘window without slider’, mean time taken by age group (7–15 years) was − 103.67 s lesser than the time taken by age group (60–75 years) and significant at p-value = (0.00). This occurred due to one sample of 60–75 years group is predominated by the people of age group 7–15 years. In 15–20 age group, all were athletic and agile which made easy and faster egress. For egress in case of ‘window without slider’, the mean timings are similar to the above two cases. Overall, the empirical results are in line with our proposals. The introduction of slider-type arrangement of emergency exit (window/door) is instrumental in reducing egress time of passengers. The difference obtained (Table 3) in time was significantly higher in terms of the quick egress with higher flow rate of passengers, and additionally cost saving is attained by removing the step pocket from bus body by introducing the slider-type arrangement in emergency exit in buses.

Enhancement for Easy Egress Through Emergency Gate of Bus

211

6 Conclusion This study’s goal was to better understand the introduction of slider-type (hinged from bottom position) emergency exit and its advantages over the one without slider-type (vertically hinge) emergency exit in passenger transport vehicles. This system greatly reduces the time available for safe passenger escape in case of emergency. Further, this system is more suitable for children and elderly passengers as compared to the conventional one. The newly introduced system can be implemented in either side or vehicle rear side as per customer and regulatory requirements. In case of fully built vehicle (FBV), the companies can implement the new system while manufacturing and in case of private bus body manufacturers, original equipment manufacturer. OEM must release the guidelines for new system of emergency exit. As this system is currently not available with any of the bus body manufacturers, it can be implemented only after a revision of AIS052 (Automotive Industry Standards), i.e. the bus body manufacturing guidelines and should be made mandatory. The ANOVA test found there was significant reduction in egress time during a panic situation due to the presence of slider-type arrangement in emergency exit in buses. For implementing new system, improvements in current gangway dimensions are also required to allow free movement of passengers. This may include clear marking of emergency exits, fluorescent strip installation on vehicle floor and emergency exit opening provision to be provided at multiple location through the vehicle.

7 Future Scope This study explored emergency exit slider-type arrangement at existing positions. For better egress during the emergency, rear side emergency door can be explored (Fig. 10). Fig. 10 Rear side emergency door

212

V. K. Singh et al.

• Rear side emergency exit is an option for the further enhancement of product in terms of safety, and this will add on passage easy egress during emergency because of its inline position with bus as compared to sideways position. • It will help out during worst case scenario like rollover, front head on collision and even in case of fire. • This feature can be added in certification regularity bodies through amendment in AIS052 and central motor vehicle rules (CMVR). This will be mandatory for all the similar product manufacturing companies for uniformity in this safety of passenger. • Both side emergency windows/doors can be provided for rollover of the bus on either side. Acknowledgements This paper has been prepared to present the findings carried out in scope and role of bus emergency door/window with bottom side hinged points for the newly introduced slider-type emergency door/window for reducing escape time during the emergency. The authors sincerely thank Mr. Pandurang Gore for design analysis support and Dr. Faisal Ahsan, Mr. Yasir Abbas, Mr. Ajay Singh and Mr. Rajit Ram Singh for their valuable suggestions and feedback for the time study of in emergency egress. The authors extend their regards towards publication Chair Prof. Faiz Hamid and Prof. R. N. Sengupta for his valuable comments and suggestions.

References Ardelean, F. A. (2017). Case study using analysis of variance to determine groups’ variations. In MATEC Web of Conferences (Vol. 126, p. 04008). EDP Sciences. Bentama, A., Khatory, A., & Millot, M. (2017, April). Spatial analysis of bus accidents in France. In 2017 International Colloquium on Logistics and Supply Chain Management (LOGISTIQUA) (pp. 124–128). IEEE. Chafekar, A., Sivasubrahmaniyan, A., & Faustino, V. (2013). Evolution of Bus Design in India (No. 2013–01–2764). SAE Technical Paper. Hofmann, A., & Dülsen, S. (2012). Fire safety performance of buses. Fires in Vehicles-FIVE, 2012, 147. Johansson, P., & Axelsson, J. (2006). Fire safety in buses WP2 report: Fire safety review of interior materials in buses. Maputi, E. S. (2010). Conceptual design of a bus emergency exit ramp. Emergency, 120(150secs), 240secs. Mohan, D., Tsimhoni, O., Sivak, M., & Flannagan, M. J. (2009). Road safety in India: challenges and opportunities. Ravishankar, S., Vaidya, N. V., & Gogate, V. S. (2012). Accident research. Pilot study on an Indian express highway. Robert, K. (1951). U.S. Patent No. 2,565,856. Washington, DC: U.S. Patent and Trademark Office. Sapper, D. B., & Reep, A. (2006). Bus Accident Reporting, Tracking and Analysis System (No. BD 549–15). Florida. Dept. of Transportation. Singh, S. K., & Misra, A. (2004). Road accident analysis: A case study of Patna City. Urban Transport Journal, 2(2), 60–75. Young, T., Blustein, J., Finn, L., & Palta, M. (1997). Sleep-disordered breathing and motor vehicle accidents in a population-based sample of employed adults. Sleep, 20(8), 608–613.