Emerging Frontiers in Operations and Supply Chain Management: Theory and Applications (Asset Analytics) 9811627738, 9789811627736

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

B. Vipin C. Rajendran Ganesh Janakiraman Deepu Philip   Editors

Emerging Frontiers in Operations and Supply Chain Management Theory and Applications

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

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B. Vipin · C. Rajendran · Ganesh Janakiraman · Deepu Philip Editors

Emerging Frontiers in Operations and Supply Chain Management Theory and Applications

Editors B. Vipin Department of Industrial Management and Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India Ganesh Janakiraman Naveen Jindal School of Management The University of Texas at Dallas Richardson, TX, USA

C. Rajendran Department of Management Studies Indian Institute of Technology Madras Chennai, Tamil Nadu, India Deepu Philip Department of Industrial Management and Engineering Indian Institute of Technology Kanpur Kanpur, Uttar Pradesh, India

ISSN 2522-5162 ISSN 2522-5170 (electronic) Asset Analytics ISBN 978-981-16-2773-6 ISBN 978-981-16-2774-3 (eBook) https://doi.org/10.1007/978-981-16-2774-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 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

Preface

In the era of advanced technological capabilities and sophisticated tools, managing the transformation process in the manufacturing and service industries is increasingly challenging due to the presence of uncertainty. Operations and supply chain management play a critical role in achieving a better match of supply with demand in volatile and uncertain conditions. With high volumes of data generated, data analytics, operations research, game theory, and other advanced optimization tools and techniques, the opportunity to improve the performance of operations and supply chain management is huge. A challenging issue in managing a supply chain in the present world is to manage the uncertainty associated with globalization. For example, the disruptions of low likelihood and high impact such as COVID-19 reinforce the need to advance the body of knowledge in the field of operations and supply chain management to achieve resilience. This edited book covers various quantitative models on operations and supply chain management such as inventory optimization, machine learning-operations research integrated model for in health care operations, game-theoretic analysis of review strategies in truthful information sharing, design of contracts in supply chains, supply chain optimization, and shop floor scheduling. In addition to the quantitative models, several innovative heuristics are proposed for different problems. This book explores qualitative models on improving the performance of small and medium enterprises and petroleum industries and a simulation model for staff allocation in the information technology industry. Finally, this book provides comprehensive review articles on vaccine supply chains and behavioral operations management. The contributions of the chapters for this book are the submissions from the Society of Operations Management Conference (SOM XXIII) held at the Indian Institute of Technology Kanpur during December 19–21, 2019, and the invited research articles. The chapters are contributed by authors from India, United States, Germany, Singapore, and United Arab Emirates. This book has thirteen chapters allocated to three broad domains. In the first part of the book, quantitative models on various operations and supply chain management are presented. Chapter “A Comparative Study on Classical and New Hybrid Continuous-Review Inventory Ordering Policies in a Supply Chain Using Mathematical Models” presents new hybrid inventory policies by combining the traditional periodic v

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Preface

and continuous review policies under stochastic demand and compares the performance of the developed policies with that of the traditional inventory policies. Chapter “Truthful Information Sharing in a Multiretailer Supply Chain: Role of Review Strategies” presents review strategies using repeated games to achieve truthful information sharing in a multi-retailer supply chain setting. In chapter “Mathematical Models and Heuristics for an Inventory Routing Problem Without Split Deliveries”, mathematical models and heuristics are proposed to address the inventory routing problem without split deliveries. Modeling of the supply chain under a price-sensitive demand subjected to service level constraints in the presence of discrete transportation lead time is given in chapter “Modelling a Supply Chain with Price-Dependent Stochastic Demand and Discrete Transportation Lead Time”. In chapter “Optimal and Heuristic Profit Sharing Using Sales Rebate Contract in a Multi-level Supply Chain”, the sales rebate contract is explored in a multi-level supply chain incorporating the fairness concern using heuristics based on genetic algorithm and simulated annealing. Chapter “A Deterministic Heuristic Algorithm to Minimize the Length of a Manufacturing Line in Transformation of Jobshops into Flowshops” focuses on the development of a deterministic heuristic algorithm for solving the problem of transforming jobshop into flowshop to minimize a manufacturing line length. The second part of the book explores data analytics, qualitative, and simulation approaches to address different operations and supply chain issues. Chapter “A Classification Algorithm Based on Linear Regression and Linear Programming for Predicting the Breast Cancer” presents a classification algorithm based on linear regression and linear programming to predict breast cancer using the data from the Wisconsin Diagnostic Breast Cancer data set. Evaluation of sustainable supply chain using artificial neural network is discussed in chapter “Predicting Sustainable Supply Chain Performance Based on GRI Metrics and Multilayer Perceptron Neural Networks”. In chapter “Green Supply Chain Management in the Indian Petroleum Industry Using AHP-VIKOR Approaches”, a multi-criteria decisionmaking approach is used to prioritize the green practices in the petroleum industry. A simulation-based approach is employed to investigate the staff allocation problem in the information technology industry in chapter “Staff Allocation for Projects in IT Service Industries: A Simulation-Based Approach”. The third part of the book is on the review of the literature and conceptual framework development on issues related to operations and supply chain management. Chapter “Supply Chain Management Practice and Supply Chain Performance: A Conceptual Systematization of Terminology and Proposed Framework for SCMP Implementation in Indian Manufacturing SMEs” proposes a conceptual framework for supply chain management practice implementation in the Indian manufacturing industry. Chapter “Behavioural Operations Management: Trends and Insights” presents a review of literature on behavioral operations management, analyzes the trends, and provides insights from the literature. Chapter “Investigating the Vaccine Supply Chain: A Review” investigates the literature on vaccine supply chain management with a special focus on the Indian context primarily from the supply chain coordination and supply chain network design perspectives.

Preface

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We gratefully acknowledge the merit of all submissions to the book which could not successfully go through the review process. We sincerely thank the reviewers G. Anand (IIM Kozhikode), Deepak Eldho Babu (MACE Kothamangalam), Mukesh Kumar Barua (IIT Roorkee), S. G. Deshmukh (IIT Delhi), M. S. Gajanand (IIM Tiruchirappalli), P. Kalpana (IIITDM Kancheepuram), Arshinder Kaur (IIT Madras), T. V. Krishna Mohan (IIM Bodh Gaya), C. Rajendran (IIT Madras), R. Rajesh (IIIT Gwalior), A. Ramesh (IIT Roorkee), Sakthivel Madankumar (Trimble Information Technologies, Chennai), Ramakrishnan Ramanathan (University of Bedfordshire), Usha Ramanathan (Nottingham Trent University), R. Sridharan (NIT Calicut), B. Srirangacharyulu (IIM Visakhapatnam), Godwin Tennyson (IIM Tiruchirappalli), G. Thangamani (IIM Kozhikode), S. Venkataramanaiah (IIM Lucknow), B. Vipin (IIT Kanpur), and S. Yamini (NIT Tiruchirappalli) who extended their support in upholding the quality of this book by providing insightful review comments. We thank Raghu Nandan Sengupta, IIT Kanpur, and Nupoor Singh, Springer for the initiative, help, and support extended in materializing this edited book. We hope that this edited book will advance the body of knowledge in the field of operations and supply chain management and serve as a valuable resource for academicians and practitioners. We believe that the book will be a good reference for academicians, practitioners, and students who are keen to learn the recent advancements in operations and supply chain management. Kanpur, India Chennai, India Richardson, USA Kanpur, India

B. Vipin C. Rajendran Ganesh Janakiraman Deepu Philip

Contents

Quantitative Models on Operations and Supply Chain Management A Comparative Study on Classical and New Hybrid Continuous-Review Inventory Ordering Policies in a Supply Chain Using Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bharathan Santhanam, P. V. R. Sethupathi, Chandrasekharan Rajendran, and Hans Ziegler

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Truthful Information Sharing in a Multiretailer Supply Chain: Role of Review Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Kavitha, R. K. Amit, and B. Vipin

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Mathematical Models and Heuristics for an Inventory Routing Problem Without Split Deliveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. R. Preethi, Chandrasekharan Rajendran, and S. Viswanathan

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Modeling a Supply Chain with Price-Dependent Stochastic Demand and Discrete Transportation Lead Time . . . . . . . . . . . . . . . . . . . . . Susheel Yadav, Anil K. Agrawal, and Manu K. Vora

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Optimal and Heuristic Profit Sharing Using Sales Rebate Contract in a Multi-level Supply Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Thomson, Raichel Elsa Thomas, K. Susithra, S. Shobana, Arshinder Kaur, and Chandrasekharan Rajendran

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A Deterministic Heuristic Algorithm to Minimize the Length of a Manufacturing Line in Transformation of Jobshops into Flowshops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 J. Krishnaraj

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Contents

Data Analytics, Qualitative, and Simulation Approaches to Operations and Supply Chain Issues A Classification Algorithm Based on Linear Regression and Linear Programming for Predicting the Breast Cancer . . . . . . . . . . . . . . . . . . . . . . 139 Sakthivel Madankumar Predicting Sustainable Supply Chain Performance Based on GRI Metrics and Multilayer Perceptron Neural Networks . . . . . . . . . . . . . . . . . 159 Devendra Singh, Krishnanand Lanka, and P. R. C. Gopal Green Supply Chain Management in the Indian Petroleum Industry Using AHP-VIKOR Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Sourabh Kumar and Mukesh Kumar Barua Staff Allocation for Projects in IT Service Industries: A Simulation-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 V. V. Rajarajan and M. S. Gajanand Literature Review and Conceptual Framework on Operations and Supply Chain Management Aspects Supply Chain Management Practice and Supply Chain Performance: A Conceptual Systematization of Terminology and Proposed Framework for SCMP Implementation in Indian Manufacturing SMEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Rohit Kumar and Manish Gupta Behavioural Operations Management: Trends and Insights . . . . . . . . . . . . 233 S. Yamini Investigating the Vaccine Supply Chain: A Review . . . . . . . . . . . . . . . . . . . . 251 Dheeraj Chandra and B. Vipin Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Editors and Contributors

About the Editors B. Vipin is an Assistant Professor at the Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur. He obtained his PhD from Indian Institute of Management Madras, Chennai, India. He has published his research findings in leading journals and he is a regular participant in leading operations management conferences. His areas of research interest are decision theory, behavioral operations management, sustainable operations, and healthcare operations management. C. Rajendran is a Professor and RAGS Family Foundation Institute Chair at the Department of Management Studies, Indian Institute of Technology Madras, Chennai, India. He obtained his PhD in industrial engineering and management from the same institute. He has been a recipient of Alexander von Humboldt Research Fellowship more than 10 times since 1996 and has published in top journals of operations management. He has been awarded Dr. rer. pol. h. c. (Honorary Doctorate) from the University of Passau in 2017, and elected Fellow of Indian National Academy of Engineering (FNAE) in 2018. His areas of research interest include scheduling in manufacturing and service systems, logistics management, inventory optimization in deterministic systems, economic lot sizing and capacitated lot sizing problems, heuristics & metaheuristics, and their applications in operations and supply chain management, total quality management and service quality. Ganesh Janakiraman is Ashbel Smith Professor of Operations Management at the Naveen Jindal School of Management, The University of Texas at Dallas, USA. He obtained his PhD from Cornell University, Ithaca, USA. He has numerous articles in high ranking journals of operations management and has also attended several conferences and seminars as an invited speaker. His areas of interest include stochastic inventory theory, and, more broadly, decision making under uncertainty.

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Editors and Contributors

Deepu Philip is a Professor at the Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur. He obtained his PhD from MSU Bozeman, USA. He is currently working on several research projects funded by national and international agencies and has papers published in well-established research journals. His areas of research interest include production and operations management, systems engineering and simulation, and local search and optimization.

Contributors Anil K. Agrawal Department of Mechanical Engineering, Indian Institute of Technology (BHU), Varanasi, India R. K. Amit Department of Management Studies, Indian Institute of Technology Madras, Chennai, India Mukesh Kumar Barua Indian Institute of Technology Roorkee, Haridwar, India Dheeraj Chandra Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India M. S. Gajanand Faculty of Operations Management and Decision Sciences, Indian Institute of Management, Tiruchirappalli, Tamil Nadu, India P. R. C. Gopal Department of School of Management, National Institute of Technology Warangal, Warangal, Telangana, India Manish Gupta Department of Mechanical Engineering, Motilal Nehru National, Institute of Technology Allahabad, Prayagraj, India Arshinder Kaur Department of Management Studies, Indian Institute of Technology Madras, Chennai, India C. Kavitha Department of Management Studies, Indian Institute of Technology Madras, Chennai, India J. Krishnaraj Department of Mechanical Engineering, MLR Institute of Technology, Hyderabad, Telangana, India Rohit Kumar Department of Mechanical Engineering, Motilal Nehru National, Institute of Technology Allahabad, Prayagraj, India Sourabh Kumar Indian Institute of Technology Roorkee, Haridwar, India Krishnanand Lanka Mechanical Engineering Department, National Institute of Technology Warangal, Warangal, Telangana, India Sakthivel Madankumar Trimble Information Technologies India Private Limited, Chennai, India K. R. Preethi Indian Institute of Technology, Madras, India

Editors and Contributors

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V. V. Rajarajan Manager - Business Analytics, ZoomRx Healthcare Solutions Pvt Ltd, Chennai, Tamil Nadu, India Chandrasekharan Rajendran Department of Management Studies, Indian Institute of Technology Madras, Chennai, India Bharathan Santhanam Department of Management Studies, Indian Institute of Technology Madras, Chennai, India P. V. R. Sethupathi Department of Management Studies, Indian Institute of Technology Madras, Chennai, India S. Shobana Department of Industrial Engineering, College of Engineering Guindy, Anna University, Chennai, India Devendra Singh Mechanical Engineering Department, National Institute of Technology Warangal, Warangal, Telangana, India K. Susithra Department of Industrial Engineering, College of Engineering Guindy, Anna University, Chennai, India Raichel Elsa Thomas Department of Mechanical Engineering, Indian Institute of Technology BHU, Varanasi, India I. Thomson Reckitt Benckiser Arabia, Dubai, UAE B. Vipin Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, India S. Viswanathan Nanyang Technological University, Nanyang Ave, Singapore Manu K. Vora Business Excellence Inc, Naperville, IL, USA Susheel Yadav Department of Mechanical Engineering, Indian Institute of Technology (BHU), Varanasi, India S. Yamini Faculty of Operations and Analytics, National Institute of Technology Tiruchirappalli, Tiruchirappalli, Tamil Nadu, India Hans Ziegler School of Business, Economics and Information Systems, University of Passau, Passau, Germany

Quantitative Models on Operations and Supply Chain Management

A Comparative Study on Classical and New Hybrid Continuous-Review Inventory Ordering Policies in a Supply Chain Using Mathematical Models Bharathan Santhanam, P. V. R. Sethupathi, Chandrasekharan Rajendran, and Hans Ziegler Abstract One of the most challenging tasks in managing a supply chain is to select an efficient inventory ordering policy, i.e., deciding on ‘when to order’ and ‘how much to order’ while minimizing the total cost and maximizing the service level. Classical inventory policies based on periodic and continuous-review are generally implemented in practice. In our work, we attempt to combine the characteristics of both periodic-review order-up-to (R, S) policy and continuous-review (s, Q)/(s, S) policies to propose two new hybrid ordering policies, namely continuous-review (s, Q*) and continuous-review (s, OQ∗ ) hybrid policies. We further develop mixed integer linear programming (MILP) models to obtain the optimal policy parameters by considering a single-stage and two-stage supply chain with discrete, deterministic demand over a finite planning horizon. The proposed policies are benchmarked against the existing order policies, namely (R, S), (s, Q), (s, S) and hybrid (R, S, Qmin ) policies. From results, we observe that the performance of the proposed hybrid policies outperforms the existing classical and hybrid policies in terms of total supply chain cost. Keywords Supply chain · Continuous-review · Hybrid inventory ordering policy · Finite time horizon · Mathematical models · Comparative study

B. Santhanam (B) · P. V. R. Sethupathi · C. Rajendran Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India 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. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_1

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1 Introduction The worldwide out-of-stock rate is 7–8%. About 45% of the customers, who experience a stock-out would buy a different product, 31% of the customers buy elsewhere, 9% do not buy the product at all, and only 15% the customers would wait for the product (Gruen et al., 2002). The comprised inventory can cost 20–40% of the product price (Manatkar et al., 2016). Efficient management of inventory in a supply chain improves the service offered to the consumers (Lee and Billington, 1992). Deciding on ‘when to order’ and ‘how much to order’ is critical for minimizing the total cost and maximizing the service level. One of the problem areas in operational decisionmaking is inventory management (Ganeshan et al., 1999), and it depends on the nature of demand (deterministic or stochastic), kind of decision control (decentralized system or centralized) and inventory control policy. Inventory control policies are classified as time and event-based. In time-based models, the inventory position is reviewed in a fixed interval of time R, and the inventory position is raised to S. This policy is referred to as (R, S) policy. In event-based models, the inventory position is continuously reviewed, and whenever the inventory position equals or falls below a predetermined reorder level, s either a fixed batch of Q units is ordered or the inventory position is raised to S. The former is referred to as (s, Q) policy and the latter as (s, S) policy. The existing research on inventory systems is quite extensive. Hence, we discuss only the most closely relevant papers related to the use of mathematical models for inventory order policies and the hybrid order policies. Daniel and Rajendran (2006) considered a supply chain in series with backorders allowed at each member and proposed genetic algorithms (GA) to find the base-stock for each member. Later, Sethupathi and Rajendran (2010) extended the work to determine the optimal review period and base-stock for a supply chain controlled by periodic-review base-stock policy. Further, Sethupathi et al. (2014) proposed mathematical models to evaluate the performance of periodic-review order-up-to (R,S) policy and (s, S) policy. They found (s, S) policy to perform superior when compared to (R, S) policy in most settings for a supply chain in series with backorders. Movahed and Zhang (2015) proposed a MILP model with uncertain lead time and demand to determine the optimal order parameters for (s, S) policy. The (R, s, S) policy is one of first hybrid policy proposed in literature. In this policy, the inventory position is reviewed in a fixed interval of time R, and the inventory position is raised to S if the inventory position equals or falls below a predetermined reorder level s. Popp (1965) extended the (R, s, S) policy in which, if the demand is small, then the retailer orders using (s, S) policy and if demand is large with a lower probability, then demand is directly delivered from the manufacturer. This kind of hybrid ordering is ideal for scenarios when holding costs are very high when compared to ordering cost and lead time is zero. Zhou et al. (2007) proposed a new policy called (R, s, t, Qmin ) policy, in which the inventory position is reviewed every R periods, and if it is lower than or equal to the reorder level s, an order is placed to raise the inventory position to s + Qmin, and when the inventory position is above s but lower than threshold t, then Qmin is

A Comparative Study on Classical and New Hybrid … Table 1 Classification of inventory ordering policy papers from literature

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Author

Parameters

Classification

Popp (1965)

(R, s, S)

Time

Zhou et al. (2007)

(R, s, t, Qmin ) Time

Sethupathi and Rajendran (2010) (R, S)

Time

Kiesmüller et al. (2011)

(R, S, Qmin )

Time

Sethupathi et al. (2014)

(s, S)

Event

ordered. Kiesmüller et al. (2011) proposed (R, S, Qmin ) policy, in which the inventory position is reviewed every R period, and if it is less than S, an order is placed to raise the inventory to S. However, if this order is smaller than Qmin , the order quantity is set to Qmin . The policies proposed in literature can be summarized as given in Table 1. From literature review, we observe that the hybrid policies proposed in the literature are time-based, i.e., the inventory position is reviewed in a fixed interval of time. There seems to be no event-based hybrid order policy. Also, to the best of our knowledge, no study has been undertaken on the relative assessment and comparison of various time and event-based inventory order policies. The salient features of our work are as follows. • New hybrid inventory order policies: Two new hybrid inventory ordering policies that combine the benefits from continuous-review policies and periodic-review policies are proposed. When the inventory is continuously reviewed, it is likely to lead to less shortage cost and ordering a fixed order quantity. Q possibly minimizes the long-term holding and ordering costs when demand is uniform. The benefit from periodic-review order-up-to (R, S) policy or continuous-review (s, S) policy is to raise the inventory position to S when there is an order placement, resulting in lower shortage costs, especially when there is high fluctuation in demand. • Development of mathematical models: Mathematical models are developed for the proposed and the existing policies to determine the optimal policy parameters by considering a single-stage and two-stage supply chain operating with discrete, deterministic demand over a finite planning horizon. • Supply chain with order costs: Optimal control of inventory in supply chain is difficult when order costs are applicable, and the conventional relationship between the order policies in the case of a single installation might not be applicable for the case of a supply chain (refer to Wang, 2011, for details). In addition, Wang found that the performance of stock-based (s, Q) policy is superior than the timebased (R, S) policy for a single installation and expected this dominance to carry over to supply chain. This observation was more of a hypothesis, and there is no experimental study to prove it. This is experimentally investigated in this work. • Comparative evaluation of inventory order policies: For the first time in literature, a relative comparison of six inventory control policies, namely the proposed continuous-review (s, Q* ) hybrid policy and continuous-review (s,OQ* ) hybrid policy, and the existing policies, namely (R, S), (s, Q), (s, S) and hybrid (R, S, Qmin )

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policies, are presented by considering a two-stage supply chain operating with backorders and discrete, deterministic demand. Kiesmüller et al. (2011) observed that the (R, S, Qmin ) policy performs better than hybrid (R, s, S) policy, and hence, it is not considered in this study. The remainder of the paper is organized as follows. Section 2 describes the assumptions, notations and the sequence of events with an example. Section 3 discusses the proposed mathematical models. Section 4 describes the data sets and compares the performance of proposed versus existing inventory policies. Section 5 presents limitations and future work. Finally, Sect. 6 discusses the summary of the work.

2 Assumptions, Notations and Sequence of Events This work considers a two-stage supply chain with a distributor (DC) and a retailer (R). Both members operate with their respective local/installation holding cost per unit per day, backorder cost per unit per day and an order placement cost per order. The cost of obsolescence and damage are considered in local/installation holding cost per unit per day, and costs of loading/unloading are considered as part of ordering cost per order. The processing lead time is integrated with transportation lead time to arrive at a replenishment lead time for each member. The retailer faces the discrete customer demand which is known a priori. The excess demand is assumed to be backordered. The unit of time is one day and is assumed to be discrete. The notations used in the mathematical model are as below: Notation TSC T N i T ROLi

SUMDEMi,t SUMORDi,t−1 Si OQi hi Oi bi LTi

Description total supply chain cost planning horizon in days over which TSC is optimized number of members in the supply chain index for members in the supply chain current day reorder level or reorder point for member i /*referred to as ‘s’ in continuous-review (s, Q), continuous-review (s, Q*) and continuous-review (s, OQ∗ ) policies. However, for better clarity, s is represented as ROL and order-up-to-level as S*/ total demand received by member i up to t sum of orders triggered at member i up to t−1 order-up-to-level at member i notional fixed order quantity for member i cost of holding a product per day for member i ordering cost per order for memberi cost incurred due to product shortage for member i replenishment lead time for member i

A Comparative Study on Classical and New Hybrid …

OIi,t Ii,t Bi,t Di,i+1,t ∗ Q i,t ∗ OQi,t

θi,t ∗ θi,t

QSi,i−1,t QS N +1,N ,t M

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on-order inventory for member i on day t on-hand inventory for member i on day t backorder cost per product per day at member i on day t demand by member i to upstream member (i + 1) at t variable order quantity which is defined as the maximum of   (SUMDEMi,t − SUMORDi,t−1 ); OQi for member i on day t variable order quantity which is defined as maximum of orderup-to level minus the inventory position and notional fixed order quantity for member i on day t a binary variable to decide on the order quantity if there is an order triggered at member i on day t; else it takes the value 0 a binary variable with a value 1, if there is an order triggered at member i on day t; else it takes the value 0 quantity shipped from member i to (i−1) on day t quantity of products that is shipped to member N on day t a large positive integer value

The order of events on each day for each member is as follows: (1)

(2)

(3)

(4)

The quantity shipped (QSi+1,i,t−L T i ) from the immediate upstream member is received if the quantity is to arrive by considering the replenishment lead time for that member. The distributor receives demand (Di−1,i,t ) from the retailer, and the retailer receives the customer demand. Demand is fulfilled based on the available on-hand inventory (Ii,t ) and the order quantity received from the upstream member (QSi+1,i,t−L T i ). Any excess demand is backordered (Bi,t ). The onhand inventory (Ii,t ) or backorder (Bi,t ) information is updated. The inventory position of the member is defined as the sum of pipeline inventory and the inventory level. Pipeline inventory is the difference between previous day’s on-order inventory (OIi,t−1 ) and quantity shipped (QSi+1,i,t−L T i ) from upstream member. Inventory level is calculated based on the difference between the on-hand inventory (Ii,t ) and the backorder (Bi,t ). If the inventory position is equal to or less than the reorder level (ROLi ) at the member, an order is triggered to the upstream member. No order placement takes place otherwise. The order quantity depends on the policy under study. Finally, the quantity shipped (QSi,i−1,t ) between members and the on-order information (OIi,t ) are updated.

The behavior of different continuous-review policies, namely (s, Q), (s, S), (s, Q ∗ ) and (s, OQ∗ ) is explained with an example. In all policies, the inventory position is continuously reviewed, and when it equals or falls below the reorder level, an order is placed. The order quantity depends on the policy. Here, we consider a single-stage system with randomly distributed demand between 20 and 60 and assume the fixed order quantity and order-up-to level as 40 and reorder level as 20. The lead time is fixed as 2 days. Holding, backorder and ordering costs are set as 8 per unit per day, 16 per unit per day and 640 per order, respectively.

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From Table 2, we observe that whenever there is unexpected rise or dip in demand, the order quantity is based on the cumulative demand since the last order or the order-up-to level minus the inventory position in the continuous-review (s, Q ∗ ) and (s, OQ∗ ) hybrid policies, respectively. If there is dip in demand or when the demand is uniform, then order quantity is based on the fixed order quantity. On comparing the holding cost, it is the least for continuous-review (s, Q) policy. This is not unexpected considering the fixed order quantity. The ordering cost is the least for continuousreview (s, S) policy while shortage cost is the least for the proposed continuous-review (s, OQ∗ ) policy. In terms of the total cost, (s, OQ∗ ) performs the best followed by (s, Q ∗ ), (s, S) and (s, Q) policies.

3 Mathematical Models Mathematical models for the proposed continuous-review (s, Q ∗ ) hybrid policy, continuous-review (s, O Q ∗ ) hybrid policy and existing continuous-review (s, Q) policy are given in Sects. 3.1, 3.2 and 3.3, respectively. The mathematical models are generic and can be run for a supply chain with any number of members.

3.1 Mathematical model for the Proposed Continuous-Review (s, Q ∗ ) Hybrid Policy In the proposed continuous-review (s, Q ∗ ) hybrid policy, when the inventory position for a member is equal to or less than the reorder level s, a replenishment order equal to the maximum of the notional fixed order quantity and the difference between total demand received by the member up to current day t and total order quantity up to day (t-1) are placed to the next upstream member. The order quantity is referred as the notional order quantity because it is fixed (time invariant) and the actual order placement reckons with this order quantity as well as with the total demand received by the member up to the current day t and the total order quantity up to day t-1. When there is an abrupt rise or dip in demand, the order quantity will be the total demand received by the member up to the current day t minus the total order quantity up to day t-1. However, when there is uniform demand with less variations, the order quantity will be equal to the notional fixed order quantity. This flexibility in order quantity aims to reduce the shortage cost when there is variation in demand, and the holding cost when there is uniform demand or dip in demand. Also, the notional fixed order quantity can be used to define the minimum order quantity (if any) that exists with the upstream member.

15

20

30

15

10

15

5

15

10

50

45

55

50

45

55

25

25

25

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

40

40

40

40

40

40

40

40

0

40

0

0

40

0

40

40

0

40

0

50

35

0

95

70

40

85

45

20

30

0

0

30

0

40

30

20

25

0

(s, S)

40

40

40

55

45

50

55

40

0

40

0

0

40

0

40

40

0

40

0

(s, Q*)

Total cost (holding + shortage + ordering costs)

Total

10

2

(s, Q)

Demand Order quantity

1

Day

40

0

70

60

45

90

45

40

40

0

0

0

40

0

50

40

0

40

0

2240

0

0

0

0

0

0

0

0

40

440

520

320

360

160

0

40

0

120

240

(s,OQ*) (s, Q)

2840

400

600

40

0

0

0

0

0

0

320

400

280

320

120

0

0

0

120

240

(s, S)

Holding cost

2280

40

0

0

0

0

0

0

0

40

440

520

320

360

160

0

40

0

120

240

(s, Q*)

3560

640

280

0

0

0

0

0

0

120

520

600

400

440

160

0

40

0

120

240

7280

640

880

1120

1360

1120

1040

880

0

0

0

0

0

0

0

160

0

80

0

0

(s,OQ*) (s, Q)

3920

0

0

0

640

400

1040

960

400

160

0

0

0

0

0

80

160

80

0

0

(s, S)

Shortage cost

4800

0

160

640

960

880

1040

880

0

0

0

0

0

0

0

160

0

80

0

0

(s, Q*)

3520

0

0

0

320

880

880

720

480

0

0

0

0

0

0

160

0

80

0

0

(s, OQ*)

17,840

8320

640

640

640

640

640

640

640

640

0

640

0

0

640

0

640

640

0

640

0

(s, Q)

8320

640

640

640

640

640

640

640

640

0

640

0

0

640

0

640

640

0

640

0

(s, Q*)

7680

640

0

640

640

640

640

640

640

640

0

0

0

640

0

640

640

0

640

0

(s, OQ*)

15,720 15,400 14,760

8960

640

640

0

640

640

640

640

640

640

640

0

0

640

0

640

640

640

640

0

(s, S)

Ordering cost

Table 2 Order policy behavior (in terms of order quantity) and associated costs for (s, Q), (s,S), (s, Q*) and (s, OQ*) policy for a sample demand stream

A Comparative Study on Classical and New Hybrid … 9

10

B. Santhanam et al.

Objective function: The objective is to minimize the overall cost that includes the holding, backorder/shortage and ordering costs for all members over T days. Minimize TSC =

T  N  

∗ h i Ii,t + bi Bi,t + Oi θi,t



(1)

t=1 i=1

Constraints: for t =1, 2,…T [ for i =1, 2,…N { The on-hand inventory or backorder for the current day is calculated using Eq. (2) based on the available on-hand inventory at member i on previous day, the member’s backorder as on the previous day, the demand received from the downstream member (i-1) and the order quantity received at t considering the replenishment lead time of member i. Retailer receives the customer demand. Ii,t − Bi,t = Ii,t−1 + QSi+1,i,t−LTi − Bi,t−1 − Di−1,i,t

(2)

If adequate on-hand inventory is not available with member i to meet the accumulated demand, then the excess demand is backordered. The assumption is bi + h i ≥ h i−1 , i ≥ 2, and if that does not hold, then Ii,t and Bi,t will coexist. To prevent this, Expression (3) needs to be added as an additional constraint.   ∗∗ ∗∗ , and Bi,t ≤ M 1 − θi,t Ii,t ≤ M × θi,t ∗∗ with θi,t ∈ {0, 1}, for ∀i ≤ N and ∀t ≤ T

(3)

Expression (4) updates the sum of demand received at member i till day t. SUMDEMi,t = SUMDEMi,t−1 + Di−1,i,t

(4)

Constraints (5) and (6) monitor the inventory position at member i and trigger an order if the inventory position is equal to or less than the reorder level at i. No order placement takes place otherwise.   ∗ Ii,t − Bi,t + OIi,t−1 − QSi+1,i,t−LTi ≤ ROLi + M 1 − θi,t

(5)

∗ Ii,t − Bi,t + OIi,t−1 − QSi+1,i,t−LTi ≥ ROLi + 1 − Mθi,t

(6)

A Comparative Study on Classical and New Hybrid …

11

Constraints (7)–(13) set the order quantity exactly equal to the maximum of notional order quantity OQi and the total demand minus the cumulative total order quantity SUMDEMi,t − SUMORDi,t−1 at the member i if there is an order place∗ ment at i. An order placement is indicated by θi,t when the inventory position is equal to or less than the ROLi . When the inventory position is greater than or equal to the ∗ becomes 0 indicating no order placement, and Di,i+1,t becomes equal to ROLi , θi,t zero. ∗ ≥ SUMDEMi,t − SUMORDi,t−1 Q i,t

(7)

∗ Q i,t ≥ OQi

(8)

∗ Q i,t ≤ M × θi,t + SUMDEMi,t − SUMORDi,t−1

(9)

  ∗ Q i,t ≤ M 1 − θi,t + OQi

(10)

  ∗ ∗ Di,i+1,t ≤ Q i,t + M 1 − θi,t

(11)

  ∗ ∗ Di,i+1,t ≥ Q i,t − M 1 − θi,t

(12)

∗ Di,i+1,t ≤ M × θi,t

(13)

Equation (14) calculates the quantity shipped between members based on the on-hand inventory and quantity received from the upstream member. QSi,i−1,t = Ii,t−1 − Ii,t + QSi+1,i,t−LTi

(14)

Equation (15) updates the on-order inventory at member i on day t. OIi,t = OIi,t−1 − QSi+1,i,t−LTi + Di,i+1,t

(15)

Equation (16) updates the sum of orders at i at the close of day t. SUMORDi,t = SUMORDi,t−1 + Di,i+1,t

(16)

} Equation (17) shows the quantity of finished products shipped immediately to the distributor. QS N +1,N ,t = D N ,N +1,t

(17)

12

B. Santhanam et al.

] The sequence of events repeats on each day for all members in the supply chain. Initial conditions: For all members, the initial inventory is initialized to be notional order quantity, the reorder level should be less than the notional order quantity, and all other variables related to inventory position are initialized to zero. Ii,0 = OQi , ROLi ≤ OQi , Bi,0 = 0, OIi,0 = 0, SUMDEMi,0 = 0, SUMORDi,0 = 0

(18)

QSi+1,i,t−LTi = 0, ∀t ≤ LTi and ∀i ≤ N

(19)

Binary variables and non-negativity constraints: ∗ ∈ {0, 1}, all other variables ≥ 0, ∀i ≤ N and ∀t ≤ T. θi,t , θi,t

(20)

3.2 Mathematical Model for the Proposed Continuous-Review (s, O Q * ) Policy In the proposed continuous-review (s, OQ∗ ) policy, whenever the inventory position equals or falls below the reorder level s, an order equal to the maximum of the notional order quantity (adapted from conventional (s, Q) policy) and the order-upto-level minus the inventory position (adapted from (R, S) and (s, S) policies) is placed to the upstream member. By making the order quantity dynamic based on the inventory position, we aim to handle variation in demand. For example, if there is uniform demand with less variation, then the order quantity will be based on the notional fixed order quantity, but when there is abrupt rise or dip in demand, the order quantity will be based on the difference between the order-up-to-level and the inventory position. This flexibility in order quantity reduces the shortage cost when there is more variation in demand and the holding cost when there is less variation in demand. The notations and the objective functions are the same as used in Sect. 3.1. Constraints: for t =1, 2,…T [ for i =1, 2,…N {

A Comparative Study on Classical and New Hybrid …

13

In addition to the Constraints (2)–(6) and (13)–(17), and omitting others, mentioned in Sect. 3.1, the following constraints are added. Constraints (21)–(26) set the order quantity exactly equal to the maximum minus inventory posiof {notional   fixed order quantity (OQi ); order-up-to-level  tion Si − Ii,t − Bi,t + OIi,t−1 − QSi+1,i,t−LTi if there is an order placement at ∗ set to one when the inventory posimember i, (an order placement is indicated by θi,t tion at the current day t is equal to or less than ROLi ). Note that when the inventory position at the current day t is greater than ROLi , no order takes place, and hence Di,i+1,t becomes equal to zero. ∗ ≥ Si − OQi,t



Ii,t − Bi,t + OIi,t−1 − QSi+1,i,t−LTi



(21)

∗ OQi,t ≥ OQi

(22)

  ∗ OQi,t ≤ M × θi,t + Si − Ii,t − Bi,t + OIi,t−1 − QSi+1,i,t−LTi

(23)

  ∗ OQi,t ≤ M 1 − θi,t + OQi

(24)

∗ ∗ Di,i+1,t ≤ OQi,t + M(1 − θi,t )

(25)

∗ ∗ Di,i+1,t ≥ OQi,t − M(1 − θi,t )

(26)

} ] The sequence of events repeats on each day for all members in the supply chain. Initial conditions: For all members, the initial inventory is initialized to be the maximum of the notional fixed order quantity and the order-up-to-level, the reorder level should be less than the notional fixed order quantity, and all other variables related to inventory position are initialized to zero. Ii,0 ≥ OQi , Ii,0 ≥ Si , ROLi ≤ OQi , Bi,0 = 0∀i ≤ N ;

(27)

  Ii,0 ≤ Si + M × θi,0 , Ii,0 ≤ OQi + M 1 − θi,0 , ∀i ≤ N ;

(28)

QSi+1,i,t−L Ti = 0, ∀t ≤ LTi and ∀i ≤ N .

(29)

Binary variables and non-negativity constraints:

14

B. Santhanam et al. ∗ θi,t , θi,t ∈ {0, 1}, all other variables ≥ 0, ∀i ≤ N , all ∀t ≤ T.

(30)

3.3 Mathematical Model for Continuous-Review (s,Q) Policy The same notations and objective function are the same as used in Sect. 3.1. Constraints: for t =1, 2,…T [ for i =1, 2,…N { In addition to the Constraints (2)–(6), (13)–(17), and omitting others, mentioned in Sect. 3.1, the following constraints are added: Constraints (31) and (32) force the order quantity to the upstream member to be equal to the fixed order quantity if there is an order placement or else it is 0.   ∗ Di,i+1,t ≤ OQi + M 1 − θi,t

(31)

  ∗ Di,i+1,t ≥ OQi − M 1 − θi,t

(32)

} ] The sequence of events repeats on each day for all members in the supply chain. Initial conditions: For all members, the initial inventory on-hand is initialized to be the fixed order quantity, the reorder level should be less than the fixed order quantity, and all other variables related to inventory position are initialized to zero. Ii,0 = OQi , ROLi ≤ OQi , Bi,0 = 0, ∀i ≤ N ;

(33)

QSi+1,i,t−LTi = 0, ∀t ≤ LTi and ∀i ≤ N .

(34)

Binary variables and non-negativity constraints: ∗ ∈ {0, 1}, all other variables ≥ 0, ∀i ≤ N and ∀t ≤ T. θi,t

(35)

A Comparative Study on Classical and New Hybrid …

15

4 Experimental Evaluation Section 4.1 discusses the data sets and the experimental settings considered in this work. In Sect. 4.2, the performance evaluation of proposed and existing policies is presented.

4.1 Data Sets and Experimental Settings The proposed as well as the existing inventory policies are evaluated by considering 640 supply chain test problem instances (2 lead time settings × 2 scenarios × 16 cost settings × 10 demand streams). Two lead time settings, namely LT1 and LT2, with replenishment lead time of one and two days, respectively, for each member, are considered. In the scenario 1, a single-stage supply chain is considered where the retailer has unlimited availability of finished products from its upstream member. In the scenario 2, a two-stage supply chain is considered where the distributor has unlimited availability of finished products from its upstream member and follows the same ordering policy with cost of holding, shortage and ordering. The different cost settings considered in this work are reported in Table 3. Table 3 Supply chain setting with respect to supply chain members Supply chain settings

Member holding cost

Member shortage cost

Member ordering cost

DC

R

DC

R

DC

R

CS1

4

8

8

16

320

640

CS2

4

8

16

32

320

640

CS3

4

8

32

64

320

640

CS4

4

8

64

128

320

640

CS5

4

8

8

16

640

1280

CS6

4

8

16

32

640

1280

CS7

4

8

32

64

640

1280

CS8

4

8

64

128

640

1280

CS9

4

8

8

16

1280

2560

CS10

4

8

16

32

1280

2560

CS11

4

8

32

64

1280

2560

CS12

4

8

64

128

1280

2560

CS13

4

8

8

16

2560

5120

CS14

4

8

16

32

2560

5120

CS15

4

8

32

64

2560

5120

CS16

4

8

64

128

2560

5120

16

B. Santhanam et al.

Ten uniformly distributed customer demand streams are sampled between 20 and 60. The different cost settings are assumed as follows: shortage cost = C × holding cost where C = 2, 4, 8, 16 and ordering cost =K × E(D)× holding cost where E(D) is the expected retail demand with respect to the demand distribution and K = 2, 4, 8, 16. The rationale for this is that we relate the holding cost as the base for the ordering cost because we assume a supply chain with a very high service level (see Silver et al., 1998).

4.2 Comparison on the Performance of Proposed Hybrid Policies versus Existing Policies The objective of this study is to make a comparative evaluation of proposed continuous-review hybrid policies against continuous-review (s, Q) policy, continuous-review (s, S) policy, periodic-review order-up-to (R, S) policy, (R, s, S) policy and (R, S, Qmin ) policy. The (R, S) order policy is considered for comparison because it is generally used in real life, especially when products are ordered from the same supplier so that coordination of replenishments of related products is achieved (Silver et al., 1998). We adapt the mathematical model from Sethupathi and Rajendran, (2010) for the (R, S) policy and Sethupathi et al. (2014) for (s, S) policy and studied the same for a two-stage supply chain. For the (R, S, Qmin ) policy, the (R, S) model, with necessary modifications in equations related to order quantity is run with different review periods varying from 1 to 5 and the best optimal solution is considered. The mathematical models are run for 40 days for single- and two-stage supply chain using ILOG CPLEX solver for different supply chain settings. The relative percentage deviation for an order policy is calculated as described in Eq. (36):   TCpolicy − TCbest × 100, Relative percentage deviation = TCbest

(36)

where TCpolicy is the optimal total cost obtained by the respective policies for a particular lead time and cost settings, and TCbest is the best total cost obtained across all policies for a particular lead time and cost settings. The average relative percentage deviation obtained for each policy in terms of total supply chain cost is given in Table 4 for a single-stage supply chain and Table 5 for a two-stage supply chain. In Table 6, the total supply chain cost for a sample demand stream is given for a single-stage supply chain. From results, we see that the (s, OQ∗ ) performs the best in terms of total supply chain cost, followed by (s, Q ∗ ), (s, Q), (s, S), (R, S, Qmin ) and (R, S) policies in same order. (R, S, Qmin ) policy performs better than (R, S) policy and continuous-review policies outperform the existing hybrid (R, S, Qmin ) policy where the inventory control is time-based. (s, S) policy performs better than (R, S). From the results, we clearly see the superior performance of continuous-review policies

9.52

13.73

14.87

12.72

10.88

16.31

14.10

13.05

11.90

13.74

10.21

12.60

CS7

CS8

CS9

CS10

CS11

CS12

CS13

CS14

CS15

CS16

Avg

14.35

CS4

12.24

11.45

CS3

CS6

12.47

CS2

CS5

9.98

(R, S)

LT1

CS1

Supply chain setting

3.13

2.11

1.57

2.00

3.97

3.83

3.34

2.69

2.55

6.89

5.02

4.24

4.86

1.62

1.67

2.30

1.48

(s, S)

1.37

1.02

0.68

0.88

1.81

0.76

1.20

1.22

1.11

1.16

2.16

1.90

2.01

1.13

1.83

1.56

1.51

(s, Q)

1.97

2.04

2.18

2.72

2.11

1.38

1.79

1.97

1.9

0.33

1.77

1.93

1.86

2.64

2.36

1.93

2.59

(R, S, Qmin )

1.26

0.81

0.59

1.00

1.97

1.28

1.05

0.76

2.43

0.91

0.79

1.54

1.96

0.98

1.59

1.18

1.32

(s,

Q∗)

0.10

0.00

0.00

0.25

0.07

0.43

0.00

0.00

0.14

0.13

0.00

0.00

0.00

0.62

0.00

0.00

0.00

(s,

O Q∗)

11.30

11.74

11.22

11.25

11.94

10.01

12.51

12.31

10.82

11.8

12.1

12

12.14

10.67

10.39

9.08

10.02

(R, S)

LT2

3.76

4.09

3.92

3.42

2.96

3.93

3.01

4.41

3.0

4.69

2.79

2.95

3.61

3.87

4.01

4.86

4.7

(s, S)

1.96

2.6

1.39

2.5

1.72

2.04

2.41

2.31

2.17

1.78

2.33

1.52

1.33

2.06

2.01

1.42

2.21

(s, Q)

2.11

2.06

1.5

1.73

2.38

2.84

2.6

2.03

2.30

1.8

1.61

1.85

1.57

1.96

2.38

3.01

2.64

(R, S, Qmin )

1.21

1.33

1.14

1.03

1.05

1.45

0.93

1.40

1.22

0.94

1.13

1.13

1.39

1.16

1.31

1.04

1.46

(s, Q ∗ )

0.46

0.73

0.39

0.38

0.14

0.1

0.86

0.90

0.04

0.50

0.60

0.21

0.02

0.82

0.29

0.51

0.88

(s, O Q ∗ )

Table 4 Performance comparison of inventory order policies in respect of average relative percentage deviation for total supply chain cost for single-stage supply chain

A Comparative Study on Classical and New Hybrid … 17

10.90

10.17

CS16

Avg

10.98

CS13

9.07

10.61

CS12

10.42

10.72

CS11

CS15

10.91

CS10

CS14

9.21

10.62

CS9

10.73

CS7

CS8

9.06

9.22

CS4

10.66

9.38

CS3

CS6

10.73

CS2

CS5

9.47

(R, S)

LT1

CS1

Supply chain setting

4.38

4.99

5.11

4.44

4.34

4.16

4.41

4.74

5.14

4.88

3.65

3.78

3.56

4.38

3.85

4.89

3.75

(s, S)

2.44

2.33

3.16

2.45

2.77

2.56

1.53

3.11

1.66

2.12

2.13

2.73

2.38

3.29

2.39

1.82

2.67

(s, Q)

3.43

3.84

3.47

3.87

3.28

3.91

3.34

2.64

3.87

3.73

3.13

3.07

3.13

3.84

2.64

3.81

3.25

(R, S, Qmin )

1.74

1.52

1.13

2.27

1.21

1.58

2.48

2.16

2.30

1.11

2.27

1.10

2.47

1.49

1.68

1.29

1.79

(s,

Q∗)

0.27

0.19

0.34

0.43

0.42

0.41

0.15

0.04

0.30

0.41

0.24

0.20

0.44

0.05

0.28

0.03

0.39

(s, OQ∗ )

10.12

10.23

10.22

9.90

10.66

10.42

10.05

9.78

9.76

10.29

9.84

10.25

9.76

10.41

9.84

10.47

10.08

(R, S)

LT2

4.15

5.17

3.59

3.69

3.96

4.57

3.60

3.98

3.73

3.68

4.99

4.07

3.65

3.87

4.88

4.49

4.45

(s, S)

2.35

3.13

2.07

2.97

2.97

1.68

2.77

2.96

2.21

1.64

3.02

1.72

1.95

2.09

1.75

2.57

2.06

(s, Q)

3.17

2.81

3.20

2.73

3.26

2.72

3.25

3.04

3.58

2.69

3.32

2.91

3.21

3.30

3.73

3.62

3.40

(R, S, Qmin )

1.76

1.53

2.49

1.35

1.56

2.37

1.45

1.70

2.00

1.96

1.30

1.62

1.49

1.77

2.13

2.08

1.35

(s, Q ∗ )

0.21

0.12

0.31

0.39

0.06

0.09

0.10

0.04

0.14

0.15

0.01

0.33

0.34

0.18

0.42

0.29

0.33

(s, OQ∗ )

Table 5 Performance comparison of inventory control policies in respect of average relative percentage deviation for total supply chain cost for two-stage supply chain

18 B. Santhanam et al.

A Comparative Study on Classical and New Hybrid …

19

Table 6 Performance comparison of inventory control policies in respect of total supply chain cost for a sample demand stream in a single-stage supply chain Supply chain setting

(R, S)

(s, S)

(s, Q)

(R, S, Qmin )

(s, Q ∗ )

(s, OQ∗ )

CS1

22,124

21,520

20,512

20,884

20,505

20,496

CS2

24,296

23,784

22,416

22,564

22,416

22,416

CS3

26,656

25,552

24,688

24,908

24,161

23,376

CS4

28,856

26,928

25,632

26,734

24,753

24,136

CS5

30,716

29,472

28,896

30,188

28,800

28,824

CS6

34,620

32,240

32,288

33,822

32,174

31,504

CS7

37,080

35,152

33,728

35,870

33,712

33,520

CS8

40,276

36,624

34,744

35,501

34,657

34,376

CS9

41,964

39,368

39,536

39,664

39,368

39,216

CS10

47,284

44,352

44,384

46,064

44,149

44,096

CS11

49,880

46,960

46,992

48,894

46,586

46,168

CS12

53,076

48,184

47,888

51,402

47,460

47,040

CS13

55,552

57,288

54,804

55,335

57,288

54,804

CS14

62,184

62,032

60,880

61,591

61,869

60,704

CS15

68,064

64,880

65,144

66,860

65,128

64,880

CS16

72,616

66,104

66,160

69,099

65,848

65,720

over the periodic-review policies in terms of total supply chain cost. On comparing the (s, S) and (s, Q) policies, (s, Q) policy seemed to be superior to the (s, S) policy in terms of holding and ordering costs. On comparing the (s, Q) and hybrid policies, we see that the (s, Q) policy had relatively lower holding cost. However, in the proposed hybrid policies, since order quantities are made dynamic, the system adapts quickly to dynamic demand patterns, thereby balancing both holding and backorder costs across the members. The dynamically varying actual order quantity helps a member to adjust to any surge or dip in demand since the last order, thereby resulting in less backorder costs. Overall, the proposed hybrid policies outperform the existing (R, S), (s, S), (s, Q), and (R, S, Qmin ) policies in terms of total cost. We find that the performance of proposed policies is quite robust when evaluated across different cost settings (e.g., high ratio of shortage to holding cost as well as in presence of relative low and high ordering costs). It is evident that, on the whole, the (s, OQ∗ ) policy with the continuous-review system emerges to be the best, followed by the (s, Q ∗ ) policy, (s, Q) policy, (s, S) policy, (R, S, Qmin ) and then (R, S) policy. The (s, OQ∗ ) policy takes advantage of the tight control of (s, Q) policy, and it is also able to exploit the dynamic order quantities similar to the (R, S) policy by setting the actual order quantity equal to the maximum of notional fixed order quantity and the order-up-to-level minus the inventory position, and thereby this hybrid policy yields the least total supply chain cost in almost all the settings. The (s, Q ∗ ) policy capitalizes on the tight control of (s, Q) policy and is able to adapt to dynamic changes in demand as the order quantity is

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B. Santhanam et al.

set to maximum of the notional fixed order quantity and the difference between total demand received by the member up to current day t and total order quantity up to day (t-1). The performance of the continuous-review (s, Q) policy and continuousreview (s, S) policy is superior to the periodic-review (R, S) policy, and this can be attributed to the continuous-review system followed in (s, Q) and (s, S) policy. This experimentally confirms the hypothesis made by Wang (2011) that the performance of stock-based (s, Q) policy is superior than the time-based (R, S) policy for a single installation and that dominance may carry over to supply chains.

5 Limitations and Future Work The main purpose of this paper is to propose new hybrid inventory ordering policies, develop mathematical models and compare its performance against existing order policies in literature. To evaluate the polices, we considered a two-stage system with discrete deterministic demand and treated any unfulfilled demand as backordered with no service level constraints. However, in current business environment with availability of alternate products and impatient behavior of customers, we may need to consider unfulfilled demand as lost demand. As future work, we wish to comprehend the performance of proposed policies when unfulfilled demand is treated as lost sales, when there are more than two stages in the supply chain, when there is lead time variability and understand the behavior of the policies when service level constraints exist.

6 Summary In this work, two new hybrid inventory ordering policies that combine the benefits and characteristics of periodic-review (R, S) policy, continuous-review (s, Q) policy and continuous-review (s, S) policy are proposed and studied in the context of single and two-stage supply chain operating under backorders. Mathematical models are proposed to determine the optimal inventory order policy parameters with the consideration of discrete, deterministic demand over a finite planning horizon. A relative evaluation of the proposed two hybrid policies (continuous-review (s, Q ∗ ) policy and continuous-review (s, OQ∗ ) policy) with the existing policies such as periodic-review order-up-to-S (R, S) policy, continuous-review (s, Q) policy, continuous-review (s, S) policy and hybrid (R, S, Qmin ) policy is presented. Based on the computational results, the proposed continuous-review (s, OQ∗ ) policy performs the best because it attempts to exploit the characteristics of both (R, S) and (s, Q) policies, followed by continuous-review (s, Q ∗ ) because it is able to adapt to dynamic changes in consumer demand, followed by the existing policies. To the best of our knowledge, it is perhaps for the first time a comparative study on the performance of the hybrid

A Comparative Study on Classical and New Hybrid …

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and classical order policies are undertaken by considering deterministic demand over a finite planning horizon. Acknowledgements The support from Alexander von Humboldt Foundation and the University of Passau is gratefully acknowledged. The authors are grateful to the reviewers and editors for their valuable feedback that have helped us improve the earlier version of the work.

References Daniel, J. S. R., & Rajendran, C. (2006). Heuristic approaches to determine base-stock levels in a serial supply chain with a single objective and with multiple objectives. European Journal of Operational Research, 175, 566–592. Ganeshan, R., Jack, E., Magazine, M. J., & Stephens P. (1999). A taxonomic review of supply chain management research. In Quantitative models for supply chain management (pp. 839–879). Kluwer Academic Publishers Gruen, T. W., Corsten, D. S., & Bharadwaj, S. (2002). Retail out-of-stocks: A worldwide examination of extent, causes and consumer responses. Grocery Manufacturers of America. Kiesmüller, G. P., De Kok, A. G., & Dabia, S. (2011). Single item inventory control under periodic review and a minimum order quantity. International Journal of Production Economics, 133(1), 280–285. Lee, H. L., & Billington, C. (1992). Managing supply chain inventory: Pitfalls and opportunities. Sloan Management Review, 33, 65–72. Manatkar, R. P., Karthik, K., Kumar, S. K., & Tiwari, M. K. (2016). An integrated inventory optimization model for facility location-allocation problem. International Journal of Production Research, 54(12), 3640–3658. Movahed, K. K., & Zhang, Z. H. (2015). Robust design of (s, S) inventory policy parameters in supply chains with demand and lead time uncertainties. International Journal of Systems Science, 46(12), 2258–2268. Popp, W. (1965). Simple and combined inventory policies, production to stock or to order? Management Science, 11(9), 868–873. Radhakrishnan, P., Prasad, V. M., & Gopalan, M. R. (2009). Inventory optimization in supply chain management using genetic algorithm. International Journal of Computer Science and Network Security, 9(1), 33–40. Sethupathi, P. 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. 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 planning horizon. In U. Ramanathan & R. Ramanathan (Eds.), Supply chain strategies (pp. 113–152). Issues and Models Springer. Shang, K. H., & Zhou, S. X. (2010). Optimal and heuristic echelon (r, nQ, T ) policies in serial inventory systems with fixed costs. Operations Research, 58(2), 414–427. Silver, E., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling (3rd ed.). John Wiley and Sons. Wang, Q. (2011). Control policies for multi-echelon inventory systems with stochastic demand. Supply chain coordination under uncertainty (pp. 83–108). Springer-Verlag. Zhou, B., Zhao, Y., & Katehakis, M. N. (2007). Effective control policies for stochastic inventory systems with a minimum order quantity and linear costs. International Journal of Production Economics, 106(2), 523–531.

Truthful Information Sharing in a Multiretailer Supply Chain: Role of Review Strategies C. Kavitha, R. K. Amit, and B. Vipin

Abstract Eliciting truthful forecast information from the supply chain partners is one of the major challenges in a supply chain. In this study, we consider a demand forecast sharing situation in a supply chain between many retailers and a supplier. The retailers wish to procure capacity from the supplier before the demand is realized and therefore share forecast with the supplier. Retailers may give an excessively optimistic demand forecast as they do not pay to establish the capacity at the supplier. We construct review strategies for both the supplier and the retailers to ensure credible information transmission. We study the repeated game scenario in such a forecast information sharing. Credibility tests are constructed based on multivariate statistics, to infer from the history of information available with the supplier. We prove that, in the repeated game, there exist bounds for the duration of review phase, credibility index, and discount rate above which truth telling results in a perfect public equilibrium which is Pareto optimal for all the parties.

1 Introduction Forecasting plays a crucial role in managing the supply chain efficiently. Forecast communications are generally costless, non-binding, and non-verifiable. In many supply chains, upstream members often do not forecast their own demand and they rely on downstream members to furnish the forecasts. This may be due to unavailability of data or lack of accuracy. To ensure abundant supply, the downstream member often has an incentive to inflate her forecast information. Dissemination of credible C. Kavitha (B) · R. K. Amit Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] B. Vipin Department of Industrial and Management Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_2

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information is often more crucial in settings wherein the sender of the information has an incentive to misreport the same. Forecast manipulation in the form of reporting overoptimistic forecasts is pervasive across industries from electronics and semiconductors to medical equipment and commercial aircraft (Lee et al., 1997; Cohen et al., 2003). Cisco, a major networking equipment supplier, had to write off $2.25 billion in excess inventory in 2001 because of inflated customer forecasts (CNET, 2002). In order to ensure credible information sharing, numerous contracts exist. Ren et al. (2006) suggest that repeated interactions lead to truthful information sharing. The authors also state that the repeated sourcing decisions will form implicit contracts between the supply chain members which is termed as relational contracts. This can be viewed as the feasible substitute for complex pricing contracts that ensure coordination in supply chain. Relational contracts are vastly studied in sociology, law, and economics but are relatively new to supply chain management. According to Helper and Henderson (2014), General Motors, a firm which was once recognized as the most successful, suffered a huge market loss with their US market share falling from 62.6% in 1980 to 19.8% in 2009, mainly due to the failure to develop and implement relational contracts with their suppliers. This is largely attributed to General Motors’ “arms’ length” approach to suppliers, whereas Toyota’s success is attributed to its “relational procurement.” Thus, relational contracts become mandatory to modern manufacturing and to survive in the competitive market. There is an emerging trend among global organizations to shift toward platform based business models to run their operations. Since majority of their supply chains are multiagents ones, it becomes crucial to study the information sharing among the supply chain agents. Owing to the importance relational contracts have gained in modern operations management, it becomes essential that we study problems associated with such contracts and ways to overcome them. We address the issue imperfect monitoring in a build-to-order supply chain with multiple retailers and supplier. Since the lead time of the product is very high, the supplier has to plan his capacity well in advance so that he could fulfill the retailers’ orders. To help supplier plan his capacity, the retailers forecast the demand and share them with the supplier. Anticipating their demand to be high, there is incentive for each retailer to over-forecast their corresponding demand as there is no cost for overcasting. Once the supplier allocates the capacity for the retailers, the final order is placed by the retailers. The suppliers produce minimum of the retailer’s allocated capacity and the final orders. This is considered to be a stage game. This game is repeated for indefinite duration. Consider a two-tier supply chain where there is a sole supplier of a long lead time product and multiple retailers who buy from this supplier. We study repeated game scenario (where a stage game is iterated or repeated number of times). The following events form a stage game: A stage game starts with retailers privately observing the value of their demand. The retailers share the forecast to the supplier which is a rough estimate of demand being high or low. The supplier tries to allocate capacity that maximizes his expected profit given the forecast. The true demands are then realized, and the final orders are placed with the supplier. The supplier satisfies the orders accordingly and generates revenue for each unit of order satisfied. The demand

Truthful Information Sharing in a Multiretailer …

25

distributions of each retailer are known, and it is a common knowledge between the retailer and the supplier. The demand for each retailer is correlated to the demand of other retailers and the exact nature of this correlation is unknown. Each retailer is non-strategic with other participating retailers. The time is subdivided into a series of review phases and punishment phases. Credibility tests are constructed based on multivariate statistics, to infer from the history of information available with the supplier. Throughout the review phase, each retailer is maintained a credibility index which is either incremented or kept unchanged based on the results in the credibility tests. If the credibility index exceeds the predefined threshold value, the retailer passes the review, and a new review phase is initiated. Otherwise, the game enters the punishment phase in which the supplier disregards the forecasts shared. After a particular punishment period, a new review phase is started. In this research work, we prove that the retailers share the forecast information truthfully with the supplier notably in the initial period of the review phase and the supplier allots the optimal capacity for each retailer assuming the information received from retailer is credible. We show that there exist bounds for the duration of review phase, credibility index and discount rate above which truth telling results in a perfect public equilibrium which is Pareto optimal for all the parties. We examine the sensitivity of the review phase with respect to the changes in the parameters of the games and provide the justification of the observed behavior.

2 Literature Review We review the literature on relational contracts and repeated games. Any economic activity will often span through various organizations and individuals. Agents follow different policies and procedures to achieve their objective. Self-interest and rationality do not allow them to share common objective (Amit and Mehta, 2010). Inefficiencies arise in the system due to these conflicting objectives. To overcome this inefficiency, numerous coordination mechanisms such as contracts, information technology, information sharing and joint decision making has been discussed in literature. Myerson (1991) suggests that coordination theory is built from game theory, where payoffs for participating agents not only depend on their actions alone, but also on other agents actions. Contracts can be viewed as one of the governance structure where it prescribes the rights and duties of each agent for each and every future state of the world. It should also suggest the actions in case of uncertainty, whereas it is impractical to state every possible future state of world due to limited time or bounded rationality of individuals as stated by Hohn (2010). Moreover, in case of indefinite contract duration, to overcome the dilemma between offering a means of flexibility and providing a means of commitment to the participating agents (Baird, 1990) suggests to create long-term contractual relationship. These long-term relationships are often called relational contracts and are informal quid pro quos between the contracting parties.

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Since the actions performed by agents are not verifiable by non contracting parties, they cannot be enforced by the third party or the court and has to be self enforcing. A key feature in many strategic situations like buyer–supplier interaction is that players interact repeatedly over time. Radner (1986) states that if an organization has short-lived players, then the Nash equilibrium of the game is Pareto inefficient, i.e., there exist payoffs which are greater than the Nash equilibrium payoffs. He further states that at least two approaches are possible to achieve Pareto optimal payoffs. The first approach is suggested by Aumann (1967), and it states that agents form binding agreements to achieve efficient payoffs. The overview and classification of supply chain contracts can be found in Govindan et al. (2013). Another approach to achieve efficient outcomes that is very common in literature is that players repeat this game for indefinite times. By doing so, the Nash equilibrium of the supergame (i.e., the repeated game) is efficient. This is put forth as Folk theorem which postulates that whenever the players are sufficiently patient, any Pareto superior payoff vector to one period Nash equilibrium payoff can be sustained as the Nash equilibrium of the repeated game. The assurance of reward and the threat of punishment are the key drivers that sustain the relationship in a repeated game and ensure desirable behavior in the present situation. Information sharing is essential to any supply chain which aims to integrate the activities of its constituent agents to achieve efficient performance. According to Chopra and Meindl (2007), information is one of the six drivers of supply chain management. Information had been identified as a chief efficiency in competitive markets by Hayek (1945). Terwiesch et al. (2005) report that big retail players such as Walmart and Best Buy along with their suppliers, have all benefited from collaborative planning, forecasting, and replenishment policies. Among all the information that is shared and received, forecast information of retailers is of utmost importance to the supplier. It is with this information that the supplier will plan and build his capacity to maximize his profit. Several authors, including (Cachon and Fisher, 2000; Lee et al., 2000; Chen, 2003; Iyer and Villas-Boas, 2003), explore the value of information shared in a supply chain. Gavirneni et al. (1999) discuss the value of information shared in a capacitated system. Cachon and Lariviere (2001) have explained how contracts can be used for credible information sharing. Lack of information has been a roadblock in reaching agreements. Crawford and Sobel (1982) in their classic paper asserts that sharing of information will lead to better agreements, but at the same time, it also leads to suspicion that revealing all the information will result in less advantageous position. There exists a chance that the other player may misuse the information shared to his advantage leaving the sender of the information worse off. The authors also state that even a completely self-interested agent will reveal some information in order to attain benefits. Demand forecast manipulation has been reported in Lee et al. (1997) and Cohen et al. (2003). According to BusinessWeek (2001), a similar situation was faced by Flextronics while receiving forecast information from Ericsson and insists that “forecasts from telecom and electronics companies are often inflated.” The financial impact of wrong forecast sharing by American Airlines costed Boeing around 2.7 million dollars as the supplier in 2004 (WallStreetJournal, 2004). In order to overcome these problems

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27

with forecast information sharing, there are various contracts that are suggested to ensure truthful exchange of information. There are certain situations where an agent takes decisions on behalf of other agent. In these cases, the actions of one contracting party are not verifiable by the other contracting agents. This gives rise to the situation of imperfect monitoring where each player receives an imperfect signal of the strategies and actions of rest of the players. Hence to coordinate the supply chain, the better informed sender should send some information to the uninformed receiver. Imperfect monitoring arises due to asymmetric information, where one party misuses this informational advantage to exploit the other party (the sender of information manipulates the information to his advantage). This class of problems is discussed by Abreu et al. (1990) and Fudenberg et al. (1994). The above scenario is described as cheap talk, as it does not directly affect the payoffs of the individual players. The term “cheap talk” refers to direct and costless communication among players. Incentive schemes and monitoring are used to reduce this type of agency problem. Thus, credibility and credulity of the information being shared are the key factors in information exchange. Testing the credibility of the information received is often more ambiguous. The examples on forecast manipulation fall under this category of cheap talk games. According to Roberts (2004), for a successful partnership, the threat of punishment should be such that the affected party must be able to retaliate (i.e., by ending the relationship). Cooperation is sustained through two means: the larger the gains, the stronger the ties and the worser the punishment, the better the relationship. The parties should be more dependent on each other so that the net gains from outside options is almost negligible. The author states that at any point in time the present value of cooperation should be greater than present value of defection for a relationship to continue. The pioneering work on strategic communication model on information sharing between agents is described by Crawford and Sobel (1982). They considered a scenario where there are two agents, one of them (Sender) has private information about a random variable. The sender introduces noise into the signal, i.e., he sends a noisy estimate of the random variable to the uninformed receiver. The receiver then takes action based on the information sent by the sender, which affects the payoffs of both the agents. Therefore, sender’s signal is expected utility maximizing given receiver’s action. The receiver will update his prior based on sender’s signal using Bayes’ rule given sender’s signaling strategy. The authors conclude that direct communication is possible only if the agents’ interests coincide. Several authors including (Radner, 1985; Stocken, 2000) have discussed simple strategy pairs which ensure truthful information transmission between the principal and the agent. These strategy pairs are called review strategies and are characterized by duration of review phase, penalty phase, and criteria to pass the review. Ren et al. (2010) devise credibility tests for a single supplier-single retailer forecast information sharing game and proposed a multiperiod review strategy profile for both the players to ensure truth telling by the retailer. Radner (1985) describes review strategies in the context of principal–agent interaction with separate discount factors for the principal and the agent. The author finds

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that for all discount factors above some critical values, there are equilibria in review strategies which yields expected utilities greater than one period expected utilities for the principal and the agent, respectively. In addition, he asserts that if the principal stops paying the reward as promised, it initiates the punishment phase. The agent is shown to optimize myopically in this phase until the principal reverts to the constant reward function. Hence, the author finds optimal review strategy profiles for both the principal and agent in the supergame. Stocken (2000) describes a situation where a corporate manager observes imperfect information on the nature of environment as favorable or unfavorable for investment. He then reveals it to the investor, who has to decide on whether to fund the project or not. This decision is based on the following information: the information given by the corporate manger and the credibility of the corporate manager. After this stage of the game, the accounting report is released and is publicly observable. The investor then observes the accounting report and updates his belief on the corporate manager and thereby evaluates his credibility. To motivate the manager to voluntarily disclose the information, the manager’s payoff is assumed to be proportional to the expected contribution on the project. The author found that in equilibrium the manager truthfully discloses the information on environment in the review phase, whereas in punishment phase he does not do so. Rubinstein (1979) studies optimal conviction policies for offenses that have been committed not deliberately. He designed review strategies to ensure that taxpayers do not conceal some part of their respective incomes to the tax authorities to evade taxes. Penalties are imposed in such a way that tax authorities can distinguish between deliberate tax evasion and offenses that have committed by accident. The author suggested that there exists a pair of strategies for taxpayer and the tax authorities in a repeated game which are jointly optimal. It is now clear that the partners in a supply chain often rely on informal or unwritten agreements, in addition to contracts. Typically, a supplier must invest in capacity far in advance of the selling season. For example, semiconductor equipment industries commit to prior production capacity before having final orders in hand. It is well documented that the downstream buyer gives the forecast of the demands in terms of “soft orders.” With this information, the supplier builds his capacity as reported in Terwiesch et al. (2005). The literature on relational contracts does not take into account a realistic case where there is a supply chain with multiagents at a specific level. Motivation for this works comes from the fact that most of the multinational companies suffer from huge loss due to incorrect forecasts. With this work, it becomes increasingly helpful for the top management of companies which primarily relies on relational procurement, to guarantee credible information sharing by its downstream supply chain members. In this study, we consider a demand forecast sharing situation in a supply chain with multiple retailers and a single supplier. There is no direct communication between the retailers. Due to information asymmetry between the supplier and the retailers, the supplier does not have sufficient information to forecast the demand as accurately as the retailers. Since the demand is a function of the random state of nature, once the final order is received supplier may wish to find the reason behind demand mismatch

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29

(of forecast and final order), whether it is due to nature of environment or intentional lie on the part of the retailers. The retailer may give an excessively optimistic initial demand forecast so as to induce him to build more capacity. The retailers do not pay to establish the supplier capacity, and he prefers more capacity, anticipating demand to be high. When the final order is low, it affects the supplier’s profits in the form of excess inventory. We study the review strategies in such a setting under relational contract for truthful information sharing.

3 Model Background The practice of forecast information sharing by the customer to the supplier and subsequent supply chain coordination using a game-theoretical model in the presence of information asymmetry was studied by Ren et al. (2010). The retailer being close to the market is able to forecast the demand more accurately. The customer wishes to acquire a part of capacity from the supplier before the actual demand is realized by him. The demand is assumed to be a scaled random variable θ.X , where X ∼ N (μ, σ ) which is positive. θ is assumed to be a random variable which take only two values either high (θh ) or low (θl ). The customer then forecasts the demand to be high or low based on market conditions. Let, Probability(θh ) = α Probability(θl ) = 1 − α,

where α ∈ (0, 1)

θ is assumed independent of X . The distributions of θ and X are common knowledge among agents. The expected utilities for the supplier and the buyer are as follows: u(θi , K) = r min(K, Di ) − h(K − Di )+ − cK

(1)

v(θi , K) = (p − r)min(K, Di ) − g(K − Di )+

(2)

where K is the supplier’s capacity, c is the supplier’s unit capacity cost, r is the price the customer pays for unit capacity allocated and utilized, and p is the per unit revenue for the customer. Overage cost for the supplier is h, whereas g is the lost sales for the customer. He then reveals it to the supplier (m = L or H ) who makes capacity allocation accordingly. The supplier can honor the information given by the customer (where he will allot the system optimal capacity to the customer) or betray the customer (where supplier will allot optimal capacity given no truthful information is shared) with the belief that information shared is untrue and hence omits it. If m = H (m = L) the supplier believes the customer that demand is DH (DL ), then he will allocate KH (KL ) that maximizes his expected utility given by

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KH = argmaxE[u(θH , K)]

(3)

K

KL = argmaxE[u(θL , K)]

(4)

K

whereas if the supplier does not believe the customer, then he will allocate K0 as if there was no meaningful information on demand. K0 = argmax(1 − α)E[u(θH , K) + (α)[u(θH , K)]]

(5)

K

The real demand is then realized, and final order is placed. At this point in time, the supplier will come to know whether the forecast which the customer shared matches with the final order placed by the customer. Subsequently, the customer will come to know whether capacity allocated was optimal according to his forecast. In a single stage version of the game, the customer will anticipate high demand even when the forecast is low and tend to inflate demand forecast. Foreseeing the customer will inflate the demand, supplier does not allocate system optimal capacity. Thus, Pareto optimal capacities are not reached because of misaligned incentives in a one-shot interaction. In case of repeated interaction between the supply chain agents, the author describes two strategies: • Trigger strategy: Credibility check is triggering, and punishment starts in subsequent phase. • Review strategy: To ensure that supplier is not punishing a truth telling customer (as the demand is a random variable and could not be accurately predicted), authors have designed review strategies. In the first R review phases, the supplier will maintain a scorecard It and a threshold for credibility γt . At the end of each time period between t=0 and t=R, once the demand is realized and orders placed, the supplier will evaluate the credibility of the customer through a series of tests and updates the score It . The credibility tests are explained as follows: 1. If the customer reports the demand to be high, i.e., mt = H at any time period t ∈ [1, R], then the customer’s score is subject to two credibility tests. a. Proportion of periods test: This test evaluates the proportion of time the customer has reported mt = H . The probability of high demand happening is 1 − α in each period, so in long run the proportion of times high demand is realized is approximately 1 − α. Let Nt be the proportion of periods customer forecasted and declared high demand √ so far in the review phase. The samfor 95% pling distribution of α follows N (α, α(1 − α)/(t − Nt ). Therefore √ confidence level Nt should not be greater than (1 − α) + zα σ α(1 − α)/Nt . This test is performed to discourage the customer to over-forecast and subsequently report high demand. b. Order quantities test: If the customer reports the demand to be high, i.e., mt = H at any time period t ∈ [1, R], then the customer’s score is subject to

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another test of truthfulness. If the customer forecasts truthfully, then sample √ mean of demand realizations follows N (θH μ, θH σ ). Therefore, the lower bound or threshold for the demand values at high demand periods is  d H ,t = θH μ − zH σ θH /Nt If the customer passes both the above tests, then his score It = It−1 + 1, otherwise the score remains unchanged It = It−1 2. If the customer reports the demand to be low, i.e., mt = L at any time period t ∈ [1, R], then the customer’s score is subject to only one test of truthfulness. If the customer √ orders truthfully, then sample mean of demand realizations follows N (θL μ, θL σ ). Therefore, the upper bound or threshold for the demand values at lower demand periods is  d L,t = θL μ + zL σ θL /(t − Nt ) If the customer clears the above test, then his score It = It−1 + 1, otherwise It = It−1 While in review phase, the customer is always trusted and supplier builds the system optimal capacity. In case the agent passes the review, then the review phase is restarted. Whereas, if the customer fails the review, then for M subsequent periods the supplier will allocate optimal capacity given no truthful information is shared. Hence after R+M phase, same process is repeated. Thus authors conclude by saying, in repeated game more efficient equilibrium is achieved through truth telling, leaving both supplier and customer better off compared to the one shot game. As discussed earlier, most of the real-world supply chains will have multiple retailers buying from a single supplier. When we try to model such case, the credibility tests devised as above may not be relevant. There are multiple sources of information from the retailers, and the supplier wishes to construe the actual forecast scenario from these sources. Therefore, we resort to multivariate statistical models in place of uni-variate models which are discussed in Ren et al. (2010). We try to implement Hotelling’s T 2 methodology of hypothesis testing for formulating the credibility tests which is discussed in the following section.

4 The Model Consider a 2-level supply chain where there is a sole supplier of a long lead time product and m retailers who buy from this supplier. The retailers owing to the proximity to their customers will have initial forecast of the demand for the product during certain time period. The supplier do not have access to the market information, and he relies on the retailer for the forecast. Since the retailers want to acquire some capacity from the supplier before the actual demand is realized, they share this

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forecast information to the supplier. The supplier then plans and builds the capacity accordingly. The demand for the product for each retailer is in some way correlated to the demand of the other retailer. We use similar notations as in Ren et al. (2010). Let the demand for each retailer be Di , where i = 1 to i = n refers to Retailer-1 to Retailer-n, respectively. Let the demand be a random variable Di = θi,j . The retailer’s message to the supplier is mi , and it is the demand size parameter, referring to demand being low and high, respectively. It is assumed that probabilities of high and low demands for each retailer i are known and the distribution of θ is common knowledge. Table 1 provides the notations used in this work. In order to model a truthful forecast information sharing situation involving many retailers, we resort to multivariate statistical models. We model a situation with m retailer (m ≥ 2). Let us assume that we are on the nth time period of the review phase. Taking into account the forecast history of m players, we have n × m matrix of forecast data, where the rows represent the n time periods and the columns denote m retailers. Therefore, the forecast matrix F if given by: ⎡ ⎤ f11 f12 f13 . . . f1m ⎢f21 f22 f23 . . . f2m ⎥ ⎢ ⎥ fjk ∈ IR (6) F =⎢ . . . . . ⎥, ⎣ .. .. .. . . .. ⎦ fn1 fn2 fn3 . . . fnm

In the above matrix, j represents the n-th time period and k represent the m-th retailer. With this information furnished by the retailer, the supplier will try to allocate the capacity to each retailer that maximizes his expected profit. After the supplier allots specific capacity, the actual demand is realized by the retailer. In order to fulfill the demand, the retailers place the final order with the supplier. This information is represented as the order quantity matrix which is also n × m and is denoted by X: ⎡

x11 ⎢x21 ⎢ X =⎢ . ⎣ ..

x12 x22 .. .

x13 x23 .. .

... ... .. .

⎤ x1m x2m ⎥ ⎥ .. ⎥ , . ⎦

xjk ∈ IR

(7)

xn1 xn2 xn3 . . . xnm

In order to find the defecting retailer across each period, we define a matrix =(F − X ). This matrix will reveal those retailers who have over-forecasted or under-forecasted. ⎡ ⎤ 11 12 13 . . . 1m ⎢21 22 23 . . . 2m ⎥ ⎢ ⎥ jk ∈ IR =⎢ . (8) .. .. . . .. ⎥ , ⎣ .. . . . . ⎦ n1 n2 n3 . . . nm

Let  denote the average of the differences across time periods for each retailer.

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Table 1 Notations Notations Explanations ci ri pi hi gi u vi mi Ki,j vic vi0 vi (I0i ) vi (IRi i −ni ) vi (θil ) βi ηi i τ and ρ

Capacity cost per unit for the supplier Price which retailer i pays to get one unit allocated and utilized Unit revenue retailer i gets by selling the product Unit overage cost for supplier for capacity allocated and not utilized Unit loss of goodwill cost for retailer when capacity is not sufficient to satisfy the realized demand Utility or profit function of the supplier Utility or profit function of the retailer i Forecast message sent by retailer i to the supplier Capacity allocated by the supplier for the retailer i for demand state j One stage expected payoff for retailer i with system optimal capacity One stage expected payoff for retailer i under non-cooperation, i.e., ignoring retailers’ forecast Normalized and discounted expected profit at the beginning of the game for retailer i Normalized expected profit when there are ni periods remaining in the review phase Ri Expected one stage payoff when retailer i observes θil and defects Conditional probability that retailer i is assessed to be truthful given he is actually truthful Conditional probability that retailer i is assessed to be truthful given he defects vi (θil ) − vi (θil , Kilc ) Review size parameters

 = 1 2 3 . . . m ,

k ∈ IR

(9)

Thus, we could test this matrix ij using Hotelling’s T 2 method. We have p retailers which we take as variables and n time units. The following subsection explains the procedure to conduct Hotelling’s T 2 test.

4.1 Hotelling’s T 2 The following are the steps involved in Hotelling’s T 2 . • Let us define the null hypothesis that the mean vector is zero. H0 : μ = 0 H1 : μ = 0

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• The test statistic which we need to calculate is T2 = n ×  × S−1 ×  • We reject the null hypothesis at 95% confidence if T2 > F(m−1),(n−m−1),95% ×

(n − m − 1) (n − 1)(m − 1)

(10)

4.2 Confidence Ellipsoids The above test helps us to understand whether the given mean vector is equal to the hypothesized mean vector. But our objective is to find which variable mean or means in the vector has contributed to the rejection of null hypothesis. This would help the supplier to find and punish the deviating retailer. In order to find that, confidence ellipsoids are constructed around the mean. If the mean vector lies outside the confidence ellipsoid, then we reject the null hypothesis. We could plug in appropriate values, in order to find the deviating retailer. The equation for confidence ellipsoid is p(n − 1)  (11) Fp,n−p,95% (X − μ) S −1 (X − μ) ≤ n(n − p) Therefore, generalizing for p retailers, the confidence ellipsoid can be written as a11 (X 1 ) + a22 (X 2 ) + ... + app (X p ) + 2 2

2

2

p p  

aij X i X j ≤

i=1 j=1

p(n − 1) Fp,n−p,95% n(n − p)

(12) If the hypothesized mean does not satisfy this equation, then we reject the null hypothesis. However, we wish to get simultaneous confidence intervals for p retailers, which is given by



 p(n − 1) Fp,n−p,95% Sii /n n(n − p) (13) The above equation will give the confidence interval for each of the p retailers. If the hypothesized difference lies in the confidence region, then the credibility index is incremented by one. If it does not lie in the confidence range, then the score is not incremented. To summarize, the review phase works as follows: Xi −

 p(n − 1) Fp,n−p,95% Sii /n ≤ μi ≤ X i + n(n − p)

• At any time period in the review phase t, the retailers owing to proximity to customers will share the demand forecast with the supplier. This is done in order to help the supplier to build his capacity. • The supplier plans and builds his capacity accordingly. • Once the final demand is realized, the retailers place the order with the supplier.

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35

• The above-mentioned credibility check is performed at each stage, and the retailers’ credibility index is updated. • The updated index is compared with the moving threshold given by γt = ((R.S − J )/R)t where, J = τ Rρ , τ > 0 and 0.5 < ρ < 1. • If the retailer’s index is less than the threshold, the review phase is terminated and punishment phase is started for this retailer. The punishment phase extends to M = XR where, X > 0. While in punishment phase, the forecast of the corresponding retailer is ignored and the system optimal capacity given no truthful forecast shared is only allocated for the retailer irrespective of the forecast. The review phase is terminated for the rest of the retailers, and a new review phase is started. • If the retailer’s index exceeds the threshold, yet less than the overall threshold given by γ = R.S − J t is incremented and the whole procedure is repeated. If the index of any one retailer exceeds the overall threshold, then review phase is terminated for all the retailers and a new review phase is started. This is done to discourage those retailers who have acquired enough credibility index and can coast for rest of the time periods in the current review phase.

5 Results Theorem 1 For every retailer, ∃ a discount rate δ, a review phase of length Ri and a credibility assessment value S such that for ∀δ > δ, Ri > Ri , S > S, > 0, there exists a perfect public equilibrium of the repeated game in which vi (I0i ) > vic − , provided the efficient capacity levels Kijc are Pareto improving for all the parties compared to Kin . In the equilibrium, the retailers always share truthful forecast and the supplier always allocate the system optimal capacity Kijc Proof The outline of the main idea for proof of the theorem is as follows: 1. The players do not have strict incentives to deviate from their designated equilibrium strategies. 2. In Lemma 2, it is shown that the retailer i will fail the review without doubt at any time ti < Ri , when Itii < max(0, ti − (Ri − qi )). 3. In Lemma 3, it is proved that the probability the players fail the review when they use prescribed review strategies is less than the likelihood of retailer failing when he is truth telling and is evaluated at the end of review phase Ri only. 4. We also prove that the characterized equilibrium ensures almost complete revelation of the private information of each retailer.

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Lemma 1 Assuming the retailer i is always truth telling, then n i −1

vi (IRi −n ) = (1 − δ)vic i

i

(Sδ)k + vi (I0i )(Sδ)ni + [(1 − δ Mi )vi0 + δ Mi vi (I0i )]

k=0

ni 

(1 − S)δ k S k−1

k=1

(14) Proof When ni = 1, then vi (IRi i −1 ) = (1 − δ)vic + δSvi (I0i ) + δ(1 − S)[(1 − δ Mi )vi0 + δ Mi vi (I0i )]

(15)

By induction, for ni vi (IRi i −ni ) = (1 − δ)vic + δSvi (IRi i −(ni −1) ) + δ(1 − S)[(1 − δ Mi )vi0 + δ Mi vi (I0i )] (16) By repeated substitution, we get vi (IRi i −ni ) = (1 − δ)vic

n i −1

ni  (1 − S)δ k S k−1

k=0

k=1

(Sδ)k + vi (I0i )(Sδ)ni + [(1 − δ Mi )vi0 + δ Mi vi (I0i )]

(17) Lemma 2 The retailer would invariably fail the review phase if Itii < max(ti − (Ri − qi ), 0), specifically when Itii is such that the likelihood of the retailer failing the review is 1. Proof To prove the above Lemma, we have to show that there exist parameters of the game where the retailer will truthfully reveal his private information when IRi i −ni = qi − ni For any Ri , when IRi i −ni = qi − (ni − 1), then the retailer has incentive to always tell the truth whenever (1 − δ)vi (θil ) + δηi vi (IRi i −(ni −1) ) + (1 − ηi )[δ(1 − δ Mi )vi0 + δ Mi +1 vi (I0i )] (1 −

δ)vi (θil , Kilc )

+

δβi vi (IRi i −(ni −1) )

+ (1 − βi )[δ(1 − δ

Mi

)vi0

+δ

Mi +1

≤

vi (I0i )]

Rearranging, i ≤

(βi − ηi ) δ[vi (IRi i −(ni −1) ) − (1 − δ Mi )vi0 + δ Mi +1 vi (I0i )] (1 − δ)

(18)

Substituting equation (17) in equation (18) yields i ≤

 n n i −2 i −1 (βi − ηi ) δ (1 − δ)vic (Sδ)k + vi (I0i )(Sδ)ni −1 + [(1 − δ Mi )vi0 + δ Mi +1 vi (I0i )]( ((1 − S)δ k S k−1 ) − 1) (1 − δ) k=0

Using L’Hopital rule as δ approaches 1,

k=1

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37

n −2

i ≤

i  (βi − ηi )  (S)k + (ni − 1)vi (I0i )(S)ni −1 + vi (Ioi )S ni −1 + [(−1)vic (−1)

k=0



(−Mi vi0 + Mi vi (I0i ) + vi (Ioi ))(

n i −1

((1 − S)S k−1 ) − 1) +

k=1

vi (I0i )

n i −1

K(1 − S)(S)k−1 + vi (I0i )S ni −1 +

k=1

(

n i −1

(1 − S)(S)k−1 − 1)vi (I0i )] (19)

k=1

We know that

n i −1

 K(S)k−1 =

k=1 n i −1

1 − Sin − ni S ni −1 (1 − S) (1 − S)2 

(S)k−1 =

k=1

1 − (S)ni −1 (1 − S)

 (20)

 (21)

Using the above equations in Eq. (20) and simplifying,  i ≤ (βi − ηi )

1 − S ni −1 1−S

 (vic

−

vi (I0i ))

+ Mi S

ni −1

 (vi (I0i )

−

vi0 )

(22)

When n is very large, then  i ≤ (βi − ηi )

1 1−S



 (vic

− 

vi (I0i ))

(23) 

From the above equation, it is clear that for some δ ≤ δ ≤ 1, ∃ S and Ri such that    1  c i (vi − vi (I0 )) will exceed δi . The for all S ≥ S and Ri ≥ Ri , (βi − ηi ) 1−S i retailer’s incentive is violated when Iti < max(0, ti − (Ri − qi )). Lemma 3 The probability the retailers fail the review phase when the agents use the prescribed review strategies is less than or equal to the likelihood of failing the review when the retailer reports truthful forecasts and is assessed only at the end of the review phase. Therefore, the retailer will fail the review if IRi i ≤ qi and when IRi i ≥ qi , the retailer will pass the review. Proof We can show that there exist histories such that the retailer can pass the review before the Ri , i.e., review period ends but fails when he is assessed only at the end of review phase Ri .

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Proposition 1 There exists a specified equilibrium which ensures nearly complete revelation of the each of retailer’s private information. Proof To show that the specified equilibrium yields practically complete revelation of private information of each retailer, we need to obtain sufficient conditions for lower bound of retailer’s normalized expected profit. Since the review phase can be terminated before Ri , assume that if the retailer fails the review, he fails at the earliest possible time ti and suppose he passes the review, he passes at the earliest possible  time t . We know that vi (I0i ) ∈ [vi0 , vic ]. By the Markovian property of the repeated game, we find the lower bound for the retailer’s normalized discounted expected profit vi (I0i )

vi (I0i )

≥ λi (1 − δ)



Ri −qi



δ k−1 vic

+δ

Ri −qi

k=1

(1 − δ

Mi

)vi0

⎡ (1 − λi ) ⎣(1 − δ)

+δ

Ri −qi +Mi

vi (I0i )

⎤



t 

+



δ k−1 vic + δ t vi (I0i )⎦ (24)

k=1

Simplifying the above equation, we get 

vi (I0i ) ≥

λi vic (1 − δ Ri −qi ) + λi δ Ri −qi (1 − δ Mi )vi0 + (1 − λi )vic (1 − δ t ) 1 − δ Ri −qi +Mi λi − (1 − λi )δ t



(25)

As δ → 1 , using L’Hopital rule Mi vi0 )λi − Ri − qi Mi λi λi + + Ri − qi

(vic +

(1 − λi ) c  v (t ) Ri − qi i  (1 − λi )t Ri − qi

(26)

When we substitute Mi = χ Ri , lim

Ri →∞

M χ = Ri − qi (1 − S).

We know that the conditional probability that the retailer fails the review at the start of the game is λi . The upper bound for this probability is denoted by λi (IRi i < qi ) = λi (IRi i < Ri S − Ji ). If each retailer reports the demand truthfully, then IRi is a binomially distributed random variable. Therefore, using Chebyshev’s inequality, λi (IRi i < qi ) =

Ri S(1 − S) Ji2

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39

ρ

Since Ji = τ Ri , when Ri → ∞ =⇒ λi → 0. Therefore, equation (11) simplifies to vi (I0i ) ≥ vic . This can be rewritten as, vi (I0i ) > vic − . Hence, for any > 0, ∃ a discount factor δ such that ∀ δ ≥ δ and ∃ a Ri such that ∀ Ri > Ri , vi (I0i ) > vic − , which is a lower bound for retailer’s expected profit.

6 Numerical Illustration In order to illustrate the above model, we give numerical example for how the effective review length changes with the parameters of the model such as τ , ρ and the maximum specified review length. We also compare the review length of trigger strategy and review strategy. The forecast and the demand matrix are obtained by random number generation. The forecast and the demand are assumed to follow normal distribution N (100, 75). We have constructed graph for the effective review length for the retailers when the maximum review period is changed (Fig. 1a). We find that the effective review length increases when the maximum review length increases. For increase in the maximum review period specified, by the threshold equation, the overall threshold also increases. Hence, the retailers require more time periods to pass the review phase and move on to the next review phase. From the graph 1b, we study how the review length changes with τ . We find that as τ increases, the review length decreases. This is in accordance with our intuition

(a) Maximum review period

(c)

Fig. 1 Impact of different parameters on review length

(b)

(d) Number of retailers

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that as τ increases the allowable margin increases, thereby decreasing the number of time periods required to pass the review. The variation in review length when ρ changes is studied as shown in Fig. 1c. It is evident that as the ρ increases, then the effective review length decreases. According to the threshold equation, as ρ increases, the number of time periods required to pass the review phase and restart the next review decreases. It is quite interesting to study how the review phase changes with increase in the number of retailers (Fig. 1d). Our intuition is that as the number of retailers increases, the supplier may get more information of the same demand scenario, so that he can easily detect the defecting retailers. But, the graph is seen to increase with the number of retailers. This can be attributed to the fact that, as the number of retailers increases, there increased distortion to the forecast information. Hence, the supplier may not be able to detect the truth telling as against the defecting retailer in lesser time frame. Hence, the review phase increases with the increase in number of retailers.

7 Managerial Implications and Conclusions Manipulation of demand forecasts has mostly been observed whenever there are multiple agents in a supply chain. A relevant case is the cloud kitchen inventory pooling promoted by delivery platforms Swiggy and Zomato. There is a single kitchen which houses cooking facilities of competing restaurants. They agree to pool their inventories but may not always be willing to share the actual demand information. The problem of inflated forecasts is especially relevant in the current pandemic scenario as the uncertainties in supply and demand create inconsistent orders in supply chain. AmerisourceBergen, a pharmaceutical distributor in the USA, experiences an inflated demand from the acute healthcare providers for increasing their inventory (Zenk, 2020; FiercePharma, 2020). Our review strategies can address the challenges in information sharing in such scenarios. This research work studies the demand forecast sharing scenario between a single supplier of a long lead time product and multiple retailers who buy from the supplier. We develop review strategies for both the supplier and the retailers to ensure credible information transmission. Credibility tests for the retailers which guarantees truthful forecast sharing are developed. We perform sensitivity analysis with respect to the parameters in the repeated game. Though these strategies can achieve credible forecast sharing in a multiretailer setting, increasing the number of retailers results in increased review length—an undesirable feature, observed in our study. This result warrants the use of relational contract as an alternative to the classical mechanism design approach in a supply chain setting with high number of agents for truthful information sharing. This work aims to provide insights into the review strategies and significance of them in real-life supply chain to ensure truthful forecast sharing. We consider repeated interactions among the agents in the form of a relational contract in an effort to ensure credible forecast sharing. A repeated and long-term relational contract

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between supply chain parties gives each agent “opportunities to review the credibility of the other party, reward truth telling, punish otherwise, and therefore provides the right incentive for truthful information sharing” as asserted by Ren et al. (2010). This repeated interactions resulting in relational contract between supply chain agents can be considered as a feasible alternative to traditional approaches in ensuring truthful information sharing. Acknowledgements We thank the editors and two reviewers for their comments and suggestions which helped us improve the quality of the manuscript.

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Mathematical Models and Heuristics for an Inventory Routing Problem Without Split Deliveries K. R. Preethi, Chandrasekharan Rajendran, and S. Viswanathan

Abstract We consider a finite horizon Inventory Routing Problem in which the vendor monitors the inventory at the retailers and makes replenishment decisions for them. A single vendor manages several retailers who face deterministic but dynamic demands. The vendor decides the delivery schedule, delivery quantities, and vehicle routes that minimize the total supply chain cost. For practical purposes, we assume that split deliveries to a retailer are not allowed and that the backorders are allowed at the retailers. Two mixed integer linear programming models are proposed to solve the specified Inventory Routing Problem (IRP). Since IRP is NP-hard, two heuristics are also developed for the same problem. Numerical experiments were conducted on randomly generated data instances to evaluate the proposed models. The numerical study reveals that the proposed models are computationally efficient. We observed that the optimality gap of the solutions from the mathematical models was no more than 3% in all the tested cases. We also show that the proposed heuristics provide consistent near-optimal solutions. The proposed models and heuristics could serve as efficient decision-making tools to help supply chain managers make quick tactical level decisions. Keywords Supply chain · Logistics · Vehicle routing · Inventory routing · No split delivery · Integer program · Heuristics

1 Introduction Vendor managed inventory (VMI) is a supply chain coordination mechanism in which a vendor monitors and manages the inventory at the retailers. In a traditional K. R. Preethi (B) · C. Rajendran Indian Institute of Technology, Madras, India C. Rajendran e-mail: [email protected] S. Viswanathan Nanyang Technological University, Nanyang Ave, Singapore e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_3

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order-based replenishment, the vendor fulfils the retailer orders that she receives. In VMI, the retailers cooperate with the vendor. The vendor has access to the demand and inventory information of the retailers using electronic data interchange (EDI). The vendor then makes replenishment decisions for the retailers such that the total supply chain cost is minimized. In a VMI system, the vendor can consolidate and coordinate multiple deliveries to the retailers, thus minimizing transportation costs. Here, the retailers also benefit from not allocating resources for inventory management (Coelho et al., 2013). For a detailed discussion on the benefits of VMI, see Waller et al. (1999). Proctor & Gamble and Walmart successfully implemented VMI in their distribution networks in the 1980s. Companies are increasingly adopting VMI due to advancements in data analysis techniques and information system technologies. The vendor faces three main decisions in a VMI setting (Campbell et al., 1998): When to serve a customer? How much to deliver? Which delivery routes to follow? The vendor must simultaneously determine the delivery schedule, the delivery quantities, and the vehicle routes. This combined inventory and distribution problem is termed as the Inventory Routing Problem (IRP). It achieves a trade-off between frequent, small-sized deliveries (low inventory costs but high transportation costs), and large shipments (low transportation but high inventory holding costs). Since IRP is a modified vehicle routing problem (Campbell & Savelsbergh, 2004), it is NP-hard. Industrial applications are driven by the ability to generate good quality solutions rapidly. Therefore, solving real-life sized instances in a computationally efficient manner is very important. In our paper, we propose two efficient mixed integer linear programming (MILP) models to find optimal solutions for the IRP with backorders and no split deliveries. We also propose two heuristics that give near-optimal solutions to the problem. We report additional numerical experiments to evaluate the performance of the MILP models and the heuristics. We ran the MILP models for a maximum of 2 h (7200 secs) on randomly generated data instances. The numerical experiments show that the proposed MILP models are computationally efficient. The optimality gap is no more than 3% in all the tested cases. We also show that the proposed heuristics provide near-optimal solutions. In Sect. 1.1, we discuss some of the literature on IRP and how our paper is different from the existing papers.

1.1 Literature Review Infinite horizon IRP determines inventory replenishment and vehicle routing decisions over the infinite horizon for a continuous review system with a constant demand rate. Replenishment strategies that can be repeated over time are developed to minimize the long-run average costs (Refer to Anily & Federgruen, 1990; Viswanathan & Mathur, 1997). In the finite horizon IRP literature, a wide range of solution methodologies (exact methods and heuristics) has been employed. Some of the papers are discussed below.

Mathematical Models and Heuristics …

45

Archetti et al. (2007) were the first to develop a branch and cut algorithm for a single product, single-vehicle IRP with two different inventory policies at the retailer. Coelho and Laporte (2013a) proposed a branch and cut algorithm for solving a multiproduct IRP. They also discussed exact methods for three different variants of IRP (Coelho & Laporte, 2013b). Bertazzi et al. (2002) developed a two-phase heuristic for a deterministic demand, finite horizon IRP with an order-up to-level inventory policy at the retailer. Campbell and Savelsberg (2004) developed a decomposition approach for solving a rolling horizon framework IRP with a constant demand rate. We note that the papers assume that some kind of inventory replenishment policy exists at the retailer. This assumption may not be true in practice. However, it reduces the search space considerably. Also, a periodic review framework with deterministic but dynamic demands is more relevant for the finite horizon IRP, especially in an enterprise resource planning (ERP) system. Therefore, we propose a finite horizon IRP with a periodic review inventory system and deterministic, dynamic retailer demands. We do not assume any replenishment policy at the retailers. The papers discussed above also assume that the backorders are not allowed at the retailers. However, Abdelmaguid et al. (2009) argued that backordering is beneficial when savings in transportation cost can be realized by delaying distribution. Backorders might become inevitable especially when there are capacity constraints, Preethi et al. (2018) proposed mathematical models for a finite horizon IRP with backorders. They showed that their mathematical model performs better than the model proposed by Abdelmaguid et al. (2009). Both the above-mentioned papers assume that split deliveries are allowed in the IRP. However, split deliveries will lead to problems in terms of inventory traceability. In practice, a single vehicle is sent out for delivery from the supplier’s end keeping the customer’s operational convenience in mind. To this end, conventional vehicle routing problems also assume that split deliveries by multiple vehicles are not allowed (see Laporte, 1992; Toth & Vigo, 2002). Therefore, we consider a finite horizon IRP with backorders and no split deliveries in our study. The paper is organized as follows. We discuss the problem specification and the two proposed MILP model formulations in Sect. 2. The computational experiments for the optimal MILP models are given in Sect. 3. In Sect. 4, we present the two proposed heuristics, their computational performance, and solution quality compared against MILP model 1 from Sect. 2. Section 5 summarizes the contribution of our work and explores further possible extensions.

2 Problem Specification We consider a finite horizon, two-echelon, Inventory Routing Problem with deterministic but dynamic retailer demands. A single vendor serves N number of retailers using V heterogeneous vehicles each with a capacity of Capv . The length of the discrete planning horizon is T .

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K. R. Preethi et al.

The sequence of events in any period t is as follows: The vehicles assigned for delivery are dispatched from the vendor. Retailers that are scheduled for delivery receive their orders. Then, a demand Dem i,t is realized at each retailer i for the current period t. The retailer satisfies the demand using on-hand inventory Ii,t . Excess demands Bi,t are backordered. Retailer i incurs a holding cost of h i per unit per period and a backordering cost of πi per unit per period. The vendor incurs a fixed transportation cost of Fv if vehicle v is used in any period. Without loss of generality, we assume that the variable transportation cost per unit distance is 1 for all the vehicles. The vendor determines the routes, delivery quantities, and schedules which minimizes the total supply chain cost. The supply chain cost is the sum of fixed and variable transportation costs, inventory holding costs, and backordering costs.

2.1 Mixed Integer Linear Programming Model 1 The parameters of the finite horizon IRP are given in Table 1. The proposed model consists of integer decision variables and continuous decision variables. v,t are indicator variables which take a value 1 if vehicle v is used in period t. We call them the vehicle assignment variables because they contain the information on which vehicles are dispatched for delivery in each period. The binary integer variables δi,v,t are equal to 1, if retailer i is served by vehicle v in period t. They are called retailer allocation variables because they determine the set of retailers allocated to each vehicle in every period. Also, these variables provide us with delivery schedules for retailers. X i, j,v,t are indicator variables associated with the arc between i and j. X i, j,v,t is equal to 1 if vehicle v visits retailer j immediately after visiting retailer i in period t. They help us determine the routes taken by the vehicles. Subscript B is used to denote the vendor node and i = 1, 2, …, N is used to denote the retailer nodes. We also use dummy nodes to indicate the vendor in this model. There is a total of 2V dummy nodes for the vendor, starting from N + 1 to N + V + V . Imagine that at the beginning of every period, vehicle 1 is stationed at node N + 1. It leaves from the node N + 1 when it leaves for distribution. After delivery, vehicle 1 returns to N + V + 1th node at the end of the period. Similarly, any vehicle ν ∈ 1, 2, . . . V is stationed at the node N + v at the beginning of a period. It leaves from node N + v and returns to node N + V + v after the deliveries are made. Now, we define binary routing variables from and to the vendor node using the dummy nodes (vendor). x N +v,i,t = 1 means that the vehicle v travels from the vendor  to retailer i in period t. xi,N +V +v,t = 1 means that the vehicle v returns to the vendor from retailer i in period t. The dummy nodes break the vendor to vendor circuits/routes into paths. This enables us to solve a Hamiltonian path problem for the vehicles instead of solving a Hamiltonian cycle. Consider a delivery route B-1–3-5-B. This is a closed route that takes more computational effort to solve. The presence of our dummy nodes

Mathematical Models and Heuristics … Table 1 Parameters of the IRP

47

Subscripts i, j

Retailers

v

Vehicles

t

Periods

B

The vendor

N + 1 to N + 2V

Dummy nodes for the vendor

Notations

Definitions (Parameters)

N

Total number of retailers

V

Total number of vehicles available for delivery in each period

T

Number of discrete periods

Dem i,t

Demand at retailer i in period t

Capv

The capacity of vehicle v

d j,i

Distance travelled to reach retailer i from retailer j

d B,i

Distance travelled to reach retailer i from the vendor

di,B

Distance travelled to reach the vendor from retailer i

hi

Inventory holding cost per unit per period at retailer i

πi

Backordering cost per unit per period at retailer i

Ci

Inventory storage capacity at retailer i

Fv

Fixed cost of operating vehicle v

breaks this circuit into a path between the dummy nodes N + v and N + V + v where v is the vehicle assigned for delivering in this route. This is illustrated in Fig. 1. The dummy nodes also prevent the formation of any sub-tours containing the vendor node. Therefore, formulating the IRP with dummy nodes saves considerable computational time. All the decision variables including the inventory-related variables are explained in Table 2.

Fig. 1 Illustration of dummy nodes extending the routes into paths

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K. R. Preethi et al.

Table 2 Decision variables used in MILP model 1 Notations

Description  1, If vehicle v is dispatched in period t

v,t

0, otherwise 

δi,v,t

1, if retailer i is served by vehicle v in period t 0, otherwise



X i, j,v,t

1, if vehicle v visits retailer j immediately after visiting retailer i in period t 0, otherwise









x N +v,i,t

1, if vehicle v travels from the vendor to retailer i in period t 0, otherwise

xi,

N +V +v,t

1, if vehicle v travels from retailer i to the vendor in period t 0, otherwise

Q i,v,t

Quantity delivered to retailer i using vehicle v in period t

Ii,t

Inventory level at retailer i in period t

Bi,t

Number of back ordered units at retailer i in period t

CQi,t 

CQv,t

The cumulative load carried by any vehicle when it leaves retailer i in period t Overall (cumulative) load carried by vehicle v while leaving the vendor in period t

Minimize Total Supply Chain Cost (Z 1 ) =

T  V 

Fv v,t +

t=1 v=1

+

+

T  N  

h i Ii,t + πi Bi,t



t=1 i=1

T  V  N 

N 

t=1 v=1 i=1

j =1 j = i

T  V  N 

X i, j,v,t di, j +

T  V  N 



x N +v,i,t d B,i

t=1 v=1 i=1



xi,N +V +v,t di,B

(1)

t=1 v=1 i=1

Subject to V   v=1

V     X i, j,v,t + X j,i,v,t ≤ δi,v,t + δ j,v,t /2 v=1

∀i = 1, 2.., N − 1, j = i + 1, i + 2, .., N , ∀t

(2)

Mathematical Models and Heuristics …

49

V 

δi,v,t ≤ 1 ∀i, t

(3)

δi,v.t ≤ N v,t ∀v, t

(4)

v=1 N  i=1 N 

X j,i,v,t + x N +v,i,t = δi,v,t ∀i, v, t

(5)

 X i, j,v,t + xi,N +V +v,t = δi,v,t ∀i, v, t

(6)

j = 1, j = i N 

j = 1, j = i N 

x N +v,i,t = v,t ∀v, t

(7)

 xi,N +V +v,t = v,t ∀v, t

(8)

i=1 N  i=1

Q i,v,t ≤ Mi δi,v,t ∀i, v, t N 

(9)

Q i,v,t ≤ Capv v,t ∀v, t

(10)

i=1 V  T 

Q i,v,t ≤

v=1 t=1

T 

Demi,t ∀i

(11)

t=1

Ii,t − Bi,t = Ii,t−1 − Bi,t−1 − Demi,t +

V 

Q i,v,t ∀i, t

(12)

v=1

Ii,t ≤ Ci ∀i, t CQv,t =

N 

Q i,v,t ∀v, t

(13)

(14)

i=1

CQi,t ≤ maxcap

V  v=1

δi,v,t ∀i, t

(15)

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K. R. Preethi et al.

  CQi,t ≤ CQv,t − Q i,v,t + maxcap 1 − x N +v,i,t ∀i, v, t CQi,t ≤ CQ j,t −

V  v=1

 Q i,v,t + maxcap 1 − 

CQi,t ≤ maxcap 1 −

V 

 X j,i,v,t

v=1 V 

(16)

 xi,N +V +v,t

∀i, j, j = i, t

(17)

 ∀i, t

(18)

v=1

where maxcap and Mi are very large numbers (Big M).  They are tightened using the following expressions: maxcap = max Capv , Mi = v 

 T min t=1 Demi,t , maxcap . The constraints are explained below. The total supply chain cost consists of fixed transportation cost, inventory holding, and backordering cost, and variable transportation cost. Equation (1) is the objective function that represents the total supply chain cost. The routing variables between any two retailers can be active only if both the retailers are scheduled for delivery in a period (Eq. 2). A retailer is visited by no more than one vehicle in any period (Eq. 3). A vehicle could visit a maximum of N retailers in a period (Eq. 4). When a retailer is scheduled for delivery by a vehicle, one of the arcs that end at the retailer node is active (Eq. 5). At the same time, one of the arcs starting from the retailer node is active in the solution (Eq. 6). If a vehicle is scheduled for dispatch in a period, routing variables from and to the corresponding dummy vendor nodes should be active in that period (Eqs. 7 and 8). The retailer receives a positive quantity only when it is allocated to a vehicle in a period (Eq. 9). The total load that a vehicle carries should be less than or equal to its capacity (Eq. 10). The total quantity that a retailer receives over the entire time horizon should be less than or equal to the retailer’s total demand in that horizon (Eq. 11). A set of constraints (Eqs. 2 and 11) are used to tighten the model. Equation (12) represents the inventory balance constraint. The inventory level at a retailer should be less than its inventory storage capacity (Eq. 13). Then, we have sub-tour elimination constraints (Eqs. 14–18). The sub-tour elimination in this model works in the following manner. The cumulative load is highest when the vehicle leaves the vendor. It keeps decreasing as the vehicle makes the deliveries and becomes zero when the vehicle returns to the vendor. So, the cumulative load when the vehicle leaves any given node should be less than or equal to the cumulative load when the vehicle leaves the preceding node. Consider a sub-tour 2-4-5-2. According to constraints (14–18), for any such sub-tour to exist, CQ2,t ≥ CQ4,t ≥ CQ5,t ≥ CQ2,t which is infeasible. Therefore, sub-tours will be eliminated in the optimal solution.

Mathematical Models and Heuristics …

51

2.2 Mixed Integer Linear Programming Model 2 In this subsection, we propose an alternate MILP model for the Inventory Routing Problem. We retain vehicle assignment and retailer allocation variables from MILP model 1. However, we reduce the number of decision variables in model 2 by using three-dimensional routing variables X i, j,t , unlike the previous model which uses four-dimensional X i, j,v,t . Therefore, the number of such routing variables becomes N 2 T instead of N 2 TV. However, we need additional constraints that link the routing variables X i, j,t with retailer allocation variables δi,v,t (Eqs. 20 and 21), to capture the complete information on each vehicle’s routes. We have other network flow constraints (Eqs. 22, 23, 26–29), vehicle capacity constraints (Eq. 31), inventory balance constraints (Eq. 33), inventory storage capacity constraints (Eq. 34), subtour elimination constraints (Eqs. 35–37), and no split deliveries constraint (Eq. 24) in this model. We use cumulative distance (C Di,t ) variables to eliminate sub-tours. The cumulative distance travelled by a vehicle when it reaches a node should be greater than or equal to the cumulative distance at the preceding node. Therefore, a subtour 2-4-5-2 will be rendered infeasible as the constraints would specify that CD2,t ≤ CD4,t ≤ CD5,t ≤ CD2,t . The new decision variables in MILP model 2 are given in Table 3. Minimize Total Supply Chain Cost (Z 2 ) =

T  v 

Fv v,t +

t=1 v=1

+

T  V  N 

T  N 

(h i Iit + πi Bit )+

t=1 i=1

x N +v,i,t d B,i

T  N  N 

X i, j,t di, j

t=1 i=1 j=1

+

t=1 v=1 i=1

T  V  N 

 xi,N +V +v,t d B,i

(19)

t=1 v=1 i=1

Subject to X i, j,t + X j,i,t ≤

 V  δi,v,t + δ j,v,t ∀ i = 1, 2, .., N − 1, 2 v=1

j = i + 1, i + 2, .., N , ∀t

(20)

Table 3 Decision variables in MILP model 2 Notations Definitions (decision variables)  X j,i,t 1, if any vehicle moves from retailer j to retailer iin period t 0, otherwise CDi,t

The cumulative distance travelled by any vehicle when it reaches retailer i in period t

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K. R. Preethi et al.

δ j,v,t − δi,v,t ≤ 1 − X j,i,t

∀v, t, i, j and j = i

x N +v,i,t ≤ δi,v,t

∀i, v, t

 xi,N +V +v,t ≤ δi,v,t V 

(22)

∀i, v, t

δi,v,t ≤ 1

(21)

(23)

∀i, t

(24)

v=1 N 

δi,v.t ≤ N v,t ∀ v, t

(25)

i=1 N 

X j,i,t +

V 

=

v=1

j = 1, j = i N 

x N +v,i,t

X i, j,t +

V 

N 

δi,v,t ∀i, t

(26)

v=1

 xi,N +V +v,t =

v=1

j =1 j = i

V 

V 

δi,v,t ∀i, t

(27)

v=1

x N +v,i,t = v,t ∀v, t

(28)

 xi,N +V +v,t = v,t ∀v, t

(29)

i=1 N  i=1

Q i,v,t ≤ Mi δi,v,t ∀i, v, t N 

(30)

Q i,v,t ≤ Capv v,t ∀v, t

(31)

i=1 V  T  v=1 t=1

Q i,v,t ≤

T 

Demi,t

∀i

(32)

t=1

Ii,t − Bi,t = Ii,t−1 − Bi,t−1 − Demi,t +

V 

Q i,v,t , ∀i, t

(33)

v=1

Ii,t ≤ Ci ∀i, t

(34)

Mathematical Models and Heuristics …

53

CDi,t ≤

MV

V 

δi,v,t

∀i, t

(35)

v=1

CDi,t ≥

V  

d B,i x N +v,i,t



∀ i, t

(36)

v=1

  CDi,t ≥ CD j,t + d j,i − MV 1 − X j,i,t

∀i, t, j and j = i

(37)

where maxcap, M j and Mv are very large numbers (big M). maxcap and Mi s are 

N  defined as before and MV = i=1 d B,i + max j, j =i d j,i .

3 Computational Experiments The computational performance of the MILP models is tested on randomly generated data instances. The retailer node locations are generated randomly on a 20 × 20 square. The vendor is assumed to be located at the centre of the square. The Euclidean distance between the nodes gives the distance between the nodes. Multiple levels of N and T are used to create different problem sizes, and three different data instances are created for each problem size. The retailer demands and inventory cost parameters are generated from standard distributions while other parameters take fixed values. The details of the data generation process are given in Table 4. Table 4 Parameter values used in the computational analysis Parameters

Data used/generated

Initial inventory

Ii,0 = 0 ∀i

Initial backorders

Bi,0 = 0 ∀i

Retailer demands

Demi,t ∼ Uniform(50, 100)∀i, t

Inventory holding costs/unit/period

h i ∼ Normal(0.1, 0.02) ∀i

Backordering cost/unit/period

πi ∼ Normal(5, 0.5) ∀i

Inventory storage capacity

Ci = 120 ∀i

Vehicle capacity

Capv = 250 ∀v

Fixed transportation cost

Fv = 10 ∀v

Variable transportation cost per unit distance

1 ∀v

Parameters

Levels used

Number of retailers

N = 5, 6, 7, 8, 9, 10

Periods

T = 3, 5

Number of vehicles

V =3

54

K. R. Preethi et al.

The proposed MILP models 1 and 2 were run on a computer with Windows 7 OS and an Intel i5 processor and 8 GB RAM. Each instance was run for a maximum CPU time of 2 h (7200 s). We have reported the upper and lower bounds obtained from CPLEX’s implementation of the branch and cut algorithm in Table 5. As the program runs, the upper and lower bounds converge to an optimum. The optimality gap is a measure of the relative difference between the bounds and is defined as (Upper Bound − Lower Bound)/Upper Bound. The algorithm stops and gives the optimal solution when this gap becomes close to zero (≤10−6 ). If an optimal solution is reached within the 2 h run time, we report the optimal solution (Lower bound = Upper bound) and the CPU run time in seconds. If the run time exceeds 2 h, we report the best bounds that were obtained within the 2 h. In Table 5, when the optimality gap is greater than 0, it means that an optimal solution was not reached before 2 h. We find that the models were able to obtain optimal solutions for most of the test cases. Both the models reach optimal solutions for 26 out of the 30 tested instances within 2 h of run time. As the problem size increases, the computational time to reach the optimum increases as well. We have reported that the optimality gap is not more than 3% (0.03) in all the tested cases. We also observe that the models do not dominate each other. They are rather complementary in terms of their computational performance. Both models offer competitive solutions in a computationally efficient manner.

4 Heuristics for the Inventory Routing Problem We know that IRP is an NP-hard problem. As evidenced by the computational experiments above, the CPU time increases as the problem size increases. It takes a huge computational effort to find the optimal solution to the problem. Therefore, we propose two simple heuristics that provide near-optimal solutions to the IRP. Both the proposed heuristics are two-phased. The first phase finds feasible delivery schedules and delivery quantities. In the second phase, delivery routes are obtained. The details of the heuristics are presented below.

4.1 Heuristic 1 The proposed heuristic is two-phased. In the first phase, we find the vehicle assignments and retailer allocations using an MILP model. We obtain the information on when the deliveries should be made, by which vehicle, and what the corresponding delivery quantities are. In the second phase, we find the routes that should be followed by the vehicles when making those scheduled deliveries. Given the delivery schedules and quantities from the first phase, we just need to solve a travelling salesman problem (TSP) for each assigned vehicle in every period. We know that the optimization of routing cost is computationally costly. Therefore, a combination of simple

T

3

3

3

5

5

5

3

3

3

5

5

5

3

3

3

5

5

5

3

3

N

5

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7

8

8

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

V

287.23

243.87

440.81

430.69

390.25

261.21

251.34

251.67

392.27

408.40

309.54

220.65

250.63

202.72

338.29

338.99

341.33

217.48

175.53

207.65

287.23

243.87

440.81

430.69

390.25

261.21

251.34

251.67

392.27

408.40

309.54

220.65

250.63

202.72

338.29

338.99

341.33

217.48

175.53

207.65

19.64

55.05

543.98

66.44

253.60

79.20

18.10

17.21

415.24

339.88

2872.32

8.39

1.34

5.29

2.29

196.98

65.57

0.72

0.69

0.34

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

287.23

243.87

440.81

430.69

390.25

261.21

251.34

251.67

392.27

408.40

309.54

220.65

250.63

202.72

338.29

338.99

341.33

217.48

175.53

207.65

MILP model 2 Optimality gap

Lower bound

CPU time (Sec)

Lower bound

Upper bound

MILP model 1

Table 5 Computational performance of the proposed MILP models

287.23

243.87

440.81

430.69

390.25

261.21

251.34

251.67

392.27

408.40

309.54

220.65

250.63

202.72

338.29

338.99

341.33

217.48

175.53

207.65

Upper bound

139.75

125.94

1276.45

151.46

412.94

129.98

107.16

26.05

6597.86

339.57

1282.44

7.41

1.65

1.56

2.96

291.18

44.60

0.56

0.25

0.31

CPU time (Sec)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(continued)

Optimality gap

Mathematical Models and Heuristics … 55

T

3

5

5

5

3

3

3

3

3

3

N

8

8

8

8

9

9

9

10

10

10

3

3

3

3

3

3

3

3

3

3

V

753.92

1027.31

362.09

330.24

312.51

353.08

394.55

441.51

418.08

269.46

764.50

1031.91

362.09

330.24

321.88

353.08

394.55

446.03

424.72

269.46

7202.41

7200.21

448.71

168.09

7202.24

1299.27

1480.17

7200.00

7200.00

26.33

0.01

0

0

0

0.03

0

0

0.01

0.02

0

764.50

1019.92

362.09

330.24

321.88

353.08

389.17

441.15

411.91

269.46

MILP model 2 Optimality gap

Lower bound

CPU time (Sec)

Lower bound

Upper bound

MILP model 1

Table 5 (continued)

764.50

1031.91

362.09

330.24

321.88

353.08

394.55

446.03

424.72

269.46

Upper bound

1167.93

7202.94

597.03

76.53

1635.56

162.37

7200.55

7200.00

7200.00

141.81

CPU time (Sec)

0

0.01

0

0

0

0

0.01

0.01

0.03

0

Optimality gap

56 K. R. Preethi et al.

Mathematical Models and Heuristics …

57

travelling salesman problem (TSP) heuristics is used to find the routes in the second phase.

4.1.1

Phase 1

The following MILP gives delivery schedules and delivery quantities for the entire planning horizon. Minimize Z H 1 =

T  V 

T  N  

Fv v,t +

t=1 v=1

δi,v,t di∗ +

t=1 v=1 i=1 V 



t=1 i=1

T  V  N 

+

h i Ii,t + πi Bi,t

T  V 

v,t d B∗

(38)

t=1 v=1

δi,v,t ≤ 1, ∀ i, t

(39)

v=1 N 

δi,v.t ≤ N v,t ,

∀v, t

(40)

∀i, v, t

(41)

∀v, t

(42)

i=1

Q i,v,t ≤ Mi δi,v,t , N 

Q i,v,t ≤ Capv v,t

i=1 V  T  v=1 t=1

Q i,v,t ≤

T 

Demi,t ∀i

(43)

t=1

Ii,t − Bi,t = Ii,t−1 − Bi,t−1 − Demi,t +

V 

Q i,v,t , ∀i, t

(44)

v=1

Ii,t ≤ Ci ∀i, t

(45)

where d B∗ is the minimum   distance travelled to reach the vendor from any retailer, d B∗ = mini = 1,..,N di,B , and di∗ is the minimum distance travelled  to reach retailer d j,i . i from either the vendor or other retailers, di∗ = min j = B,1,..,N j = i In this phase, we retain the vehicle assignment and retailer allocation variables which determine which retailers are scheduled for delivery and which vehicles are making the deliveries in each period. The inventory-related variables and the delivery quantities are also included in the first phase. However, the precedence-based routing

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variables X i, j,v,t and hence the routing decisions are eliminated from this part of the analysis. Here, the objective function (Z H 1 ) is the sum of fixed transportation cost, inventory holding, and backordering costs and an approximation of the variable transportation cost. The approximation is the sum of the minimum distance travelled to reach each active node over the entire finite time horizon. It is given by the last two terms of the objective function. It is worthwhile to note that the first phase also gives a meaningful lower bound for the IRP.

4.1.2

Phase 2

We have the vehicle retailer assignments and the quantities to be delivered from phase 1. Now, constructing the tours for the vehicles in each period will give a feasible solution to the IRP. Tours are constructed using two simple travelling salesman problem (TSP) heuristics. First, we use the nearest neighbourhood heuristic to form the tours. The tours are then improved by using the pairwise interchange heuristic. The routing costs from this phase combined with the fixed transportation cost and the inventory costs from phase 1 give a feasible solution (upper bound) for the IRP. The nearest neighbourhood algorithm is implemented as follows. Let us just consider a single vehicle and the set of retailers allocated to it, in the first period. The vehicle starts at the vendor node, visits the nearest retailer node, and makes the delivery. From there, the vehicle visits another unvisited retailer node which is closest to the current node. The vehicle returns to the vendor node after all the allocated deliveries are made. Repeat this for all the vehicles assigned for delivery in each period. Once we have the nearest neighbourhood solution, pairwise interchanges of the retailers are made to search for an improved solution.

4.2 Heuristic 2 4.2.1

Phase 1

In this heuristic, the first phase is MILP based in which we find the delivery schedules and quantities. We extend the MILP formulation from Heuristic 1 by including additional variables and constraints. The additional variables help us keep track of the position in which the retailers are visited by a vehicle. This provides extra information on the routes and therefore, a better approximation for the variable transportation cost. The additional variables in the model are presented in Table 6. Like Heuristic 1, the phase 1 objective function consists of the fixed cost of transportation, inventory holding, and backordering costs and a better approximation of the variable transportation cost. This approximation given by the last part of the objective function consists of two parts. For retailers that are visited either in the first position or in the last position,

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Table 6 Decision variables used in Heuristic 2–Phase 1 Notations λi,1,v,t

Definitions  1, if vehicle v delivers first to retailer i in period t 0, otherwise

λi,2,v,t



1, if vehicle v does not visit retailer i in the first position in period t 0, othherwise

λi,3,v,t



1, if vehicle v delivers last to retailer i in period t 0, otherwise



v,t



1, if vehicle v serves only one retailer in period t 0, otherwise

the distance between the vendor and the retailer is added to the variable cost approximation. For the remaining active retailers, the least distance travelled to reach that retailer from any other retailer is added  to the variable cost component. We have d j,i . defined di∗∗ = min j = 1, .., N j = i Consider a situation in which a vehicle v visits only one retailer in a given period t. Here, the same retailer is visited in the first and the last position by the vehicle. Therefore, we need both the values of λi,1,v,t and λi,3,v,t to be equal to 1 in that  case. To ensure this, we introduce another binary variable v,t in the model. This variable takes a value of 1 if vehicle v serves only one retailer in period t. Minimize Z H 2 =

T  V 

Fv v,t +

t=1 v=1

T  N  

h i Ii,t + πi Bi,t



t=1 i=1

T  V  N    + d B,i λi,1,v,t + di,B λi,3,v,t + λi,2,v,t di∗∗

(46)

t=1 v=1 i=1

In addition to the constraints (39) to (45), we add the following constraints to the MILP model. N 

λi,1,v,t = v,t ∀v, t

(47)

i=1

λi,1,v,t + λi,2,v,t = δi,v,t ∀i, v, t

(48)

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λi,3,v,t = v,t ∀v, t

(49)

δi,v,t ≥ 2 − (N + 1)v,t ∀v, t

(50)

  δi,v,t ≤ 1 + (N + 1) 1 − v,t ∀v, t

(51)

i=1 N  i=1 N  i=1

λi,1,v,t + λi,3,v,t ≤ δi,v,t + v,t ∀i, v, t

(52)

N N     δi,v,t + v,t ∀v, t λi,1,v,t + λi,2,v,t + λi,3,v,t =

(53)

i=1

i=1

λi,1,v,t + λi,2,v,t + λi,3,v,t ≤ 2δi,v,t ∀i, v, t

(54)

A vehicle should visit exactly one retailer in the first position if the vehicle is assigned for delivery (Eq. 47). A retailer allocated for delivery is visited either in the first position or other positions by a vehicle (Eq. 48). A vehicle should visit exactly  one retailer in the last position if it is assigned for delivery (Eq. 49). v,t takes a value 1 if the vehicle v serves only one retailer in period t (Eqs. 50 and 51). Both λi,1,v,t and λi,3,v,t are equal to 1 if retailer i is the only retailer visited by vehicle v in period t (Eqs. 52 and 53). This enables us to add the distance from the vendor to retailer i and back to the vendor to the variable cost approximation. To further tighten the model, we add constraints (Eq. 54) in the model. Phase 1 of Heuristic 2 will also provide a meaningful lower bound to the problem.

4.2.2

Phase 2

Given the delivery schedules and quantities from Phase 1, we obtain the routes using the nearest neighbourhood heuristic followed by the implementation of the pairwise interchange heuristic. The routes together with the phase 1 delivery schedules and delivery quantities provide a heuristic solution to the Inventory Routing Problem. The routing cost from Phase 2 plus the fixed transportation cost and the inventory costs from Phase 1 is the total supply chain cost and is a feasible solution (upper bound) to the IRP.

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4.3 Computational Analysis and Discussion The performance of the proposed heuristics is presented in Table 7. We used the same set of generated data instances from Sect. 3 to evaluate the computational efficiency Table 7 Computational performance of the heuristics N

T

V

Upper bound fromMILP model 1

Heuristic 1

Heuristic 2

Feasible solution

Deviation (in %)

Feasible solution

Deviation (in %)

5

3

3

207.65

264.54

27

262.06

26

5

3

3

175.53

199.14

13

197.24

12

5

3

3

217.48

246.35

13

246.31

13

5

5

3

341.33

367.47

8

379.78

11

5

5

3

338.99

373.74

10

373.74

10

5

5

3

338.29

361.52

7

369.14

9

6

3

3

202.72

238.37

18

238.75

18

6

3

3

250.63

287.76

15

284.41

13

6

3

3

220.65

239.43

9

246.46

12

6

5

3

309.54

323.08

4

344.49

11

6

5

3

408.40

483.26

18

441.34

08

6

5

3

392.27

406.77

4

430.94

10

7

3

3

251.67

322.8

28

302.45

20

7

3

3

251.34

306.93

22

299.04

19

7

3

3

261.21

303.17

16

311.73

19

7

5

3

390.25

448.92

15

444.68

14

7

5

3

430.69

503.19

17

537.18

25

7

5

3

440.81

481.93

9

517.33

17

8

3

3

243.87

287.21

18

278.73

14

8

3

3

287.23

341.22

19

335.18

17

8

3

3

269.46

286.85

6

295.87

10

8

5

3

424.72

515.79

21

519.23

22

8

5

3

446.03

549.1

23

575.19

29

8

5

3

394.55

469.11

19

488.62

24

9

3

3

353.08

428.02

21

445.57

26

9

3

3

321.88

383.81

19

352.34

9

9

3

3

330.24

406.16

23

390.36

18

10

3

3

362.09

439.53

21

436.34

21

10

3

3

1031.91

1075.58

4

10

3

3

764.50

834

9

1099.4 848.64

7 11

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of the proposed heuristics. We show that both the heuristics generate near-optimal solutions to the IRP. Heuristic 1 gives good quality feasible solutions to the problem for all the tested instances within a few seconds of run time (not more than 10 s). We compared the performance of the heuristics against the upper bounds obtained from MILP model 1 (described in Sect. 2). The percentage deviation of the heuristic solutions from the upper bounds obtained using MILP model 1 is reported in Table 7. We observe that the maximum deviation is 28% among the tested cases. The computational efficiency of heuristic 1 is more evident in larger problem sizes. Heuristic 2 also obtains good quality solutions within reasonable CPU time. It takes not more than 2 min (except in 1 case) to obtain the solution to the problem. The maximum deviation of heuristic 2 solutions from the MILP model’s upper bound is 29%. Again, the advantage of heuristic 2 over the optimal MILP model is more pronounced in larger problems. The quality of heuristic 1 solutions is comparable to that of heuristic 2’s solution, but the latter runs longer. This is expected as the first phase of heuristic 2 has more decision variables and constraints than heuristic 1. However, it is worthwhile to note that the lower bounds from heuristic 2 (first phase) will be tighter. Both the heuristic solutions can be used as initial seeding solutions for a metaheuristic.

5 Conclusion 5.1 Theoretical Contributions We developed two efficient MILP models to find optimal solutions to the IRP with backorders and no split deliveries. We showed that the models are computationally efficient and can solve smaller instances easily. For larger instances, we were able to obtain close bounds within 2 h of run time. The optimality gap was no more than 3% in the tested data instances. We further proposed two MILP-based heuristics that give near-optimal solutions to the NP-hard IRP. The heuristics can be used to generate initial solutions to a metaheuristic.

5.2 Future Research We do not account for delivery times in the model. We assume that all the deliveries could be made within a given period. Therefore, there is potential for future research which includes delivery times and delivery time windows. Future research may also incorporate the vendor’s production decisions in the analysis. Alternatively, the vendor’s inventory planning problem for ordering from an external supplier (a three-echelon supply chain) can also be considered.

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Acknowledgements Note from the first author—I am grateful to Indian Institute of Technology, Madras for funding the research. I would also like to thank Nanyang Technological University, Singapore, for its support in carrying out the research.

References Abdelmaguid, T. F., Dessouky, M. M., & Ordóñez, F. (2009). Heuristic approaches for the inventoryrouting problem with backlogging. Computers & Industrial Engineering, 56(4), 1519–1534. Anily, S., & Federgruen, A. (1990). One warehouse multiple retailer systems with vehicle routing costs. Management Science, 36(1), 92–114. Archetti, C., Bertazzi, L., Laporte, G., & Speranza, M. G. (2007). A branch-and-cut algorithm for a vendor-managed inventory-routing problem. Transportation Science, 41(3), 382–391. Bertazzi, L., Paletta, G., & Speranza, M. G. (2002). Deterministic order-up-to level policies in an inventory routing problem. Transportation Science, 36(1), 119–132. Campbell, A. M., & Savelsbergh, M. W. (2004). A decomposition approach for the inventory-routing problem. Transportation Science, 38(4), 488–502. Campbell, A., Clarke, L., Kleywegt A., & Savelsbergh, M. (1998). The inventory routing problem. Fleet management and logistics (pp. 95–113) Coelho, L. C., & Laporte, G. (2013a). A branch-and-cut algorithm for the multi-product multivehicle inventory-routing problem. International Journal of Production Research, 51(23–24), 7156–7169. Coelho, L. C., & Laporte, G. (2013b). The exact solution of several classes of inventory-routing problems. Computers & Operations Research, 40(2), 558–565. Coelho, L. C., Cordeau, J. F., & Laporte, G. (2013). Thirty years of inventory routing. Transportation Science, 48(1), 1–19. Laporte, G. (1992). The vehicle routing problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(3), 345–358. Preethi, K. R., Rajendran, C., & Viswanathan, S. (2018). Lower bounding methods for an inventory routing problem with backorders. In: X. Xu, et al. (Ed.), Proceedings of International Conference on Computers and Industrial Engineering 48th 2018 (Vol. 3, p. 2022). Toth, P., & Vigo, D. (2002). The vehicle routing problem. In Society for industrial and applied mathematics. Viswanathan, S., & Mathur, K. (1997). Integrating routing and inventory decisions in one-warehouse multiretailer multiproduct distribution systems. Management Science, 43(3), 294–312. Waller, M., Johnson, M. E., & Davis, T. (1999). Vendor-managed inventory in the retail supply chain. Journal of Business Logistics, 20, 183–204.

Modeling a Supply Chain with Price-Dependent Stochastic Demand and Discrete Transportation Lead Time Susheel Yadav, Anil K. Agrawal, and Manu K. Vora

Abstract This article deals with the modeling of a supply chain where a single manufacturer sells its product through multiple buyers. All the buyers face a pricedependent stochastic demand. The manufacturer sends the item to the buyers in multiple shipment of equal-sized sub-batches. A third-party logistics ships the item to the buyers. The different types of the vehicles provided by this third party have buyerspecific transportation cost per shipment and transportation lead time per shipment. Under these conditions, the safety stock of the retailers and the discrete transportation lead time become crucial for the supply chain, especially when a minimum service level is defined for each retailer. The problem is formulated as a Mixed Integer Nonlinear Programming (MINLP) model and is solved for the maximum supply chain profit. Finally, important managerial insights are drawn for the safety factor and service levels of the buyers. Keywords Price-sensitive stochastic demand · Discrete transportation lead time · Service level · MINLP · SCM

1 Introduction Today, the businesses are competing along the supply chain (Li et al., 2006) and this makes supply chain management (SCM) an important task to be handled judiciously. The presence of a large number of supply chain partners associating themselves into various activities and functions makes the chain complex (Arshinder et al., 2008). Efficient integration among the supply chain partners emerged as an important strategy for SCM as the competition proliferated (Ahmadizar et al., 2015). Arshinder et al. (2011) mentioned joint consideration of the functions and processes by supply S. Yadav (B) · A. K. Agrawal Department of Mechanical Engineering, Indian Institute of Technology (BHU), Varanasi 221005, India e-mail: [email protected] M. K. Vora Business Excellence Inc, Naperville, IL 60567-5585, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_4

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chain partners as an important aspect of the supply chain. Joint consideration of the inventories across the supply chain is one such school of thought where the inventory policies of the supply chain partners are decided by keeping the total supply chain in view and not the one of them.

2 Literature Review 2.1 Joint Economic Lot Size (JELS) Models Goyal (1976) proposed Joint Economic Lot Size (JELS) model that aimed to minimize the joint inventory-related costs for the case of a single vendor and single buyer (SVSB). He showed that a considerable saving can be achieved by determining the economic lot size jointly. Later, this model was extended along different considerations. This paper was the building block for many integrated inventory models that were developed by various researchers along different considerations. Banerjee (1986) in his JELS model considered the supplier as a manufacturer supplying product to a buyer on lot-for-lot policy. Goyal (1988) extended this and modified the lot-for-lot supply policy to the policy of supplying multiple sub-batches of equal size after the lot has been completely produced. Lu (1995), and Agrawal and Raju (1996) proposed the strategy of supplying equal lot size batches even during the production, to further reduce the joint inventory cost. Hill (1997, 1999), Goyal (2000), and Goyal and Nebebe (2000) extended the basic JELS model with the consideration of the shipment in unequal-sized batches or in combination of equal- and unequal-sized sub-batches. A lot of research work is available on joint consideration of inventory of supply chain partners. A review of the same is available in the work of Goyal and Gupta (1989), Ben-Daya et al. (2008), and Glock (2012a).

2.2 Inventory Models with Deterministic Demand The classical inventory models assume the demand as constant (Baker and Urban, 1988); however, it need not be so in practice (Gupta, 1994). There are certain factors that affect the demand; one such factor is the selling price of the item which regulates the customer’s demand (Das Roy and Sana, 2011; Agrawal and Yadav, 2020; Yadav et al., 2020). In the early work of Whitin (1955), the demand of the product was taken as a function of the product price. The models which assume the price–demand relationship with certainty are known as deterministic inventory models (Bushuev et al., 2011). This price–demand relationship serves as the input parameter in proposing various kinds of inventory policies. Later, Urban (1992), Abad (1994), Goyal and Gunasekaran (1995), Petruzzi and Dada (1999), You (2005), and Wu et al. (2009) considered the price-dependent demand in their studies. These studies in essence

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determine the inventory policies as well as the pricing decisions for the supply chain. Initial research in JELS problem was concentrated on production shipment schedule in terms of number and size of the batches, mostly considering the demand as exogenous where it does not depend on any other parameter (Jokar and Sajadieh, 2009). Later on, Sajadieh and Jokar (2009), Jokar and Sajadieh (2009), Zanoni et al. (2014), Lin and Lin (2015), Taleizadeh et al. (2016), Taleizadeh and Noori-Daryan (2016), and Chen and Sarker (2017) considered a deterministic price–demand relationship in their study of JELS inventory problems.

2.3 Inventory Models with Stochastic Demand When the deterministic demand assumption is relaxed, the model turns out to be stochastic in nature (Whitin, 1954). Stochastic demand assumption has extensively followed in single-period inventory models over the time. Sharafali and Co (2000) are the first who considered the stochastic demand function in integrated inventory model of a single vendor and a single buyer. Later on, deterministic demand assumption is also relaxed by Ben-Daya and Hariga (2004) in JELS model as they considered the lead time demand to follow normal distribution making the model stochastic in nature. The stochastic demand assumption in JELS model is also followed by Taleizadeh et al. (2010), Jha and Shanker (2013), AlDurgam et al. (2017), Tiwari et al. (2018), and Wangsa and Wee (2018).

2.4 Inventory Models with Price-Dependent Stochastic Demand Though, the retail demand is not only price-sensitive but also stochastic (Ray et al., 2005). Mills (1959) was the first to study the effect of price and randomness on the demand. Later, several authors have studied the effect of price-dependent stochastic nature of demand on pricing and inventory control policies (Petruzzi and Dada, 1999; Ray et al., 2005; Lau et al., 2007; Jadidi et al., 2017). But there is hardly any literature available on the effect of a price-dependent stochastic nature of demand on pricing and inventory policies in the integrated vendor–buyer environment. This study tries to fill this research gap by considering the demand as price-dependent stochastic JELS problem environment.

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2.5 Inventory Models with Lead Time, Transportation Time, and Their Control When the demand is assumed as stochastic in nature, lead time becomes an important concern and its control leads to several benefits (Ouyang et al., 2004). Giri and Roy (2015) asserted that lead time reduction can bring lower safety stock level, low stock-out loss, and improved customer service level, leading toward an efficient supply chain. Liao and Shyu (1991) were the first who argued that lead time can be controlled at an additional cost and considered lead time as a variable rather than assuming it as uncontrollable parameter. Ben-Daya and Raouf (1994) considered lead time and ordered quantity as variable and further classified the additional cost on reducing the lead time into three categories: administrative cost, transportation cost, and supplier’s speedup cost. Liao and Shyu (1991), and Ben-Daya and Raouf (1994) both agreed that transportation time is an important factor in such studies; however, an explicit consideration of transportation time was not included in their models. Glock (2012b) studied the lead time reduction strategies for a single-vendor and single-buyer integrated inventory model with explicit consideration of transportation time. He considered transportation time is related to the total nonproductive time and expressed it as a fraction of the total nonproductive time. Mou et al. (2017) revisited Glock (2012b) and relaxed the assumption that the transportation time is a fraction of total nonproductive time. They expressed the transportation time and total nonproductive as two independent variables. Chang and Lo (2009) considered delivery lead time explicitly in their study; they considered that the total lead time is composed of continuous lead time and discrete lead time. They assumed that delivery time depends on the mode of transportation and can be reduced only by changing the mode of transportation making this reduction as discrete. Multanen (2011) studied empirically the effect of reduction in transportation time on inventory levels and costs. He found that it is not necessary that increasing the transportation cost to reduce the transportation time will increase the total cost of the supply chain because at the same time due to the reduction in the transportation time, savings can be gained in inventory cost. Ng et al. (1997) emphasized any attempt to reduce the cycle time must consider transportation time also, and suggested a number of ways to reduce it. Lead time can be reduced by investing more in advanced transportation systems. The transportation time is a major factor in lead time reduction issues and requires more recognition. Braglia et al. (2016) studied the safety stock for a single-vendor and single-buyer supply chain for a case of controllable lead time. They developed both exact and approximate algorithms.

2.6 Inventory Models with Service-Level Constraint In the situations when the demand exceeds the quantity available, stock-out occurs. Ouyang et al. (1996) considered this stock-out in their inventory model by assuming

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that a fraction of the unmated demand can be backordered at an additional cost while the remaining will be lost. Thus, a stock-out cost is introduced in the inventory model. However in practice, it is very difficult to estimate a stock-out cost, so dealing stockout through the consideration of a service-level constraint is an alternative (Aardal et al., 1989). Ouyang and Wu (1997) also considered a service-level constraint in their inventory model to deal with stock-out. Jha and Shanker (2009a, b) considered a service-level constraint as a substitute of stock-out cost in a single-vendor and single-buyer integrated inventory problem for decaying and non-decaying items, respectively. Li et al. (2011) analyzed two inventory models for a single-vendor and single-buyer supply chain with the consideration of service-level constraint; the first model is a decentralized model based on Stackelberg game, and the second is an integrated model. Jha and Shanker (2013) considered the minimum service-level constraint in a single-vendor and multiple buyers’ integrated inventory problem also. A large number of inventory models could be found in the literature, and most of them are bounded by their defined conditions. Thus, a model defined for one environment cannot be applied in a different environment. This study is inspired by the problem of a manufacturing company which sells its product through its retailers and uses a third party for the transportation of the item. Each of the vehicle types provided by the third-party logistics has a buyer-specific transportation cost and a transportation lead time. In the literature, a number of inventory models are available which considers the lead time as continuous and can be reduced at the expense of crashing cost. But in this problem, the transportation lead time is considered as discrete and different lead time values can be selected for a specific buyer at the expense of a different transportation cost. Additionally, a minimum service level is defined for each of the retailers; i.e., each of the retailers has to maintain a certain level of safety stock in order to fulfill the minimum service-level constraint. The demand of the retailers is defined as price-dependent stochastic. We present this problem as an Mixed Integer Nonlinear Programming (MINLP) mathematical model and solve for the maximum supply chain profit. The organization of the remaining paper is as follows: Section 3 describes the notations and assumptions involved. The model considered has been described in Sect. 4. The mathematical formulation of the problem is presented in Sect. 5. Section 6 presents illustrative numerical examples and their results. Sensitivity analysis has been presented in Sect. 7. Section 8 summarizes the results, provides some important managerial insights, and suggests future research directions.

3 Assumptions and Notations The problem also gets characterized by the following assumptions and notations. 1. 2. 3.

The planning has been considered for infinite time. Replenishment cycle for each sub-batch for all the buyers is of same length. Each of the buyers has a minimum service-level constraint.

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4.

Transportation lead time for a buyer depends upon the type of the vehicle and on the buyer itself. The vehicle type selected by the buyer has a transportation lead time L i j , and the demand during this period willbe normally distributed with mean D L i = Di L i j and a standard deviation of σεi L i j . Thus, the transportation    lead time demand can be expressed as X ∼ N Di L i j , σεi L i j . Third-party logistics provider can provide any number of vehicles to the buyers. For shipment of a sub-batch from the manufacturer to the buyer, any number of a particular type of vehicle can be used by the buyers. Inventory carrying cost rate for a supply chain partner may be different from others.

5.

6.

αi

Proportion of the demand not fulfilled from the stock, thus (1 − αi ) service level

c

Per unit cost incurred by the manufacturer ($)

i

Index for a buyer

j

Index for a vehicle type

N

Total number of buyers

P

Annual production rate of manufacturer (units/year)

hi

Annual inventory holding cost rate incurred for ith buyer

hm

Annual inventory holding rate incurred for manufacturer

r pi

Reorder point

si

Safety stock

Cj

Capacity of vehicle type j (units)

Fi j

Per shipment transportation cost for transporting the items to ith buyer using single unit of vehicle type j ($)

m

Number of sub-batches during production cycle of the manufacturer

w

Unit purchase price to buyers ($)

x

Ratio of annual production rate to annual cumulative demand rate

D

Annual demand for the manufacturer being equal to cumulative demand of all the buyers (units/year)

T

Total annual supply chain profit ($/year)

X C¯

Demand during the transportation lead time Total annual cost to manufacturer ($/year)

P¯ Pˆ

Total annual profit of manufacturer ($/year)

Ai

Ordering cost per order of ith buyer ($/order)

Total annual profit of all the buyers ($/year)

Am Setup cost per setup of manufacturer ($/setup) α

Profit margin of the manufacturer as a fraction of unit cost of the manufactured item

ki

Safety factor

n

Number of equal-sized sub-batches per order

t

Common replenishment cycle time for all the buyers (year)

Q

Production lot size of the manufacturer (units) (continued)

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71

(continued) pi

Unit selling price of ith buyer ($)

Ni j Number of the vehicles of type j employed in supplying the item to the ith buyer

4 The Model The supply chain considered in this study is a single-manufacturer and multiple buyers’ supply chain which deals in one single item. The cost of third-party logistics service provider is borne by the buyers. The third-party logistics provider offers a number of vehicle types with different capacities to the buyers and charges the transportation cost from the particular buyer on the basis of the vehicle type selected. Each vehicle of a particular type provided by the third-party logistics provider has a transportation cost and a transportation time specific to the buyer, and thus a same type of vehicle may have a different transportation cost and a different transportation time for different buyers. Due to this, the model assumes the nonidentical transportation lead time for the buyers. Inventory at the buyers’ end is reviewed continuously, and for each of the buyers, a service-level constraint is specified; as in many practical situations, the shortage cost cannot be exactly quantified due to intangible losses (Choudhary and Shankar, 2013). Service-level constraint also limits the stock-out level per cycle for each buyer. The buyers sell the item in the market where the ith buyer faces an average demand rate of (Di ), per unit time which is price-sensitive and stochastic in nature, and since the manufacturer the demand of all the   fulfills N Di per unit time. buyers so the manufacturer faces a demand rate of D = i=1 The demand function of the ith buyer can be expressed as Di = y( pi ) + εi

(1)

where y( pi ) is a linear decreasing function of the per unit price pi and can be expressed as y( pi ) = ai − bi pi , where ai , bi , and pi , in Eq. (1), respectively, represent the maximum demand, price elasticity constant, and selling price of the item and εi is a continuous random variable for the ith buyer. The above expression is in the same line with that of Petruzzi and Dada (1999), Ray et al. (2005), and Jadidi et al. (2017). Thus, Di = ai − bi pi + εi .

(2)

This is also assumed that εi follows a normal distribution with mean μεi and variance as σε2i , i.e., N μεi , σε2i . Thus, the demand rate of the ith buyer can be   expressed as Di ∼ N λ( pi ), σε2i , where λ( pi ) = y( pi ) + μεi , and if μεi = 0, ∀i, then λ( pi ) = y( pi ).

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A synchronization policy as of the second model of Hoque (2008), and Jha and Shanker (2013) is followed in this study to achieve the synchronization in delivery and replenishment among the manufacturer and the buyers. The manufacturer produces the item at a finite production rate (P) and produces a lot of size of Q units in one setup. However, these Q units are shipped to all the buyers in n number of equal-sized sub-batches. These sub-batches are shipped to the buyers even during the production of the lot. A buyer can select any number of the vehicles of a particular type for the transportation of the sub-batch from the manufacturer to it. Although the shipment from the manufacturer to all the buyers is made at the same time due to the vehicle and buyer-specific transportation lead times, the buyers may receive the orders at different times (Fig. 1). From the sub-batch of size q = Qn , the ith buyer receives a quantity qi = DDi q which is in the ratio of its annual demand (Di ) to the total annual   N demand D = i=1 Di of all the buyers. The ith buyer consumes this quantity qi at a rate of Di in time t = Dqii . For each buyer in the network, the replenishment cycle time will be t = Dq11 = Dq22 = . . . = Dq NN . The above philosophy for the production and shipment to the buyers can be understood from Fig. 1 for a single-manufacturer

Fig. 1 Inventory pattern of the single-manufacturer and multiple buyers’ supply chain

Modeling a Supply Chain with Price-Dependent Stochastic Demand …

73

and three-buyer case. It can be observed from Fig. 1 that t=

Q nD

(3)

5 Mathematical Model Formulation 5.1 Annual Profit of the Manufacturer The manufacturer charges a profit markup of α on its incurred total annual costs and supplies to the buyers at a price of w. The manufacturer incurs the following costs.

5.1.1

Annual Cost on Finished Products Incurred by the Manufacturer

Since the manufacturer faces a total annual demand of D units and incurs a per unit cost c which include all the related costs such as unit raw material purchase cost, unit raw material holding cost, and unit processing cost, thus the manufacturer will incur an annual cost of Pm = cD.

5.1.2

(4)

Annual Inventory Holding Cost Incurred by Manufacturer

This cost is taken in proportion to the average inventory that can be determined by dividing the manufacturer’s inventory cycle area by the cycle length. Inventory held by the manufacturer in one cycle can be calculated by determining the area under the inventory cycle curve of the manufacturer. From Fig. 1, the area under the inventory cycle curve of the manufacturer (Cm ) can be determined as 

 q P q P q q q + +q −1 Cm = 2P 2DD D D

  P q P q q +q − 1 (m − 1) + ... + 2DD D D



P Q − (m − 1)q D − q Q − (m − 1)q DP − q + 2 P    Q − (m − 1)q DP − q q P − + Q − (m − 1)q − q D D P

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 q   . + {1 + 2 + 3 + . . . + (n − m − 1)} q D Average inventory of the manufacturer (Im ) can be calculated by dividing the area under the inventory cycle curve of the manufacturer (Cm ) or inventory held by the manufacturer in one cycle by the cycle time (D/Q). Hence, average inventory of the manufacturer will be 



q q q P q P q + +q −1 Im = 2P 2DD D D



q P q P q + ... + +q − 1 (m − 1) 2DD D D



P Q − (m − 1)q D − q Q − (m − 1)q DP − q + 2 P    Q − (m − 1)q DP − q q P − + Q − (m − 1)q − q D D P   q  D (5) + [1 + 2 + 3 + . . . + (n − m − 1)] q D Q After simplification,   1 Q (n − 1)2 + (n − m − 1)(n − m) + m(2n − m − 1) − . Im = 2 2n P/D P/D

(6)

Therefore, the annual inventory holding cost of the manufacturer (Hm ) can be calculated by multiplying the average inventory by inventory holding cost rate (rm ) and per unit cost (c). Thus, Hm = rm cIm After putting the value of Im from Eq. (6) in the above, we would get   1 Q (n − 1)2 Hm = 2 + (n − m − 1)(n − m) + m(2n − m − 1) − rm c 2n P/D P/D (7)

5.1.3

Annual Setup Cost of the Manufacturer

Since the manufacturer produces Q units per setup and satisfies an annual demand of D units annually, so the setup cost can be expressed as

Modeling a Supply Chain with Price-Dependent Stochastic Demand …

Sm =

D Am Q

75

(8)

¯ will be the sum of annual purchase So, the total annual cost of the manufacturer (C) cost (Pm ), annual inventory holding cost (Hm ), and annual setup cost (Sm ). Thus, C¯ = Pm + Hm + Sm Substituting the values of Pm , Hm , and Sm into the above equation from their respective Eqs. (4), (7), and (8), we get  1 Q + (n − m − 1)(n − m) C¯ = cD + 2 2n P/D  D (n − 1)2 +m(2n − m − 1) − cm c + Am . P/D Q

(9)

Since the manufacturer charges a markup (α) on his total incurred cost and supplies the item to the buyers at per unit selling price of w, thus the selling price (w) of the manufacturer will be w = (1 + α)

C¯ D

(10)

¯

where CD would represent per unit production cost on the item by the manufacturer. Manufacturer’s profit can be calculated by multiplying the markup factor to his ¯ will be incurred total cost. Therefore, the annual profit of the manufacturer ( P) ¯ P¯ = α C.

(11)

5.2 Annual Profit of the ith Buyer Profit of the ith buyer can be calculated by subtracting the total annual cost incurred by the buyer from his total annual revenue (Ri ). Various costs incurred by this buyer will be annual purchase cost (Pi ), inventory holding cost (Hi ), ordering cost (Oi ), and transportation cost (Ti ). So, the annual profit of ith buyer will be: 





Pi = Ri − (Pi + Hi + Oi + Ti ).





Pi = Di pi − (Pi + Hi + Oi + Ti ).

(12)

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Various cost elements experienced by the buyers are described and developed below.

5.2.1

Annual Purchase Cost to the ith Buyer

Buyer i purchases annually a quantity (Di ) equal to its annual demand and pays w per unit to the manufacturer; thus, annual purchase cost for the ith buyer will be Pi = Di w.

5.2.2

(13)

Annual Inventory Holding Cost to ith Buyer

The buyer takes the services of the third-party logistics provider which offers a number of types of vehicle to transport the units from the manufacturer to the buyer where each vehicle takes a buyer-specific transportation lead time and has a buyerspecific cost per shipment. A buyer can select any number of the vehicles of a particular type for transportation. The reorder point for the ith buyer is r pi which is equal to the sum of demand during the transportation lead time and the safety stock. Thus, r pi = demand during the transportation lead time + safety stock or r pi = Di L i j + σεi



Li j

where L i j is the transportation lead time for the ith buyer transporting the items with the jth type of vehicle. The average inventory of the ith buyer  will be the mean of the inventory just before the receipt of the ordered quantity  r pi − Di L i j and just after the receipt of the ordered quantity qi + r pi − Di L i j . Thus, average inventory of the ith buyer is   r pi − Di L i j + qi + r pi − Di L i j Ibi = 2     Di L i j + σεi L i j − Di L i j + qi + Di L i j + σεi L i j − Di L i j Ibi = 2     σεi L i j + qi + σεi L i j Ibi = 2

Modeling a Supply Chain with Price-Dependent Stochastic Demand …

Ibi =

q

i

2

+ σεi



77

 Li j

Annual holding cost for the ith buyer is Hi = Ibi ri w Hi = Since qi =

i

2

+ σεi



 L i j ri w

Q Di n D

Hi =

5.2.3

q

 Q Di + σεi L i j ri w 2n D

(14)

Annual Ordering Cost to the ith Buyer

Annual ordering cost to the ith buyer can be expressed as Oi =

5.2.4

D Ai Q

(15)

Annual Transportation Cost to the ith Buyer

The buyer takes the services of the third-party logistics provider which offers a number of types of the vehicle where each vehicle takes a specific transportation lead time to transport the units from manufacturer to the buyer and has a specific cost per trip. A buyer can select any number of the vehicles of a particular type for transportation. However, each of the buyers has a minimum service-level constraint (1 − αi ), where αi is the proportion of the demand not fulfilled from the stock. Thus, a buyer has to select a vehicle keeping its cost, capacity, and transportation lead time in view. Thus, the transportation cost of the ith buyer will be D  Ti = n Ni j Fi j ∀i Q j=1



N

(16)

The vehicle type selected by the buyer has a transportation lead time L i j , and the demand during this period  will be normally distributed with mean D L i = Di L i j and a standard deviation of σεi L i j . Thus, the transportation lead time demand can be    expressed as X ∼ N Di L i j , σεi L i j . Since in this study, any stock-out cost has not been considered at the buyers’ end, rather a service-level constraint is defined

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for each of the buyers. The stock-out cost at the buyers’ end can be replaced by a service-level constraint assuming that the buyer has to fulfill a minimum fill rate (Ouyang et al., 1996; Jha and Shanker, 2009a, b, 2013). Thus, the expected shortages at the buyer’s end cannot exceed the given value αi . Shortages at the end of the inventory cycle of ith buyer ≤ αi Ordered quantity per cycle of ith buyer The expected shortages at the end of the cycle of ith buyer in such environment are also given as σεi L i j ψ(ki ) by Ravindran et al. (1987), and Jha and Shanker (2013). where ψ(ki ) = ϕ(ki ) − ki [1 − φ(ki )] Here, ϕ, φ represent the standard normal probability density function and cumulative distribution function, respectively, and ki is the safety factor of the ith buyer. or  σεi L i j ψ(ki ) ≤ αi ∀i qi where qi =

Q Di n D

 Dσεi L i j ψ(ki ) ≤ αi ∀i Q Di /n

(17)

5.3 Annual Profit of the Supply Chain Total supply chain profit is the sum of manufacturer’s profit and the profit of all the buyers. Thus, the total supply chain profit can be computed from the following T = P¯ +

N  i=1



Pi .

(18)

Modeling a Supply Chain with Price-Dependent Stochastic Demand …

79

5.4 Formulation as a Mathematical Programming Problem Because of integer requirements on the values of ‘m’ and ‘n’, the objective function is discontinuous and classical optimization approach cannot be used. Thus, the problem of maximization of overall supply chain profit is mathematically expressed as N Maximize T = P¯ + i=1 Pi 

subject to Equations (1), (2), (3), (4), (5), (6), and Ni j C j ≤ qi ∀i∀ j

(19)

ai bi

∀i

(20)

pi ≥ w

∀i

(21)

Di

(22)

pi ≤

D= σi

√

N  i=1

L i ψ(ki ) ≤ αi ∀i qi

¯ C, ¯ Im ≥ 0 α, t, Q, D, w, P, 

(23) (24)



ai bi , Di , Hi , Oi , pi , Pi , Ri , Pi , Ti ≥ 0 ∀i

(25)

n, m, ≥ 0 and integer

(26)

Ni j ≥ 0 and integer ∀i∀ j

(27)

Equation 19 represents the constraint that for each of the buyers, the capacity of the selected vehicles must be equal to or larger than the size of its sub-batch. Equation 20 represents the maximum selling price of the retailers so that the demand cannot be negative, while Eq. 21 represents that this selling price must be greater than the wholesale price of the manufacturer. Equation 22 represents that the demand faced by the manufacturer is the sum of the individual demands of the retailers. Equation 23 represents the service-level constraint for each of the buyers. Finally, Eqs. 24, 25, 26, and 27 represent the non-negativity and integer constraint for various parameter and decision variables.

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The objective function of the problem is a nonlinear function and involves a large number of continuous and discrete variables for which it is very difficult to investigate the concavity property analytically (Mandal and Giri, 2015), although several numerical methods can be used to optimize the such problems (Pattnaik, 2013). In this paper, we propose the use of optimization software package LINGO 17.0 to solve the developed mathematical model. Hariga et al. (2007) also emphasized to make use of the high speed of computers and efficient solvers to such complex problems.

6 Illustrative Example The use of proposed model is being illustrated through an illustrative numerical problem. The example problem is solved using an optimization software LINGO 17.0 on an Intel (R) Core (TM) i7-6700 CPU @ 3.40 GHz processor system with 8 GB RAM. The illustrative example considered here is of a single manufacturer and three retailers (N = 3). The manufacturer-related data are: P = 12,000 units/year, c = $ 60, Am = $ 1000, and rm = 0.10. The buyers-related data are given in Table 1. The data for the third-party logistics are given in Tables 2 and 3. Third-party logistics service provider offers three types of vehicles, namely Types I, II, and III, with capacities as 20, 40, and 60 units, respectively. Their buyer-specific cost is given in Table 2, while the transportation lead time data for the same are provided in Table 3. Table 4 summarizes the results obtained from the solution from the LINGO 17.0 solver. Table 5 shows the vehicle mix selected by the buyers for the maximum supply Table 1 Example problem data for buyers Buyer (i)

ai

bi

σεi

Ai

ri

αi

1

1500

10

400

25

0.20

0.05

2

1800

12

450

23

0.20

0.08

3

1400

11

300

22

0.20

0.06

Table 2 Transportation cost data for example problem

Vehicle Type I ( j = 1)

Type II ( j = 2) Type III ( j = 3)

Buyer (i)

Transportation cost

Transportation cost

Transportation cost

1

16

30

42

2

20

25

30

3

12

22

30

Modeling a Supply Chain with Price-Dependent Stochastic Demand … Table 3 Transportation lead time data for example problem

81

Vehicle Type I ( j = 1)

Type II ( j = 2) Type III ( j = 3)

Buyer (i) Transportation Transportation Transportation lead time (days) lead time (days) lead time (days) 1

4

6

8

2

5

7

10

3

3

5

7

Table 4 Optimal values of the variables from the model Variable

Value

Variable

Value

Variable

Value

Variable

Value

T P¯

35,768.47

w

62.08

α

0

q3

40

0

D

1991.85

p2

85.04

k1

1.13

11,307.07

D1

692.82

p3

80.03

k2

0.95

16,071.00

D2

779.42

Q

920.00

k3

0.96



P1 

P2 

P3

8390.39

D3

519.61

q1

53.33

t

0.0769

n

6.00

q2

60

Table 5 Number of vehicles with their types selected by the buyers

Buyer (i) Vehicles Type I ( j = 1) Type II ( j = 2) Type III ( j = 3) 1

3

0

0

2

3

0

0

3

2

0

0

chain profit. The prices of all the buyers are shown in Table 4 along with the safety factors of all the buyers with the selected vehicles. The sub-batch sizes for the buyers are also in the same table which shows that FTL is not necessary for the coordinated supply chain because one of the buyers (buyer 1) selects the capacity of the vehicles which is slightly greater than its sub-batch size. The model also shows that the buyers prefer faster mode of transportation in order to reduce the transportation lead time.

7 Sensitivity Analysis 7.1 Sensitivity Analysis with Respect to Service Level In this section, the effect of the service level on the inventory policy is analyzed by changing the service level. For this analysis, we have used the same data as of given in

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Table 6 Effect of service level on safety factor and total supply chain profit Service level (%)

Safety factor

q1

q2

q3

n

D

T

1.06

53

60

40

6

1920

35,579

1.40

53

60

40

6

1920

35,088

1.73

1.62

53

60

40

6

1920

34,758

1.84

1.89

1.79

53

60

40

6

1920

34,512

2.11

2.15

2.05

53

60

40

6

1920

34,119

k1

k2

k3

95.0

1.14

1.19

97.5

1.46

1.51

98.5

1.68

99.0 99.5

Table 7 Effect of active and inactive service level on transportation lead time Service level

Safety factor

Transportation lead time

Buyer 1 (%)

Buyer 2 (%)

Buyer 3 (%)

k1

k2

k3

Buyer 1

Buyer 2

Buyer 3

95.0

92.0

94.0

1.14

0.95

0.97

4

5

3

95.0

50

94.0

1.14

0.00

0.97

4

10

3

50

50

94.0

0.00

0.00

0.97

8

10

3

the example section except that for the service level. First, all the buyers are assumed to have a minimum service level of 95%; then, it is gradually increased for all the buyers up to a value of 99.5%. It can be observed from Table 6 that as the service level of the buyers is increased the safety factor for the buyers is also increased which is in the line of the available literature that when the service level is increased the buyers have to maintain a higher safety inventory. The result of maintaining a higher safety inventory can be seen on the total supply chain profit which is decreasing with the increase of the service level. However, the lot size and number of deliveries remain constant. Buyers of a supply chain may have different service levels; in that scenario, a service level may not be active for all the buyers. This analysis has been conducted by considering different service levels for different buyers as given in Table 7. The first scenario where all the buyers have a nonzero safety factor shows that the servicelevel constraint is active for all the buyers and all the buyers choose the transportation lead time considering this constraint. However, the service-level constraint is inactive for buyer 2 in scenario 2 and for buyers 1 and 2 in scenario 3. The safety factor for the inactive constraint condition is zero. Also, for inactive constraint the buyers choose comparatively cheap and slow transportation modes.

7.2 Sensitivity Analysis with Respect to σεi The effect of demand uncertainty will be another important analysis for this study from the realistic applications. For this analysis, the values of σεi for different buyers

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83

Table 8 Effect of σεi on safety factor and total supply chain profit Change in σεi (%)

Safety factor

q1

q2

q3

n

D

T

0.95

53

60

40

5

1593

47,677

0.85

71

80

53

4

1793

42,906

1.11

0.88

71

80

53

4

1892

39,570

1.19

1.06

53

60

40

6

1992

35,579

1.16

1.22

1.09

53

60

40

6

2091

31,019

+10

1.04

1.18

0.96

71

80

53

5

2191

25,930

+20

0.90

1.05

0.83

100

113

75

4

2390

13,784

k1

k2

k3

−20

1.02

1.08

−10

0.93

1.08

−5

0.96

0

1.14

+5

have been changed by the amount shown in Table 8. Thus, the first scenario of Table 8 shows that the values of σεi have been reduced by 20% for all the buyers, while the last scenario shows that the same has been increased by 20% in the last scenario. A larger value of σεi indicates the larger demand uncertainty. Thus, the model predicts a lower total supply chain profit for higher demand uncertainty and vice versa. Regarding the safety factor, there is no such trend that can be seen directly from the table because of the change in the values of sub-batch size and the number of sub-batches in one lot. Thus, the true effect of the σεi can be seen collectively on the safety factor, sub-batch size, and the number of sub-batches in one lot which is increasing with the increase in σεi values. Thus, a higher demand uncertainty results in higher safety stock and a lower total supply chain profit and vice versa.

8 Conclusion The paper deals with the problem of the case company and develops the mathematical model for its integrated inventory and pricing problem. The model also takes care for the transportation of the item, its lead time, and service-level requirement. The developed model is an MINLP which is very complex and is solved by LINGO 17.0 solver. The results show that a higher service level requires higher safety factor and faster modes of transportation. The model is also validated for different service levels for different buyers. Another analysis shows that a higher demand uncertainty is not in favor of the supply chain. Efficient heuristics can be developed in the future to solve the model. The model can be extended with perishable product. The transportation of the perishable product with the different technology-based refrigerated trucks (different deterioration rates) will be an interesting study. Acknowledgements The authors are grateful to the editors Dr. Rajendran C and Dr. Vipin B and two anonymous referees for their valuable and constructive comments and suggestions on the earlier

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versions of this article. The same has helped the authors significantly in improving the readability and the contents of the present paper considerably.

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Optimal and Heuristic Profit Sharing Using Sales Rebate Contract in a Multi-level Supply Chain I. Thomson, Raichel Elsa Thomas, K. Susithra, S. Shobana, Arshinder Kaur, and Chandrasekharan Rajendran

Abstract A supply chain (SC) is an interdependent network of many SC members who add value to the product successively until it gets to the final stage and attains utility. One of the most important problems in any supply chain (SC) consisting of organizations is the sub-optimization due to the fact that the SC decision making is distributed over various players. An interesting solution to the issue, which leaves the decision power and supply chain structure intact, is the use of contract mechanisms. Ideally, contract mechanisms ensure that the whole SC is optimized as if it were a single unit (coordination) and is designed such that all the players benefit from working together through the coordination mechanism (win-win). Once coordination is in place, the next challenge is to devise some mechanism to share the profits gained due to coordination. To enable coordination among the SC members, sales rebate contracts in a multi-level SC using fair share mechanism have been devised. In the sales rebate contract, the supplier charges the buyer wholesale price per unit I. Thomson Reckitt Benckiser Arabia, Dubai 119481, UAE e-mail: [email protected] R. E. Thomas Department of Mechanical Engineering, Indian Institute of Technology BHU, Varanasi 221005, India e-mail: [email protected] K. Susithra · S. Shobana Department of Industrial Engineering, College of Engineering Guindy, Anna University, Chennai 600025, India e-mail: [email protected] S. Shobana e-mail: [email protected] A. Kaur · C. Rajendran (B) Department of Management Studies, Indian Institute of Technology Madras, Chennai 600036, India e-mail: [email protected] A. Kaur e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_5

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purchased but then gives rebate to the buyer per unit sold above some threshold. The objective of this paper is to determine the optimal rebates given to downstream SC members and threshold value of sales, which maximizes the SC profit as well as fairly shares the profits among the SC members. This objective can be achieved by minimizing the deviation between proportion of each member’s value addition and proportion of each member’s respective share in the total profit by coordinating through sales rebate contract. An analytical model is developed to formulate the supply chain coordination problem and simultaneously formulate the optimization problem to achieve fair distribution of profit among the SC members. Three heuristic search techniques based on genetic algorithm and simulated annealing have been proposed to solve the problem. Keywords Multi-level supply chain · Supply chain coordination · Sales rebate contracts · Profit sharing · Analytical model · Optimization · Genetic algorithm · Simulated annealing

1 Introduction A supply chain is termed a coordinated supply chain, when all the supply chain members work toward a common objective of maximizing the total supply chain profitability. Generally, the supply chain members being primarily concerned with optimizing their own objectives tend to sell/buy an order quantity which is optimum for them and that self-serving focus often results in poor performance of the whole supply chain. Supply chain management (SCM) is a set of approaches utilized to effectively integrate suppliers, manufacturers, warehouses and retailers which are interdependent entities. The current challenges of globalization and product proliferation in supply chain demands supply chains to be considered as a single system to effectively manage the flow of resources and information between supply chain members. The supply chain members are expected to align their respective objectives with that of the whole supply chain to minimize the system-wide cost and satisfy the servicelevel requirement (Simchi-Levi et al., 2008). SCM is a philosophy of managing the supply chain (externally) as well as managing a company (internally). Nalla (2008) observed SCM as the coordination of different business entities in the supply chain to reduce waste (costs), create value to customers and thus enhance revenues. Coordination in the above definition refers to managing challenges due to inter-dependencies among business entities by aligning goals and process/functions. According to Nalla (2008), organizations seek to achieve coordination through different approaches. The first step toward establishing coordination might be to share information between the entities in the SC. The sharing of information is indeed a necessary condition but may not be sufficient to achieve coordination and improve overall SC performance. Besides sharing of information, Nalla (2008) observed organizations could use two other approaches to achieve coordination. The first approach is to modify the

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governance structure of the trading relationship, for example, by modifying the ownership (i.e., “who owns what”) and/or by modifying decision rights (i.e., “who decides what”). Modifying the governance structure works only when the owner gets the decision rights over the functional people. This approach is most difficult to implement within and across companies in SC. The second approach for achieving coordination within the SC is to modify the terms of trade. The modification of the terms is achieved through incentive schemes or contracts over certain trade parameters (variables). This approach aims to achieve coordination among business entities by providing incentives to share risks and/or rewards. There are several contract mechanisms that can be designed and used to make sure that the independent decisions made by business entities optimize the overall performance of the whole chain (in such a case, we say that the mechanism coordinates the chain). The contracts must enable all independent business entities to improve their performance when compared to an uncoordinated situation. In such a case, we say that the contract mechanism leads to a win-win situation. In supply chain coordination, the whole supply chain arrives at a consensus of how much to order (optimal order quantity of whole supply chain). However, firms lack the incentive to implement those actions. To create that incentive, the firms can adjust their terms of trade via a contract that establishes a transfer payment scheme. This paper explores profit sharing between supply chain members in a centralized supply chain. A systematic framework for an analytical model is built to find out the optimal rebate and threshold values that ensure a transparent and trustworthy supply chain coordination. Contrary to the initial agreement where a decentralized supply chain exists, here a centralized supply chain is considered. The profit is being distributed among the supply chain members based on the value they add to the product. The remainder of the paper is organized as follows: In Sect. 2, we give details of various coordination mechanisms available in the literature and introduce the research problem. In Sect. 3, an analytical model of multi-level (stage) supply chain is introduced and a three-level supply chain is discussed in detail. In Sect. 4, we present the overall problem of determining the decision variables and sharing rewards among supply chain members using a heuristic technique. In Sect. 5, meta-heuristic search techniques such as genetic algorithm and simulated annealing are discussed in order to solve the problems. In Sect. 6, various experiment settings are discussed, and finally in Sect. 7 the study is concluded.

2 Literature Review Cachon (2003) defined supply chain coordination as follows: A contract is said to coordinate the supply chain if the set of supply chain optimal actions is in Nash equilibrium; i.e., no firm has a profitable unilateral deviation from the set of supply chain optimal actions. Tsay (1999) defined supply chain contracts formally as the rules for transactions between the supply chain actors and later utilized

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incentives (risks and rewards) to make the SC members’ decisions coherent with each other. A number of different contract types are identified in the literature such as buyback contracts (Sang, 2016), revenue sharing contracts, quantity flexibility contracts and sales rebate contracts (Emmons & Gilbert, 1998; Koulamas, 2006; Lian & Deshmukh, 2009’ Krishnan et al., 2004; Taylor, 2002). Generally, the analysis of contracts offers guidance in negotiating the terms of the relationship between the supplier and the buyer. The contracts are designed to sort out conflicts that may crop up in the future. The focus of this paper is sales rebate contract where the supplier charges a wholesale price per unit purchased and gives the retailer a rebate per unit sold above a threshold fixed by the supplier. The objective of this paper is to explore the applicability of sales rebate contract in a multi-level supply chain. Sales rebate contracts can be implemented as promotional trade incentives and especially as alternative sales promotion to price discounts (Arcelus et al., 2008). Rebates in a multi-level supply chain have been less explored, as increase in the number of SC members increases the complexity in optimizing the key parameters that are involved for the intermediate members in the SC. A rebate is different from an order quantity discount as it only applies to items sold to end users. Hence, a rebate contract is more efficient than an order quantity discount because it provides a direct incentive for retailers to increase sales (Wong et al., 2009). The supplier needs to know the exact quantity sold by the retailer in order to pay the rebate, but difficulties arise when the supplier cannot acquire the retailer’s sales data directly. On the other hand, the data obtained from the retailer may not be authentic as the retailer may claim more rebates than what the actual sales allow. The ability of the manufacturer to arrive at the best price and rebate is conditioned by the level of information that the manufacturer has about the uncertainty faced by the retailer as well as the retailer’s cost and demand function (Arcelus, 2007). Krishnan et al. (2004) proposed a buyback with the sales rebate contract to coordinate the news vendor with a fixed price but effort-dependent demand. The rebate induces the retailer to price too low (in an effort to generate sales above the rebate threshold), but a buyback induces the retailer to price too high, so it is possible to counteract the deleterious effects of the rebate on price. Taylor (2002) discussed a rebate contract for a supply chain composed of one supplier and one retailer, in which retailers were allowed to determine their order quantities consistent with the optimal order quantities so as to optimize the global supply chain’s profit. In the auto industry, channel rebates are termed as dealer incentives. In the literature on supply chain contract mechanisms, the main focus seems to be on a situation with only two decision makers: a buyer and a supplier, however in reality a SC usually consists of more than two entities. Whereas, in reality a SC usually consists of more than two entities. A natural question therefore is how the fundamental ideas underlying contract mechanisms can be generalized to settings with a multi-echelon supply chain (van der Rhee et al., 2010). There is a need to capture the problem of multi-level supply chains and to analyze how contract decision variables can be determined at each interface between supply chain entities. The complexity of managing multi-level supply chain and hence the contract decision

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variables at two interfaces has been captured in the analytical model proposed in the next section. In the literature, the emphasis has been given to achieve supply chain coordination with uniform order quantity, with increase in profits of whole supply chain as a performance indicator of coordinated supply chain. Even if the supply chain contract helps in improving the system-wide (total supply chain) profits, the challenge still remains in determining the fair share of the benefits among supply chain entities. In this paper, we have analyzed the sales rebate contracts in multi-level supply chain considering all the members to be a part of the supply chain. To analyze the utility of sales rebate contracts, we compare the expected profits of all supply chain members as well as total expected supply chain profits with the case when supply chain members act independently. The savings gained by adopting sales rebate contracts have to be shared fairly among supply chain members based on the value addition done by respective supply chain members. The model proposed in this paper simultaneously determines the decision variables of the sales rebate contracts and the fair share of savings among supply chain members. Further, the fair share of profits is determined along with the decision variables of rebate contracts in a three-level supply chain using meta-heuristic techniques; genetic algorithm (GA) and simulated annealing (SA) have been adopted. Genetic algorithms have the ability to handle nonlinear interactions in the model to arrive at the near-optimum values of the decision variables. GAs have been applied in various supply chain problems like supply chain design problems (Chang, 2010), determination of base stock values in serial supply chain (Daniel & Rajendran, 2006), information sharing in supply chain, vendor managed inventory (VMI) in supply chain (Nachiappan & Jawahar, 2007) and comparison of periodic review order up to policy and continuous review policy comparison in a serial supply chain (Sethupathi et al., 2014). The application of GAs in SC contract area seems to be less explored in the literature. Simulated annealing (SA) heuristic generates near-optimal solutions to combinatorially intractable problems. The heuristic, however, is generic and has to be modified in the context of the specific problem under study. The main reason for which SA heuristic is supposed to give a good solution is due to the fact that it also accepts inferior solutions (of course, with a certain acceptance criterion) in search of a good solution. Many papers have reported successful applications of simulated annealing in scheduling, and the same can be extended to SC contracts.

3 Model for Sales Rebate Contracts in a Multi-level Supply Chain Contracts have been widely explored, and they help in coordinating the supply chain as well as improving the performance of the whole supply chain. Most of the contracts have been explored for two-level supply chain. In general, a typical supply chain has more than two members dealing with each other. When we consider more than two

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Fig. 1 Supply chain model

members in the supply chain system, the decisions at one interface are related to the decision at the other interface of the supply chain (Fig. 1).

3.1 Assumptions The sequence of events in end customer supply chain model are as follows; the most downstream member entity 1 of the supply chain based on the end customer demand decides to order q units from entity 2. The same order size is passed on from entity 2 to entity 3 all the way to entity N . Subsequently, the amount q is shipped from entity N all the way to entity 1. Finally, the customer demand occurs and the buyer sells the amount min(q, customer demand) to the end customer. We now build an N-echelon model where we can imagine the entities in the supply chain to be retailer, distributor, manufacturer, supplier and so on. We consider the following basic assumptions: • An N-level supply chain is considered comprising a supplier, a manufacturer, a distributor, a retailer and so on. • A single period model with short life cycle product is assumed where the end of period inventory must be disposed off. • The demand is stochastic in nature. • Cost/price data are known in advance. • A single product flows through the supply chain. • Information lead time is negligible or zero. • Base stock level at every installation takes discrete integer values. • There is no lot size or discount policy for any installation. • There are no holding cost, shortage cost and transportation cost. • All installations have infinite capacity. • The most upstream installation is linked to an infinite supply source. • The end customer demand information is shared across the SC.

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3.2 Notations and Terminologies All prices are expressed in standard unit cost P: αi P: s: qd : qd∗ : qc : qc∗ : ti : ri : X: πiX : πX: i: P Si : Vi : S(q):

Purchase cost Price at which units are sold by entity i to entity i − 1 such that αi−1 ≥ αi ∀i ∈ {1, 2, 3...N } Salvage value at retailer Quantity ordered by entity 1 to entity 2, which is passed to entities upstream Decentralized optimal order quantity; optimal order quantity of entity 1 Order quantity of SC, where all the supply chain members are considered to be a part of one system Centralized optimal order quantity; optimal order quantity of SC system Threshold fixed by entity i to entity i − 1 Rebate paid by entity i to entity i − 1 for each unit sold above the threshold Denotes the following scenarios—Case A—decentralized , Case B—centralized , Case C—coordinated Expected profit of entity i in case X {X ∈ A, B, C} Expected profit of whole supply chain in case X {X ∈ A, B, C} Entity in the supply chain {i ∈ 1, 2, 3 . . . N } Proportion of increase in profit of entity i due to coordination w.r.t. total profit gain of SC Proportion of value addition by the SC entity i w.r.t. the total value addition of SC Expected sale.

3.3 Analytical Model The expected sales S(q) as per Cachon (2003) are given as follows: q S(q) =

∞ x · f (x)dx +

q f (x)dx q

0

q x f (x)dx

= q(1 − F(q)) + 0

q S(q) = q −

F(x)dx 0

S  (q) = 1 − F(q) = F(q),

(1)

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where x is stochastic end customer demand, f (x) is probability density function of end customer demand and F(x) is cumulative distribution function of end customer demand. The demand is assumed to follow a uniform distribution with parameters a, b, though not restrictive. This is done for generalized understanding. The cumulative distribution is, without loss of generality, assumed to be the following:

F(x) =

⎧ ⎪ ⎨0

if x < a if a ≤ x ≤ b if x > b

(x−a) ) ⎪ (b−a)

⎩

1

The expected sales is q S(q) = q −

F(x)dx

a q

x −a dx b−a a   (q − a)2 =q− 2(b − a) =q−

Therefore, the expected sales, when there is decentralized decision making, qd , are written as follows:   (qd − a)2 (2) S(qd ) = qd − 2(b − a) Similarly, the expected sales when there is centralized decision making of the order quantity from supply chain perspective, qc can be written as in Eq. (3)  S(qc ) = qc −

(qc − a)2 2(b − a)

 (3)

3.4 Model Formulation In this paper, the proposed model describes three cases: First is a case where there is a decentralized decision making, wherein the retailer decides the order quantity, i.e., the decentralized optimal order quantity qd∗ ; second where there is a centralized decision making wherein the entities in the supply chain order a common quantity, i.e., centralized optimal order quantity qc∗ ; and thirdly where supply chain coordination is facilitated with the installation of sales rebate contracts between every consecutive pair of entities to order the centralized optimal order quantity. The expected profits

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of each member and the whole supply chain for three different cases are analyzed below.

3.4.1

Case A—Decentralized Supply Chain

The profit optimization of entity 1 alone is considered in this case. This function is proved to be an increasing concave function in (qd ) using the first- and second-order derivatives. The optimum order quantity is computed from this function. Each entity of the supply chain orders the optimal order quantity of entity 1 which is called decentralized order quantity qd∗ . The profit function of retailer is π1A (qd ) = α1 P S(qd ) − α2 Pqd + s(qd − S(qd )) It can be observed that the function in equation is concave. The optimum order quantity of entity 1, qd∗ is computed from the above function. Entity 2, entity 3, all the way up to entity N ship qd∗ , and their profits are based on this order quantity. qd∗

=F

−1



α1 P − α2 P α1 P − s

 (4)

where qd∗ is positive. The profit function of entity 1 with qd∗ of entity 1 can be written as shown below π1A (qd∗ ) = α1 P S(qd∗ ) − α2 Pqd∗ + s(qd∗ − S(qd∗ )) The profit functions w.r.t. entities 2 to N can be written as follows: π2A (qd∗ ) = α2 Pqd∗ − α3 Pqd∗ π3A (qd∗ ) = α3 Pqd∗ − α4 Pqd∗ , so on upto, π NA−1 (qd∗ ) = α N −1 Pqd∗ − α N Pqd∗ π NA (qd∗ ) = α N Pqd∗ − Pqd∗ Total expected profit of the supply chain in Case A is as follows: N i=1 A

πiA (qd∗ ) = π1A (qd∗ ) + π2A (qd∗ ) + π3A (qd∗ ) · · · + π NA−1 (qd∗ ) + π NA (qd∗ )

π (qd∗ ) = α1 P S(qd∗ ) − Pqd∗ + s(qd∗ − S(qd∗ ))

(5)

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In this case of decentralized decision making, the optimal order quantity is determined as per the interest of entity 1 and not in the interest of whole supply chain. In the next subsection, another case is analyzed, where each SC member is considered to be a part of a single supply chain system.

3.4.2

Case B—Centralized Supply Chain

In this case, the SC members can be considered as a part of one organization and work to maximize the total supply chain profit. It is assumed in this case that SC members will coordinate and determine the centralized optimal order quantity, which is optimal to the whole supply chain system. The expected profit of entities 1 to N (when they all agree to order (qc )) is as follows: π1B (qc ) = α1 P S(qc ) − α2 Pqc + s(qc − S(qc )) π2B (qc ) = α2 Pqc − α3 Pqc π3B (qc ) = α3 Pqc − α4 Pqc , so on upto, π NB−1 (qc ) = α N −1 Pqc − α N Pqc π NB (qc ) = α N Pqc − Pqc The total expected profit of the supply chain will be as presented in the below equation N

πiB (qc ) = π1B (qc ) + π2B (qc ) + π3B (qc ) · · · + π NB−1 (qc ) + π NB (qc )

i=1

π B (qc ) = α1 P S(qc ) − Pqc + s(qc − S(qc )) In Case B, we have an increasing concave function. The optimum order quantity of whole supply chain is given in Eq. (6). Each member of the supply chain orders the optimal order quantity for the whole supply chain given by the below equation qc∗

=F

−1



α1 P − P α1 P − s

 (6)

where qc∗ (optimal order quantity of the supply chain) is positive. Substituting qc∗ in all the profit equations, we get the expected profits of all SC entities. The expected profit functions for entities in the supply chain are given in the following equations. π1B (qc∗ ) = α1 P S(qc∗ ) − α2 Pqc∗ + s(qc∗ − S(qc∗ ))

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π2B (qc∗ ) = α2 Pqc∗ − α3 Pqc∗

π3B (qc∗ ) = α3 Pqc∗ − α4 Pqc∗ , so on upto, π NB−1 (qc∗ ) = α N −1 Pqc∗ − α N Pqc∗ π NB (qc∗ ) = α N Pqc∗ − Pqc∗ Total expected supply chain profit is N

πiB (qc∗ ) = π1B (qc∗ ) + π2B (qc∗ ) + π3B (qc∗ ) · · · + π NB−1 (qc∗ ) + π NB (qc∗ )

i=1

π B (qc∗ ) = α1 P S(qc∗ ) − Pqc∗ + s(qc∗ − S(qc∗ ))

(7)

Lemma 1 The centralized optimal order quantity qc∗ is greater than the decentralized optimal order quantity qd∗ . We build the following expression/constraints α1 P − P α1 P − α2 P > since P < αi P for i ∈ {1, . . . N } α1 P − s α1 P − s Using Eqs. (4), (5), (6) and (7), we get f (qc∗ ) > f (qd∗ ) Therefore, when we calculate the quantities using inverse, we will have qc∗ > qd∗

(8)

Thus, the qc∗ is greater than the qd∗ . Lemma 2 The expected sale of centralized optimal order quantity S(qc∗ ) is greater than the expected sale of decentralized optimal order quantity S(qd∗ ). We build the following inequality which is valid from Lemma 1: qc∗ −



(qc∗ − a)2 2(b − a)



> qd∗ −



(qd∗ − a)2 2(b − a)



since qc∗ > qd∗

In the above inequality, substitutions can be made using Eqs. (2) and (3), resulting in the following relationship:

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S(qc∗ ) > S(qd∗ )

(9)

Lemma 3 The optimal order quantity qx∗ is greater than or equal to the expected sale of optimal order quantity S(qx∗ ) where x is either c, centralized order quantity qc∗ , or d, decentralized order quantity qd∗ . This can be inferred from the following equations: qc∗

>

S(qc∗ )

considering

S(qc∗ )

=

qc∗

 −

qd∗ > S(qd∗ ) considering S(qd∗ ) = qd∗ −



(qc∗ − a)2 2(b − a) (qd∗ − a)2 2(b − a)

 

Therefore, we have qc∗ > S(qc∗ ) qd∗ > S(qd∗ )

(10) (11)

The expected sales of SC are greater with qc∗ than the expected sales with the qd∗ . On comparing the expected profits of all the SC members for Case A and Case B, we note that the retailer may incur losses as per Lemma 1, but there is some scope for improving the sales as seen from Lemma 2. Whereas the expected profits of the entity 2 and the entity 3 may improve in Case B as compared to Case A, the expected profits of the whole supply chain is also more in Case B. The retailer may not agree to coordinate with other SC members as she/he may incur losses by ordering qc∗ . To encourage the retailer to order qc∗ , the upstream members can devise some mechanism so that no SC member would incur losses and at the same time centralized supply chain system should also not be compromised. In the next scenario Case C, sales rebate contracts will be introduced and the coordination achieved will be analyzed.

3.4.3

Case C—Supply Chain Coordination with Centralized Optimal Order Quantity and Sales Rebate Contracts

In this case, each member of the supply chain orders qc∗ . The upstream member will fix some threshold value of sales above which the downstream member can avail rebates from the upstream member. If entity i − 1 sells more than the threshold ti fixed by entity i, entity i pays to entity i − 1 a rebate for each unit sold that is above the threshold limit. The profit functions with sales rebate contracts for the entities in the SC are as follows. The profit functions are as follows:

+

 π1C (qc∗ ) = α1 P S(qc∗ ) − α2 Pqc∗ + s qc∗ − S(qc∗ ) + r2 S(qc∗ ) − t2

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+

+ π2C (qc∗ ) = α2 Pqc∗ − α3 Pqc∗ − r2 S(qc∗ ) − t2 + r3 S(qc∗ ) − t3

+

+ π3C (qc∗ ) = α3 Pqc∗ − α4 Pqc∗ − r3 S(qc∗ ) − t3 + r4 S(qc∗ ) − t4 , so on up to,

+ π NC −1 (qc∗ ) = α N −1 Pqc∗ − α N Pqc∗ − r N −1 (S(q∗c ) − t N −1 )+ + r N S(qc∗ ) − t N

+ π NC (qc∗ ) = α N Pqc∗ − Pqc∗ − r N S(qc∗ ) − t N Total expected profit of the supply chain is N

πiC (qc∗ ) = π1C (qc∗ ) + π2C (qc∗ ) + π3C (qc∗ ) · · · + π NC −1 (qc∗ ) + π NC (qc∗ )

i=1

 π C (qc∗ ) = α1 P S(qc∗ ) − Pqc∗ + s qc∗ − S(qc∗ )

(12)

An interesting observation here is that the expected profits in scenarios B and C are equal. Here, the total supply chain profit is the same as that obtained in Eq. (7). The transfer pay between the supply chain entities in this case will be related to sales rebates and threshold values, which are internal to the supply chain. Hence, it shows that total expected supply chain profits will be the same, but the way that supply chain entities share risks and rewards is the question. In this case, we have the following decision variables: ri ∀{i ∈ 2, 3, . . . N }, ti ∀{i ∈ 2, 3, . . . N }. Another decision variable in this problem is to determine the share of each supply chain entity, due to gain in the expected profits of the whole supply chain. In this paper, we propose a model which determines the decision variables pertaining to the sales rebate contracts (rebates and threshold values) as well as a fair share of gain in the total SC expected profits among SC members due to the rebates. Each supply chain entity adds value to the product in different forms throughout the supply chain. One of the methods to fairly share the gain in the total supply chain expected profit is based on the value addition to the product (difference between selling price and cost incurred) by the respective SC entities. The profit sharing objective is to minimize the deviation between the proportion of profit gained by SC entities and the proportion of value addition by the same SC entities.

3.5 Objective Function The profits gained by using sales rebate contracts for the whole SC must be fairly shared among the entities based on their respective value addition to the product (difference between selling price and cost incurred).

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Objective: Minimize Z =

N

|P Si − Vi |

i=1

= |P S1 − V1 | + |P S2 − V2 | + |P S3 − V3 | · · ·

+ |P S N −1 − VN −1 | + |P S N − VN | (13)

where π1C − π1A π2C − π2A π3C − π3A , P S = , P S = ..., 2 3 πC − π A πC − π A πC − π A π C − π NA−1 π NC − π NA P S N −1 = N −1 , P S = N πC − π A πC − π A α1 P − α2 P α2 P − α3 P α3 P − α4 P , V2 = , V3 = ..., V1 = α1 P − P α1 P − P α1 P − P α N −1 P − α N P αN P − P VN −1 = , VN = α1 P − P α1 P − P P S1 =

3.6 Constraints Since the SC entities will order the qc∗ only if their profits are more in Case C than in Case A and the rebates help improve the profits of each entity while ordering qc∗ , we have the following inequalities which have to be satisfied to coordinate the supply chain.

πiC

≥

πiA

πC ≥ π A

(14)

where i ∈ {1, . . . N }

(15)

From constraint 14, we have πC ≥ π A



 α1 P S(qc∗ ) − Pqc∗ + s qc∗ − S(qc∗ ) ≥ α1 P S(qd∗ ) − Pqd∗ + s qd∗ − S(qd∗ ) α1 P(S(qc∗ ) − S(qd∗ )) − P(qc∗ − qd∗ ) + s(qc∗ − S(qc∗ ) − qd∗ + S(qd∗ )) ≥ 0 From constraint 15, for entity 1 we have

Optimal and Heuristic Profit Sharing Using Sales Rebate …

π1C ≥ π1A α1 P S(qc∗ ) − α2 Pqc∗ + s(qc∗ − S(qc∗ ))

+ +r2 S(qc∗ ) − t2 ≥ α1 P S(qd∗ ) − α2 Pqd∗ + s(qd∗ − S(qd∗ )) α1 P(S(qc∗ ) − S(qd∗ )) − α2 P(qc∗ − qd∗ ) + s(qc∗ − S(qc∗ ) − qd∗ + S(qd∗ ))

+ +r2 S(qc∗ ) − t2 ≥ 0

+ r2 S(qc∗ ) − t2 ≥ (α2 P − s)(qc∗ − qd∗ ) − (α1 P − s)(S(qc∗ ) − S(qd∗ )) From constraint 15, for entity 2 we have π2C ≥ π2A

+

+ α2 Pqc∗ − α3 Pqc∗ − r2 S(qc∗ ) − t2 + r3 S(qc∗ ) − t3 ≥ α2 Pqd∗ − α3 Pqd∗

+

+ α2 P{qc∗ − qd∗ } − α3 P(qc∗ − qd∗ ) − r2 S(qc∗ ) − t2 + r3 S(qc∗ ) − t3 ≥ 0

+

+ r3 S(qc∗ ) − t3 ≥ r2 S(qc∗ ) − t2 − (α2 P − α3 P)(qc∗ − qd∗ ) From constraint 15, for entity 3 we have π3C ≥ π3A

+

+ α3 Pqc∗ − α4 Pqc∗ − r3 S(qc∗ ) − t3 + r4 S(qc∗ ) − t4 ≥ α3 Pqd∗ − α4 Pqd∗

+

+ α3 P(qc∗ − qd∗ ) − α4 P(qc∗ − qd∗ ) − r3 S(qc∗ ) − t3 + r4 S(qc∗ ) − t4 ≥ 0

+

+ r4 S(qc∗ ) − t4 ≥ r3 S(qc∗ ) − t3 − (α3 P − α4 P)(qc∗ − qd∗ ) From constraint 15, for entity N−1 we have π NC −1 ≥ π NA−1

+ α N −1 Pqc∗ − α N Pqc∗ − r N −1 S(qc∗ ) − t N −1

+ +r N S(qc∗ ) − t N ≥ α N −1 Pqd∗ − α N Pqd∗

+ α N −1 P(qc∗ − qd∗ ) − α N P(qc∗ − qd∗ ) − r N −1 S(qc∗ ) − t N −1

+ +r N S(qc∗ ) − t N ≥ 0

+

+ r N S(qc∗ ) − t N ≥ r N −1 S(qc∗ ) − t N −1 −(α N −1 P − α N P)(qc∗ − qd∗ )

From constraint 15, for entity N we have

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π NC ≥ π NA

+ α N Pqc∗ − Pqc∗ − r N S(qc∗ ) − t N ≥ α N Pqd∗ − Pqd∗

+ α N P(qc∗ − qd∗ ) − P(qc∗ − qd∗ ) − r N S(qc∗ ) − t N ≥ 0

+ r N S(qc∗ ) − t N ≤ (α N P − P)(qc∗ − qd∗ ) From the above inequalities, we get (α2 P − s)(qc∗ − qd∗ ) − (α1 P − s)(S(qc∗ ) − S(qd∗ ))

+ ≤ r2 S(qc∗ ) − t2 ≤ (α2 P − P)(qc∗ − qd∗ )

(16)

(α3 P − s)(qc∗

≤ r3 S(qc∗ )

(17)

− qd∗ ) − (α1 P − s)(S(qc∗ ) − S(qd∗ )) + − t3 ≤ (α3 P − P)(qc∗ − qd∗ ) (α N −1 P − s)(qc∗ − qd∗ ) − (α1 P − s)(S(qc∗ ) − S(qd∗ ))

+ ≤ r N −1 S(qc∗ ) − t N −1 ≤ (α N −1 P − P)(qc∗ − qd∗ ) (α N P − s)(qc∗ − qd∗ ) − (α1 P − s)(S(qc∗ ) − S(qd∗ ))

+ ≤ r N S(qc∗ ) − t N ≤ (α N P − P)(qc∗ − qd∗ )

(18) (19)

It is evident that the generalized expression for the above is (αi P − s)(qc∗ − qd∗ ) − (α1 P − s)(S(qc∗ ) − S(qd∗ ))

+ ≤ ri S(qc∗ ) − ti ≤ (αi P − P)(qc∗ − qd∗ )∀i ∈ {2, 3, . . . N }

(20)

In order to meet the above-said objective, subject to the given constraints, an appropriate technique is required to find the values of the decision variables ri ∀{i ∈ 2, 3, . . . N }, ti ∀{i ∈ 2, 3, . . . N } and determine the fair share of profits gained due to coordination by sales rebate contracts. It can be observed from the above equations and inequalities that the objective function is linear but the constraints are nonlinear in nature. Considering computational complexity of determining rebate values and threshold values in N-level supply chain, meta-heuristic techniques, namely genetic algorithm (GA) and simulated annealing (SA), are adopted. The objective of the problem is to find the rebate and threshold values which, when installed in the supply chain, will provide a fair share of profit to all the SC members involved.

4 Proposed Heuristic Researchers have confirmed through experimentation that it is important and good to start the search technique with a solution constructed by a heuristic than to start the search technique with a randomly generated solution because such a heuristic seed

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solution serves to enhance the process of convergence in the search process (Daniel & Rajendran, 2006). Therefore, the choice of an appropriate starting solution is important due to the computational complexity and because convergence to an extent depends on the initial solution. Bound-Based Sampling Heuristic In the bound-based sampling heuristic, we sample a value of threshold from between the lower and upper bounds of the order quantity; i.e., we chose a random value based on the bounds. Entity 2 has a special constraint as threshold limit has to be set less than the expected value of sales. For other entities since the quantity shipped between them is fixed, the threshold limit can be up to to the maximum order quantity. The lower limit for threshold can be greater than or equal to zero. A value of ri is sampled in the interval (riLL , riUL ) tiLL ≥ 0 tiU L ≤ S(q∗sc ) for all {i ∈ 3, 4 ... N }

(21) (22)

The upper and lower limits for thresholds and sample values {t2 , t3 , . . . t N } are used to evaluate the upper and lower bounds for rebates represented by riU L and riL L respectively. The rebate values ri ∀{i ∈ 2, 3, . . . N } thus sampled, coupled with the threshold limits ti ∀{i ∈ 2, 3, . . . N }, constitute the initial seed solution for heuristic search algorithms. The solution obtained has relatively higher fitness than the solution obtained by the random procedure and is definitely a feasible solution.

5 Proposed Meta-Heuristics Three different meta-heuristic algorithms have been proposed to identify the sales rebate and threshold values for all the supply chain entities to equally share the supply chain profit: 1. Parallel weighted genetic algorithm 2. Parallel simulated annealing 3. Parallel weighted GA followed by parallel SA. The mechanics of each of these algorithms are modeled to be able to generate the sales rebate and threshold values for the N entities in the supply chain. An initial feasible solution is generated by using the bound-based sampling heuristic proposed in the previous section.

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Fig. 2 Chromosome structure

5.1 Parallel Weighted Genetic Algorithm GA is a search algorithm based on the mechanics of natural selection and natural genetics. Daniel and Rajendran (2006) developed different variants of GA for a supply chain. Each solution is represented in the form of a chromosome. A chromosome uses value encoding to represent the threshold and rebate values. Figure 2 shows the representation of a chromosome and an example. The length of the chromosome depends on the number of entities in the supply chain; i.e., if N is the supply chain length, then chromosome length is 2(N − 1). The genes in positions 1 to (N − 1) represent the rebate given to the supply chain entities. r2 which is in the first position denotes the rebate given by entity 2 to entity 1 and so on. The genes in positions N to 2(N − 1) represent the threshold for the supply chain entities. t2 which is in the N position denotes the threshold set by entity 2 to entity 1 and so on. In a parallel GA, two sets of initial populations A and B each containing equal number of parent chromosomes are generated and mutated until the termination criterion is attained. For each of A and B: • All the chromosomes in the initial population are evaluated, and their respective objective function values (Z ) are obtained by evaluating Eq. (13). Fitness value 1 (since Z is to ( f ) of each chromosome is computed using the formula f = 1+Z be minimized). • A child chromosome (offspring) is created by taking the weighted mean (with respect to relative fitness values of chromosomes) of the values of respective genes of parent chromosomes. In addition to this, all the parent chromosomes in the initial population as well as the new child chromosome are mutated with a perturbation proportion PP. The best chromosomes according to their fitness function from the set of (initial parent population, mutated parent population and mutated child chromosome) are chosen as the next parent population. The best chromosome from the next generation of both A and B replaces the worst chromosome of the other one, and next parent population is formed. This ensures communication between the two parallel populations A and B. The above-mentioned steps are repeated. After a certain number of generations, the algorithm is terminated and the best chromosome at the end of termination provides the rebates and the threshold values for the respective members in the supply chain (Fig. 3).

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Fig. 3 Parallel weighted GA illustration

5.1.1

Notations

no_gen: len: pop_size: ini_pop: ri : ti : riUL : riLL : tiUL : tiLL : f: r el f : u: P P: c_chrom: m_pop:

5.1.2

Number of generations Length of the chromosome (number of genes) Population size Sik :] ith gene of the kth chromosome for i ∈ {1, 2, 3, . . . len} and k ∈ {1, 2, 3, . . . pop_size} Initial population Rebate given by entity i to entity i − 1, i ∈ {2, 3, . . . N } Threshold limit set by entity i to entity i − 1 i ∈ {2, 3, . . . N } Upper limit of rebate given by entity i to entity i − 1, i ∈ {2, 3, . . . N } Lower limit of rebate given by entity i to entity i − 1, i ∈ {2, 3, . . . N } Upper limit of rebate set by entity i to entity i − 1, i ∈ {2, 3, . . . N } Lower limit of rebate set by entity i to entity i − 1, i ∈ {2, 3, . . . N } Fitness value of the chromosome Relative fitness value of the chromosome Uniform random number in the interval (0, 1) Perturbation proportion Offspring obtained from the weighted mean of parent population Resultant population consisting of offspring after mutation.

Step-by-Step Procedure of the Parallel Arithmetic GA (Considering a 3-Member Supply Chain)

• Step 1 : Initialize no_gen = 0. • Step 2 : Generate two sets A and B of initial population (ini_pop) each of size (pop_size) equal to 5 (set arbitrarily for the purpose of demonstrating the applicability of GA to the current problem) with each chromosome representing a set of rebates and thresholds of distributor and manufacturer. The first two genes carry

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the rebate values, and the last two carry the threshold values. For set A: if i ∈ {1, 2} Sik = riLL + (riUL − riLL ) ∗ u if i ∈ {3, 4} Sik = tiLL + (tiUL − tiLL ) ∗ u For set B, the same is repeated, but instead of the random number u its antithetic 1 − u is used. Note that ini_pop w.r.t A and B refers to populations A and B at the beginning of every generation. • Step 3: For each of the sets A and B, do the following: – 3.1 Evaluate every chromosome in ini_pop (by evaluating the objective function Eq. 13), and obtain the value of Z . Obtain the fitness value f k for every chromosome k. – 3.2 Relative fitness of each of the parent chromosomes is calculated as f f where  f is the sum of the fitness functions of all the parent chromosomes. – 3.3 A child chromosome is formed by taking the weighted mean of the ini_pop. c_chromi = 5k=1 (rel f k ∗ Sik ) for i ∈ {1, 2, 3, . . . len} – 3.4 All the chromosomes in ini_pop and c_chrom are mutated with a perturbation proportion P P = 0.2 m_popik = min(max((Sik ∗ (1 − P P) + Sik ∗ 2 ∗ P P ∗ u), riLL [or tiLL ]), riUL [or tiUL ]) for i ∈ {1, 2, 3, . . . len} c_chromi = min(max((c_chromi ∗ (1 − P P) + c_chromi ∗ 2 ∗ P P ∗ u), riLL [or tiLL ]), riUL [or tiUL ]) for i ∈ {1, 2, 3, . . . len} – 3.5 The best 5 chromosomes from ini_pop, m_pop and c_chrom are chosen and set as the next ini_pop. • Step 4: The best chromosome from the ini_pop of both sets A and B combined is chosen, and it replaces the worst chromosome of the ini_pop of the other population. This communication establishes parallelism between the two sets A and B. • Step 5: Set no_gen = no_gen + 1. • Step 6: Repeat steps 2, 3, 4 and 5 until no_gen = 500 which is the termination criterion. • Step 7 : The best chromosome among the resultant 10 chromosomes of A and B constitutes the solution to the problem.

5.2 Parallel Simulated Annealing Algorithm In the proposed parallel SA, the idea of parallelism is drawn from Janakiram et al. (1996) and the implementation is drawn from Daniel and Rajendran (2005). Janakiram et al. (1996) suggested a clustering algorithm for SA in which there are n nodes of the network that run using different initial solutions. After a fixed number of iterations, they exchange their partial results to get the best one. All the nodes

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accept the best partial solution and start applying the SA technique for that best partial result. This process is repeated until the termination criterion is attained. Approaches of these kinds are often highly problem-dependent, and one variant of this algorithm that suites the problem discussed here is the parallel SA algorithm. In parallel SA, there are two parallel nodes A and B corresponding to two different solutions. For each of A and B, the following are done: • The initial node is perturbed, and the objective function (Z ) is obtained by evaluating Eq. (13). • If (Z ) of the perturbed solution is better than the (Z ) of the initial solution, then the initial solution is updated, else the initial solution is updated with a predefined probability. • If the solution is not improving even after two consecutive iterations, the following are done in a cyclic manner: – Reheating is done. Temperature T is increased to 500. – The so far best solution of the parallel node replaces the current best solution of the node under consideration. – Both reheating and adoption of solution from the parallel node take place. The above steps are repeated until the termination criterion is reached which is the minimum temperature possible (Fig. 4).

5.2.1 N: Z: S: S: rf:

Notations Number of supply chain members Objective function value (Eq. 13) Initial solution sequence Perturbed solution sequence Temperature reduction factor (0 < r f < 1)

Fig. 4 Parallel SA illustration

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T: len: ri : ti : riUL : riLL : tiUL : tiLL : it_count: : u:

5.2.2

Temperature (T > 0) Length of the sequence:]- 2*(N-1) Rebate given by entity i to entity i − 1, i ∈ {2, 3, . . . N } Threshold limit set by entity i to entity i − 1, i ∈ {2, 3, . . . N } Upper limit of rebate given by entity i to entity i − 1, i ∈ {2, 3, . . . N } Lower limit of rebate given by entity i to entity i − 1, i ∈ {2, 3, . . . N } Upper limit of threshold set by entity i to entity i − 1, i ∈ {2, 3, . . . N } Lower limit of threshold set by entity i to entity i − 1, i ∈ {2, 3, . . . N } Iteration count absolute percentage change in the objective value function Uniform random number in the interval (0, 1).

Step-by-Step Procedure of the Parallel Simulated Annealing Algorithm (Considering a 3-Member Supply Chain)

• Step 1: Initialize it_count = 0. • Step 2: Generate two nodes (candidate solution) A and B with initial solutions (S), each of length (len = 4). The first two values are the rebate values, and the last two are the threshold values. For set A: if i ∈ {1, 2} Si = riLL + (riUL − riLL ) ∗ u if i ∈ {3, 4} Si = tiLL + (tiUL − tiLL ) ∗ u For set B, the same is repeated, but instead of the random number u its antithetic 1 − u is used. • Step 3: Set T = 500 and i = 0 and r f = 0.2. • Step 4: For 500 ≥ T ≥ 17, do the following: • 4.1: For each of the nodes A and B and for 0 ≤ j ≤ 10, do: – 4.1.1: Perturb S as Si = min(max(Si ∗ (1 − r f ) + 2 ∗ r f ∗ u, riLL ), riUL ) ∀ i ∈ {1, 2} Si = min(max(Si ∗ (1 − r f ) + 2 ∗ r f ∗ u, tiLL ), tiUL ) ∀ i ∈ {3, 4}  )−z(S) ∗ 100 – 4.1.2: Compute  = z(S z(S) if  < 0: Update S = S  . − else if e T ≥ u: Update S = S  . – 4.1.3: If the value of the objective function is not improving even after two iterations and it_count ≤ 10, 000, do the following in a cyclic manner: · Reheating: Set T = 500 and it_count = it_count + 1 · Update the best solution S (obtained so far) with respect to the current node with the best solution S (obtained so far) of the parallel node. Set it_count = it_count + 1

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· Do both the above steps: reheating and adoption of the best solution (obtained so far) from the parallel node. – 4.1.4: Set j = j + 1. – 4.2: Set T = T ∗ r . • Step 5: The node with a smaller objective function Z , between A and B, constitutes the solution to the problem.

5.3 Parallel Genetic Algorithm Followed by Parallel Simulated Annealing Algorithm Experiments show that a good initial solution for simulated annealing improves the quality of the solution. Genetic algorithm alone often fails to provide optimal solutions as its quality is heavily dependent on the quality of the initial population. Considering these two concepts, an algorithm is proposed that feeds the best chromosomes from the final child population of a parallel GA as the initial solutions of the two nodes for a parallel SA. By doing so, the quality of the solution is tremendously increased.

5.3.1

Step-by-Step Procedure of the Parallel GA-SA Algorithm

• Step 1: Do steps 1 to 6 of the parallel GA listed in 1. • Step 2: Set the best chromosome from population A as the initial solution for node A and the best chromosome from population B as the initial solution for node B of the parallel SA. • Step 3: Do steps 1 to 5 of the parallel SA listed in 2.

6 Experiments and Discussion To compare the efficiency of various algorithms proposed above, we take a three-stage supply chain and determine the objective function (equation 13) and the computational time for each algorithm. For a more real-time perspective of the model, we refer to entity α_1 as retailer, entity α_2 as distributor and entity α_3 as manufacturer and discuss the expected profits and contract parameter decisions (i.e., rebates and thresholds). We perform the two sets of experiments on the model described in Sect. 5. Experimental settings have been identified by varying the cost settings for the exchange of goods between the entities and also varying the demand patterns of the end customer.

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Table 1 Supply chain experiment 1—fixed cost parameter subjected to varied demand

In the first experiment, we have a fixed cost setting between the three entities, namely retailer, distributor and manufacturer. The costs in experiments E1 are fixed; assuming the purchase cost P for manufacturer is 40, she/he sells it to the distributor at 52 (adds 30% value over the purchase cost). The distributor purchases it from the manufacturer at 52 and sells it at 64 (adds 60% value over the purchase cost). Finally, the retailer purchases it from the distributor at 64 and sells it at 76 (adds 90% value over the purchase cost). They exchange the goods at a predetermined price but are subjected to varied demand. The demand patterns used in this particular test are 1– 100, 101–200, 201–300 and 301–400. These model settings are described in Table 1. The terms α3 , α2 and α1 signify the quantum of value addition that has been done by manufacturer, distributor and retailer, respectively. Table 1 presents the experimental setting E1_1, E1_2 and E1_3 in detail. Tables 1.1, 1.2 and 1.3 give the results obtained for the setup in experiment 1 using the three different meta-heuristics. All the meta-heuristic algorithms were run for a computational time of around 1 s. The results show that the best value of the objective function is attained using parallel GA-SA. Parallel weighted GA is the least accurate due to a restricted computational time, while parallel SA performs considerably well and produces solutions comparable to that of parallel GA-SA. Based on the experimental results, the profits of the supply chain entities when a GA-based heuristic was used are presented in Figs. 5, 6 and 7. First observation that

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Fig. 5 Results of supply chain experiment 1

we are able to make is that there is a significant increase in the total profit in Case B and Case C compared to Case A. This is primarily due to the increase in supply chain order quantity to centralized order quantity qc from the decentralized order quantity qd . We will notice that as the demand increases/stabilizes, the profits in Case B and Case C tend to match that in Case A. We notice that in Case B of E1_2, the retailer goes into a loss of 312 units. In Case B of E1_2, profits are not evenly shared and the retailer faces most of the risk. To enable the retailer to participate in ordering the higher-order quantity, the sales rebate contract plays a pivotal role. We notice in Case C that the profits have been shared between various supply chain entities based on the value addition to the product, thereby ensuring that no member faces an excessive loss while trying to increase the overall supply chain profit. In experiment 2, we have a varied cost setting with a fixed demand. In experiments E2_1, assuming the purchase cost P for manufacturer is 50, she/he sells it to the distributor at 75 (adds 50% value over the purchase cost). The distributor purchases it from the manufacturer at 75 and sells it at 90 (adds 80% value over the purchase cost). Finally, the retailer purchases it from the distributor at 90 and sells it at 95 (adds 90% value over the purchase cost). Experiments E2_2 and E2_3 also drive scenarios where the value addition can be read in the above manner. Table 2 presents the experimental setting E2_1, E2_2 and E2_3 in detail. Tables 2.1, 2.2 and 2.3 give the results obtained for the setup in experiment 2 using the three different meta-heuristics. These results show that in terms of giving the optimum rebate and threshold values, both parallel SA and parallel GA-SA perform well but as the number of members in the supply chain increases, parallel SA-GA is expected to give a better solution as compared to parallel SA algorithm. Parallel GA requires a greater computational time to produce efficient results; hence in this time setting, it is not performing well.

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Fig. 6 Results of supply chain experiment 1

Fig. 7 Results of supply chain experiment 1

Presented in Figs. 8, 9 and 10 are the profits obtained by various entities when a GA-based heuristic was used to generate the rebate and threshold values of the sales contract for experiment 2. In this experiment, we notice that the retailer is faced with a loss in Case B where we have a centralized decision making but no coordination. In Case C, we notice that coordination enables profit sharing between all the supply chain entities, thereby encouraging the retailer to take the risk by ordering more quantity. An interesting observation in this experiment is with regard to the profits shared in Case C. The total profits remain same in Case B and Case C, but in Case C as we have installed sales rebate contract to ensure coordination, the profits are shared

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Fig. 8 Results of supply chain experiment 2

Fig. 9 Results of supply chain experiment 2

Fig. 10 Results of supply chain experiment 2

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Table 2 Supply chain experiment 2—varied cost parameters subjected to constant demand

based on the value added by the entity with respect to the purchase price. Considering the example of setting E2_1, we notice that the SC is able to generate an additional profit of 3000 units in Case B and Case C as compared to Case A, which is split between the three entities based on their value addition in Case C, while in Case B the retailer is at a loss.

7 Conclusion This paper presented a profit sharing mechanism using sales rebate contracts in a multistage coordinated supply chain. The generic N-echelon model was mathematically described, and the reasons why it cannot be solved have been listed. Several meta-heuristics have been developed and have been tested on a 3-echelon model. It was observed that the total SC profit has increased in Case B when compared to Case A because of centralization, but the increase in profits is not proportionally shared by every member according to their respective value addition. A single member (retailer) is subjected to more risk. The introduction of contracts has compensated for the retailer’s risk, and the increased profit due to supply chain coordination is

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fairly shared among the members, as proportional to their value addition to the product. Here, the deviation between the proportion of profit shared and the proportion of their value addition is minimized using genetic algorithm and simulated annealing. Analyzing the quality of the optimum solution obtained, it can be concluded that both parallel GA-SA and parallel SA meta-heuristic techniques can be used and the better of the two can be chosen for determining the sales rebate and threshold values. Future work can be done on the application of particle swarm optimization and ant colony optimization algorithms in determining the rebate and threshold values. Acknowledgements This work becomes a reality with the kind support and help of many individuals. We would like to extend our sincere thanks to all of them. We are overwhelmed by the quick review and response we received related to the publication of this work. Our heartfelt gratitude to the editor and the reviewers for their diligent work.

References Arcelus, F. J., Kumar, S., & Srinivasan, G. (2007). Pricing and rebate policies for the newsvendor problem in the presence of a stochastic redemption rate. International Journal of Production Economics, 107(2), 467–482. Arcelus, F. J., Kumar, S., & Srinivasan, G. (2008). Pricing and rebate policies in the two-echelon supply chain with asymmetric information under price-dependent, stochastic demand. International Journal of Production Economics, 113(2), 598–618. Cachon, G. P. (2003). Supply chain coordination with contracts. Handbooks in Operations Research and Management Science, 11, 227–339. Chang, Y.-H. (2010). Adopting co-evolution and constraint-satisfaction concept on genetic algorithms to solve supply chain network design problems. Expert Systems with Applications, 37(10), 6919–6930. Daniel, J. S. R., & Rajendran, C. (2005). Determination of base-stock levels in a serial supply chain: a simulation-based simulated annealing heuristic. International Journal of Logistics Systems and Management, 1(2/3), Daniel, J., & Rajendran, C. (2006). Heuristic approaches to determine base-stock levels in a serial supply chain with a single objective and with multiple objectives. European Journal of Operational Research, 175(1), 566–592. Emmons, H., & Gilbert, S. M. (1998). Note the role of returns policies in pricing and inventory decisions for catalogue goods. Management Science, 44(2), 276–283. Janakiram, D., Sreenivas, T.H., & Subramaniam, K. G. (1996). Parallel simulated annealing algorithms. Journal of Parallel and Distributed Computing, 37(121), 207–212 (1996) Koulamas, C. (2006). A newsvendor problem with revenue sharing and channel coordination. Decision Sciences, 37(1), 91–100. Krishnan, H., Kapuscinski, R., & Butz, D.A. (2004). Coordinating contracts for decentralized supply chains with retailer promotional effort. Management Science, 50(1), 48–63 (2004) Lian, Z., & Deshmukh, A. (2009). Analysis of supply contracts with quantity flexibility. European Journal of Operational Research, 196(2), 526–533. Nachiappan, S. P., & Jawahar, N. (2007). A genetic algorithm for optimal operating parameters of vmi system in a two-echelon supply chain. European Journal of Operational Research, 182(3), 1433–1452. Nalla, V.R. (2008). Contract mechanisms for coordinating operational and marketing decisions in a supply chain: models and analysis. Ph. D. Thesis, Nyenrode Business Universiteit, The Netherlands.

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Sang, S. (2016). Buyback contract with fuzzy demand and risk preference in a three level supply chain, 46, 518–526. 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 planning horizon. Supply Chain Strategies: Issues, and Models. Springer. Simchi-Levi, D., Kaminsky, P., & Simchi-Levi, E. (2008). Designing and managing the supply chain. Managing the Supply Chain. Taylor, T. A. (2002). Supply chain coordination under channel rebates with sales effort effects. Management Science, 48(8), 992–1007. Tsay, A. A. (1999). The quantity flexibility contract and supplier-customer incentives. Management Science, 45(10), 1339–1358. van der Rhee, B., van der Veen, J. A. A., Venugopal, V., & Nalla, V. R. (2010). A new revenue sharing mechanism for coordinating multi-echelon supply chains. Operations Research Letters, 38(4), 296–301. Wong, W.-K., Qi, J., & Leung, S. Y. S. (2009). Coordinating supply chains with sales rebate contracts and vendor-managed inventory. International Journal of Production Economics, 120(1), 151– 161.

A Deterministic Heuristic Algorithm to Minimize the Length of a Manufacturing Line in Transformation of Jobshops into Flowshops J. Krishnaraj Abstract This research paper focuses on the development of a deterministic heuristic algorithm for solving the problem of transforming jobshops into flowshops with the aim of minimizing a manufacturing line length. The considered problem is a variant of the familiar stringology problem of the shortest common supersequence (SCS) which finds the shortest string length. The proposed deterministic heuristic algorithm generates seed solutions from the concatenation pattern of forward reduction, inverse reduction, and machine elimination techniques, in addition to a new local search scheme called two-pair adjacent interchange scheme (TPAIS). The best seed sequence is improved by three concatenation patterns of the Job Index-Based Insertion Scheme (JIS) and Modified Job Index-Based Swap Scheme (MJSS), thereby leading to three variants of the suggested heuristic algorithm (called HAV1 to HAV3). An extensive computational analysis is carried out to illustrate the performance of the three variants of the proposed deterministic heuristic algorithm. The computational results show that the HAV3 performs well in terms of minimizing a manufacturing line length and number of duplicate machines saved in comparison with the existing algorithm.

1 Introduction In jobshop scheduling problems, the process sequence (machine routings) of the jobs is not the same, and machines that can able to perform similar operations are grouped (French, 1982). Hence, the flow of each job in jobshop scheduling is not unidirectional. But in case of a flowshop, jobs follow a unidirectional flow as per the respective job sequence through the shop. As per Knolmayer et al. (2002), the conversion of jobshops into flowshops is significantly important in the situation of reducing the manufacturing flow line (i.e., to achieve an efficient supply chain management). It is noteworthy that the minimization of manufacturing flow line length (i.e., flowshop length) leads to a reduced inventory level (Kimms, 2000). J. Krishnaraj (B) Department of Mechanical Engineering, MLR Institute of Technology, Hyderabad, Telangana, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_6

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The minimization of a manufacturing flow line length is related to the familiar classical SCS stringology problem (Framinan & Ruiz-Usano, 2002). Framinan (2005) stated that although the SCS problem and jobshop transformation problem appear similar, they are not in terms of assumptions made in the problem statement. Timkovsky (1989) stated that in overall SCS problem is identified to be NP-hard. From the literature studies, it is observed that some researchers (to cite a few, Framinan and Ruiz-Usano (2002), Framinan (2005), Rajendran et al. (2010)) attempted the problem of transmuting jobshops into flowshops with the aim of minimizing the manufacturing flow line length. Mousavi et al. (2012), Dondi (2013), and Saifullah and Islam (2016) attempted the SCS stringology problem; however, they are not directly related to the transformation of jobshops into flowshops. The SCS stringology problem has many applications in the fields of DNA analysis, planning, artificial intelligence, query optimization, data compression in the database, and various bioinformatics activities (Saifullah and Islam 2016). For a given set of strings L, the SCS problem leads to find a minimal string length (X) that is a supersequence of each string in L (strings in L are embedded). A string is a set of zero or more characters from  (an alphabet). For example, consider a set of strings L = {cba, abba, abc} over an alphabet  = {a, b, c}. The shortest supersequence for L is abcba (i.e., X = 5). Exact algorithms like branch-and-bound (Fraser & Irving, 1995) and dynamic programming (DP) (Jiang & Li, 1995) are proposed for solving the SCS problem. These algorithms execute optimally in polynomial time for small instances. But for large instances, DP uses lots of memory and branch-andbound consumes exponential time (Barone et al., 2001). Because of NP-hardness, the exact algorithms failed to compute SCS for larger problem instances. Many heuristic algorithms and meta-heuristic approaches were attempted to solve this problem. The heuristic approaches include majority merge (Branke et al., 1998), reduce, and expand (Barone et al., 2001) and deposition and reduction (DR) (Ning & Leong, 2006). The meta-heuristic approaches include an ant colony optimization algorithm (Michel & Middendorf, 1999), an artificial bee colony algorithm (Noaman & Jaradat, 2011), enhanced beam search algorithm (Mousavi et al., 2012), and chemical reaction optimization algorithm (Saifullah & Islam, 2016). This work focuses on the development of a deterministic heuristic algorithm to transform jobshops into flowshops with the aim of minimizing the manufacturing line length. Note that deterministic heuristics are computationally faster than metaheuristics always executes the same way and produces the same solution. The minimization of manufacturing line is more relevant and meaningful for today’s dynamic production environment which reduces the number of duplicate machines in the manufacturing line (i.e., a sequence of machine routings corresponds to the job sequence) and helps to minimize the lead times during the production of jobs which also reduces the inventory levels. This problem was attempted by very few researchers, Framinan (2005) and Rajendran et al. (2010), and reported the best string length for 70 well-known benchmark problem instances. Rajendran et al. (2010) employed a meta-heuristic algorithm which generally consumes higher computational time. Hence in this work, a deterministic heuristic algorithm is proposed for the minimization of length of the manufacturing line for the problem of transforming jobshops

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into flowshop, and results are compared with the results yielded by Framinan (2005) through deterministic heuristic. To illustrate the effectiveness and performance of the proposed deterministic heuristic, the well-known benchmark problems (considered in Framinan, 2005) of 40 instances (LA01 ~ LA40) of eight different sizes owing to Lawrence (1984), 10 problems (ORB1 ~ ORB10) due to Applegate and Cook (1991), and 20 problems (SWV1 ~ SWV20) of Storer et al. (1992) are considered for the analysis of the transformation of jobshops into flowshops. An extensive performance analysis on these 70 jobshop benchmark problems is carried out to evaluate the proposed deterministic heuristic algorithm with the aim of minimizing the manufacturing line length on the jobshop transformation problem.

2 Problem Description The transformation of jobshops into flowshops with the aim of minimizing the length of a manufacturing line may be described as follows: Consider a jobshop of where n jobs have to be processed according to   m machines a routing matrix M := Mi j , where Mi j indicates the jth operation to be performed on job i. In this problem, the objective is to find a flowshop with the minimum number of machines where all job operations can be completed in the order indicated in M. The obtained solution, say shortest common supersequence, should be a feasible one with a minimum length of manufacturing line. Let us consider a job ordering J = { j1 − j2 − . . . − jn } of a jobshop is transformed into a following flowshop:   S = M j1,1 − M j2,1 − . . . − M jn,1 − M j1,2 − M j2,2 − . . . − M j1,m − . . . − M jn,m The above flowshop is a feasible one because the machines of all jobs are sequenced as per the job ordering as well as satisfies the machine routing of jobs as specified in the routing matrix M. In this case, the length of manufacturing line is found as follows: L = n × m. Let us consider a jobshop problem with 6 jobs and 4 machines (i.e., n = 6 and m = 4) and its machine routings as given in Table 2 (refer Appendix 1). Assume the job ordering as J= {1-2-3-4-5-6} gives S = {1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-41-3-3-4-1-2} and its length of the manufacturing line, L = 24. The flowshop, S, is a feasible solution, but it is not a shortest common supersequence. For instance, let us consider S = {1-2-3-4-1-3-2-4-3}. It is a feasible solution which follows all job operations performed as per the order specified in machine routings. In this case, the length of the manufacturing line, L = 9 and S, is one of the supersequence which saved 15 duplicated machines of flowshop.

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3 Proposed Deterministic Heuristic Algorithm In this research work, three variants of the deterministic heuristic algorithm, namely heuristic algorithm variant-1 (HAV1), heuristic algorithm variant-2 (HAV2), and heuristic algorithm variant-3 (HAV3), are proposed. The notations and terminologies used in the heuristic algorithm are first discussed in Sect. 3.1, and then the step-by-step procedure of the heuristic algorithm and its variants are presented in Sect. 3.2.

3.1 Notations and Terminology n m S1 , S2 , S3 , S4 L 1, L 2, L 3, L 4 BS BL

number of jobs number of machines a sample of shortest common super seed sequences of machines length of manufacturing line corresponds to S1 , S2 , S3 , S4 best shortest common supersequence found in a HAV shortest string length (length of manufacturing line) corresponds to BS found in a HAV.

3.2 The Step-by-Step Procedure of the Proposed Deterministic Heuristic Algorithm and Its Variants (HAV1 to HAV3)

Step 1: Generation of shortest common super seed sequence of machines 1. , based on the machine routing correStep 1.1: Initialize a super sequence, sponding to a job sequence {1-2-3-4-5-6-…-n} and assign the length of manufacturing line (string length) to . Step 1.2: Apply Forward Reduction Technique (proposed by Framinan (2005); See Appendix 1 for details) on , assign the obtained super sequence of machines and its string length to . to then assign the super sequence of machines

to

and

to

else and . do not update Step 1.3: Apply Inverse Reduction Technique (proposed by Framinan (2005); See Appendix 2 for details) on , assign the obtained super sequence of machines to and its string length to . then

A Deterministic Heuristic Algorithm to Minimize the Length …

assign the super sequence of machines

to

123

and

to

else do not update and . Step 1.4: Apply Machine Elimination Technique (proposed by Framinan and Ruiz-Usano (2002); See Appendix 3 for details) on , assign the obtained super sequence of machines to and its string length to . then assign the super sequence of machines

to

and

to

else do not update and . Step 1.5: Invoke Two-Pair Adjacent Interchange Scheme (TPAIS) (proposed in this work; See Appendix 4 for details) on , assign the obtained super sequence and its string length to . of machines to then assign the super sequence of machines

to

and

to

.

else and . do not update Step 2: Generation of shortest common super seed sequence of machines 2. differ from those in Step 1 in and its string length Steps in the generation of terms of applying first the inverse reduction technique then forward reduction technique and thereafter the machine elimination technique and TPAIS. Step 3: Generation of shortest common super seed sequence of machines 3. based on the machine routing Step 3.1: Initialize a super sequence of machines, corresponds to a job sequence {n-…-6-5-4-3-2-1} (antithetic job sequence) and assign the length of the manufacturing line (string length) to . Step 3.2 to Step 3.5: Correspond to the application of forward reduction, inverse reduction, machine elimination techniques, and TPAIS (see Steps 1.2 to 1.5). Step 4: Generation of shortest common super seed sequence of machines 4. and its string length differ from those in Step 3 in Steps in the generation of terms of applying first the inverse reduction technique then forward reduction technique and thereafter the machine elimination technique and TPAIS. based on Step 5: Choose the best seed super sequence of machines among to to . Assign the best seed super sequence of the lowest string length among machines to BS and its string length to BL. Step 6: Apply concatenation of JIS (used by Rajendran et al. (2010); See appendix 5 for details) and MJSS (modified job-index based swap scheme, originally used by Rajendran and Ziegler (2004); See appendix 6 for details) such as JIS-MJSSJIS-MJSS-JIS-MJSS in case of HAV1, concatenation of JIS-JIS-MJSS-JIS-MJSSJIS in case of HAV2 and concatenation of JIS-JIS-JIS-MJSS-JIS-MJSS in case of HAV3 on an initial job sequence {1-2-3-4-5-6-…-n}; Return the best shortest common super sequence of machines (BS) and its string length (BL). STOP.

Note that on every perturbed job sequence obtained during the application of JIS and MJSS, the supersequence of machines based on its machine routings is initialized

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and the set of procedures followed in Step 1 to Step 5 is executed to update the best job sequence, best shortest common supersequence of machines (BS), and shortest string length (BL). In the exploration of minimization of the length of manufacturing line, the proposed variants (HAV1 to HAV4) employ the new local search scheme TPAIS for the interchanging of adjacent machines to overcome the drawback present in the earlier approaches like forward reduction, inverse reduction, and machine elimination techniques. Hence, the order of forward reduction–inverse reduction–machine elimination–TPAIS is used in case of HAV1 and HAV3 and the order of inverse reduction–forward reduction–machine elimination–TPAIS is used in case of HAV2 and HAV4.

4 Performance Evaluation of the Proposed Deterministic Heuristic Algorithm The performance of the proposed three variants of the deterministic heuristic algorithm, namely HAV1, HAV2, and HAV3, is evaluated on the seventy well-known jobshop benchmark instances (see Sect. 1 for details) considered by Framinan (2005) to solve the jobshop problems by transforming them into flowshops with the aim of minimizing the manufacturing line length. The results (best string length) of the proposed heuristic algorithm variants are compared with the results reported by Framinan (2005). It is to be noted that Framinan (2005) had consolidated and reported the best string length as an outcome of the application of algorithms such as H2 and H3 employed by Branke et al. (1998), BS by Framinan and Ruiz-Usano (2002), and TS by Framinan (2005). Note that the proposed deterministic heuristic algorithm variants (HAV1 to HAV3) generate only 12n2 job sequences (including antithetic job sequences) per variant during its search process to find the best string length which corresponds to the machine routing or manufacturing flow line. The results of the proposed heuristic algorithm variants HAV1 to HAV3 are presented in Table 1. It is found that the proposed heuristic algorithm variants, HAV1, HAV2, and HAV3, together yield better results than those reported by Framinan (2005) on 46 instances out of 70 benchmark jobshop problem instances considered. Note that on remaining instances, the proposed HAV variants yield the same best manufacturing line length as reported by Framinan (2005). By considering all the results (best manufacturing line length) reported by Framinan (2005), the proposed variants HAV1, HAV2, and HAV3 produce best string length on 30, 31, and 33 instances, respectively, out of 70 benchmark jobshop problem instances considered. Also, note that the proposed HAV1, HAV2, and HAV3, respectively, saved a total of 89, 95, and 97 duplicate machines in the manufacturing flow line for the entire set of problem instances. Based on the number of duplicate machines saved in the manufacturing line (for all 70 benchmark problem instances considered) and the number of best string length, the proposed HAV3 shows the better performance (refer to Table 1).

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Table 1 Computational results Jobshop problem instance

n

La01

10

La02

10

La03

m

String length reported by Framinan (2005)

String length obtained by proposed HAV1

String length obtained by proposed HAV2

String length obtained by proposed HAV3

5

13

13(0)

13(0)

13(0)

5

12

12(0)

12(0)

12(0)

10

5

12

12(0)

12(0)

12(0)

La04

10

5

13

13(0)

13(0)

13(0)

La05

10

5

12

12(0)

12(0)

12(0)

La06

15

5

14

14(0)

14(0)

14(0)

La07

15

5

14

14(0)

14(0)

14(0)

La08

15

5

13

13(0)

13(0)

13(0)

La09

15

5

14

14(0)

14(0)

14(0)

La10

15

5

14

14(0)

14(0)

14(0)

La11

20

5

14

14(0)

14(0)

14(0)

La12

20

5

14

14(0)

14(0)

14(0)

La13

20

5

15

15(0)

15(0)

15(0)

La14

20

5

15

15(0)

15(0)

15(0)

La15

20

5

14

14(0)

14(0)

14(0)

La16

10

10

35

33(2)

33(2)

33(2)

La17

10

10

35

34(1)

34(1)

34(1)

La18

10

10

38

36(2)

36(2)

36(2)

La19

10

10

35

32(3)

32(3)

32(3)

La20

10

10

35

35(0)

35(0)

35(0)

La21

15

10

42

39(3)

39(3)

39(3)

La22

15

10

41

39(2)

39(2)

39(2)

La23

15

10

42

39(3)

39(3)

39(3)

La24

15

10

43

41(2)

41(2)

41(2)

La25

15

10

42

39(3)

39(3)

38(4)

La26

20

10

43

42(1)

43(0)

43(0)

La27

20

10

44

43(1)

42(2)

42(2)

La28

20

10

46

44(2)

44(2)

44(2)

La29

20

10

45

44(1)

44(1)

44(1)

La30

20

10

45

43(2)

44(1)

44(1)

La31

30

10

49

47(2)

46(3)

44(5)

La32

30

10

49

48(1)

46(3)

47(2)

La33

30

10

49

48(1)

45(4)

48(1)

La34

30

10

49

47(2)

48(1)

49(1)

La35

30

10

49

49(0)

48(1)

48(1)

La36

15

15

79

78(1)

76(3)

76(3) (continued)

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Table 1 (continued) Jobshop problem instance

n

m

String length reported by Framinan (2005)

String length obtained by proposed HAV1

String length obtained by proposed HAV2

String length obtained by proposed HAV3

La37

15

15

79

71(8)

73(6)

74(5)

La38

15

15

80

76(4)

73(7)

73(7)

La39

15

15

80

74(6)

75(5)

71(9)

La40

15

15

78

75(3)

74(4)

76(2)

orb01

10

10

28

27(1)

27(1)

27(1)

orb02

10

10

34

32(2)

32(2)

33(1)

orb03

10

10

20

20(0)

20(0)

20(0)

orb04

10

10

34

33(1)

33(1)

33(1)

orb05

10

10

26

26(0)

26(0)

26(0)

orb06

10

10

28

27(1)

27(1)

27(1)

orb07

10

10

34

32(2)

32(2)

33(1)

orb08

10

10

20

20(0)

20(0)

20(0)

orb09

10

10

34

33(1)

33(1)

33(1)

orb10

10

10

26

26(0)

26(0)

26(0)

swv01

20

10

29

28(1)

29(0)

29(0)

swv02

20

10

28

27(1)

27(1)

27(1)

swv03

20

10

28

27(1)

28(0)

27(1)

swv04

20

10

29

29(0)

29(0)

29(0)

swv05

20

10

30

29(1)

29(1)

29(1)

swv06

20

15

61

59(2)

58(3)

58(3)

swv07

20

15

59

57(2)

57(2)

56(3)

swv08

20

15

60

56(4)

55(5)

55(5)

swv09

20

15

57

58(−1)

56(1)

54(3)

swv10

20

15

57

57(0)

58(−1)

58(−1)

swv11

50

10

32

32(0)

32(0)

32(0)

swv12

50

10

33

32(1)

32(1)

32(1)

swv13

50

10

32

32(0)

32(0)

32(0)

swv14

50

10

33

32(1)

32(1)

32(1)

swv15

50

10

33

32(1)

32(1)

32(1)

swv16

50

10

52

51(1)

53(−1)

53(−1)

swv17

50

10

54

50(4)

51(3)

52(2)

swv18

50

10

54

53(1)

52(2)

52(2)

swv19

50

10

53

50(3)

51(2)

51(2)

swv20

50

10

55

53(2)

53(2)

52(3)

The improvement of HAV1, HAV2, and HAV3 with respect to the results reported by Framinan (2005) is indicated in the bracket, and it shows the number of duplicate machines saved

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127

5 Summary The problem of transforming jobshops into flowshops is considered to minimize the length of the manufacturing line and solved by using the three variants of deterministic heuristic algorithms (HAV1 to HAV3) in this work. The proposed algorithm employs the concatenation of forward reduction, inverse reduction, and machine elimination techniques along with a new local search scheme called two-pair adjacent interchange scheme during seed solution generation on machine routings (manufacturing flow line) and during the main search process which uses the concatenation patterns of JIS and MJSS on job sequence to obtain the reduced string length. The computational evaluation of the three variants of the proposed deterministic heuristic algorithm is carried out by comparing it with the existing reported work. The computational result shows that the HAV3 has performed well in terms of minimizing the length of the manufacturing line (string length) and the number of duplicate machines saved in most of the jobshop benchmark problem instances. In this work, the effectiveness of the developed approach is tested with respect to minimization of the length of manufacturing line corresponding to the problem of transforming jobshops into flowshops, but its impact on the performance measures like makespan and total flow time of jobs for the jobshop problems is not studied. Acknowledgements The author is very much obliged to the reviewers and the editors for their valuable suggestions and observations to improve the previous version of the paper. The author is thankful to the Indian National Academy of Engineering (INAE) for the opportunity given to do this research work under the mentoring of engineering teachers by the INAE Fellows scheme and under Prof. C. Rajendran, INAE Fellow in IIT Madras.

Appendix 1 Forward Reduction Technique (Framinan, 2005) Let us consider a jobshop problem with 6 jobs and 4 machines (i.e., n = 6 and m = 4) and its machine routings as given below to illustrate various string reduction techniques used in this work. The initial job sequence is obtained by considering the jobs in the order as {1-23-4-5-6} (or {6-5-4-3-2-1} in case of antithetic) in the generation of seed sequence, and a supersequence of machines (S) is formed. In this forward reduction technique, the supersequence of machines is scanned from left to right to check the machines which are required to process the chosen job one at a time from the job sequence. The feasibility of the supersequence of machines or manufacturing flow line is ascertained for machine routings of every job, and reduced string length (L) is obtained. The forward reduction technique is demonstrated in Table 3 through an example of jobshop problem as given in Table 2.

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Table 2 Machine routings of a jobshop (n = 6 and m = 4) Job

Machine routing

1

{1

–

2

–

3

–

4}

2

{1

–

3

–

2

–

4}

3

{1

–

4

–

2

–

3}

4

{2

–

3

–

1

–

4}

5

{2

–

4

–

1

–

3}

6

{3

–

4

–

1

–

2}

Table 3 Demonstration of forward reduction technique Job sequence

Supersequence of machines

{1-2-3-4-5-6}

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job

Corresponding machines as per the respective machine routing of the job

Job 1

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 2

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 3

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 4

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 5

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 6

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job sequence

Shortest supersequence of machines

{1-2-3-4-5-6}

{1-2-3-4-1-3-2-4-3}

Note that after the application of the forward reduction technique, the string length (or manufacturing line) of the job sequence {1-2-3-4-5-6} is reduced to 9 from 24, and hence the saved number of duplicated machines is 15

Appendix 2 Inverse Reduction Technique (Framinan, 2005) The initial job sequence {1-2-3-4-5-6} (or {6-5-4-3-2-1}, in case of antithetic) is considered in the generation of seed sequence, and a supersequence of machines (S) is formed. In this inverse reduction technique, the supersequence of machines is scanned from right to left to check the machines (i.e., machines in the order of right to left from machine routings) which are required to process the chosen job one at a time from reverse direction of the job sequence. The feasibility of the supersequence of machines or manufacturing flow line is ascertained for machine routings of every job, and reduced string length (L) is obtained. The inverse reduction technique is demonstrated in Table 4 through an example of jobshop problem as given in Table 2.

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Table 4 Demonstration of inverse reduction technique Job sequence

Supersequence of machines

{1-2-3-4-5-6}

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job

Corresponding machines as per the respective machine routing of the job

Job 6

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 5

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 4

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 3

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 2

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job 1

{1-2-3-4-1-3-2-4-1-4-2-3-2-3-1-4-2-4-1-3-3-4-1-2}

Job sequence

Shortest supersequence of machines

{1-2-3-4-5-6}

{1-2-3-1-4-2-4-1-3-4-1-2}

Note that after the application of the inverse reduction technique, the string length (or manufacturing line) of the job sequence {1-2-3-4-5-6} is reduced to 12 from 24, and hence the saved number of duplicated machines is 12. Also, note that the resultant supersequence of machines yielded from the inverse reduction technique need not be the same as that of the resultant supersequence of machines yielded by the forward reduction technique

Appendix 3 Machine Elimination Technique (Framinan & Ruiz-Usano, 2002) This technique helps in further reduction of string length (L) by removing one machine at a time from the supersequence of machines (S), and checking of feasibility is made for the resultant supersequence of machines in terms of fulfilling the machine routings of each job in the considered job sequence (say, {1-2-3-4-5-6}). Let us consider the supersequence of machines {1-2-3-4-1-3-2-4-3} obtained from the forward reduction technique for the application of the machine elimination technique which is demonstrated in Table 5 through an example of jobshop problem as given in Table 2.

Appendix 4 Proposed Two-Pair Adjacent Interchange Scheme (TPAIS) A new local search, two-pair adjacent interchange scheme (TPAIS), is proposed in this work to explore a further decrease in string length. This local search is employed for interchanging of adjacent machines found in the input supersequence of machines (say, supersequence obtained from machine elimination technique employed during seed generation, i.e., during Step 1 to Step 4 in Sect. 3.2). The step-by-step procedure involved in TPAIS is explained below.

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Table 5 Demonstration of machine elimination technique Step no.

Eliminated machine

Resultant supersequence of machines, S

Remarks

1

Machine 1 found in position 1

{1-2-3-4-1-3-2-4-3}

S does not fulfill the machine routings of every jobs; hence, machine 1 is retained

2

Machine 2 found in position 2

{1-2-3-4-1-3-2-4-3}

S does not fulfill the machine routings of every jobs; hence, machine 2 is retained

3

Machine 3 found in position 3

{1-2-3-4-1-3-2-4-3}

S does not fulfill the machine routings of every jobs; hence, machine 3 is retained

4

Machine 4 found in position 4

{1-2-3-4-1-3-2-4-3}

S does not fulfill the machine routings of every jobs; hence, machine 4 is retained

5

Machine 1 found in position 5

{1-2-3-4-1-3-2-4-3}

S does not fulfill the machine routings of every jobs; hence, machine 1 is retained

6

Machine 3 found in position 6

{1-2-3-4-1-3-2-4-3}

S fulfills the machine routings of every jobs. Hence, machine 3 is eliminated and supersequence of machines, S, is updated with {1-2-3-4-1-2-4-3}, L = 8

7

Machine 2 found in position 6

{1-2-3-4-1-2-4-3}

Updated S does not fulfill the required machine routings of every jobs; hence, machine 2 is retained

8

Machine 4 found in position 7

{1-2-3-4-1-2-4-3}

S does not fulfill the required machine routings of every jobs; hence, machine 4 is retained

9

Machine 3 found in position 8

{1-2-3-4-1-2-4-3}

S does not fulfill the required machine routings of every jobs; hence, machine 3 is retained and L =8

Note that after the application of the machine elimination technique, the string length (or manufacturing line) of the job sequence {1-2-3-4-5-6} is further reduced to 8 and hence the saved number of duplicated machines is 16. Also, note that in some cases, for example, consider the supersequence of machines {1-2-3-1-4-2-4-1-3-2}, and machine elimination technique does not reduce the supersequence further

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131

Step 1: Let us consider a supersequence of machines, obtained after machine elimination technique during Step 1 to Step 4 in Section 3.2 (i.e., Assign the supersequence of machines in case of Step 1; in case of Step 2; in case of Step 3 and in case of Step 4) and its corresponding string length as (say, in in case of Step 2; in case of Step 1; case of Step 3 and in case of Step 4). Step 2: Set . Step 3: then go to Step 10 else proceed to Step 4. Step 4: then proceed to Step 5 else go to Step 6. Step 5: Step 5.1: Do the following for { Swap the machines appeared in positions and in ; Apply forward reduction, inverse reduction, and machine elimination techniques on the perturbed sequence. Call the resultant supersequence of machines obtained as and its string length as . } Step 5.2: set , where, Step 5.3: then to and its assign the reduced supersequence of machines string length to , and proceed to step 6 else go to step 6.

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Step 6: Swap the machines found in positions and in sequence ; Apply forward reduction, inverse reduction, and machine elimination techniques on the perturbed sequence. Call the resultant supersequence of machines obtained as and its string length as . then go to step 8 else proceed to step 7. Step 7: Step 7.1: Do the following for { Swap the machines appeared in positions and in ; Apply forward reduction, inverse reduction, and machine elimination techniques on the perturbed sequence. Call the resultant supersequence of machines obtained as and its string length as . } , where, Step 7.2: set Step 7.3: then assign the reduced supersequence of machines to and , and proceed to Step 8 else go to Step 8. Step 8: then assign the reduced supersequence of machines to and its string length to , and proceed to Step 9 else go to Step 9. Step 9: Set and return to Step 3. Step 10: Assign the finally returned supersequence of machines to and its in case of Step 1; similarly, to corresponding string length to in case of Step 2; to and to and to in case of Step 3; to and to in the case of Step 4.

Appendix 5 Job Index-Based Insertion Scheme (JIS) (used by Krishnaraj et al., 2012, 2014, 2019)

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133

To explore further reduction in supersequence of machines, the initial job sequence (consider J S = {1 − 2 − 3 − . . . − n}) is subjected to JIS with a performance measure of shortest string length. Note that whenever a perturbed job sequence is obtained (i.e., supersequence of machines is initialized based on the perturbed job sequence), the full set of procedures followed in Step 1 to Step 5 in Sect. 3.2 is applied to get the reduced supersequence of machines and its string length which in turn maximize the number of duplicate machines saved. Let [k] represent the index of the job at position k of the initial job sequence J S under consideration. The working of local search scheme is given below:

{ { then insert job in position of the job sequence under consideration and adjust the sequence accordingly by not changing the relative positions of the other jobs; execute the set of procedures followed in Step 1 to Step 5 as in Section 3.2 by initializing the supersequence of machines based on the perturbed job sequence to explore the decreased supersequence of machines, and its string length, . } sequences based on select the best job sequence among these minimum string length; if the string length is less than or equal to the string length of the considered sequence (i.e., sequence ), then interchange the sequence by the best one obtained and update the supersequence of machines, and its string length, . Also update the best shortest common supersequence of machines ) and its shortest string length if necessary (i.e., }

Appendix 6 Modified Job Index-Based Swap Scheme (MJSS) (originally used by Rajendran and Ziegler 2004) Let JS be the job sequence under consideration and [k] be the index of the job at position k of this sequence. The working of local search scheme is given below:

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{ { then swap job with job , with remaining jobs in unaltered; execute the set of procedures followed in Step 1 to Step 5 as in Section 3.2 by initializing the supersequence of machines based on the perturbed job sequence to explore the reduced supersequence of machines, and its string length, . if the string length corresponds Also update the job sequence to the supersequence based on the perturbed job sequence is less than the string length, . (Note that this updating of job sequence is new in this work) } sequences based on select the best job sequence among these minimum string length; if the string length is less than or equal to the string length of the considered sequence (i.e., sequence ), then interchange the sequence by the best one obtained and update the supersequence, and its string length, . and its shortAlso update the best shortest common supersequence ) est string length if necessary (i.e., }

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Kimms, A. (2000). Minimal investment budjets for flow line configuration. IIE Transactions, 32(4), 287–298. Knolmayer, G., Mertens, P., & Zeier, A. (2002). Supply chain management based on SAP systems. Berlin: Springer. Krishnaraj, J., Pugazhendhi, S., Rajendran, C., & Thiagarajan, S. (2012). A modified ant-colony optimization algorithm to minimize the completion time variance of jobs in flowshops. International Journal of Production Research, 50(20), 5698–5706. Krishnaraj, J., Pugazhendhi, S., Rajendran, C., & Thiagarajan, S. (2014). A heuristic algorithm to minimize the total flowtime of jobs in permutation flowshops. International Journal of Industrial and Systems Engineering, 4(4), 511–532. Krishnaraj, J., Pugazhendhi, S., Rajendran, C., & Thiagarajan, S. (2019). Simulated annealing algorithms to minimize the completion time variance of jobs in permutation flowshops. International Journal of Industrial and Systems Engineering, 31(4), 425–451. Lawrence, S. (1984). Supplement to resource constrained project scheduling: An experimental investigation of heuristic scheduling techniques. Pittsburgh, PA: Graduate School of Industrial Administration, Carnegie-Mellon University. Michel, R., & Middendorf, M. (1999). An ACO algorithm for the shortest common super-sequence problem. Mousavi, S. R., Bahri, F., & Tabataba, F. S. (2012). An enhanced beam search algorithm for the shortest common supersequence problem. Engineering Applications of Artificial Intelligence, 25(3), 457–467. Ning, K., & Leong, H. W. (2006). Towards a better solution to the shortest common super-sequence problem: The deposition and reduction algorithm. BMC Bioinformatics, 7(Suppl 4), S12). Noaman, M. M., & Jaradat, A. S. (2011). Solving shortest common supersequence problem using artificial bee colony algorithm. International Journal of ACM Jordan., 2(3), 180–185. Rajendran, S., Rajendran, C., & Ziegler, H. (2010). An ant-colony algorithm to transform jobshops into flowshop: A case of shortest-common-supersequence stringology problem. In 5th International ICST conference, BIONETICS 2010, Boston, USA, December 2010. Rajendran, C., & Ziegler, H. (2004). Ant-colony algorithms for permutation flowshop scheduling to minimize makespn/total flowtime of jobs. European Journal of Operational Research, 155(2), 426–438. Saifullah, K. C. M., & Islam, M. R. (2016). Chemical reaction optimization for solving the shortest common supersequence problem. Computational Biology and Chemistry, 64, 82–93. Storer, R. H., Wu, S. D., & Vaccari, R. (1992). New search spaces for sequencing problems with applications to job-shop scheduling. Management Science, 38(10), 1495–1509. Timkovsky, V. G. (1989). Complexity of common subsequences and supersequences problems and related problems. Cybernetics and Systems Analysis, 25(5), 565–580.

Data Analytics, Qualitative, and Simulation Approaches to Operations and Supply Chain Issues

A Classification Algorithm Based on Linear Regression and Linear Programming for Predicting the Breast Cancer Sakthivel Madankumar

Abstract In this study, we consider the classification problem in the healthcare domain, and the objective of the problem is to diagnose the breast cancer (binary classification: malignant/benign) based on the number of different features with respect to tumor characteristics. We propose an algorithm to solve this classification problem. The proposed algorithm first eliminates the linear dependency between the input features and then focuses on enriching the feature set based on these independent attributes in order to capture the curvilinear function between the classes. The proposed algorithm, subsequently, uses the multiple linear regression (MLR) model on the training data set to capture the relationship between the response/class variable (malignant/benign) and the set of features (from the enriched feature set). Thereafter, the proposed linear programming (LP) model makes use of the set of relatively influential attributes from the MLR based on the absolute values of the coefficients, to find the classification function/expression for predicting the breast cancer using the training data set. As a part of this process, LP model also finds the initial thresholds where b and b + 1 are thresholds for malignant and benign classes, respectively. In the final phase, the proposed algorithm fine-tunes the thresholds obtained in the LP model through a search process, to determine the exact boundary between the target classes (malignant/benign) using the validation data set. In order to evaluate the performance of the algorithm, we use the Wisconsin diagnostic breast cancer (WDBC) data set from the University of California—Irvine machine learning repository, and we compare the performance with the existing algorithm in the literature. From the results, we observe that the proposed algorithm performs better in terms of the accuracy to predict the breast cancer. Keywords Cancer diagnosis · Classification algorithm · Linear regression · And linear programming

S. Madankumar (B) Trimble Information Technologies India Private Limited, Chennai 600113, India © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_7

139

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S. Madankumar

Notations M: N0 : N1 :

i: j, c: ai j : fi j :

total number of instances/observations. total number of input features/attributes in each observation. total number of features retained after the elimination of multicollinearity among the standardized features/attributes. total number of features (after the augmentation) at the end of transformation using the exponential function. total number of features (after the augmentation) at the end of transformation using the logistic function. total number of features (after the augmentation) at the end of transformation using the cosine function. total number of features after considering the contributions of attributes from their transformations using higher order polynomials. total number of features after considering the contributions based on the interaction among original as well as transformed attributes with higher order polynomials. index for observations. indices for features. value of attribute/feature j for observation i. value of feature j for observation i after the transformation.

xj:

mean of attribute j; where x j =

N2 : N3 : N4 : N5 : N6 :

sj: τ: λ: |λ|: λk : ε: P: Q:

pk : qk : |P|: |Q|: β0 :

 N0

i=1

ai j

.

√ M i=1 (ai j −x j )×(ai j −x j ) standard deviation of the attribute j; where s j = . M threshold value (upper limit) for the Variance Inflation Factor (VIF). set of multipliers to be used in the exponential and logistic functions, and for this study we assumed the following values for λ; λ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. total number of elements in set λ. an element from set λ; and λk ∈ λ; for example, when k assumes value 1 then λk = λ1 = 1. a constant with a very small value 0.00001. set of power terms considered with respect to the higher-order polynomials, and for this study we assumed the following values for P; P = {2, 3, 4}. set of power terms considered with respect to the interaction effect of attributes, and for this study we assumed the following values for Q; Q = {1, 2, 3, 4}. an element from set P; and pk ∈ P; for example, when k assumes value 1 then pk = p1 = 2. an element from set Q; and qk ∈ Q; for example, when k assumes value 1 then qk = q1 = 1. total number of elements in set P. total number of elements in set Q. an intercept of the estimated regression expression. N0

A Classification Algorithm Based on Linear Regression …

βj: yi : 

yi : yi : i : α: D: γj: errori : b:

141

coefficient of each attribute j in the estimated regression expression. value of the response variable for observation i (binary classification: 0 for malignant and 1 for benign). expression to predict the response for observation i. value of the estimated response for observation i. error/deviation of yi from yi in MLR. value for the regularization parameter; and in this study, we have set the value as 10 for α. a set of relatively-influential attributes based on the absolute value of the coefficients (from MLR). a real (unrestricted) variable to capture the coefficient for attribute j in the LP model. a real (non-negative) variable to capture the error in the expression in the LP model. a real (unrestricted) variable to capture the boundary/threshold in the LP model.

1 Introduction Health care is a field/domain that focuses on improving the health of the people through the application of preventive measures, and the procedures/measures to diagnose the diseases at the right (or early) stage, and provide the treatments to cure the diseases, in order to recover the infected or diseased people. The advancement in the field of artificial intelligence (AI) and machine learning (ML) has given the hope to leverage the respective methods and technologies in the healthcare field, in order to aid the doctors/physicians to identify/predict the diseases at the right stage by analyzing the medical records of the patients; then, the physicians can make use of this prediction as an input to further analyze as well as provide treatments at the right stage to the patients. Cancer is a critical disease with high mortality rate in health care; Siegel et al. (2012) studied the cancer statistics and estimated the total number of new cases (1,639,910) and deaths (577,190) due to cancer in the United States for the year 2012. In their study, they also estimated that breast cancer was accounting for 29% out of the new female cancer cases. Typically, mammograms are used by physicians and radiologists to predict the cancer, but in general, there is a high variability in the prediction by different radiologists. This was also evident in the study conducted by Elmore et al. (1994), and only 3% of the cancers were identified. The advancement in technologies (in the medical field and also in the data analytics) and the computing power has led us to extract more tumor features and also apply the ML algorithms to treat the diagnosis problem as a classification problem to predict the breast cancer. Studies by earlier researchers (Wolberg et al., 1995; Pena-Reyes &

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Sipper, 1999) proved that data mining techniques can be used to solve this problem as a classification problem. Researchers have traditionally used support vector machine (SVM) techniques to solve this classification problem of predicting breast cancer (Bennett & Blue, 1998; Huang et al., 2008; Akay, 2009; Prasad et al., 2010). Zheng et al. (2014) studied the Wisconsin diagnostic breast cancer (WDBC) data set from the University of California—Irvine machine learning repository, and they applied K-means algorithm to reduce the number of features and then applied the SVM algorithm on the reduced feature set to predict the breast cancer. In this work, we apply operations research (OR) techniques to solve the classification problem for predicting the breast cancer, and our proposed algorithm comprises multiple linear regression (MLR) and linear programming (LP) techniques to solve the classification problem. This is one of the few attempts in applying OR techniques to solve the classification problem. Madankumar et al. (2017) proposed an LP-based classifier for solving the classification problem, and the respective algorithm may not work well on the high-dimensional data set due to the fact that the respective LP classifier was defining a decision variable (in the LP model) for each feature, and when the number of features is high, then the resulting number of LP decision variables will also be high which would make the model to be unsolvable in reasonable CPU time. In this study, we introduce a step where we apply the MLR to identify the features that are promising, and then, we apply the LP on the reduced feature set to find the boundary between the classes. We compare the efficiency of the proposed algorithm with the results obtained by Zheng et al. (2014) using their K-SVM algorithm (Kmeans followed by SVM), and we observe that our algorithm performs better in terms of the accuracy to predict the breast cancer. Please note that the terms “features” and “attributes” are used interchangeably in this paper, and they both represent the measured inputs (the independent variables) of the problem. This chapter is organized as follows: In Sect. 2, we present the summary of the proposed algorithm, in Sect. 3, we present the algorithm in detail with the respective implementation details for each step, and in Sect. 4, we present the experimental results of the proposed algorithm where we compare the accuracy of the proposed algorithm with the ML algorithm available in the literature for predicting the breast cancer.

2 Proposed Methodology In this study, we consider the classification problem in the healthcare domain. The objective of the classification problem is to diagnose breast cancer (binary classification: malignant/benign) based on the number of different features with respect to tumor characteristics. A set of instances are given with the respective diagnostic results, and for each instance, the values with respect to the input features (different tumor characteristics) and the corresponding classification of the tumor (whether it is

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benign and malignant) are available in the data set. We propose an algorithm to solve this classification problem (binary classification: malignant/benign), the summary of the proposed algorithm is presented in this section, and we also present the overview of the algorithm as a sequence of steps in Fig. 1. The proposed algorithm first eliminates the linear dependency between the input features/attributes with the help of Variance Inflation Factor (VIF) (Step 2 in Sect. 3); and then focuses on enriching the feature set based on the resulting independent attributes, by transforming the independent attributes using non-linear functions such as exponential (Step 3.1 in Sect. 3), logistic (Step 3.2 in Sect. 3) and cosine (Step 3.3 in Sect. 3) functions; and then further transformation of attributes through higherorder polynomials (Step 3.4 in Sect. 3), and then further enrichment by having a set of additional features to track the interaction (Step 3.5 in Sect. 3) among the

Fig. 1 Overview of the proposed algorithm

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independent attributes from their polynomials. These steps are carried out mainly to capture the nonlinear/curvilinear boundary between the classes (two predetermined classes: malignant and benign). Then, the proposed algorithm splits the data set in to training data set, validation data set, and test data set. Subsequently, the proposed algorithm uses the multiple linear egression (MLR) (see Step 5 in Sect. 3) model on the training data set, in order to capture the relationship between the response/class variable and the set of features (from the enriched feature set). As a part of this step, the MLR also finds the coefficient for each attribute based on its contribution toward the response variable. The proposed algorithm uses the ridge regularization in order to avoid the effect of over fitting on the training data set, so that the model can even work well for the new/unseen samples when it is tested against test and validation data set. Thereafter, the proposed linear programming (LP) model (see Step 6 in Sect. 3) makes use of the set of relatively influential attributes from the MLR (based on the absolute values of the coefficients) in order to find the classification function/expression for predicting the breast cancer using the training data set, and as a part of this process, LP model also finds the initial thresholds where b and b + 1 are thresholds for malignant and benign classes, respectively. In the final phase, the proposed algorithm fine-tunes the thresholds obtained in the LP model through a search process (see Step 7 in Sect. 3), to determine the exact boundary between the target classes (malignant/benign) using the validation data set. Finally, the proposed algorithm makes use of the test data set to evaluate the model in terms of accuracy, and the respective results are presented in Sect. 4.

3 Proposed Algorithm Step 1: Standardization The algorithm first standardizes each feature to transform the mean of the attribute to 0 and standard deviation of the respective attribute to 1.   f i, j = ai j − x j /s j , i = 1, 2, . . . , M; and j = 1, 2, . . . , N0 .

(1)

A Numerical Illustration We consider a sample data set presented in Table 1 to numerically illustrate Step 1, and this data set consists of four features/attributes (N0 = 4) and one target/dependent variable for binary classification. As prescribed by the algorithm, we apply Step 1 to this data set, and the corresponding standardized features are presented in Table 2; Figure 2 also depicts the respective computation for a sample value.

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Table 1 Sample data set for numerical illustration Classification (yi )

Feature 1

Feature 2

Feature 3

Feature 4

0

3.13

36.00

108.63

19.25

0

5.90

41.00

124.18

33.60

0

4.42

29.00

87.88

25.00

0

8.72

24.00

73.74

46.00

1

8.13

21.00

64.63

42.75

1

4.06

33.00

99.81

23.60

1

6.92

46.00

139.38

39.20

1

8.06

10.00

31.61

41.30

1

4.98

42.00

127.00

29.10

1

6.88

49.00

148.38

39.30

Table 2 Values of the standardized attributes

Feature 1 −1.64

Feature 2

Feature 3

0.25

0.23

Feature 4 −1.68

−0.12

0.68

0.68

−0.04

−0.93

−0.35

−0.36

−1.02

1.43

−0.78

−0.77

1.39

1.10

−1.04

−1.03

1.01

−1.13

−0.01

−0.02

−1.18

0.44

1.11

1.12

1.06

0.61

−1.99

−1.99

−0.63

0.77

0.76

−0.55

0.42

1.37

1.38

0.62

0.85

Fig. 2 Effect of applying the standardization step for a sample value {0.25} with x j = 6.12 and s j = 1.82

Step 2: Elimination of multicollinearity In order to eliminate the multicollinearity among the standardized features/attributes, this step calculates the variance inflation factor (VIF) for each feature/attribute, and VIF describes the factor by which a respective attribute’s estimated coefficient is inflated because of the correlation with other attributes (O’brien, 2007). In this study, we make use of the Python library “statsmodels.stats.outliers_influence.variance_inflation_factor” to calculate the VIF

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for each attribute, and we follow an iterative procedure to eliminate the multicollinearity. In this process, we choose the first attribute which has the VIF greater than the threshold, and we first eliminate the attribute from the feature set. Then, we calculate the VIF for the remaining attributes, and we repeat the same process till the remaining attributes satisfy the condition that the corresponding VIFs are less than the threshold. In this study, we have considered the threshold (τ ) as 5 for VIF so that an attribute is eliminated for further analysis when the respective attribute is dependent on other remaining attributes with R 2 (coefficient of determination) value greater than 0.8. Let us denote the number of features retained after the elimination of multicollinearity among the attributes as N1 , and these set of features are used for further analysis and consideration. This step is mainly carried out to reduce the change in estimated regression coefficients when there is a linear dependency among the attributes/features. A Numerical Illustration In this numerical illustration, we eliminate the multicollinearity among the standardized attributes as prescribed in Step 2 for the standardized data set presented in Table 2; this step eliminates the standardized features (Feature-1 and Feature-2) through the iterative procedure described in Step 2 because of their corresponding linear dependency with Feature-3 and Feature-4, then, the remaining set of features (N1 = 2) are considered for further analysis, and the respective attributes are presented in Table 3. Step 3: Enrichment of feature set The procedure then augments the feature set (by using the reduced set of features N1 obtained from Step 2) by considering the following aspects: • Contribution of attributes from their transformations using nonlinear functions such as exponential, logistic and cosine functions. • Contributions of attributes from their transformations using higher-order polynomials. Table 3 Retained features (N1 = 2) after the elimination of multicollinearity

Feature 3 0.23

Feature 4 −1.68

0.68

−0.04

−0.36

−1.02

−0.77

1.39

−1.03

1.01

−0.02

−1.18

1.12

0.61

−1.99

0.85

0.76

−0.55

1.38

0.62

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Fig. 3 Effect of applying the exponential function on the standardized feature

• Contributions based on the interaction among the attributes with their higher-order polynomials. Step 3.1: Transformation of attributes using the exponential function Our procedure applies the exponential transformation for the standardized features (N 1 ) retained in Step 2, with the consideration of set of λ values, and the function used for this transformation is defined in Eq. (2); the total number of features (after the augmentation) at the end of this transformation would be N2 , and it is defined in Eq. (3); the effect of applying this transformation on the standardized attribute is captured in Fig. 3 with varying values (from −3.0 to 3.0) of the standardized attribute. f i,c = e−λk × fi, j i = 1, 2, . . . , M; j = 1, 2, . . . , N1 ; λk ∈ λ; k = 1, 2, . . . , |λ|; and c = N1 + ( j − 1) × |λ| + k

(2)

N2 = N1 + N1 × |λ|.

(3)

A Numerical Illustration As prescribed in Step 3.1, we apply the exponential transformation for the standardized features retained (N1 = 2) in Step 2 (for the data set presented in Table 3). But for ease of understanding, we only apply the transformation for the first observation in the sample data set with the consideration of subset of λ = {1, 5, 10} values, the corresponding transformed values with respect to first observation are presented in Table 4, and if we consider set λ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then the total number of features at the end of this transformation is N2 = 22; Fig. 4 also depicts the respective computation for a sample value.

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Table 4 Effect of exponential transformation on the sample data set Standardized feature

Transformed value when λ1 = 1

Transformed value when λ2 = 5

Transformed value when λ3 = 10

0.23

0.79

0.32

0.10

−1.68

5.38

4488.29

20, 144, 720.26

Fig. 4 Effect of applying the exponential function for a sample value {0.23} with λ1 = 1

Step 3.2: Transformation of attributes using the logisticfunction The procedure applies the logistic transformation for the standardized features (N 1 ) retained in Step 2, with the consideration of set of λ values, and the function used for this transformation is defined in Eq. (4); the total number of features (after the augmentation) at the end of this transformation would be N3 , and it is defined in Eq. (5); the effect of applying this transformation (with λ1 = 1) on the standardized attribute is captured in Fig. 5 with varying values (from −3.0 to 3.0) of the standardized attribute. f i,c =

1

 1+e

−λk ×

f i, j 6



i = 1, 2, . . . , M; j = 1, 2, . . . , N1 ; λk ∈ λ; k = 1, 2, . . . , |λ| and c = N2 + ( j − 1) × |λ| + k. N3 = N2 + N1 × |λ|.

Fig. 5 Effect of applying the logistic function on the standardized feature

(4)

(5)

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Table 5 Effect of logistic transformation on the sample data set Standardized feature

Transformed value when λ1 = 1

Transformed value when λ2 = 5

Transformed value when λ3 = 10

0.23

0.51

0.55

0.60

−1.68

0.43

0.20

0.06

Fig. 6 Effect of applying the logistic function for a sample value {0.23} with λ1 = 1

A Numerical Illustration As prescribed in Step 3.2, we apply the logistic transformation for the standardized features retained (N1 = 2) in Step 2 (which are presented in Table 3). But for ease of understanding, we only apply the transformation for the first observation in the sample data set with the consideration of subset of λ = {1, 5, 10} values, and the corresponding transformed values with respect to first observation are presented in Table 5; if we consider set λ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then the total number of features at the end of this transformation is N3 = 42; Fig. 6 also depicts the respective computation for a sample value. Step 3.3: Transformation of attributes using the cosine function The procedure applies the cosine transformation for the standardized features (N 1 ) retained in Step 2, and in order to perform this transformation, this step first scales the attribute to take the value between 0 and 1 and then multiply the respective value by π (radians), and the function used for this transformation is defined in Eq. (6); the total number of features (after the augmentation) at the end of this transformation would be N4 , and it is defined in Eq. (7). The effect of applying this transformation is captured in Fig. 7 with varying values (from −3.0 to 3.0) of the standardized attribute.   ⎧ ⎫⎞ ⎛ f i j − min f i j ⎨ ⎬ 1≤i≤M ⎠     f ic = cos⎝π × ⎩ max f i j − min f i j + ε ⎭ 1≤i≤M

1≤i≤M

i = 1, 2, . . . , M; j = 1, 2, . . . , N1 ; and c = N3 + j. N4 = N3 + N1 .

(6) (7)

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Fig. 7 Effect of applying the cosine function on the standardized attribute

A Numerical Illustration As prescribed in Step 3.3, we apply the cosine transformation for the standardized features retained (N1 = 2) in Step 2 (which are presented in Table 3). But for ease of understanding, we only apply the transformation for the first observation in the sample data set, the corresponding transformed values with respect to first observation are presented in Table 6, and the total number of features at the end of this transformation is N4 = 44; Fig. 8 also depicts the respective computation for a sample value. Step 3.4: Transformation of attributes using higher-order polynomials The procedure applies this transformation for the standardized features (N 1 ) retained in Step 2, in order to capture the contribution of attributes from their higher-order polynomials, and the function used for this transformation is defined in Eq. (8); the total number of features (after the augmentation) at the end of this transformation Table 6 Effect of cosine transformation on the sample data set

Standardized feature 0.23 −1.68

Transformed value −0.48 1.00

  Fig. 8 Effect of applying the cosine function for a sample value {0.23} with min f i j = −1.99 1≤i≤M   and max f i j = 1.38 1≤i≤M

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would be N5 , and it is defined in Eq. (9).  p f ic = f i j k ∀i = 1, 2, . . . , M; j = 1, 2, . . . , N1 ; k = 1, 2, . . . , |P|; pk ∈ P; and c = N4 + ( j − 1) × |P| + k.

(8)

N5 = N4 + N1 × |P|.

(9)

A Numerical Illustration As prescribed in Step 3.4, we transform the values of the first observation in the sample data set through their respective higher-order polynomials for the standardized features retained (N1 = 2) in Step 2 (which are presented in Table 3), the corresponding transformed values with respect to first observation are presented in Table 7, the total number of features at the end of this transformation is N5 = 50; and Fig. 9 also depicts the respective computation for a sample value. Step 3.5: Transformation of attributes using the interaction between the attributes The procedure applies this transformation, in order to capture the contribution of attributes from the interaction effect between the attributes/features (N 1 ) retained in Step 2, and the function used for this transformation is captured in Eq. (10); the total number of features (after the augmentation) at the end of this transformation would be N6 , and it is defined in Eq. (11).  q  q  f ic = f i j k × f i j  k ∀i = 1, 2, . . . , M; j = 1, 2, . . . , (N1 − 1); ∀ j  = ( j + 1), ( j + 2), . . . , N1 where j < j  ; k = 1, 2, . . . , |Q|; k  = 1, 2, . . . , |Q|; qk and qk  ∈ Q.

(10)

Table 7 Effect on the contribution of attributes from their higher-order polynomials on the sample data set Standardized feature

Transformed value when p1 = 2

Transformed value when p2 = 3

Transformed value when p3 = 4

0.23

0.05

0.01

0.00

−1.68

2.83

−4.76

8.00

Fig. 9 Effect of higher-order polynomials for a sample value {0.23} with p1 = 2

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  (N1 − 1) × |Q| × |Q|. N6 = N5 + N1 × 2

(11)

A Numerical Illustration As prescribed in Step 3.5, we transform the values of the first observation in the sample data set based on the interaction effect between the standardized features retained (N1 = 2) in Step 2 (which are presented in Table 3), the corresponding transformed values with respect to first observation are presented in Table 8, and the total number of features at the end of this transformation is N6 = 66; Fig. 10 also depicts the respective computation for a sample set of values. Step 4: Scaling The algorithm finally scales each feature (including the features obtained in Step 3: enrichment of the feature set) to take the value between 0 and 1, and the respective function is defined in Eq. (12):

fi j =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

f i j − min



1≤i≤M

 max

1≤i≤M



fi j



⎫ ⎪ ⎪ ⎬

 ∀i = 1, 2, . . . , M; and j = 1, 2, . . . , N6 .    ⎪ ⎭ f i j − min f i j + ε ⎪ 1≤i≤M

(12) Step 5: Multiple Linear Regression(MLR) using the training data set (Mtrain ) In this step, the algorithm applies MLR on the training data set to fit a regression line between N6 independent attributes (that are obtained/transformed/scaled in the previous steps: Step 2 to Step 4) and the dependent/response variable y (malignant or benign). In order to avoid the effect of overfitting, the algorithm considers ridge regularization as defined in Eq. (15). So, the objective function (zlr ) focuses on minimizing the magnitude of the coefficients (β j ) in addition to the minimization Table 8 Effect on the contribution of features based on their interactions on the sample data set Standardized feature ( j = 1)

Standardized feature ( j = 2)

Power term qk

Power term qk 

Transformed Value

0.23

−1.68

1.00

1.00

−0.39

0.23

−1.68

2.00

4.00

0.44

Fig. 10 Effect of interaction between the values {0.23, 1.681} with qk = 1 and qk  = 1

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of errors (i ). The MLR finds the coefficient (β j ) for each attribute j based on the respective attribute’s contribution toward target variable. Finally, the value of the expression yi is captured in variable yi , and the value of variable yi and the values of the coefficients β j are used as inputs in the LP model (in Step 6). yi = β0 +

N5    β j × f i j , ∀i = 1, 2, . . . , Mtrain .

(13)

j=1

i = yi − yi , ∀i = 1, 2, . . . , Mtrain Minimize : zlr = i2 + α ×

P 

β 2j

(14)

(15)

j=1

yi = yi

(16)

Step 6: Linear Programming(LP) model using the training data set (Mtrain ) In this step, we make use of the coefficients (β j ) obtained from the MLR (in Step 5), and then, we select a set of D relatively influential attributes based on the absolute values of the coefficients (β j ) for further analysis. In order to select D relatively influential attributes, we sort the values of coefficients β j in descending order, and then, we select the first D attributes where |D| is min {N6 , Mtrain }. In addition, we also use the value yi obtained from the MLR (in Step 5), and then, we solve the following minimization problem to minimize the errors (defined in Eq. (17)). So as a part of this process, the proposed LP model finds the classification function/expression for predicting the breast cancer using the training data set, by varying/searching the coefficients (γ j ) of the relatively influential attributes in set D. In this process, LP model also finds the initial thresholds where b and b + 1 are thresholds for malignant and benign classes, respectively. Minimize : zlp =

M train 

errori

(17)

i=1

For each observation in training data set, and  when the observation i is of cate    gory/class 0 (malignant tumor), then expression yi + j∈D γ j × f i j − errori should be less than the boundary (b): yi +

  γ j × f i j − errori ≤ b.

(18)

j∈D

when the observation i is of category/class 1 (benign tumor), then expression       yi + j∈D γ j × f i j + errori should be greater than the boundary (b + 1):

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yi +

  γ j × f i j + errori ≥ (b + 1).

(19)

j∈D

Step 7: Determination of exact boundary for the target classes using the validation data set (Mval ) In this step, the algorithm fine-tunes the thresholds obtained in the LP model (Step 6) through a search process to determine the exact boundary between the categories/classes (using validation data set Mval ). So, the algorithm increments the value b from 0 to 1 with the help of step size (0.05), and for each value of b , the algorithm performs the following: 1. 2.

3.

For each observation in validation data set (Mval), the algorithm calculates the     value for the expression yi + j∈D γ j × f i j . If the calculated value is less than (b + b ), then the corresponding observation is classified as class/category 0 (malignant tumor), and if the calculated value is greater than (b + b ), then the corresponding observation is classified as class/category 1 (benign tumor). Calculate the accuracy value for the given (b + b ) for the validation data set (Mval ), choose the best (b + b ) based on the accuracy value for the validation data set (Mval ), and it is defined as follows: Accuracy (number of observations predicted correctly in validation data set) × 100. = (number of observations in validation data set) (20)

So through Steps 5 to 7, the classification model is trained and fine-tuned using the training and validation data set, and then with the help of the classification func   tion/expression yi + j∈D γ j × f i j obtained in Step 6 and the exact boundary (b + b ) obtained in Step 7, the algorithm can classify/predict new observation to its respective target class (malignant/benign tumor). Step 8: Evaluation of the classifier using the test data set (Mtest ) In this step, algorithm makes use of the unseen/test data set (Mtest ) to evaluate the model in terms of the accuracy to predict the target classes  (and it is defined  in Eq. (21)). It makes use of the classification expression (yi + j∈D γ j × f i j ) obtained in Step 6 and the exact boundary (b + b ) obtained in Step 7, to classify/predict each observation in the test data set to corresponding target class (malignant/benign tumor). Accuracy =

(number of observations predicted correctly in test data set) × 100 (number of observations in test data set) (21)

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4 Experimental Results The proposed algorithm is evaluated with the help of the WDBC data set, and we compare the performance of the proposed algorithm with the results of the K-SVM algorithm proposed by Zheng et al. (2014). Please find the link to the data set: UCI Machine LearningRepository: Breast Cancer Wisconsin (Diagnostic) Data Set. The WDBC data set contains a set of instances/observations (569 observations) for which the corresponding diagnostic results/classification of the tumor (whether it is benign and malignant) are given; each observation contains 31 different tumor characteristics (features). In order to have a fair comparison of the proposed algorithm with the K-SVM by Zheng et al. (2014), in this study, the proposed algorithm is also validated with the help of tenfold cross validation as validated by Zheng et al. (2014). First, we apply tenfold cross validation on the given data set to split the data into ten equal-sized groups, and of the 10 groups, one group is treated as the test data set, and then, we again apply tenfold cross validation to split the remaining groups (9 groups) into training data set and validation data set. The same steps are repeated ten times/iterations so that each of ten groups are used exactly once as the test data set. We present the number of features that gets augmented at each stage of featureextraction process in Table 9, and in total, 31 features are enriched into 1573 features through various steps (from Step 1 to Step 4). In Step 5, we apply the MLR technique on these 1573 attributes to find the values of the variables yi and the coefficients β j . These values are used in the LP model (in Step 6); as a part of this step, the algorithm selects 460 relatively influential attributes (|D| = min{N6 , Mtrain } = min (1573, 460) = 460), and from this set D, we present the first ten relatively influential attributes (out of 460 attributes) in Table 10. Zheng et al (2014) followed a different approach, their algorithm first focused on reducing the number of features (from 31 to 6 features), and this was achieved with the help of K-means algorithm by finding the representative clusters (with the respective centroids) for both type of tumors; then, a membership function was applied on the observations to find the similarity between each observation and the identified centroids (for the representative clusters). Thus, the reduced number of Table 9 Number of features over various stages of feature enrichment phase Stages of feature enrichment

N0

N1

N2

N3

N4

N5

N6

Number of features

31

13

143

273

286

325

1573

Table 10 Coefficients (β j ) of the first ten relatively influential attributes from the set D Attributes in set D

1

2

3

4

5

6

7

8

9

10

Coefficients 0.198 0.171 0.144 −0.132 0.118 0.092 −0.083 0.071 −0.070 0.067 (β j )

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Table 11 Accuracy of the proposed classifier (for test data set) over the iterations in tenfold cross validation Iterations 1

2

3

4

5

6

7

8

9

10

Accuracy 96.55% 96.55% 98.24% 98.24% 98.24% 98.24% 100% 100% 100% 96.42%

features for each observation represented the similarity between this sample and the identified clusters, these features then made use of to diagnose the cancer (through SVM), and overall accuracy of their algorithm was 97.38%. We also present the results of the proposed classifier for the test data set (Step 8), across ten iterations (for tenfold cross validation) in Table 11, and we observe that on average (across ten iterations of tenfold cross validation), our algorithm performs better with the accuracy of 98.25%, whereas the K-SVM algorithm by Zheng et al. (2014) reported 97.38% (overall accuracy) which is around 0.87% improvement, and even 0.1% improvement is critical in healthcare domain (and particularly in breast cancer diagnosis).

5 Conclusion In this study, we proposed an algorithm for solving the classification problem which internally used multiple linear regression (MLR) and linear programming (LP) to find the nonlinear boundary between the target classes. We evaluated the performance of the algorithm in terms of the accuracy, and the proposed algorithm was able to perform better with the accuracy of 98.25%, while the K-SVM algorithm by Zheng et al. (2014) reported 97.38% (overall-accuracy) for predicting the breast cancer. Thus, the current work proposed an algorithm for solving the classification problem using the OR techniques, which is in comparable with the ML algorithm available in the literature for solving the classification problem (for predicting the breast cancer). Future work can be in the direction of exploring the options to partially/completely use the proposed algorithm in combination with the existing ML algorithms, to study and improve the effect of the same in various problem domains and the respective problem instances. Acknowledgements I thank Professor C. Rajendran, Indian Institute of Technology Madras, India, for his input and support. I am thankful to the reviewers and the editors for their valuable comments and suggestions to improve the paper.

References Akay, M. F. (2009). Support vector machines combined with feature selection for breast cancer diagnosis. Expert Systems with Applications, 36(2), 3240–3247.

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Bennett, K. P., & Blue, J. A. (1998). A support vector machine approach to decision trees. In 1998 IEEE International Joint Conference on Neural Networks Proceedings. IEEE World Congress on Computational Intelligence (Cat. No. 98CH36227) (Vol. 3, pp. 2396–2401). IEEE. Elmore, J. G., Wells, C. K., Lee, C. H., Howard, D. H., & Feinstein, A. R. (1994). Variability in radiologists’ interpretations of mammograms. New England Journal of Medicine, 331(22), 1493–1499. Huang, C. L., Liao, H. C., & Chen, M. C. (2008). Prediction model building and feature selection with support vector machines in breast cancer diagnosis. Expert Systems with Applications, 34(1), 578–587. Madankumar, S., Navya, P., Rajendran, C., Gupta, N. S., & Valarmathi, B. (2017). Linear programming (LP)-based two-phase classifier for solving a classification problem with multiple objectives. In Big Data Analytics Using Multiple Criteria Decision-Making Models (pp. 87–112). CRC Press. O’brien, R. M. (2007). A caution regarding rules of thumb for variance inflation factors. Quality and Quantity, 41(5), 673–690. Prasad, Y., Biswas, K. K., & Jain, C. K. (2010). SVM classifier based feature selection using GA, ACO and PSO for siRNA design. In International conference in swarm intelligence (pp. 307–314). Springer. Pena-Reyes, C. A., & Sipper, M. (1999). A fuzzy-genetic approach to breast cancer diagnosis. Artificial Intelligence in Medicine, 17, 131–155. Siegel, R., Naishadham, D., & Jemal, A. (2012). Cancer statistics, 2012. CA: A Cancer Journal for Clinicians, 62(1), 10–29. Wolberg, W. H., Street, W. N., & Mangasarian, O. L. (1995). Image analysis and machine learning applied to breast cancer diagnosis and prognosis. Analytical and Quantitative Cytology and Histology, 17, 77–87. Zheng, B., Yoon, S. W., & Lam, S. S. (2014). Breast cancer diagnosis based on feature extraction using a hybrid of K-means and support vector machine algorithms. Expert Systems with Applications, 41(4), 1476–1482.

Predicting Sustainable Supply Chain Performance Based on GRI Metrics and Multilayer Perceptron Neural Networks Devendra Singh, Krishnanand Lanka, and P. R. C. Gopal

Abstract The sustainable supply chain performance prediction model anticipates the values of key performance indicators also known lagging metric, based on the leading metrics. The prediction system helps decision-makers to identify the performance gaps and helps to take necessary action plans to minimize the deviation between the targets set and the outputs that are estimated by the model. This study uses a worldwide accepted Global Reporting Initiative (GRI) metrics and evaluates values of level 1 metrics that are effects of level 2 metrics by artificial neural networks(ANN). The literature presents several sustainable supply chain performance evaluation models; however, a prediction model based on a combination of GRI and ANN is a fairly unexplored area. Multilayer perceptron neural network model has the ability to adjust with the environment of use with the help of past performance data unlike models present in the literature that require manual parameterization for updating and implementing them. MATLAB is used for the computational implementation of ANN models. From the results of Pearson correlation coefficient, the correlation obtained among the targeted and forecasted performance values for the GRI level 1 metrics is positive. Keywords Sustainable supply chain management · GRI · Multilayer perceptron

D. Singh (B) · K. Lanka Mechanical Engineering Department, National Institute of Technology Warangal, Warangal, Telangana 506004, India e-mail: [email protected] K. Lanka e-mail: [email protected] P. R. C. Gopal Department of School of Management, National Institute of Technology Warangal, Warangal, Telangana 506004, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_8

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1 Introduction Sustainable supply chain management (SSCM) is defined as the material and information flow management and coordination between firms across the supply chain while considering goals from all three dimensions of sustainable development, i.e., economic, environmental and social (Erol et al., 2011). Sustainability has become the mainstream of the business world today (Gopal & Thakkar, 2015). The combination of economic, social and environmental performances of a practice is defined as its sustainable performance (Bauman & Genoulaz, 2014). Sustainable supply chain performance (SSCP) assessment deals with the practices taking place throughout the supply chain during operations of an organization and reflects how well it is doing economically, environmentally and socially. Firms started to move toward sustainable development because of increased environmental concerns and labor exploitation. A sustainable development, by definition, is ‘a development that meets the needs of the present without compromising the ability of future generations to meet their own needs’ (The World Commission on Environment and Development, 1987). For sustainable development, Elkington introduced the concept of the triple bottom line, TBL in 1994 (Elkington, 2004) that consists of three dimensions which are economy, environment and society. These are also known as 3Ps of TBL meaning profit, people, and planet. Profit: The traditional measure of corporate profit. People: Measures how socially responsible an organization has been throughout its operations. Planet: Measures how environmentally responsible a firm has been. In TBL, the profits of an organization do matter but just not at the expense of social and environmental concerns. Ignoring TBL can lead to the destruction of the rainforest, exploitation of labor, damage to the ozone layer and other negative impacts. Profit of an organization is a monetary term that means it can be calculated, whereas social and environmental responsibilities of an organization are subjective terms and are not quantities. So Global Reporting Initiative, an international independent standards organization founded in 1997, introduced GRI standards in 2000 to measure sustainability. GRI standards quantified the environmental effects as GHG emissions, waste disposed of, etc., and social effects as accidents reported in the firm, training hours provided, etc. GRI standards are a systematic set of standards of economic, environmental and social aspects. A GRI model can be constructed by arranging these standards in a hierarchical manner into two levels: level 1 metrics and level 2 metrics. Level 1 metrics can be termed as lagging metrics, and level 2 metrics are also known as leading metrics. In organizations, the term, key performance indicators (KPIs), is used instead of level 1 metrics. This GRI model recommends to analyze the relationship of cause and effect between the different levels of the metrics as a way of trying to find underlying causes for bad performance. For example, performance in respect of economic value distributed, a metric of level 1, is an outcome of multifactor performance such as payments to providers of capital,

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operating cost, payments to government by country and employee wages and benefits. Therefore, the performance of level 1 metrics can be evaluated by using level 2 metrics. This study proposes the idea to build a model that can estimate the cause and effect relationship accurately among the different levels of metrics. This model can be used for the prediction of lagging metrics (output variable) with the help of leading metrics (input variable). This prediction model will help the decision-makers in the organization to compare the targets they have set and values predicted by the model. The deviation, if any, between the targets and predicted outcomes can be analyzed, and action plans can be taken to minimize or eliminate it to achieve desired targets by changing their inputs. To estimate the causal relationship, the artificial neural network has been used as it has the ability to adjust with a specific environment of use with the help of past performance data. ANN has the advantage of modeling linear as well as nonlinear relationships. The literature presents various sustainable supply chain performance measurement models (Ding, 2005; Erol et al., 2011, Buyukozkan & Çifçi, 2012, Uysal, 2012, Govindan et al., 2013, Jakhar, 2015, Marioka & Carvahlo, 2016) based upon different multi-criteria decision-making techniques, but the area of sustainable supply chain performance prediction system is still untouched. The performance measurement systems proposed in the literature reviewed have a drawback that corresponds to the necessity of collecting the specialists’ judgments for parameterizing the decision rules and linguistic terms of both the variables: input and output. One of the limitations of these systems is that they are unable to review and adjust the relationship by using past performance data between metrics. Secondly, parameterizing the system with the help of judgments from specialists consumes a lot of time, whereas, these limitations of the system that predict performance can be overcome by using it with artificial neural networks. So, in this study, a new prediction system for sustainable supply chain performance has been proposed which estimates the lagging metrics values of the GRI model based on leading metrics values by using artificial neural networks (as illustrated in Fig. 4).

2 Sustainable Supply chain Assessment and Management Information, material and capital flow link suppliers, focal companies and customers in a supply chain. In accordance with the product’s value, the burden of the environment and society comes resulting from various production stages. In light of this, the supply chain’s focal companies might be held responsible for their suppliers’ environmental and social performance (Seuring & Muller, 2008). Improvement in a supply chain’s service and goods quality can considerably be achieved by performance management of the organization and its suppliers (Sweeney, 2011; SCC, 2012).

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There are reasons given by Lima Jr and Carpinetti (2019), based on Parker (2000), why organizations should measure performance: “(1) for success identification, (2) to check whether customer requirements are met, (3) to help them understand their processes, (4) to identify where problems bottlenecks, waste, etc., exist and where improvements are necessary, (5) to make sure decisions are fact-based and not supposition, emotion, faith or intuition-based, (6) to show whether planned improvements actually happened.” Performance measurement is an important aid for making judgments and decisions (Parker, 2000).

2.1 GRI Model GRI, acronyms for Global Reporting Initiative, is a set of standards developed in the year 2000 by Global Reporting Initiative, an international standards organization formed in 1997, to help businesses, governments and other organizations grasp and convey their influence on issues like human rights, corruption and climate change. The GRI framework enables organizations to identify, gather and report their environmental, economic and social impacts. Annual sustainability reports are published by many firms in order to be transparent about the effects of their processes on the environment, society and economy. Reporting frameworks of GRI launched in 2018 are the most recent GRI standards. These GRI standards have a modular and interrelated structure (GRI, 1997). In light of this, these GRI standards have been arranged in a hierarchical structure into two levels (level 1 and level 2) having a causal relationship to create a GRI model for all three attributes of sustainability: economic, environmental and social. That means for each attribute there is a set of level 1 metrics, and corresponding to each level 1 metric, there are several level 2 metrics (Fig. 1). Level 1 metrics are used as outputs, and level 2 metrics are used as inputs. The GRI model involves both linear as well as nonlinear cause and effect relationships. Therefore, the artificial neural network (ANN) has been used as a technique to estimate this relationship so that the whole arrangement (GRI model and ANN) can be used as a performance prediction system. The performance regarding level 1 metrics can be achieved by using level 2 metrics as inputs. Studies reviewed in the literature have not used GRI standards so far and that too with ANN technique so it remains a new idea. ANN can obtain a relationship between cause and effect in any environment with the help of past performance data which is also known as the ability to adjust to a particular environment of use.

2.2 Multilayer perceptron Neural Networks Artificial neural networks (ANNs) are models of artificial intelligence (AI) inspired by structure of the human brain and its learning skills. From literature, it is evident

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Fig. 1 GRI model

that wide applications of ANNs in sustainable supply chain management are present. Out of various ANN models, majorly multilayer perceptron (MLP) neural networks are used because of its ability to deal with time forecasting and function approximation problems accurately. MLP networks have the ability to adjust to the required environment with the help of previous data of performance by using a supervised learning algorithm that utilizes input and output variables as a set of samples. In the majority of applications, backpropagation algorithm is used. Synaptic connections by interconnecting a set of artificial neurons form an MLP network. It consists of input variables in an input layer, output variables in an output layer and hidden neurons in one or more hidden layers. For each input variable, there is a neuron in the input layer, and similarly each output variable there is a neuron in the output layer. Data characteristics are extracted and stored by neurons in hidden layer. Also, most of the operations of data processing are done by the hidden layer. Figure 2 shows a general structure of an MLP network. In total, seven elements together compose an artificial neuron: inputs, weights, activation threshold, linear combiner, activation function, activation potential and outputs. The learning of a perceptron is a process of iterations. The learning starts by assigning each weight a random value. It then calculates the error function by comparing the corresponding outputs with expected values and modifies the weights to minimize the error function. As the training stage completes, an optimal weight matrix is so obtained that the mean square error (MSE) among the values that are expected and the outputs that are estimated by the network is minimized. Obtaining this optimal weight matrix is the main objective of training. Learning rate, number of hidden layer neurons, hidden layer count and activation function applied in output and hidden layers are some of the parameters that influence the prediction accuracy of the models directly, and they have to be chosen empirically. Studies (Lenard et al., 1995) suggest to use either three-fourths of input variables or half of the combined output and input variables as the count of neurons in hidden layer. Epochs count which is defined as the number

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Fig. 2 Multilayer perceptron neural networks

of times a network computes the subset of training samples is considered as both efficiency parameter as well as network training stopping criteria. An MLP network with a single hidden layer has the ability to estimate both linear and nonlinear relationships, and also it can adjust to the required environment of use by the help of past performance data. Because of its abilities, an integration of GRI metrics with MLP neural networks can contribute to the literature as well as to develop ANN systems that can help practitioners to enhance sustainable supply chain performance.

3 Sustainable Supply Chain performance Prediction System A system for predicting performance allows the approximation of the results of lagging metrics (output variables) with the help of leading metrics’ (input variables) performance levels. The levels of performance regarding leading metrics are set either by actual performance data or by estimation. Systems for prediction are based majorly on the technique of artificial intelligence which estimates the cause and effect relationship among metrics by using learning algorithms. The prediction systems help managers to create systematic ways of predicting the performance of various metrics and identifying issues in the company. Unahabhokha et al. (2007) developed a performance prediction system and applied it in an industrial printer manufacturer and in a textile company. Prediction systems can also support decision-making in the context of the supply chain (Lima Jr & Carpinetti, 2019). Though the literature available on performance prediction systems for the supply

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Fig. 3 Sustainable supply chain performance prediction system

chain is less, the area of performance prediction systems for the sustainable supply chain is yet to be discovered. The performance prediction system proposed in this study is a combination of GRI metrics and ANN (Fig. 3). Based on past data of performance, it can adjust to a particular environment of use. It also does not require any judgments from the specialists for parameterization of various decision rules unlike multi-criteria decision-making techniques (Lima Jr & Carpinetti, 2019). Based on Lima Jr and Carpinetti (2019), a theoretical framework was developed for performance prediction. The whole idea is to estimate the deviation between predicted lagging metrics values and the target values; if any deviation is present, then development and execution of action plans can be carried out in advance to prevent or mitigate deviation from desired outcomes. Figure 4 illustrates the proposed system for predicting performance which consists of ten models of MLP neural network implemented on the hierarchically arranged GRI standards into two levels. The output of ANN model 1 is used as an input to ANN model 2 and also by studying the mathematical expressions present in the GRI model to obtain level 1 metrics, it can be realized that nonlinear relationship among input and output variables is present only in ANN model 3. It must be noted that the whole idea proposed in this study was applied to the Tata Motors to put forward an example (Sects. 3.1 and 3.2). The GRI reports of Tata Motors from 2010 to 2018 were analyzed, and only those metrics were extracted out which could be arranged in two hierarchical levels. Hence, the number of parameters is different for the three dimensions of sustainability. The process of performance management presented in Fig. 3 can be carried out by firstly selecting the set of GRI metrics that are in line with the company’s competitive strategy and performance goals. Secondly, for each selected metric data must be

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Fig. 4 Sustainable supply chain performance prediction system

collected by the focal company. That means, the current values of input variables (level 2 metrics) must be collected or estimated, and also performance targets must be set for chosen metrics (level 1) for performing performance prediction tests. Now, the normalization of collected values should be done by taking the universe of discourse limits and Eq. 1. xi =

X i − X min X max − X min

(1)

After normalizing the metrics values of level 2, the values are supplied to the system for prediction, and by using them the trained ANN models measure the expected level 1 metrics performance values. These measured values are obtained as normalized values so they have to be converted back to the original form. This can be done by taking the universe of discourse limits defined during the training stage for each output metric. The targeted performance values can be compared with these estimated performance outputs, and hence managers can analyze the deviation if any, and therefore can identify the areas of gaps to put efforts on to ultimately minimize or avoid this deviation.

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For the training purpose of ANN models, the numerical values of metrics of both the levels must be gathered by the focal company so that the prediction system can be made adjustable to a particular environment of use. This data should be arranged as input and output variables in a table for each ANN model. For each ANN model, a set of networks depending upon various parameters (hidden and output layer activation functions, neurons count in hidden layer) should be characterized as candidates. The learning rate and training epochs count must also be selected. The total samples should be divided into two sets: 70% of them are used for training purposes and 30% for testing as well as validating purposes. For every candidate network, the accuracy level is obtained by analyzing the mean square error (MSE) between the validation subset and system estimated output values. The network with the least MSE should be selected as the best network among all the candidate networks for respective ANN model. In case the input values of variables lie outside the universe of discourse limits defined earlier during the process of training, then the ANN models need updating of new limits and hence retraining.

3.1 Illustrative Application To evaluate the proposed system’s prediction accuracy along with the process of grabbing the best candidate network for each ANN model, the development of an illustrative application case was done. The GRI metrics adopted for the proposed system are shown in Table 1 with measurement units. For each ANN model, six candidate networks were defined by varying various parameters as shown in Table 2. Neurons count in hidden layer was defined as per the quantity of variables used for input (x n ) for every ANN model. Thus, x n , x n – 1, x n + 1 are the number of neurons for which candidate networks were tested. A single hidden layer is sufficient to map all linear as well as nonlinear relationships (Haykin, 2010; Silva et al., 2017). Hyperbolic tangent and logistic in the hidden layer neurons are the two nonlinear activation functions that were checked. Both of these functions are widely used in ANNs. The learning procedure of neurons is different for different activation functions. Hence, the trained networks will provide different accuracies for different activation functions. As there is only one output in each ANN model, therefore a single neuron is used in the output layer. The adoption of activation function as a linear function was done for output layer neuron so that the output signals from the neurons of hidden layer form a linear combination and it becomes easy to compute. Therefore, based upon all these parameters (Table 2), 60 candidate networks (six for each ANN model) were created. For validation stage, maximum value of 10E−03 was defined for MSE. Using MATLAB, these 60 candidate networks in total were implemented, trained and validated. After training, the candidate network with least MSE at validation stage out of six candidate networks for each ANN model was selected as the best network for ANN model, hence for performance prediction.

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Table 1 Metric definitions Model

Variable

Brief Description

Measurement unit

Universe of discourse

ANN Model 1

x1

Operating cost: An INR Billion organization can measure the operational costs as payments for purchased goods, product parts, facilities and services rendered outside the organization

[350, 600]

x2

Employee wages and INR Billion benefits: An organization can measure it as overall payroll (which includes salaries of employees and amounts paid on behalf of the workers to government entities) plus net benefits (which excludes training, safety gear expenses or other things specifically relevant to the job function of the employee)

[20, 50]

x3

Payments to providers of capital: A company can measure the payments made to capital providers as dividends to all shareholders, plus interest fees paid to loan providers

INR Billion

[13, 35]

x4

Payments to INR Billion government by country: A company can measure government payments as all taxes of the company plus related sanctions paid at the international, national and local level. Company taxes can include corporate, income and property

[30, 60]

(continued)

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Table 1 (continued) Model

ANN Model 2

ANN model 3

Variable

Brief Description

Measurement unit

Universe of discourse

y1

Economic value INR Billion distributed: It is the sum of payments to capital providers, operating costs, benefits and wages of employee, community investments and payments to government by country

[413, 745]

x5

Economic value INR Billion generated: A company can measure economic value generated as total sales plus revenues from asset sales and investments in financing

[400, 600]

x6

Economic value INR Billion distributed: It is the sum of payments to capital providers, operating costs, benefits and wages of employee, community investments and payments to government by country

[418.3056403, 736.8343429]

y2

Economic value retained: Direct economic value generated is the sum of economic value distributed and economic value retained

INR Billion

[−283.13384, 143.86489]

x7

Direct energy generated: Energy generated inside the organization

GJ

[900,000, 3,000,000]

x8

Indirect energy GJ generated: Energy purchased from outside the organization

[1,200,000, 2,500,000]

(continued)

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Table 1 (continued) Model

ANN model 4

ANN model 5

Variable

Brief Description

Measurement unit

Universe of discourse

x9

Renewable energy: GJ Renewable energy is the energy generated from renewable sources such as hydro, biomass, wind, geothermal, solar

[110,000, 500,000]

y3

% renewable energy of GJ total energy: How much part of total energy consists of renewable energy

[2.17277119, 18.08650957]

x 10

Scope 1: The tCO2 e greenhouse gas (GHG) emissions produced by the consumption of non-renewable fuels are known as direct (Scope 1) GHG emissions

[50,000, 250,000]

x 11

Scope 2: The use of purchased electricity, heating, cooling and steam adds to the indirect energy (Scope 2) emissions from the company

tCO2 e

[250,000, 500,000]

y4

GHG Emissions: Total GHG emissions produced by the organization, i.e., both direct and indirect

tCO2 e

[300,000, 750,000]

x 12

SOx : Oxides of sulfur resulted from the manufacturing processes of the organization

Ton

[50, 650]

x 13

NOx : Oxides of nitrogen resulted from the manufacturing processes of the organization

Ton

[90, 250]

x 14

SPM: Particulate matter Ton is the sum of all solid and liquid particles suspended in air many of which are hazardous

[120, 350]

(continued)

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Table 1 (continued) Model

ANN model 6

ANN model 7

Variable

Brief Description

Measurement unit

Universe of discourse

y5

Air emissions: Manufacturing processes contribute to air emissions, namely oxides of nitrogen (NOx), sulfur dioxide (SO2 ) and particulate matter

Ton

[260, 1250]

x 15

Municipal: Water supplied from municipal (third party) suppliers

m3

[2,000,000, 8,500,000]

x 16

Surface water: Water m3 that is naturally found on the surface of the earth in icebergs, rivers, bogs, ice caps, glaciers, ice sheets, ponds, streams and lakes

[750,000, 1,600,000]

x 17

Groundwater: Water that is held in an underground formation and that can be recovered from there

m3

[650,000, 1,800,000]

x 18

Rainwater: Water m3 harvested from rain and is used

y6

Water abstracted: The total water used for operations in the organization

m3

[3,440,000, 12,550,000]

x 19

Hazardous waste generated: Hazardous waste is waste that has substantial or potential threats to public health or the environment

Ton

[5000, 7500]

x 20

Non-hazardous waste Ton generated: All waste materials not specifically deemed hazardous under federal law are considered nonhazardous wastes

[40,000, 650,000]

[70,000, 150,000]

(continued)

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Table 1 (continued) Model

ANN model 8

ANN model 9

ANN model 10

Variable

Brief Description

Measurement unit

Universe of discourse

y7

Waste generation: Total Ton waste generated in the organization (hazardous and non-hazardous)

[75,000, 157,500]

x 21

Hazardous waste Ton disposed of: Complete weight of hazardous waste includes reuse, incineration, deep well disposal, recycle, composting, landfill, onsite storage, recovery, etc.

[4500, 10,000]

x 22

Non-hazardous waste Ton disposed of: Complete weight of non-hazardous waste includes reuse, incineration, deep well disposal, recycle, composting, landfill, onsite storage, recovery, etc.

[45,000, 150,000]

y8

Total waste disposed of: Ton Total weight of waste disposed of (hazardous and non-hazardous)

[49,500, 160,000]

x 23

Male hires: Total number of male employees hired in the organization

Dimensionless

[900, 1500]

x 24

Female hires: Total number of female employees hired in the organization

Dimensionless

[65, 160]

y9

Total hires: Total Dimensionless number of employees hired (male and female) in the organization

[965, 1660]

x 25

Male attrition: Total number of male employees laid off in the organization

Dimensionless

[1500, 3000]

x 26

Female attrition: Total number of female employees laid off in the organization

Dimensionless

[100, 180]

(continued)

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Table 1 (continued) Model

Variable

Brief Description

Measurement unit

Universe of discourse

y10

Total attrition: Total number of employees (male and female) laid off in the organization

Dimensionless

[1600, 3180]

Table 2 Candidate networks Network

Hidden neurons

Activation function (Hidden neuron)

Activation function (Output neuron)

Network 1

xn − 1

Logistic function

Linear function

Network 2

xn

Logistic function

Linear function

Network 3

xn + 1

Logistic function

Linear function

Network 4

xn − 1

Hyperbolic tangent function

Linear function

Network 5

xn

Hyperbolic tangent function

Linear function

Network 6

xn + 1

Hyperbolic tangent function

Linear function

3.2 Data Generation Process The following processes were done for sample generation to train and validate the models: • By taking into consideration the universe of discourse, the random values were generated for level 2 metrics using MS Excel. • With the help of equations presented in the GRI model, the values of level 1 metrics were obtained by using level 2 metrics. • Normalization of all the input and output values was done using Eq. 1. As the random values generated between the given universe of discourse do not include both extreme values, we have separately added a set of lower and a set of upper bound values in each of the respective data sets. For data collection, the annual GRI reports of Tata Motors from 2010 to 2018 (nine data sets) were analyzed, and metrics were extracted out and were arranged in two levels. But nine data sets would not have been sufficient for ANNs as ANNs require big data sets for training. So, by using MS excel, data for all the metrics was generated within the range we defined by analyzing the GRI reports.

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3.3 Training and Validation Results During the training and validation stage, the learning rate value was set to 0.03 initially according to Bilgehan (2011) and training epochs were set to 8000. The values of MSE obtained during the training procedure of all the candidate networks are presented in Table 3. During the training stage, the cause and effect relationships get tuned among input and output metrics and after that validation takes place. The MSE values generated during the validation stage of all the candidate networks are also presented by Table 3. The candidate network with the lowest MSE at the validation stage is considered as the network having the best prediction accuracy for each model (Table 4). The best candidate networks are highlighted in bold as shown in Table 3. Pearson correlation coefficient (R) indicates the relationship degree among sample values and output values at the validation stage. For selected candidate networks (having least MSE), values of R were also obtained for evaluating prediction accuracy in addition to MSE at the validation stage. R varies in [−1, 1] interval with − 1 representing a correlation which is perfectly negative, 1 representing a perfectly positive correlation, and 0 representing correlation does not exist among the variables (Montgomery & Runger, 2012). The pattern of association can be obtained among the two variables as an equation. So in order to do that, linear regression tests were also done using MATLAB. Results obtained for regression analysis and correlation coefficient are shown in Fig. 5. The targeted performance values are referred by the horizontal axis, and the ANN model estimated values are referred by the vertical axis.

4 Conclusion A prediction model of supply chain performance composed of MLP networks and GRI metrics is proposed in this study. Sixty candidate networks, in total, were implemented, and all the candidate networks were evaluated depending upon the validation stage prediction accuracy. By analyzing the results, Fig. 5 and Tables 3 and 4, it can be concluded that the performance of level 1 metrics can be estimated with high accuracy by using MLP networks. The validation stage MSE was recorded the lowest (8.0238E−07) for ANN model 10 and the highest (2.8760E−05) for ANN model 2. The R values (Fig. 5) indicate that between predicted and expected values of the level 1 metrics, the correlation is highly positive for every ANN model. The results so obtained show that use of GRI metrics and MLP networks for prediction purpose of a sustainable supply chain‘s performance is an effective method. Through a prospective evaluation of sustainable supply chain efficiency, the system proposed in the study enhances rational decision-making. The managers can use it to acquire a prospective scenario of the performance of business by comparing

Predicting Sustainable Supply Chain Performance Based … Table 3 MSE obtained in training and validation stage

175

ANN model

Network

MSE

ANN model 1

Network 1

1.157E−04

1.148E−04

Network 2

1.797E−05

2.405E−05

Network 3

2.929E−05

3.741E−05

Network 4

1.892E−04

1.007E−04

Network 5

3.377E−05

1.078E−04

Network 6

4.453E−05

3.95E−05

Network 7

7.225E−04

1.398E−03

Network 8

9.22E−05

1.029E−04

Network 9

7.101E−04

9.958E−04

Network 10 8.158E−04

1.969E−04

Network 11 9.379E−05

2.876E−05

Training stage Validation stage

ANN model 2

ANN model 3

ANN model 4

ANN model 5

ANN Model 6

Network 12 1.201E−04

1.028E−04

Network 13 3.296E−05

2.7206E−05

Network 14 5.993E−04

7.9323E−04

Network 15 2.834E−04

2.0192E−04

Network 16 7.157E−05

3.589E−05

Network 17 4.973E−04

3.3838E−04

Network 18 3.891E−05

9.0673E−05

Network 19 1.08E−04

3.653E−05

Network 20 1.589E−05

3.53E−04

Network 21 1.161E−05

1.186E−04

Network 22 1.66E−04

1.0469E−04

Network 23 3.986E−05

1.633E−05

Network 24 1.816E−05

1.692E−05

Network 25 8.785E−05

7.909E−05

Network 26 2.468E−06

2.972E−06

Network 27 9.825E−06

9.742E−04

Network 28 1.05E−04

7.229E−05

Network 29 1.463E−05

9.055E−05

Network 30 1.989E−04

8.063E−05

Network 31 1.533E−04

7.6763E−03

Network 32 9.926E−06

1.549E−04

Network 33 1.264E−04

1.2889E−04

Network 34 7.695E−06

4.3953E−06

Network 35 1.949E−05

4.2195E−05

Network 36 2.673E−06

1.8737E−05 (continued)

176 Table 3 (continued)

D. Singh et al. ANN model

Network

MSE

ANN model 7

Network 37 1.314E−04

Training stage Validation stage

ANN model 8

ANN model 9

1.0625E−04

Network 38 1.15E−05

4.2342E−05

Network 39 1.487E−04

8.237E−05

Network 40 1.768E−04

2.1278E−04

Network 41 1.856E−05

2.8435E−05

Network 42 2.028E−05

1.299E−05

Network 43 1.308E−04

1.2793E−04

Network 44 1.357E−05

1.0609E−05

Network 45 1.203E−05

7.3038E−06

Network 46 1.621E−04

1.2607E−04

Network 47 1.983E−04

2.1786E−04

Network 48 1.458E−04

1.5091E−04

Network 49 1.199E−04

7.9029E−05

Network 50 2.572E−05

1.7218E−05

Network 51 1.778E−06

1.1025E−06

Network 52 1.661E−04

6.5288E−05

Network 53 1.43E−04

1.0528E−04

Network 54 2.056E−04

2.4005E−04

ANN model 10 Network 55 1.131E−04

5.4813E−05

Network 56 1.113E−06

8.0238E−07

Network 57 7.477E−05

7.9082E−05

Network 58 1.591E−04

1.7954E−04

Network 59 3.21E−06

5.3957E−06

Network 60 1.764E−05

6.4184E−05

the predicted performance values with defined targets for all the level 1 metrics. This comparison will enable managers to get a better insight of the ramifications of underperformance or over performance in various operational facets. For example, managers can assess environment dimension performance based on calculations of the “air emissions” (level 1 metric) as a result of the model level 2 metrics (input variables). In this way, the wide range of the GRI metrics leads to the identification of critical areas of business in order to put preventive action plans at the input stage for improvements. Apart from all its advantages, this study lacks whenever a metric is removed or added because in that case the whole model needs retraining. Another limitation is that as ANNs are environment specific so as the work environment changes the input values also change and hence the model will require retraining. ANNs produce highly accurate results for complex data; however, the training time of models increases with

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Table 4 Network candidate selected for every ANN model in the prediction systems ANN model

Chosen network

MSE (Validation stage)

R (Validation stage)

Number of hidden layer neuron

Activation function of the hidden layer

Learning rate

ANN model 1

Network 2

2.4050E−05

0.99988

4

Logistic Function

0.03

ANN model 2

Network 11

2.8760E−05

0.99981

2

Hyperbolic tangent

0.03

ANN model 3

Network 13

2.7206E−05

0.99974

2

Logistic Function

0.03

ANN model 4

Network 23

1.6330E−05

0.99980

2

Hyperbolic tangent

0.03

ANN model 5

Network 26

2.9720E−06

0.99997

3

Logistic function

0.03

ANN model 6

Network 34

4.3953E−06

0.99995

3

Hyperbolic tangent

0.03

ANN model 7

Network 42

1.2990E−05

0.99993

3

Hyperbolic tangent

0.03

ANN model 8

Network 45

7.3038E−06

0.99997

3

Logistic function

0.03

ANN model 9

Network 51

1.1025E−06

1

3

Logistic function

0.03

ANN model 10

Network 56

8.0238E−07

1

2

Logistic function

0.03

high complexity of the data. This model helps to original equipment manufacturer (OEMs) to assess the sustainable performance metrics such as percentage renewable energy used of total energy (GJ), total water abstraction (m3 ), GHG emissions (tCO2 ), air emissions (Ton), waste generation (Ton) and waste disposed (Ton) effectively which are useful to make internal and internal benchmarking. Further, it also helps to initiate policy decisions on improvement of sustainability performance. Further studies can use different artificial intelligencey techniques and algorithms to develop comparative studies in this area and compare the advantages and disadvantages of the techniques. Also the idea of performance prediction model can be implemented to agile, flexible or lean supply chain.

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Fig. 5 Regression analysis and correlation coefficient

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Acknowledgements I would like to thank Francisco Rodrigues Lima Junior (Assistant Teacher at Federal University of Technology, Brazil) for his assistance and support. I am grateful that the annual GRI reports of Tata Motors for collection of data were available on Global Reporting Initiative. Finally, I must express my very profound gratitude to my parents and to my friends for providing me with unfailing support and continuous encouragement throughout the process of researching and writing this paper.

References Acquaye, A., Ibn-Mohammad, T., Genovese, A., et al. (2018). A quantitative model for environmentally sustainable supply chain performance measurement. 49, 63–66 Agami, N., Saleh, M., & Rasmy, M. (2014). An innovative fuzzy logic based approach for supply chain performance management. IEEE Systems Journal, 8, 336–342. Bilgehan, M. (2011). Comparison of ANFIS and NN models—with a study in critical buckling load estimation. Applied Soft Computing, 11, 3779–3791. Buyukozkan, G., & Çifçi, G. (2012). A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers. Expert Systems with Applications, 39, 3000–3011. Chardine-Bauman, E., & Botta-Genoulaz, V. (2014). A framework for sustainable performance assessment of supply chain management practices. Computers and Industrial Engineering, 76, 138–147. Chen, L., Feldmann, A., & Tang, O. (2015). The relationship between disclosures of corporate social performance and financial performance: Evidences from GRI reports in manufacturing industry. International Journal of Production Economics, 170, 445–456. Ding, G. K. (2005). Developing a multicriteria approach for the measurement of sustainable performance. Building Research and Information, 33, 3–16. Erol, I., Sencer, S., & Sari, R. (2011). A new fuzzy multi-criteria framework for measuring sustainability performance of a supply chain. Ecological Economics, 70, 1088–1100. Ganga, G. M. D., & Carpenitti, L. C. R. (2011). A fuzzy logic approach to supply chain performance management. International Journal of Production Economics, 134, 177–187. Global Reporting Initiative. (1997). https://www.globalreporting.org/Pages/default.aspx. Accessed 15 September 2019. Gopal, P. R. C., & Thakkar, J. (2015). Development of composite sustainable supply chain performance index for the automobile industry. International Journal of Sustainable Engineering, 8, 366–385. Govindan, K., Khodaverdi, R., & Jafarian, A. (2013). A fuzzy multi criteria approach for measuring sustainability performance of a supplier based on triple bottom line approach. Journal of Cleaner Production, 47, 345–354. Haykin, S. (2010). Neural networks and learning machines (3rd ed.). Pearson Education India. Jakhar, S. K. (2015). Performance evaluation and a flow allocation decision model for a sustainable supply chain of an apparel industry. Journal of Cleaner Production, 87, 391–413. Elkington, J. (2004). https://www.johnelkington.com/archive/TBL-elkington-chapter.pdf. Enter the triple bottom line. Lenard, M. J., Alam, P., & Madey, G. R. (1995). The application of neural networks and a qualitative response model to the auditor’s going concern uncertainty decision. Decision Sciences, 26, 209– 227. Li, Y., & Mathiyazhagan, K. (2017). Application of DEMATEL approach to identify the influential indicators towards sustainable supply chain adoption in the auto components manufacturing sector. Journal of Cleaner Production, 172, 2931–2941.

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Lima-Junior, F. R., & Carpinetti, C. R. (2019). Predicting supply chain performance based on SCOR® metrics and multilayer perceptron neural networks. International Journal of Production Economics, 212, 19–38. Montgomery, D. C., & Runger, G. C. (2012). Applied statistics and probability for engineers, 5th ed. Willey Morioka, S. N., & Carvalho, M. M. (2016). Measuring sustainability in practice: Exploring the inclusion of sustainability into corporate performance systems in Brazilian case studies. Journal of Cleaner Production, 136, 123–133. Parker, C. (2000). Performance measurement. Work study. SCC. (2012). Supply chain council. Supply Chain Operation Reference Model 12.0 (SCOR model). SCC. (2012). Supply chain council. Supply Chain operation Reference Model. Supply Chain Council 11.0. Seuring, S., & Müller, M. (2008). From a literature review to a conceptual framework for sustainable supply chain management. Journal of Cleaner Production, 16, 1699–1710. Silva, I. N., Spati, D. H., Flauzino, R. A., Liboni, L. H. B., & Alves, S. F. R. (2017). Artificial neural networks: A practical course. Springer. Su, D., Horng, C., Tseng, M., et al. (2016) Improving sustainable supply chain management using a novel hierarchical grey-DEMATEL approach. Journal of Cleaner Production, 134, 469–481. Sweeney, E. (2011). Towards a unified definition of supply chain management: The four fundamentals. International Journal of Applied Logistics, 2, 30–48. The World Commission on Environment and Development. (1987). https://sustainabledevelopment. un.org/content/documents/5987our-common-future.pdf. Report of the World Commission on Environment and Development: Our Common Future. Unahabhokha, C., Platts, K., & Tan, K. H. (2007). Predictive performance measurement system: A fuzzy expert system approach. Benchmarking: An International Journal, 14(1), 77–91. Uysal, F. (2012). An integrated model for sustainable performance measurement in supply chain. Procedia-Social and Behavioral Sciences, 62, 689–694.

Green Supply Chain Management in the Indian Petroleum Industry Using AHP-VIKOR Approaches Sourabh Kumar and Mukesh Kumar Barua

Abstract Petroleum industry operations have positively and negatively impacted a range of areas covered by sustainable developments, including communities, economies, and ecosystems. Therefore, the petroleum industry has redefined its business development strategy for achieving the green sustainable goal. This study aims to identify green supply chain management practices and prioritize these to know their level of importance in petroleum industries. The literature review was scrutinized, and the main criteria were identified, which were further filtered by the expert panel. Two criteria, basic and advanced green practices and the seven subcriteria, were evolved to assess green practices indices identification. This study employs multicriteria decision-making (MCDM) techniques AHP and VIKOR using R programming to prioritize the green practice indices. The analysis shows that investing in and implementing renewable energy, risk prevention systems to cover possible environmental accidents, and preventing oil and methane leaks are the top three green practices. The finding helps practitioners and policymakers of the petroleum companies to implement effective green practices to sustain the environment. Keywords Green practices · R- language · Oil and gas · Sustainability

1 Introduction The supply and demand for crude oil and petroleum products are the key factors determining the world economy’s status (Hamilton, 2009; Krichene, 2002). Petroleum companies provide more than 50% of global fuel consumption and are expected to remain by 2035. On the one hand, global demands for fossil fuels are still rising. On the other hand, companies face complex investment challenges due to the severe operating environment of exploration, production, and refining activities (Asif & Muneer, 2007). The 2030 Agenda for sustainable development also requires support from the S. Kumar (B) · M. K. Barua Indian Institute of Technology Roorkee, Haridwar, India M. K. Barua e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3_9

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oil and gas industries (Yusuf et al., 2013). The petroleum industry is a key pillar of the global energy system and one of the drivers of economic and social development (Omer, 2008). Environmental sustainability effects in greater amounts by the release of waste in the environment. In the exploration and production segment, drilling starts with an appraisal phase that explores hydrocarbon presence and identifies the reservoir’s size and nature. The location of the drilling site depends on the characteristics of geological formation and depends on surface constraints. It is because to balance environmental sustainability with logistical constraints without affecting the efficiency of drilling. Most of the oil reservoirs have already been developed, and new reservoirs can harness environment conditions if proper green practices are not implemented. The petroleum industry projects are crucial because they directly enhance the world economy and national GDPs. It plays a big economic impact, so it is also a key player in politics and strategic decision making in government. The petroleum industry often accuses of greenhouse gas emission in upstream, midstream, and downstream segments. As per Shell oil company sustainability report, they fixed the greenhouse gas intensity such as upstream segment GHD intensity ≤0.164, refining GHG ≤1.05 tonnes, chemical plant GHG ≤0.97, and reduce flaring in upstream business 0, then a traditional wholesale price is not able to coordinate the supply chain because the government orders less, and  f N /θ n ∗ dG(y) = Dc . In addition, the supplier produces less than the optimum level 0 for δ > 0, if the government charges penalty Cs to the supplier due to late delivery, then it helps to coordinate the supply chain. Thus, the wholesale price contract with penalty cost due to shortage coordinates the supply chain and new objective functions become M F = E[cn + D(( f N /Y ) − n)+ − Pr f N + Cs ( f N − nY )+ ] and government objective function G F = E[sT ( f ) + Ca ( f N + δ( f N − nY )+ ) + Pr f N − Cs ( f N − nY )+ ]. Dai et al. (2016) consider demand uncertainty and propose a contract between the supplier and retailer to improve on-time delivery performance of influenza vaccine. First, they design the delivery time-dependent quantity flexibility (D-QF) contract and late rebate (LR) contract and observe that these contracts are not able to provide incentives and motivation to the supplier who bears the risk of the early production of influenza vaccine to improve on-time delivery. Then, the authors improve their model with buyback-and-late-rebate (BLR) contract, which assist to remove double marginalization. Suppose (1 − α) is the probability of late delivery, β is the probability of influenza vaccine strain similar to the composition approved by the authority, cr and ce are the unit production cost in regular and early production mode, ca is administrative cost, b is buyback credit, w is wholesale price, p is the selling price per unit to retailer, and ρ is the proportion of wholesale price that supplier rebates for late delivery, then the authors propose that BLR coordinates the supply chain if, w−ce ) ( p−w−ca )(ce −β cr ) and ρ ∗ = w(1−α){β . This contract integrates the properties b∗ = β(p(β p−ca )−ce ( p−ca )−ce } of buyback and late rebate contracts to provide credit for leftover inventory and rebate for late delivery.

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Arifoglu and Tang (2020) propose a two-sided incentive scheme to coordinate the influenza VSC. The model considers yield uncertainty from the supply side and positive and negative externalities from the demand side. Positive externalities are defined as factors that have a positive effect on society, for instance, an individual getting vaccination also has a benefit for the society by protecting them against the infection due to herd immunity. Likewise, negative externalities have a cost to society, for instance, any individual getting vaccinated reduces the chance of others getting vaccination and increases chances of infection due to the usage of a vaccine dose. In general, two sets of incentive programs are designed to coordinate the supplier and customer’s incentives with a social optimum level: (i) coordinating vaccination incentive r (Q r ) and (ii) coordinating transfer payment T (Q r ). It is observe that when vaccine supply is high, the customer’s demand in the decentralized supply chain is relatively low than the social optimum due to positive externalities, therefore the model works in a way that it generates a positive incentive (r (Q r ) > 0) to customers so that more customers can come for vaccination. When supply is low, the demand is more than the social optimum level due to negative externalities; therefore, to restrain the high vaccine demand and to ensure that the customer’s having high infection costs gets a vaccination, the model generates a negative vaccination incentive r (Q r ) < 0. To coordinate the transfer payment, a transfer amount of T (Q r ) is given to the supplier so that supplier produces an optimal production quantity Q c , which can help to coordinate the marginal benefits of the supplier and the society. Apart from the above-mentioned studies, other researchers also contribute to designing effective VSC contracts. For example, Yan et al. (2017) consider a health authority, vaccine supplier and population to propose a set of contracts to optimize the influenza VSC. First, the authors design a payment scheme contract in which the payment is made on the order quantity Q from the health authority. Such payment scheme contracts improve the efforts decisions of suppliers and health authorities. Next, they configure contracts in which extra installment is an element of the number of infections. Such contracts help healthcare organizations to minimize costs due to the payments based on infections, deaths, etc. Shamsi et al. (2018) develop an option contract for vaccines required in emergencies from two suppliers. The model plans to limit the acquirement and social costs utilizing the susceptible-infected-recovered (SIR) model. Mamani et al. (2012) consider an oligopoly market and design a subsidy contract for vaccine market coordination and socially optimal vaccine coverage. Mamani et al. (2013) study the the inefficiencies in allocation of influenza vaccines and design a game theoretical model to reduce inefficiencies. Yarmand et al. (2014) consider an infection outbreak and propose a two-phase vaccine allocation model to different locations to minimize vaccine production and administration cost. Adida et al. (2013) consider a monopoly business model for an imperfect vaccine with operational issues and system impacts and propose a two-section menu of endowments that prompts a socially effective degree of coverage. Chandra and Vipin (2021) propose a subsidy contract to coordinate the VSC of child immunization India.

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5 Supply Chain Distribution Supply chain distribution is one of the important phases in the VSC of developing countries. It determines how the vaccines should be distributed to the health centers, how many levels are required to distribute the vaccines, points of distributing the vaccines, how much inventory should be kept at each level, etc. (Lotty & van Jaarsveld, 2018). Since last-mile delivery of vaccines is one of the complex stages in the VSC, therefore, various issues often arise in delivering the vaccine during the endpoint. Tana et al. (2014) point out that issues such as handling of vaccines, cold chain monitoring, storage of vaccines are common issues faced by the immunization programs in vaccines distribution. Gupta et al. (2013) mention that in India the coordination issues in VSC have prompted issues concerning the supply chain and normal accessibility of immunizations supply at outreach centers. Their study also indicates that precise and timely forecasting of the materials required for immunizations is a significant bottleneck in the distribution of vaccines on time with required doses. As VSC in India comprises multiple levels and due to the large population and complex demography, the VSC issues are inevitable. Hence, the supply chain design in India should be robust that can deliver vaccines efficiently and effectively to the delivery locations. Thus, the supply chain designers can rethink of redesigning the existing VSC. For example, is it necessary to store vaccines at divisional vaccine stores as the vaccines can be supplied directly to district vaccine stores from state vaccine stores? In this regard, Chen et al. (2014) propose a planning model to improve the vaccine distribution performance in developing nations. Their model consists of various parameters such as demand of vaccines at each location, capacity at each stage, inventory lost at each location and doses administered. The objective function     in their model is defined as j nd j +  i j t ( pi,R j,t + pi,F j,t ). The first term in the expression ensures the number of full vaccination of children at a location j, while the second term represents i vaccines used from a refrigerator at location j in time t and likewise i vaccines used from a freezer at location j in time t. The model assists to capture the current practice of procurement and distribution of vaccines in developing nations and ensures that distribution efficiency is improved. This model has been adopted by the immunization programs of Niger, Vietnam and Thailand, and the implementation has shown positive results. Ng et al. (2018) design a multi-criterion approach to improve vaccine distribution efficiency by minimizing cost and reproduction number, improving social benefits of influenza vaccination. functions to minimize vaccination cost  of their  objective  One φ φ φ φ is formulated as g∈ G φ∈ q∈ Q g qg r g sg l gq . Here qg represents the number of φ

people that can be immunized by following a strategy φ, r g represents the costs of vaccination a single person in any risk group, sg is the percentage of people in risk φ group g in the entire population, and l gq is the percentage of immunized people in the risk group g by following a vaccination plan q under an allocation strategy φ. They consider the SIR model and divide the population into risk groups like seniors, infants, babies, etc. Allocation strategy is the type of vaccination strategy followed by the immunization program to vaccinate a group of peoples. For example, they may

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follow a different strategy to vaccinate infant who needs more precaution during vaccine administration as compared to senior citizens. The proposed model can help to reduce immunization costs by 10% than the average annual costs of immunization programs. Another instance of redesigning the VSC is in the case of an emergency or outbreak. Today, due to the rise in infectious diseases, it is imperative that the supply chain be adequately responsive so that the need for vaccines in outbreaks/ emergencies can be met with short delivery times. Therefore, the VSC in India should have the capacity and capability to fulfill the demand of vaccines of any state with a outbreak/emergency by their neighboring states. In such scenarios, the logistics of private companies may help distribute vaccines to the children. Yarmand et al. (2014) propose a two-phase vaccination policy model in case of an emergency or an outbreak. The model helps to minimize production costs by reducing the usage of vaccine doses. Such models should be designed in the scenario of India so that deaths due to outbreaks can be reduced. Yadav et al. (2014) suggest that integration of VSC with the supply chains of various health commodities may help to minimize costs incurred due to storage, transportation, warehousing and distribution. They propose that the VSC designers can integrate activities such as procurement, transport, storage and ordering with other health products supply chain to improve VSC efficiency and effectiveness. However, it requires proper planning at each level of the supply chain as various uncertainties may affect the overall performance of the VSC. Outreach can be another solution to improve distribution performance. In India, the majority of the population resides in the remote and rural areas. Therefore, due to transportation and other complexities, bringing children to immunization health centers becomes difficult for the parents. Hence, outreach can be an effective measure to distribute vaccines to the desired locations. Few studies propose mathematical models to address outreach location problems. For instance, Lim et al. (2016) develop various coverage models to find the optimal outreach location to improve vaccination coverage. In one of their binary coverage models, the problem is formulated as follows: A population residing in a radius of R will be vaccinated through outreach, and the remaining cannot be vaccinated through outreach. Let k is the total number of villages that will be covered by outreach, pi is the population in each village, ci is the cost at each outreach, di j is the distance between two villages i, j, N is maximum outreach permissible to set up, ai ∈ {0, 1}, where 1 belongs to the village i if it has been finalized as outreach else 0, bi ∈ {0, 1}, where 1 indicates if village i is covered else 0. So the objective n function is to maximize pi bi , with constraints the number ofpeople to be vaccinated through outreach i=1  n ai ≤ N . Hence, such as bi ≤ j∈ Si ai for Si = { j : di j ≤ R, j, i = 1, ..., n}, i=1 such coverage models can be helpful in UIP to cover more population in rural and remote areas. Inventory management is also important in the vaccine distribution phase. Proper inventory management ensures that vaccines do not go waste, and the demand can be fulfilled with the available stock. In this aspect, Hovav et al. (2015) propose a

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inventory model for influenza vaccine distribution. Tebbens et al. (2010) design an optimal vaccine inventory tool for the control of eradicated diseases.

6 Observations and Insights of the Study We perform a systematic literature review in VSC and observe that various contracts and distribution models have been proposed by various authors that can help to improve coordination and delivery performance. One major gap we identified that with respect to child immunization in India is that no such models have been developed to improve coordination and distribution. Therefore, by looking at the results of the VSC performance improvements of other nations through VSC contracts, it can be summarized that such contracts can surely help child immunization in India to optimize their supply chain performance. In VSC, as the supplier alone faces the risk of supply uncertainty, therefore, a contract in which the government is incentivizing the supplier in case of the overproduction can be helpful to achieve channel coordination. In this context, a subsidy contract in the presence of supply uncertainty can be helpful to the UIP India to coordinate the supply chain. In a subsidy contract, UIP may provide financial support to the supplier if the supplier’s production quantity is greater than the UIP order quantity. This can motivate the supplier to produce more and fulfill the order in each batch. Other contracts such as cost sharing will also be helpful to UIP India. A late rebate contract in which the supplier gives a rebate if the delivery period is late can be also appropriate in the VSC of UIP. UIP lacks on-time financial support from the government which affects the release in the purchase order; therefore, suppliers and UIP should also consider such aspects while designing the contracts. For example, the supplier may fulfill the demand of UIP even if payment is late and may charge a small amount of penalty if such a case happens. This may help the UIP to meet the child immunization demand. Contracts such as D-QF or option contract may pose some challenges in UIP due to the nature of the contracts. In D-QF, the retailer can return some leftover inventory to the supplier at a full price up to a certain level; therefore, in the case of child immunization, implementing the D-QF can be difficult because for child immunization vaccines there is less number of suppliers and returning the leftover stock with the full price condition may not be acceptable by the supplier due to factors such as supply uncertainty, high-order volume and low-profit margins. UIP may also consider some distribution models to tackle uncertainty such as COVID-19 and other outbreaks. For example, due to COVID-19, immunization rates have been affected drastically in major states of India. Therefore, models in which such disruptions can be handled will be really helpful. Models with outreach centers may help to improve immunization rates in India.

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7 Discussion and Implications of the Study An efficient and effective VSC can help improve child immunization program performance; however, relatively low volume of literature on VSC highlights the fact that the field is still to gain wide popularity among the researchers. In this paper, we perform a thorough literature review on VSC and identify that only 263 documents in four international citation indexes since 1965 have addressed the issue of VSC in their study. We also report that only 18 papers targeted VSC of India—a low volume of research on the child immunization program performance of India, indicating that a significant research opportunities exists in VSC especially in the context of ongoing pandemic. From Table 3, it can be observed that 18 documents, which focus on VSC of India, have primarily focused on performance measurement, vaccine wastage, economic characteristics, vaccine introduction, etc., however important aspects such as vaccine demand forecasting and shortages have not been addressed in any of the studies. Hence, researchers and academicians can focus on addressing forecasting and shortage issues in VSC to assist UIP to improve vaccine availability. The literature review reveals that the primary focus of researchers is to study vaccines used for any age group, for illustration, Lin et al. (2020) study the cold chain transportation decision improvements in VSC, and Hovav et al. (2015) design a network flow model for inventory management of influenza vaccine and distribution. Only a few studies focused on basic vaccines and their modeling and analysis to improve child vaccination performance. For instance, Chandra and Kumar (2021) evaluate the effect of KPIs of the VSC on the sustainable development of a health operation of UIP India as Mission Indradhanush, Chen et al. (2014) study the planning model for the WHO-EPI vaccine, and Tebbens et al. (2010) study the optimal polio vaccine stockpile design. Focusing on other aspects such as decision-making problems in VSC, Arji et al. (2019) report that around 40 papers (2005–2019) employ the fuzzy concept as a decision-making tool in human infectious diseases, but in the field of VSC for child immunization in developing countries like India, only a few studies have considered the application of fuzzy for decision-making such as Chandra et al. (2018) and Chandra and Kumar (2019). We further discuss the importance of coordination in the supply chain and how different contracts achieve coordination in different VSC settings. As compared to the immunization programs of developed nations, coordination of VSC is not been given substantial importance by the UIP India. Establishing coordinating contracts in the vaccination programs can reduce cost of the supply chain and improve vaccines delivery performance. We also discuss various issues in VSC distribution in, India and how a well-designed distribution channel can help to improve VSC performance.

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8 Conclusions, Limitations and Future Scope In this study, we review the literature on VSC and investigate the literature addressing the supply chain of vaccination programs in India. We identify 263 documents, of which, 18 concentrated on VSC of India. VSC is undoubtedly one of the topics in which a significant amount of research gaps exists. India runs the largest child immunization program in the world reinforcing the need to investigate the issues in vaccine supply chain provide solutions. Further, we provide comprehensive review on the models on supply chain coordination and supply chain network design for vaccines in a global setting. In the current COVID-19 scenario with multiple organizations focusing on developing vaccines, governments need to ensure efficient and effective supply chains of to be available vaccines by ensuring proper contractual terms and optimal network design for the supply chain. One of the major limitations of our study is that the research papers are drawn from SCIE, SSCI, A&HCI and ESCI indexed journals. Future research review can consider other indexed journals and look into other issues and aspects of VSC than the contract design and network design. Based on the analysis of literature on VSC, we find the need for better modeling and analysis of VSC considering the region-specific and product-specific characteristics. Incorporating the stochastic nature of different parameters such as demand and supply is essential in designing the VSC distribution network. Future research can look into the different supply and demand uncertainties to be addressed while designing contracts for VSC coordination. SIR modeling approach to better capture the demand for vaccines can be considered as an extension to the proposed models. Acknowledgements The authors would like to thank all the anonymous reviewers for the constructive comments and insightful suggestions to improve the quality of the paper.

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Index

A Agri-food supply chain, 235, 244 Analytical model, 90, 91, 93, 95, 234, 235 Artificial intelligence, 120, 141, 162, 164, 177 Artificial neural network, 159, 161, 162

B Backorder cost, 6, 7, 19 Base-stock level, 68, 94 Behavioral experiments, 237, 238, 240 Behavioural operations management, 233– 235, 244 Bound-based sampling heuristic, 105 Buyer-supplier relationship, 235, 237, 238, 242

C Cancer diagnosis, 156 Capacity, 23–26, 28–35, 45, 47, 50, 51, 53, 94, 207, 242, 253, 262, 263 Centralized supply chain, 91, 98, 100 Chebyshev’s inequality, 38 Cold chain transportation, 265 Comparative evaluation, 16 Confidence ellipsoids, 34 Consistency index, 187 Continuous-review policy, 16, 93 Contract mechanism, 89, 91, 92 Cost-sharing contract, 259 Credibility index, 23, 25, 34, 35 Credibility test, 23, 25, 27, 30, 31, 40

Cumulative distance, 51

D Decentralized supply chain, 91, 97, 261 Deterministic heuristic, 119–122, 124, 127 Digital technology, 226, 228, 229 Discount rate, 23, 25, 35 Discrete transportation lead time, 65 Distribution, 16, 25, 29, 30, 32, 39, 44–46, 53, 90, 96, 189, 205, 208, 212, 236, 252, 253, 255–257, 262–266 Dummy node, 46, 47

E Electronic data interchange, 44 Enterprise resources planning, 218 E- procurement, 220, 222, 224, 228 Equilibrium, 23, 25, 28, 31, 35, 38, 238 Expected payoff, 33 Expected sales, 95, 96, 99, 100

F Finite horizon, 43–46 Finite-time horizon, 58 Fixed cost, 47, 58, 112 Flowshop, 119–121, 124, 127 Forecast information sharing, 23, 27, 29, 32 Forecast matrix, 32 Forward reduction technique, 127–129

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 B. Vipin et al. (eds.), Emerging Frontiers in Operations and Supply Chain Management, Asset Analytics, https://doi.org/10.1007/978-981-16-2774-3

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270 G Game theory, 25, 237–239 Game-theoretical model, 29, 261 Genetic algorithm, 4, 90, 91, 93, 104–106, 111, 117, 204 Global reporting initiative matrix, 159, 164, 165, 167, 176 Goodwill cost, 33 Green supply chain management, 181

H Hamiltonian cycle, 46 Hamiltonian path, 46 Health care management, 243 Heuristic, 43–45, 54, 57–62, 90, 91, 93, 104, 105, 112, 114, 119, 120, 122, 124, 235 Holding cost, 4, 6, 8, 12, 15, 16, 19, 44, 46, 47, 53, 94 Hotelling’s T 2 , 31, 33 Humanitarian operations, 243 Human resource allocation problem, 204 Hybrid inventory ordering policy, 5, 20

I Immunization health center, 253, 263 Infinite horizon, 44 Influenza vaccine, 259–261, 264, 265 Information and communication technology, 217, 228 Information asymmetry, 28, 29, 259 Information quality, 219, 222, 224, 225 Information sharing, 24–29, 40, 41, 93, 218– 222, 224, 225, 228, 229, 237, 239, 240 Information technology, 25, 201 Internal Lean practices, 219, 222, 224, 225, 228, 229 Inventory, 3–8, 10–20, 24, 29, 40, 43–48, 50, 51, 53, 57, 58, 60, 62, 93, 94, 119, 120, 225, 227, 235, 237–239, 245, 260, 262–265 Inventory replenishment, 44, 45 Inventory routing problem, 43–45, 51, 54, 60 Inverse reduction technique, 128, 129

J Job-index based insertion scheme, 119

Index Jobshop, 119–121, 124–129 Joint economic lot size, 66

K K-means algorithm, 142, 155

L Lead-time, 4, 24, 31, 40, 68, 94 Lean supply chain, 177 L’Hopital rule, 36, 38 Linear programming, 139, 142, 144, 156 Linear regression, 174 Logistics, 140, 143, 146, 148, 149, 167, 173, 177, 189, 203, 218, 224, 228, 235–237, 239, 241, 263 Logistics management, 235–237, 239

M Machine learning, 139, 141, 142 Machine-routing, 119–121, 123, 124, 127– 130 Manufacturing line, 119–122, 124, 127–130 Manufacturing resources planning, 218 Manufacturing SME, 217, 219, 221 Material requirement planning, 218 Mathematical models, 4–6, 8, 12, 14, 16, 20, 43, 45, 203, 239, 263 Meta-heuristic, 91, 93, 104, 105, 112, 113, 116, 117, 120 Mixed integer linear programming, 3, 43, 44, 46, 51 Mixed integer non-linear programming, 65, 69 Multi-criteria decision-making, 161, 165, 181, 183, 184, 186, 193 Multi-echelon supply chain, 62, 92 Multilayer perceptron, 159, 163, 164 Multi level supply chain, 92, 93 Multiple linear regression, 139, 142, 144, 156 Multi-retailer, 40 Multivariate statistics, 23, 25

N Nash equilibrium, 26, 91 Nearest neighbourhood algorithm, 58 No split-delivery, 44, 45, 51, 62

Index NP-hard, 43, 44, 54, 62, 120

O On-hand inventory, 7, 10, 11, 46 On-order inventor, 7, 11 Operations research, 142, 235, 236 Optimal allocation, 204, 207 Optimal order quantity, 91, 92, 95–100 Ordering cost, 4–8, 10, 15, 16, 19, 46, 47, 50, 58 Order quantities test, 30 Order-up-to-level, 6, 12, 13, 19 Outsourcing, 220, 222, 224, 228, 239

P Parallel simulated annealing, 105, 108, 110, 111 Parallel weighted genetic algorithm, 106 Pareto optimal, 23, 25, 26 Pearson correlation coefficient, 159, 174 Perfect public equilibrium, 23, 25, 35 Periodic-review order-up-to policy, 3–5, 16, 20 Petroleum industry, 181–185, 189, 192, 193 Pipeline inventory, 7 Positive externalities, 261 Postponement, 219, 220, 222, 224, 225, 228, 229 Price-dependent stochastic demand, 65, 67 Price-sensitive stochastic demand, 67 Process sequence, 119 Profit margin, 70, 264 Project management, 235, 236, 244 Proportion of periods test, 30 Punishment phase, 25, 28, 35

Q Quantitative modelling, 237, 238 Quantity discount, 92 Quantity flexibility, 92, 260

R Regression analysis, 174, 178 Relational contract, 24, 25, 28, 29, 40, 41 Renewable energy, 170, 177, 181, 183, 192– 194 Reorder point, 6

271 Repeated game, 23–26, 28, 31, 35, 38, 40 Replenishment, 6–8, 10, 15, 16, 26, 43–45 Replenishment cycle, 69, 70, 72 Resource allocation, 202–204, 210, 213, 245 Review length, 39, 40 Review strategy, 23, 27–30, 35, 37, 39, 40

S Safety factor, 65, 70, 78, 81–83 Safety stock, 220, 222, 228 Sales rebate contracts, 89, 90, 92, 93, 96, 100, 101, 104, 113, 114, 116 Scheduling, 93, 119, 203, 204 Service integrator, 242 Service level constraint, 20 Service provider, 204–206, 241–243 Service supply chain, 235, 241, 242 Setup cost, 70, 74, 75 Shortage cost, 5, 8, 12, 15, 16, 94 Simulation, 201, 203, 207, 210, 211, 213 Simulation-based optimization, 207, 210 Staff allocation, 203, 212 Statement of work, 204, 212 Strategic collaboration and lean practices, 220, 222 Strategic planning, 220, 221, 223–225, 229 Subcontracting, 220, 223, 224, 228 Sub tour elimination, 50, 51 Supersequence, 119–121, 123, 127–130 Supply chain, 3–6, 8, 12–20, 23, 24, 26–31, 40, 41, 43, 44, 46, 50, 60, 62, 89– 102, 104–106, 111–116, 160, 161, 164, 165, 174, 182, 217–222, 226– 229, 234–242, 244, 245, 251, 252, 254–256, 258–260, 262–266 Supply chain coordination, 43, 90, 91, 93, 96, 100, 116, 252, 256, 266 Supply chain management, 24, 26, 90, 119, 184, 233, 235–239 Supply chain management practice, 181, 226 Supply chain performance, 159–161, 164– 166, 174, 217, 239, 252, 264 Support vector machine, 142 Sustainability, 160, 162, 165, 177, 182–184, 189, 193, 219, 244 Sustainable supply chain management, 160, 163

272 T Transportation cost, 44–46, 50, 53, 58, 60, 94 Travelling salesman problem, 54, 57, 58 Trigger strategy, 30, 39 Truthful information sharing, 24, 29, 40, 41 Two-pair adjacent interchange scheme, 119, 127, 129 Two-sided incentive scheme, 261

Index U Universal immunization program, 251

V Vaccine supply chain, 251, 266 Variance inflation factor, 140, 143, 145 Vendor managed inventory, 43, 93