Labor and Supply Chain Networks 3031208544, 9783031208546

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Springer Optimization and Its Applications 198

Anna Nagurney

Labor and Supply Chain Networks

Springer Optimization and Its Applications Volume 198 Series Editors Panos M. Pardalos , University of Florida My T. Thai , University of Florida Honorary Editor Ding-Zhu Du, University of Texas at Dallas Advisory Editors Roman V. Belavkin, Middlesex University John R. Birge, University of Chicago Sergiy Butenko, Texas A&M University Vipin Kumar, University of Minnesota Anna Nagurney, University of Massachusetts Amherst Jun Pei, Hefei University of Technology Oleg Prokopyev, University of Pittsburgh Steffen Rebennack, Karlsruhe Institute of Technology Mauricio Resende, Amazon Tamás Terlaky, Lehigh University Van Vu, Yale University Michael N. Vrahatis, University of Patras Guoliang Xue, Arizona State University Yinyu Ye, Stanford University

Aims and Scope Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences. The series Springer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS.

Anna Nagurney

Labor and Supply Chain Networks

Anna Nagurney Isenberg School of Management University of Massachusetts Amherst, MA, USA

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-031-20854-6 ISBN 978-3-031-20855-3 (eBook) https://doi.org/10.1007/978-3-031-20855-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to essential workers and to those fighting for freedom in Ukraine, with deep gratitude for your heroism.

Preface

Supply chains have become front page news in the past several years driven, in large part, by disruptions in the COVID-19 pandemic, which have led to various shortages of products as well as delays in their deliveries. In addition to the pandemic, climate change, conflicts, and wars are further exacerbating supply chain problems. Behind supply chains, whether those associated with your favorite food products, or life-saving medicines and supplies, or high tech products that make innovation and connectivity possible, are workers. This book was written to bring the laborers behind our supply chain networks to the forefront. In this book, I explore prominent issues concerning supply chain networks and labor theoretically, qualitatively, and computationally, with the support of numerous case studies based on solved numerical examples. The supply chain network models with labor range from perishable product food ones to models for blood service organizations and disaster management. The network topologies of the various problems provide graphical representations, and the underlying behavior of the cognizant decision-makers and stake-holders is captured along with their interactions. Both optimization models and game theory models are constructed with insights as to the impacts of various bounds on labor availability as well as wages. A plethora of disruptions is considered and metrics for supply chain network resilience introduced. For easy reference, this book includes an Appendix that provides the foundations of the underlying mathematical methodologies used in this book. The audience for this book includes researchers, practitioners, students, and policy makers, in different disciplines, such as operations research and economics, who are interested in wide-ranging issues surrounding supply chain networks and labor that the world is now facing. I hope that this book demonstrates that workers must be valued, safeguarded, and compensated for their labor appropriately. Amherst, MA, USA September 2022

Anna Nagurney

vii

Acknowledgments

This book is the product of years of rumination and research with a heightened focus on the book’s theme in the pandemic because of the numerous supply chain disruptions and the escalation of the need for labor in many sectors from agriculture to healthcare. The book would not have been possible without the assistance and encouragement of many organizations and individuals. I acknowledge the Isenberg School of Management Dean Anne P. Massey for her leadership, support, and understanding. I thank my colleagues in the Department of Operations and Information Management at the Isenberg School at the University of Massachusetts Amherst for their collegiality as well as the Department Chairs during the preparation of this book, especially Professors Bob Nakosteen and Senay Solak. I also thank the vice provost for global affairs at the University of Massachusetts Amherst, Kalpen Trivedi, for his excellent leadership of international programs and support of international partnerships. The issues considered in this book are global in nature, and my work with the Kyiv School of Economics (KSE) in Ukraine over the past few years, as a member of both its Board of Directors and its International Academic Board, has enriched my connections to Ukraine. Both my parents were born in Ukraine and left their birthplaces as refugees during WWII. The determination, stamina, courage, and creativity of educators in Ukraine during Russia’s unprovoked war against it are a lofty bar that I aspire to attain. My colleagues at KSE are heroes, continuing to teach, to conduct research, to advise the government, and to inform the world. I am deeply honored to now be serving as a co-chair of the Board of Directors of KSE. I thank the president of KSE, Tymofiy Mylovanov, and the rector of KSE, Tymofii Brik, plus KSE Professors Oleg Nivievskyi and Nataliia Shapoval for their collegiality and impact. Special thanks to my collaborators on problems related to supply chains and networks: Professor Ladimer S. Nagurney of the University of Hartford, Professor Patrizia Daniele of the University of Catania in Italy, Professor Tina Wakolbinger of the Vienna University of Economics and Business in Austria, Professor Deniz Besik of the University of Richmond, Professor Sara Saberi of the Worcester Polytechnic Institute, Professor Pritha Dutta of Pace University, and Professor ix

x

Acknowledgments

Mojtaba Salarpour of Texas A&M Commerce. This past year, we marked the 20th year of the founding of the Virtual Center for Supernetworks, which I have directed since its founding. I am proud of the Center Associates who continue to push the boundaries of research frontiers. I thank my current doctoral student Dana Hassani for his careful reading of earlier versions of this book and for his conscientious attention to detail. I am grateful to Professor Panos M. Pardalos of the University of Florida and to Professor Ilias S. Kotsireas of Wilfrid Laurier University in Canada for our wonderful collaborations in organizing several Dynamics of Disasters conferences and for co-editing with me the conference proceedings. Many thanks also to all my colleagues and friends around the globe who provided support, whether virtual or in person, over the past several very challenging years for our planet. Your stories and humor provided sustenance when it was badly needed. As an academic, I find teaching energizing and extremely rewarding. I thank all my students at the University of Massachusetts Amherst who stayed with me, whether I instructed via Zoom or, more recently, in person. Ideas generated through discussions and reflections in my classes on transportation and logistics, humanitarian logistics and healthcare, and the graduate seminar on networks, game theory, and variational inequalities continue to inspire and to sustain. Special appreciation to the guest lecturers who shared their experiences on healthcare, emergency preparedness, and disaster response. Since the onset of the pandemic, I have responded to numerous requests from journalists and the media because of the growing recognition of the importance of supply chains by the public and by government decision makers and policy makers. The journalists stimulated many research ideas, and I am deeply grateful to them for the intellectual exchanges and for keeping the world informed, including now with Russia’s war against Ukraine. I acknowledge the professional society of INFORMS (Institute for Operations Research and the Management Sciences) and its staff for creating networking and educational opportunities during the pandemic and also for publicizing the research and the practical importance of the research findings of its members. Many thanks to Elizabeth Loew of Springer, the publisher of this book, for her shepherding of this book project, and to the three anonymous reviewers who provided constructive comments. I am forever grateful to my family for their patience and understanding throughout this book project. Finally, my sincerest thanks to all those who provide essential products and services and to those fighting for the freedom of Ukraine and the world.

Contents

Part I Labor and Supply Chains 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Organization of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4

2

Perishable Food Supply Chain Networks with Labor . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Perishable Food Supply Chain Network Model with Labor . . 2.2.1 Variational Inequality Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Illustrative Examples 2.1 and 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Food Supply Chain Network Numerical Examples. . . . . . . . 2.4 Summary, Conclusions, and Suggestions for Future Research. . . . . 2.5 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 10 15 16 19 20 27 28 29

3

Optimization of Supply Chains Under Different Labor Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Supply Chain Network Models with Labor . . . . . . . . . . . . . . . . . . . . 3.2.1 Variational Inequality Formulations of the Elastic Demand Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Variational Inequality Formulations of the Fixed Demand Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Scenario 1 Healthcare Product Supply Chain Elastic Demand Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Scenario 3 Healthcare Product Supply Chain Elastic Demand Examples: Reduction of Labor Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 36 42 44 46 48 49

53 xi

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3.4

Supply Chain Network Efficiency and Resilience to Labor Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Efficiency of a Supply Chain Network and Importance Identification of a Network Component . . . . . . 3.4.2 Resilience with Respect to Labor Disruptions . . . . . . . . . . . . . 3.4.3 Supply Chain Network Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary, Conclusions, and Suggestions for Future Research. . . . . 3.6 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Game Theory Modeling of Supply Chains and Labor Disruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Supply Chain Network Game Theory Modeling Under Labor Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Governing Equilibrium Conditions and Variational Inequality Formulations . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Scenario 1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Scenario 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Supply Chain Network Economy Efficiency and Importance Identification of Components. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary, Conclusions, and Suggestions for Future Research. . . . . 4.6 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 56 59 60 61 65 65 68 73 77 78 85 87 88 89 89

Part II Endogenous Wages and Productivity Investments 5

6

Wages and Labor Productivity in Supply Chains with Fixed Labor Availability on Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Supply Chain Network Game Theory Models with Wage-Responsive Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Model Without Wage Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Model with Wage Bounds plus Lagrange Analysis . . . 5.3 The Algorithm and Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 High Value Supply Chain Numerical Examples . . . . . . . . . . . 5.4 Summary, Conclusions, and Suggestions for Future Research. . . . . 5.5 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 98 99 102 106 108 118 119 119

Wage-Dependent Labor and Supply Chain Networks . . . . . . . . . . . . . . . . . 121 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2 The Supply Chain Network Game Theory Models with Wage-Dependent Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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6.2.1

Equilibrium Conditions and Variational Inequality Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Algorithm and Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Numerical Results for a Single Firm . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Numerical Results for Multifirm Examples . . . . . . . . . . . . . . . . 6.4 Summary, Conclusions, and Suggestions for Future Research. . . . . 6.5 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8

Investments in Labor Productivity: Single Period Model . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Labor Productivity Investment Supply Chain Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Lagrange Analysis and Alternative Variational Inequality Formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Alternative Variational Inequality Formulations . . . . . . . . . . . 7.3.2 Additional Lagrange Analysis with Interpretations. . . . . . . . 7.4 Computational Procedure and Numerical Examples . . . . . . . . . . . . . . . 7.4.1 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary, Conclusions, and Suggestions for Future Research. . . . . 7.6 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiperiod Supply Chain Network Investments in Labor Productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Multiperiod Supply Chain Network Optimization Model with Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Algorithm and Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Examples 8.1, 8.2, and 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Summary, Conclusions, and Suggestions for Future Research. . . . . 8.5 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126 129 130 138 144 145 145 149 149 152 157 159 161 163 163 164 176 178 178 181 181 184 188 190 197 198 199

Part III Advanced Supply Chain Network from Profit to Non-Profit Organizations 9

Multitiered Supply Chain Networks with Labor . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The Multitiered Supply Chain Network Equilibrium Model with Labor and No Bounds on Labor Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 205

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9.2.2

The Multitiered Supply Chain Network Equilibrium Model with Labor and Link Bounds on Labor Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Algorithmic Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Multitiered Supply Chain Numerical Examples . . . . . . . . . . . . . . . . . . . . 9.5 Summary, Conclusions, and Suggestions for Future Research. . . . . 9.6 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

11

12

215 217 220 227 228 229

International Migrant Labor and Supply Chains. . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Supply Chain Network Model with Investments in Attracting Migrant Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 High Value Food Product Numerical Examples . . . . . . . . . . . . . . . . . . . . 10.5 Summary, Conclusions, and Suggestions for Future Research. . . . . 10.6 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233 233

Labor and Blood Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Supply Chain Network Model of the Blood Service Organization with Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Summary, Conclusions, and Suggestions for Future Research. . . . . 11.5 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265

Disaster Management and Labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Multiproduct Supply Chain Network Model with Labor for Disaster Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Algorithm and Humanitarian Organization Supply Chain Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Summary, Conclusions, and Suggestions for Future Research. . . . . 12.5 Sources and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 283

237 245 247 261 262 262

268 278 279 280 281

286 292 298 299 299

Glossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 A

Optimization Theory, Variational Inequalities, and Game Theory . . . A.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Karush–Kuhn–Tucker Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . A.3 Variational Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Qualitative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

305 305 308 312 315

Contents

A.4

xv

The Relationships Between Variational Inequalities and Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 A.4.1 An Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

List of Figures

Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 5.1 Fig. 5.2 Fig. 5.3

Fig. 5.4

Fig. 5.5

Fig. 6.1

The perishable food supply chain network topology . . . . . . . . . . . . . . . Supply chain network topology for two illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for the numerical Example 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for Example 2.6 . . . . . . . . . . . . . . . The supply chain network topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for the illustrative examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for Examples 3.1 and 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for Examples 3.3 and 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology of the game theory model with labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for Examples 4.1–4.3 . . . . . . . . The supply chain network topology for the numerical Examples 4.4 through 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for the numerical examples. . . Sensitivity analysis for different wage bounds on the supply chain networks of both firms and effects on the firms’ profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity analysis for different wage bounds on the supply chain networks of both firms, with firm 1 having a lower bound than firm 2, and effects on the firms’ profits . . . . . . . . . . Sensitivity analysis for different wage bounds on the supply chain networks of both firms, with firm 1 having a higher bound than firm 2, and effects on the firms’ profits . . . . . . . . . The supply chain network topology of the Game theory models with wage-dependent labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 17 21 26 37 44 49 51 68 79 82 98 108

113

117

117 124 xvii

xviii

Fig. Fig. Fig. Fig.

List of Figures

6.2 6.3 6.4 6.5

Fig. 6.6 Fig. 7.1 Fig. Fig. Fig. Fig.

7.2 7.3 7.4 8.1

Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5

Supply chain network topology for Examples 6.1 and 6.2 . . . . . . . . . Supply chain network topology for Examples 6.3 and 6.4 . . . . . . . . . Supply chain network topology for Examples 6.5, 6.6, and 6.7 . . . . The supply chain network topology for the numerical Examples 6.8 through 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity analysis for different wage ceilings and effects in the firms’ profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology for optimization of product path flows and investments in labor productivity . . . . . . . . . . Supply chain network topology for Examples 7.1, 7.2, and 7.3 . . . . Supply chain network topology for Examples 7.4, 7.5, and 7.6 . . . . Supply chain network topology for Examples 7.7, 7.8, and 7.9 . . . . The supernetwork structure of the multiperiod supply chain network model with labor productivity enhancements . . . . . . . Supernetwork topology of the multiperiod supply chain network numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity analysis for different discount rates r and effects on the firm’s NPV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity analysis for different values of α for managerial links o and n and effects on the firm’s NPV . . . . . . . . . . . . Sensitivity analysis for different values of factor  preceding a∈Lˆ va1 expression in all the demand price

130 133 136

functions and effects on the firm’s NPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The supply chain network topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supply Chain Network for Numerical Examples 9.1 and 9.2 . . . . . . Supply chain network for numerical Examples 9.3 and 9.4 . . . . . . . . The supply chain network topology for the model . . . . . . . . . . . . . . . . . . The supply chain network topology for the numerical examples. . . Effect on profit of the firm when the demand price function intercepts are doubled, tripled, and so on, with Example 10.1 being the baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect on optimal total labor hours of domestic labor and of international migrant labor in the supply chain network when the demand price function intercepts are doubled, tripled, and so on, with Example 10.1 being the baseline. . . . . . . . . . . Blood service organization supply chain network topology . . . . . . . The supply chain network topology of the humanitarian organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The humanitarian organization supply chain network topology for the numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric Interpretation of VI(F, K ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 206 221 224 237 248

139 144 153 165 169 174 185 189 195 196

1

Fig. Fig. Fig. Fig. Fig. Fig.

9.1 9.2 9.3 10.1 10.2 10.3

Fig. 10.4

Fig. 11.1 Fig. 12.1 Fig. 12.2 Fig. A.1

257

257 269 287 294 313

List of Tables

Table 2.1 Table 2.2

Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 4.1 Table 4.2 Table 4.3 Table 5.1 Table 5.2 Table 5.3 Table 6.1 Table 6.2 Table 6.3

Notation for the perishable food supply chain network model with labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decay rate, time duration, productivity factor, throughput, hourly wage, labor link bound, operational cost, and discarding cost functions for Example 2.1 . . . . . . . . . . . . . . Notation for the supply chain network models with labor . . . . . . . . Optimal product flows for Examples 3.5–3.7 representing Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal link labor values for Examples 3.5–3.7 representing Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Efficiency and resilience measures for Examples 3.8 through 3.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the supply chain game theory modeling framework with labor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium product path flows for Examples 4.7 through 4.11 representing Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link labor values for Examples 4.7 through 4.11 representing Scenario 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the models with wage-dependent labor productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows, Lagrange multipliers, and hourly wages for Examples 5.1 through 5.4 . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows, Lagrange multipliers, and hourly wages for Examples 5.9 and 5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the supply chain game theory model with wage-dependent labor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows, labor values, and hourly wages for Examples 6.1 and 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows, labor values, and hourly wages for Examples 6.3 and 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

22 39 54 54 58 69 85 86 99 112 116 125 132 135

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xx

Table 6.4 Table 6.5 Table 7.1 Table 7.2 Table 7.3 Table 7.4 Table 7.5 Table 7.6 Table 7.7 Table 8.1 Table 8.2 Table 8.3 Table 10.1 Table 10.2 Table 10.3 Table 10.4 Table 10.5

Table 10.6 Table 10.7

Table 10.8 Table 10.9

Table 11.1

List of Tables

Equilibrium link flows, labor values, and hourly wages for Examples 6.5, 6.6, and 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows, labor values, and hourly wages for Examples 6.8 through 6.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the supply chain network models with labor . . . . . . . . Equilibrium link flows and labor values for Examples 7.1, 7.2, and 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link productivity investments and hourly wages for Examples 7.1, 7.2, and 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows and labor values for Examples 7.4, 7.5, and 7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link productivity investments and hourly wages for Examples 7.4, 7.5, and 7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link flows and labor values for Examples 7.7, 7.8, and 7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link productivity investments and hourly wages for Examples 7.7, 7.8, and 7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the multiperiod supply chain network model . . . . . . . . Equilibrium link flows and labor values for Examples 8.1, 8.2, and 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium link productivity enhancements and hourly wages for Examples 8.1, 8.2, and 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the supply chain network model with international migrant labor—parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the supply chain network model with international migrant labor—variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the supply chain network model with international migrant labor—functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal link flows and domestic and international migrant labor values for Examples 10.1, 10.2, 10.3 . . . . . . . . . . . . . . . Optimal link international migrant attraction investments and domestic labor bound Lagrange multipliers for Examples 10.1, 10.2, and 10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal link flows and domestic and international migrant labor values for Examples 10.4 through 10.7 . . . . . . . . . . . . Optimal link international migrant attraction investments and domestic labor bound Lagrange multipliers for Examples 10.4 through 10.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal link flows and domestic and international migrant labor values for Examples 10.8 and 10.9 . . . . . . . . . . . . . . . . . Optimal link international migrant attraction investments and domestic labor bound Lagrange multipliers for Examples 10.8 and 10.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation for the blood services supply chain network model—parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

138 141 154 168 169 172 173 176 177 186 193 194 238 239 239 251

252 255

256 259

260 270

List of Tables

Table 11.2 Notation for the blood services supply chain network model—variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 11.3 Notation for the blood services supply chain network model—functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 12.1 Definition of links, the link upper bounds, and associated total cost, and other functions for Examples 12.1 . . . . . . . . . . . . . . . . . Table 12.2 Optimal link flows and Lagrange multipliers for Examples 12.1, 12.2, and 12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part I

Labor and Supply Chains

Chapter 1

Introduction

Abstract In this chapter, the background and motivation for this book are provided, along with the organization of the book into parts and chapters.

1.1 Background and Motivation People, and, in particular, workers, are essential to supply chain networks that form the critical infrastructure for the functioning of our economies. Without workers, crops cannot be planted and harvested; products from Personal Protective Equipment (PPE) to life-saving medicines and vaccines cannot be manufactured, and products from consumer goods to high tech products, including computer chips, cannot be produced. Furthermore, labor is needed for the transportation of products and for their storage and ultimate distribution to demand markets. Labor is also essential for healthcare provision and other service industries and critical in all phases of disaster management. The COVID-19 pandemic, coupled with the impacts of climate change, plus conflicts and wars, including Russia’s war against Ukraine, which commenced on February 24, 2022, has vividly and dramatically demonstrated our dependence on the effective and efficient functioning of supply chain networks, which, in turn, requires labor availability and productivity. This challenging global scenario provides strong evidence for the need for a holistic, system-wide perspective for the modeling, analysis, and solution of supply chain problems that include the labor resources. The purpose of this book is to construct such a formalism through the prism of networks, which yields a graphic representation of supply chains, consisting of multiple stakeholders. The book considers both single decision-makers and multiple decision-makers engaged in supply chain activities of production, transportation, storage, and distribution and captures their behavior and interactions. Importantly, it handles many realistic constraints faced by firms today, as they seek to produce and deliver products, while dealing with competition, various constraints on labor, a variety of disruptions, labor shortages, challenges associated with proper wage determination, plus the computation of optimal investments in labor productivity subject to budget constraints. Such issues are also highly relevant © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_1

3

4

1 Introduction

now due to climate change, as workers deal with more extreme weather and work conditions, with the agricultural sector being a prominent example. Not only are profit-maximizing firms considered in this book (with relevant associated issues such as waste management in the case of the food sector, for example) but also non-profits, as in blood services as well as in humanitarian organizations engaged in disaster relief. This book provides prescriptive suggestions for ameliorating negative impacts of labor disruptions and demonstrates the benefits of appropriate wage determination. It also describes how the integration of international migration networks can assist in reducing labor shortages in supply chains and the fair wages that should be paid. Importantly, it provides models of multitiered supply chains with labor since a disruption in a particular tier can have propagating impacts on other tiers. Finally, the impacts of policy interventions, in the form of wage bounds, and their ramifications, in terms of the volume of attracted labor, product prices, product volumes, as well as profits are explored. The book is filled with many network figures, graphs, and tables with data, both input and output. This book is inspired by the incredible contributions, selflessness, and support of essential workers in the pandemic from farmers, freight service providers, and supermarket workers to healthcare workers and educators, whose dedication is deeply appreciated. They, along with all those living, struggling, and working in regions affected by conflicts and wars, as well as climate change, represent the faces of humanity that await peace and prosperity for all. In the end, it is really all about people and our planet.

1.2 Organization of This Book This book is organized into three parts, with each part consisting of four chapters, for a total of twelve chapters. Each chapter is as self-contained as feasible with Sources and Notes as well as references included. This book is based on refereed publications and on new results, not previously reported. The notation is standardized and references are updated, with the existing global scenario serving as a backdrop to bring currency to the insights gained from both the qualitative analyses and the numerical supply chain network examples with labor herein. Part I, Labor and Supply Chains, includes this chapter on background and motivation as well as a series of chapters of increasing generality and a spectrum of constraints on labor. Chapter 2 focuses on a perishable product supply chain network optimization model for food with bounds on labor on supply chain links. The model is a generalized network model with arc multipliers associated with the links. Chapter 3 presents supply chain network optimization models, with applications to healthcare products, under three distinct sets of labor constraints. Both elastic demand and fixed demand formulations are given. In addition, a supply chain network performance measure is introduced along with the quantification of resilience with respect to labor availability and productivity. Chapter 4 develops

1.2 Organization of This Book

5

a game theory supply chain network model under three distinct sets of labor constraints. Supply chain network efficiency in a game theory context is also discussed. The numerical examples are drawn from a food application. Part II, Endogenous Wages and Productivity Investments, formulates supply chain network models that capture highly relevant issues such as the wages that should be paid to workers as well as the impacts of investments in labor productivity. Chapter 5 presents a game theory model in which the amounts of labor on supply chain network links are fixed and wages are endogenous. A model without wage bounds as well as one with wage bounds is constructed. Numerical examples are solved for a high value product. Chapter 6 develops a supply chain network game theory model with labor being wage-dependent. Chapters 7 and 8, in turn, present supply chain network optimization models, single period and multiperiod ones, respectively, in which investments in labor productivity are captured. Labor availability is responsive to wages. Numerical examples in each of these chapters yield valuable managerial insights and also prescriptions for decision-makers and policy. Part III, Advanced Supply Chain Networks from Profit to Non-Profit Organizations, presents additional novel models that include labor. Chapter 9 proposes multitiered supply chain network equilibrium models, without and with link labor bounds, consisting of competing manufacturers and competing retailers, with distinct wage settings, and multiple demand markets. Chapter 10 then describes a model that integrates human migration networks with supply chains with labor as a means to assess the potential of reducing labor shortages through investments in attracting migrant labor. Different wage settings are allowed for domestic versus migrant labor as well as the possibility of information asymmetry associated with wages. Numerical examples are for a high value food product. Chapters 11 and 12 focus on non-profit organizations with objective functions that are different from those selected by for-profit firms. Chapter 11 unveils a supply chain network optimization model with labor for a blood service organization. Since blood is perishable, a generalized network framework is utilized, as it was for food in Chapter 2. In this chapter, demand uncertainty is included. This model is, hence, for a supply chain network for services and, in particular, one for healthcare services. The final chapter in this book, Chapter 12, proposes a supply chain network optimization model for disaster management, which includes labor. Laborers can be paid or they can be volunteers, under availability constraints. The model captures the phases of disaster preparedness and response, and both demand and cost uncertainty are included. The numerical examples are inspired by the need for humanitarian relief because of refugee crises due to real-world events. The Appendix contains some fundamentals to support the understanding of the mathematical models and the methodology of variational inequalities used in this book.

Chapter 2

Perishable Food Supply Chain Networks with Labor

Abstract On March 11, 2020, the World Health Organization officially declared the COVID-19 pandemic. This major healthcare disaster transformed our daily lives and the operations of governments, businesses, educational institutions, and, of course, healthcare. It elevated and expanded the roles of essential workers in healthcare as well as in the food industry. The food industry underwent significant disruptions in the pandemic for reasons that included compromised labor resources. Labor is needed in all food supply chain network activities from production through distribution. With climate change and increasing geopolitical risks, one can expect additional disruptions to our food supply chains and an increase in food insecurity. The major invasion of Russia in Ukraine on February 24, 2022 is further exacerbating food shortages and also pushing food prices higher. In this chapter, a supply chain generalized network optimization framework focused on perishable food is constructed that includes the critical resource of labor. The model captures labor availability and labor productivity in order to quantify the impacts of disruptions due to illnesses, social distancing requirements, decreases in labor productivity, and even reductions in worker availability during times of conflicts and war. Theoretical results and numerical examples for a fresh produce product are detailed with quantification of a spectrum of disruptions on product flows, demands, prices, and the profits of the food firm.

2.1 Introduction Healthy, high-quality food is essential to nutrition and the well-being of humanity. Many nutritious foods, however, are perishable and include: fresh produce, dairy items, meat, and fish. During the COVID-19 pandemic, such vital food supply chains have been critically stressed for several reasons, since workers became ill from the SARS-Cov-2 coronavirus with some tragically succumbing to the disease and with others reluctant to work because of fear of contagion (Bjattarai and Reiley 2020). Some workers could not travel to assist with food production and harvesting because of travel restrictions. In addition, the coronavirus amplified the risk of contagion for workers engaged in supply chain network activities of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_2

7

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2 Perishable Food Supply Chain Networks with Labor

production, transportation, processing, storage, and distribution that had difficulty socially distancing and/or did not have ideal ventilation in their work areas. In the first spring of the COVID-19 pandemic, the severe impacts of the pandemic on meat supply chains in the USA were notable (see Hirtzer and Skerrit 2020 and Scheiber and Corkery 2020). Many of the meat supply chains for pork, beef, and chicken make use of processing plants belonging to large agribusinesses located in different states (Little 2020; Rosane 2020; Reiley 2020). Close to two dozen of such plants had to shut down due to illnesses of the workers in March and April 2020, impacting farmers and consumers alike (Corkery and Yaffe-Bellany 2020). At a time when there is growing food insecurity (Morath 2020), as well as other global challenges, some farmers even had to resort to culling their animals when they could not be processed (Polansek and Huffstutter 2020). Also, with schools, restaurants, and hotels closed at the height of the pandemic, the demand not only for meat but also for fresh produce as well as dairy changed (Schrotenboer 2020). Some potato farmers, unable to get their produce processed, resorted to discarding their ripe vegetables (Associated Press 2020). The dumping of food results in an immense amount of waste, a reduction in the farmers’ incomes, and also decreases the amount of food available for consumers (see, e.g., Wootson Jr. 2020). Dairy farmers were also hit early in the pandemic, and many had to throw away milk from their cows due to processing challenges (Huffstutter 2020). There have also been disruptions associated with freight service provision, with truckers fearing contracting the coronavirus (CBSSacramento 2020). The alteration of the product demand landscape, with many shocks, resulted even in port congestion at various ports in the USA, with shortages of labor also playing a role (Goodman 2021). With China even in the spring of 2022 battling the coronavirus, traffic jams of container ships are an issue in Shanghai, the world’s largest port (Barrett 2022). In the summer of 2020, the harvesting of many fresh fruits and vegetables in the USA was challenged, in part, since migrant labor was in short supply for picking the harvest (Shoichet 2020 and Nickel and Walljasper 2020). Much of the seasonal fresh produce harvesting in the USA is done by migrant workers. This is also the case in other parts of the world, including Europe as well as Australia. Clearly, labor is a crucial resource in perishable food supply chain networks, demonstrated vividly and dramatically under COVID-19. Agribusiness firms and even farmers began to reevaluate their food supply chains, considering and investigating possible new distribution channels and demand markets. This became a global issue since the COVID-19 pandemic was not and is not limited to a time window or to a specific region (see Cullen 2020). Interestingly, although economists have tackled the use of factors of production, notably, capital and labor, in production functions (cf. Mishra (2007) and the references therein), the explicit incorporation of labor into a complete supply chain network model for perishable products has not been thoroughly investigated until the pandemic. Only when a system-wide perspective is taken can one identify the impacts of labor availability and labor productivity and the impacts of disruptions during a pandemic or even a war on profits, costs, product waste, and consumer prices. Having such a framework in times of peace and prosperity is also very

2.1 Introduction

9

valuable. Furthermore, when it comes to perishable food supply chains, there is also a quality issue since delays in a cold chain can impact the quality of the perishable product (cf. Yu and Nagurney 2013; Besik and Nagurney 2017; Nagurney and Li 2016; Nagurney et al. 2018). As emphasized in Yu and Nagurney (2013), food supply chains are distinct from other supply chains in that there is a continuous and significant change in the quality of the food products as they move through the pathways of the supply chain network to points of demand and consumption. Hence, the quality of food products decreases over time, even under the best cold chain processes (see Sloof et al. 1996; Zhang et al. 2003; Rong et al. 2011). Moreover, many food supply chains are global in structure, due to consumers’ interests and demands throughout the year. Hence, a disruption to a node or link in a food supply chain network thousands of miles away from demand markets may negatively impact availability of desired food products as well as prices. Furthermore, now with climate change and heightened geopolitical risk, vividly demonstrated by Russia’s war against Ukraine, with the major invasion of February 24, 2022, one can expect additional stressors to critically important food supply chains (see Reed et al. 2022; Food and Agricultural Organization of the United Nations 2022). Labor will continue to be essential to food supply chain networks and all of its associated activities from production and processing to transportation, storage, and distribution. In this chapter, a fundamental supply chain generalized network optimization model for perishable food products is constructed that: (1) Includes labor on all the associated supply chain network economic activities (2) Can be utilized to quantify impacts of labor disruptions (3) Can be applied to different food products, with appropriate adaptations This model builds on the model of Yu and Nagurney (2013) to include labor and its associated levels of availability. Although the literature on supply chain network optimization is rich (cf. Geunes and Pardalos 2003; Nagurney 2006; Wu and Blackhurst 2009; Nagurney et al. 2013; Nagurney and Li 2016, and the references therein), it has not integrated labor into a rigorous mathematical framework for product perishability (cf. Nagurney et al. 2013). Such an integration can provide valuable insights for the management and analysis of perishable food supply chains during the pandemic and even in times when the world is not faced with a global healthcare catastrophe or a major war. This chapter is organized as follows. In Section 2.2, the perishable food supply chain network model with labor is constructed and the variational inequality formulation given, along with the theoretical analysis. In Section 2.3, an algorithm is outlined, which resolves the problem into closed form expressions for the food product flows on the supply chain network paths at each iteration, along with the Lagrange multipliers associated with the labor availability link capacities. The algorithm is then applied to compute solutions to a series of numerical examples consisting of a fresh produce product—cantaloupe. The numerical examples quantify the impacts of a variety of disruptions, notably, reductions in labor availability, a decrease in labor productivity, and a freight service disruption, on the food firm’s

10

2 Perishable Food Supply Chain Networks with Labor

optimal sales, profits, labor resources, as well as the consumer demand. Section 2.4 concludes and summarizes the contents of this chapter. It also includes suggestions for future research. Section 2.5 contains the Sources and Notes for this chapter.

2.2 The Perishable Food Supply Chain Network Model with Labor In the mathematical model, there is a single food firm, which, depending on the application, can be a farm or an agribusiness. There is a single perishable food product that is produced, which can be meat or a dairy product, fresh fruit or vegetable, etc. The food firm is a profit-maximizer, and its supply chain network is depicted in Figure 2.1. This topology may be modified according to the specific case under investigation. The supply chain topology is denoted by the graph G = [N , L ], where N is the set of nodes and L is the set of links. The top node 1 in Figure 2.1 corresponds to the food firm, and the bottom nodes: 1, . . . , J correspond to the demand markets. The demand markets can be grocery stores, organizations (such as hospitals, schools, businesses, restaurants, or even food banks), and/or direct consumers. There is assumed to exist one directed path (or more) joining node 1 with each demand market node. The curved links in the supply chain network topology in Figure 2.1 denote direct sales, which can capture sales at the farm, sales at farmers’ markets, or sales direct to the demand markets, without going through storage. Direct sales by farmers, for example, were highlighted as being of increasing importance in the pandemic. Furthermore, direct sales may enhance fresh food product quality since less transportation may be needed. As illustrated in Figure 2.1, the food firm has nM production sites, nC processors, and nD distribution centers and serves the J demand markets. The top set of links connecting the top two tiers of nodes corresponds to the food production at each of the production sites of the firm. Multiple possible links connecting node 1 with its production facilities, M1 , . . . , MnM , are allowed to capture different production technologies at different costs. The second set of links in Figure 2.1 from the production site nodes, connected to the processors of the firm, is denoted by: C1,1 , . . . , CnC ,1 . These links correspond to the shipment links between the production sites and the processors. Different links represent different possible modes of transport. The third set of links connecting nodes: C1,1 , . . . , CnC ,1 to C1,2 , . . . , CnC ,2 , represents the processing activity of the perishable food product. The next set of nodes in Figure 2.1 denotes the distribution centers. Hence, the fourth set of links, which connect the processor nodes to the distribution centers, is the set of shipment links. The distribution nodes are: D1,1 , . . . , DnD ,1 . Here, multiple modes of transportation are also permitted, with faster ones being more costly than slower ones.

2.2 The Perishable Food Supply Chain Network Model with Labor

11

Fig. 2.1 The perishable food supply chain network topology

The fifth set of links in Figure 2.1 connects nodes: D1,1 , . . . , DnD ,1 to nodes: D1,2 , . . . , DnD ,2 , and corresponds to the storage links. Different technologies, at associated costs, may be available for the storage network economic activity. Preservation of freshness and quality of food adds to its value.

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2 Perishable Food Supply Chain Networks with Labor

The final group of links in Figure 2.1 joining the two bottom tiers of the supply chain network corresponds to distribution links over which the perishable food product items are shipped from the distribution centers to the demand markets. As highlighted previously, the curved links in Figure 2.1 joining the upstream production nodes directly with demand market nodes allow for the possibility of on-site production and processing and direct availability, with the latter representing demand market nodes located at the farms, or at farmers’ markets, or transported to consumers or other demand points. A path p in the perishable food supply chain network joins node 1, the origin node, to a demand market node, a destination node. The paths are acyclic and consist of a sequence of links reflecting the supply chain network activities associated with producing the perishable product and having it sold at the demand markets. Let Pk denote the set of paths, which represent alternative associated possible supply chain network processes, joining the pair of nodes (1, k). P denotes the set of all paths joining the origin node 1 to the demand market nodes. In the perishable food supply chain network, there are nP paths and nL links. A typical demand market node is denoted by k and a typical link by a. The notation for the food supply chain network model with labor is given in Table 2.1. All vectors are assumed to be column vectors. Table 2.1 Notation for the perishable food supply chain network model with labor Notation xp fa la dk Notation βa μp αa l¯a wa Notation zˆ a (fa ) cˆa (f ) ρk (d)

Variables The product flow on path p; group all the path flows into the vector nP x ∈ R+ The product flow on link a; group all the link flows into the vector n f ∈ R+L The labor hours available for link a activity, ∀a ∈ L ; group all the n labor hours available into the vector l ∈ R+L The demand for the product at demand market k, k = 1, . . . , J ; group J the demands into the vector d ∈ R+ Parameters The throughput factor on link a, which lies in the range (0, 1], ∀a ∈ L  The throughput on path p, where μp = a∈p βa ; p ∈ P Positive factor relating inputs of labor to product flow on link a, ∀a ∈ L The upper bound on the availability of labor on link a, ∀a ∈ L The hourly wage paid to a laborer on link a, ∀a ∈ L Functions The discarding cost function associated with link a, ∀a ∈ L The operational cost function associated with link a, ∀a ∈ L The demand price for the product at demand market k, k = 1, . . . , J

2.2 The Perishable Food Supply Chain Network Model with Labor

13

The product path flows must be nonnegative, that is, xp ≥ 0,

∀p ∈ P.

(2.1)

The perishability of the food product is handled through the use of arc multipliers in a generalized network framework. Associated with each arc is an implicit time duration for completion, which can depend on the labor availability and is incorporated in the multiplier βa for each link a. Under ideal conditions, one would expect full labor availability and efficient processing resulting in lower food waste. The arc multipliers describe the decrease in quantity, which allows for the capture of the discarding of spoiled products along the paths to the demand markets. For example, if βa = .9 on link a, then, beginning with a volume of product link flow on a of fa , one would be left with .9fa at the end of the link. Here it is assumed that the arc multiplier βa on production links is equal to 1. The definition of arc path multipliers was introduced for food supply chains in Yu and Nagurney (2013), where the multiplier, βap , which is the product of the multipliers of the links on path p that precede link a in that path, is defined as

βap ≡

⎧ δap ⎪ ⎪ ⎨ ⎪ ⎪ ⎩



βb , if {a  < a}p = Ø,

b∈{a  0, solving the equation: wa ∂ Cˆ p1 (x ∗ ) ∂ ρˆ1 (x ∗ ) ∗ + δap1 − ρˆ1 (x ∗ ) − x ∂xp1 αa ∂xp1 p1 a∈L

= 5.8xp∗1 − 164.30 = 0, where δap1 = 1 if link a is contained in path p1 or 0, otherwise. The solution to the above equation is: xp∗1 = 28.32. Of course, one also has that: fa∗ = . . . = ff∗ = xp∗1 = 28.32 and d1∗ = xp∗1 = 28.32. The second illustrative example has the same data as the first illustrative example except that now there is a loss associated with the distribution link with βf = .8, with zˆ f (ff ) = .5ff2 and μp1 = βf = .8. Applying a solution procedure similar to the one used for the first illustrative example, from variational inequality (2.12), under the assumption that xp∗1 > 0, the solution: xp∗1 = 19.42 is obtained. Comparing the results for illustrative Examples 2.1 and 2.2 reveals the fact that when perishability is taken into consideration, with βf = .8, and with the above data, the food firm chooses to produce and distribute a smaller quantity of the fresh produce product so as to minimize the discarding cost of the waste. Remark 2.2 In addition to the already noted pharmaceutical product supply chains that may need a cold chain and be subject to perishability, vaccines and many blood products are also perishable. The supply chain generalized network optimization model in this chapter, hence, with suitable modifications, is amenable to handling such important applications as well; see also Nagurney et al. (2013). Here the focus has been on perishability in which arc multipliers are never greater than 1. In the case that a multiplier has a value greater than 1, this would correspond to the associated link having a gain in the volume of the product as it moved across the link. Of course, one could also have the arc multipliers be a function of the product flows on the links as discussed in Nagurney and Besik (2022).

2.3 The Computational Procedure

19

2.3 The Computational Procedure The algorithm that is used in Section 2.3.1 to compute solutions to numerical examples, whose solutions satisfy variational inequality (2.12), is the modified projection method of Korpelevich (1977). Each of the algorithm’s two fundamental steps at an iteration yields closed form expressions for the computation of the product path flows as well as the Lagrange multipliers associated with the link labor capacities. The algorithm is, therefore, relatively easy to implement, even in the case of a generalized network as in our perishable food product supply chain network optimization model. The steps of the modified projection method are given below, with τ denoting an iteration counter. The Modified Projection Method Step 0: Initialization Initialize with X0 ∈ K . Set the iteration counter τ := 1, and let η be a scalar such that 0 < η ≤ θ1 , where θ is the Lipschitz constant (cf. (2.19) below). Step 1: Computation Compute X¯ τ by solving the variational inequality subproblem:

X¯ τ + ηF (Xτ −1 ) − Xτ −1 , X − X¯ τ  ≥ 0,

∀X ∈ K .

(2.16)

Step 2: Adaptation Compute Xτ by solving the variational inequality subproblem:

Xτ + ηF (X¯ τ ) − Xτ −1 , X − Xτ  ≥ 0,

∀X ∈ K .

(2.17)

Step 3: Convergence Verification If |Xτ − Xτ −1 | ≤ , with  > 0, a pre-specified tolerance, then stop; otherwise, set τ := τ + 1 and go to Step 1. The modified projection method is guaranteed to converge to a solution of variational inequality (2.14) provided that the function F (X) is monotone and Lipschitz continuous (and that a solution exists). Recall that the function F (X) is said to be monotone, if

F (X1 ) − F (X2 ), X1 − X2  ≥ 0,

∀X1 , X2 ∈ K ,

(2.18)

and the function F (X) is Lipschitz continuous, if there exists a constant θ > 0, known as the Lipschitz constant, such that F (X1 ) − F (X2 ) ≤ θ X1 − X2 ,

∀X1 , X2 ∈ K .

(2.19)

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2 Perishable Food Supply Chain Networks with Labor

These conditions can be expected to hold in many reasonable applications of the supply chain generalized network optimization model with labor.

2.3.1 Food Supply Chain Network Numerical Examples The modified projection method is implemented in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the computation of solutions to the following numerical examples. The numerical examples are inspired by a fresh produce application, that of cantaloupes, which are a rich source of nutrients. Cantaloupes consumed in the USA are produced in California and in Mexico and parts of Central America. Here the focus is on production in the USA. The food firm has two production sites, a single processor, two distribution centers, and two demand markets, all of which are located in the Unites States. The numerical examples, hence, have the supply chain network topology depicted in Figure 2.3, except where noted. The links are labeled numerically. As noted in Section 2.1, perishable food products deteriorate over time even under the best conditions. Here, as in Yu and Nagurney (2013), from which our dataset is adapted, it is assumed that the temperature and other environmental conditions associated with each post-production activity/link are given and fixed. Hence, as in Nahmias (1982), each perishable food unit has a probability of e−γ t to survive another t units of time, where γ is the decay rate, which is given and fixed. Let N0 denote the quantity at the beginning of the time interval (link). Then, the quantity surviving at the end of the time interval, which is implicit for each link in the supply chain network, follows a binomial distribution with parameters n = N0 and probability = e−γ t . Hence, the expected quantity surviving at the end of the time interval (specific link), denoted by N(t), can be expressed as N(t) = N0 e−γ t .

(2.20)

In the cantaloupe application, the throughput factor βa for a post-production link a becomes βa = e−γa ta ,

∀a ∈ L ,

(2.21)

where γa and ta are the decay rate and the time duration associated with the link a, respectively. These are given and fixed, but with the latter also adapted to factor in labor. In some cases, food deterioration follows the zero-order reactions with linear decay (see Tijskens and Polderdijk 1996; Rong et al. 2011 and Besik and Nagurney 2017). In that case, βa = 1 − γa ta for a post-production link. According to Suslow et al. (1997), usually, cantaloupes can be stored for 12 through 15 days at 36 to 41 ◦ F.

2.3 The Computational Procedure

21

Fig. 2.3 The supply chain network topology for the numerical Example 2.1

The algorithm is deemed to have converged if the absolute value of the difference between each computed successive iterate is less than or equal to 10−7 . Example 2.1 (Baseline Example) The input data for Example 2.1 are reported in Table 2.2. The decay rates reported in Table 2.2 are per day, and the time duration is in days. As noted in Yu and Nagurney (2013), the cost functions are constructed utilizing the data on the average costs available on the Web (see, e.g., Meister 2004a,b and LeBoeuf 2002), but here the labor costs are handled separately. The demand price functions are ρ1 (d) = −.001d1 + 4,

ρ2 (d) = −.0001d2 + 6.

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2 Perishable Food Supply Chain Networks with Labor

Table 2.2 Decay rate, time duration, productivity factor, throughput, hourly wage, labor link bound, operational cost, and discarding cost functions for Example 2.1 Link a 1 2 3 4 5 6 7 8 9 10 11 12 13

γa – – 0.150 0.150 0.040 0.015 0.015 0.010 0.010 0.015 0.015 0.015 0.015

ta – – 0.20 0.25 0.50 1.50 3.00 3.00 3.00 1.00 3.00 3.00 1.00

αa βa 1.00 2000.00 1.00 3000.00 0.970 3000.00 0.963 3000.00 0.980 3000.00 0.978 4000.00 0.956 4000.00 0.970 10000.00 0.970 10000.00 0.985 8000.00 0.956 8000.00 0.956 9000.00 0.985 9000.00

wa 100.00 100.00 150.00 150.00 110.00 180.00 180.00 120.00 120.00 170.00 190.00 180.00 200.00

l¯a 2000.00 2000.00 3000.00 3000.00 4000.00 2000.00 2000.00 3000.00 3000.00 20000.00 20000.00 20000.00 20000.00

cˆa (f ) 0.005f12 + 0.03f1 0.006f22 + 0.02f2 0.003f32 + 0.01f3 0.002f42 + 0.02f4 0.002f52 + 0.05f5 0.005f62 + 0.01f6 0.01f72 + 0.01f7 0.004f82 + 0.01f8 0.004f92 + 0.01f9 2 + 0.01f 0.005f19 10 2 + 0.1f 0.015f11 11 2 + 0.1f 0.015f12 12 2 + 0.01f 0.005f13 13

zˆ a (fa ) 0.00 0.00 0.00 0.00 0.001f52 + 0.02f5 0.00 0.00 0.001f82 + 0.02f8 0.001f92 + 0.02f9 2 + 0.02f 0.001f10 10 2 + 0.02f 0.001f11 11 2 + 0.02f 0.001f12 12 2 + 0.02f 0.001f13 13

The paths for Demand Market 1 are: p1 = (1, 3, 5, 6, 8, 10), p2 =(1, 3, 5, 7, 9, 12), p3 = (2, 4, 5, 6, 8, 10), and p4 = (2, 4, 5, 7, 9, 12). There are also four paths for Demand Market 2: p5 = (1, 3, 5, 6, 8, 11), p6 = (1, 3, 5, 7, 9, 13), p7 = (2, 4, 5, 6, 8, 11), and p8 = (2, 4, 5, 7, 9, 13). Since Example 2.1 serves as the baseline example, the labor bounds on the links are set very high for all links a ∈ L in order to see what the food product flows, demands, prices, and profit of the food firm would be in the non-disaster situation with many workers willing and able to work on the supply chain network economic activities. The algorithm converges to the following optimal product path flow pattern: xp∗1 = 4.52, xp∗5 = 27.28,

xp∗2 = 0.00,

xp∗3 = 4.81,

xp∗6 = 38.10,

xp∗7 = 27.91,

xp∗4 = 0.00, xp∗8 = 38.15.

The Lagrange multiplier λ∗a = 0.00 for all links a ∈ L . The demands are: = 8.26 and d2∗ = 113.86 with prices at the demand markets of: ρ1 = 3.99 and = 5.89. These prices are reasonable for cantaloupes, a popular fruit in the USA. The food firm earns a profit of 329.52. Note that the data for this example are on a daily basis. d1∗ ρ2

Example 2.2 (Example with a Freight Service Disruption) In Example 2.2, the following scenario is considered: The freight service providers associated with link 13 have become ill. That link for transport of the cantaloupes is, in effect, no longer available, and it is removed from the supply chain network topology of Figure 2.3. All of the other data in this example remain as in Example 2.1. Note that paths p6

2.3 The Computational Procedure

23

and p8 for Demand Market 2 now no longer exist. The other path definitions are as in Example 2.1. The new optimal product path flow pattern is xp∗1 = 8.71,

xp∗2 = 13.28,

xp∗3 = 8.95,

xp∗5 = 32.28,

xp∗7 = 32.41.

xp∗4 = 13.50,

The Lagrange multipliers for the twelve links remain all equal to 0.00. The demand price now decreases at Demand Market 1 but increases at Demand Market 2 with ρ1 = 3.96 and ρ2 = 5.94, at the respective demands: d1∗ = 38.12 and d2∗ = 55.57. The demand at Demand Market 2 drops by over 50% as compared to the demand in Example 2.1. The food firm now earns a profit of only 219.03, a 33% drop from the profit it earns in Example 2.1. This example demonstrates quantitatively how the lack of labor on a single link, which is a freight one, may significantly negatively impact a food firm. During the pandemic, it has been noted that not only labor associated with food production and processing has been impacted but freight service provision has as well. It is also important to recognize the importance of freight for the transport of agricultural commodities overall. For example, in the war of Russia against Ukraine, many of the Black Sea ports are blocked and some even damaged. Plus, roads have been compromised and damaged resulting in additional challenges in getting agricultural products to markets. Production is not sufficient—one must get the products to consumers, and with disruptions to critical infrastructure as well as to labor, one can expect further increases in food insecurity globally (Schreck 2022). Example 2.3 (Example with a Freight Service Disruption and Loss of Productivity) Example 2.3 has the same data as Example 2.2 except that now even greater disruptions are considered. The disruptions affect the speed of processing due to the institution of social/physical distancing among the workers as well as the aftereffects of some having experienced the illness, so that some workers may be less productive than before. In Example 2.3, the αa values for all a ∈ L are set to one-tenth of their respective value in Table 2.2. The computed optimal path flow pattern for Example 2.3 is xp∗1 = 0.00,

xp∗2 = 1.17,

xp∗3 = 0.00,

xp∗5 = 21.38,

xp∗7 = 26.18.

xp∗4 = 6.14,

The Lagrange multipliers for the twelve links are, again, equal to 0.00. One can see the big decrease in the cantaloupe product paths flows in Example 2.3, as compared to the values in Example 2.2. In contrast to Example 2.1, now paths p1 and p3 are not utilized for Demand Market 1. The demand prices increase to ρ1 = 3.99 and ρ2 = 5.96 at the demands of: d1∗ = 6.12 and d2∗ = 40.84. The

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2 Perishable Food Supply Chain Networks with Labor

food firm only earns a profit of 72.96. This example emphasizes the importance of productivity in all supply chain network economic activities and the impact of a drastic reduction. Clearly, this example is also relevant in times of conflict, strife, and war. For example, farmers in parts of Ukraine are having to clear their agricultural lands of mines in Russia’s war against it (MacDonald 2022). Example 2.4 (Example with a Freight Service Disruption, Loss of Productivity, but Increase in Price Consumers Are Willing to Pay) Example 2.4 has the same data as Example 2.3 except that the food firm is very concerned about the loss of profits and has increased marketing so that consumers are now willing to pay a higher price for the cantaloupes at both demand markets. The fixed term in each demand price function has now doubled. The demand price functions in Example 2.4 are ρ1 (d) = −.001d1 + 8,

ρ2 (d) = −.0001d2 + 12.

The rest of the data are as in Example 2.3. The computed optimal path flow pattern for this example is xp∗1 = 4.46,

xp∗2 = 18.52,

xp∗3 = 7.72,

xp∗5 = 59.31,

xp∗7 = 62.22.

xp∗4 = 21.71,

The Lagrange multipliers for the links are equal to 0.00. The demands are now: d1∗ = 44.54 and d2∗ = 104.38 with the demand prices: 7.96 for Demand Market 1 and 11.90 for Demand Market 2. In the COVID-19 pandemic, we saw the escalation in prices of many perishable food products. The firm now earns a profit of: 608.70, over eight times of the profit that it earns in Example 2.3. Example 2.5 (The Cantaloupe Supply Chain Under Further Stress Because of the Pandemic) Example 2.5 represents the most stressed supply chain network example. The data for Example 2.5 are as in Example 2.4 except for the following. The 1 availability of labor is now severely compromised so that the l¯a values are 1000 times the respective value in Example 2.4; that is, l¯1 = 2.00, l¯2 = 2.00, and so on. 1 Also, the link labor productivity factors are now 10 times their respective values in Example 2.4. One now has: α1 = 20.00, α2 = 30.00, and so on. With the demand price functions as in Example 2.4, the solution results in all cantaloupe product flows and Lagrange multipliers being identically equal to 0.00. The food firm is very concerned for its viability and business sustainability in the pandemic. With extraordinary, subsequent marketing efforts, the firm has influenced consumers’ willingness to pay higher prices for their nutritious product. Now the demand price functions are ρ1 (d) = −.001d1 + 40,

ρ2 (d) = −.001d2 + 60.

2.3 The Computational Procedure

25

The remainder of the data remain as immediately above. The optimal product path flows are xp∗1 = 0.00,

xp∗2 = 0.00,

xp∗3 = 0.00,

xp∗5 = 24.97,

xp∗7 = 59.84.

xp∗4 = 0.00,

The Lagrange multipliers are all equal to 0.00 except that now: λ∗2 = 30.8309,

λ∗5 = 65.5255.

The second production site and the storage facility are utilizing the labor at their respective bounds. Observe that the food firm has no product consumed at Demand Market 1. At Demand Market 2, however, there is a positive demand, d2∗ = 72.75. The demand price at Demand Market 2 is: 31.93. The firm, by having consumers willing to pay a higher price, now enjoys a profit of 407.54, even under very restricted labor and impaired productivity. Example 2.6 (Example with Added Direct Sale Demand Markets) Given the results in Example 2.5, the food firm has decided to investigate the possibility of direct sales as depicted in Figure 2.4. There are two added demand markets: 3 and 4, with added links 14 and 15. Path p9 = (1, 14) and path p10 = (2, 15). The operational cost data on the direct demand market links are 2 cˆ14 (f ) = .0025f14 + .01f14 ,

2 cˆ15 (f ) = .0025f15 + .02f15 ,

and the waste disposal cost functions are 2 zˆ 14 (f14 ) = .0005f14 ,

2 zˆ 15 (f15 ) = .0005f15 .

The new link data parameters and labor bounds are α14 = 40.00,

α15 = 40.00;

β14 = .99,

β15 = .99;

w14 = 120.00, l¯14 = 5.00,

w15 = 120.00; l¯15 = 5.00.

The demand price functions at the new direct demand markets are ρ3 (d) = −.001d3 + 18,

ρ4 (d) = −.001d4 + 20.

The rest of the data remain as in Example 2.5. The modified projection method yields the following optimal solution: The optimal product flows are

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2 Perishable Food Supply Chain Networks with Labor

Fig. 2.4 The supply chain network topology for Example 2.6

xp∗1 = 0.00, xp∗5 = 0.00,

xp∗2 = 0.00, xp∗7 = 0.00,

xp∗3 = 0.00, xp∗9 = 38.93,

xp∗4 = 0.00, xp∗10 = 59.63,

and the optimal Lagrange multipliers are λ∗1 = 180.76,

λ∗2 = 368.18,

with all other Lagrange multipliers identically equal to 0.00.

2.4 Summary, Conclusions, and Suggestions for Future Research

27

The demand for the cantaloupes is: 0.00 at Demand Markets 1 and 2. All sales are at the new direct demand markets with d3∗ = 38.54 and d4∗ = 59.04 at prices: ρ3 = 17.96 and ρ4 = 19.94. The profit of the food firm rises to 1, 131.31. This example is illuminating and reveals that direct sales, whether at farmers’ markets or nearby farm stands may help a food firm in a pandemic or in another disaster. Many perishable product firms are now investigating and constructing new distribution channels.

2.4 Summary, Conclusions, and Suggestions for Future Research The COVID-19 pandemic dramatically revealed the importance of labor to supply chains, including perishable food supply chains. The pandemic, as a major healthcare disaster, transformed our daily lives and the operations of governments, businesses, healthcare, and educational institutions. It demonstrated the importance of essential workers, including farmers, food processors, and grocery workers. At a time when consumers need nutritious foods more than ever, there have been serious disruptions to food supply chains due, in part, to the reduction of labor capacity as well as labor productivity. The reduction is occurring for multiple reasons, including COVID-19 illness, loss of life, fear of going to work, and the closure of food facilities due to the need for sanitization and even redesign because of the importance of social distancing. Furthermore, many food items, including fresh produce, meat, fish, and dairy, are perishable food products, and their quality deteriorates even under the best conditions. The negative impacts of labor shortfalls and decreases in productivity are being felt in all supply chain network economic activities from production to distribution. Furthermore, with increasing strife, conflicts, and wars globally, coupled with climate change, we are seeing an escalation in food insecurity with major ramifications. In this chapter, a rigorous supply chain generalized network optimization framework that explicitly includes labor and bounds on labor on links for perishable food is constructed. The food firm is interested in maximizing profits (for its business sustainability) with the objective function including revenue, with the demand price functions being a function of the demand, and with operational and discarding costs as well as costs of labor. Linear production functions that map labor on a link to product flow are utilized. A variational inequality formulation of the problem is derived, which enables the effective computation of the solution consisting of food product flows and Lagrange multipliers associated with the capacities on labor. Numerical examples are presented based on a fresh produce product— cantaloupes, in which the quality deterioration is also captured. The impacts of labor disruptions in terms of availability as well as productivity are considered and the potential of adding direct demand markets on the food firm’s profit, demand market prices, product flows, and demands.

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The supply chain generalized network optimization model can be adapted to specific perishable products and scenarios, whether in times of peace or wartime and under different disaster settings. The model has the flexibility to handle different supply chain network topologies, costs (operational and discarding ones), perishability, and productivity factors, as well as demand price functions, along with bounds on labor availability on supply chain network links. Having a rigorous mathematical framework for perishable food products can assist in decision-making and policy-making and allows for opportunities to enhance food security. An extension of the model to a game theory setting would be interesting as well as making the arc multipliers flow-dependent as well as, possibly, dependent on labor. Some research regarding the former, in the context of spatial price equilibrium models, as noted earlier, can be found in Nagurney and Besik (2022).

2.5 Sources and Notes This chapter is based on Nagurney (2021) with new examples, expanded and updated discussion, as well as standardized notation used in this book. Yu and Nagurney (2013) utilized a generalized network approach for their competitive supply chain network equilibrium model for perishable food products, which has influenced this work. Such an approach has origins in the work of Nahmias (1982) in studies on perishable inventory. For example, in the case of fresh produce items, such as fruits and vegetables, exponential time decay is often used. For further background on food science and food deterioration, I refer the interested reader to Thompson (2002) and Gustavsson et al. (2011). In a series of papers (see Besik and Nagurney 2017; Nagurney et al. 2018), explicit food decay formulae were used to capture food quality deterioration. Similar types of multipliers have also been used in other perishable product supply chain models for pharmaceuticals by Masoumi et al. (2012) and for blood supply chains by Nagurney et al. (2012) and Nagurney and Dutta (2019). Ahumada and Villalobos (2009) reviewed studies of agricultural supply chains, whereas Higgins et al. (2010) focused on practice and network analysis. The review of Bjorndal et al. (2012) describes operations research applications that include agriculture and fisheries. The meat industry and, specifically, meat processing plant activities have received attention from operations researchers from the modeling perspective dating to 1990 (Whitaker and Cammel 1990). Albornoz et al. (2015) developed a mixed integer programming model focusing on meat packing operations for pork at the operational level and included worker daily hours. The authors noted that they did not handle distribution. Rodriguez et al. (2014) identified gaps in the literature. The volume edited by Baourakis et al. (2004) contains a collection of articles on supply chains and finance, revealing methodologies used and tools for risk management. Several articles therein are on food. Vlontzos and Pardalos (2017) discuss data mining and optimization issues in the food industry and highlight a range of successful case studies on fresh produce as well as processed foods.

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Additional insights and perspectives on COVID-19 and the impacts of the pandemic on food supply chains can be found in Aday and Aday (2020), Anderson et al. (2021), and Barman et al. (2021). For information on the major disruptions to food supply chains by the war of Russia against Ukraine, often referred to as the breadbasket of Europe, if not the world, see: D’Agostino (2022), Hussein and Wiseman (2022), and Quinn and Verbyany (2022). For a personal account of a farmer in Ukraine, struggling with labor shortages and the destruction of his agricultural land, as well as critical infrastructure, including that of electric power provision and water, see Gall (2022).

References Aday, S., Aday, M.S., 2020. Impact of COVID-19 on the food supply chain. Food Quality and Safety, 4(4), 167–180. Ahumada, O., Villalobos, J., 2009. Application of planning models in the agri-food supply chain: A review. European Journal of Operational Research, 196(1), 1–20. Albornoz, V.M., Gonzalez-Araya, M., Gripe, M.C., Rodriguez, S.V., 2015. A mixed integer linear program for operational planning in a meat packing plant. In: Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pp. 254–261. Anderson, J.D., Mitchell, J.L., Maples, J.G., 2021. Invited review: Lessons from the COVID-19 pandemic for food supply chains. Applied Animal Science, 37(6), 738–747. Associated Press, 2020. Coronavirus pandemic leads to Idaho potato market woes. From CNN.com, April 27. Baourakis, G., Migdalas, A., Pardalos, P.M., Editors, 2004. Supply Chain and Finance. World Scientific Publishing Co., Singapore. Barman, A., Das, R., De, P.K., 2021. Impact of COVID-19 in food supply chain: Disruptions and recovery strategy. Current Research in Behavioral Sciences. November 2, 100017. Barrett, E., 2022. China’s COVID-19 lockdown is inflaming the world’s supply chain backlog, with 1 in 5 container ships stuck outside congested ports. Fortune, April 21. Besik, D., Nagurney, A., 2017. Quality in competitive fresh produce supply chains with application to farmers’ markets. Socio-Economic Planning Sciences, 60, 62–76. Bjattarai, A., Reiley, L., 2020. The companies that feed America brace for labor shortages and worry about restocking stores as coronavirus pandemic intensifies. The Washington Post, March 13. Bjorndal, T., Herrero, I., Newman, A., Romero, C., Weintraub, A., 2012. Operations research in the natural resource industry. International Transactions in Operational Research, 19, 39–62. CBSSacramento, 2020. Trucking through coronavirus pandemic: Drivers describe new changes on the road. April 7. Corkery, M., Yaffe-Bellany, D., 2020. The food chain’s weakest link: Slaughterhouses. The New York Times, April 18. Cullen, M.T., 2020. COVID-19 and the risk to food supply chains: How to respond? Food and Agricultural Organization of the United Nations. March 29, Rome, Italy. D’Agostino, S., 2022. Global hunger crisis looms as war in Ukraine sends food prices soaring. Bulletin of Atomic Scientists, April 12. Food and Agricultural Organization of the United Nations, 2022. Assessing investment needs in Ukraine’s agricultural reconstruction and recovery. April 7. Gall, C., 2022. A farmer holds on, a fraying lifeline for a besieged corner of Ukraine. The New York Times, June 4.

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Geunes, J., Pardalos, P.M., 2003. Network optimization in supply chain management and financial engineering: An annotated bibliography. Networks, 42(2), 66–84. Goodman, P.J., 2021. ‘It’s not sustainable’: What America’s port crisis looks like up close. The New York Times, October 21. Gustavsson, J., Cederberg, C., Sonesson, U., van Otterdijk, R., Meybeck, A., 2011. Global food losses and food waste. The Food and Agriculture Organization of the United Nations, Rome, Italy. Higgins, A., Miller, C., Archer, A., Ton, T., Fletcher, C., McAllister, R., 2010. Challenges of operations research practice in agricultural value chains. Journal of the Operational Research Society, 61(6), 964–973. Hirtzer, M., Skerrit, J., 2020. Americans on cusp of meat shortage with food chain breaking down. Bloomberg, April 27. Huffstutter, P.J., 2020. U.S. dairy farmers dump milk as pandemic upends food markets. Reuters, April 7. Hussein, F., Wiseman, P., 2022. Finance heads urged to boost fight against food insecurity. Associated Press, April 19. Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York. Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. LeBoeuf, J., 2002. Crop time line for cantaloupes, honeydews, and watermelons in California. AgriDataSensing, Inc., October 1. Little, A., 2020. A pork panic won’t save our bacon. Bloomberg Opinion, April 30. MacDonald, A., 2022. After Russian retreat, Ukraine’s farmers discover fields full of mines. The Wall Street Journal, April 28. Masoumi, A.H., Yu, M., Nagurney, A., 2012. A supply chain generalized network oligopoly model for pharmaceuticals under brand differentiation and perishability. Transportation Research E, 48, 762–780. Meister, H.S., 2004a. Sample cost to establish and produce cantaloupes (slant-bed, spring planted). U.C. Cooperative Extension – Imperial County Vegetable Crops Guidelines, August. Meister, H.S., 2004b. Sample cost to establish and produce cantaloupes (mid-bed trenched). U.C. Cooperative Extension – Imperial County Vegetable Crops Guidelines, August. Mishra, S.K., 2007. A brief history of production functions. MPRA Paper No. 5254. Morath, E., 2020. How may U.S. workers have lost jobs during coronavirus pandemic? There are several ways to count. The Wall Street Journal, June 3. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A. 2006. Supply Chain Network Economics: Dynamics of Prices, Flows and Profits. Edward Elgar Publishing, Cheltenham, England. Nagurney, A., 2021. Perishable food supply chain networks with labor in the Covid-19 pandemic. In: Dynamics of Disasters - Impact, Risk, Resilience, and Solutions. I.S. Kotsireas, A. Nagurney, P.M. Pardalos, and A. Tsokas, Editors, Springer Nature Switzerland AG, pp. 173– 193. Nagurney, A., Besik, D., 2022. Spatial price equilibrium networks with flow-dependent arc multipliers. Optimization Letters, 26, 2483–2500. Nagurney, A., Besik, D., Yu, M., 2018. Dynamics of quality as a strategic variable in complex food supply chain network competition: The case of fresh produce. Chaos, 28, 043124. Nagurney, A. Dutta, P., 2019. Supply chain network competition among blood service organizations: A Generalized Nash Equilibrium framework. Annals of Operations Research, 275(2), 551–586. Nagurney, A., Li, D., 2016. Competing on Supply Chain Quality: A Network Economics Perspective. Springer International Publishing Switzerland.

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Nagurney, A., Masoumi, A.H., Yu, M., 2012. Supply chain network operations management of a blood banking system with cost and risk minimization. Computational Management Science, 9(2), 205–231. Nagurney, A., Yu, M., Masoumi, A.H., Nagurney, L.S., 2013. Networks Against Time: Supply Chain Analytics for Perishable Products. Springer Science+Business Media, New York, New York. Nahmias, S., 1982. Perishable inventory theory: A review. Operations Research, 30(4), 680–708. Nickel, R., Walljasper, C., 2020. Canada, U.S. farms face crop losses due to foreign worker delays. Reuters, April 6. Polansek, T., Huffstutter, P.J., 2020. Piglets aborted, chickens gassed as pandemic slams meat sector. Reuters, April 27. Quinn, A., Verbyany, V., 2022. Ukraine’s farmers fight on the front line of global food crisis. Bloomberg, April 24. Reed, J., Terazono, E., Geal, A., Joiner, S., Clark, D., Learner, S., 2022. How Russia’s war in Ukraine upended the breadbasket of Europe. Fortune, April 27. Reiley, L., 2020. Meat processing plants are closing due to Covid-19 outbreaks. Beef shortfalls may follow. The Washington Post, April 16. Rodriguez, S., Pla, L., Faulin, J., 2014. New opportunities in operations research to improve pork supply chain efficiency. Annals of Operation Research, 219, 5–23. Rong, A., Akkerman, R., Grunow, M., 2011. An optimization approach for managing fresh food quality throughout the supply chain. International Journal of Production Economics, 131(1), 421–429. Rosane, O., 2020. Meat processing plants close as working conditions encourage spread of coronavirus. EcoWatch, April 14. Samuelson, W.F., Marks, S.G., 2012. Managerial Economics, seventh edition. John Wiley & Sons, Inc., Hoboken, New Jersey. Scheiber, N., Corkery, M., 2020. Missouri pork plant workers way they can’t cover mouths to cough. The New York Times, April 24. Schreck, A., 2022. AP interview: UN chief says Mariupol is starving. Associated Press, April 14. Schrotenboer, B., 2020. US agriculture: Can it handle coronavirus, labor shortages and panic buying? USA Today, April 4. Shoichet, C.E., 2020. The farmworkers putting food on America’s tables are facing their own coronavirus crisis. CNN.com, April 11. Sloof, M., Tijskens, L.M.M., Wilkinson, E.C., 1996. Concepts for modelling the quality of perishable products. Trends in Food Science & Technology, 7(5), 165–171. Suslow, T.V., Cantwell, M., Mitchell, J., 1997. Cantaloupe: Recommendations for maintaining postharvest quality. Department of Vegetable Crops, University of California, Davis, California. Thompson, J.F., 2002. Waste management and cull utilization. In: Postharvest Technology of Horticultural Crops, third edition. A.A. Kader, Editor, University of California Agriculture & Natural Resources, Publication 3311, Oakland, California, pp. 81–84. Tijskens, L.M.M., Polderdijk, J.J., 1996. A generic model for keeping quality of vegetable produce during storage and distribution. Agricultural Systems, 51(4), 431–452. Vlontzos, G., Pardalos, P.M., 2017. Data mining and optimisation issues in the food industry. International Journal of Sustainable Agricultural Management and Informatics, 3(1), 44–64. Whitaker, D., Cammel, S., 1990. A partitioned cutting stock problem applied on the meat industry. Journal of the Operational Research Society, 41(9), 801–807. Wootson, Jr., C.R., 2020. As produce rots in the field, one Florida farmer and an army of volunteers combat ‘a feeling of helplessness’ one cucumber at a time. The Washington Post, April 30. Wu, T., Blackhurst, J., Editors, 2009. Managing Supply Chain Risk and Vulnerability: Tools and Methods for Supply Chain Decision Makers. Springer, London, England. Yu, M., Nagurney, A., 2013. Competitive food supply chain networks with application to fresh produce. European Journal of Operational Research, 224(2), 273–282. Zhang, G., Habenicht, W., Spiess, W.E.L., 2003. Improving the structure of deep frozen and chilled food chain with tabu search procedure. Journal of Food Engineering, 60, 67–79.

Chapter 3

Optimization of Supply Chains Under Different Labor Constraints

Abstract In this chapter, supply chain network optimization models are constructed, which include labor as an important variable in the network economic activity links, along with associated capacities under distinct sets of constraints. Labor is a critical resource in supply chains from production to transportation, storage, and distribution. In a pandemic, or other disaster settings, and even in peace time, the availability of labor for different supply chain network activities may be disrupted due to illness, fear of contagion, morbidity, necessity of social distancing, unwillingness or inability to work, childcare or caretaker issues, a strike, etc. The modeling framework in this chapter considers, first, elastic demands for a product and, subsequently, fixed demands, since some products may be more or less price elastic. The supply chain network framework, which includes electronic commerce, is relevant to many products such as protective personal and medical equipment, as well as to particular nonperishable food items. It is also relevant to disaster scenarios outside of pandemics, such as conflicts and wars, where labor may need to be reallocated to different supply chain network activities. Theoretical results as well as computed numerical examples are presented. In addition, a supply chain network efficiency measure is constructed that allows for the importance identification (and ranking) of nodes, links, or a combination thereof. Resilience measures are proposed and applied to quantify the resilience of a supply chain network to disruptions in labor.

3.1 Introduction The COVID-19 pandemic has transformed our world, presenting immense challenges to governments, businesses, medical and educational institutions, organizations, including humanitarian ones, as well as to consumers. The urgent need for a plethora of products and supplies, including those in healthcare, has demonstrated the criticality of supply chain networks. In the pandemic, there have been shortages of numerous products from critical needs ones such as pharmaceuticals and medical equipment, including personal protective equipment (PPE) (cf. Ranney et al. 2020) and sanitation supplies (Morrison 2020), to even the more plebeian, yet essential © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_3

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ones—toilet paper (Fisher 2020). As the pandemic progresses and evolves from country to country, newspapers around the globe continue to highlight empty store shelves of certain products (Swanson 2020) including baby formula in the USA (Horsley 2022). Many of the news articles are explicitly emphasizing bottlenecks and disruptions associated with the lack of labor (Bhattarai and Reiley 2020). Some reports note the inability to secure inputs of material resources for production, but these also may have deficiency of labor undertones (Rabouin et al. 2020). Many factories in China were shut down in early 2020 for weeks under a lockdown since the coronavirus that causes the COVID-19 illness in this pandemic is believed to have originated in Wuhan, China (Mistreanu 2020). Some factories gradually reopened, resulting in shortages of pharmaceuticals, many of which are produced in China, along with face masks, etc. (cf. Harney 2020). Furthermore, freight services have been negatively impacted as well due to workers’ illnesses, fear of contracting the coronavirus, and even border closures in certain regions resulting in transport delays (cf. Saul et al. 2020). Even in the spring of 2022, there are lockdowns in China, affecting global supply chains and products (Hetzner 2022). The congestion associated with many ports has become a reality (Jones 2022). In this pandemic, major electronic commerce retailers, such as Amazon, experienced labor shortages, due, in part, to escalating demand for online deliveries as many consumers began to work from home (Del Ray 2020). The pandemic is making companies completely reevaluate their supply chain networks (Shih 2020). In fact, Amazon even eliminated its freight division that competed with FedEx and UPS (Ziobro 2020). At the same time, there have been reports that Amazon workers in certain distribution centers have been concerned about their health and with contracting the coronavirus (Heater 2020). Clearly, this new world commercial landscape is being deeply affected by the availability of labor to contribute to each link in a supply chain network, from production to the ultimate distribution of products to points of demand (Bhattarai and Reiley 2020). Interestingly, prior to the COVID-19 pandemic, the inclusion of labor as a vital resource in supply chain networks had not attained much attention in the literature. Typically, a cost associated with production, transportation, etc., is noted, but the actual needs of labor for production, etc., are not explicitly quantified, although, in the economics literature, two major factors of production are capital and labor. For an excellent history of production functions used in economics, along with a discussion of some of the controversies, see Mishra (2007). However, in the economics literature, the full richness of supply chain network topologies (see, e.g., Nagurney et al. 2013; Nagurney and Li 2016, and the references therein) and associated issues are still virgin territory. It is, nevertheless, worth noting that product assembly processes associated with multitiered supply chain networks, along with the importance of specific suppliers, have been researched (cf. Li and Nagurney 2017). There is a literature on manpower planning and scheduling, but this literature does not capture the full supply chain (see, e.g., Jaillet et al. 2022). The COVID-19 pandemic has created novel pressures on supply chains, which include tackling potential decreases in labor resources. And, since labor is an essential

3.1 Introduction

35

input into each supply chain network economic activity, this can result in increasing costs, lower profits for firms, higher prices for consumers, and unfulfilled demand. Furthermore, due to the global nature of many supply chains, a disruption in one part of the globe can propagate internationally across borders and oceans. In parallel, the theme of resilience of supply chains has become a dominant one in the pandemic. Now, with Russia’s war against Ukraine raging (Zaliska et al. 2022), and additional threats to supply chains, due to climate change and the increasing number of disasters, as well as people affected by them (Kotsireas et al. 2021; Nagurney 2021; Novoszel and Wakolbinger 2022), supply chain resilience, with fundamental contributions by: Kleindorfer and Saad (2005), Wagner and Bode (2006), Nagurney (2006), Tang (2006), Tang and Tomlin (2008), Nagurney and Qiang (2009), Nagurney et al. (2013), and Sheffi (2015), has attracted renewed interest (Ivanov and Das 2020; Ivanov and Dolgui 2020; Sodhi and Tang 2021; Ozdemir et al. 2022; Ramakrishnan 2022). A tool that can quantify the resilience of a supply chain network to labor disruptions is timely and needed. In this chapter, such a tool is constructed. This chapter builds on the work of Nagurney and Qiang (2009) in quantifying performance/efficiency of critical infrastructure networks as well as those of Qiang et al. (2009) and Li and Nagurney (2017) in assessing the performance of supply chain networks. The framework constructed here, however, allows one to quantify the resilience of a supply chain network subject to the reduction of labor availability (capacities) or under a reduction in labor productivity. The former situation can arise, for example, as a consequence of illness, deaths, being unavailable to work or refusing to, labor strikes, being called to war or other types of service, etc. The latter situation can arise because of the need for social distancing, a decrease in productivity due to long COVID or other illness or stressors, fatigue, etc. In this chapter, a supply chain network optimization model is constructed, in which the firm is a profit-maximizing one, and seeks to determine the optimal path flows of the product from its production sites through the supply chain network to the demand markets. The demands for the product at the demand markets are assumed to be elastic, that is, the consumers are sensitive to the price of the product. Electronic commerce is also allowed, since that is an important feature of the commercial landscape during the COVID-19 pandemic, with emphasis on “social distancing” and, hence, the reduction of shoppers in grocery stores, pharmacies, and other essential retail outlets. The model considers three sets of labor constraints, of increasing flexibility of movement of labor for the supply chain network economic activities. In the first set, each supply chain network link has an upper bound of available labor, as was the case in the model in Chapter 2. In this scenario, labor is not free to move to other production sites, nor to other distribution centers, or to assist in freight service provision. In the second set of constraints, the labor is free to move across a supply chain tier of network economic activities (such as production, or transportation, or storage, and, finally, distribution). There is a capacity of labor associated with each tier of supply chain network links. Hence, those who have skills in production, or in distribution, or freight service provision may be reallocated according to their specific skills. This has been happening

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3 Optimization of Supply Chains Under Different Labor Constraints

in freight service provision, for example, during the COVID-19 pandemic (see CBSSacramento 2020). In the case of the third set of labor constraints, which corresponds to the most flexible scenario, labor is free to move across all the supply chain network economic activities, and there is a single capacity. Shifting employees among different tasks during the COVID-19 pandemic has been noted by McKinsey & Company as a means toward resilience and returning the supply chain to effectiveness while re-envisioning and reforming supply chain operations for enhanced performance (Aryapadi et al. 2020). Subsequently, the fixed demand supply chain network optimization analogue is constructed for the three labor availability scenarios, which is a special case of the elastic demand model version. Examples of fixed demand products include some medicines as well as baby formula. The variational inequality formulations of the elastic demand and fixed demand versions, under the three scenarios, are provided. In particular, the variational inequality formulations incorporate the Lagrange multipliers associated with the labor capacity constraints under each of the three scenarios. This gives us a unified formulation for computational purposes. This chapter is organized as follows. In Section 3.2, the supply chain network optimization modeling framework with labor is presented, along with theoretical foundations and illustrative examples. In Section 3.3, a computational procedure is proposed and differences in the implementation for the case of elastic demands versus fixed demands highlighted. In Section 3.3, additional numerical examples are presented and their computed solutions, which are obtained algorithmically, to further demonstrate the relevance of the modeling framework to assess needs and flexibility in supply chains. The numerical examples focus on life-saving healthcare supplies. Section 3.4 then turns to the quantification of supply chain network performance and proposes an efficiency measure, which allows for the ranking of the importance of nodes and links or a combination thereof. In addition, resilience measures that quantify the resilience of a supply chain network to labor disruptions are provided. Section 3.5 summarizes the results and presents suggestions for future research. Section 3.6 is the sources and notes section.

3.2 The Supply Chain Network Models with Labor The supply chain network models with labor under three different sets of labor constraints, with increasing degree of flexibility, are presented in this section. First, the elastic demand case is considered, and then the special fixed demand one. The profit-maximizing firm’s supply chain network is depicted, for definiteness, in Figure 3.1. The topology can be adapted according to the specific application, and hence, the framework is not limited to such a network topology. The top node 1 corresponds to the firm, and the bottom nodes: 1, . . . , J correspond to the demand markets. The demand markets can be institutions, such as healthcare ones (depending on the specific application), retailers, and/or direct consumers. It is

3.2 The Supply Chain Network Models with Labor

37

Fig. 3.1 The supply chain network topology

assumed that there exists one path (or more) joining node origin 1 with each demand market node. According to Figure 3.1, the firm is considering nM production sites; nD distribution centers and must serve the J demand markets. A set of links from the top-tiered node 1 is connected to the manufacturing nodes of the firm, which are denoted, respectively, by: M1 , . . . , MnM , and these links correspond to the production links. The links from the production nodes are joined with the distribution

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3 Optimization of Supply Chains Under Different Labor Constraints

center nodes of the firm and are denoted by: D1,1 , . . . , DnD ,1 . These links represent the transportation options between the production sites and the distribution centers where the product can be stored. The links joining nodes: D1,1 , . . . , DnD ,1 with nodes: D1,2 , . . . , DnD ,2 correspond to the storage options. The distribution links connect the nodes: D1,2 , . . . , DnD ,2 with the demand market nodes: 1, . . . , J . In order to capture electronic commerce in the form of direct shipments from the production sites to the demand markets, I draw the corresponding links joining the M1 , M2 , . . ., MnM nodes with the demand market nodes. The supply chain network consists of the graph G = [N , L ], where N denotes the set of nodes and L the set of links. In the case of Scenario 2, the possible reallocation of labor across a tier of supply chain network economic activities is considered. Define L 1 as the set of links corresponding to production, L 2 as the set of links corresponding to transportation, and so on. Since the supply chain network modeling framework is flexible (and not limited to the topology in Figure 3.1), the final (non-electronic commerce) tier of links is denoted by L T and the electronic commerce links by L T +1 , if such an option exists. A path p in the supply chain network joins origin node 1 to a demand market node, which is a destination node. The paths are acyclic and consist of a sequence of links representing the supply chain network activities associated with producing the product and having it ultimately delivered to the demand markets. Let Pk denote the set of paths, which represent alternative associated possible supply chain network processes, joining the pair of nodes (1, k). P denotes the set of all paths joining node 1 to the demand market nodes. There are nP paths in the supply chain network and nL links. A typical demand market node is denoted by k and a typical link by a. The additional notation for the model is given in Table 3.1. All vectors are assumed to be column vectors. In terms of the capacities on labor, three different sets of constraints are considered with the associated notation below, as it is needed. The conservation of flow equations is as follows. The demand at each demand market must be satisfied by the product flows of the firm to each demand market, that is, xp = dk , k = 1, . . . , J. (3.1) p∈Pk

In addition, the product flow on a link is equal to the sum of flows on paths that contain that link, that is, fa = xp δap , ∀a ∈ L , (3.2) p∈P

where δap = 1, if link a is contained in path p, and is 0, otherwise. The path flows must be nonnegative, that is, xp ≥ 0,

∀p ∈ P,

(3.3)

3.2 The Supply Chain Network Models with Labor

39

Table 3.1 Notation for the supply chain network models with labor Notation xp fa la dk Notation αa wa l¯a l¯t

l¯ Notation cˆa (f ) ρk (d)

Variables The product flow on path p; group all the path flows into the vector n x ∈ R+P The product flow on link a; group all the link flows into the vector n f ∈ R+L The labor available for link a activity, ∀a ∈ L The demand for the product at demand market k, k = 1, . . . , J ; group J the demands into the vector d ∈ R+ Parameters Positive factor relating inputs of labor to product flow on link a, nL ∀a ∈ L ; group the αs into the vector α ∈ R+ The hourly wage of labor on link a, ∀a ∈ L ; group all wages into the n vector w ∈ R+L The upper bound on the availability of labor on link a under Scenario 1, ∀a ∈ L The upper bound on labor availability for tier t activities under Scenario 2, with tier t = 1 being production; tier t = 2 being transportation, and so on until t = T , which corresponds to distribution. Here, T + 1 corresponds to the electronic commerce tiered links The upper bound on labor availability under Scenario 3 Functions The operational cost associated with link a, ∀a ∈ L The demand price for the product at demand market k, k = 1, . . . , J

since the product will be produced in nonnegative quantities. In addition, and, as in Chapter 2, the following relationship between link flows and labor holds: fa = αa la ,

∀a ∈ L .

(3.4)

According to (3.4), the output on each link of product is a linear function of the labor input. Hence, in terms of economics, this is a linear production function. Note that the above conservation of flows corresponds to a pure network flow problem, as opposed to a generalized network flow problem, which was the topic of Chapter 2, in the context of perishable product food supply chains. In the models in this chapter, the volume of product flow that arrives at the end of a link is the same as that started on the link. Hence, there are no losses (or gains) in terms of the product flow. The firm seeks to maximize its profits. The objective function faced by the firm is the difference between the revenue denoted by the sum over all the demand markets of the price the consumers are willing to pay for the product at a demand market times the demand there minus the total costs consisting of the operational costs associated with the links and the wages paid out to labor on the links:

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3 Optimization of Supply Chains Under Different Labor Constraints

Maximize

J

ρk (d)dk −



cˆa (f ) −

a∈L

k=1



w a la .

(3.5)

a∈L

The optimization problem is subject to constraints (3.1) through (3.4) and the following sets of constraints depending on the specific labor scenario. Labor Scenario 1: Bound on Labor on Each Link The first labor scenario is the most restrictive. Labor is not transferable from link to link, which may reflect inability and/or unwillingness to move, as well as skills geared toward a specific activity and location. The additional constraints relevant to this scenario are la ≤ l¯a ,

∀a ∈ L .

(3.6)

Labor Scenario 2: Bound on Labor on Each Activity Tier of the Supply Chain Network The second labor scenario, in turn, considers the following situation. Workers involved in production may be reallocated to other production sites; the same holds for transportation service provision, since freight service providers are expected to have similar skills. Also, distribution center workers may be free to move from one distribution center to another; since they have similar skills, this is not unreasonable. Hence, the additional constraints in Scenario 2 are la ≤ l¯1 ,

(3.7,1)

a∈L 1



la ≤ l¯2 ,

(3.7,2)

a∈L 2

and so on until

la ≤ l¯T +1 .

(3.7,T +1)

a∈L T +1

Labor Scenario 3: Single Bound on Labor for the Full Supply Chain Network Finally, the third scenario offers the most flexibility. This is being done now in practice as reported by McKinsey & Company (cf. Aryapadi et al. 2020). In this scenario, the workers are able to do all the tasks associated with the supply chain network activities, that is, production, transportation, storage, and distribution. In this scenario, the additional constraint to (3.1) through (3.4) is

3.2 The Supply Chain Network Models with Labor



41

¯ la ≤ l.

(3.8)

a∈L

In view of (3.1), (3.2), and (3.4), one can express objective function (3.5) exclusively in terms of path flows by incorporating these constraints directly into the objective function, with the proviso that the following functions are defined: c˜a (x) ≡ cˆa (f ), ∀a ∈ L ; ρ˜k (x) ≡ ρk (d), ∀k. Hence, the objective function (3.5) now becomes the following in path flows: Maximize

J

ρ˜k (x)

p∈Pk

k=1



xp −

c˜a (x) −

a∈L

wa xp δap . αa

a∈L

(3.9)

p∈P

Notice that I also make use in (3.9) of the substitution for Equation (3.4) of: la =  p∈P xp δap

, for all a ∈ L . It is assumed that the objective function in (3.9) is αa concave, which will hold if the total revenue component is concave and the total link cost functions are convex. It is also assumed that the revenue functions and the total link cost functions are continuously differentiable. Since (3.1), (3.2), and (3.4) are directly incorporated into the objective function (3.9), the nonnegativity assumption on the path flows (3.3) is retained. The constraints (3.6) through (3.8), corresponding to Scenarios 1 through 3, respectively, are now re-expressed in path flows. Specifically, under Scenario 1, (3.6), in path flows is 

xp δap

p∈P

αa

≤ l¯a ,

∀a ∈ L .

Under Scenario 2, the set of constraints (3.7) becomes  p∈P xp δap ≤ l¯1 , α a 1

(3.10)

(3.11,1)

a∈L





p∈P

xp δap

αa

a∈L 2

≤ l¯2 ,

(3. 11,2)

and, so on, until a∈L T +1



p∈P

xp δap

αa

≤ l¯T +1 .

(3. 11,T +1)

Finally, under Scenario 3, constraint (3.8) is equivalent to the following constraint in path flows:

42

3 Optimization of Supply Chains Under Different Labor Constraints

a∈L



p∈P

xp δap

αa

¯ ≤ l.

(3.12)

3.2.1 Variational Inequality Formulations of the Elastic Demand Case Now, variational inequality formulations of the above supply chain network optimization model under the three distinct scenarios for labor availability are provided. The solutions to the supply chain network optimization model with labor under each of the scenarios are guaranteed to exist since the feasible sets are all bounded due to capacities on the availability of labor (albeit of different forms), and therefore, the product flows are also bounded. The proofs of the below formulations follow immediately from the classical theory of variational inequalities (Kinderlehrer and Stampacchia 1980 and Nagurney 1999) with related applications to supply chains and derivations given in Nagurney (2006, 2010) and Nagurney and Li (2016). Indeed, the feasible set underlying each of the labor scenarios is convex since the constraints are linear. Furthermore, since the objective function is concave, by assumption, the Karush–Kuhn–Tucker conditions (see the Appendix) are both necessary and sufficient for optimality, and these can be formulated directly as the variational inequality problems below. The variational inequality formulations enable the implementation of an effective computational procedure. Labor Scenario 1 Variational Inequality Formulation For Labor Scenario 1, associate the nonnegative Lagrange multiplier λa with the link labor constraint for each link a given by (3.10) and group these Lagrange multipliers n n +n into the vector λ ∈ R+L . Define the feasible set K 1 ≡ {(x, λ) ∈ R+P L }. The solution to the Scenario 1 optimization problem with objective function (3.9) is equivalent to the solution of the variational inequality: Determine (x ∗ , λ∗ ) ∈ K 1 such that J k=1 p∈Pk

⎡

⎤ J ∗ ∗ ∗ ˆ ∂ C (x ) wa ∂ ρˆv (x ) λa ⎣ p + δap − ρˆk (x ∗ ) − xq∗ + δap ⎦ ∂xp αa ∂xp αa a∈L

v=1

q∈Pv

   ∗δ   x   ap p p∈ P × xp − xp∗ + l¯a − × λa − λ∗a ≥ 0, αa a∈L

where

a∈L

∀(x, λ) ∈ K 1 ,

(3.13)

3.2 The Supply Chain Network Models with Labor

∂ cˆb (f ) ∂ Cˆ p (x) ≡ δap , ∂xp ∂fa

∀p ∈ P ,

a∈L b∈L

43

∂ ρˆk (x) ∂ρk (d) ≡ , ∂xp ∂dk

∀p ∈ Pk , ∀k.

(3.14)

Labor Scenario 2 Variational Inequality Formulation

For Labor Scenario 2, associate the nonnegative Lagrange multiplier μt with labor constraint (3.11, t), for t = 1, . . . , T + 1, and define the vector of Lagrange n +T +1 T +1 and the feasible set K 2 ≡ {(x, μ) ∈ R+P }. The multipliers μ ∈ R+ variational inequality formulation, whose solution corresponds to the supply chain network optimization problem with objective function (3.9) and with labor under Scenario 2, is: Determine (x ∗ , μ∗ ) ∈ K 2 , such that: J

⎡ ⎣

k=1 p∈Pk

+

T +1 t=1

+

J wa ∂ Cˆ p (x ∗ ) ∂ ρˆv (x ∗ ) ∗ + δap − ρˆk (x ∗ ) − xq ∂xp αa ∂xp a∈L

μt∗

q∈Pv

  1 δap × xp − xp∗ αa t

a∈L

 T +1

l¯t −

t=1

v=1







p∈P

xp∗ δap

αa

a∈Lt



  × μt − μt∗ ≥ 0,

∀(x, μ) ∈ K 2 .

(3.15)

Labor Scenario 3 Variational Inequality Formulation

Finally, the variational inequality formulation for the solution of the supply chain network optimization problem under Labor Scenario 3 is provided. Recall that this scenario is the most generous in terms of the movement of labor across the links of the supply chain network (cf. (3.12)). Associate the nonnegative Lagrange multiplier γ with constraint (3.12), and define the appropriate feasible set K 3 ≡ {(x, γ ) ∈ n +1 R+P }. The variational inequality for Scenario 3 with objective function (3.9) is: Determine (x ∗ , γ ∗ ) ∈ K 3 , such that: J k=1 p∈Pk

⎡

⎤ J ∗) 1 wa ∂ Cˆ p (x ∗ ) ∂ ρ ˆ (x v ⎣ + δap − ρˆk (x ∗ ) − xq∗ + γ ∗ δap ⎦ ∂xp αa ∂xp αa a∈L

v=1

   ∗     p∈P xp δap ∗ ¯ × γ − γ ∗ ≥ 0, × xp − xp + l − αa a∈L

q∈Pv

a∈L

∀(x, γ ) ∈ K 3 .

(3.16)

44

3 Optimization of Supply Chains Under Different Labor Constraints

3.2.2 Illustrative Examples Several examples corresponding to Scenarios 1 and 3 above are now presented for illustrative purposes. The examples are solved algebraically. In Section 3.4, additional, more complex, numerical examples are solved, algorithmically. The supply chain network topology is depicted in Figure 3.2. It consists of the firm, its two production facilities, a single distribution center, and a single demand market. There is no electronic commerce. The example is inspired by an expensive item for the treatment of patients with COVID-19, and the demand point corresponds to hospitals that the firm is considering delivering the healthcare item to. The hospitals are in the same metropolitan region, such as Boston or NYC, both of which have been severely impacted by the coronavirus. The data are as follows. The operational costs on the links (without the labor costs) are cˆa (f ) = 2fa2 ,

cˆb (f ) = 2fb2 , cˆe (f ) = fe2 + 2fe ,

cˆc (f ) = .5fc2 ,

cˆd (f ) = .5fd2 ,

cˆf (f ) = .5ff2 ;

the wages on the links are: wa = 10,

wb = 10,

wc = 4,

wd = 4,

we = 2,

Fig. 3.2 The supply chain network topology for the illustrative examples

wf = 6.

3.2 The Supply Chain Network Models with Labor

45

First, Scenario 1 is considered, with the linear expressions relating labor to product flows being: fa = la ,

fb = lb ,

fc = 10lc ,

fd = 10ld ,

fe = 10le ,

ff = 20lf ,

which corresponds to: αa = αb = 1; αc = αd = 10; αe = 10, and αf = 20. The bounds on the links’ labor availability are l¯a = 20,

l¯b = 20,

l¯c = 30,

l¯d = 30,

l¯e = 100,

l¯f = 120.

The demand price function at the demand market is ρ1 (d) = −d1 + 80000.

Define path p1 = (a, c, e, f ) and path p2 = (b, d, e, f ). Due to the supply chain network topology in Figure 3.2 and the fact that the operational cost function on link a and the wage is equal to the operational cost function on link b and the wage is as well, and, similarly, the operational cost on link c is equal to the operational cost on link d (as are the wages), one knows that: xp∗1 = xp∗2 . One can, hence, solve variational inequality (3.13) algebraically, noting that the optimal path flows are positive. Furthermore, it is reasonable to assume that the optimal labor values on links a and b will be at the imposed bounds, since they are low. A straightforward computation then yields xp∗1 = 20.00

xp∗2 = 20.00,

and, hence, dw∗ 1 = 40.00 with the demand price: 79, 960.00. In addition, it follows that fa∗ = fb∗ = fc∗ = fd∗ = 20.00,

fe∗ = ff∗ = 40.00,

and la∗ = lb∗ = 20.00,

lc∗ = ld∗ = 2.00,

le∗ = 4.00,

lf∗ = 2.00.

Since the upper bound on labor is met on link a , and on link b, that is, these constraints are tight, one can also compute the associated Lagrange multipliers: λ∗a = λ∗b = 79, 687.10. All other Lagrange multipliers associated with the links and labor constraints on them are equal to 0.00. The profit of the firm corresponding to the value of the objective function (3.5), equivalently (3.9), is 3, 195, 644.00. Now, Scenario 3 is considered. I retain the data as above except that I set l¯ =  ¯ a∈L la = 320. This represents the situation that the firm has the same amount of available labor as in Scenario 1, but the labor is free to move among the supply chain network economic activities. The solution to this supply chain network optimization problem is governed by variational inequality (3.16). It is easy to see that the optimal

46

3 Optimization of Supply Chains Under Different Labor Constraints

solution is now: xp∗1 = xp∗2 = 160, with the optimal Lagrange multiplier being: γ ∗ = 4, 469.70. The demand d1∗ = 320.00 and the demand price is now: 79, 680.00. In addition, one has that fa∗ = fb∗ = fc∗ = fd∗ = 160.00,

fe∗ = ff∗ = 320.00,

and la∗ = lb∗ = 160.00,

lc∗ = ld∗ = 16.00,

le∗ = 32.00,

lf∗ = 16.00.

The firm, under much greater flexibility, now enjoys a profit of: 25, 223, 040.00, and the hospitals obtain 320 of the healthcare items, as opposed to only 40 in Scenario 1 above, and at a lower price. Even this simple set of illustrative examples demonstrates the need for flexibility for labor movement in a pandemic situation or even in other disaster settings, given the advantages. Of course, in the case of certain products that are quite complex, such a transferral/reallocation of labor between/among tasks may not be possible or may require additional training, which may be costly. Interestingly, as reported by Kallingal (2020), many airlines early on in the COVID-19 pandemic grounded planes and temporarily laid-off workers, with some airlines, notably, in Sweden and in the United Kingdom, encouraging flight attendants to retrain to help hospitals in the coronavirus pandemic. These examples also provide insights under times when crisis management is not needed since a firm may be interested in evaluating the benefits of flexibility of its labor assets in its supply chain. In addition, the values of the Lagrange multipliers provide valuable information as well, since they act as shadow prices reflecting the added contribution to profits of an additional unit of the associated labor resource with respect to the relevant constraint(s).

3.2.3 Variational Inequality Formulations of the Fixed Demand Case The above supply chain network optimization model with labor, under different scenarios, assumes that the demand for the product at the demand markets is elastic, that is, that the consumers are sensitive to the price. In the case of certain products during the COVID-19 pandemic, and other crisis situations, demand for certain products may be inelastic, that is, fixed. Observe that a special case of each of the above scenarios can be constructed for the fixed demand case. Indeed, one now has the following constraints, where, without loss of generality, dk is assumed to be known and fixed for all k :

3.2 The Supply Chain Network Models with Labor



xp = dk ,

47

k = 1, . . . , J.

(3.17)

p∈Pk

The feasible sets for the three scenarios, respectively, in the case of fixed n +n demands are: Kˆ 1 ≡ {(x, λ) ∈ R+P L such that (3.17) holds}; Kˆ 2 ≡ {(x, μ) ∈ nP +T +1 n +1 R+ such that (3.17) holds}, and Kˆ 3 ≡ {(x, γ ) ∈ R+P such that (3.17) holds}. In the case of fixed demands, the objective function (3.9) simplifies to Minimize



c˜a (x) +

a∈L

wa xp δap . αa

a∈L

(3.18)

p∈P

As in the elastic demand case, each set of labor constraints corresponds to a different feasible set. Observe that, in the fixed demand case, the labor capacities may be such that the demand cannot be satisfied and, hence, the problem is infeasible. Of course, this may actually occur in the case of a pandemic or in a crisis situation, including a war, since demand for a product, especially a critical needs product, may exceed the availability of labor to produce it because of labor shortfalls. In order to check if the demands can be satisfied under a scenario, and, as noted in Qiang and Nagurney (2012), where a bicriteria supply chain network performance measure was introduced in the case of disasters, one can first solve the maximum flow problem (cf. Ahuja et al. 1993), which is a well-known classical network optimization problem in operations research. Here, the assumption is made that the demands can be satisfied. The analogous variational inequalities for the fixed demand case under the three scenarios are now given. The results are immediate by incorporating the fixed demand constraints (3.17) and simplifying the respective scenarios variational inequalities: (3.13), (3.15), and (3.16), accordingly, as done below. Labor Scenario 1 Variational Inequality Formulation for the Fixed Demand Case

The solution to the Labor Scenario 1 supply chain network optimization problem for the fixed demand case is equivalent to the solution of the variational inequality: Determine (x ∗ , λ∗ ) ∈ Kˆ 1 such that J k=1 p∈Pk

+

a∈L



   wa λ∗ ∂ Cˆ p (x ∗ ) a + δap + δap × xp − xp∗ ∂xp αa αa



 l¯a −

a∈L

p∈P

xp∗ δap

αa



a∈L

  × λa − λ∗a ≥ 0,

∀(x, λ) ∈ Kˆ 1 .

(3.19)

Labor Scenario 2 Variational Inequality Formulation for the Fixed Demand Case

The variational inequality formulation for the Labor Scenario 2 supply chain network optimization problem in the case of fixed demands is, in turn: Determine (x ∗ , μ∗ ) ∈ Kˆ 2 , such that:

48

3 Optimization of Supply Chains Under Different Labor Constraints



J k=1 p∈Pk

 T +1   1 wa ∂ Cˆ p (x ∗ ) t∗ + δap + μ δap × xp − xp∗ ∂xp αa αa t a∈L

a∈L

t=1

   T +1 ∗   p∈P xp δap t ¯ l − + × μt − μt∗ ≥ 0, αa t

∀(x, μ) ∈ Kˆ 2 .

(3.20)

a∈L

t=1

Labor Scenario 3 Variational Inequality Formulation for the Fixed Demand Case

The variational inequality for Labor Scenario 3 in the fixed demand case is, in turn: Determine (x ∗ , γ ∗ ) ∈ Kˆ 3 , such that J



k=1 p∈Pk



+ l¯ −

   1 wa ∂ Cˆ p (x ∗ ) ∗ + δap + γ δap × xp − xp∗ ∂xp αa αa

a∈L



a∈L

∗ p∈P xp δap

αa



a∈L

  × γ − γ ∗ ≥ 0,

∀(x, γ ) ∈ Kˆ 3 .

(3.21)

All of the above six variational inequalities (three for the elastic demand case and three for the fixed demand case) can be put into standard variational inequality form (cf. (2.14) or the Appendix).

3.3 The Computational Procedure Since the supply chain network optimization model with labor, under its realizations in the case of both elastic demands and fixed demands, and under all three labor capacity scenarios, admits a variational inequality formulation, the modified projection method of Korpelevich (1977) is proposed for computational purposes. In the case of elastic demands, each iteration of the modified projection method will yield closed form expressions for the path flows and for the associated Lagrange multipliers. This is a nice feature for implementation. On the other hand, in the case of fixed demands, in order to guarantee that the fixed demand is satisfied at each demand market, it is recommend that the equilibration algorithm of Dafermos and Sparrow (1969), which has been used in a variety of network and supply chain settings (cf. Nagurney and Zhang 1996; Nagurney 1999), will be applied. The conditions for convergence of the modified projection method are that the function F (X) that enters the variational inequality (cf. (2.14)) is Lipschitz continuous and monotone (refer to Chapter 2). These are reasonable conditions for the supply chain network optimization model with labor under its demand realizations as well as scenarios. In the case of fixed demands, one still has that at each iteration the relevant Lagrange multipliers can be computed exactly and in closed form. The statement of this algorithm can be found in Chapter 2 and in the Appendix.

3.3 The Computational Procedure

49

The modified projection method is implemented in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the computation of solutions to the subsequent numerical examples. Elastic demand examples are solved, first for Scenario 1 and then for Scenario 3. The algorithm is initialized as follows. The elastic demand for each demand market is set to 40 and equally distributed among the paths connecting each demand market from the origin node 1 (The Firm). The Lagrange multipliers are initialized to 0. The algorithm is deemed to have converged if the absolute difference of the path flows differs by no more than 10−7 and the same for the Lagrange multipliers.

3.3.1 Scenario 1 Healthcare Product Supply Chain Elastic Demand Examples Examples 3.1 and 3.2 have the supply chain network topology given in Figure 3.3, whereas Examples 3.3 and 3.4 have electronic commerce included and, hence, have the supply chain network topology depicted in Figure 3.4. Example 3.1 (Baseline) Example 3.1 is constructed from the Scenario 1 Illustrative Example in Section 3.2.2 with the addition of a new Demand Market 2. This example has the same data as that example with the following additions for the new link g: cˆg (f ) = .5fg2 ,

wg = 6.00,

αg = 20.00,

so that fg = 20lg .

Fig. 3.3 The supply chain network topology for Examples 3.1 and 3.2

50

3 Optimization of Supply Chains Under Different Labor Constraints

The demand price function at the second demand market is: ρ2 (d) = −d2 + 80500. Observe that those at Demand Market 2 are willing to pay a higher price for the healthcare product than those at Demand Market 1. Since Example 3.1 serves as the baseline for subsequent examples, I am interested in determining what would be the optimal flows and optimal levels of labor if the bounds on labor are quite high (and, thus, the associated optimal Lagrange multipliers would all be 0.00). Thus, the link labor bounds are set as l¯a = 200000.00,

l¯b = 20000.00,

l¯e = 100000.00,

l¯c = 30000.00,

l¯f = 120000.00,

l¯d = 30000.00,

l¯g = 120000.00.

Paths p1 and p2 remain as in Section 3.2.2 with the new paths p3 and p4 associated with the Demand Market 2 being: p3 = (a, c, e, g), p4 = (b, d, e, g). The modified projection method converges to the following solution: xp∗1 = xp∗2 = 3301.55,

xp∗3 = xp∗4 = 3384.88.

The optimal link labor values are la∗ = 6686.43,

lb∗ = 6686.43,

le∗ = 1337.29,

lc∗ = 668.64,

lf∗ = 330.16,

ld∗ = 668.64,

lg∗ = 338.49.

As expected, all the Lagrange multipliers are equal to 0.00: λ∗a = λ∗b = λ∗c = λ∗d = λ∗e = λ∗f = λ∗g = 0.00. The demand price at Demand Market 1 is: 73, 396.90 and at Demand Market 2: 73, 730.25. The computed respective demands are: 6, 603.10 and 6, 769.75. The profit of the firm is 536, 520, 192.00. Example 3.2 (A Much Tighter Labor Bound on a Manufacturing Link) The situation is now considered in which the pandemic has had an impact on labor especially in the community where the first manufacturing plant in Figure 3.3 is located. Hence, the labor bound on link a is now significantly reduced to l¯a = 5000.00. The rest of the data is identical to that in Example 3.1. The modified projection method converges to the following solution: xp∗1 = 2458.39,

xp∗2 = 3648.73,

The optimal link labor values are

xp∗3 = 2541.71,

xp∗4 = 3732.07.

3.3 The Computational Procedure

la∗ = 5000.00,

51

lb∗ = 7380.79,

le∗ = 1238.01,

lc∗ = 500.00,

lf∗ = 305.36,

ld∗ = 738.08,

lg∗ = 313.69.

Since the bound on link a is now tight, with la∗ = 5000.00, the associated Lagrange multiplier is positive and is equal to 11903.52. All other Lagrange multipliers are equal to 0.00. The demand price at Demand Market 1 is now: 73,892.89, and at Demand Market 2, it is: 74,226.22. One can see the increase in the price for the healthcare product, as compared to the values in Example 3.1. The computed optimal demands at the demand markets are: 6107.11 for Demand Market 1 and 6273.78 for Demand Market 2. The firm now obtains a profit of: 526,483,680.00, a decrease of over 10,000,000 from that in Example3.1. Example 3.3 (Introduction of Electronic Commerce) Example 3.3 introduces electronic commerce to Example 3.2 with the underlying supply chain network topology as in Figure 3.4. The electronic commerce links in Figure 3.4 are link h joining the first manufacturing plant node with Demand Market 1 and link i joining the second manufacturing node with Demand Market 2.

Fig. 3.4 The supply chain network topology for Examples 3.3 and 3.4

52

3 Optimization of Supply Chains Under Different Labor Constraints

There are now two additional paths defined as p5 = (a, h),

p6 = (b, i).

The data in Example 3.3 are as in Example 3.2 with the following additions associated with the electronic commerce links: cˆh (f ) = fh2 ,

cˆi (f ) = fi2 ,

and wh = 10.00,

wi = 10.00,

αh = 1.00,

αi = 1.00,

with labor bounds on the electronic links of l¯h = 100000.00,

l¯i = 100000.00.

The modified projection method converges to the following solution: xp∗1 = 0.00,

xp∗2 = 2967.86,

xp∗3 = 0.00,

xp∗4 = 1501.03,

xp∗5 = 5000.00,

xp∗6 = 7450.30.

The optimal link labor values are la∗ = 5000.00,

lb∗ = 11919.19,

le∗ = 446.89,

lc∗ = 0.00,

lf∗ = 446.89,

lh∗ = 5000.00,

ld∗ = 446.89,

lg∗ = 75.05,

li∗ = 7450.30.

For both demand markets, the paths p5 and p6 with the electronic commerce link garner the majority of the healthcare product flows. In the case of electronic commerce, one still has freight service delivery, but, in our framework, there is no transportation to and from a distribution center. Paths p1 and p3 have zero product flow. The demand price at Demand Market 1 is now: 72,032.09, whereas at Demand Market 2, it is 71,548.67. With the introduction of electronic commerce, the consumers at the demand markets enjoy lower prices. The computed optimal demands at the two demand markets are now, respectively, 7967.91 and 8951.33. The firm has a profit of: 763,964,416.00, which exceeds the profits garnered in Examples 3.1 and 3.2. In Example 3.3, unlike in Example 3.1, since there is a much tighter bound on the labor available on link a (as was also the case in Example 3.2), the optimal labor value on link a is at the bound of 5000.00, and the Lagrange multiplier is positive and now equal to: 34,043.8672. All other Lagrange multipliers are equal to 0.00.

3.3 The Computational Procedure

53

Example 3.4 (Closure of a Manufacturing Plant) Example 3.4 has the same data as Example 3.3 except that now I consider an even more disruptive scenario (as has occurred during the COVID-19 pandemic). Specifically, I consider the situation that, due to illnesses, employees’ fear of contagion, etc., the first manufacturing plant is shut down. Hence, now: l¯a = 0.00. The modified projection method converges to the solution: xp∗1 = 0.00,

xp∗2 = 4932.13,

xp∗4 = 72.47,

xp∗5 = 0.00,

xp∗3 = 0.00,

xp∗6 = 7539.58.

The computed optimal link labor values are la∗ = 0.00,

lb∗ = 12544.18,

le∗ = 500.46,

lc∗ = 0.00,

lf∗ = 246.61,

lh∗ = 0.00,

ld∗ = 500.46,

lg∗ = 362.00,

li∗ = 7539.58.

Observe that for Demand Market 1 only path p2 has positive flow with the other two paths not used, which makes sense since essentially link a is unavailable due to the closure of the manufacturing plant. The Lagrange multiplier λ∗a = 70,114.92 with all other Lagrange multipliers equal to 0.00. The demand price at Demand Market 1 is: 75,067.77 at a computed optimal demand of: 4932.23. At Demand Market 2, the demand price is: 72,887.95 at a demand of: 7612.06. With closure of the plant, the firm’s profit is only 503,570,432.00, and it suffers immense losses in profits (over 250,000,000.00) as compared to Example 3.3. This profit is the lowest of all the computed numerical examples to this point. This example vividly illustrates the importance of keeping operations running during the pandemic and having appropriate healthcare pandemic mitigation processes and procedures in place. With one of the two manufacturing plants closed, the prices rise at both demand markets. As another wave of the coronavirus hit China in the spring of 2022, some manufacturing plants there were shut down with ripple effects felt globally (Associated Press 2022).

3.3.2 Scenario 3 Healthcare Product Supply Chain Elastic Demand Examples: Reduction of Labor Availability The computed solutions to elastic demand examples for Scenario 3 are now reported. The supply chain network topology remains as in Figure 3.4, and the data are identical to those in Examples 3.3 and 3.4 except that, instead of having link labor bounds, there is a single upper bound on labor l¯. Specifically, three examples under this scenario are solved. Example 3.5 has the upper bound l¯ = 50000.00;

54

3 Optimization of Supply Chains Under Different Labor Constraints

Table 3.2 Optimal product flows for Examples 3.5–3.7 representing Scenario 3 Optimal product flows xp∗1 xp∗2 xp∗3 xp∗4 xp∗5 xp∗6

Example 3.5 1313.60 1294.37 1380.91 1361.68 7998.25 8046.33

Example 3.6 1785.36 1766.13 1852.67 1833.44 214.22 262.30

Example 3.7 958.35 958.35 1041.69 1041.69 0.00 0.00

Table 3.3 Optimal link labor values for Examples 3.5–3.7 representing Scenario 3 Optimal link labor values la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗ lh∗ li∗

Example 3.5 10692.76 10702.38 269.45 265.61 535.06 130.40 137.13 7998.25 8046.33

Example 3.6 3852.25 3861.87 363.80 359.96 723.76 177.57 184.31 214.22 262.30

Example 3.7 2000.00 2000.00 200.00 200.00 400.00 95.84 104.17 0.00 0.00

Example 3.6 has l¯ = 10, 000.00, and Example 3.7 has l¯ = 5000.00. These examples illustrate the impact of reduction of labor availability (whether due to a pandemic or other crisis such as a major conflict or war). The computed optimal product flows are reported in Table 3.2. For Example 3.5, since the labor upper bound constraint is not tight, γ ∗ = 0.00. In Example 3.6, on the other hand, the constraint is tight and γ ∗ = 28,305.57. In Example 3.7, γ ∗ = 44,989.31. The computed optimal link labor values are reported in Table 3.3. In Example 3.5, the computed demand at Demand Market 1 is: 10,606.22, whereas the demand at Demand Market 2 is: 10,788.91. The computed demand in Example 3.6 is: 3765.71 for Demand Market 1 and 3948.40 for Demand Market 2. In Example 3.7, the computed demand for the healthcare product is: 1916.71 for Demand Market 1, and it is: 2083.37 for Demand Market 2. In Example 3.5, the demand price at Demand Market 1 is: 69,393.78, and at Demand Market 2, it is: 69,711.09. In Example 3.6, the demand price at Demand Market 1 is: 76,234.29, and at Demand Market 2, it is: 76,551.59. Finally, in Example 3.7, the demand price at Demand Market 1 is: 78,073.30, and at Demand Market 2, it is: 78,416.62. As the labor resources become more constrained, the prices rise, which is very reasonable, and we see it happening now for various products during the COVID-19 pandemic as well as during Russia’s war against Ukraine.

3.4 Supply Chain Network Efficiency and Resilience to Labor Disruptions

55

In terms of profits, the firm earns in Example 3.5 a profit of: 858,307,968.00, a profit of: 451,028,736.00 in Example 3.6, and a profit of: 272,973,568.00 in Example 3.7. The profit earned in Example 3.5 is the highest of all the examples, and that in Example 3.7 is the lowest. Interestingly, in Example 3.7, the paths with the electronic commerce links are not used; that is, the paths p5 and p6 have zero flow. At first, this may seem puzzling; however, if one looks at the cognizant variational inequality (3.16) and the term:  γ ∗ a∈L α1a δap , this result is clarified. In the dataset, the links on the other paths, corresponding to those post the common manufacturing link(s), have αa s higher than the αh and αi , signifying greater productivity among the storage workers and the associated freight service providers.

3.4 Supply Chain Network Efficiency and Resilience to Labor Disruptions Now, the important topic of supply chain network efficiency (performance) is discussed. The focus here is on the elastic demand supply chain network optimization models.

3.4.1 Efficiency of a Supply Chain Network and Importance Identification of a Network Component Here, without loss of generality, the efficiency measure can handle any of the three elastic demand models corresponding to distinct sets of labor constraints. Let l¯ now correspond to the labor bounds associated with the specific model, associated with variational inequality (3.13), (3.14), or (3.16), respectively. The resilience of a supply chain network is with respect to a baseline measure of performance, which is referred to as efficiency, E , and is defined as ¯ ≡ E = E (G , c, ˆ ρ, w, α, l)

k

dk∗ ρk (d ∗ )

J

,

(3.22)

with the demands, d ∗ , and the incurred demand market prices in (3.22), evaluated at the solution to the corresponding variational inequality. According to (3.22), given a supply chain network topology, and the various parameters and functions, a supply chain with labor is evaluated as performing better if, on the average, it can handle higher demands at lower prices. Using ideas in Nagurney and Qiang (2009) and in Nagurney and Li (2016) for supply chains, one can then define the importance of a supply chain network component g (node, link, or a combination of nodes and links), I (g), which

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represents the efficiency drop when g is removed from the supply chain network, as I (g) =

¯ − E (G − g, c, ¯ E (G , c, ˆ ρ, w, α, l) ˆ ρ, w, α, l) ΔE = . ¯ E E (G , c, ˆ ρ, w, α, l)

(3.23)

One can rank the importance of nodes or links, using (3.23). This is valuable for decision-making and policy-making. For example, those supply chain network components that are of higher importance should be the subject of greater investment in terms of maintenance.

3.4.2 Resilience with Respect to Labor Disruptions ¯ denote the reduction of labor I now introduce the two resilience measures. Let lγ availability with γ ∈ (0, 1] so that, if, for example, γ = .8, this means that the labor availability associated with the labor constraints is now 80% of the original labor availability as in E . Resilience Measure Capturing Labor Availability ¯ One then can construct R lγ , where ¯ ¯ ¯ = R lγ = R lγ (G, c, ˆ ρ, w, α, l)

¯

E lγ × 100%, E

(3.24)

with E as in (3.22). The expression in (3.24) captures the resilience of the supply chain network subject to reduction of labor availability. The closer the value is to 100%, the greater the resilience. Resilience Measure Capturing Labor Productivity

Analogously, one can construct a resilience measure, R αγ , that captures the resilience of the supply chain network with respect to the reduction in productivity of the laborers, as reflected by the change in the vector of link productivity factors α with γ , again, lying in the range: (0, 1], where ¯ = R αγ = R αγ (G , c, ˆ ρ, w, α, l)

E αγ × 100%. E

(3.25)

3.4.3 Supply Chain Network Data I now describe the data for the supply chain network examples for which I calculate ¯ ¯ the above efficiencies: E , E lγ , E αγ , and the resilience measures R lγ and R αγ , for γ = .9, .7, .5, .3, .1.

3.4 Supply Chain Network Efficiency and Resilience to Labor Disruptions

57

The first example in this set, Example 3.8, has the identical data to that in Example 3.1 with the exception that the link labor bounds l¯a = 10,000 for all links a ∈ L . The topology of the supply chain network is given in Figure 3.3. The second example in this set, Example 3.9, has the same data as Example 3.8, except that now there is a single bound on labor l¯ = 70,000. Hence, labor is free to work on any link, provided that the sum of the labor hours does not exceed 70,000. Note that 70,000 is the sum of the labor bounds on all the links in Example 3.8. The third example, Example 3.10, has the identical data to that of Example 3.8 except that now I add electronic commerce links h and i as in Figure 3.4. The additional data for this example associated with the electronic commerce links are cˆh (f ) = fh2 ,

cˆi (f ) = fi2 ,

and wh = 10.00,

wi = 10.00,

αh = 1.00,

αi = 1.00,

with labor bounds on the electronic links of l¯h = 10,000.00,

l¯i = 10,000.00.

The fourth example in this set, denoted by Example 3.11, has the same topology and data as Example 3.10 except that the labor availability constraint is for the entire supply chain with l¯ = 90,000. I choose the value of 90,000 since there are nine links in Example 3.10, with each link having a bound of 10,000, and hence, there would be a total labor availability of 90,000 under the assumption that laborers would be free and interested in doing whichever tasks needed in the supply chain network (production, transportation, storage, or distribution) with the productivity factors being as in Example 3.10. The final example, Example 3.12, is a type of stress test. Example 3.12 has the same data as Example 3.11, but now the labor hours available are no longer 90,000, but, rather, there are only 70,000 hours available. Hence, the results for Example 3.12 allow us to make a comparison with Example 3.9 in terms of the impact of adding electronic commerce and having the same total amount of labor in the supply chain network available as before but having additional supply chain activities of electronic commerce. The efficiency and resilience measures are reported in Table 3.4 for the five supply chain network examples in this set. The data in Table 3.4 reveal interesting results. One sees, from comparing results for Example 3.8 and 3.9, and also for Examples 3.10 and 3.11, that, consistently, by having labor be free to move across the supply chain network, as is the case for Example 3.9, versus Example 3.8, and for Example 3.11, versus Example 3.10, one can attain a higher efficiency of the supply chain (with the same total number of labor hours available) as well as a higher resilience with respect to both resilience measures and at different values of γ .

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Table 3.4 Efficiency and resilience measures for Examples 3.8 through 3.12 Efficiency and resilience E ¯ E l.9 ¯ l.7 E ¯

E l.5 ¯ E l.3 ¯ l.1 E E α.9 E α.7 E α.5 E α.3 E α.1 ¯ R l.9 ¯

R l.7 ¯ R l.5 ¯

R l.3 ¯ R l.1 R α.9 R α.7 R α.5 R α.3 R α.1

Example 3.8 0.0667 0.0658 0.0505 0.0357 0.0212 0.0071 0.0658 0.0505 0.0355 0.0210 0.0069 0.9872 0.7571 0.5351 0.3178 0.1064 0.9870 0.7566 0.5327 0.3149 0.1035

Example 3.9 0.0909 0.0909 0.0909 0.0909 0.0909 0.0909 0.0909 0.0909 0.0909 0.0909 0.0362 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3977

Example 3.10 0.1424 0.1263 0.0956 0.0665 0.0388 0.0126 0.1263 0.0956 0.0665 0.0388 0.0126 0.8872 0.6712 0.4467 0.2727 0.0886 0.8872 0.6712 0.4467 0.2727 0.0886

Example 3.11 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1092 0.0470 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.7098 0.3054

Example 3.12 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1538 0.1391 0.0877 0.0362 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9044 0.5704 0.2350

Now, I consider whether resilience with respect to labor availability yields similar answers to resilience with respect to labor productivity. From Table 3.4, one sees ¯ is very similar that, for each supply chain network example, the value of R lγ to the corresponding R αγ for the same value of γ for each example, in fact, identical in many cases, until the value of γ gets reduced to .5 or lower. Also, one sees that, when this does occur, as in Example 3.8, Example 3.9, Example 3.11, and Example 3.12, the resilience of the respective supply chain network with respect to labor availability exceeds the resilience with respect to labor productivity. This suggests that the firms should take care of their workers since a reduction in labor productivity can impact the supply chain network performance and the corresponding resilience. In order to investigate impacts of a modification in the supply chain network topology, as in the addition of electronic commerce, for example, one can compare the results in Table 3.4 for Example 3.12 and Example 3.9. Clearly, the efficiency of the supply chain network with electronic commerce options (Example 3.12) is consistently higher than that for the supply chain network without electronic commerce (Example 3.9) at the same value of labor availability and disruption and at the same level of disruption to labor productivity on the links with the exception

3.5 Summary, Conclusions, and Suggestions for Future Research

59

of the respective values of E .3α (and those respective values are equivalent to two decimal points). In addition, both these supply chain networks, under the specific data, retain their efficiency under even restrictive disruptions to labor availability. However, that is not the case when there are disruptions to labor productivity. Interestingly, as can be seen from the values of R αγ , the supply chain network in Example 3.9, which is without electronic commerce, is more resilient than the supply chain network in Example 3.12 for γ = .5, .3, .1. This can be explained by noting that the available labor hour amount is divided among fewer supply chain network economic activities in the case of Example 3.9. Again, one sees, from the investigation of results for Example 3.12 versus those for Example 3.9, that labor productivity on the links, when disrupted, can have even a bigger impact on resilience than a disruption to labor availability. The above supply chain network examples are stylized, but they, nevertheless, reveal the types of questions that can be addressed using the efficiency and resilience measures proposed here for supply chain networks in which labor is a critical resource.

3.5 Summary, Conclusions, and Suggestions for Future Research The COVID-19 pandemic has impacted the globe, causing great personal and economic strife and uncertainty. Healthcare has been one of the major sectors impacted since this sector is at the forefront of the battle against the coronavirus. In addition, the food sector, with food essential to well-being and health, has been challenged, with many reluctant to go to grocery stores and even hygiene and sanitation supplies experiencing shortages. Importantly, products for essential workers, including healthcare ones, such as PPEs, have been in short supply in the USA during various times in the COVID-19 pandemic. This is, due, in part, to much of the manufacturing of such products (and many other, including pharmaceuticals) having been done originally in China, where the coronavirus was believed to have originated, specifically, in the city of Wuhan. With factories shut down and also the demand growing globally, prices of such products, when they can be acquired, have grown. Even states in the USA have been competing for such supplies and scrambling to acquire them for the hospital workers. With people getting sick, some tragically perishing, and others experiencing anxiety and fear, plus the acknowledgment of social distancing to mitigate the spread, supply chains are trying to adapt to this new world scenario. Critical to such supply chains is the resource of labor, and during the pandemic, the need for this resource has become vividly apparent. Some factories and processing plants have actually been hot spots of spread of the coronavirus since workers are in proximity to do their tasks. With labor less available, some facilities

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have had to shut down, creating further disruptions to needed supplies and products. Some businesses have even reallocated their workers to different tasks. In this chapter, a supply chain network optimization framework is developed that explicitly contains labor as a variable in supply chain network economic activities of production, transportation, storage, and distribution. Electronic commerce is allowed, since it has been a lifeline for many in obtaining products, with the goal of reducing exposure in grocery stores, for example. First, profit-maximizing models with elastic demands are proposed in the case of three labor scenarios, of increasing flexibility, as to reallocation. The first scenario has labor bounds on the links; the second scenario allows for transfer of labor across a tier of activity, whereas the third, and most flexible scenario, allows for labor reallocation across the supply chain, and there is a single bound on labor. Illustrative examples are provided. Subsequently, fixed demand analogues for the three scenarios are constructed. The methodological tool for the formulation, analysis, and solution of the models is that of variational inequality theory. In addition to the models and theoretical constructs, solutions to computed numerical examples are provided, motivated by the healthcare disaster caused by the pandemic. The results reveal the impacts on demands, product flows, prices, as well as profits of the reduction of available labor, the introduction of electronic commerce, and also a plant closure. Furthermore, a supply chain network efficiency measure is delineated, along with the definition of the importance of a supply chain network node, link, or combination thereof. The efficiency measure is then used to construct resilience measures for a supply chain network associated with disruptions in labor, in the form of either labor availability or labor productivity. Solution of a series of supply chain network examples yields managerial insights of relevance also to policy-makers and decision-makers. This work serves as the foundation for the explicit incorporation of labor and the quantification of the impacts of labor availability in different scenarios, for other applications, with appropriate adaptations and extensions. Future research can include incorporating the costs of training and re-educating workers to assume different supply chain network tasks.

3.6 Sources and Notes This chapter is based primarily on the paper by Nagurney (2021), with Section 3.4 drawn from Nagurney and Ermagun (2022). Here the discussion has been amplified and the notation standardized. The World Health Organization declared the COVID-19 pandemic on March 11, 2020 (see WHO 2020). Given the disruptions that occurred in a spectrum of supply chain networks, researchers have been responding through relevant publications. For example, Queiroz et al. (2020) described a research agenda via a structured literature review of COVID-19 related work and supply chain research on earlier

References

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epidemics. Ivanov (2020a) overviewed simulation-based research focusing on the potential impacts on global supply chains of the COVID-19 pandemic. Ivanov and Dolgui (2020) emphasized the importance of a novel perspective through the application of intertwined supply networks (ISNs). Currie et al. (2020) identified multiple, complex challenges due to the COVID-19 pandemic and clarified how simulation modeling can assist in supporting enhanced decision-making. Ivanov (2020b) proposed a new concept—that of a viable supply chain (VSC), in which viability is considered as an underlying supply chain property spanning: agility, resilience, and sustainability. Such a perspective can aid firms in their decisions on the recovery and re-building of their supply chains after crises of long duration such as the COVID-19 pandemic. Ivanov and Das (2020) captured the ripple effect of an epidemic outbreak in global supply chains in their model, with the inclusion of the velocity of pandemic propagation, the duration of production, distribution and market disruption, and a demand decline. The authors analyzed pandemic supply risk mitigation measures and associated recovery paths and included a discussion of prospective global supply chain (re)-designs. Clearly, the pandemic has driven the importance and relevance of supply chains to the forefront in both practice and in the news and is now stimulating the investigation of theoretical as well as empirical constructs. For example, Craighead et al. (2020) highlighted a spectrum of theories that they consider powerful tools for illuminating impacts of the pandemic on supply chains, how organizations responded, and also how supply chains and associated processes can be adjusted if and when another pandemic arrives. Among the theories, they highlight game theory, which we turn to in Chapter 4. van Hoek (2020), in turn, discussed an initial empirical exploration of supply chain risks experienced in the context of COVID-19 and approaches in practice in order to enhance supply chain resilience and argued that such research can assist in closing the gap between supply chain resilience research and efforts in industry in this domain. Paul and Chowdhury (2021) built a mathematical model that can handle both supply and demand disruptions. The model optimizes the revised production plan in the recovery window and can be solved analytically. Using a numerical example, the authors also showed how the model is capable of optimizing the recovery plan in order to better address the disruptions in the pandemic.

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Chapter 4

Game Theory Modeling of Supply Chains and Labor Disruptions

Abstract The COVID-19 pandemic has brought attention to supply chain networks due to disruptions for many reasons, including that of labor shortages as a consequence of illnesses, death, risk mitigation, as well as travel restrictions. Many sectors of the economy from food to healthcare have been competing for workers, as a consequence. In this chapter, a supply chain game theory network framework is constructed that captures labor constraints under three different scenarios. The appropriate equilibrium constructs are defined, along with their variational inequality formulations. Computed solutions to numerical examples inspired by shortages of migrant labor to harvest fresh produce; specifically, blueberries, in the United States, reveal the impacts of a spectrum of disruptions to labor on the product flows and the profits of the firms in the supply chain network economy. A supply chain network efficiency measure is proposed along with the identification of the importance of supply chain network components. This research adds to the literature in both economics and operations research.

4.1 Introduction The COVID-19 pandemic has shown dramatically the importance of the health of workers to the global economy (World Health Organization 2020). Product supply chains as varied as those for Personal Protective Equipment (PPE) (Burki 2020) and other medical supplies (Ranney et al. 2020), toilet paper and cleaning supplies (Gao 2020), meat (Corkery and Yaffe-Bellany 2020) and fresh produce (Laborde et al. 2020; Knight 2020), and even blood (Nagurney 2020 have been disrupted for reasons including that of the reduction of labor availability because of illnesses and death of workers, along with workers’ fear of contracting the disease. Some laborers have not been able to travel for seasonal employment because of travel restrictions to mitigate the spread of the coronavirus that causes the COVID-19 disease, resulting in both economic and personal losses (IHS Markit 2020). According to Adam (2022), some 15 million people have died in the first two years of the COVID19 pandemic. The negative impact of this global healthcare and economic crisis has

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affected different tiers of supply chains and associated network economic activities of production, transportation, storage, and distribution. Labor has emerged as essential in the functioning of supply chains. Without farmers, food cannot be produced. Without workers to harvest, food may end up rotting in the fields or be discarded (see Associated Press 2020); without freight service provision, food cannot be delivered to food processors; and without food processors, food cannot be processed, resulting in additional waste (see Polansek and Huffstutter 2020). Ultimately, freight service provision is also essential for deliveries to retail outlets for purchase by consumers or even for direct transport to consumers using, for example, electronic commerce. Freight service workers have also been impacted by COVID-19 (Parker 2020). Employees at distribution centers have been felled by COVID-19 (Daniels 2020), exacerbating worker shortages at a time of greater demand because of electronic commerce (see Hardwick 2020). Shortages of labor in manufacturing facilities were an issue even earlier on in February 2020 in China, since that country was impacted first by the coronavirus, which was to have originated in Wuhan, with implications for global supply chains (see Bloomberg 2020). Labor shortages arose due to illnesses, the need for social distancing in facilities, as well as adherence to quarantines. Each link of a product supply chain requires labor, and, therefore, reduction in labor capacity can propagate through paths of a supply chain network, affecting product flows as well as prices (see, e.g., OECD 2020). Furthermore, unhealthy workers cannot be fully productive. Plus, as the war of Russia against Ukraine rages, with major disruptions to agricultural production as well as the distribution of food products to markets, food insecurity and food prices are rising (Treisman 2022). This chapter builds on the findings in Chapters 2 and 3, but, in contrast, the focus here is on game theory. Specifically, a supply chain network game theory framework is constructed for the modeling of competition among firms that produce a differentiated but substitutable product. The novelty of the framework lies in that it explicitly includes labor availability. Prior to the COVID-19 pandemic, the incorporation of labor into a competitive supply chain network for differentiated products had not been addressed. This is an important area of research since only through a full supply chain network perspective one can identify the impacts of labor availability and possible capacity disruptions on profits, costs, and consumer prices and the impacts of competition. As done in Chapter 3, but in an optimization setting, three different scenarios of labor availability are considered in the modeling framework, with accompanying constraints. In the first scenario, there is a bound on labor availability associated with each link of the supply chain network of each firm in the supply chain network economy. This is the most restrictive scenario in that labor capacities are imposed on individual links, and, hence, there is not the freedom of movement from firm to firm and across tiers as in the other scenarios. In the second scenario, there is a bound on the labor availability associated with each activity tier of the supply chain networks, such as production, transportation, storage, and distribution. This scenario is relevant to the farming sector, since farmers compete for seasonal migrant workers for harvesting the products, with the COVID-19 creating shortfalls of such labor in

4.1 Introduction

67

many parts of the globe (see, e.g., Corbishley 2020). In the third scenario, there is a single bound on labor availability in the supply chain network economy, and labor is free to move across a tier or between tiers. Barrero et al. (2020) argue for the reallocation of labor because of the economic impacts of the COVID-19 pandemic, which this scenario enables the evaluation and quantification of. The governing equilibrium conditions for the first scenario correspond to a Nash (1950, 1951) Equilibrium, whereas those for the second and third scenarios correspond to a Generalized Nash Equilibrium (GNE) (cf. Debreu 1952 and Arrow and Debreu 1954), since the feasible sets associated with the firms’ strategies are common, that is, shared, because of the respective labor constraints. Although game theory supply chain network models are now fairly well established (cf. Nagurney et al. 2002; Nagurney 2006; Qiang et al. 2013; Toyasaki et al. 2014; Nagurney and Li 2016; Saberi 2018; Saberi et al. 2018; Yu et al. 2019, and the references therein) and advances continue to be made (see, e.g., Gupta et al. 2021 and Gupta and Ivanov 2020), Generalized Nash Equilibrium models have only recently been applied to supply chains. For example, Nagurney et al. (2017) introduced a competitive supply chain network equilibrium model with outsourcing in which firms competed for limited capacity at shared distribution facilities. However, labor was not explicitly considered. Also related to our theme here, in part, is the application of GNE supply chain models in disaster relief (cf. Nagurney et al. 2016, 2018, 2019; and Nagurney et al. 2020, 2021) and in healthcare (Nagurney and Dutta 2019). Both Nash Equilibrium and Generalized Nash Equilibrium are applied in this chapter for supply chain network competition with labor. In Chapter 2, the introduction of labor into a supply chain generalized network optimization model for perishable food in the COVID-19 pandemic was formulated and studied. Therein, arc multipliers captured perishability, akin to the work of Yu and Nagurney (2013), but only labor bounds on links were considered. In Chapter 3, a nonperishable product was considered and an optimization perspective taken for the supply chain network in the case of labor under different scenarios and in the case of elastic or fixed demands for the firm’s product. In Chapters 2 and 3 of this book, there was no competition among firms and no game theory concepts were utilized. The pandemic has disrupted lives globally and the economy. The contributions in this chapter can assist firms in guiding their decisions on the recovery and rebuilding of their supply chains after crises of long duration such as the COVID-19 pandemic as well as conflicts, strife, and wars. This chapter is organized as follows. In Section 4.2, the supply chain network game theory framework for differentiated products under three distinct labor scenarios is built. The supply chain network structure is identified, along with the behavior of the profit-maximizing firms and the underlying constraints. For each scenario, two alternative variational inequality formulations of the governing equilibrium conditions are presented. For Scenarios 2 and 3, which are Generalized Nash Equilibrium problems, the concept of a Variational Equilibrium is used to enable the variational inequality formulations. The existence results are also discussed. The alternative variational inequality formulations allow for the implementation

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

of an effective algorithm, which is recalled in Section 4.3, along with computed solutions to a series of numerical examples. The numerical examples are motivated by shortages of migrant labor to harvest fresh produce, in particular, blueberries in the United States. Section 4.4 then turns to the assessment of supply chain network efficiency in a game theory context and the identification of the importance of supply chain network components, such as nodes and links. The results of the chapter are summarized in Section 4.5, along with the conclusions and suggestions for future research. Section 4.6 is the Sources and Notes Section for this chapter.

4.2 Supply Chain Network Game Theory Modeling Under Labor Constraints There are I firms in the supply chain network economy that produce a substitutable product and compete noncooperatively in the production, transportation, storage, and distribution of their products to demand markets. The firms also compete with one another for labor, since labor is essential to the above network economic activities. Each firm is represented as a supply chain network of its economic activities as drawn in Figure 4.1. Note that, according to Figure 4.1, the supply chain networks of the individual firms do not have any links in common. Table 4.1 contains the basic notation for the model. All vectors are column vectors.

Fig. 4.1 The supply chain network topology of the game theory model with labor

Each firm i, i = 1, . . . , I , owns niM production facilities, can make use of niD distribution centers, and can provide its product to the J demand markets. Let L i denote the links comprising the supply chain network of firm i, i = 1, . . . , I , that

4.2 Supply Chain Network Game Theory Modeling Under Labor Constraints

69

it owns/controls, with a total of nL i elements. The links of L i include firm i’s links to its production nodes, the links from production nodes to the distribution centers, the storage links, and the links from the distribution centers to the demand markets. L then denotes the full set of links in the supply chain network economy with L = ∪Ii=1 L i with a total of nL elements. Let G = [N , L ] denote the graph consisting of the set of nodes N and the set of links L in Figure 4.1. Each firm seeks to determine its optimal product quantities that maximize its profits by using Figure 4.1 as a schematic, coupled with the labor volumes, for which the relevant constraints are discussed below. Let Pki denote the set of paths in firm i’s supply chain network terminating in demand market k, i = 1, . . . , I and k = 1, . . . , J , and let P i denote the set of nP i paths of firm i, i = 1, . . . , I . Then, P denotes the set of all nP paths in the supply chain network economy. Table 4.1 Notation for the supply chain game theory modeling framework with labor Notation xp ; p ∈ Pki

fa dik

la Notation αa wa l¯a l¯t

l¯ Notation cˆa (f ) ρik (d)

Variables The nonnegative flow on path p originating at firm node i and terminating at demand market k, i = 1, . . . , I and k = 1, . . . , J ; n i group firm i’s product path flows into the vector x i ∈ R+P . Note that x i is the vector of strategic variables of firm i; group all the nP firms’ product path flows into the vector x ∈ R+ The nonnegative flow of the product on link a, ∀a ∈ L ; group all the n link flows into the vector f ∈ R+L The demand for the product of firm i at demand market k, i = 1, . . . , I and k = 1, . . . , J ; group the {dik } elements for firm i J and all the demands into the vector d ∈ R I ×J into the vector d i ∈ R+ + The labor on link a (usually denoted in person hours), ∀a ∈ L Parameters Positive factor relating input of labor to output of product flow on n link a, ∀a ∈ L ; group the αs into the vector α ∈ R+L Hourly wage on link a, ∀a ∈ L ; group all wages into the vector nL w ∈ R+ The upper bound on the availability of labor on link a under Scenario 1, ∀a ∈ L The upper bound on labor availability for tier t activities under Scenario 2, with tier t = 1 being production; tier t = 2 refers to transportation, and, so on, until t = T , which corresponds to distribution. Here, T + 1 corresponds to the electronic commerce tier The upper bound on labor availability under Scenario 3 Functions The operational cost associated with link a, ∀a ∈ L The demand price function for the product of firm i at demand market k, i = 1, . . . , I and k = 1, . . . , J

The production links from the top-tiered nodes i, i = 1, . . . , I , representing firm i, in Figure 4.1 are connected to the production nodes of firm i, which are denoted,

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

respectively, by M1i , . . . , M i i . The links from the production nodes, in turn, are nM

joined with the distribution center nodes of each firm i, i = 1, . . . , I , and correi , . . . , Di . The spond to transportation links. These nodes are denoted by D1,1 i links joining nodes

i , . . . , Di D1,1 niD ,1

with nodes

i , . . . , Di D1,2 niD ,2

nD ,1

correspond to the

i , . . . , Di storage links. There are also distribution links connecting the nodes D1,2 i

nD ,2

for i = 1, . . . , I , with the bottom-tiered demand market nodes 1, . . . , J . Finally, there are links joining the production nodes with the demand market nodes and these links correspond to direct shipments to the demand markets. For example, in the case of demand markets corresponding to homes, such links can capture electronic commerce. They can also denote direct deliveries to consumers at demand markets where the producers are farms and such distribution channels have been initiated because of the pandemic (see, for example, Shea 2020). Of course, the supply chain network topology in Figure 4.1 can be adapted for the specific application under consideration, with the appropriate addition/deletion of nodes, links, and/or supply chain network tiers. The demand for each firm’s product at each demand market must be satisfied by the product flows from the firm to that demand market. Hence, the following conservation of flow equations must hold for each firm i: i = 1, . . . , I :

xp = dik ,

k = 1, . . . , J.

(4.1)

p∈Pki

Moreover, the path flows must be nonnegative; that is, for each firm i, i = 1, . . . , I : xp ≥ 0,

∀p ∈ P i .

(4.2)

The link flows of each firm i, i = 1, . . . , I , are related to the path flows by the expression:

fa =

xp δap ,

∀a ∈ L i ,

(4.3)

p∈P

where δap = 1, if link a is contained in path p and 0, otherwise. According to (4.3), the flow of a firm’s product on a link is equal to the sum of that product’s flows on paths that contain that link. As in Chapters 2 and 3, it is assumed that the product output on each link is a linear function of the labor input. This corresponds to what is known as a linear production function in economics. Hence, one has that fa = αa la ,

∀a ∈ L i ,

i = 1, . . . , I.

The greater the value of αa , the more productive the labor on the link.

(4.4)

4.2 Supply Chain Network Game Theory Modeling Under Labor Constraints

71

The utility function of firm i, U i , i = 1, . . . , I , is the profit, given by the difference between its revenue and its total costs: Ui =

J



ρik (d)dik −

cˆa (f ) −

a∈L i

k=1



w a la .

(4.5a)

a∈L i

The first expression after the equal sign in (4.5a) is the revenue of firm i. The second expression in (4.5a) is the operational costs for the supply chain network L i of firm i and the third expression captures the total labor costs of firm i. The functions Ui : i = 1, . . . , I , are assumed to be concave, with the demand price functions being monotone decreasing and continuously differentiable and the total link cost functions being convex and also continuously differentiable. The optimization problem of each firm i: i = 1, . . . , I , is, hence, for firm i: Maximize

J k=1



ρik (d)dik −

cˆa (f ) −

a∈L i



w a la ,

(4.5b)

a∈L i

subject to (4.1), (4.2), (4.3), and (4.4). Three distinct scenarios are now considered in terms of the labor availability in the supply chain network economy and the associated bound(s) on specific links. Labor Scenario 1: A Bound on Labor on Each Supply Chain Network Link In Scenario 1, the additional constraints on the fundamental model described above are la ≤ l¯a ,

∀a ∈ L .

(4.6)

According to (4.6), there is an upper bound on labor associated with each link in the supply chain network of Figure 4.1. In this scenario, unlike the subsequent two scenarios, the feasible sets of the individual firms will depend only on their specific strategies and not on the strategies of the other firms, as will be shown below. Labor Scenario 2: A Bound on Labor on Each Tier of Links in the Supply Chain Network In Scenario 2, there is a bound on labor associated with each activity tier in the supply chain network economy; that is, in addition to the original constraints (4.1) through (4.4), the firms are now faced with the following constraints: a∈L 1

la ≤ l¯1 ,

(4.7,1)

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4 Game Theory Modeling of Supply Chains and Labor Disruptions



la ≤ l¯2 ,

(4.7,2)

a∈L 2

and so on, until

la ≤ l T +1 .

(4.7,T +1)

a∈L T +1

Observe that, in Scenario 2, unlike in Scenario 1, the firms now have shared, that is, common, labor constraints. Hence, their underlying feasible sets will no longer be disjoint. This will result in a Generalized Nash Equilibrium, rather than a Nash Equilibrium. I will elaborate further on this when I present the variational inequality formulations of the model under the different scenarios. Hence, in this scenario, labor is movable across a tier within a firm or across firms. Scenario 2 allows for the movement of labor across the firms’ different production facilities, across the distinct storage facilities, etc. Since one can expect that the skill set of a worker will be transferable across production facilities of a differentiated, but substitutable product; similarly, across distribution facilities or freight provision services, this scenario is quite reasonable in the pandemic or even during wartimes. Such a scenario was also considered in an optimization, single firm context in Chapter 3, the same for Scenario 3 below. Labor Scenario 3: A Single Labor Bound on Labor for All the Links in the Supply Chain Network Scenario 3 may be interpreted as being the least restrictive of the scenarios considered here in that labor is transferable across different activities of production, transportation, storage, and distribution. In the pandemic, we are seeing that some employees are assuming different tasks in supply chain networks from they had been doing previously. This may enhance agility and flexibility, provided that labor has the requisite skills or the skills can be acquired fairly readily. For example, farmers now in the United States have been innovating in terms of direct sales to consumers and taking on different tasks, such as deliveries of their farmed products to homes (see Woolever 2020). This can benefit both producers and consumers. In Scenario 3, in addition to constraints (4.1) through (4.4), the firms are now faced with the following single constraint:

¯ la ≤ l.

(4.8)

a∈L

The objective function of each firm i: i = 1, . . . , I , given by (4.5b), is now formulated in path flow variables exclusively. One is able to do this because of expressions (4.1), (4.2), and (4.4), which, recall, relates labor to product flow. Specifically, one can redefine the operational cost link functions as c˜a (x) ≡ cˆa (f ),

4.2 Supply Chain Network Game Theory Modeling Under Labor Constraints

73

∀a ∈ L , and the demand price functions as ρ˜ik (x) ≡ ρ ik (d), ∀i, ∀k. In addition, as xp δap

, for all a ∈ L . discussed in Chapter 3, in view of (4.3) and (4.4) la = p∈P αa Also, recall that, according to Table 4.1, x i denotes the vector of strategies, which are the path flows, for each firm i: i = 1, . . . , I . One can redefine the utility/profit functions U˜ i (x) ≡ U i : i = 1 . . . , I , and group the profits of all the firms into an I -dimensional vector U˜ , such that U˜ = U˜ (x).

(4.9)

Objective function (4.5b), in lieu of the above, can now be expressed as Maximize

U˜ i (x) =

J k=1

ρ˜ik (x)

p∈Pki



xp −

c˜a (x) −

a∈L i

wa xp δap . αa i

a∈L

p∈P

(4.10) It follows that constraint (4.6) for Scenario 1 can be re-expressed exclusively in path flows; the same holds for constraints (4.7,1) through (4.7,T +1) for Scenario 2 and for constraint (4.8) for Scenario 3.

4.2.1 Governing Equilibrium Conditions and Variational Inequality Formulations The governing equilibrium conditions for the different scenarios are now stated and alternative variational inequality formulations for each scenario provided.

4.2.1.1

Scenario 1 Nash Equilibrium Conditions and Variational Inequality Formulations {x i |x i

n i R+P ,

 i

xp δap

p∈P Define the feasible set Ki for firm i thus Ki ≡ ∈ ≤ αa  I l¯a , ∀a ∈ L i }, for i = 1, . . . , I . Also, define K ≡ i=1 Ki . In Scenario 1, each firm competes noncooperatively until the following equilibrium is achieved.

Definition 4.1 (Supply Chain Network Nash Equilibrium for Scenario 1) A path flow pattern x ∗ ∈ K is a supply chain network Nash Equilibrium if for each firm i: i = 1, . . . , I : U˜ i (x i∗ , xˆ i∗ ) ≥ U˜ i (x i , xˆ i∗ ), where xˆ i∗ ≡ (x 1∗ , . . . , x i−1∗ , x i+1∗ , . . . , x I ∗ ).

∀x i ∈ Ki ,

(4.11)

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

According to (4.11), a supply chain Nash Equilibrium is established if no firm can improve upon its profits unilaterally. The feasible set K is a convex set. Applying the classical theory of Nash equilibria and variational inequalities, under the imposed assumptions on the underlying functions, it follows that (cf. Gabay and Moulin 1980 and Nagurney 1999) the solution to the above Nash Equilibrium problem (see Nash 1950, 1951) coincides with the solution of the variational inequality problem: determine x ∗ ∈ K, such that −

I

∇x i U˜ i (x ∗ ), x i − x i∗  ≥ 0,

∀x ∈ K,

(4.12)

i=1

where ·, · represents the inner product in the corresponding Euclidean space, which here is of dimension nP , and ∇x i U˜ i (x) is the gradient of U˜ i (x) with respect to x i . The existence of a solution to variational inequality (4.12) is guaranteed since the feasible set K is compact and the utility functions are continuously differentiable, under our imposed assumptions (cf. Kinderlehrer and Stampacchia 1980). An alternative variational inequality to the one in (4.12) over a simpler feasible set is now provided. Introduce Lagrange multiplier λa associated with the constraint (4.6) for each link a ∈ L and group the Lagrange multipliers for each firm i’s supply chain network L i into the vector λi . Then, group all such vectors n for the firms into the vector λ ∈ R+L . Also, define the feasible sets: Ki1 ≡  n i +n i {(x i , λi )|(x i , λi ) ∈ R+P L }: i = 1, . . . , I , and K 1 ≡ Ii=1 Ki1 . Using similar arguments as in Theorem 1 in Nagurney et al. (2017), the following result is immediate. Theorem 4.1 (Alternative Variational Inequality Formulation of Nash Equilibrium for Scenario 1) The supply chain network Nash Equilibrium satisfying Definition 4.1 is equivalent to the solution of the variational inequality: determine the vector of equilibrium path flows and the vector of optimal Lagrange multipliers, (x ∗ , λ∗ ) ∈ K 1 , such that ⎡ ⎤ J I J ∗) ⎢ ∂ C˜ p (x ∗ ) λ∗ wa ∂ ρ ˜ (x il ⎥ a + δap + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ ⎣ ∂x α α ∂x p a a p i i i i a∈L

i=1 k=1 p∈P k

× [xp − xp∗ ] +



 l¯a −

a∈L



p∈P

a∈L

xp∗ δap

αa



q∈Pl

l=1

  × λa − λ∗a ≥ 0,

∀(x, λ) ∈ K 1 ,

(4.13)

where ∂ cˆb (f ) ∂ C˜ p (x) ≡ δap , ∂xp ∂fa a∈L b∈L

∀p ∈ P ,

∂ ρ˜il (x) ∂ρil (d) ≡ , ∂xp ∂dik

∀p ∈ Pki , ∀i, ∀k.

(4.14)

4.2 Supply Chain Network Game Theory Modeling Under Labor Constraints

75

The above feasible set K 1 is the nonnegative orthant. This feature enables the implementation of an algorithm, which is described in the next section, which is an iterative procedure that yields closed form expressions at an iteration for the path flows and the link Lagrange multipliers.

4.2.1.2

Scenario 2 Generalized Nash Equilibrium Conditions and Variational Inequality Formulations

In Scenarios 2 and 3, on the other hand, not only do the utility functions of the firms depend on their own strategies and on those of the other firms, but the feasible sets do as well. This happens because of the corresponding labor constraints, which capture that labor is “shared.” Therefore, in Scenarios 2 and 3, the governing concept is no longer that of Nash Equilibrium, but, rather, it is that of Generalized Nash Equilibrium (see Debreu 1952 and Arrow and Debreu 1954). This presents additional challenges, since, as noted in Nagurney et al. (2017), Generalized Nash Equilibrium problems cannot be directly formulated as variational inequality problems, but, instead, quasi-variational inequalities are often used as the formulation (see, e.g., Facchinei and Kanzow 2010). It is well-known (cf. Luna 2013 and the references therein) that quasi-variational inequality problems are, nevertheless, much harder to solve than finite-dimensional variational inequality problems. One can utilize a refinement of the Generalized Nash Equilibrium (GNE) in order to secure a variational inequality formulation. The refinement is a Variational Equilibrium and it is a specific type of GNE (see Kulkarni and Shanbhag 2012). In a Generalized Nash Equilibrium defined by a Variational Equilibrium, the Lagrange multipliers associated with the common/shared constraints are all the same. As noted in Nagurney et al. (2017), this provides a fairness interpretation and is reasonable from an economic standpoint. The set of shared constraints under Scenario 2 is denoted by S 1 , where S 1 ≡ {x|all constraints (4.7,1)–(4.7,T +1) hold}. Definition 4.2 (Generalized Nash Equilibrium Under Scenario 2) A product flow vector x ∗ ∈ K ∩ S 1 is a Generalized Nash Equilibrium Under Scenario 2, if for each firm i: i = 1, . . . , I : U˜ i (x i∗ , xˆ i∗ ) ≥ U˜ i (x i , xˆ i∗ ),

∀x i ∈ Ki ∩ S 1 ,

(4.15)

where, as defined previously, xˆ i∗ ≡ (x 1∗ , . . . , x i−1∗ , x i+1∗ , . . . , x I ∗ ). Definition 4.3 (Variational Equilibrium Under Scenario 2) A strategy vector x ∗ is said to be a Variational Equilibrium of the above Generalized Nash Equilibrium according to Definition 4.2 if x ∗ ∈ K ∩S 1 is a solution of the variational inequality: −

I

∇xi U˜ i (x ∗ ), x i − x i∗  ≥ 0, i=1

∀x ∈ K ∩ S 1 .

(4.16)

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

Clearly, a solution x ∗ to variational inequality (4.16) exists. For Scenario 2, associate the nonnegative Lagrange multiplier μt with labor constraint (4.7, t), for t = 1, . . . , T + 1. Define the vector of Lagrange multipliers n +T +1 T +1 μ ∈ R+ and the feasible set K 2 ≡ {(x, μ)|(x, μ) ∈ R+P }. An alternative variational inequality formulation to that of (4.16), whose solution corresponds to the Variational Equilibrium under Scenario 2, is: determine (x ∗ , μ∗ ) ∈ K 2 , such that ⎡ J I J ∗ wa ∂ ρ˜il (x ∗ ) ∗ ⎢ ∂ C˜ p (x ) ∗ + δ − ρ ˜ (x ) − xq ⎣ ap ik ∂xp αa ∂xp i i i a∈L

i=1 k=1 p∈P k

+

T +1 t=1

+

T +1 t=1

μt∗ ⎡

q∈Pl

l=1

⎤

1 δap ⎦ × [xp − xp∗ ] αa t

a∈L

⎣¯l t −

a∈L t

 p∈P

xp∗ δap

αa

⎤

  ⎦ × μt − μt∗ ≥ 0,

∀(x, μ) ∈ K 2 . (4.17)

Variational inequality (4.17) will allow for the application of an effective and efficient computational procedure.

4.2.1.3

Scenario 3 Generalized Nash Equilibrium Conditions and Variational Inequality Formulations

In Scenario 3, there is a single labor constraint, and labor is allowed to move freely among the supply chain network economic activities across the firms. Proceeding in a similar manner as for Scenario 2, define S 2 , where S 2 ≡ {x| (4.8) holds}. Then, one can state the following definitions: Definition 4.4 (Generalized Nash Equilibrium Under Scenario 3) A product flow vector x ∗ ∈ K ∩ S 2 is a Generalized Nash Equilibrium Under Scenario 3, if for each firm i: i = 1, . . . , I : U˜ i (x i∗ , xˆ i∗ ) ≥ U˜ i (x i , xˆ i∗ ),

∀x i ∈ Ki ∩ S 2 .

(4.18)

Definition 4.5 (Variational Equilibrium Under Scenario 3) A strategy vector x ∗ is said to be a Variational Equilibrium of the above Generalized Nash Equilibrium according to Definition 4.3 if x ∗ ∈ K ∩S 1 is a solution of the variational inequality:

4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples

−

I

∇xi U˜ i (x ∗ ), x i − x i∗  ≥ 0,

∀x ∈ K ∩ S 2 .

77

(4.19)

i=1

Of course, a solution to variational inequality (4.19) is guaranteed to exist since the underlying feasible set K ∩ S 2 is compact. As for Scenarios 1 and 2, alternative variational inequality formulations are now provided. Denote the Lagrange multiplier associated with the labor constraint (4.8) n +1 by γ . Also, define the feasible set K 3 ≡ {(x, γ )|(x, γ ) ∈ R+P }. An alternative ∗ ∗ variational inequality to that of (4.16) is determine (x , γ ) ∈ K 3 such that ⎡ I J J wa ⎢ ∂ C˜ p (x ∗ ) ∂ ρ˜il (x ∗ ) ∗ + δap − ρ˜ik (x ∗ ) − xq ⎣ ∂xp αa ∂xp i i i i=1 k=1 p∈P

+γ ∗

k

a∈L



l=1



1 δap × [xp − xp∗ ] + l¯ − αa

a∈L

a∈L

 p∈P

xp∗ δap

αa



q∈Pl

  × γ − γ ∗ ≥ 0, ∀(x, γ ) ∈ K 3 .

(4.20) Observe that the feasible set K 3 is also the nonnegative orthant and this allows for an effective solution of variational inequality (4.20). It is important to emphasize that the Lagrange multipliers associated with the respective constraint sets in Scenarios 1, 2, and 3 provide valuable information, upon problem solution. They quantify the value of an additional unit of the labor resource associated with the respective constraint. Having a computational procedure that can compute both the equilibrium  product path flows (from which the labor values are xp δap

then obtained from la = p∈P , for all links a ∈ L ) and the equilibrium αa Lagrange multipliers enriches the toolbox for managerial insights and decisionmaking. Such an algorithm is proposed in Section 4.3. All of the above six variational inequalities can be put into standard variational inequality form (2.14).

4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples As in Chapters 2 and 3, the modified projection method of Korpelevich (1977) is used for computations. In this chapter, however, the supply chain network game theory model under different labor scenarios is the focus. For the alternative variational inequality formulations of Scenarios 1, 2, and 3, given, respectively, by (4.13), (4.17), and (4.20), each iteration of the modified projection method yields closed form expressions for the product path flows and for the associated Lagrange multipliers. Recall that modified projection method is guaranteed to converge if

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

the function F (X) that enters the variational inequality is Lipschitz continuous and monotone. These are reasonable conditions for the supply chain network game theory model with labor under the scenarios considered here. The supply chain network numerical examples are now presented. The modified projection method is implemented in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the computation of solutions. The algorithm is initialized as follows. The elastic demand for each demand market is initialized at 40 and equally distributed among the paths connecting each demand market from each origin node (firm). The Lagrange multipliers are initialized to 0.00. The modified projection method is deemed to have converged if the absolute difference of the path flows differs by no more than 10−7 and the same for the Lagrange multipliers. The numerical examples are inspired by the summer harvest season for fresh produce and, specifically, the need to harvest berries, including blueberries. Much of the seasonal fresh produce harvesting in the United States is done by migrant workers. This is the case also in other parts of the world, including Europe. The harvesting of blueberries in the northeastern United States, with major growing areas being New Jersey and Maine, has been especially challenged because of the COVID-19 pandemic (see Tully 2020 and Russell 2020). Interestingly, although operations researchers have been tackling human migration networks for several decades now, it is only recently that regulations have been incorporated in such network-based models (see, e.g., Nagurney and Daniele 2021 and Nagurney et al. 2021). Here, the focus is on seasonal migrants, rather than those who wish to relocate permanently to new locations. The numerical examples are stylized, but Internet available resources are utilized to obtain blueberry price and picking data in the United States (see Galinato et al. 2016 and howmuchisit.org 2018). The flow variables are in pounds of blueberries; the prices are in dollars per pound, and labor is in person hours. In Section 4.3.1, the numerical examples are of Scenario 1 with the governing variational inequality (4.13). In Section 4.3.2, the examples are of Scenario 3, with the governing variational inequality (4.20).

4.3.1 Scenario 1 Examples Examples 4.1, 4.2, and 4.3 have the supply chain network topology given in Figure 4.2. In these examples, there is a single firm, with Example 4.1 serving as the baseline, which is a blueberry farm. It has two planting sites (production locations), access to a single distribution center, and the blueberries are distributed to two demand markets. Example 4.1 (Baseline Example) The operational cost functions on the links are cˆa (f ) = .0006fa2 ,

cˆb (f ) = .0007fb2 ,

cˆc (f ) = .001fc2 ,

cˆd (f ) = .001fd2 ,

4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples

cˆe (f ) = .002fe2 ,

cˆf (f ) = .005ff2 ,

79

cˆg (f ) = .005fg2 .

The wages are wa = 10,

wb = 10,

wc = 15,

wd = 15,

we = 20,

wf = 17,

wg = 18.

Fig. 4.2 The supply chain network topology for Examples 4.1–4.3

The link labor productivity factors are αa = 24,

αb = 25,

αc = 100,

αd = 100,

αe = 50,

αf = 100,

αg = 100.

l¯d = 300,

l¯e = 100,

l¯f = 120,

l¯g = 120.

The bounds on labor are l¯a = 200,

l¯b = 200,

l¯c = 300,

The labor bounds are set high since this is the baseline example. The demand price functions are ρ11 (d) = −.0001d11 +6, ρ12 (d) = −.0002d12 +8. The paths are p1 = (a, c, e, f ), p2 = (b, d, e, f ), p3 = (a, c, e, g), and path pr = (b, d, e, g).

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

The modified projection method yields the following solution. The equilibrium path flows are xp∗1 = 91.46,

xp∗2 = 85.77,

xp∗3 = 185.40,

xp∗4 = 179.77.

The equilibrium link labor values are la∗ = 11.54,

lb∗ = 10.62,

lc∗ = 2.77,

lf∗ = 1.77,

ld∗ = 2.66,

le∗ = 10.85,

lg∗ = 3.65.

As expected, all the Lagrange multipliers are equal to 0.00: λ∗a = λ∗b = λ∗c = λ∗d = λ∗e = λ∗f = λ∗g = 0.00.

The demand price at Demand Market 1 is: 5.98 and at Demand Market 2 7.93. The computed respective demands are 177.23 and 365.16. The profit of the firm is 1, 684.47. Example 4.2 (Disruption in Labor Availability on Link a) Example 4.2 has the same data as Example 4.1, except that now the following situation is considered. Workers have been getting sick and are not as available to pick the blueberries at the first harvesting site. Specifically, now the bound on the labor on link a has been drastically reduced to l¯a = 10. The modified projection method yields the following equilibrium solution: the equilibrium path flow pattern is xp∗1 = 72.99,

xp∗2 = 99.19,

xp∗3 = 167.01,

xp∗4 = 193.21.

The equilibrium link labor values are la∗ = 10.00,

lb∗ = 11.70,

lc∗ = 2.40,

lf∗ = 1.72,

ld∗ = 2.92,

le∗ = 10.65,

lg∗ = 3.60.

The Lagrange multipliers are all equal to 0.00, except for λ∗a = 5.0253. The demand price at Demand Market 1 is 5.98 and at Demand Market 2: 7.93 with the computed respective demands being, respectively, 172.18 and 360.22. The profit of the firm is 1, 680.61. Since the labor amount is at the bound on link a, the associated Lagrange multiplier is positive. Due to the labor disruption, the profit of the firm decreases. The demands also decrease.

4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples

81

Example 4.3 (No Labor Availability on Link a) Example 4.3 is a Variant of Example 4.2 and has the same data except that now, due to illnesses, there is no labor available on link a; hence, l¯a = 0.00. The modified projection method converges to the following equilibrium solution. The equilibrium path flow pattern is xp∗1 = 0.00,

xp∗2 = 139.38,

xp∗3 = 0.00,

xp∗4 = 327.99.

The equilibrium link labor values are la∗ = 0.00,

lb∗ = 18.69,

lc∗ = 0.00,

lf∗ = 1.39,

ld∗ = 4.67,

le∗ = 9.35,

lg∗ = 3.28.

The Lagrange multipliers are all equal to 0.00, except for λ∗a = 162.6470. One can see from this example the value of increasing labor availability on link a, as revealed through the significantly higher value of the Lagrange multiplier λ∗a . The profit of the firm now drops to 1,466.78. The demand price at Demand Market 1 is 5.99 and at Demand Market 2 it is 7.93. The computed respective demands are 139.38 and 327.99. The demand for blueberries drops at each demand market. Clearly, the farmer should do everything possible to secure the health of the workers at his production/harvesting facilities, so that the blueberries can be harvested in a timely manner and so that profits do not suffer. Example 4.4 (Addition of a Competitor) In Examples 4.4 through 4.6, I evaluate the impact of a competitor, that is, there is now another farm growing blueberries in the general area of the first farm. The underlying supply chain network topology is now as in Figure 4.3. The new firm also has two production sites, a single distribution center, and serves the same two locations as demand markets as Firm 1 does. The data for the first firm are identical to the data in Example 4.2 except with a modification of the firm’s demand price functions in order to capture competition with the new firm. In particular, the demand price functions of Firm 1 are now ρ11 (d) = −.0001d11 − .00005d21 + 6,

ρ12 (d) = −.0002d12 − .0001d22 + 8.

The demand price functions of Firm 2 are ρ21 (d) = −.0003d21 + 7,

ρ22 (d) = −.0002d22 + 7.

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

Fig. 4.3 The supply chain network topology for the numerical Examples 4.4 through 4.6

The operational costs associated with Firm 2’s supply chain network L 2 are cˆh (f ) = .00075fh2 ,

cˆi (f ) = .0008fi2 ,

cˆl (f ) = .0015fl2 ,

cˆj (f ) = .0005fj2 ,

cˆm (f ) = .01fm2 ,

cˆk (f ) = .0005fk2 ,

cˆn (f ) = .01fn2 .

The wages of labor for Firm 2 are wh = 11,

wi = 22,

wj = 15,

wk = 15,

wl = 18,

wm = 18,

wn = 18.

The link labor productivity factors for Firm 2 on its supply chain network are αh = 23,

αi = 24,

αj = 100,

αk = 100,

αl = 70,

αm = 100,

αn = 100.

l¯k = 200,

l¯l = 300,

l¯m = 100,

l¯n = 100.

The bounds on labor, in turn, are l¯h = 800,

l¯i = 90,

l¯j = 200,

4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples

83

The four new paths associated with Firm 2 are p5 = (h, j, l, m),

p6 = (i, k, l, m),

p7 = (h, j, l, n),

p8 = (i, k, l, n).

The modified projection method yields the following equilibrium solution. The equilibrium path flow pattern is xp∗1 = 73.23,

xp∗2 = 98.85,

xp∗5 = 142.85,

xp∗6 = 53.08,

xp∗3 = 166.77,

xp∗4 = 192.38,

xp∗7 = 143.81,

xp∗8 = 54.04.

The equilibrium link labor values are la∗ = 10.00,

lb∗ = 11.65,

lc∗ = 2.40,

ld∗ = 2.91,

le∗ = 10.62,

lf∗ = 1.72,

lg∗ = 3.59,

lh∗ = 12.46,

li∗ = 4.46,

lj∗ = 2.87,

lk∗ = 1.07,

ll∗ = 5.63,

∗ lm = 1.96,

ln∗ = 1.98.

The Lagrange multipliers are all equal to 0.00 except for λ∗a = 4.9295. The product prices at equilibrium are ρ11 = 5.97,

ρ12 = 7.91,

ρ21 = 6.94,

ρ22 = 6.96,

with the equilibrium demands: ∗ d11 = 172.07,

∗ d12 = 359.15,

∗ d21 = 195.94,

∗ d22 = 197.86.

The profit for Firm 1 is 1,671.80 and the profit for Firm 2 is 1,145.06. With the addition of a competitor, the prices of Firm 1 for its blueberries drop at the demand markets and its profit is also negatively impacted. Example 4.5 (Modification of Demand Price Functions) Example 4.5 has the same data as Example 4.4 except that now the demand price functions ρ21 (d) and ρ22 (d) include a cross-term, so that ρ21 (d) = −.0003d21 − .0001d11 + 6,

ρ22 (d) = −.0002d22 − .0001d12 + 7.

The equilibrium path flow is now xp∗1 = 73.22, xp∗5 = 142.62,

xp∗2 = 98.85, xp∗6 = 52.86,

xp∗3 = 166.78, xp∗7 = 143.12,

xp∗4 = 192.38, xp∗8 = 53.36.

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

The equilibrium link labor values are la∗ = 10.00,

lb∗ = 11.65,

lc∗ = 2.40,

ld∗ = 2.91,

le∗ = 10.62,

lf∗ = 1.72,

lg∗ = 3.59,

lh∗ = 12.42,

li∗ = 4.43,

lj∗ = 2.86,

lk∗ = 1.06,

ll∗ = 5.60,

∗ lm = 1.95,

ln∗ = 1.96.

The Lagrange multipliers are all equal to 0.00 except for λ∗a = 4.93. The product prices at equilibrium are ρ11 = 5.97,

ρ12 = 7.91,

ρ21 = 6.92,

ρ22 = 6.92,

with the equilibrium demands: ∗ d11 = 172.07,

∗ d12 = 359.16,

∗ d21 = 195.48,

∗ d22 = 196.48.

The profit for Firm 1 is 1,671.86 and the profit for Firm 2 is 1,134.61. The profit for Firm 1 rises ever so slightly, whereas that for Firm 2 decreases. Example 4.6 (Disruptions in Storage Facilities) Example 4.6 has the same data as Example 4.5 except that now there is a major disruption in terms of the spread of COVID-19 at the distribution centers of both firms with the bounds on labor corresponding to the associated respective links now being l¯e = 5,

l¯l = 5.

The modified projection method computes the following equilibrium path flow pattern: xp∗1 = 15.65,

xp∗2 = 14.38,

xp∗5 = 131.97,

xp∗6 = 42.63,

xp∗3 = 110.60,

xp∗4 = 109.35,

xp∗7 = 132.36,

xp∗8 = 43.02.

The equilibrium link labor values are la∗ = 5.26,

lb∗ = 4.95,

lc∗ = 1.26,

ld∗ = 1.24,

le∗ = 5.00,

lf∗ = 0.30,

lg∗ = 2.20,

lh∗ = 11.49,

li∗ = 3.57,

lj∗ = 2.64,

lk∗ = 0.86,

ll∗ = 5.00,

∗ lm = 1.75,

ln∗ = 1.75.

All computed equilibrium Lagrange multipliers are now equal to 0.00 except for those associated with the distribution center link labor bounds which are

4.3 The Algorithm and Seasonal Fresh Produce Supply Chain Network Examples

λ∗e = 157.2138,

85

λ∗l = 43.6537.

The product prices at equilibrium are now ρ11 = 5.99,

ρ12 = 7.94,

ρ21 = 6.94,

ρ22 = 6.94,

with the equilibrium demands: ∗ d11 = 30.03,

∗ d12 = 219.96,

d21 = 174.61,

d22 = 175.39.

The profit for Firm 1 is now drastically reduced to 1,218.74, and the profit for Firm 2 also declines but by a much smaller amount to 1,126.73.

4.3.2 Scenario 3 Examples In this subsection, the modified projection method is applied to compute solutions to numerical examples of Scenario 3. The supply chain network topology remains as in Figure 4.3. The data for Examples 4.7 through 4.11 in this set are identical to that in Example 4.6, except ¯ there are no longer any link bounds but, rather, only a single labor bound l. ¯ ¯ Example 4.7 has l = 80, Example 4.8 has l = 70, Example 4.9 has l¯ = 60, Example 4.10 has l¯ = 30, and Example 4.11 has l¯ = 10. The computed equilibrium path flows for these examples are reported in Table 4.2 with the equilibrium labor values given in Table 4.3. Table 4.2 Equilibrium product path flows for Examples 4.7 through 4.11 representing Scenario 3 Equilibrium path flows xp∗1 xp∗2 xp∗3 xp∗4 xp∗5 xp∗6 xp∗7 xp∗8

Example 4.7 91.35 85.67 184.84 179.18 142.63 52.84 143.14 53.34

Example 4.8 83.08 78.87 176.75 172.54 136.59 48.42 137.04 48.84

Example 4.9 42.05 45.09 136.58 139.60 106.62 26.50 106.81 26.69

Example 4.10 1.03 11.31 96.40 106.66 76.65 4.58 76.58 4.51

Example 4.11 0.00 0.00 24.01 60.82 20.51 0.00 20.30 0.00

The computed associated equilibrium Lagrange multiplier is as follows. For Example 4.7, the labor bound is not tight and, therefore, γ ∗ = 0.00. For Example 4.8, γ ∗ = 3.99, for Example 4.9: γ ∗ = 23.79, for Example 4.10: γ ∗ = 43.59, and for Example 4.11: γ ∗ = 68.00.

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

Table 4.3 Equilibrium link labor values for Examples 4.7 through 4.11 representing Scenario 3 Equilibrium labor values la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗ lh∗ li∗ lj∗

lk∗ ll∗ ∗ lm ln∗

Example 4.7 11.51 10.59 2.76 2.65 10.82 1.77

Example 4.8 10.83 10.06 2.60 2.51 10.22 1.62

Example 4.9 7.44 7.39 1.79 1.85 7.27 0.8

Example 4.10 4.06 4.72 0.97 1.18 4.31 0.12

Example 4.11 1.00 2.43 0.24 0.61 1.70 0.00

3.64 12.42 4.42 2.86

3.49 11.90 4.05 2.74

2.76 9.28 2.22 2.13

2.03 6.66 0.38 1.53

0.85 1.77 0.00 0.41

1.06 5.60 1.95 1.96

0.97 5.30 1.85 1.86

0.53 3.81 1.33 1.34

0.09 2.32 0.81 0.81

0.00 0.58 0.21 0.20

As for the profits, it is interesting to see how the disruptions in labor availability impact the profits negatively. In Example 4.7, Firm 1 earns a profit of 1675.58, whereas Firm 2 earns a profit of 1134.42. In Example 4.8, with labor being at the capacity, Firm 1 earns a profit of 1671.23, whereas Firm 2 earns a profit of 1131.76, which is only a small decrease. However, there is a big decrease in profit for both firms in Example 4.9. In Example 4.9, Firm 1 earns a profit of 1506.94, whereas Firm 2 earns a profit of 1022.56. In Example 4.10, with the labor capacity further reduced, Firm 1 earns only a profit of 1105.06, whereas Firm 2 earns a profit of only 753.56. And, in the final example, Example 4.11, with the labor bound l¯ = 10, Firm 1’s profit drops further to 523.19, whereas Firm 2’s profit drops to 228.91. The above examples are stylized, and, clearly, examples of Scenario 2 can also be effectively computed using the proposed algorithm. The supply chain network game theory framework provides decision-makers with the flexibility of evaluating impacts of labor reductions due to COVID-19 (or during other types of disasters including wars). Of course, this framework also permits the assessment of investments in terms of maintaining labor bound values at high levels. In the case of the pandemic, the latter can be achieved, in part, through social distancing, proper sanitation, and providing uncrowded living facilities for migrant workers. The numerical examples are inspired by crises faced by farmers due to labor shortages as a consequence of the COVID-19 pandemic but are also relevant to war situations as in Ukraine. The supply chain network game theory framework can be adapted to different industrial sectors where labor plays an important role, such as, for example, in medical supply production from PPEs to ventilators. Of course, the framework can also be adapted for medical treatments and even for vaccines.

4.4 Supply Chain Network Economy Efficiency and Importance Identification. . .

87

The computed solutions demonstrate that the proposed algorithm is effective for the game theory supply chain network model with labor constraints and that the model is illuminating. Furthermore, having such an algorithm allows for the calculation of the quantitative impacts on equilibrium product flows, product prices, and firm profits of labor availability and possible disruptions under different scenarios. Finally, the computed equilibrium Lagrange multipliers associated with the various labor constraints provide the value in relaxing a specific labor constraint.

4.4 Supply Chain Network Economy Efficiency and Importance Identification of Components In Section 3.4, a supply chain network efficiency measure was proposed for the elastic demand supply chain network optimization models in Chapter 3. In this section, an extension of the efficiency measure (3.22) and the importance identification of a supply chain network component (3.23) are presented for the game theory models in this chapter. For definiteness, the efficiency measure here is denoted by E 2 . Here, with multiple firms, the efficiency is for the supply chain network economy associated with the differentiated products. Without loss of generality, let l¯ now correspond to the labor bounds associated with the specific model, associated with variational inequalities (4.13), (4.17), or (4.20), respectively. The efficiency E 2 is defined as ¯ ≡ E = E (G , c, ˆ ρ, w, α, l) 2

2

J I i=1 k=1

∗ dik ρik (d ∗ )

IJ

,

(4.21)

with the demands, d ∗ , and the incurred demand market prices in (4.21), evaluated at the solution to the corresponding variational inequality. According to (4.21), given a supply chain network topology of multiple firms, and the various parameters and functions, a supply chain network economy with labor is evaluated as performing better if, on the average, it can handle higher demands for the differentiated products at lower prices. Similar to the results in Chapter 3, for the single firm case, I (g), which represents the efficiency drop when g is removed from the supply chain network economy, is defined as I (g) =

¯ − E 2 (G − g, c, ¯ ˆ ρ, w, α, l) ˆ ρ, w, α, l) ΔE 2 E 2 (G , c, = . 2 2 ¯ E E (G , c, ˆ ρ, w, α, l)

(4.22)

One can rank the importance of nodes or links, or a combination thereof, using (4.22). Resilience measures akin to (3.24) and (3.25) for the single firm case can also be applied for the competitive multifirm case with the efficiency of the supply chain

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4 Game Theory Modeling of Supply Chains and Labor Disruptions

network economy as in (4.21). In addition, one can construct resilience measures for a specific firm in the supply chain network economy by considering disruptions to labor for the particular firm.

4.5 Summary, Conclusions, and Suggestions for Future Research In this chapter, a supply chain network game theory framework was developed, consisting of multiple profit-maximizing firms competing noncooperatively in the production, storage, and ultimate distribution of their differentiated products in the presence of labor constraints. Three different scenarios of labor constraints consisting of bounds on labor availability on supply chain links in the first scenario, bounds on labor availability across a tier of the supply chain networks in the second scenario, and, finally, in the third scenario, a single bound on the labor in the supply chain network economy were considered. The framework is a contribution to the operations research and to the economic literatures and is inspired by the challenges associated with labor and accompanying shortages in the COVID-19 pandemic. It is also of relevance in the case of labor disruptions that are happening around the globe for reasons that include wars and strife as well as climate change. The governing concept in the case of labor Scenario 1 is that of Nash Equilibrium, whereas, because of the shared constraints in terms of labor, the governing concept in the case of labor Scenarios 2 and 3 is that of a Generalized Nash Equilibrium. For each labor scenario, two variational inequality formulations are provided, with the variational inequality that includes the Lagrange multipliers associated with the labor constraints being especially amenable to solution via the proposed algorithm, since both the equilibrium product flows and the equilibrium Lagrange multipliers can be computed iteratively using explicit formulae. Qualitative results are presented in terms of existence of the equilibrium patterns, along with solutions to a series of numerical examples. The numerical examples are grounded in disruptions taking place in terms of shortages of migrant workers to harvest fresh produce, in particular, berries, and, specifically, blueberries in parts of the United States. Such labor shortfalls are not only a critical issue now in the United States but also in other parts of the globe, as a consequence of the COVID19 pandemic. Furthermore, labor shortages in many important economic sectors are a reality now. In addition, an efficiency measure is proposed, extending the supply chain network efficiency measure of Chapter 3, which was for a single firm. Here, since the model is a multifirm game theory one, the efficiency measure is for the supply chain network economy associated with the differentiated products. The framework constructed in this chapter is a partial equilibrium one since the focus is on oligopolistic competition in supply chain networks of relevance to the pandemic. In future research, it is expected that the supply chain network game

References

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theory model(s) herein will be adapted to specific industrial sectors and the data fitted empirically. It is also expected that different labor constraints may be identified and perhaps even different functions from the linear ones that are utilized here to relate labor to product flow. Also, interesting avenues for future research are the use of cooperative game theory as well as the incorporation of costs associated with training/education to enable the reallocation of labor with appropriate skills where there may be shortfalls. This is happening in practice (cf. Reuters 2020), where former airline workers are being retrained as healthcare workers. Finally, it would be interesting to construct a general equilibrium model.

4.6 Sources and Notes In recent years, the upheavals on our planet have been major, with the COVID19 pandemic, climate change, as well as numerous conflicts and wars, including Russia’s war against Ukraine, on the continent of Europe. Such challenging times have demonstrated that people, as workers, are essential to all links in supply chain networks. In an increasingly competitive world, firms are competing with one another and also even for workers, as shortages of laborers, from those on farms and various manufacturing sites, to freight service workers, and even retail workers, become increasingly commonplace. This chapter is based principally on the paper by Nagurney (2021). Here, as in other modeling chapters in this book, the notation has been standardized and relevant additional research as well as world events included to provide the proper setting for the methodological contributions as well as the tools that can be harnessed for enhanced decision-making and policy-making as well as insights. The efficiency and resilience of supply chain networks will continue as extremely important features of the global economy. The network efficiency measure in this chapter is adapted from Nagurney (2022), where additional results can be found. Therein the focus is on defense critical supply chain networks with the inclusion of risk management and labor. Supply chain networks are critical infrastructure for numerous products and, hence, essential for the well-being of societies and the functionality of economies. Their importance and relevance have been dramatically highlighted in the past several years.

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Nagurney, A., 2022. Defense critical supply chain networks and risk management with the inclusion of labor: Dynamics and quantification of performance and the ranking of nodes and links. Accepted in: Handbook for Management of Threats, Security and Defense, Resilience and Optimal Strategies. K. Balomenos, A. Fytopoulos, and P.M. Pardalos, Editors, Springer Nature. Nagurney, A., Alvarez Flores, E., Soylu, C., 2016. A Generalized Nash Equilibrium model for post-disaster humanitarian relief. Transportation Research E, 95, 1–18. Nagurney, A., Daniele, P., 2021. International human migration networks under regulations. European Journal of Operational Research, 291(3), 894–905. Nagurney, A., Daniele, P., Alvarez Flores, E., Caruso, V., 2018. A variational equilibrium network framework for humanitarian organizations in disaster relief: Effective product delivery under competition for financial funds. In: Dynamics of Disasters: Algorithmic Approaches and Applications, I.S. Kotsireas, A. Nagurney, and P.M. Pardalos, Editors, Springer International Publishing Switzerland, pp. 109–133. Nagurney, A., Daniele, P., Cappello, G., 2021. Human migration networks and policy interventions: Bringing population distributions in line with system-optimization. International Transactions in Operational Research, 28(1), 5–26. Nagurney, A., Dong, J., Zhang, D., 2002. A supply chain network equilibrium model. Transportation Research E, 38(5), 281–303. Nagurney, A., Dutta, P., 2019. Competition for blood donations. Omega, 212, 103–114. Nagurney, A., Li, D., 2016. Competing on Supply Chain Quality: A Network Economics Perspective. Springer International Publishing Switzerland. Nagurney, A., Salarpour, M., Daniele, P., 2019. An integrated financial and logistical game theory model for humanitarian organizations with purchasing costs, multiple freight service providers, and budget, capacity, and demand constraints. International Journal of Production Economics, 212, 212–226. Nagurney, A., Salarpour, M., Dong, J., Dutta, P., 2021. Competition for medical supplies under stochastic demand in the Covid-19 pandemic: A Generalized Nash Equilibrium framework. In: Nonlinear Analysis and Global Optimization, T.M. Rassias and P.M. Pardalos, Editors, Springer Nature Switzerland AG, pp. 331–356. Nagurney, A., Salarpour, M., Dong, J., Nagurney, L.S., 2020. A stochastic disaster relief game theory network model. SN Operations Research Forum, 1, 10. Nagurney, A., Yu, M., Besik, D., 2017. Supply chain network capacity competition with outsourcing: A variational equilibrium framework. Journal of Global Optimization, 69(1), 231–254. Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49. Nash, J.F., 1951. Noncooperative games. Annals of Mathematics, 54, 286–298. OECD, 2020. COVID-19 and the food and agriculture sector: Issues and policy responses. April 29. Parker, W., 2020. As COVID-19 cases rise, truckers share their stories of diagnosis and recovery. Land Line, July 30. Polansek, T., Huffstutter, P.J., 2020. Piglets aborted, chickens gassed as pandemic slams meat sector. Reuters, April 27. Qiang, Q., Ke, K., Anderson, T., Dong, J., 2013. The closed-loop supply chain network with competition, distribution channel investment, and uncertainties. Omega, 41, 186–194. Ranney, M.L., Griffieth, V., Jha, A.K., 2020. Critical supply shortages – The need for ventilators and Personal Protective Equipment during the Covid-19 pandemic. The New England Journal of Medicine, April 30. Reuters, 2020. Laid-off SAS airline staff offered fast-track healthcare training. March 19. Russell, E., 2020. COVID-19 cases among blueberry workers worsen farm labor shortage. Portland Press Herald, Portland, Maine, August 3. Saberi, S., 2018. Sustainable, multiperiod supply chain network model with freight carrier through reduction in pollution stock. Transportation Research E, 118, 421–444.

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Saberi, S., Cruz, J.M., Sarkis, J., Nagurney, A., 2018. A competitive multiperiod supply chain network model with freight carriers and green technology investment option. European Journal of Operational Research, 266(3), 934–949. Shea, A., 2020. From dirt to doorstep, during coronavirus crisis Massachusetts farms find ways to come to you. WBUR, March 31. Toyasaki, F., Daniele, P., Wakolbinger, T., 2014. A variational inequality formulation of equilibrium models for end-of-life products with nonlinear constraints. European Journal of Operational Research, 236, 340–350. Treisman, R., 2022. Global food prices hit their highest recorded levels last month, driven up by the war. NPR, April 8. Tully, T., 2020. How you get your berries: Migrant workers who fear virus, but toil on. The New York Times, July 5. Woolever, L., 2020. Local farms embrace change in the face of coronavirus. Baltimore Magazine, April 6. World Health Organization, 2020. WHO Director-General’s opening remarks at the media briefing on COVID-19 - 11 March 2020. Geneva, Switzerland. Yu, M., Cruz, J.M., Li, D., 2019. The sustainable supply chain network competition with environmental tax policies. International Journal of Production Economics, 217, 218–231. Yu, M., Nagurney, A., 2013. Competitive food supply chain networks with application to fresh produce. European Journal of Operational Research, 224(2), 273–282.

Part II

Endogenous Wages and Productivity Investments

Chapter 5

Wages and Labor Productivity in Supply Chains with Fixed Labor Availability on Links

Abstract The ongoing challenges on our planet have dramatically demonstrated the importance of labor to supply chain network activities from production to distribution with shortfalls in labor availability, for numerous reasons, resulting in product shortages and the reduction of profits of firms. Even as progress is being made, issues associated with labor are still arising. Increasing wages is a strategy to enhance labor productivity and, also to ameliorate, in part, labor shortages but has not been fully explored in a supply chain network context. In this chapter, a game theory supply chain network model is constructed of firms competing in producing a substitutable, but differentiated, product, and seeking to determine their equilibrium product path flows, as well as hourly wages to pay their workers, under fixed labor amounts associated with links, and wage-responsive productivity factors. The theoretical and computational approach utilizes the theory of variational inequalities. First, a model without wage bounds on links is introduced, which is then extended to include wage bounds. Lagrange analysis is conducted for the latter model, which yields interesting insights, as well as an alternative variational inequality formulation. A series of numerical examples reveals that firms can gain in terms of profits by being willing to pay higher wages, resulting in benefits also for their workers, as well as consumers, who enjoy lower demand market prices for the products. However, sensitivity analysis should be conducted to determine the range of such wage bounds. Ultimately, it is observed that the profits may decrease and then stabilize. This work adds to the literature on the integration of concepts from economics and operations research for supply chain networks and also has policy implications.

5.1 Introduction The past few years, with the COVID-19 pandemic, increasing ramifications of climate change, as well as heightened geopolitical risks and wars, have dramatically shown the importance of labor to global supply chains. Disruptions associated with lack of labor have negatively impacted the production, transportation, storage, and distribution of numerous products from PPEs and other healthcare products © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_5

95

96

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

to cleaning supplies and even food items from meat to fresh produce (cf. Burki 2020; Corkery and Yaffe-Bellany 2020; Nickel and Walljasper 2020; Schrotenboer 2020). Lack of labor due to COVID issues has resulted in port congestion, at times when there was insufficient labor to unload containers (db group 2021), produce getting spoiled since it could not be harvested in a timely manner (Nagurney 2021; Luckstead et al. 2021), the culling of animals due to shortfalls in processing (Polansek and Huffstutter 2020), and products not delivered in time due to unavailability of freight service providers, especially truckers (Cardona 2021). Now, as more and more individuals are getting vaccinated against COVID-19 and many areas are “opening up,” many industries continue to deal with labor shortages and this is affecting the economy as well as the availability of goods and their prices (Conerly 2021). There has been much media attention noting that many companies are raising wages to attract workers with such a strategy being taken by firms in the hospitality industry, including restaurants, such as Chipotle and Starbucks, as well as online retailers such as Wayfair and also Walmart, Target, and Costco (Reuter and Winck 2021). Even Amazon has been raising wages in order to attract workers (Palmer 2021). Interesting, and also highly relevant, is recent evidence from practice that raising wages may enhance labor productivity, including that in manufacturing (Karp 2021). This, in turn, may alleviate some of the labor shortages. Wolters and Zilinsky (2015) reviewed the literature and theory on how wage increases influence productivity. They state that higher wages motivate employees to work harder, which is supported by research done by Yellen (1984) and by Levine (1992) in the case of large manufacturing companies. Wolters and Zilinsky (2015) also note that higher wages are associated with better health and concomitantly greater stamina and less illness, which enhance worker productivity. These authors also emphasize that laborers who are preoccupied with income security perform less well at work and cite a World Bank (2015) development report that references many field studies, acknowledging that poverty taxes one’s mental capacities and self-control. Fisman and Luca (2018), more recently, state that paying workers’ wages that are above the market rate can also be an important motivating force for a company’s employees. Lolla and O’Rourke (2020), in their two-year quasi-experiment in an apparel factory, note the positives of higher wages on both productivity of the workers and on profits of the manufacturer. Van Biesebroeck (2015) surveyed the empirical and theoretical literature on wages and productivity and provided an extensive list of references. Strain (2019) argues that, based on empirical evidence, the link between productivity and wages is strong. In this chapter, the impacts of wage-responsive productivity of labor in supply chain networks on product consumer prices and the profits of competing firms are explored. As in Chapter 4, a game theory perspective grounded in the theory of variational inequalities is used. This chapter, however, in contrast to Chapter 4, considers the following:

5.1 Introduction

97

1. Each link productivity factor is an increasing function of the wage on the link (and not fixed). Hence, the productivity factors are wage-responsive. 2. The amount of labor available on each link is fixed (and is not a variable). 3. There is an upper bound on the wage on each link that a firm desires to pay. The previously noted work considered bounds on labor and not on wages. 4. Lagrange analysis is conducted on the model, which yields an alternative variational inequality, amenable for elegant solution, plus managerial insights. 5. Sensitivity analysis is also conducted in order to ascertain the impacts of changes to wage-responsive productivity as well as to bounds on wages that firms are willing to pay their employees. The Nobel laureate Joseph Stiglitz (1982) also considered wage-responsive (dependent) productivity of labor but not in a supply chain network context as is done here, wherein different links of a firm associated with production, transportation, storage, and distribution and different sites can have distinct wageresponsive productivity factors, and, of course, these can differ also across the supply chain links of the competing firms. This chapter is organized as follows. Section 5.2 constructs the supply chain network game theory models with wage-responsive productivity factors and fixed labor on links. First, for definiteness, the model without wage bounds is presented and then the model with wage bounds on the links. The governing equilibrium conditions are stated and variational inequality formulations provided. Then, for the model with wage bounds, Lagrange analysis is conducted, which yields an alternative variational inequality with nice features for algorithmic computations. The Lagrange analysis also provides managerial insights. In Section 5.3, the computational procedure is outlined and the closed form expression structure of the resulting algorithmic steps at each iteration given. Conditions for convergence are recalled. I then report on the results for the application of the algorithm in solving a series of supply chain numerical examples. The numerical examples reveal the benefits of firms being willing to pay higher wages to their workers in terms of profits for the firms, higher wages for the workers, and lower demand market prices for consumers. However, it is imperative that a holistic approach be taken and that the full picture of supply chain competition among firms is captured. I also find that, in the case of fixed labor associated with the links, and wage-responsive productivity factors, after a particular bound on wages is achieved, the profit of a firm may decrease, and, ultimately, profits of firms may stabilize. The results of this chapter are summarized in Section 5.4, where some suggestions for future research are also outlined. Section 5.5 is the Sources and Notes Section for this chapter.

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5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

5.2 The Supply Chain Network Game Theory Models with Wage-Responsive Productivity In the supply chain network economy, there are I firms, with a typical firm denoted by i. Each firm produces a product, which is substitutable but differentiated. The firms compete in an oligopolistic manner, each seeking to maximize its profits. Each firm has its own supply chain network, with the supply chain networks of all firms, as depicted in Figure 5.1. The supply chain network of a given firm consists of production links, transportation links, and the storage and distribution links. Observe that a firm, according to Figure 5.1, can have multiple production sites and multiple storage facilities. The firms sell their products at the same demand markets. The supply chain network topology in Figure 5.1 is only representative and can be modified according to the specific application under study. The notation for the model follows closely that in Chapter 4.

Fig. 5.1 The supply chain network topology

Let G = [N , L ] represent the graph consisting of the set of nodes N and the set of links L as in Figure 5.1. Pki denotes the set of paths in firm i’s supply chain network with a path p originating in the top tier node i and ending in a demand market k, with i = 1, . . . , I and k = 1, . . . , J . P i is then the set of all paths of firm i, i = 1, . . . , I . Group all the latter paths into the set P with nP elements. Observe from Figure 5.1 that the supply chain networks of the firms have no links in common. Table 5.1 contains the basic model notation. All vectors are column vectors.

5.2 The Supply Chain Network Game Theory Models with Wage-Responsive. . .

99

5.2.1 The Model Without Wage Bounds First, the conservation of flow equations are presented and then the equations relating labor on each link to the product output on the link. Here, as in preceding modeling chapters, a linear production function, as in economics (cf. Mishra 2007; Samuelson and Marks 2012), is assumed, but one that is wage-responsive, as in Table 5.1. Table 5.1 Notation for the models with wage-dependent labor productivity Notation xp ; p ∈ Pki

fa wa dik

Notation f ixed la w¯ a αa wa

Notation cˆa (f ) ρik (d)

Variables nonnegative product flow on path p beginning at firm node i and ending at demand market k: i = 1, . . . , I and k = 1, . . . , J . Firm i’s product path n i flows are grouped into the vector x i ∈ R+P . All the firms’ product path nP flows are grouped into the vector x ∈ R+ Nonnegative flow of the product on link a, ∀a ∈ L ; group the link flows into nL the vector f ∈ R+ Wage for a unit of labor on link a per hour the cognizant firm is willing to pay, on links a ∈ L i for i = 1, . . . , I Demand for the product of firm i at demand market k, i = 1, . . . , I and J. k = 1, . . . , J ; group the {dik } elements for firm i into the vector d i ∈ R+ I ×J All the demands are grouped into the vector d ∈ R+ Parameters Fixed amount of labor on link a (typically denoted in person hours), ∀a ∈ L Upper bound on wage on link a that the firm responsible for the link is willing to pay, for a ∈ L i for i = 1, . . . , I Productivity factor relating input of labor to output of product flow on link a, where αa is given ∀a ∈ L and is positive and is referred to as the wageresponsiveness productivity factor Functions Operational cost associated with link a, ∀a ∈ L Demand price function for firm i’s product at demand market k, i = 1, . . . , I and k = 1, . . . , J

The model without wage bounds is presented first and then extended to include wage bounds in Section 5.2.2, where the Lagrange analysis is conducted. All the product path flows must be nonnegative; that is, xp ≥ 0,

∀p ∈ P i ,

∀i.

(5.1)

Also, the demand for each product must be satisfied at each demand market; that is, for each firm i: i = 1, . . . , I : xp = dik , k = 1, . . . , J. (5.2) p∈Pki

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5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

The link flows of each firm i, i = 1, . . . , I , are related to the product path flows as follows: fa = xp δap , ∀a ∈ L i , (5.3) p∈P

where δap = 1, if link a is contained in path p and 0, otherwise. One of the novel features of the model is the use of the following equations: f ixed

fa = αa wa la

∀a ∈ L i ,

,

i = 1, . . . , I,

(5.4)

so that the greater the value of the wage wa on link a, the more productive the labor on the link. The utility function of each firm i,  U i , i = 1, . . . , I , represents the profit, which is the difference between its revenue, Jk=1 ρik (d)dik , and its operational costs and   f ixed all the wages paid for labor, a∈Li cˆa (f ) − a∈L i wa la , as follows: Ui =

J

ρik (d)dik −



cˆa (f ) −

f ixed

w a la

.

(5.5)

a∈L i

a∈Li

k=1



The utility functions Ui , i = 1, . . . , I , are assumed to be concave, with the demand price functions being monotone decreasing and continuously differentiable and with the operational link cost functions being convex and also continuously differentiable. Due to (5.2), one can define demand price functions ρ˜ik (x) ≡ ρik (d), ∀i, ∀k, and, due to (5.3), one can define the operational link cost functions c˜a (x) ≡ cˆa (f ), ∀a ∈ L . Also, using (5.4) and, subsequently, (5.3), one can conclude that f ixed w a la

=

fa

f ixed l f ixed a αa la

=

(



p∈P

xp δap )

αa

∀a ∈ L .

,

(5.6)

Also, define U˜ i (x) ≡ Ui , i = 1, . . . , I , and, by making use of (5.2) again, one obtains U˜ i (x) =

J k=1 p∈P i k

ρ˜ik (x)xp −

a∈L i

c˜a (x) −

( a∈L i



p∈P

xp δap )

αa

,

i = 1, . . . , I.

(5.7) n i The feasible set Ki for firm i is defined as Ki ≡ {x i |x i ∈ R+P , for i =  1, . . . , I }. Also, K ≡ Ii=1 Ki . Each firm i, i = 1, . . . , I , seeks to determine its vector of strategies consisting n i of its product path flows x i ∈ R+P that maximizes its profits, U˜ i (x), satisfying the Nash (1950, 1951) equilibrium conditions in the definition below.

5.2 The Supply Chain Network Game Theory Models with Wage-Responsive. . .

101

Definition 5.1 (Supply Chain Network Nash Equilibrium for the Game Theory Model Without Wage Bounds) A path flow pattern x ∗ ∈ K is a supply chain network Nash Equilibrium if for each firm i, i = 1, . . . , I : U˜ i (x i∗ , xˆ i∗ ) ≥ U˜ i (x i , xˆ i∗ ),

∀x i ∈ Ki ,

(5.8)

where xˆ i∗ ≡ (x 1∗ , . . . , x i−1∗ , x i+1∗ , . . . , x I ∗ ). According to (5.8), a Nash equilibrium is achieved when no firm, acting unilaterally, can improve upon its profits.

5.2.1.1

Variational Inequality Formulations

Using the classical theory of Nash equilibria and variational inequalities, since, under the imposed assumptions on the underlying functions, the utility functions for each firm are concave and continuously differentiable (cf. Gabay and Moulin 1980 and Nagurney 1999), it follows that the solution to the above Nash Equilibrium problem (see Nash 1950, 1951) coincides with the solution of the variational inequality problem: determine x ∗ ∈ K, such that −

J I ∂ U˜ i (x ∗ ) × (xp − xp∗ ) ≥ 0, ∂x p i

∀x ∈ K,

(5.9)

i=1 k=1 p∈P

k

which, in expanded form, is: determine x ∗ ∈ K, such that ⎡ ⎤ J I J ∗ ∗ 1 ∂ ρ˜il (x ) ⎢ ∂ C˜ p (x ) ⎥ + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ ⎣ ∂xp αa ∂xp i i i a∈L

i=1 k=1 p∈P

k

× [xp − xp∗ ] ≥ 0,

l=1

∀x ∈ K,

q∈Pl

(5.10)

where ∂ cˆb (f ) ∂ C˜ p (x) ∂ ρ˜il (x) ∂ρil (d) ≡ δap , ∀p ∈ P i , ∀i, ≡ , ∀p ∈ Pki , ∀i, ∀k. ∂xp ∂f ∂x ∂d a p ik i i a∈L b∈L

(5.11)

Once the equilibrium is computed—as is discussed in the next section—the wages on the links can be determined by using (5.4).

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5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

5.2.2 The Model with Wage Bounds plus Lagrange Analysis The above model is now extended to include upper bounds on wages that the firms are willing to pay their workers per hour. Distinct upper limits on different links are allowed, for the sake of flexibility. The model remains as above except for the addition of the following constraints: wa ≤ w¯ a ,

∀a ∈ L .

(5.12)

Making use of (5.3) and (5.4), (5.12) can be re-expressed as

f ixed

xp δap ≤ w¯ a αa la

,

∀a ∈ L .

(5.13)

p∈P

Define the feasible set K1i ≡ {x i ≥ 0, and (5.13) holds for all a ∈ L i }, with  K1 ≡ Ii=1 K1i . With the wage link upper bounds (5.13), the statement of the Nash Equilibrium according to Definition 5.1 is still valid but over the feasible set K1 . The variational inequality (5.10) also holds but with the new feasible set K1 . Note that, since there are bounds on the wages, the link flows are also bounded, as are the path flows; hence, the feasible set K1 is compact. Since all the functions in (5.10) are continuous, under the imposed assumptions, it follows from the classical theory of variational inequalities (see Kinderlehrer and Stampacchia 1980) that a solution exists. Define V (x) as ⎡ ⎤ J I J ∗ ∗ ˜ 1 ∂ ρ˜il (x ) ⎢ ∂ Cp (x ) ⎥ V (x) ≡ + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ ⎣ ∂x α ∂x p a p i i i a∈L

i=1 k=1 p∈P k

l=1

× [xp − xp∗ ]

q∈Pl

(5.14)

and observe that the variational inequality with wage bounds can be rewritten as the following minimization problem: min V (x) = V (x ∗ ) = 0.

(5.15)

K1

In order to construct the Lagrange function, I reformulate the constraints as below, with the associated Lagrange multiplier next to the corresponding constraint: ea =



f ixed

xp δap − w¯ a αa la

≤ 0,

gp = −xp ≤ 0,

p , ∀p,

λa , ∀a,

p∈P

(5.16)

5.2 The Supply Chain Network Game Theory Models with Wage-Responsive. . .

103

and Γ (x) = (ea , gp )a∈L ;p∈P .

(5.17)

The Lagrange function L(x, λ, ) is now constructed, where λ is the vector of all λa s and  is the vector of all p s: ⎡ ⎤ J I J ∗) ∗) ˜ (x ∂ C 1 ∂ ρ ˜ (x il ⎢ p ⎥ L(x, λ, ) = + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ ⎣ ∂xp αa ∂xp i i i a∈L

i=1 k=1 p∈P

k

× [xp − xp∗ ] +



ea λa +

a∈L



l=1

q∈Pl

(5.18)

gp p ,

p∈P

nP nL nP ∀x ∈ R+ , ∀λ ∈ R+ , ∀ ∈ R+ .

The feasible set K1 is convex, and the Slater condition is satisfied. Indeed, we n know that Γ (x) is convex and ∃x¯ ∈ R+P : Γ (x) ¯ < 0, since one can always construct a small enough path flow pattern. Hence, if x ∗ is a minimal solution to problem n n (5.15), there exist λ∗ ∈ R+L and  ∗ ∈ R+P such that the vector (x ∗ , λ∗ ,  ∗ ) is a saddle point of the Lagrange function (5.18): L(x ∗ , , λ) ≤ L(x ∗ ,  ∗ , λ∗ ) ≤ L(x,  ∗ , λ∗ )

(5.19)

and ea∗ λ∗a ,

∀a,

gp∗ p∗ = 0,

∀p.

(5.20)

From the right-hand side of (5.19), it follows that x ∗ ∈ R+P is a minimal point of the function L(x,  ∗ , λ∗ ) in the whole space R nP , and, therefore, one had that for all p ∈ Pki , ∀i, k: n

⎤ J ∗ ∗ ˜ 1 ∂ ρ˜il (x ) ⎥ ⎢ ∂ Cp (x ) =⎣ + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ ∂xp α ∂x a p i i ⎡

∂L(x ∗ ,  ∗ , λ∗ ) ∂xp

a∈L

+



λ∗a δap − p∗ = 0,

a∈L

together with conditions (5.20). The following theorem can now be stated.

l=1

q∈Pl

(5.21)

104

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

Theorem 5.1 Conditions (5.20) and (5.21) correspond to an equivalent variational inequality to the one in (5.10), but over the feasible set K1 , given by: determine 2n +n (x ∗ ,  ∗ , λ∗ ) ∈ R+ P L such that ⎡ J I

nR ⎢ ∂ C˜ p (x ∗ ) 1 ∂ ρ˜il (x ∗ ) ∗ + δap − ρ˜ik (x ∗ ) − xq ⎣ ∂xp αa ∂xp i i i a∈L

i=1 k=1 p∈P k

+

a∈L i

+



l=1

λ∗a δap − p∗ ⎦ × [xp − xp∗ ] ⎡ ⎤     ⎣w¯ a αa laf ixed − xp∗ × p − p∗ + xp∗ δap ⎦ × λa − λ∗a ≥ 0,

p∈P

a∈L 2n

∀(x, , λ) ∈ R+ P

+nL

p∈P

(5.22)

,

or simplified as: determine (x ∗ , λ∗ ) ∈ R+P n

⎡

J I

⎣

i=1 k=1 p∈P i k

−

∂xp

l=1

+

+nL

such that

1 ∂ C˜ p (x ∗ ) + δap + λ∗a δap − ρ˜ik (x ∗ ) ∂xp α a i i a∈L

⎡

⎥ xq∗ ⎦ × [xp − xp∗ ]

q∈Pli

⎣w¯ a αa laf ixed

a∈L

a∈L

⎤

J ∂ ρ˜il (x ∗ )



q∈Pl

⎤

−



⎤ xp∗ δap ⎦ ×

  λa − λ∗a ≥ 0,

n

∀(x, λ) ∈ R+P

+nL

.

p∈P

(5.23) +n

Proof First, note that a vector (x ∗ ,  ∗ , λ∗ ) ∈ R+ P L satisfying (5.20) and (5.21) also satisfies variational inequality (5.22). Now it is proved that such a vector also satisfies variational inequality (5.10). If one multiplies (5.21) by (xp − xp∗ ) and sums up the resultant with respect to all paths p ∈ P, one gets 2n

⎡ ⎤ J I J   ∗) ∗) ˜ (x ∂ C 1 ∂ ρ ˜ (x il ⎢ p ⎥ + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ × xp − xp∗ ⎣ ∂xp αa ∂xp i i i a∈L

i=1 k=1 p∈P

=

p∈P

k

[−

a∈L

λ∗a δap + p∗ ] × [xp − xp∗ ].

l=1

q∈Pl

(5.24)

5.2 The Supply Chain Network Game Theory Models with Wage-Responsive. . .

105

Expanding the right-hand side of (5.24) yields =−



λ∗a δap xp +

p∈P a∈L

=





λ∗a δap xp∗ +

p∈P a∈L

⎡ ⎣w¯ a αa laf ixed −

a∈L



p∗ xp −

p∈P

⎤ xp δap ⎦ λ∗a +

p∈P







p∗ xp∗ ,

(5.25a)

p∈P

p∗ xp −

p∈P



p∗ xp∗ ,

(5.25b)

p∈P

by applying (5.20) and (5.13). The third term in (5.25b) is clearly nonnegative, whereas the fourth term in (5.25b), due to (5.20), is equal to zero. The term in brackets in (5.25b) is nonnegative due to the feasible set K1 . The conclusion that variational inequality (5.22) is equivalent to variational inequality (5.10) with the feasible set K1 follows. Variational inequality (5.23) then follows from (5.22) since the nonnegativity of the product path flows x is guaranteed by the feasible set in (5.23). The Lagrange analysis is now conducted. Note that Lagrange analysis has been a topic for other network-based equilibrium problem (cf. Daniele 2001, 2004, 2006; Barbagallo et al. 2012; Toyasaki et al. 2014; Colajanni et al. 2018; Caruso and Daniele 2018, and by Nagurney 2021). I utilize (5.21) and consider the case where a path p has a positive product flow at the solution, so that xp∗ > 0, in which case, due to (5.20), p∗ = 0. Assume that for this path none of the link flows are at their upper bounds, in which case, according to (5.20), all the λ∗a s on links on the path are equal to zero. Then, (5.21) yields for such a path p ∈ Pki , ∀i, k: ⎡ ⎤ J ∗) 1 ∂L(x ∗ ,  ∗ , λ∗ ) ⎢ ∂ C˜ p (x ∗ ) ∂ ρ ˜ (x il ⎥ =⎣ + δap − ρ˜ik (x ∗ ) − xq∗ ⎦ = 0 ∂xp ∂xp α ∂x a p i i a∈L

q∈Pl

l=1

(5.26) or J 1 ∂ C˜ p (x ∗ ) ∂ ρ˜il (x ∗ ) ∗ + δap = ρ˜ik (x ∗ ) − xq . ∂xp αa ∂xp i i a∈L

l=1

(5.27)

q∈Pl

Expression (5.27) has the interpretation that the marginal operational costs on the path with the inclusion of the sum of the inverses of the wage-responsive productivity factor on links comprising the path are equal to the marginal revenue associated with the product flow on the path. This is a good situation and can be interpreted as the marginal revenue being equal to the full marginal costs, which means that the profit is maximized. Furthermore, from (5.27), one sees that the greater the wage-responsiveness productivity factors on links, making up such a path, the “closer” the marginal revenue on the path to the marginal total operational costs on the path.

106

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

On the other hand, assume now that the flow on the path is still positive, but that one or more of the link flows on links making up the path are at their respective upper bounds, in which case, one or more of the λ∗a s will be greater than zero on such links according to (5.20). Then, (5.21) implies that J ∂ C˜ p (x ∗ ) 1 ∂ ρ˜il (x ∗ ) ∗ + δap + λ∗a δap = ρ˜ik (x ∗ )+ xq . ∂xp αa ∂xp i i i a∈L

a∈L

l=1

(5.28)

q∈Pl

From equation (5.28), one sees that, with one or more link flows at their upper bounds, the marginal revenue associated with the path exceeds the marginal operational costs on the path with the inclusion of the sum of the inverses of the wage-responsive productivity factor on links comprising the path by an amount  ∗ a∈L i λa δap , which is not a good situation, since profits could be higher if the bounds were loosened. Also, clearly, the greater the αa s on links a comprising a path, then in both (5.26) and (5.28), the marginal revenue on the path is closer to the marginal total operational costs on the path. This reflects that a path is comprised of links on which workers produce more for a given wage (are more productive). The Lagrange multipliers, the λ∗a s, provide valuable information and may also be interpreted as shadow prices. For example, if one looks back at constraints (5.13), with which these Lagrange multipliers are associated, one sees that a unit increase in the right-hand side of (5.13) for link a will result in a λ∗a increase in profits for the firm. This can be achieved by the cognizant firm (cf. (5.13)) by raising the f ixed wage bound w¯ a and/or increasing labor la and/or even incentivizing workers to be more sensitive to wages through raising αa . Furthermore, by comparing the positive λ∗a s, a firm can see on which link(s) it makes the most sense to invest in terms of loosening the bounds, with those links with the highest values of their λ∗a s, ultimately, being the most important to invest in, from a profit-enhancing perspective.

5.3 The Algorithm and Numerical Examples Before presenting the algorithm, variational inequality (5.23) is put into standard form (cf. (2.14)): determine X∗ ∈ K such that

F (X∗ ), X − X∗  ≥ 0,

∀X ∈ K ,

(5.29)

where F is a given continuous function from K to R N , K is a given closed convex set, and ·, · denotes the inner product in N-dimensional Euclidean space. n +n Set K ≡ R+P L , X ≡ (x, λ), and N = nP + nL . Also, define the vector F ≡ (F1 , F2 ), where the components of F1 consist of the elements:

∂ C˜ p (x) ∂xp

+

5.3 The Algorithm and Numerical Examples





a∈L i

λa δap − ρ˜ik (x) −

J

∂ ρ˜il (x) ∂xp



i q∈Pli xq , ∀p ∈ Pk ,  f ixed ∀i, ∀k, and the components of F2 consist of the elements: w¯ a la − p∈P xp δap , 1 a∈L i αa δap

+

107

l=1

∀a ∈ L . Then, clearly, variational inequality (5.23) coincides with variational inequality (5.29), with the above definitions. As in Chapters 2, 3, and 4, I apply the modified projection method of Korpelevich (1977) for the computation of solutions to the supply chain network numerical examples. Please refer to Chapter 2 or to the Appendix for this algorithm. For illustrative purposes, I present the explicit formulae that its Step 1 takes for the model(s). Step 2 can be then easily adapted. The modified projection method is guaranteed to converge if the function F (X) that enters the variational inequality problem (5.29) is monotone and Lipschitz continuous, provided that a solution exists. The elegance of this procedure for the computation of solutions to both the supply chain network game theory models with wage-dependent labor can be seen in the following explicit formulae. Explicit Formulae at Iteration τ for the Product Path Flows in Step 1 In particular, one has the following closed form expressions for the path flows in Step 1 for the solution of variational inequality (5.23): ⎧ ⎪ ⎨

x¯pτ = max 0, xpτ −1 + η(ρ˜ik (x τ −1 ) + ⎪ ⎩ −

a∈L i

∀p ∈

Pki ;

λτa −1 δap −

J ∂ ρ˜il (x τ −1 ) l=1

⎫ ⎬

∂xp

q∈Pli

xqτ −1 −

∂ C˜ p (x τ −1 ) ∂xp

1 δap ) , ⎭ αa i

a∈L

i = 1, . . . , I ; k = 1, . . . , J.

(5.30)

Explicit Formulae at Iteration τ for the Lagrange Multipliers in Step 1 Similarly, one has the following closed form expressions for the Lagrange multipliers in Step 1 at an iteration τ : ⎧ ⎨

⎛

λτa = max 0, λτa −1 + η ⎝ ⎩



p∈P

⎞⎫ ⎬ f ixed ⎠ xpτ −1 δap − w¯ a αa la , ⎭

∀a ∈ L .

(5.31)

As for the solution of variational inequality (5.10), governing the model without wage upper bounds, (5.31) is no longer needed, whereas  (5.30) still holds for Step 1 of the modified projection method but with the term: − a∈L λaτ −1 δap removed.

108

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

5.3.1 High Value Supply Chain Numerical Examples The modified projection method is coded in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the numerical computations. The algorithm is initialized with a demand of 40 for each firm–demand market pair, with the demand then equally divided among the associated path flows. The Lagrange multipliers are all initialized to zero. The convergence tolerance is 10−7 in that the absolute difference between two successive variable iterates differs by no more than this amount.

Fig. 5.2 The supply chain network topology for the numerical examples

There are two firms and two demand markets in the numerical examples. Each firm has at its disposal two production plants, a single distribution center, and sells its product at two demand markets, as depicted in Figure 5.2. The operational link cost functions are cˆa (f ) = 2fa2 ,

cˆb (f ) = 2fb2 ,

cˆe (f ) = fe2 + 2fe , cˆh (f ) = 1.5fh2 ,

cˆc (f ) = .5fc2 ,

cˆf (f ) = .5ff2 ,

cˆi (f ) = 1.5fi2 + fi ,

cˆd (f ) = .5fd2 ,

cˆg (f ) = .5fg2 ,

cˆj (f ) = fj2 + 2fj ,

cˆk (f ) = fk2 ,

5.3 The Algorithm and Numerical Examples

cˆl (f ) = .5fl2 ,

109

cˆm (f ) = .5fm2 + fm ,

cˆn (f ) = fn2 + 2fn .

The demand price functions are ρ11 (d) = −5d11 − 2d21 + 800, ρ21 (d) = −3d21 − d11 + 700,

ρ12 (d) = −5d12 − d22 + 850, ρ22 (d) = −5d22 − .5d12 + 750.

The operational cost functions and demand price functions are constructed to reflect a fairly high value product that is not that expensive to produce, transport, store, and distribute. f ixed The αa and the la parameters (cf. (5.4)) are, for a ∈ L , as follows: αa = 0.5, αb = 0.5, αc = 0.3, αd = 0.3, αe = 0.4, αf = 0.5, αg = 0.3, αh = 0.3, αi = 0.4, αj = 0.3, αk = 0.3, αl = 0.5, αm = 0.3, αn = 0.3, f ixed

la

f ixed

lh

f ixed

= 10, lc

f ixed

= 3, lj

= 10, lb

= 3, li

f ixed

f ixed

f ixed

= 9, ld

f ixed

= 9, lk

f ixed

= 7, le

f ixed

= 9, ll

f ixed

= 8, lf

f ixed

= 8, lm

f ixed

= 6, lg

f ixed

= 7, ln

= 8,

= 8.

The paths are defined as path p1 = (a, c, e, f ), path p2 = (b, d, e, f ), path p3 = (a, c, e, g), path p4 = (b, d, e, g), path p5 = (h, j, l, m), path p6 = (i, k, l, m), path p7 = (h, j, l, n), and path p8 = (i, k, l, n, ). Series 1: Examples 5.1 Through 5.4 In this series of examples, the impacts on product flows, consumer prices, and firm profits of raising the link wage bounds are investigated. Example 5.1 has all link wage bounds w¯ a = 10, Example 5.2 has all wage bounds w¯ a = 15, Example 5.3 has all wage bounds w¯ a = 20, and Example 5.4 has all wage bounds w¯ a = 25. The remaining data for Examples 5.1 through 5.4 are as described above. Example 5.1 (Results) The modified projection method converges to the following equilibrium path flow pattern: xp∗1 = 7.19,

xp∗2 = 7.19,

xp∗3 = 8.81,

xp∗4 = 8.81,

xp∗5 = 0.59,

xp∗6 = 2.09,

xp∗7 = 8.41,

xp∗8 = 9.91.

The demand market prices are ρ11 = 722.75,

ρ12 = 743.56,

ρ21 = 677.59,

ρ22 = 722.86.

110

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

The profit for Firm 1 is 20,530.75 and the profit for Firm 2 is 13,628.85. Example 5.2 (Results) The modified projection method converges to the following equilibrium path flow pattern: xp∗1 = 11.19,

xp∗2 = 11.19,

xp∗5 = 1.94,

xp∗6 = 4.19,

xp∗3 = 12.81, xp∗7 = 11.56,

xp∗4 = 12.81, xp∗8 = 13.81.

The demand market prices are now ρ11 = 675.81,

ρ12 = 696.54,

ρ21 = 659.22,

ρ22 = 711.82.

The profit for Firm 1 is now 26,605.95, whereas the profit for Firm 2 is 19,213.26. With the wage bounds raised, reflecting that the firms are willing to pay their workers more for their labor, the profit for each firm increases, while the demand market prices that consumers pay decrease, signaling a win–win situation. Example 5.3 (Results) The modified projection method for Example 5.3, with all wage bounds set to 20, converges to the following equilibrium path flow pattern: xp∗1 = 15.20,

xp∗2 = 15.20,

xp∗5 = 3.29,

xp∗6 = 6.29,

xp∗3 = 16.81, xp∗7 = 14.71,

xp∗4 = 16.81, xp∗8 = 17.71.

The demand market prices are now ρ11 = 628.88,

ρ12 = 649.53,

ρ21 = 640.86,

ρ22 = 700.78.

The profit for Firm 1 is now 29,897.83 and the profit for Firm 2 is 24,012.63. With the wage bounds further increased, the profits of both firms increase (as compared to their respective values in Examples 5.1 and 5.2) and, again, the demand market prices decrease at all demand markets under more generous upper bounds on wages. Example 5.4 (Results) The algorithm for Example 5.4, with all wage bounds now set to 25, converges to the following equilibrium path flow pattern: xp∗1 = 18.69,

xp∗2 = 18.69,

xp∗5 = 4.67,

xp∗6 = 8.42,

xp∗3 = 20.30, xp∗7 = 17.83,

xp∗4 = 20.30, xp∗8 = 21.58.

The demand market prices are ρ11 = 586.95,

ρ12 = 607.61,

ρ21 = 623.37,

ρ22 = 690.28.

5.3 The Algorithm and Numerical Examples

111

The profit for Firm 1 is 30,425.58 and the profit for Firm 2 is 28,060.79. One sees that the results are quite robust and reveal that raising wages can benefit both firms and consumers. The computed equilibrium link flows, Lagrange multipliers, and wages (cf. (5.4)) for Examples 5.1 through 5.4 are reported in Table 5.2. From Table 5.2, one sees that, as the wage upper bounds are raised, as one proceeds from Examples 5.1 through 5.4, the wages for labor on all the supply chain network links increase, and this, as discussed above, benefits consumers in terms of lower prices, as well as firms, in terms of higher prices. Furthermore, since, in the model, the labor is fixed on all the links, the workers also benefit. Also, one sees that, in Examples 5.1 through 5.4, the Lagrange multipliers associated with links h and i are all positive and that the wages on these links are at the upper bounds. The Lagrange multiplier on link e, on the other hand, is positive in Examples 5.1 through 5.3 but is equal to 0.00 in Example 5.4 since in Example 5.4 the wage on link e is not at the upper bound of 25. The firms may wish to invest in the links with positive Lagrange multipliers, since doing so, with the large values of the Lagrange multipliers, they can enhance their profits significantly. I now proceed to conduct additional sensitivity analysis and, in Figure 5.3, report on the profits of the two firms as the link wage bounds are raised, while also graphing these results for Examples 5.1 through 5.4. Something interesting is happening, and this further emphasizes the importance of having a rigorous theoretical and computational framework for conducting such exercises. At a wage bound of 30, Firm 1 now has a lower profit than it had at a wage bound of 25, whereas the profit of Firm 2 continues to increase, but at a decreasing rate. Plus, Firm 2 now has a profit exceeding that of Firm 1. And, for link wage bounds of 65 or higher, the profit of Firm 1 stabilizes at 27,225.99 and that of Firm 2 at 39,800.45. It is important to emphasize that the model has, in effect, link production functions that relate labor, which is fixed, and the wage-responsiveness productivity factor and wage to the product output on each link. However, the product flows are associated with paths, since the product requires multiple supply chain links, beginning from production to ultimate distribution, and the latter, for each firm and demand market pair, sum up to the demand. All these are intricately related. Of course, the solution of the supply chain network model without bounds for this dataset would yield the same profits (and equilibrium pattern) as obtained for wages on the supply chain links of 65 or above. Series 2: Examples 5.5 Through 5.8 In this series of examples, a sensitivity analysis on the αa parameters is done. Specifically, the data in these examples are as in Example 5.4 (with the wage bounds all equal to 25), but Example 5.5 has all the αa s increased fourfold; Example 5.6 has all the αa s doubled. Example 5.7, on the other hand, has the αa s in Example 5.4 halved, whereas Example 5.8 has the αa s in Example 5.4 multiplied by 14 . Example 5.5 (Results) The demand market prices for Example 5.5 are

112

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

Table 5.2 Equilibrium link flows, Lagrange multipliers, and hourly wages for Examples 5.1 through 5.4 Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fh∗ fi∗ fj∗

fk∗ fl∗ fm∗ fn∗ λ∗a λ∗b λ∗c λ∗d λ∗e λ∗f λ∗g λ∗h λ∗i λ∗j

λ∗k λ∗l λ∗m λ∗n wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗ wh∗ wi∗ wj∗

wk∗ wl∗ ∗ wm wn∗

Equilibrium value Example 5.1 Example 5.2 16.00 24.00 16.00 24.00 16.00 24.00 16.00 24.00 32.00 48.00 14.38 22.38

Example 5.3 32.00 32.00 32.00 32.00 64.00 30.39

Example 5.4 38.99 38.99 38.99 38.99 77.97 37.38

17.62 9.00 12.00 9.00

25.62 13.50 18.00 13.50

33.61 18.00 24.00 18.00

40.60 22.50 30.00 22.50

12.00 21.01 2.68 18.33 0.00 0.00 0.00 0.00 480.64 0.00

18.00 31.50 6.13 25.37 0.00 0.00 0.00 0.00 313.67 0.00

24.00 42.00 9.58 32.42 0.00 0.00 0.00 0.00 146.70 0.00

30.00 52.50 13.08 39.42 0.00 0.00 0.00 0.00 0.00 0.00

0.00 585.85 572.69 0.00

0.00 520.68 500.02 0.00

0.00 455.51 427.34 0.00

0.00 391.02 355.35 0.00

0.00 0.00 0.00 0.00 3.20 3.20 5.93 7.62 10.00 4.79

0.00 0.00 0.00 0.00 4.80 4.80 8.89 11.43 15.00 7.46

0.00 0.00 0.00 0.00 6.40 6.40 11.85 15.24 20.00 10.13

0.00 0.00 0.00 0.00 7.80 7.80 14.44 18.56 24.37 12.46

7.34 10.00 10.00 3.33

10.67 15.00 15.00 5.00

14.00 20.00 20.00 6.67

16.91 25.00 25.00 8.33

4.45 4.67 1.28 7.74

6.67 7.00 2.92 10.57

8.89 9.33 4.56 13.51

11.11 11.67 6.23 16.42

5.3 The Algorithm and Numerical Examples

ρ11 = 551.09,

ρ12 = 574.63,

113

ρ21 = 555.96,

ρ22 = 650.81.

The profit for Firm 1 is now 27,741.21 and the profit for Firm 2 is 40,774.57. The demand market prices are lower for consumers at the demand markets as compared to their respective values in Example 5.4. Firm 1 has a lower profit, whereas Firm 2 has a greater profit in Example 5.5 than it did in Example 5.4. 40 Firm 1 Profit Firm 2 Profit

35

103

30

Profits

+

25

20

15

10 10

20

30 40 50 60 Wage Bound On Supply Chain Links

70

80

Fig. 5.3 Sensitivity analysis for different wage bounds on the supply chain networks of both firms and effects on the firms’ profits

Example 5.6 (Results) The demand market prices for Example 5.6 are ρ11 = 557.52,

ρ12 = 580.64,

ρ21 = 567.58,

ρ22 = 657.62.

The profit for Firm 1 is 28,096.79 and the profit for Firm 2 is 39,826.35. In Example 5.6, the demand market prices are lower than in Examples 5.4 and 5.5. The profit for Firm 1 is greater than it earned in Example 5.5, whereas the profit of Firm 2 is lower than it earned in Example 5.4. Example 5.7 (Results) The demand market prices for Example 5.7 are ρ11 = 698.99,

ρ12 = 720.35,

ρ21 = 668.37,

ρ22 = 717.36.

The profit for Firm 1 is now 23,494.23 and the profit for Firm 2 is 16,217.42. With αa s lower than in Example 5.4, the prices are higher in Example 5.7 than in Example 5.4 and also higher (with smaller αa s) than in Example 5.6. Both firms now suffer a drop in their profits as compared to their profits in Example 5.6.

114

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

Example 5.8 (Results) The demand market prices for Example 5.8 are ρ11 = 757.08,

ρ12 = 779.71,

ρ21 = 691.25,

ρ22 = 731.21.

The profit for Firm 1 is 13,512.69 and the profit for Firm 2 is 8,471.89. With even smaller values of αa s, for all links a ∈ L , the prices increase further at the demand markets, and the profits of both firms decrease substantially. The general progression of demand market prices and firm profits as the αa s decrease makes sense, since one can see from (5.4) that, for a fixed amount of labor and wage, a lower αa on a link a would result in a lower product output volume on the link. Series 3: Examples 5.9 and 5.10 In this series of numerical examples, the impacts of one firm having higher wage bounds on its supply chain network activities than the other firm are studied. Example 5.9 (Results) Example 5.9 has the same data as Example 5.1, except that w¯ a = 10, for all links a ∈ L 1 , whereas w¯ a = 15, for all links a ∈ L 2 . The modified projection method converges to the following equilibrium path flow pattern: xp∗1 = 7.17, xp∗5 = 2.13,

xp∗2 = 7.17, xp∗6 = 4.38,

xp∗3 = 8.83,

xp∗4 = 8.83,

xp∗7 = 11.38,

xp∗8 = 13.63.

The demand market prices are ρ11 = 715.33,

ρ12 = 736.66,

ρ21 = 666.16,

ρ22 = 716.16.

The profit for Firm 1 is 20,303.18 and the profit for Firm 2 is 19,364.23. As compared to the results in Example 5.1, Firm 1 now loses in the competition with Firm 2, which is willing to pay its workers higher wages. It has a lower profit than it had in Example 5.1, whereas Firm 2 enjoys a substantially increased profit, showing, again, that firms can earn greater profits, while paying their workers more. For the computed equilibrium link flows, Lagrange multipliers, and wages for Examples 5.9 and 5.10, please refer to Table 5.3. Example 5.10 (Results) Example 5.10 also has the same data as Example 5.1, but now I switch and have Firm 1 having the higher wage bounds, in that w¯ a = 15, for all a ∈ L 1 , with Firm 2 now having wage bounds of w¯ a = 10, for all links a ∈ L 2 . The modified projection method converges to the equilibrium path flow pattern: xp∗1 = 11.21,

xp∗2 = 11.21,

xp∗3 = 12.79,

xp∗5 = 0.40,

xp∗6 = 1.90,

xp∗7 = 8.60,

xp∗4 = 12.79, xp∗8 = 10.10.

5.3 The Algorithm and Numerical Examples

115

The demand market prices are ρ11 = 683.24,

ρ12 = 703.44,

ρ21 = 670.64,

ρ22 = 718.52.

The profit for Firm 1 is 26,948.15 and the profit for Firm 2 is 13,535.19. I now compare the profits of the two firms to the results in Example 5.2, where all the wage bounds were set to 15. In Example 5.10, Firm 1, by being willing to pay workers more than Firm 2, earns a higher profit than it did in Example 5.2, whereas Firm 2 has a substantial decrease in profits, as compared to what it earned in Example 5.2, where it was willing to pay workers a maximum of 15 (rather than 10 as in Example 5.10). The above supply chain network examples provide useful insights for firms, workers, consumers, as well as policy-makers. Of special significance and relevance is that firms can garner enhanced profits in paying their workers. Of course, the framework assumes wage-responsive productivity factors, but, as noted in the Section 5.1, economists, including the Nobel laureate Joseph Stiglitz (1982), have argued that many workers will be more productive under higher wages. However, it is imperative to conduct sensitivity analysis to see the quantifiable impacts on profits of wage bound increases since, after a point, there may be diminishing returns. Additional sensitivity analysis is now conducted. In particular, I investigate the impacts of increasing the wage bounds, using Example 5.9 as the baseline where the wage bound of Firm 1 is lower than that of Firm 2. I proceed with the following increments of wage bounds for Firm 1 and Firm 2 in parentheses: (10, 15), (15, 20),. . ., (70, 75), on the profits of the firms. The results are reported in Figure 5.4. Firm 1 has profits higher than Firm 2 until the wage bounds of (25, 30), after which the profit of Firm 1 decreases, but that of Firm 2 increases, but, at a decreasing rate, until it stabilizes at 38,800.45 and the profit of Firm 1 stabilizes at 27,255.99. I continue the sensitivity analysis, but now Firm 1 has a higher wage bound than Firm 2. I proceed with computing the solution to Example 5.9 but now with wage bounds of (15, 10), (20, 15), (25, 20), . . ., (75, 70). The profits of the two firms are displayed in Figure 5.5. Firm 1 now enjoys a higher profit at (15, 10) than it did at (10, 15) in Figure 5.4, the same for (20, 15) than at (15, 20) in Figure 5.4, and so on, suggesting, again, that lifting wage bounds can have positive effects on a firm’s profits. Firm 2, on the other hand, now has consistently lower profit at (15, 10) than it did at (10, 15) in Figure 5.4 the same for (20, 15) than at (15, 20), and so on. At (70, 65), as also seen in previous results, the profits reach 27,225.99 for Firm 1 and 39,800.45 for Firm 2 and remain there for (75, 70) (and so on). Firm 1, while enjoying increasing profits at the first three points, then encounters a reduction in profit at (30, 25) and that trend continues until its profit of 27,225.99. Firm 2 enjoys an increase in profits as one moves along the x axis in Figure 5.5, with stabilization at the profit level of 39,800.45.

116

5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

Table 5.3 Equilibrium link flows, Lagrange multipliers, and hourly wages for Examples 5.9 and 5.10

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fh∗ fi∗ fj∗

fk∗ fl∗ fm∗ fn∗ λ∗a λ∗b λ∗c λ∗d λ∗e λ∗f λ∗g λ∗h λ∗i λ∗j

λ∗k λ∗l λ∗m λ∗n wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗ wh∗ wi∗ wj∗

wk∗ wl∗ ∗ wm wn∗

Equilibrium value Example 5.9 Example 5.10 16.00 24.00 16.00 24.00 16.00 24.00 16.00 24.00 32.00 48.00 14.33 22.43 17.67 13.50 18.00 13.50

25.57 9.00 12.00 9.00

18.00 31.50 6.50 25.00 0.00 0.00 0.00 0.00 473.48 0.00

12.00 21.01 2.31 18.70 0.00 0.00 0.00 0.00 320.83 0.00

0.00 526.13 505.48 0.00

0.00 580.40 567.23 0.00

0.00 0.00 0.00 0.00 3.20 3.20 5.93 7.62 10.00 4.78

0.00 0.00 0.00 0.00 4.80 4.80 8.89 11.43 15.00 7.48

7.36 15.00 15.00 5.00

10.66 10.00 10.00 3.33

6.67 7.00 3.10 10.42

4.45 4.67 1.10 7.79

5.3 The Algorithm and Numerical Examples

117

40 Firm 1 Profit Firm 2 Profit

35

103

30

Profits

+

25

20

15 (10,15) (15,20)(20,25)(25,30)(30,35)(35,40)(40,45)(45,50)(50,55) (55,60)(60,65)(65,70)(70,75) Wage Bounds On Supply Chain Links(Wage Bound on Links of Firm 1, Wage Bound on Links of Firm 2)

Fig. 5.4 Sensitivity analysis for different wage bounds on the supply chain networks of both firms, with firm 1 having a lower bound than firm 2, and effects on the firms’ profits

40 Firm 1 Profit Firm 2 Profit

103

35

30

Profits

+

25

25

15 (15,10)(20,15) (25,20)(30,25)(35,30) (40,35)(45,40)(50,45) (55,50)(60,55) (65,60) (70,65)(75,70) Wage Bounds on Supply Chain Links (Wage Bound on Links of Firm 1, Wage Bound on Links of Firm 2)

Fig. 5.5 Sensitivity analysis for different wage bounds on the supply chain networks of both firms, with firm 1 having a higher bound than firm 2, and effects on the firms’ profits

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5 Wages and Labor Productivity in Supply Chains with Fixed Labor Availability. . .

5.4 Summary, Conclusions, and Suggestions for Future Research The pandemic has demonstrated the importance of labor and its productivity to supply chains of numerous different products. Disruptions to labor have negatively impacted production, the transportation of goods, and even storage, as workers got ill, and many perished. In this chapter, a novel game theory framework is presented that integrates labor into supply chains and that captures wage-responsive productivity associated with the various supply chain network activities. First, a model without wage bounds on links, representing the maximum hourly wages that firms are willing to pay, is presented, and then the model is extended to include such wage bounds. The link productivity factors are linearly increasing in wages. The governing equilibrium concept in the competitive game theory models is that of Nash Equilibrium. The theory of variational inequalities is utilized to formulate the equilibrium conditions and, for the model with wage bounds, a Lagrange analysis is conducted, which also yields an alternative variational inequality formulation with nice features for computations. The numerical results reveal that firms that are willing to pay their workers higher hourly wages can enjoy higher profits, and can, hence, beat their competitors, in terms of lower prices for consumers at the demand markets, higher wages for the workers, and higher profits. The framework also has relevance for addressing, in part, the labor shortage in various industrial sectors and even in freight services since it is shown that, even with fixed labor amounts associated with various supply chain network economic activities, having more productive labor, that is wage-sensitive, can yield financial gains for firms. However, it is important to conduct sensitivity analysis, since, after a point of increases in the wage bounds, a firm may experience a decline in profits. Furthermore, ultimately, the profits of competing firms may stabilize and there will be no change with increases in the wage bounds, which can also be considered to be wage ceilings. This is also interesting, since it suggests a natural type of wage “cap.” This work adds to the literature on the integration of principles from economics and operations research perspectives for supply chains. Future research directions can include extending some of the concepts formulated herein to various service sectors, including the healthcare sector, which has also faced many disruptions in the COVID-19 pandemic, along with decreases in productivity, as well as shortfalls and shortages in labor. I turn to this sector, in the context of blood supply chains, in Chapter 11 of this book.

References

119

5.5 Sources and Notes This chapter is based on the paper by Nagurney (2022). In this chapter, the standardized notation used in this book is used. The results in this chapter demonstrate that taking care of one’s workers, in terms of higher wages, can yield benefits for both workers and firms who are engaged in competition in supply chain networks. The results here further support the importance of having a holistic, network approach to capturing the behavior of firms engaged in competition and demonstrate also the richness of modeling that can be achieved through game theory and the theory of variational inequalities. The work herein also supports what is being seen in practice in that raising wages can have positive ramifications in a supply chain network economy.

References Barbagallo, A., Daniele, P., Maugeri, A., 2012. Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula. Nonlinear Analysis, Theory, Methods and Applications, 75(3), 1104–1123. Burki, T., 2020. Global shortage of personal protective equipment. Lancet Infectious Diseases, 20(7), 785–786. Cardona, C., 2021. Nationwide shortage of truck drivers impacting Central Florida. Orlando.com. Caruso, V., Daniele, P., 2018. A network model for minimizing the total organ transplant costs. European Journal of Operational Research, 266(2), 652–662. Colajanni, G., Daniele, P., Giuffre, S., Nagurney, A., 2018. Cybersecurity investments with nonlinear budget constraints and conservation laws: Variational equilibrium, marginal expected utilities, and Lagrange multipliers. International Transactions in Operational Research, 25, 1443–1464. Conerly, B., 2021. The labor shortage is why supply chains are disrupted. Forbes, July 7. Corkery, M., Yaffe-Bellany, D., 2020. The food chains weakest link: Slaughterhouses. The New York Times, April 18. Daniele, P., 2001. Variational inequalities for static equilibrium market, Lagrangian function and duality. In: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. F. Giannessi, A. Maugeri, and P.M. Pardalos, Editors, Kluwer Academic Publishers, Amsterdam, pp. 43–58. Daniele, P., 2004. Time-dependent spatial price equilibrium problem: existence and stability results for the quantity formulation model. Journal of Global Optimization, 28(3–4), 283–295. Daniele, P., 2006. Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar Publishing, Cheltenham, United Kingdom. db group, 2021. Port congestion - drivers and impacts on international freight forwarding. May 31. Fisman, R., Luca, M., 2018. How Amazon’s higher wages could increase productivity. Harvard Business Review, October 10. Gabay, D., Moulin, H., 1980. On the uniqueness and stability of Nash equilibria in noncooperative games. In: Applied Stochastic Control of Econometrics and Management Science. A. Bensoussan, P. Kleindorfer, and C.S. Tapiero, Editors, North-Holland, Amsterdam, The Netherlands, pp. 271–294. Karp, E., 2021. The case for raising wages in manufacturing. Forbes, March 2. Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York.

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Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. Levine, D.I., 1992. Can wage increases pay for themselves? Test with a production function. Economic Journal, 102, 1102–115. Lolla, N., O’Rourke, D., 2020. Factory benefits to paying workers more: The critical role of compensation systems in apparel manufacturing. PLOS ONE, February 20. Luckstead, J., Nayga, Jr., R.M., Snell, H.A., 2021. Labor issues in the food supply chain amid the COVID-19 pandemic. Applied Economic Perspectives and Policy, 43(1), 382–400. Mishra, S.K., 2007. A brief history of production functions. MPRA Paper No. 5254. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A., 2021. Perishable food supply chain networks with labor in the Covid-19 pandemic. In: Dynamics of Disasters - Impact, Risk, Resilience, and Solutions. I.S. Kotsireas, A. Nagurney, P.M. Pardalos, and A. Tsokas, Editors, Springer Nature Switzerland AG, pp. 173– 193. Nagurney, A., 2022. Supply chain networks, wages, and labor productivity: Insights from Lagrange analysis and computations. Journal of Global Optimization, 83, 615–638. Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49. Nash, J.F., 1951. Noncooperative games. Annals of Mathematics, 54, 286–298. Nickel, R., Walljasper, C., 2020. Canada, U.S. farms face crop losses due to foreign worker delays. Reuters, April 6. Palmer, A., 2021. Amazon to hike wages for over 500,000 workers. CNBC.com, April 28. Polansek, T., Huffstutter, P.J., 2020. Piglets aborted, chickens gassed as pandemic slams meat sector. Reuters, April 27. Reuter, D., Winck, B., 2021. Under Armour is the latest company to raise wages during the pandemic. Here are the major firms that hiked pay over the past year. Business Insider, May 19. Samuelson, W.F., Marks, S.G., 2012. Managerial Economics, seventh edition. John Wiley & Sons, Inc., Hoboken, New Jersey. Schrotenboer, B., 2020. US agriculture: Can it handle coronavirus, labor shortages and panic buying? USA Today, April 4. Stiglitz, J.E., 1982. Alternative theories of wage determination and unemployment: The efficiency wage model. In: The Theory and Experience of Economic Development - Essays in Honor of Sir W. Arthur Lewis. M. Gersovitz, C.F. Diaz-Alejandro, G. Ranis, and M.R. Rosenzweig, Editors, George Allen & Unwin, London, England, pp. 78–106. Strain, M.A., 2019. The link between wages and productivity is strong. The Economic Strategy Group. The Aspen Institute, February 4. Toyasaki, F., Daniele, P., Wakolbinger, T., 2014. A variational inequality formulation of equilibrium models for end-of-life products with nonlinear constraints. European Journal of Operational Research, 236(1), 340–350. Van Biesebroeck, J., 2015. How tight is the link between wages and productivity? A survey of the literature. International Labour Office, Geneva, Switzerland. Wolters, J., Zilinsky, J., 2015. Higher wages for low-income workers lead to higher productivity. Peterson Institute for International Economics, January 13. World Bank, 2015. Poverty. In: World development report 2015: Mind, society, and behavior. Washington DC. Yellen, J.L., 1984. Efficiency wage models of unemployment. American Economic Review, 74(2), 200–205.

Chapter 6

Wage-Dependent Labor and Supply Chain Networks

Abstract Disruptions to labor in the past few very challenging years have highlighted its importance. Many companies are now increasing wages in order to attract workers. In this chapter, a computable supply chain network framework is presented that includes labor as a resource, along with (hourly) wages that should be paid. Both a single firm and multiple firms are considered with the latter engaged in competition with respect to substitutable products. Variational inequality formulations for the models without wage ceilings and with wage ceilings are given, along with a computational procedure. Supply chain numerical examples demonstrate the impacts of the addition of electronic commerce: the use of outsourcing, and competition, along with increases, as well as the tightening, of wage ceilings on supply chain network links. The general, flexible, holistic framework reveals some unexpected results and also shows the benefits, in terms of profits, of the free market, with no imposed wage upper bounds.

6.1 Introduction Labor is critical for supply chain functionality from the production of goods to their transport via freight service provision, their storage, and the final distribution to consumers at demand markets. Throughout the COVID-19 pandemic, the world has witnessed the impacts of labor disruptions due to workers becoming ill, with many dying. Related issues of labor availability and decreases in labor productivity have also affected our societies and economies globally (see, e.g., Bhattarai and Reiley 2020). The news has been filled with reports of worker shortages in various sectors, not only early on in the pandemic but even 18 months since the declaration of the pandemic, as the vaccination campaigns proceed and economies begin to open up. Examples of labor shortages in the USA, for example, have been noted in manufacturing, and restaurants, as they have started to reopen, as well as in construction (see Morath 2021). The transportation and logistics sector in the United States is also struggling to find workers with surging demand from consumers of various products from household goods to recreation products (DC Velocity Staff 2021). Some companies are raising wages in order to attract workers, including © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_6

121

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6 Wage-Dependent Labor and Supply Chain Networks

Walmart, Amazon, and Costco (Casselman 2021) as are restaurant chains such as McDonald’s and Chipotle (Creswell 2021). According to Associated Press News (2021), wages and benefits grew quickly for workers in the United States in January, February, and March of 2021, the largest gain in more than 13 years, as noted by the United States Labor Department. Labor shortages, exacerbated in the pandemic, and occurring for a plethora of reasons, are also prevalent in the United Kingdom (see Coles 2021), with the compounding of problems due to Brexit, the European Union (cf. Adascalitei and Weber 2021), and even parts of Asia, including China (Yifan Xie and Qi 2021) as well as in Canada and Australia (Wolf 2021). Indeed, labor shortages are now a global phenomenon. Shortages of labor for different supply chain network activities, from production, to freight service provision, and even retail, are significantly impacting global supply chains with port congestion resulting from a lack of workers leading to further product delivery delays (cf. Caminiti 2021). The healthcare sector, in addition to other sectors in the United States, is contending with labor shortages, with hospitals competing for nurses and hundreds reporting critical staff shortages (Frey 2021). This is also the situation in parts of the European Union (European Commission 2021). Clearly, addressing labor issues, within a supply chain context, along with wages of workers, is of societal importance. Indeed, even in the pandemic, there has been much discussion as to possible impacts of raising the minimum wage (see Leonhardt 2021). Companies are raising pay in order to attract workers (Casselman 2021). According to VacasSoriano and Aumayr-Pintar (2021), between January 2020 and January 2021, among 21 European Union countries and the United Kingdom with statutory minimum-wage systems, rates have risen in all but the following countries: Belgium, Spain, Greece, and Estonia. However, as noted in Chapters 4 and 5, there has been only limited work, to date, that integrates labor as an important resource in supply chain networks and also addresses competition for labor. Although the economics literature considers labor, also in a competitive setting (see, e.g., Mikesell 1940; Okuguchi 1993; Card and Kruger 1994; de Pinto and Goerke 2020, and the references therein), that literature does not capture supply chains in a general, holistic manner as is done in this book, utilizing the rich conceptual framework of supply chain networks. In this chapter, a supply chain network game theory modeling framework is proposed in which labor on supply chain links is an increasing function of the hourly wage. In Chapters 3 and 4, the hourly wage of labor is assumed to be fixed. Having labor be elastic, as a function of wages, in a computable mathematical model, enables decision-makers to determine the wages that should be paid. In Chapter 5, the wages are elastic, but the amount of labor on each supply chain network link is fixed. In this chapter, an extension of the model is also built where each firm imposes an upper bound or ceiling on the wages that it is willing to pay and the wage ceiling can be distinct for different links on a firm’s supply chain network. One can then determine the impacts of wage bounds not only on the specific firm’s profits but also on the profits of the other competing firms.

6.2 The Supply Chain Network Game Theory Models with Wage-Dependent. . .

123

The chapter is organized as follows. In Section 6.2, the supply chain network model with wage-dependent labor, without wage bounds, is constructed and then the one with wage bounds. The variational inequality formulations of the governing equilibrium conditions are provided. In Section 6.3, an algorithm is proposed and closed form expressions provided for the computation of the variables for both models, at each iteration, along with conditions for convergence. It is then applied to solve a series of supply chain network numerical examples for a single firm and for multiple firms, in order to investigate such scenarios as the introduction of electronic commerce and the outsourcing of production to a lower wage location. The impacts on profits of lowering (and raising) the wages that the firms are willing to pay are also studied. In Section 6.4, the results are summarized and the conclusions presented, along with suggestions for future research. Section 6.5 is the Sources and Notes Section for this chapter.

6.2 The Supply Chain Network Game Theory Models with Wage-Dependent Labor The supply chain network game theory modeling framework with wage-dependent labor is now introduced. The model without wage bounds on links is first presented and then the model with such bounds. The notation follows closely the notation in Chapter 5 but is detailed here for easy reference. There are I firms that are engaged in competition as they produce a substitutable product and provide for its subsequent transportation, storage, and distribution to demand markets. Each firm has its supply chain network as depicted in Figure 6.1 and the firms sell their products at common demand markets. According to Figure 6.1, the supply chain networks of the individual firms do not have any links in common. Each firm i, i = 1, . . . , I , has available niM production facilities, niD distribution centers, and can sell its product at J demand markets. L i denotes the links making up the supply chain network of firm i, i = 1, . . . , I , with nL i elements. The links of L i include firm i’s links to its production nodes, the links from production nodes to the distribution centers, the storage links, and the links from the distribution centers to the demand markets. L is the complete set of links in the supply chain network economy, where L = ∪Ii=1 Li , consisting of nL elements. Let G = [N , L ] represent the graph made up of the set of nodes N and the set of links L as in Figure 6.1. Figure 6.1 also includes direct links from production sites to demand markets since electronic commerce may be available and has clearly been of significance in the past few years. The supply chain network topology in Figure 6.1 can be adapted to the specific application under investigation with links and nodes added and/or removed accordingly. Each firm competes noncooperatively with the other firms and seeks to determine its optimal product quantities on its supply chain network pathways that maximize its profits, along with the labor volumes, which are a function of the hourly wages that each of the firms is willing to pay for its

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6 Wage-Dependent Labor and Supply Chain Networks

Fig. 6.1 The supply chain network topology of the Game theory models with wage-dependent labor

supply chain network economic activities of production, transportation, storage, and distribution. Table 6.1 contains the basic notation for the model. All vectors are column vectors. First, the conservation of flow equations are given and then the equations relating link flows to labor and also the equations relating labor on the links to the wages associated with the respective links. The path flows must be nonnegative; that is, for each firm i, xp ≥ 0,

∀p ∈ P i ,

∀i.

(6.1)

Also, the demand for each firm’s product at each demand market must be equal to the sum of the product flows from the firm to that demand market. Hence, for each firm i: i = 1, . . . , I , xp = dik , k = 1, . . . , J. (6.2) p∈Pki

The link flows of each firm i, i = 1, . . . , I , depend on the path flows as follows: fa =

p∈P

xp δap ,

∀a ∈ L i ,

(6.3)

6.2 The Supply Chain Network Game Theory Models with Wage-Dependent. . .

125

where δap = 1, if link a is contained in path p and 0, otherwise. Hence, the flow of a firm’s product on a link is equal to the sum of that product’s flows on paths that use that link. Having introduced the conservation of flow equations, I now turn to describing the important relationships between labor and the product link flows and also wages and labor. Specifically, as is being done throughout this book, I assume a linear Table 6.1 Notation for the supply chain game theory model with wage-dependent labor Notation Variables xp ; p ∈ Pki The nonnegative flow on path p originating at firm node i and terminating at demand market k, i = 1, . . . , I and k = 1, . . . , J . Firm i’s product path flows are n i grouped into the vector x i ∈ R+P . The vector x i is the vector of strategic variables of firm i. All the firms’ product path flows are grouped into the vector nP x ∈ R+ fa The nonnegative flow of the product on link a, ∀a ∈ L ; group the link flows into n the vector f ∈ R+L la The labor on link a (usually denoted in person hours), ∀a ∈ L wa The hourly wage paid to a laborer on link a, ∀a ∈ L dik The demand for the product of firm i at demand market k, i = 1, . . . , I and J and all k = 1, . . . , J ; group the {dik } elements for firm i into the vector d i ∈ R+ I ×J the demands into the vector d ∈ R+ Notation Parameters Pki The set of paths in firm i’s supply chain network terminating in demand market k, i = 1, . . . , I and k = 1, . . . , J Pi The set of all nP i paths of firm i, i = 1, . . . , I P The set of all nP paths in the supply chain network economy αa Positive factor relating input of labor to output of product flow on link a, ∀a ∈ L γa Positive factor relating input of wage to labor on link a, ∀a ∈ L w¯ a Upper bound of wage on link a that the cognizant firm is willing to pay, at its links a ∈ L i for i = 1, . . . , I Notation Functions cˆa (f ) The operational cost associated with link a, ∀a ∈ L ρik (d) The demand price function for the product of firm i at demand market k, i = 1, . . . , I and k = 1, . . . , J

production function, where fa = αa la ,

∀a ∈ L i ,

i = 1, . . . , I,

(6.4)

so that the greater the value of αa , the more productive the labor on the link. Furthermore, since here wage-dependent labor is introduced in order to ascertain what wages should be paid by the firms, it is assumed that la = γa w a ,

∀a ∈ L i ,

i = 1, . . . , I.

(6.5)

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6 Wage-Dependent Labor and Supply Chain Networks

According to (6.5), there is greater labor availability at higher wages. This is reflected also in data and policy-making (see Domm 2021). The utility functions of the firms are now given. Then, some of the above mathematical relationships are utilized to express each utility function exclusively in terms of the product path flow variables, which are the strategic variables. The utility function of firm i, U i , i = 1, . . . , I , is the profit, given by the difference between its revenue and its total costs: Ui =

J

ρik (d)dik −



cˆa (f ) −

a∈L i

k=1



(6.6)

w a la .

a∈L i

The first term after the equal sign in (6.6) is the firm’s revenue. The second term after the equal sign captures all the link operational costs of the firm, whereas the last term in (6.6) is the firm’s expenses for labor in its supply chain network. The utility functions Ui , i = 1, . . . , I , are assumed to be concave, with the demand price functions being monotone decreasing and continuously differentiable and with the operational link cost functions being convex and also continuously differentiable. In view of (6.2), one can define demand price functions ρ˜ik (x) ≡ ρik (d), ∀i, ∀k, and, in view of (6.3), we can define the operational link cost functions c˜a (x) ≡ cˆa (f ), ∀a ∈ L . Furthermore, using (6.5), (6.4), and then (6.3), observe that  l2 w a la = a = γa

fa αa

γa

2

=

(



p∈P xp δap ) αa2 γa

2

,

∀a ∈ L .

(6.7)

Define U˜ i (x) ≡ U i , i = 1, . . . , I . The utility functions U i , i = 1, . . . , I , in (6.6) can be re-expressed as U˜ i (x) =

J

ρ˜ik (x)xp −

k=1 p∈P i

k

a∈L i

c˜a (x) −

( a∈L i



p∈P xp δap ) αa2 γa

2

.

(6.8)

6.2.1 Equilibrium Conditions and Variational Inequality Formulations I now state the Nash (1950, 1951) equilibrium conditions governing this noncooperative game theory model and then I present the variational inequality formulation. n i The feasible set Ki for firm i is defined as Ki ≡ {x i |x i ∈ R+P }, for i = I 1, . . . , I . Also, K ≡ i=1 Ki .

6.2 The Supply Chain Network Game Theory Models with Wage-Dependent. . .

127

Definition 6.1 (Supply Chain Network Nash Equilibrium for Model with WageDependent Labor with No Wage Bounds) A path flow pattern x ∗ ∈ K is a supply chain network Nash Equilibrium if for each firm i, i = 1, . . . , I , U˜ i (x i∗ , xˆ i∗ ) ≥ U˜ i (x i , xˆ i∗ ),

∀x i ∈ Ki ,

(6.9)

where xˆ i∗ ≡ (x 1∗ , . . . , x i−1∗ , x i+1∗ , . . . , x I ∗ ). From (6.9), one knows that a Nash Equilibrium is established when no firm, acting unilaterally, can improve upon its utility, which represents its profits. Applying the classical theory of Nash equilibria and variational inequalities, under our imposed assumptions on the underlying functions (cf. Gabay and Moulin 1980 and Nagurney 1999), it follows that the solution to the above Nash Equilibrium problem (see Nash 1950, 1951) coincides with the solution of the variational inequality problem: determine x ∗ ∈ K, such that −

J I ∂ U˜ i (x ∗ ) × (xp − xp∗ ) ≥ 0, ∂x p i

∀x ∈ K.

(6.10)

i=1 k=1 p∈P

k

Variational Inequality Formulation for Model Without Wage Bounds An expansion of variational inequality (6.10) yields: determine x ∗ ∈ K, such that ⎡ ⎤  J I J ∗ ∗) ( p∈P xp∗ δap ) ∂ ρ ˜ (x il ⎢ ∂ C˜ p (x ) ⎥ + 2 δap − ρ˜ik (x ∗ ) − xq∗ ⎦ ⎣ 2γ ∂x ∂x α p p a a i i i a∈L

i=1 k=1 p∈P

k

× [xp − xp∗ ] ≥ 0, ∀x ∈ K,

l=1

q∈Pl

(6.11)

where ∂ cˆb (f ) ∂ C˜ p (x) ∂ ρ˜il (x) ∂ρil (d) ≡ δap , ∀p ∈ P i , ∀i, ≡ , ∀p ∈ Pki , ∀i, ∀k. ∂xp ∂f ∂x ∂dik a p i i a∈L b∈L

(6.12)

Once the equilibrium is computed—as discussed in the next section—one can determine the labor values on the links using (6.4) and also the wages on the links using (6.5). Variational Inequality Formulation for Model with Wage Bounds The above model is now extended with the introduction of upper bounds on wages that the firms are willing to pay per hour their workers. Distinct upper limits on different links are permitted. Specifically, the model remains as above except for the addition of the following constraints:

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6 Wage-Dependent Labor and Supply Chain Networks

wa ≤ w¯ a ,

∀a ∈ L .

(6.13)

Using the previous expressions (6.4) and (6.5), (6.13) becomes

xp δap ≤ αa γa w¯ a ,

∀a ∈ L .

(6.14)

p∈P

Define the feasible set K1i ≡ {x i ≥ 0, and (6.14) holds for all a ∈ L i }, and  K1 ≡ Ii=1 K1i . With the inclusion of the wage link upper bounds, the statement of the Nash Equilibrium according to Definition 6.1 is still appropriate but over the feasible set K1 . The variational inequality (6.11) also holds but with the new feasible set K1 . It is worth noting that since there are bounds on the wages, the link flows are, hence, bounded, as are the path flows and, therefore, the feasible set K1 is compact. Since all the functions in (6.11) are continuous, under our imposed assumptions, one knows then from the classical theory of variational inequalities (see Kinderlehrer and Stampacchia 1980) that a solution exists. For computational purposes, the following variational inequality in the case of wage link bounds is now proposed. Associate the Lagrange multiplier λa , with each link constraint as in (6.14) and group the Lagrange multipliers into the vector λ ∈ n R+L . Define the feasible set K2 ≡ {x|x ≥ 0, and λ ≥ 0} and, referring to (5.23), it follows that an alternative variational inequality to the one in (6.11), with bounds on wages, is: determine (x ∗ , λ∗ ) ∈ K2 such that ⎡

J I

⎣

i=1 k=1 p∈P i

( ∂ C˜ p (x ∗ ) + 2 ∂xp i a∈L

k

−

l=1

+

a∈L

⎡

∗ p∈P xp δap ) δap αa2 γa

+



λ∗a δap − ρ˜ik (x ∗ )

a∈L

⎤

J ∂ ρ˜il (x ∗ )

∂xp



⎥ xq∗ ⎦ × [xp − xp∗ ]

q∈Pli

⎣w¯ a αa γa −



⎤

  xp∗ δap ⎦ × λa − λ∗a ≥ 0,

∀(x, λ) ∈ K2 .

(6.15)

p∈P

It is notable that variational inequality (6.11) and variational inequality (6.15) have a simple feasible set, which is the nonnegative orthant but of different dimensions. This feature allows for an iterative scheme, which yields closed form expressions for the variables at each iteration and, hence, is easy to implement. I now put variational inequality (6.15) into standard form (2.14). Define X ≡ (x, λ) consisting of nP + nL elements and F (X), also consisting of nP + nL elements, with F (X) ≡ (F 1 (X), F 2 (X)), with the p-th element of   ( p∈P xp δap ) ∂ C˜ (x) F 1 (X) being ∂xp p + δap + a∈L i 2 a∈L λa δap − ρ˜ik (x) − α2 γ a a

6.3 The Algorithm and Numerical Examples

J l=1

∂ ρ˜il (x) ∂xp



129

xq and the j -th element of F 2 (X) being w¯ j αj γj − p∈P xp δjp . Also, set K ≡ K2 and N ≡ nP + nL . Then, clearly, variational inequality (6.15) can be put into standard form (2.14). q∈Pli

Remark 6.1 It is important to emphasize that the above game theory models, as a special case, include the single firm counterpart. Such models are also contributions to the literature since a firm can “independently” then investigate what wages it should pay and the potential impacts of having higher (or tighter) bounds on wages that it is willing to pay workers, whether at its various production sites, or for transportation and distribution, as well as storage. In the supply chain network numerical examples in the next section, a series of single firm examples are first presented and then results for multifirm examples.

6.3 The Algorithm and Numerical Examples The algorithm of Korpelevich (1977) is used for the computation of solutions to the numerical examples. The algorithm is given in Chapter 2 and in the Appendix. Solutions to single firm examples are given in Section 6.3.1 and to multifirm examples in Section 6.3.2. The elegance of this procedure for the computation of solutions to both the supply chain network game theory models with wage-dependent labor can be seen in the following explicit formulae. Explicit Formulae at Iteration τ for the Product Path Flows in Step 1 In particular, one has the following closed form expressions for the path flows in Step 1 for the solution of variational inequality (6.15): ⎧ ⎛ ⎪ J ⎨ ∂ ρ˜il (x τ −1 ) τ −1 ∂ C˜ p (x τ −1 ) ⎜ x¯pτ = max 0, xpτ −1 + η ⎝ρ˜ik (x τ −1 ) + xq − ⎪ ∂xp ∂xp ⎩ l=1 q∈P i l

−



λτa −1 δap −

a∈L



2

(

⎞⎫ τ −1 ⎬ p∈P xp δap ) ⎠ , ∀p ∈ Pki ; i = 1, . . . , I ; k = 1, . . . , J. δ ap ⎭ αa2 γa



a∈L i

(6.16)

Explicit Formulae at Iteration τ for the Lagrange Multipliers in Step 1 Similarly, one has the following closed form expressions for the Lagrange multipliers in Step 1 at an iteration τ : ⎧ ⎨

⎛

λτa = max 0, λτa −1 + η ⎝ ⎩



p∈P

⎞⎫ ⎬ xpτ −1 δap − w¯ a αa γa ⎠ , ⎭

∀a ∈ L .

(6.17)

130

6 Wage-Dependent Labor and Supply Chain Networks

The analogues of expressions (6.16) and (6.17) for Step 2 follow easily. As for the solution of variational inequality (6.11), for the model without wage upper bounds or ceilings, (6.17) is no longer needed, whereas (6.16)  still holds for Step 1 of the modified projection method but with the term − a∈L λτa −1 δap removed. The modified projection method is implemented in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the computations. The algorithm is initialized with a demand of 40 for each firm–demand market pair, with the demand then equally divided among the associated path flows. In the case of the examples with the wage ceilings, the Lagrange multipliers are all initialized to 0.00. The convergence tolerance is 10−7 in that the absolute difference between two successive variable iterates differed by no more than this amount. The supply chain network numerical examples are stylized but reflect reasonable wages and correspond to a product that is fairly expensive. The insights gained, nevertheless, have broader ramifications.

6.3.1 Numerical Results for a Single Firm The supply chain network topology for Examples 6.1 and 6.2 is as given in Figure 6.2. The network is as in Figures 3.3 and 4.2 but with different data. The firm has two production sites at its disposal, a single distribution center for storage, and serves two demand markets. Example 6.1 is without wage ceilings, whereas Example 6.2 is constructed from Example 6.1 but includes wage ceilings.

Fig. 6.2 Supply chain network topology for Examples 6.1 and 6.2

6.3 The Algorithm and Numerical Examples

131

Example 6.1 (Single Firm, No Wage Ceilings) The operational link cost functions are cˆa (f ) = 2fa2 ,

cˆb (f ) = 2fb2 ,

cˆe (f ) = fe2 + 2fe ,

cˆc (f ) = .5fc2 ,

cˆf (f ) = 0.5ff2 ,

cˆd (f ) = .5fd2 ,

cˆg (f ) = 0.5fg2 .

The demand price functions are ρ11 (d) = −5d11 + 800,

ρ12 (d) = −5d12 + 850.

The alpha link parameters are αa = 55,

αb = 50,

αc = 35,

αd = 35,

αe = 60,

αf = 38,

αg = 36,

and the gamma link parameters are γa = 0.1,

γb = 0.1,

γc = 0.09,

γd = 0.07,

γe = 0.08

γf = 0.06,

γg = 0.08.

The paths are path p1 = (a, c, e, f ), path p2 = (b, d, e, f ), path p3 = (a, c, e, g), and path p4 = (b, d, e, g). The modified projection method yields the following equilibrium path flow pattern: xp∗1 = 19.39,

xp∗2 = 19.36,

xp∗3 = 21.66,

xp∗4 = 21.63.

The equilibrium link flows, labor values, and hourly wages are reported in Table 6.2. The demand price at the first demand market is 606.27 and at the second demand market the price is 633.52, with the respective equilibrium demands of 38.75 and 43.30. The firm earns a profit of 33,816.98. Example 6.2 (Single Firm, Wage Ceilings) Example 6.2 has the identical data to the data in Example 6.1, except that now wage ceilings of 15 are imposed on all the links. Hence, the firm is not willing to pay more than the minimum wage that has been much discussed in the news in the United States lately (cf. Cooper et al. 2021). The equilibrium path flow pattern is now xp∗1 = 16.87,

xp∗2 = 16.85,

xp∗3 = 19.15,

xp∗4 = 19.13.

The demand price at the first demand market is 631.37 and at the second demand market the price is 658.63, with the respective equilibrium demands of 33.73 and

132

6 Wage-Dependent Labor and Supply Chain Networks

38.27. Hence, the demand prices now rise at both demand markets, hurting the consumers. The equilibrium link flows, labor values, and hourly wages are reported in Table 6.2. Observe that the equilibrium wage on link e is at the upper bound of 15. The Lagrange multiplier associated with the wage ceiling on link e is, hence, positive, and it is equal to 100.73. All other equilibrium Lagrange multipliers are equal to 0.00. It can be seen, from Table 6.2, that the hourly wages now decrease on all the supply chain network economic links as compared to their respective values in Example 6.1. The firm earns a profit of 33,311.27. Hence, the firm suffers a decrease in profits by imposing an upper bound on the wages that it is willing to pay. This result speaks to the advantage of having wages freely equilibrate without wage ceilings. Example 6.3 (Single Firm, Electronic Commerce, No Wage Ceilings) In Examples 6.3 and 6.4, the supply chain network topology of the firm is as given in Figure 6.3. The network topology is as in Figure 3.4 but with different data. In these examples, I am interested in investigating the impact of the introduction of electronic Table 6.2 Equilibrium link flows, labor values, and hourly wages for Examples 6.1 and 6.2

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗

lg∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗

Equilibrium value Example 6.1 Example 6.2 41.05 36.02 40.99 35.98 41.05 36.02 40.99 35.98 82.04 72.00 38.75 33.73 43.30 0.75 0.82 1.17 1.17 1.37 1.02

38.27 0.66 0.72 1.03 1.03 1.20 0.89

1.20 7.46 8.20 13.03 16.73 17.09 16.99

1.06 6.55 7.20 11.44 14.68 15.00 14.79

15.03

13.29

6.3 The Algorithm and Numerical Examples

133

Fig. 6.3 Supply chain network topology for Examples 6.3 and 6.4

commerce as represented by links h and i in Figure 6.3. The data for Example 6.3 are identical to the data in Example 6.1 but with the following additions. There are now two additional paths defined as p5 = (a, h),

p6 = (b, i).

The data on the electronic commerce links are cˆh (f ) = fh2 ,

cˆi (f ) = fi2 ,

αh = 40,

αi = 45,

γh = 0.1,

γi = 0.1

Also, with the advent of electronic commerce, the demand price functions are modified to ρ11 (d) = −d11 + 800,

ρ12 (d) = −d12 + 850.

The modified projection method yields the following equilibrium path flow pattern: xp∗1 = 11.20,

xp∗2 = 9.22,

xp∗3 = 17.88,

xp∗5 = 80.14,

xp∗6 = 85.03.

xp∗4 = 15.90,

134

6 Wage-Dependent Labor and Supply Chain Networks

It is clear that the greatest product volumes are on the pathways associated with the electronic commerce links. The demand price at the first demand market is now 699.43 and at the second demand market the price is 731.20, with the respective equilibrium demands of 100.57 and 118.80. The profit earned by the firm is 90,663.52. Example 6.4 (Single Firm, Electronic Commerce, Wage Ceilings) Example 6.4 has the identical data to that in Example 6.3, but now wage ceilings of 15 are imposed on all the links. The new equilibrium path flow pattern is xp∗1 = 8.48,

xp∗2 = 3.46, xp∗5 = 58.09,

xp∗3 = 15.93,

xp∗4 = 10.90,

xp∗6 = 60.64.

The profit of the firm now drops to 83,385.75. The pathways associated with the electronic commerce links are, again, the ones that have the highest volumes of product flowing on them. In Example 6.4, both links a and b have equilibrium wages at the imposed wage ceilings of 15. The associated equilibrium Lagrange multipliers are, respectively, 212.48 and 252.59. All other Lagrange multipliers are equal to 0.00. The demand price at the first demand market is now 729.97 and at the second demand market the price is 762.53, with the respective equilibrium demands of 70.03 and 87.47. The demand prices now rise (as they did in Example 6.2) signaling that the consumers are, in a sense, worse-off, under the imposed wage ceilings. The profit of the firm is also now lower than in Example 6.3, where there are no imposed wage ceilings. The equilibrium link flows, labor values, and hourly wages are reported in Table 6.3 for both Examples 6.3 and 6.4. The wage for each of the supply chain network links is lower in Example 6.4 than in Example 6.3. The workers also suffer in that their hourly wages decrease. Example 6.5 (Single Firm, Outsourcing, No Wage Ceilings) Examples 6.5 through 6.7 have the supply chain network topology depicted in Figure 6.4. In these examples, I am interested in evaluating the impacts when the firm makes use of another production site that it outsources the production to. The production site has lower manufacturing costs but is located a bit further from the firm’s distribution center.

6.3 The Algorithm and Numerical Examples Table 6.3 Equilibrium link flows, labor values, and hourly wages for Examples 6.3 and 6.4

135 Equilibrium value Example 6.3 Example 6.4 109.22 82.50 110.15 75.00 29.08 24.41 25.12 14.36 54.20 38.78 20.43 11.95

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fh∗ fi∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗

lg∗ lh∗ li∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗ wh∗ wi∗

33.77 80.14 85.03 1.99 2.20 0.83 0.72 0.90 0.54

26.83 58.09 60.64 1.50 1.50 0.70 0.41 0.65 0.31

0.94 2.00 1.89 19.86 22.03 9.23 10.25 11.29 8.96

0.75 1.45 1.35 15.00 15.00 7.75 5.86 8.08 5.24

11.73 20.03 18.90

9.32 14.52 13.47

The data for Example 6.5 are identical to the data in Example 6.1 except for the new data to represent links j and k as in Figure 6.4. The operational costs on these links are cˆj (f ) = fj2 ,

cˆk (f ) = fk ;

the alpha and gamma parameters on these two new links are αj = 45,

αk = 35,

γj = 0.07,

γk = 0.06.

The new paths are path p7 = (j, k, e, f ) and path p8 = (j, k, e, g).

136

6 Wage-Dependent Labor and Supply Chain Networks

The equilibrium path flow pattern is xp∗1 = 13.24,

xp∗2 = 13.22,

xp∗3 = 14.76,

xp∗7 = 16.65,

xp∗8 = 18.16.

xp∗4 = 14.74,

Fig. 6.4 Supply chain network topology for Examples 6.5, 6.6, and 6.7

The profit of the firm is now 37,406.73. The firm enjoys a sizeable increase in profit, as compared to that it earns in Example 6.1. The demand price at the first demand market is 584.47 and at the second demand market 611.71, with a demand at the first market of 43.11 and a demand of 47.66 at the second demand market. The demand prices drop at both demand markets, as compared to their values in Example 6.1, a plus for consumers. The equilibrium link flows, labor values, and hourly wages are given in Table 6.4. The wages decrease at the two original production sites but increase at the distribution center, which is now handling an increase in product volume. Example 6.6 (Single Firm, Outsourcing, Wage Ceilings on All Links, Tighter Ones on Outsourcing Links) Example 6.6 has the identical data to that in Example 6.5 except that now the interest is in investigating the impacts of wage ceilings. The wage ceilings on the original links remain as in Example 6.2, with a value of 15 each, but the outsourcing links have lower wage bounds (since the authorities there can pay lower wages), so we have that the wage bounds are equal to 10 on both link j and link k.

6.3 The Algorithm and Numerical Examples

137

The equilibrium path flow pattern is now xp∗1 = 12.00,

xp∗2 = 11.98,

xp∗3 = 13.52,

xp∗7 = 9.74,

xp∗8 = 11.26.

xp∗4 = 13.50,

The path flow is lower on each path as compared to the corresponding path flow in Example 6.5. The profit of the firm is 35,667.08. The profit is lower than in Example 6.5 and one sees that, consistently, lower wages result in a lower profit for the firm. The demand price at the first demand market is now 631.37 and at the second demand market 658.63. These values are higher in both cases than the demand market prices in Example 6.5. The demand at the first market is now 33.73, whereas the demand at the second demand market is 38.27. The equilibrium link flows, labor values, and hourly wages are given in Table 6.4. Wages now decrease on all the links. The Lagrange multiplier on link e is 153.52 and that on link k is 43.35 since the wages associated with these two supply chain network links are at the imposed wage ceilings associated with these links. All other link Lagrange multipliers are equal to 0.00. Example 6.7 (Single Firm, Outsourcing, Tighter Wage Ceilings on All Links) Example 6.7 has the same data as Example 6.6 except that now the firm has all of the wage ceilings set to 10 (rather than 15). Therefore, the wage ceilings on all links in the supply chain network in Figure 6.4 are equal to 10. The computed equilibrium path flow pattern is now xp∗1 = 6.65,

xp∗2 = 6.64,

xp∗3 = 8.16,

xp∗4 = 8.15,

xp∗7 = 8.45,

xp∗8 = 9.96.

Under lower wage ceilings, the path flow is lower on each path as compared to the corresponding path flow in Example 6.6. The profit of the firm is now 29,115.85. The profit is lower than in Example 6.6, further evidence that lower wage ceilings, which result in lower wages, yield a lower profit for the firm. The demand price at the first demand market is 691.36 and at the second demand market 718.73. They are higher in both cases than the demand market prices in Example 6.6. The demand at the first demand market has now dropped to 21.73, whereas the demand at the second demand market has now decreased to 26.27. The computed equilibrium link flows, labor values, and hourly wages are given in Table 6.4. The Lagrange multiplier on link e is 387.75, with all other Lagrange multipliers equal to 0.00.

138 Table 6.4 Equilibrium link flows, labor values, and hourly wages for Examples 6.5, 6.6, and 6.7

6 Wage-Dependent Labor and Supply Chain Networks

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fj∗ fk∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗ lj∗

lk∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗ wj∗ wk∗

Equilibrium value Example 6.5 Example 6.6 28.00 25.52 27.96 25.48 28.00 25.52 27.96 25.48 90.77 72.00 43.11 33.73

Example 6.7 14.81 14.79 14.81 14.79 48.00 21.73

47.66 34.81

38.27 21.00

26.27 18.41

34.81 0.51 0.56 0.80 0.80 1.51 1.14

21.00 0.46 0.51 0.73 0.73 1.2 0.89

18.41 0.17 0.30 0.42 0.42 0.80 0.57

1.32 0.77

1.06 0.47

0.73 0.41

0.99 5.09 5.59 8.89 11.41 18.91 18.91

0.60 4.64 5.10 8.10 10.40 15.00 14.79

0.53 2.69 2.96 4.70 6.04 10.00 9.53

16.55 11.05

13.29 6.67

9.12 5.84

16.58

10.00

8.77

6.3.2 Numerical Results for Multifirm Examples The multifirm numerical examples, Examples 6.8 through 6.11, have the supply chain network topology depicted in Figure 6.5. The topology is the same as in Figure 5.2, but different data are used for the examples. Observe that the supply chain network for Firm 1 is identical to its network in Examples 6.1 and 6.2 and the data are also as therein except that the demand price functions now have a term to capture competition with the other firm. Firm 2 also has at its disposal two production sites, a single distribution center, and sells its product at the same demand markets as does Firm 1.

6.3 The Algorithm and Numerical Examples

139

Example 6.8 (Two Firm Example, No Wage Ceilings) The data for Firm 1 are identical to the data in Example 6.1 but with the demand price functions now including cross-terms to capture competition with the other firm; that is, ρ11 (d) = −5d11 − 2d21 + 800,

ρ12 (d) = −5d12 − d22 + 850.

The data for the supply chain network of Firm 2 are The operational link cost functions are cˆl (f ) = 1.5fl2 ,

cˆm (f ) = 1.5fm2 + fm ,

cˆp (f ) = 0.5fp2 ,

cˆn (f ) = fn2 + 2fn ,

cˆq (f ) = 0.5fq2 + fq ,

cˆo (f ) = fo2 ,

cˆr (f ) = fr2 + 2fr .

Fig. 6.5 The supply chain network topology for the numerical Examples 6.8 through 6.11

The demand price functions are ρ21 (d) = −3d21 − d11 + 700, The alpha link parameters are

ρ22 (d) = −5d22 − 0.5d12 + 750.

140

αl = 40,

6 Wage-Dependent Labor and Supply Chain Networks

αm = 45,

αn = 30,

αo = 30,

αp = 55,

αq = 39,

αr = 35,

γo = 0.09,

γp = 0.09

γq = 0.07,

γr = 0.1.

and the gamma link parameters are γl = 0.1,

γm = 0.1,

γn = 0.09,

Observe that Firm 2 is better at attracting workers with its gamma link parameters the same or higher than those of Firm 1 for similar supply chain network activities. The new paths associated with Firm 2’s supply chain network are p9 = (l, n, p, q),

p10 = (m, o, p, q),

p11 = (l, n, q, r),

p12 = (m, o, p, r).

Note that in Example 6.8 there are no wage ceilings. In the subsequent numerical examples, there are wage ceilings on Firm 1’s network only in Example 6.9; then on Firm 2’s network only in Example 6.10, and, finally, there are wage ceilings on all the supply chain network links in Example 6.11. The modified projection method yields the following equilibrium path flow pattern: xp∗1 = 17.65, xp∗9 = 18.07,

xp∗2 = 17.63,

xp∗3 = 19.61,

xp∗4 = 19.58.

xp∗10 = 18.18,

xp∗11 = 39.66,

xp∗12 = 39.78.

The equilibrium link flows, labor values, and hourly wages are reported in Table 6.5 for this, as well as the subsequent numerical examples. The demand price of Firm 1’s product at the first demand market is 551.14 and at the second demand market the price is 574.14, with the respective equilibrium demands of 35.28 and 39.19. The demand price of Firm 2’s product at the first demand market is 555.99 and at the second demand market the price is 650.96, with the respective equilibrium demands of 36.24 and 79.44. Firm 1 earns a profit of 27,857.52 and Firm 2 earns a profit of 40,874.88. Clearly, Firm 2 is more competitive. Firm 1, now, with competition, has a much lower profit than it did in Example 6.1. Firm 2 is able to attract more labor for its supply chain network activities than Firm 1 and, on the majority of its supply chain network links, pays a higher wage than Firm 1 does (for the analogous activities). Example 6.9 (Two Firm Example, Wage Ceilings Only on Firm 1’s Supply Chain Network) Example 6.9 has the same data as Example 6.8, but now I impose wage ceilings of 15 on all the supply chain network links of Firm 1 only. The modified projection method converges to the following equilibrium path flow pattern: xp∗1 = 17.03,

xp∗2 = 17.00,

xp∗3 = 19.00,

xp∗4 = 18.97.

6.3 The Algorithm and Numerical Examples

141

Table 6.5 Equilibrium link flows, labor values, and hourly wages for Examples 6.8 through 6.11 Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fl∗ fm∗ fn∗ fo∗ fp∗ fq∗ fr∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗

lg∗ ll∗ ∗ lm ln∗ lo∗ lp∗ lq∗ lr∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗

wg∗ wl∗ ∗ wm wn∗ wo∗ wp∗ wq∗ wr∗

Equilibrium value Example 6.8 Example 6.9 37.26 36.02 37.21 35.98 37.26 36.02 37.21 35.98 74.47 72.00 35.28 34.03

Example 6.10 38.65 38.60 38.65 38.60 77.26 36.76

Example 6.11 36.02 35.98 36.02 35.98 72.00 34.13

39.19 57.73 57.96 57.73 57.96 115.69 36.24 79.44 0.68 0.74 1.06 1.06 1.24 0.93

37.97 57.80 58.03 57.80 58.03 115.83 36.35 79.47 0.66 0.72 1.03 1.03 1.20 0.90

40.50 37.02 37.23 37.02 37.23 74.25 21.75 52.50 0.70 0.77 1.10 1.10 1.29 0.97

37.87 37.02 37.23 37.02 37.23 74.25 21.75 52.50 0.66 0.72 1.03 1.03 1.20 0.90

1.09 1.44 1.29 1.92 1.93 2.10 0.93 2.27 6.77 7.44 11.83 15.19 15.51 15.47

1.05 1.46 1.29 1.92 1.92 2.11 0.93 2.27 6.55 7.20 11.44 14.68 15.00 14.93

1.12 0.93 0.83 1.23 1.24 1.35 0.58 1.50 7.03 7.72 12.27 15.76 16.10 16.12

1.05 0.93 0.83 1.23 1.24 1.35 0.58 1.50 6.55 7.20 11.44 14.68 15.00 14.97

13.61 14.43 12.88 21.38 21.47 23.37 13.28 22.70

13.18 14.45 12.90 21.41 21.49 23.40 13.32 22.71

14.06 9.25 8.27 13.71 13.79 15.00 7.97 15.00

14.15 9.25 8.27 13.71 13.79 15.00 7.97 15.00

142

6 Wage-Dependent Labor and Supply Chain Networks

xp∗9 = 18.12,

xp∗10 = 18.23,

xp∗11 = 39.68,

xp∗12 = 39.79.

The demand price of Firm 1’s product at the first demand market is now 557.13 and at the second demand market the price is 580.69, with the respective equilibrium demands of 34.03 and 37.97. The demand price of Firm 2’s product at the first demand market is 556.91 and at the second demand market the price is 651.94, with the respective equilibrium demands of 36.35 and 79.47. The demand market prices of Firm 1’s product rise whereas those of Firm 2’s product only a very small amount. Firm 1 earns a profit of 27,818.46 and Firm 2 earns a profit of 40,968.69. This result is quite interesting. Firm 1 has lower profits under wage ceilings and, in fact, “helps” Firm 2 to achieve higher profits. This example demonstrates the importance of having a holistic supply chain network framework since changes associated with a firm can actually have impacts on other firms and, sometimes, in not expected ways. The Lagrange multiplier associated with link e is 24.64, since the wage on the link is at its upper bound. All other Lagrange multipliers are equal to 0.00. Additional results are reported in Table 6.5. Firm 2, again, pays its workers more than Firm 1 (except on link q) on similar links and attracts more labor for those activities than does Firm 1. Example 6.10 (Two Firm Example, Wage Ceilings Only on Firm 2’s Supply Chain Network) Example 6.10 has the same data as Example 6.8 but now wage ceilings of 15 are imposed on all the supply chain network links of Firm 2 only. The modified projection method converges to the following equilibrium path flow pattern: xp∗1 = 18.39, xp∗9 = 10.82,

xp∗2 = 18.37,

xp∗3 = 20.26,

xp∗4 = 20.23.

xp∗10 = 10.93,

xp∗11 = 26.20,

xp∗12 = 26.30.

Firm 1 has increases of product flow on all of its paths, as compared to its flows in Example 6.9; in contrast, Firm 2 now experiences decreases in all its product path flows. The demand price of Firm 1’s product at the first demand market is now 572.69 and at the second demand market the price is 595.02, with the respective equilibrium demands of 36.76 and 40.50. The demand price of Firm 2’s product at the first demand market is 597.99 and at the second demand market the price is 677.25, with the respective equilibrium demands of 21.75 and 52.50. The demand market prices of Firm 1’s product rise whereas those of Firm 2’s product only a very small amount. Firm 1 earns a profit of 29,975.13 and Firm 2 earns a profit of 35,586.85. Firm 1 benefits profit-wise from Firm 2 imposing wage ceilings, whereas Firm 2 suffers a decrease in profits as compared to Examples 6.9 and 6.8. Firm 2, now, on three links, pays its workers lower wages than does Firm 1 for the corresponding links. The Lagrange multiplier associated with link p is 246.32 and that associated with link r is 7.32. All other Lagrange multipliers are equal to 0.00. Please refer

6.3 The Algorithm and Numerical Examples

143

to Table 6.5 for additional results on the equilibrium link flows, labor values, and hourly wages. Example 6.11 (Two Firm Example, Wage Ceilings on All Supply Chain Network Links) Example 6.11 has the same data as Example 6.8, but now I impose wage ceilings of 15 on all the supply chain network links of both Firm 1 and Firm 2. The modified projection method converges to the following equilibrium path flow pattern: xp∗1 = 17.08, xp∗9 = 10.82,

xp∗2 = 17.06,

xp∗3 = 18.94,

xp∗4 = 18.92.

xp∗10 = 10.93,

xp∗11 = 26.20,

xp∗12 = 26.30.

The demand price of Firm 1’s product at the first demand market is now 585.83 and at the second demand market the price is 608.17, with the respective equilibrium demands of 34.13 and 37.87. The demand price of Firm 2’s product at the first demand market is 600.61 and at the second demand market the price is 678.57, with the respective equilibrium demands of 21.75 and 52.50. The wages on links: e, p, and r are at the respective imposed wage ceilings with equilibrium Lagrange multipliers, respectively, of 52.73, 248.95, and 6.01. Also, in this example, Firm 2 now pays lower wages than Firm 1 on two (both only two) links for analogous activities. Firm 1 earns a profit of 29,836.55, whereas Firm 2 earns a profit of 35,713.00. Interestingly, with both firms imposing wage ceilings of the same value on all their supply chain network links, Firm 1 earns a greater profit than it did in Example 6.8, with no wage ceilings, but Firm 2, in contrast, earns a lower profit than it did in Example 6.8. It has two of its links with wages at the ceiling, whereas Firm 1 has only one of its links at the wage upper bound. Please refer to Table 6.5 for additional results for this example. The above supply chain network numerical results demonstrate the importance of computing solutions to supply chain network problems using a full network perspective so that decisions made on one part of the supply chain, and these could include those made by a specific firm and are also analyzed with respect to impacts on other firms. The above results are stylized but, nevertheless, reveal the type of information that can be obtained and assessed from product path and link flows to wages that should be paid and the labor involved. In Figure 6.6, the results of sensitivity analysis for Example 6.11 are displayed. I report the profit of Firm 1 and of Firm 2, when the wage ceilings are at 10, for all links, at 13 for all links, and then at 15, 17, 19, 21, 23, and 25 for all links. As can be seen from Figure 6.6, the profit of Firm 2 continues to increase until it reaches a plateau, which is to be expected. The profit of Firm 1, in contrast, increases, then decreases, and then plateaus. The sensitivity analysis further demonstrates the importance of a holistic approach to supply chain network modeling, analysis, and computations.

144

6 Wage-Dependent Labor and Supply Chain Networks 45 Firm 1 Profit Firm 2 Profit

103

40

Profits

+

35

30

25 10

15 20 Wage Ceilings on Supply Chain Links

25

Fig. 6.6 Sensitivity analysis for different wage ceilings and effects in the firms’ profits

6.4 Summary, Conclusions, and Suggestions for Future Research The pandemic has vividly revealed the importance of labor to the functionality of supply chain networks and the availability of associated products. As the world moves forward and, hopefully, one day before too long, past the pandemic, it is clear that labor and labor shortages are permeating the news. Many companies have started to raise wages in order to attract workers. In this chapter, a supply chain network framework is constructed consisting of one or more firms engaged in the production, transportation, storage, and distribution of a product that is substitutable. The framework assumes linear production functions as well as labor availability that is an increasing function of the wages paid. The firms are profit-maximizers, compete, and seek to determine their optimal product flows on the supply chain network pathways to the demand markets, along with the wages that should be paid on the various supply chain network links, and the associated labor values. First, a model without wage ceilings (upper bounds) is presented and then a model with wage ceilings. The work is highly relevant since there is much discussion now as to what workers should be paid in the United States as well as in many other countries. Numerical examples are presented for a single firm, followed by examples of multiple firms, which are solved using the proposed algorithm that has nice features at each iteration for implementation. The supply chain network models with wage-

References

145

dependent labor, without and with wage ceilings, are all uniformly formulated and solved as variational inequality problems. The numerical results clearly reveal the importance of a system-wide, holistic approach to supply chain network modeling since decisions made by a specific firm can have unexpected impacts on other competing firms in the supply chain network economy. Furthermore, the results strongly suggest that having wages and labor equilibrate without any price ceilings is beneficial for an individual firm and also for firms engaged in competition. This research adds to the application of variational inequalities to supply chain network problems of relevance in the pandemic and beyond and also advances the literature integrating concepts from operations research and economics. A dynamic version of this model would be worthwhile to construct. Another possible extension would be to have wages be a function (even, perhaps, a nonlinear one) of the respective productivity factors.

6.5 Sources and Notes These results in this chapter have not been reported elsewhere and are published here for the first time. This chapter is a companion chapter to Chapter 5. Recall that in Chapter 5, however, the amount of labor on each supply chain link is fixed, whereas in this chapter, the labor volumes are variables. Both chapters formulate models in which there are ceilings on wages. Related literature on game theory supply chain network models for differentiated products, as formulated and studied here, using the methodology of variational inequalities, but without the inclusion of labor, has been developed for food applications (Yu and Nagurney 2013 and Besik and Nagurney 2017) and for pharmaceuticals (Masoumi et al. 2012; Nagurney et al. 2013). Both of these sectors have been essential in the COVID-19 pandemic and have also been disrupted (see Chowdhury et al. 2021). See also Khan et al. (2022) and Chapters 2 and 4 for research on COVID-19 food-related issues, and Chapter 3, Nagurney et al. (2021), Sciacca and Daniele (2021), and Salarpour and Nagurney (2021) for research on medical product shortcomings, including shortages of PPEs.

References Adascalitei, D., Weber, T., 2021. The pandemic aggravated labour shortages in some sectors; the problem is now emerging in others. Eurofound, July 21. Associated Press News, 2021. Employers paying higher wages to attract workers back, data shows. April 30. Besik, D., Nagurney, A., 2017. Quality in competitive fresh produce supply chains with application to farmers’ markets. Socio-Economic Planning Sciences, 60, 62–76.

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Bhattarai, A., Reiley, L., 2020. The companies that feed America brace for labor shortages and worry about restocking stores as coronavirus pandemic intensifies. The Washington Post, March 13. Caminiti, S. 2021. Lack of workers is further fueling supply chain woes. CNBC.com, September 28. Card, D., Kruger, A., 1994. Minimum wages and employment: A case study of the fast-food industry in New Jersey and Pennsylvania. American Economic Review, 84(4), 772–793. Casselman, B., 2021. The U.S. economy is sending confusing signals. What’s going on? The New York Times, June 3. Chowdhury, P., Paul, S.K., Kaisar, S., Moktadir, M. A., 2021. COVID-19 pandemic related supply chain studies: A systematic review. Transportation Research E, 148, 102271. Coles, I., 2021. U.K. businesses plea for more European workers. The government says no. The Wall Street Journal, October 8. Cooper, D., Mokhiber, Z., Zipperer, B., 2021. Raising the federal minimum wage to $15 by 2025 would lift the pay of 32 million workers. Economic Policy Institute, March 9. Creswell, J., 2021. McDonald’s to increase wages as job market tightens. The New York Times, May 13. DC Velocity Staff, 2021. Labor shortage hits supply chain hard. Supply Chain Quarterly, May 12. Domm, P., 2021. Workers’ wages are rising at the fastest pace in years. Companies’ profits could take a hit. CNBC, May 22. de Pinto, M., Goerke, L., 2020 Welfare-enhancing trade unions in an oligopoly with excess entry. The Manchester School, 88(1), 60–90. European Commission, 2021. Analysis of shortage and surplus occupations 2020. European Union, Luxembourg. Frey, M., 2021. Hospitals battle burnout, compete for nurses as pandemic spurs US staffing woes. S&P Global, April 16. Gabay, D., Moulin, H., 1980. On the uniqueness and stability of Nash equilibria in noncooperative games. In: Applied Stochastic Control of Econometrics and Management Science. A. Bensoussan, P. Kleindorfer, and C.S. Tapiero, Editors, North-Holland, Amsterdam, The Netherlands, pp. 271–294. Khan, S.A.R., Razzaq, A., Yu, Z., Shah, A., 2022. Disruption in food supply chain and undernourishment challenges: An empirical study in the context of Asian countries. Socio-Economic Planning Sciences, 82, Part A, 101033. Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York. Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. Leonhardt, M., 2021. How raising minimum wage to $15 per hour could affect workers and small businesses. CNBC, February 24. Masoumi, A.H., Yu, M., Nagurney, A., 2012. A supply chain generalized network oligopoly model for pharmaceuticals under brand differentiation and perishability. Transportation Research E, 48(4), 762–780. Mikesell, R.F., 1940. Oligopoly and the short-run demand for labor. The Quarterly Journal of Economics, 55(1), 161–166. Morath, E., 2021. Millions are unemployed. Why can’t companies find workers? The Wall Street Journal, May 6. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A., Li, D., Nagurney, L.S., 2013. Pharmaceutical supply chain networks with outsourcing under price and quality competition. International Transactions in Operational Research, 20(6), 859–888. Nagurney, A., Salarpour, M., Dong, J., Dutta, P., 2021. Competition for medical supplies under stochastic demand in the Covid-19 pandemic. In: Nonlinear Analysis and Global Optimization. T.M. Rassias and P.M. Pardalos, Editors, Springer Nature Switzerland AG, pp. 331–356.

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Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49. Nash, J.F., 1951. Noncooperative games. Annals of Mathematics, 54, 286–298. Okuguchi, K., 1993. Cournot oligopoly with profit-maximizing and labor-managed firms. Keio Economic Studies, 30(1), 27–38. Salarpour, M., Nagurney, A., 2021. A multicountry, multicommodity stochastic game theory network model of competition for medical supplies inspired by the Covid-19 pandemic. International Journal of Production Economics, 235, 108074. Sciacca, D., Daniele, P., 2021. A dynamic supply chain network for PPE during the Covid-19 pandemic. Journal of Applied and Numerical Optimization, 3(2), 403–424. Vacas-Soriano, C., Aumayr-Pintar, C., 2021. Minimum wages rise, but more slowly. Social Europe, July 13. Wolf, M., 2021. The global labor shortage: How Covid-19 has changed the labor market. Deloitte Insights, August 23. Yifan Xie, S., Qi, L., 2021. Chinese factories are having labor pains– ‘We can hardly find any workers.’ The Wall Street Journal, August 25. Yu, M., Nagurney, A., 2013. Competitive food supply chain networks with application to fresh produce. European Journal of Operational Research, 224(2), 273–282.

Chapter 7

Investments in Labor Productivity: Single Period Model

Abstract Shortages of labor continue even now as economies begin to open up with progress on vaccinations. Investing in labor productivity is a possible mechanism in moderating shortfalls in labor. In this chapter, a supply chain network optimization model is constructed, whose solution yields optimal product path flows to demand markets, the optimal investments in link labor productivity, as well as labor hours needed, and the optimal wages of the workers in production, transportation, storage, and distribution. The model includes a budget constraint on the investments, along with maximum bounds on investments on the supply chain network links. The theoretical framework, which includes Lagrange analysis, and the computational approach are based on the theory of variational inequalities. Managerial insights are provided obtained via the Lagrange analysis and a series of numerical examples, which demonstrate that such investments can help both the firm and the consumers.

7.1 Introduction There are many relevant issues associated with labor in supply chains, which have been exacerbated in the pandemic. With vaccinations increasing and certain economies rebounding, labor issues continue in the pandemic. Many firms and organizations have had difficulty in attracting workers (cf. Rosenberg 2021, Morath 2021). This is not just a United States phenomenon. Weber (2021) emphasizes that the labor shortage that is impacting the USA is also coming to Europe, where it could prove even more difficult to repair. The productivity of labor has also decreased in different sectors, with additional studies being warranted (see Bloom et al. 2020). Firms are trying to identify the wages that should be paid and whether wages can serve as a mechanism to attract labor during shortages and shortfalls (cf. Sanandaji et al. 2021, Simon 2021). In the COVID-19 pandemic, major electronic commerce retailers, such as Amazon, have also experienced labor shortages, due, in part, to immense demand for online deliveries and are seeking many new employees (Del Ray 2020). They are also increasing the wages that they pay (Herrera 2021). Hintzmann et al. (2021) state that many economies had barely recovered from the last crisis of 2007–2008, which was financial in nature, and then were deeply © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_7

149

150

7 Investments in Labor Productivity: Single Period Model

affected by the COVID-19 pandemic. The authors emphasize that industries in the European Union are suffering, as many have had to shut down or reduce their production as well as to decrease their labor force, while under severe financial pressure. Furthermore, the companies have had to find strategies to survive. In their empirical study, which focused on 18 European countries between 1995 and 2017, the authors are concerned with labor productivity and industrial policy. They find that, among the variables considered significant, investments in advertising and marketing, organizational capital, R&D investment, and design are the ones that contribute individually the most to productivity growth in European manufacturing, whereas computerized information has a complementary effect with other such assets. Stundziene and Saboniene (2019) argue that increases in labor productivity are key drivers of higher welfare in every economy. The goal of their research was to test if the investment in tangible assets enhances labor productivity in the European manufacturing sector. Their results show that, with consideration of all European countries, a 1% increase in gross investment in tangible goods per person employed has a 0.0373% long-run effect on apparent labor productivity. Garton (2017) states that there is a positive cycle between productivity and people with higher levels of productivity allowing society to reinvest in human capital and with proper investments resulting in higher labor productivity. He notes that, in the years between 2005 and 2015, labor productivity in the United States, as measured by GDP per labor hour, was less than 1% for 7 of the 10 years, according to the OECD. Furthermore, he believes that productivity could be improved if we stopped the underinvestment in human capital. Chaney Cambon (2021) emphasizes that, after a decade of minimal increases in labor productivity, worker productivity might be about to accelerate, as a consequence of pandemicinduced technological adoption, which could raise economic growth and wages in coming years while keeping away inflation pressure. In her article, she highlights a study by McKinsey, wherein approximately 75% of the respondents at North American and European companies to the survey conducted in December 2020 expected to speed up investment in new technology in 2020–2024, higher than the 55% who said that they increased such investments in 2014–2019. There is a synergistic cycle between productivity and people: higher levels of productivity allow society to reinvest in human capital and smart investments result in higher labor productivity. Economists have, historically, included labor, along with capital, in the construction of production functions, but their analyses have not considered supply chains holistically. It is important to capture the latter since local disruptions can permeate much farther afield, as has been emphasized in this book. Given the timeliness, it is critical to also identify the possible benefits of investments in labor productivity. Jorgenson (1991) argues that investments in productivity of labor can take many forms from investment in tangible assets, which he terms “hardware,” to investment in intangible ones, such as R&D, which he refers to as “software,” as well as investment in human capital through the acquisition of skills and education. And, in the pandemic, investments in health and safety as to work environments can reduce stress and enhance workers’ productivity (see Igoe 2021).

7.1 Introduction

151

In this chapter, I return to a single firm optimization model, as was done in Chapters 2 and 3. Here, however, I propose a supply chain network optimization approach for identifying investments in productivity of labor in any/all supply chain network links. The firm seeks to determine its profit-maximizing product path flows from production sites to points of demand, along with the labor required, the wages that workers should be paid, and the optimal investments in labor productivity. The model builds on the optimization research on the integration of labor into supply chain networks in Chapters 2 and 3, but with a crucial distinction—that of the optimal allocation of investments, subject to a budget constraint and bounds on the link productivity investments. In the supply chain network optimization model introduced in this chapter, unlike in the models in Chapters 2 and 3, there are no bounds on labor, but labor availability is wage-dependent, as in Chapters 5 and 6. This is relevant since many companies are now looking at raising wages in order to attract workers. Furthermore, the firm can invest in labor productivity on its supply chain network, subject to a budget constraint and also a maximum on the investment allowable on each link. The latter is important since there may be a maximum that a firm may wish to allocate for productivity enhancement on a link. In addition, there may be a maximum that may be achievable, regardless of the investment, because of human limitations. Several classical models in economics that focus on productivity and growth are highlighted in Stiroh (2001). However, none of these consider a supply chain network perspective. For a recent survey on COVID-19 and supply chains, see Queiroz et al. (2020). In addition, in this chapter, as was done in Chapter 5, but in the context of a game theory model, I provide a Lagrange analysis, which yields alternative variational inequality formulations, along with deeper managerial insights. One of the variational inequality formulations is then used for computational purposes, since the proposed algorithm provides us with closed form expressions for the product path flows, the link productivity investments, and the Lagrange multiplier associated with the investment budget constraint at each iteration. This chapter is organized as follows. In Section 7.2, the supply chain network optimization model is constructed. The model consists of a single firm interested in determining its profit-maximizing optimal product path flows to the demand markets, along with the optimal investments in link productivity. The investments are subject to a budget constraint and upper bounds on each link. The link productivity factors are generalized from those in Chapters 2–6 to allow for investments. In the optimization model, labor is wage-dependent, in that the higher the wage, the greater the labor availability. The solution of the model yields the optimal product path and link flows, as well as the optimal link productivity investments, along with the labor hours needed on each link and the wages to be paid the workers. The variational inequality formulation of the optimal solution is also given. In Section 7.3, Lagrange analysis is conducted. The analysis enables the construction of alternative variational inequality formulations, one of which is over the nonnegative orthant, and very amenable to solution via the algorithm outlined in Section 7.4. Lagrange analysis is also utilized to obtain managerial insights of an economics nature. Section 7.4 contains solutions to a series of supply chain network

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7 Investments in Labor Productivity: Single Period Model

numerical examples, for which full results are reported. Section 7.5 summarizes the results in this chapter and presents the conclusions plus makes suggestions for future research. Sources and Notes for this chapter are contained in Section 7.6.

7.2 The Labor Productivity Investment Supply Chain Network Model The optimization model for labor productivity investments in supply chain networks is now described. Many firms in different industrial sectors, including agriculture and manufacturing, are dealing with shortfalls in labor, and, hence, enhancing labor productivity may be an avenue for increasing product availability. The optimization model considers a supply chain network topology, as depicted in Figure 7.1, which can be adapted, depending upon the specific application and circumstances under study. The top node corresponds to the firm, with the subsequent directed links corresponding to production in the first tier, to transportation in the second tier, storage at the distribution centers, and, with the bottom tier of links denoting the distribution to points of demand, which are represented by the bottom nodes: 1, . . . , J , corresponding to the demand markets. The supply chain network in Figure 7.1 is abstracted as the graph G = [N , L ], where N denotes the set of nodes and L denotes the set of links. A path p in the supply chain network joins the top-tiered node 1 to a bottom-tiered demand market node. The paths are acyclic and each path consists of a sequence of links representing the supply chain network activities of production, transportation, storage, and distribution to a demand market. A demand market may be a business, an organization, a retailer, or even consumers at their home. Let Pk denote the set of paths, representing alternative supply chain network processes, joining the pair of nodes (1, k), with k denoting a typical demand market node. P then denotes the set of all paths joining node 1 to the demand market nodes. There are nP paths in the supply chain network and nL links. A typical link in the supply chain network is denoted by a. The firm is interested in maximizing its profits by identifying its optimal product path flows plus the investments in labor productivity on the links. The additional notation for the model is given in Table 7.1. All vectors are assumed to be column vectors. I now provide the constraints and then construct the objective function representing the profit that the firm wishes to maximize. The path flows must be nonnegative, that is, xp ≥ 0,

∀p ∈ P.

(7.1)

7.2 The Labor Productivity Investment Supply Chain Network Model

153

Fig. 7.1 The supply chain network topology for optimization of product path flows and investments in labor productivity

The demand at each demand market must be satisfied by the sum of the product flows of the firm on paths to each demand market, that is, p∈Pk

xp = dk ,

k = 1, . . . , J.

(7.2)

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7 Investments in Labor Productivity: Single Period Model

Table 7.1 Notation for the supply chain network models with labor Notation xp fa la dk va wa γa vamax B cˆa (f, va ) ρk (d) αa + βa va

Variables nP the product flow on path p; group all the path flows into the vector x ∈ R+ nL the product flow on link a; group all the link flows into the vector f ∈ R+ the labor available for link a activity, ∀a ∈ L the demand for the product at demand market k, k = 1, . . . , J ; group the demands J into the vector d ∈ R+ the investment in labor productivity on link a, ∀a ∈ L ; group all the investments nL in links into the vector v ∈ R+ the (hourly) wage paid for a unit of labor on link a, ∀a ∈ L . Parameters positive factor relating wage to labor on link a, ∀a ∈ L the maximum investment possible in labor productivity on link a, ∀a ∈ L the budget of the firm for labor productivity investments. Functions the operational cost associated with link a, ∀a ∈ L the demand price for the product at demand market k, k = 1, . . . , J the link productivity function relating input of labor to product flow on link a, ∀a ∈ L . Let πa = αa + βa va , ∀a ∈ L .

The product flow on each link is equal to the sum of flows on paths that contain that link: fa = xp δap , ∀a ∈ L , (7.3) p∈P

where δap = 1, if link a is contained in path p and is 0, otherwise. Since the product output on a link is equal to the labor input on the link times the productivity on the link, which is no longer fixed as in Chapters 2–6, but, rather, is a function of the investment in productivity on the link, one has that fa = (αa + βa va )la ,

∀a ∈ L ,

(7.4)

where 0 ≤ va ≤ vamax ,

∀a ∈ L .

(7.5)

According to (7.5), the investment on each link in terms of labor productivity must be nonnegative and cannot exceed the imposed maximum investment desired on a link by the firm. Having the constraints in (7.5) is important since there may be a maximum achievable productivity for a given link. Equation (7.4), in turn, is an extension of a linear production function (cf. Mishra 2007) to include the labor productivity enhancement due to investment. Such an investment can be for education, improvement in hardware or technology or even software as well as

7.2 The Labor Productivity Investment Supply Chain Network Model

155

investment in health and safety in the pandemic or as a consequence of climate change. Also, the firm does not exceed the budget that it has allocated for labor productivity investments. Hence, the following constraint also applies:

va ≤ B.

(7.6)

a∈L

I emphasize the flexibility of the model in that a firm can invest in any or all of its supply chain network links. The solution of the full supply chain network optimization model will yield which links the firm should invest in and also at what level. In addition, it is assumed that the availability of labor is wage-dependent, so that la = γa w a ,

∀a ∈ L .

(7.7)

According to (7.7), the higher the wage, the greater the labor availability. This is also reasonable, and higher wages are now being used by many companies to attract workers. A similar construct is used in Chapter 6 in a game theory context. The firm seeks to maximize its profit, with the profit denoted by U , being the difference between its revenue and the total cost, with the total cost consisting of the sum of the operational costs on all the links and the investments in labor productivity on the links plus the total wages paid. Hence, the profit is expressed as U=

J

ρk (d)dk −



cˆa (f, va ) −

a∈L

k=1



va −

a∈L



w a la .

(7.8)

a∈L

The maximization problem is subject to the above constraints. I now show that (7.8) can be expressed solely in terms of product path flows and link labor productivity investments. In view of (7.3), one can define link operational cost c˜a (x, va ) ≡ cˆa (f, va ), for all links a ∈ L . Also, in view of (7.2), one can define demand price function ρ˜k (x) ≡ ρk (d), for all k. Using (7.4) and (7.7) and, then, (7.3), one deduces that  w a la =

xp δap

2

(αa + βa va )

1 , γa

∀a ∈ L .







p∈P

(7.9)

I now rewrite (7.8) as U˜ (x, v) =

J k=1

ρ˜k (x)

p∈Pk

xp −

a∈L

c˜a (x, va ) −

a∈L

va −

a∈L



p∈P xp δap

(αa + βa va )

2

1 . γa

(7.10)

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7 Investments in Labor Productivity: Single Period Model

The firm’s goal is to maximize (7.10) subject to the nonnegativity constraints (7.1), the budget constraint (7.6), and the bounds on the investments on the links (7.5) (since the constraints (7.2), (7.3), (7.4), and (7.7) have been embedded into the objective function). Define the feasible set K 1 ≡ {(x, v)| (7.1), (7.5), and (7.6) hold}. Observe that the feasible set K 1 is convex. Also, assume that the profit function U˜ (x, v) is continuously differentiable and concave. Then, it follows that an optimal solution to the above supply chain network optimization problem coincides with the solution of the following variational inequality problem (cf. Kinderlehrer and Stampacchia 1980 and Nagurney 1999): determine (x ∗ , v ∗ ) ∈ K 1 such that −

∂ U˜ (x ∗ , v ∗ ) ∂ U˜ (x ∗ , v ∗ ) ×(xp −xp∗ )− ×(va −va∗ ) ≥ 0, ∂xp ∂va

p∈P

∀(x, v) ∈ K 1 ,

a∈L

(7.11a) or, equivalently, by expanding out (7.11a), determine (x ∗ , v ∗ ) ∈ K 1 , such that J

⎡ ⎣

k=1 p∈Pk

 ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq δap + ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) a∈L

−ρ˜k (x ∗ ) −

J ∂ρl (x ∗ ) l=1

⎡

∂xp

⎤



xq∗ ⎦ × xp − xp∗

q∈Pl

⎢ ∂ c˜a (x ∗ , v ∗ ) " #−3 a + 1 − 2 αa + βa va∗ + ⎣ ∂va a∈L



⎛ ⎞2 ⎤  βa ⎝ ∗ ⎥  xp δap ⎠ ⎦ × va − va∗ ≥ 0, γa p∈P

∀(x, v) ∈ K 1 ,

(7.11b)

where ∂ cˆb (f, vb ) ∂ C˜ p (x, v) ≡ δap , ∀p ∈ P , ∂xp ∂fa a∈L b∈L

∂ ρ˜l (x) ∂ρl (d) ≡ , ∀p ∈ Pk , ∀k. ∂xp ∂dk

(7.12) A solution (x ∗ , v ∗ ) ∈ K 1 is guaranteed to the above variational inequalities since the feasible set is compact and the underlying functions, under the imposed assumptions, are continuous.

7.3 Lagrange Analysis and Alternative Variational Inequality Formulations

157

7.3 Lagrange Analysis and Alternative Variational Inequality Formulations I now turn to Lagrange analysis, which enables one to construct alternative variational inequality formulations, one of which we will use for computational purposes in the next section. Lagrange analysis also provides us with deeper insights. By setting V (x, v) =

J k=1 p∈Pk

⎡ ⎣

 ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq δap + ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) a∈L −ρ˜k (x ∗ ) −

J ∂ρl (x ∗ ) l=1

⎡

∂xp

⎤

  xq∗ ⎦ × xp − xp∗

q∈Pl

⎛ ⎞2 ⎤ ⎢ ∂ c˜a (x ∗ , v ∗ )  βa ⎝ ⎥  a + 1 − 2(αa + βa va∗ )−3 xp∗ δap ⎠ ⎦ × va − va∗ , + ⎣ ∂va γa a∈L

p∈P

(7.13) variational inequality (7.11b) can be rewritten as the following minimization problem: min V (x, v) = V (x ∗ , v ∗ ) = 0. K1

(7.14)

Given the previous assumptions, all the functions in (7.14) are convex and continuously differentiable. The constraints are reformulated as below in order to construct the Lagrange function, with the associated Lagrange multipliers stated immediately afterward. g=



va − B ≤ 0,

η1 ,

a∈L

h1a = −va ≤ 0,

λ1a , ∀a,

h2a = va − vamax ≤ 0, ep = −xp ≤ 0,

λ2a , ∀a,

μp , ∀p.

(7.15)

I now construct the Lagrange function L(x, v, η1 , λ1 , λ2 , μ), where λ1 and λ2 are, respectively, the vectors of all the λ1a s and λ2a s, and μ is the vector of all the μp s, as follows:

158

7 Investments in Labor Productivity: Single Period Model

L(x, v, η1 , λ1 , λ2 , μ) ⎡  J ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq ⎣ + δap = ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) k=1 p∈Pk

a∈L

−ρ˜k (x ∗ ) −

J ∂ρl (x ∗ ) l=1

⎡

∂xp

⎤



xq∗ ⎦ × xp − xp∗



q∈Pl

⎛ ⎞2 ⎤ ⎢ ∂ c˜a (x ∗ , v ∗ )  βa ⎝ ⎥  a + 1 − 2(αa + βa va∗ )−3 xp∗ δap ⎠ ⎦ × va − va∗ + ⎣ ∂va γa a∈L

+ gη1 +

p∈P



h1a λ1a +

a∈L n





h2a λ2a +

a∈L

(7.16)

ep μp ,

p∈P

n

n

n

n

∀x ∈ R+P , ∀v ∈ R+L , ∀η1 ≥ 0, ∀λ1 ∈ R+L , ∀λ2 ∈ R+L , ∀μ ∈ R+P . Since the feasible set K 1 is convex and the Slater condition is satisfied, if (x ∗ , v ∗ ) n n is a minimal point of (7.14), there exist η1∗ ≥ 0, λ1∗ ∈ R+L , λ2∗ ∈ R+L , ncalP ∗ ∗ ∗ 1∗ 1∗ 2∗ ∗ μ ∈ R+ , such that the vector (x , v , η , λ , λ , μ ) is a saddle point of the Lagrange function (7.16): L(x ∗ , v ∗ , η1 , λ1 , λ2 , μ) ≤ L(x ∗ , v ∗ , η1∗ , λ1∗ , λ2∗ , μ∗ ) ≤ L(x, v, η1∗ , λ1∗ , λ2∗ , μ∗ )

(7.17) and g ∗ η1∗ = 0, 1∗ h1∗ a λa = 0,

∀a ∈ L ,

2∗ h2∗ a λa = 0,

∀a ∈ L ,

ep∗ μ∗p = 0,

∀p ∈ P.

(7.18)

From the right-hand side of (7.17), it follows that x ∗ ∈ R+P and v ∗ ∈ R+L is a minimal point of L(x, v, η1∗ , λ1∗ , λ2∗ , μ∗ ) in the whole space, and, therefore, one has that, for all paths p ∈ P, n

n

∂L(x ∗ , v ∗ , η1∗ , λ1∗ , λ2∗ , μ∗ ) ∂xp ⎡  ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq ⎣ δap − ρ˜k (x ∗ ) = + ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) a∈L

7.3 Lagrange Analysis and Alternative Variational Inequality Formulations

−

J ∂ρl (x ∗ )

∂xp

l=1

159

⎤ xq∗ ⎦ − μ∗p = 0,

(7.19)

q∈Pl

and for all links a ∈ L , ∂L(x ∗ , v ∗ , η1∗ , λ1∗ , λ2∗ , μ∗ ) ∂va ⎡ ∗ ∗ ⎢ ∂ c˜a (x , va )

=⎣

∂va

⎛

+ 1 − 2(αa + βa va∗ )−3

βa ⎝ γa



⎞2 ⎤ ⎥ xp∗ δap ⎠ ⎦

p∈P

2∗ + η1∗ − λ1∗ a + λa = 0,

(7.20)

together with conditions (7.18).

7.3.1 Alternative Variational Inequality Formulations The variational inequalities that are now presented are defined on the nonnegative orthant, which enables the resolution of our proposed computational procedure into steps yielding closed form expressions in the variables. Theorem: Alternative Variational Inequality Formulations Conditions (7.18), (7.19), and (7.20) represent an alternative form of variational n n inequality (7.11b) given by: determine x ∗ ∈ R+P , v ∗ ∈ R+L , η1∗ ≥ 0, λ1∗ ∈ nL n n L P R+ , λ2∗ ∈ R+ , μ∗ ∈ R+ , such that p∈P



 ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq + δap ∗ ∂xp γa (αa + βa va )(αa + βa va∗ ) a∈L

⎤ J   ∗) ∂ρ (x l −ρ˜k (x ∗ ) − xq∗ − μ∗p ⎦ × xp − xp∗ ∂xp q∈Pl

l=1

⎡

⎤ ⎛ ⎞2 ⎢ ∂ c˜a (x ∗ , v ∗ ) " # β −3 a a 2∗ ⎥ ⎝ + + 1 − 2 αa + βa va∗ xp∗ δap ⎠ + η1∗ − λ1∗ ⎣ a + λa ⎦ ∂va γa a∈L

p∈P





      × va − va∗ + B − va∗ × η1 − η1∗ + va∗ × λ1a − λ1∗ a +

 a∈L

a∈L

vamax

− va∗



a∈L

    + xp∗ × μp − μ∗p ≥ 0, × λ2a − λ2∗ a p∈P

160

7 Investments in Labor Productivity: Single Period Model n

n

n

n

n

∀x ∈ R+P , v ∈ R+L , ∀η1 ≥ 0, λ1 ∈ R+L , λ2 ∈ R+L , μ ∈ R+P ,

(7.21)

or, more simply, determine x ∗ ∈ R+P , va∗ , where 0 ≤ va∗ ≤ vamax , ∀a, and η1∗ ≥ 0, such that n



⎡ ⎣

p∈P

 ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq δap + ∗ ∂xp γa (αa + βa va )(αa + βa va∗ ) a∈L

−ρ˜k (x ∗ ) −

⎤ J  ∂ρl (x ∗ ) ∗ ⎦  xq × xp − xp∗ ∂xp q∈Pl

l=1

⎡

⎤ ⎛ ⎞2 ⎢ ∂ c˜a (x ∗ , v ∗ )  β a ⎝ ⎥  a + 1 − 2(αa + βa va∗ )−3 xp∗ δap ⎠ + η1∗ ⎦ × va − va∗ + ⎣ ∂va γa a∈L

p∈P

⎡ + ⎣B −



⎤

  va∗ ⎦ × η1 − η1∗ ≥ 0,

a∈L n

∀x ∈ R+P , ∀va , where 0 ≤ va ≤ vamax , ∀a, ∀η1 ≥ 0.

(7.22)

Proof It follows directly from (7.18), (7.19), and (7.20) that for x ∗ ∈ R+P , v ∗ ∈ n n n n R+L , η1∗ ≥ 0, λ1∗ ∈ R+L , λ2∗ ∈ R+L , μ∗ ∈ R+P , satisfying those expressions also satisfies variational inequality (7.21). It is now established that such vectors also satisfy variational inequality (7.11b). Multiplying (7.19) by (xp − xp∗ ), one obtains n

⎡ ⎣

 ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq δap − ρ˜k (x ∗ ) + ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) a∈L

−

J ∂ρl (x ∗ ) l=1

=

μ∗p

∂xp

⎤ xq∗ ⎦ × (xp − xp∗ )

q∈Pl

× (xp − xp∗ ),

(7.23)

and, since μ∗p xp ≥ 0, for all p, and μ∗p xp∗ = 0, for all p, summation of the left-hand side of (7.23) over all paths p yields

7.3 Lagrange Analysis and Alternative Variational Inequality Formulations

p∈P

161

⎡

 ∗ 2 ∂ C˜ p (x ∗ , v ∗ ) q∈P xq δaq ⎣ δap + ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) a∈L

−ρ˜k (x ∗ ) −

⎤

J ∂ρl (x ∗ ) l=1

∂xp

xq∗ ⎦ × (xp − xp∗ ) ≥ 0,

n

∀x ∈ R+P .

q∈Pl

(7.24) Multiplying (7.20) by (va − va∗ ), in turn, yields ⎡ ∗ ∗ ⎢ ∂ c˜a (x , va )

⎣

∂va

⎞2 ⎤ βa ∗ ⎠ ⎥ + 1 − 2(αa + βa va∗ )−3 ⎝ xp δap ⎦ × (va − va∗ ) γa ⎛

p∈P

2∗ ∗ = (−η1∗ + λ1∗ a − λa ) × (va − va ).

(7.25)

Summation, in turn, over all links a ∈ L of (7.25), and the use of (7.18), gives us ⎡ ⎛ ⎞2 ⎤ ⎢ ∂ c˜a (x ∗ , v ∗ ) βa ⎝ ⎥ a + 1 − 2(αa + βa va∗ )−3 xp∗ δap ⎠ ⎦ × (va − va∗ ) ⎣ ∂va γa

a∈L

= −η1∗

p∈P

a∈L

va + η1∗ B −

a∈L

λ2∗ a va +

a∈L

max + λ2∗ a va

a∈L



λ1∗ a va −

∗ λ1∗ a va .

a∈L

(7.26) The first two terms on the right-hand side of (7.26) result in a nonnegative value, as do the second two terms. The next to the final term in (7.26) is also nonnegative, whereas the last term is equal to zero. Hence, the first conclusion follows. Furthermore, variational inequality (7.22) follows from variational inequality (7.21) since the feasible set underlying the former captures the nonnegativity assumption on the product path flows and on the productivity link investments with the latter not exceeding the respective imposed upper bounds. The proof is complete.

7.3.2 Additional Lagrange Analysis with Interpretations I now utilize the above Lagrange analysis results to obtain deeper insights. Making use of (7.19), I consider the case where the optimal product flow on a path p, p ∈ Pk , is positive; that is, xp∗ > 0, which means that μ∗p = 0. From (7.19), one then gets

162

7 Investments in Labor Productivity: Single Period Model

⎡

⎤  J ∗ ∗) 2 ∂ C˜ p (x ∗ , v ∗ ) ∂ρ (x q∈ P xq δaq l ⎣ δap = ρ˜k (x ∗ ) + + xq∗ ⎦ . ∂xp γa (αa + βa va∗ )(αa + βa va∗ ) ∂xp a∈L

l=1

q∈Pl

(7.27) Equation (7.27) has the interpretation that the marginal costs, which include what we refer to as the marginal operational cost on a path and the marginal cost associated with labor on the path (see also (7.9) and (7.10)), are precisely equal to the marginal revenue. This is a good result in terms of economics. If the optimal product flow on the path is still positive and if there is a link a on which the optimal investment on the link is neither at its upper bound nor at its lower bound and the budget is not exhausted, then one knows, from (7.20), that ⎡ ⎢ ∂ c˜a ⎣

(x ∗ , v ∗ ) ∂va

⎞2 ⎤ βa ∗ ⎠ ⎥ + 1 = 2(αa + βa va∗ )−3 ⎝ xp δap ⎦ . γa ⎛

(7.28)

p∈P

Expression (7.28) has the interpretation that the marginal cost associated with investing in the productivity of the link is equal to the marginal return of the investment. On the other hand, if the budget is exhausted, then one can conclude that the marginal cost associated with investing in productivity on the link is greater than the marginal return of the investment and that is not a good situation. The Lagrange multiplier η1∗ is then greater than zero, and its interpretation as a shadow price reflects how much can be gained in terms of the profit by increasing the budget by a unit. Of course, if the investment on the link is at its upper bound, the budget is exhausted, and the path has positive flow at optimality, then the marginal investment cost on the link exceeds the marginal return by even a greater amount. Returning to (7.19), for completeness, one sees that if, on the other hand, the optimal product flow on a path p is zero, that is, xp∗ = 0, then the marginal costs on the path exceed the marginal revenue on the path, so it makes no sense, from a profit standpoint, to use that path for product flow. Recall that a path consists of production, transportation, storage, and distribution in our basic framework. The framework, as noted earlier, can be adapted from a supply chain topological standpoint, as need be, and can even incorporate the option of electronic commerce (or direct sales) as investigated in the context of food supply chains in Chapter 2. I demonstrate this feature through numerical examples in the next section. Also, from (7.20), one can see that, if the investment on a link a is zero, that is, va∗ = 0, and the budget is not exhausted, then the marginal costs associated with investing in the productivity on the link exceeds the marginal return of investing in the link productivity.

7.4 Computational Procedure and Numerical Examples

163

7.4 Computational Procedure and Numerical Examples All of the above variational inequalities can be put into standard variational inequality form (cf. (2.14)), where the finite-dimensional variational inequality problem VI(F, K ) is to determine a vector X∗ ∈ K ⊂ R N , such that

F (X∗ ), X − X∗  ≥ 0,

∀X ∈ K ,

(7.29)

where F is a given continuous function from K to R N , K is a given closed, convex set, and ·, · denotes the inner product in N-dimensional Euclidean space. I will be solving variational inequality (7.22), since the proposed algorithm resolves the problem into a series of subproblems in the variables, for which I provide explicit formulae. Hence, for completeness, I now put variational inequality (7.22) into standard form (7.29). Define the vector X ≡ (x, v, η) and the vector F (X) ≡ (F 1 (X), F 2 (X), F 3 (X)) where the p-th component of F 1 (X), Fp1 (X), is ⎡ Fp1 (X)

=⎣

 2 ∂ C˜ p (x, v) q∈P xq δaq δap + ∂xp γa (αa + βa va )(αa + βa va ) a∈L

−ρ˜k (x) −

J ∂ρl (x) l=1

∂xp

⎤ xq ⎦ ;

q∈Pl

the a-th component of F 2 (X), Fa2 (X), is ⎡

⎛

βa ⎢ ∂ c˜a (x, v) Fa2 (X) = ⎣ + 1 − 2(αa + βa va )−3 ⎝ ∂va γa



⎞2

⎤

⎥ xp δap ⎠ + η1 ⎦ ,

p∈P

   and the single component of F 3 (X) = B − a∈L va . N = nP + nL + 1 and n K = {(x, v, η1 )|x ∈ R+P , 0 ≤ va ≤ vamax , ∀a, η1 ≥ 0}.

7.4.1 Computational Procedure As in Chapters 2–6, the modified projection method, due to Korpelevich (1977), is utilized for computational purposes. Recall that algorithm is guaranteed to converge to a solution of variational inequality (7.29) if F (X) is monotone and Lipschitz continuous, and a solution exists. The explicit formulae for all the variables for the model for Step 1 (cf. Chapter 2 or the Appendix) are now provided. The analogues of Step 2 easily follow.

164

7 Investments in Labor Productivity: Single Period Model

Explicit Formulae at Iteration τ for the Product Path Flows in Step 1 Specifically, one has the following closed form expressions for the path flows in Step 1 in the solution of variational inequality (7.22): ⎧ ⎨

⎛

x¯pτ = max 0, xpτ −1 + η ⎝ρ˜k (x τ −1 ) + ⎩

J ∂ ρ˜l (x τ −1 ) τ −1 ∂ C˜ p (x τ −1 , v τ −1 ) xq − ∂xp ∂xp

−

q∈Pl

l=1



τ −1 2 q∈P xq δaq δap γa (αa + βa va )(αa + βa va )

$ ,

∀p ∈ Pk ; k = 1, . . . , J.

(7.30)

a∈L

Explicit Formulae at Iteration τ for the Link Productivity Investments Flows in Step 1 Also, one has the following closed form expressions for the link productivity investments in Step 1 in the solution of variational inequality (7.22): ⎧ ⎪ ⎨

⎧ ⎪ ⎨



v¯aτ = max 0, min vaτ −1 + η(2 αa + βa vaτ −1 ⎪ ⎪ ⎩ ⎩ −

−3

⎛ ⎞2 βa ⎝ τ −1 ⎠ xp δap γa p∈P

∂ c˜a (x τ −1 , vaτ −1 ) − 1 − η1τ −1 , vamax ∂va

%% ,

∀a ∈ L .

(7.31)

Explicit Formula at Iteration τ for the Lagrange Multiplier in Step 1 Finally, one has the following closed form expression for the Lagrange multiplier in Step 1 at an iteration τ : ⎧ ⎨

⎛

η¯ 1τ = max 0, η1τ −1 + η ⎝ ⎩

a∈L

⎞⎫ ⎬ vaτ −1 − B ⎠ . ⎭

(7.32)

It is straightforward to adapt the above closed form expressions for the special case of the model without link investment upper bounds and/or without a budget constraint.

7.4.2 Numerical Examples The modified projection method is implemented in FORTRAN and a Linux system at the University of Massachusetts Amherst used for computing solutions to the

7.4 Computational Procedure and Numerical Examples

165

subsequent numerical examples. The computational procedure was initialized as follows. All the link investments and the Lagrange multiplier associated with the budget constraint were set to 0.00. The initial demand at each market was set to 40 with the demand equally distributed among the paths terminating in each demand market. The convergence tolerance is 10−7 ; that is, the algorithm is considered to have converged when the absolute value of the difference between each of the variables at two successive iterations differs by no more than this value. The supply chain network numerical examples are used as a “proof of concept” and are not focused on a specific application but, nevertheless, yield broader insights. Specific applications, with particular features and parameterized accordingly, can be readily solved using the algorithm.

7.4.2.1

Examples 7.1, 7.2, and 7.3

The supply chain network topology for Examples 7.1, 7.2, and 7.3 is given in Figure 7.2. The firm has two production sites, one distribution center for storage, and sells its product at two demand markets. Example 7.1 assumes that there is no investment possible and, hence, there is no investment budget and no bounds on the link productivity investments. The data for Example 7.1 are identical to the data for Example 6.1. Example 7.2 then has the identical data to the data in Example 7.1 but with the investments added and the beta values as given under Example 7.2 below. Example 7.3, in turn, has the identical data to the data in Example 7.2 but with the addition of bounds on the link productivity investments as well as a budget.

Fig. 7.2 Supply chain network topology for Examples 7.1, 7.2, and 7.3

166

7 Investments in Labor Productivity: Single Period Model

Example 7.1 (No Investment Parameters Beta, No Budget Constraint, and No Bounds on Link Productivity Investments) The operational link cost functions (since I do not consider investments until the next examples) are cˆa (f ) = 2fa2 ,

cˆb (f ) = 2fb2 ,

cˆe (f ) = fe2 + 2fe ,

cˆc (f ) = 0.5fc2 ,

cˆf (f ) = 0.5ff2 ,

cˆd (f ) = 0.5fd2 ,

cˆg (f ) = 0.5fg2 .

The demand price functions are ρ1 (d) = −5d1 + 800,

ρ2 (d) = −5d2 + 850.

The alpha link parameters are αa = 55,

αb = 50,

αc = 35,

αd = 35,

αe = 60,

βd = 0,

βe = 0,

αf = 38,

αg = 36,

the beta link parameters are βa = 0,

βb = 0,

βc = 0,

βf = 0,

βg = 0,

and the gamma link parameters are γa = 0.1, γb = 0.1, γc = 0.09, γd = 0.07, γe = 0.08 γf = 0.06, γg = 0.08. The paths are defined as path p1 = (a, c, e, f ), path p2 = (b, d, e, f ), path p3 = (a, c, e, g), and path p4 = (b, d, e, g). The modified projection method yields the following equilibrium product path flow pattern: xp∗1 = 19.39,

xp∗2 = 19.36,

xp∗3 = 21.66,

xp∗4 = 21.63.

The equilibrium link flows and labor values are reported in Table 7.2, whereas the equilibrium productivity investments and hourly wages are reported in Table 7.3. The demand price at the first demand market is 606.27 and at the second demand market the price is 633.52, with the corresponding equilibrium demands of 38.75 and 43.30. The firm earns a profit of 33,816.98. Example 7.2 (Positive Investment Parameters Beta, No Budget Constraint, and No Bounds on Link Productivity Investments) Example 7.2 has the same data as Example 7.1, but now I include investments (but no bounds). The operational costs are as in Example 7.1 in terms of the link flow dependence, but they are now extended to have an investment component as follows:

7.4 Computational Procedure and Numerical Examples cˆa (f, va ) = 2fa2 + 0.05va2 ,

cˆb (f, vb ) = 2fb2 + 0.1vb2 ,

cˆd (f, vd ) = 0.5fd2 + 0.05vd2 ,

167 cˆc (f, vc ) = 0.5fc2 + 0.05vc2 ,

cˆe (f, ve ) = fe2 + 2fe + 0.1ve2 ,

cˆf (f, vf ) = 0.5ff2 + 0.1vf2 ,

cˆg (f, vg ) = 0.5fg2 + 0.1vg2 .

The βs are no longer equal to zero, as in Example 7.1, but are now βa = 10,

βb = 10,

βc = 20,

βd = 20,

βe = 10,

βf = 10,

βg = 10.

The link productivity functions (cf. Table 7.1) are, thus, of the form: πa = αa +βa va , for all links a ∈ L with the α terms as in Example 7.1. The modified projection method yields the following equilibrium product path flow pattern: xp∗1 = 19.38,

xp∗2 = 19.36,

xp∗3 = 21.66,

xp∗4 = 21.63.

The equilibrium link flows and labor values for Example 7.2 are reported in Table 7.2, whereas the equilibrium productivity investments and hourly wages are reported in Table 7.3. The demand price at the first demand market is 606.28 and at the second demand market the price is 633.54, with the corresponding equilibrium demands of 38.74 and 43.29. The firm earns a profit of 33,868.11. The total investment outlay of the firm is 17.72. Note that there is a good return on investment, since the profit in Example 7.1 is 33,816.98, whereas now the profit is 33,868.11. I also, for completeness, report the values of the link productivity functions at the equilibrium: πa = 67.00,

πb = 64.03,

πf = 67.90,

πg = 66.32.

πc = 84.36,

πd = 90.88,

πe = 98.37,

Note that since the βs are all equal to zero in Example 7.1, the π s there are not investment-dependent and collapse to the corresponding α on the link. The labor hours needed on each link decrease, as compared to the respective result in Example 7.1, and the wage on each link also decreases. Example 7.2 demonstrates the benefits for the firm of investing in link productivity. Note that the operational link cost does depend on the investment on the link since there may be, for example, some maintenance and other related costs associated with the investment encumbered. Example 7.3 (Positive Investment Parameters Beta, Budget Constraint, and Bounds on Link Productivity Investments) The data for Example 7.3 are as in Example 7.2 with the budget B = 15 and vamax = . . . = vgmax = 3. The path flows remain essentially as in Example 7.2 since the demand price functions are not functions

168

7 Investments in Labor Productivity: Single Period Model

of the investments. The equilibrium link flows and labor values are reported in Table 7.2 with the equilibrium link productivity investments and the hourly wages reported in Table 7.3. Wages are now higher, in order to attract labor since more is needed due to a decrease in productivity as compared to that in Example 7.2. The investment on link e is at the upper bound. The total investment outlay is now 15, so the budget is exhausted with the Lagrange multiplier being positive and with a value of η1∗ = 0.29. The link productivity functions evaluated at the computed equilibrium link investments are now πa = 62.70,

πb = 60.69,

πf = 64.93,

πg = 63.41.

πc = 79.14,

πd = 85.41,

πe = 90.00,

The profit of the firm is 33,867.59. Under the imposed budget constraint and the maximum bounds on investment links, the profit of the firm decreases from 33,868.11 in Example 7.2 to 33,867.59 in Example 7.3.

7.4.2.2

Examples 7.4, 7.5, and 7.6

In the second series of examples, I consider a supply chain network topology as depicted in Figure 7.3. There is now an additional production site available to the firm, but it still has a single distribution center and serves two demand markets. These examples follow a similar pattern to that of Examples 7.1, 7.2, and 7.3. Table 7.2 Equilibrium link flows and labor values for Examples 7.1, 7.2, and 7.3

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗

Equilibrium value Example 7.1 Example 7.2 41.05 41.04 40.99 41.00 41.05 41.04 40.99 41.00 82.04 82.04 38.75 38.74

Example 7.3 41.05 41.01 41.05 41.01 82.06 38.75

43.30 0.75 0.82 1.17 1.17 1.37 1.02

43.29 0.61 0.64 0.49 0.45 0.83 0.57

43.30 0.65 0.68 0.52 0.48 0.91 0.60

1.20

0.65

0.68

7.4 Computational Procedure and Numerical Examples Table 7.3 Equilibrium link productivity investments and hourly wages for Examples 7.1, 7.2, and 7.3

Notation va∗ vb∗ vc∗ vd∗ ve∗ vf∗ vg∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗

169

Equilibrium value Example 7.1 Example 7.2 − 1.20 − 1.40 − 2.47 − 2.79 − 3.84 − 2.99

Example 7.3 0.77 1.07 2.21 2.52 3.00 2.69

− 7.46 8.20 13.03 16.73 17.09 16.99

3.03 6.13 6.40 5.41 6.44 10.42 9.51

2.74 6.55 6.76 5.76 6.86 11.40 9.95

15.03

8.16

8.54

Example 7.4 (No Investment Parameters Beta, No Budget Constraint, and No Bounds on Link Productivity Investments) Example 7.4 has the identical data to that in Example 7.1 but with the additional data for links h and i as follows:

Fig. 7.3 Supply chain network topology for Examples 7.4, 7.5, and 7.6

170

7 Investments in Labor Productivity: Single Period Model

cˆh (f, vh ) = fh2 + 0.05vh2 , αh = 45,

cˆi (f, vi ) = 0.5fi2 + 0.05vi2 ,

αi = 30,

γh = 0.1,

βh = 0,

βi = 0,

γi = 0.08,

and

since in this example, as done in Example 7.1, I consider the case that there are no investments in link productivity. Also, there are two new paths: path p5 = (h, i, e, f ) and path p6 = (h, i, e, g). The modified projection method converges to the following equilibrium product path flow pattern: xp∗1 = 11.91,

xp∗2 = 11.89,

xp∗5 = 13.41,

xp∗6 = 21.77.

xp∗3 = 20.20,

xp∗4 = 13.43,

The original paths have much lower volumes of product flow than they had in Example 7.1, with the two new paths having the largest volume of product flow. The profit is 38,138.79. The profit increases substantially with the introduction of a new production site, by more than 4000. The demand price at the first demand market is 580.02 and at the second demand market the price is 607.26, with the corresponding equilibrium demands of 44.00 and 48.55. Consumers also benefit since the prices at the demand markets decrease. The computed equilibrium link flows and labor values are given in Table 7.4 and the equilibrium link productivity investments and the hourly wages are reported in Table 7.5. Example 7.5 (Positive Investment Parameters Beta, No Budget Constraint, and No Bounds on Link Productivity Investments) Example 7.5 has the same data as Example 7.4 with the beta parameters now being positive and as in Example 7.2, with the addition of the following ones on the added two links: βh = 15,

βi = 15.

As in Example 7.2, there are no bounds on the link productivity investments and no budget. The modified projection method now converges to the following equilibrium product path flow pattern: xp∗1 = 11.88,

xp∗2 = 11.87,

xp∗5 = 13.39,

xp∗6 = 21.76.

xp∗3 = 20.24,

xp∗4 = 13.40,

The additional equilibrium results are reported in Tables 7.4 and 7.5.

7.4 Computational Procedure and Numerical Examples

171

The firm invests a total amount of 19.74. The demand price at the first demand market is 580.01 and at the second demand market 607.26 with equilibrium demands of 44.00 and 48.55, respectively. The firm earns a profit of 38,202.74. The firm gains in profit by investing in productivity in the supply chain links. In Example 7.4, the profit is 38,138.79, whereas now, with an investment of only 19.74, the profit has risen to 38,202.74. The values of the link productivity functions at the equilibrium, where recall that πa = αa + βa va , ∀a ∈ L , are πa = 55.00, πb = 50.26, πc = 62.93, πd = 67.91, πe = 104.32, πf = 72.54,

πg = 70.41,

πh = 75.98,

πi = 79.26.

Example 7.6 (Positive Investment Parameters Beta, Budget Constraint, and Bounds on Link Productivity Investments) Example 7.6 is constructed from Example 7.5 and has the same data but with the addition of the same budget and link investment bounds as in Example 7.3. Hence, the budget is 15 and all the v max s are equal to 3, including on the two added links. The modified projection method now converges to the following equilibrium product path flow pattern: xp∗1 = 11.88,

xp∗2 = 11.88,

xp∗5 = 13.39,

xp∗6 = 21.77.

xp∗3 = 20.25,

xp∗4 = 13.40,

The additional equilibrium results are reported in Tables 7.4 and 7.5. The firm invests a total amount of 15.00 and the Lagrange multiplier η∗ = 0.64. The demand price at the first demand market is 579.92 and at the second 607.18 with equilibrium demands of 44.02 and 48.56, respectively. The firm now earns a profit of 38,200.93, a decrease, but not a significant one, from the profit in Example 7.5. The values of the link productivity functions at the equilibrium, where recall that πa = αa + βa va , ∀a ∈ L , are πa = 55.00,

πb = 50.00,

πc = 54.67,

πd = 59.17,

πf = 66.38,

πg = 64.39,

πh = 66.70,

πi = 70.26.

The investment on link a is at the imposed upper bound.

πe = 90.00,

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Table 7.4 Equilibrium link flows and labor values for Examples 7.4, 7.5, and 7.6

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fh∗ fi∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗ lh∗ li∗

7.4.2.3

Equilibrium value Example 7.4 Example 7.5 25.33 25.28 25.30 25.26 25.33 25.28 25.30 25.26 92.54 92.55 44.00 44.00

Example 7.6 25.29 25.27 25.29 25.27 92.58 44.02

48.55 41.91 41.91 0.46 0.51 0.72 0.72 1.54 1.16

48.55 42.00 42.00 0.46 0.50 0.40 0.37 0.89 0.61

48.56 42.02 42.02 0.46 0.51 0.46 0.43 1.03 0.66

1.35 0.93 1.40

0.69 0.55 0.53

0.75 0.63 0.60

Examples 7.7, 7.8, and 7.9: Introduction of Electronic Commerce

In the third, final, series of numerical examples, I consider the impact of electronic commerce. Specifically, to the supply chain network topology in Figure 7.3, I now add direct links j and k to Demand Markets 1 and 2, respectively, from nodes M1 and M3 and consider numerical examples with the supply chain network topology in Figure 7.4. Example 7.7 (Positive Investment Parameters Beta, No Budget Constraint, and No Bounds on Link Productivity Investments) From the previous numerical examples, it is clear that allowing for investments in link productivity could raise profits of the firm. Example 7.7 serves as the baseline from which I then construct in this series Examples 7.8 and 7.9. Hence, Example 7.7 (unlike Examples 7.1 and 7.4) has positive beta parameters on all of its links. Example 7.7 has the identical data to that in Example 7.5 but with the additional data for links j and k as follows: cˆj (f, vj ) = 1.5fj2 + 0.05vj2 , αj = 55,

αk = 60,

cˆk (f, vk ) = 2fk2 + 0.1vk2 , γj = 0.1,

γk = 0.1,

7.4 Computational Procedure and Numerical Examples Table 7.5 Equilibrium link productivity investments and hourly wages for Examples 7.4, 7.5, and 7.6

Notation va∗ vb∗ vc∗ vd∗ ve∗ vf∗ vg∗ vh∗ vi∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗ wh∗ wi∗

173

Equilibrium value Example 7.4 Example 7.5 − 0.00 − 0.03 − 1.40 − 1.65 − 4.43 − 3.45

Example 7.6 0.00 0.00 0.98 1.21 3.00 2.84

− − − 4.61 5.06 8.04 10.33 19.28 19.30

3.44 2.07 3.28 4.60 5.03 4.46 5.31 11.09 10.11

2.84 1.45 2.68 4.60 5.05 5.14 6.10 12.86 11.05

16.86 9.31 17.46

8.62 5.53 6.62

9.43 6.30 7.48

and βj = 20,

βk = 20.

There are two new paths: path p7 = (a, j ), and path p6 = (h, k). There is no budget and no vamax on all links a ∈ L . The modified projection method converges to the following equilibrium product path flow pattern: xp∗1 = 0.00, xp∗5 = 14.09,

xp∗2 = 11.80, xp∗6 = 11.06,

xp∗3 = 8.77, xp∗7 = 34.93,

xp∗4 = 0.00, xp∗8 = 34.88.

The additional equilibrium results are reported in Tables 7.6 and 7.7. One can see that the paths with the electronic commerce links have the highest product path flows and that both paths p1 and p4 are not even used (in contrast to the results in Example 7.5) and, hence, have zero flow. The demand price at the first demand market is 522.51 and at the second 549.82 with equilibrium demands of 55.50 and 60.04, respectively. The firm earns a profit of 47,685.11. The profit is higher than in Example 7.5 by more than 8000. Electronic commerce benefits the firm in terms of profit and consumers, in terms of demand market prices, which are now lower.

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Fig. 7.4 Supply chain network topology for Examples 7.7, 7.8, and 7.9

The values of the link productivity functions at the equilibrium, where recall that πa = αa + βa va , ∀a ∈ L , are πa = 61.25,

πb = 50.89,

πc = 35.00,

πd = 68.93,

πe = 74.14,

πg = 49.85,

πh = 86.81,

πi = 50.61,

πj = 76.13,

πk = 75.06.

πf = 48.77,

Example 7.8 (Positive Investment Parameters Beta, Budget Constraint, Bounds on Link Productivity Investments, and Demand Price Function Changes) In Example 7.8, the budget of 15 remains, as in earlier examples with a budget. Also, vamax = 3 for all links that are not electronic commerce links (as in previous examples with bounds), but now the following bounds on the electronic commerce links are imposed: vjmax = 4,

vkmax = 4.

Also, the case where the consumers at the demand markets are grateful for the e-commerce option is now considered with them being willing to pay higher prices. The demand price function intercept terms are changed from 800 to 850 for the first demand market and from 850 to 900 for the second demand market. The modified projection method now converges to the following equilibrium product path flow pattern:

7.4 Computational Procedure and Numerical Examples

xp∗1 = 0.00,

xp∗2 = 12.57,

xp∗5 = 14.89,

xp∗6 = 11.69,

xp∗3 = 9.37, xp∗7 = 37.06,

175

xp∗4 = 0.00, xp∗8 = 36.96.

Additional equilibrium results are reported in Tables 7.6 and 7.7. One can see that the paths with the electronic commerce links, as in Example 7.7, have the highest product path flows and that both paths p1 and p4 have zero flow, as they did in Example 7.7. The demand price at the first demand market is 554.97 and at the second 582.31 with equilibrium demands of 59.01 and 63.54, respectively. The firm earns a profit of 53,640.90, which is higher than the profit in Example 7.7. This shows the potential benefit of having consumers being willing to pay higher prices and firms can achieve this through marketing, for example. The values of the link productivity functions at the equilibrium are πa = 63.30,

πb = 52.40,

πc = 35.00,

πd = 71.43,

πe = 76.29,

πg = 51.33,

πh = 90.00,

πi = 52.50,

πj = 78.88,

πk = 77.48.

πf = 50.46,

The total investments are 13.87. The investment on link h is at the upper bound of 3. Example 7.9 (Positive Investment Parameters Beta, Budget Constraint, Bounds on Link Productivity Investments, and Increase in Production Costs) Example 7.9 has the same data as Example 7.8, but now a production disruption occurs with the operational costs on links a and b increasing to the following, respectively: cˆa (f, va ) = 3fa2 + 0.05va2 ,

cˆb (f, vb ) = 3fb2 + 0.1vb2 .

The modified projection method converges to the following equilibrium product path flow pattern: xp∗1 = 0.00, xp∗5 = 10.83,

xp∗2 = 11.51, xp∗6 = 13.02,

xp∗3 = 13.70, xp∗7 = 31.44,

xp∗4 = 0.00, xp∗8 = 37.96.

The additional equilibrium results are reported in Tables 7.6 and 7.7. The demand price at the first demand market is 566.71 and at the second 590.97 with equilibrium demands of 56.66 and 61.81, respectively. The firm earns a profit of 51,863.57. With higher costs at two of the three production sites, the profit now decreases. The values of the link productivity functions at the equilibrium, where recall that πa = αa + βa va , ∀a ∈ L , are

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7 Investments in Labor Productivity: Single Period Model

πa = 57.74,

πb = 50.00,

πc = 35.00,

πd = 63.02,

πe = 76.70,

πg = 48.46,

πh = 90.00,

πi = 60.58,

πj = 71.49,

πk = 78.61.

πf = 50.28,

The total investments are 13.01. The investment on link h remains at the upper bound of 3. Table 7.6 Equilibrium link flows and labor values for Examples 7.7, 7.8, and 7.9

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fh∗ fi∗ fj∗ fk∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗ lh∗ li∗ lj∗ lk∗

Equilibrium Value Example 7.7 Example 7.8 34.93 37.06 25.89 27.46 0.00 0.00 25.89 27.46 45.72 48.52 20.56 21.94

Example 7.9 3‘.44 22.35 0.00 22.35 49.07 25.22

25.16 54.71 19.83 34.93

26.58 58.02 21.06 37.06

23.85 64.67 26.72 31.44

34.88 0.57 0.51 0.00 0.38 0.62 0.42

36.95 0.59 0.52 0.00 0.38 0.64 0.42

37.96 0.54 0.45 0.00 0.35 0.64 0.46

0.50 0.62 0.39 0.46

0.52 0.64 0.40 0.47

0.49 0.72 0.44 0.44

0.46

0.48

0.48

7.5 Summary, Conclusions, and Suggestions for Future Research Disruptions to labor in the past several years have been deep and widespread, affecting agriculture and manufacturing as well as freight service provision, among many other economic sectors. Many firms continue to have difficulties in attracting labor. The pandemic, climate change, and various conflicts, including wars, have affected the productivity of labor with investments in productivity being a possible avenue to enable enhanced output.

7.5 Summary, Conclusions, and Suggestions for Future Research Table 7.7 Equilibrium link productivity investments and hourly wages for Examples 7.7, 7.8, and 7.9

Notation va∗ vb∗ vc∗ vd∗ ve∗ vf∗ vg∗ vh∗ vi∗ vj∗

vk∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗ wg∗ wh∗ wi∗ wj∗ wk∗

177

Equilibrium value Example 7.7 Example 7.8 0.62 0.83 0.09 0.24 0.00 0.00 1.70 1.82 1.41 1.63 1.08 1.25

Example 7.9 0.27 0.00 0.00 1.40 1.67 1.63

1.39 2.91 1.37 1.06

1.53 3.00 1.50 1.19

1.25 3.00 2.04 0.82

0.75 5.70 5.09 0.00 5.37 7.71 7.03

0.87 5.85 5.24 0.00 5.49 7.95 7.25

0.93 5.45 4.47 0.00 5.07 8.00 7.74

6.31 6.17 4.0 4.59

6.47 6.45 5.02 4.70

6.15 7.19 5.51 4.40

4.65

4.77

4.83

In this chapter, an optimization model for supply chain networks is built that includes labor as a resource. The model allows for wage-dependent labor, as well as investments in supply chain link productivity. The investments are subject to a budget constraint and also to a bound on the investment on each link. The solution of the model yields the optimal product path flows from the firm to the demand markets, the optimal link productivity investments, as well as the labor hours needed on the supply chain network links, and the wages that should be paid to the workers on the links, which consist of production, transportation, storage, and distribution links. Alternative variational inequality formulations of the optimal solution are provided, and Lagrange analysis is conducted. The proposed algorithmic scheme has nice features for implementation since it resolves the variational inequality formulation that is utilized into subproblems which yield closed form expressions in the product path flows and the Lagrange multiplier associated with the budget constraint. The algorithm is implemented and applied to compute solutions to three sets of supply chain network numerical examples, including a set with electronic commerce. The full solutions are reported, including the labor hours and the wages that should be paid. I find that investments of link productivity can enhance profits

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7 Investments in Labor Productivity: Single Period Model

for the firm and reduce the product price at the demand markets for the consumers. Also, adding a production site can enhance profits as well as taking advantage of electronic commerce. Firms, however, should be careful in moderating their operational costs since increases can have a big impact on the bottom line. Future research can include alternative investment functions to those in (7.4).

7.6 Sources and Notes This chapter is based on the paper by Nagurney (2022). This chapter returns to optimization modeling, which was the focus in Chapters 2 and 3, but with the added variables of investments in link supply chain network productivity, subject to the firm’s budget constraint. This chapter, as Chapter 5, provides Lagrange analysis, along with managerial insights. For Lagrange analysis on other network-based applications, see Daniele (2001, 2004, 2006), Barbagallo et al. (2012), Caruso and Daniele (2018), Colajanni et al. (2018), Daniele and Sciacca (2021), and Nagurney and Daniele (2021). This model is a single period model. In Chapter 8, a multiperiod supply chain network model with investments in productivity is presented.

References Barbagallo, A., Daniele, P., Maugeri, A., 2012. Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula. Nonlinear Analysis, Theory, Methods and Applications, 75(3), 1104–1123. Bloom, N., Bunn, P., Mizen, P., Smietanka, P., Thwaites, G., 2020. The impact of COVID-19 on productivity. NBER working paper no. 28233, Cambridge, Massachusetts. Caruso, V., Daniele, P., 2018. A network model for minimizing the total organ transplant costs. European Journal of Operational Research, 266(2), 652–662. Chaney Cambon, S., 2021. U.S.’s long drought in worker productivity could be ending. The Wall Street Journal, April 4. Colajanni, G., Daniele, P., Giuffre, S., Nagurney, A., 2018. Cybersecurity investments with nonlinear budget constraints and conservation laws: Variational equilibrium, marginal expected utilities, and Lagrange multipliers. International Transactions in Operational Research, 25, 1443–1464. Daniele, P., 2001. Variational inequalities for static equilibrium market, Lagrangian function and duality. In: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. F. Giannessi, A. Maugeri, and P.M. Pardalos, Editors, Kluwer Academic Publishers, Amsterdam, pp. 43–58. Daniele, P., 2004. Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model. Journal of Global Optimization, 28(3–4), 283–295. Daniele, P., 2006. Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar Publishing, Cheltenham, United Kingdom. Daniele, P., Sciacca, D., 2021. An optimization model for the management of green areas. International Transactions in Operational Research, 28, 3094–3116. Del Ray, J., 2020. Amazon was already powerful. The coronavirus pandemic cleared the way to dominance. Vox, April 10. Garton, E., 2017. The case for investing more in people. Harvard Business Review, September 4.

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Herrera, S., 2021. Amazon raising pay for hundreds of thousands of workers. The Wall Street Journal, April 28. Hintzmann, C., Llads-Masllorens, J., Ramos, R., 2021. Intangible assets and labor productivity growth. Economies, 9(2), 82. Igoe, K.J., 2021. How COVID-19 has changed the standards of worker safety and health and how organizations can adapt. Harvard T.H. Chan School of Public Health, Cambridge, Massachusetts, March 21. Jorgenson, D.W., 1991. Investing in productivity growth. In: Technology and Economics: Papers Commemorating Ralph Landau’s Service to the National Academy of Engineering. National Academy Press, Washington DC, pp. 57–63. Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York. Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. Mishra, S.K., 2007. A brief history of production functions. MPRA Paper No. 5254, http://mpra. ub.uni-muenchen.de/5254/. Morath, E., 2021. Millions are unemployed. Why can’t companies find workers? The Wall Street Journal, May 6. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A., 2022. Optimization of investments in labor productivity in supply chain networks. International Transactions in Operational Research, 29(4), 2116–2144. Nagurney, A., Daniele, P., 2021. International human migration networks under regulations. European Journal of Operational Research, 291(3), 894–905. Queiroz, M.M., Ivanov, D., Dolgui, A., Wamba, S.F., 2020. Impacts of epidemic outbreaks on supply chains: Mapping a research agenda amid the COVID-19 pandemic through a structured literature review. Annals of Operations Research, 1–38. Rosenberg, J.M., 2021. Where are the workers? Small businesses struggle to fill jobs. Chicago Tribune, April 21. Sanandaji, T., Monte, F., Ham, A., Tarki, A., 2021. Attracting talent during a worker shortage. Harvard Business Review, June 14. Simon, R., 2021. The wage wager. The Wall Street Journal, August 7–8. Stiroh, K.J., 2001. What drives productivity growth? FRBNY Economic Policy Review, March, 37–59. Stundziene, A., Saboniene, A., 2019. Tangible investment and labour productivity: Evidence from European manufacturing. Ekonomska Istraivanja / Economic Research, 32(1), 3519–3537. Weber, A., 2021. Europe heads for jobs crunch that may be deeper than the U.S.’s. Bloomberg, May 20.

Chapter 8

Multiperiod Supply Chain Network Investments in Labor Productivity

Abstract It is a challenging time in history with the COVID-19 pandemic plus the impacts of climate change being notable with the frequency of various disasters increasing as well as their impacts. In addition, conflicts and wars around the globe are adding to disruptions of societies and economies. Nevertheless, products still must be produced, transported, and consumed in order to sustain humanity. Labor has revealed itself to be a critical resource in supply chain networks. In this chapter, a multiperiod supply chain network model is constructed, which captures labor associated with the various supply chain network activities of production, transportation, storage over time, and distribution. The model includes link labor productivity enhancements that allow for investments in enhanced safety and health of the workers. A net present value (NPV) approach is utilized with the optimization model being formulated and solved as a variational inequality problem. Complete input and solution results for a series of numerical examples are reported in terms of product flows, labor required, wages, as well as link labor productivity enhancements. The relevance of the framework is further supported through sensitivity analysis. It is found that investing in link labor productivity can benefit workers in terms of higher wages earned and the firm in terms of an increase in the optimal value of its objective function represented by the NPV, provided that consumers are aware of and responsive to such investments.

8.1 Introduction This is a historic, very challenging era with the COVID-19 pandemic and the devastating impacts of climate change, with the latter fueling an increasing number of various disasters from hurricanes to heatwaves and wildfires (Samenow and Patel 2021). In addition to such “sudden-onset” disasters, climate change is also exacerbating droughts and rainfall in different areas and affecting agriculture and the food supply (Cartier 2021). Furthermore, the increasing emissions due to the use of fossil fuels are reducing the quality of life and the health of people (Lelieveld et al. 2019). In our highly connected planet, viruses can spread easily through travel (Van Beusekom 2020), and smoke, due to wildfires, can result in unhealthy air thousands © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_8

181

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8 Multiperiod Supply Chain Network Investments in Labor Productivity

of miles from the origin locations (cf. Rott 2021). Furthermore, the growing number of conflicts around the globe as well as Russia’s war against Ukraine is fueling additional uncertainty and disruptions to supply chains (D’Agostino 2022). Many firms and organizations now realize that they have to invest in labor for health and safety reasons (International Labour Organization 2020). Not everyone can work from home and that includes essential workers from various manufacturers, farmers and food processors, freight service providers, technicians, first responders, as well as healthcare workers, among others. Furthermore, working conditions, plus higher wages, have been critical in attracting and keeping workers throughout the pandemic, even now, as more and more people are vaccinated and as economies begin to gradually open up (Rosenberg 2021). It is important to realize that climate change and COVID-19 are actually linked, in that it has been found that smoke from wildfires in the Pacific Northwest may have led to thousands more cases of COVID-19 and more deaths among those who tested positive for the coronavirus (Partlow 2021). Early on in the pandemic, the large number of cases of COVID-19, as well as deaths of workers, including those in the food sector, from workers at meat processing plants in the USA to migrant farm labor in many countries, captured the attention of many, including consumers who dealt with empty store shelves, a reduced portfolio of products, as well as higher prices of grocery items. Companies realized that they needed to invest in enhanced coronavirus mitigation procedures in their workplaces, from social distancing and the use of PPEs, to the implementation of various barriers, enhanced ventilation, etc. (Pacheco 2020). Such investments were seen in various grocery stores, manufacturing plants, restaurants, various retail outlets, and other work venues (cf. Lindsay 2020). The importance of labor to the production, transportation, storage, and distribution of products in supply chain networks has resonated with producers and consumers alike. Keeping workers healthy and safe affects profits positively as well as a company’s reputation, since the negative fallout from widely publicized illnesses at food processing plants, factories, distribution centers, and warehouses can have lasting effects. Similarly, with global warming and climate change, workers as well as firms are needing to adapt since higher temperatures are affecting the productivity of labor (see Day et al. 2019). de Lima et al. (2021) note that labor supply and labor productivity are sensitive to increasing heat stress expected under climate change, especially in economic sectors that depend heavily on outdoor work, notably, agriculture (Hertel and de Lima 2020; Sainato 2021). Being subjected to heat reduces the capability for physical activity in a plethora of work environments, and heat kills farm workers every year (Shapiro 2021). In this chapter, I continue with the development of an optimization model, as was done in Chapter 7, for labor productivity investments. A multiperiod supply chain network optimization model is constructed in which a firm seeks to determine its optimal enhancements in labor productivity associated with its supply chain network activities, at a cost, along with its optimal product path flows to the demand markets, under profit maximization, and using a net present value (NPV) approach. The operational costs depend not only on the product flow but also

8.1 Introduction

183

on the productivity enhancements, as was done in Chapter 7, since there may be maintenance costs encumbered over the time horizon. The demand price functions depend not only on the product demand at the demand markets but also on the productivity enhancements. Hence, the optimization model captures that consumers may be more (or less) sensitive to investments by a company in labor productivity. This feature has some analogues in the sustainability literature where consumers may be willing to pay higher prices for products that encumber lower emissions, that is, are more “green” (see, e.g., Saberi et al. 2018 and the references therein). Examples of enhancements associated with labor productivity can include better temperature-controlled work environments, regular access to water and restroom breaks, provision of PPEs and other types of sanitary supplies, better lighting, ventilation, and work area design, among others. The specific focus in this chapter is on labor productivity enhancements of the “hardware” variety, rather than education and teaching of new skills, since the overarching theme here is the pandemic healthcare disaster as well as the climate change one (see IPCC 2021) with a thread also including Russia’s war against Ukraine and impacts on labor. The supply chain network optimization model in this chapter, in addition to being a multiperiod one, differs from the models in the preceding chapters in the following ways: 1. A supernetwork (cf. Nagurney and Dong 2002) is used to abstract the multiperiod supply chain network optimization problem in order to precisely define the paths for the product path flows and to also capture the managerial activity associated with the management of the supply chains. 2. The link operational cost functions depend not only on the product flows but also on the link labor productivity enhancements made in the first time period, as do the demand price functions. The former capture the possible maintenance needed in subsequent time periods, whereas the latter reflect that consumers (to greater or lesser degrees) are willing to pay more for a product if workers have enhanced protection, which enables greater productivity. It is important to emphasize that the economics literature is rich with respect to labor issues (cf. Stiglitz 1982; Van Biesebroeck 2015; Wolters and Zilinsky 2015, and Karp 2021) but not in terms of supply chains and the integration of labor in supply chains, which I focus on here. This chapter also adds to the disaster literature. For collections of interesting papers on many disaster themes as well as phases of disaster management, see the edited volumes by Kotsireas et al. (2016), Kotsireas et al. (2018) and by Kotsireas et al. (2021). The chapter is organized as follows. In Section 8.2, the multiperiod supply chain network model with link labor productivity enhancements is constructed and the supernetwork topology identified. The variational inequality formulation of the problem is provided, which is used for algorithmic purposes. Section 8.3 presents the algorithm and a series of numerical examples for which the complete input and output results are reported. Section 8.3 includes additional results in the form of sensitivity analysis exercises in order to ascertain quantitatively the impact on

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8 Multiperiod Supply Chain Network Investments in Labor Productivity

the firm’s NPV of changes in the discount rate, modifications to the managerial link productivity factors (which are not subject to productivity enhancements), and changes in the demand price functions to reflect greater sensitivity in the demand price functions to the labor productivity enhancements. Section 8.4 presents a summary of the results in this chapter and the conclusions as well as suggestions for future research. Section 8.5 is the Sources and Notes Section for this chapter.

8.2 Multiperiod Supply Chain Network Optimization Model with Investments Consider a single firm, which seeks to maximize the NPV of its investments in labor productivity enhancements over a finite period planning horizon. The time horizon T is discretized, where t = 1, . . . , T . In each time period t, the firm has nM production sites available, nDC distribution centers at which storage can take place, and serves J demand markets. The multiperiod supply chain network topology is depicted in Figure 8.1, as a supernetwork, with the firm denoted as the super node 0. I make use of a supernetwork (cf. Nagurney and Dong 2002) since it makes the definitions of paths precise and allows for the inclusion of managerial links. The firm in period t is then denoted by node (0, t) with lower-tiered nodes corresponding, in time period t, respectively, to: production (M1,t , . . . , MnM ,t ), distribution (DC1,t , . . . , DCnDC ,t ), and the demand markets (1, t, . . . , J, t). Let Lt denote the set of directed links of the firm in its supply chain network, at time period t, where t = 1, . . . , T , with the corresponding link from node 0 to node (0, t) also included. Let Lˆt then denote the set of directed links of the firm in its supply chain network, at time period t, in the supernetwork in Figure 8.1 not including the top-most links joining firm node 0 to node (0, t). The links emanating from the node 0 are grouped into the set L 0 . Denote a typical link by a. The full supply chain supernetwork in Figure 8.1 is represented by the graph G = [N , L ], where N corresponds to the set of nodes and L to the set of links. Lˆ , in turn, consists of all links Lˆt , t = 1, . . . , T . The links in Lˆ represent the supply chain network operational links, whereas the links in L 0 are the managerial links. The managerial links represent the management of the supply chain networks that is needed in the different time periods. Investments can take place in any of the operational links but do not take place on the managerial links. The managerial links, however, encumber wages to be paid, as do the operational links for labor provision. Assume that investment in the supply chain links takes place in the first time period with those investments then affecting operational link costs and demand price functions in the first and subsequent time periods. Let Pk,t denote the set of paths joining the firm node 0 to demand market node (k, t) representing demand market k in period t, where t = 1, . . . , T , k = 1, . . . , J . The set of paths P, with the number of elements equal to nP , consists of all the paths Pk,t for all k and t. The paths are acyclic, and each path consists of a

8.2 Multiperiod Supply Chain Network Optimization Model with Investments

185

Fig. 8.1 The supernetwork structure of the multiperiod supply chain network model with labor productivity enhancements

sequence of links representing the supply chain network activities of production, transportation, and distribution to a demand market, within a time period, as well as storage, if the link is over two time periods. Each path also includes a managerial link. Assume that the production sites, storage sites, transportation, and distribution links, as well as the demand markets are fixed from time period to time period. Hence, the number of production sites, the number of distribution centers, and the number of demand markets as in time period 1 remain as in time period 1 throughout the time horizon. Furthermore, assume that investments are made only in the first time period. Similar assumptions in terms of supply chain investments, but not concerning labor, in multiperiod supply chain network models, have also been made by Liu and Cruz (2012), Saberi et al. (2018), Liu and Wang (2019), and Yu et al. (2022). The additional notation for the model is given in Table 8.1. All vectors are column vectors. The firm seeks to maximize the NPV of its investments in enhancing labor productivity over the planning horizon with the objective function denoted by U , that is,

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8 Multiperiod Supply Chain Network Investments in Labor Productivity

Table 8.1 Notation for the multiperiod supply chain network model Notation xp fa la dk,t

va1

Variables nP The product flow on path p; group all the path flows into the vector x ∈ R+ nL The product flow on link a; group all the link flows into the vector f ∈ R+ The labor available for link a activity, ∀a ∈ L , typically, denoted in labor hours The demand for the product at demand market k, k = 1, . . . , J , in time period t, t = 1, . . . , T ; group the demands for a time period i into the vector dt ∈ R J and TJ then group all such vectors for all the time periods into the vector d ∈ R+ ˆ The enhanced labor productivity on link a, ∀a ∈ L1 . Investments in link labor productivity enhancement only take place in the first period; group the labor nLˆ

wa Notation r γa vamax δap Notation cˆa (fa , va1 ) ρk,t (d, v 1 ) αa + va1

T C(v 1 )

Maximize

productivity enhancements into the vector v 1 ∈ R+ 1 The (hourly) wage paid for a unit of labor on link a, ∀a ∈ L Parameters NPV discount rate Positive factor relating wage to labor on link a, ∀a ∈ L The maximum investment possible in labor productivity on link a, ∀a ∈ Lˆ1 Indicator equal to 1, if link a is contained in path p and 0, otherwise Functions The operational cost associated with link a, excluding the labor cost, ∀a ∈ Lˆ ; cˆa (fa , va1 ) = 0 for all links a ∈ L 0 The demand price for the product at demand market k, k = 1, . . . , J in time period t, t = 1, . . . , T The link productivity function relating input of labor to product flow on link a, ∀a ∈ Lˆ . Let πa = αa + va1 , ∀a ∈ Lˆ . Note that πa is identical for all similar links across the time periods t = 1, . . . , T , since va1 is determined for all links a ∈ Lˆ1 in the first period. Link productivity functions for the managerial links a ∈ L 0 are as follows: πa = αa , ∀a ∈ L 0 The total cost associated with the labor productivity enhancements in vector v 1 . This function can take on different functional forms, provided that it is convex and continuously differentiable. In the numerical experiments, T C(v 1 ) =  1 2 a∈Lˆ1 (va )

U=

T t=1

−

⎧ ⎫ J ⎨ ⎬ 1 ρk,t (dt , v 1 )d(k,t) − cˆa (fa , va1 ) − wa la t ⎭ (1 + r) ⎩ a∈Lˆt

k=1



wa la − T C(v 1 )

a∈Lˆt

(8.1)

a∈L 0

subject to: p∈Pk,t

xp = dk,t ,

k = 1, . . . , J ; t = 1, . . . , T ,

(8.2)

8.2 Multiperiod Supply Chain Network Optimization Model with Investments

fa =



xp δap ,

∀a ∈ L ,

187

(8.3)

p∈P

xp ≥ 0,

∀p ∈ P, ∀a ∈ Lˆ1 ,

0 ≤ va1 ≤ vamax , fa = (αa + va1 )la ,

∀a ∈ Lˆ ,

(8.4) (8.5)

(8.6a)

fa = αa la ,

∀a ∈ L 0 ,

(8.6b)

la = γa w a ,

∀a ∈ Lˆ .

(8.7)

Constraints (8.2), (8.3), and (8.4) are the conservation of product flow equations. (8.2) guarantees that the demand for the product is met at each demand market in each time period. Equation (8.3) makes sure that the product flow on each link is equal to the sum of the product flows on paths that use that link. Equation (8.4) represents the nonnegativity assumption on the product path flows. The constraints (8.5) are the bounds on the labor productivity enhancements on the links, which take place in time period 1, and then are sustained on the identical links in each subsequent time period. There are upper bounds since there may be physical limitations as to what productivity enhancements can be achieved. The expressions (8.6a) and (8.6b), in turn, reflect that the product output is equal to the link productivity function times the labor in hours available for that link activity. Note that investments in link productivity enhancements are not allowed on managerial links. Note that by (8.6a) it is also meant that the link productivity function in each time period on a given link is the same over all time periods after time period 1 for the same link (cf. Figure 8.1) since the labor productivity link enhancement is determined in time period 1 for each link with the investment and then sustained throughout the planning horizon. In addition, here, according to (8.7), it is assumed that labor availability is linear with respect to the wage, which is consistent with the work of the Nobel Laureate Angus Deaton; cf. Deaton and Muellbauer (1981), where many references can be found as to such a frequently used structure by economists. Note that the total financial outlay, in turn, that is, the total investments in link labor productivity enhancements, is captured in the function T C(v 1 ) in (8.1). I now show that objective function (8.1) can be expressed solely in terms of product path flows and link labor productivity enhancements. Indeed, in view of (8.3), one can define link operational cost c˜a (x, va1 ) ≡ cˆa (fa , va1 ), for all links a ∈ Lˆ . Also, in view of (8.2), one can define demand price function ρ˜k,t (x, v 1 ) ≡ ρk,t (d, v 1 ), for all k and all t. Applying (8.6a), (8.6b), and (8.7) and, subsequently, (8.4), one concludes that

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8 Multiperiod Supply Chain Network Investments in Labor Productivity



xp δap

p∈P

w a la =

2

(αa + va1 ) 

w a la =

p∈P

xp δap

2

αa

1 , γa

∀a ∈ Lˆ ,

(8.8a)

1 , γa

∀a ∈ L 0 .

(8.8b)

Letting U˜ (x, v 1 ) ≡ U , one can now rewrite (8.1) as U˜ (x, v 1 ) =

T t=1

−

J 1 { ρ˜k,t (x, v 1 ) xp − c˜a (x, va1 ) t (1 + r)

a∈Lˆt

p∈Pk,t

k=1



p∈P

xp δap

2

(αa + va1 )

1 }− γa 0 a∈L

a∈Lˆt



p∈P

xp δap

2

αa

1 − T C(v 1 ), γa

(8.9) which is subject to constraints (8.4) and (8.5). The firm’s goal, hence, is to maximize (8.9) subject to constraints (8.4) and (8.5). Define the feasible set K ≡ {(x, v 1 )| (8.4) and (8.5) hold}. The feasible set K is convex. Also, assume that the NPV function U˜ (x, v 1 ) is continuously differentiable and concave. Then, it follows that an optimal solution to the above supply chain network optimization problem coincides with the solution of the variational inequality problem (cf. Kinderlehrer and Stampacchia 1980 and Nagurney 1999): Determine (x ∗ , v 1∗ ) ∈ K such that −

∂ U˜ (x ∗ , v 1∗ ) ∂ U˜ (x ∗ , v 1∗ ) × (xp − xp∗ ) − × (va1 − va1∗ ) ≥ 0, ∂xp ∂va1

p∈P

a∈Lˆ

∀(x, v 1 ) ∈ K.

(8.10)

Variational inequality (8.10) can be put into standard form (cf. (2.14)). In the supernetwork model, X ≡ (x, v 1 ) and F (X) ≡ (F 1 (X), F 2 (X)), where the p-th ˜ (x,v 1 ) ˜ (x,v 1 ) element of F 1 (X) is − ∂ U∂x and the a−th element of F 2 (X) is − ∂ U∂v . In the 1 p a model, N = nP + nLˆ and K = K. 1 The algorithm is now highlighted and then applied to compute solutions to several numerical examples.

8.3 The Algorithm and Numerical Examples The algorithm that is implemented for the solution of numerical examples is the modified projection method of Korpelevich (1977). Please refer to Chapter 2 or to

8.3 The Algorithm and Numerical Examples

189

the Appendix. It is guaranteed to converge to the solution of a variational inequality (8.10), provided that the function F (X) that enters the variational inequality, as defined above, is monotone and Lipschitz continuous and that a solution exists. The modified projection method is implemented in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the computation of solutions to the numerical examples. The modified projection method is initialized as follows. The link productivity enhancements are set to 0.00. The initial demand at each market is set to 40 with the demand equally distributed among the paths terminating in each demand market. The convergence tolerance is 10−7 . This means that the modified projection method is considered to have converged when the absolute value of the difference between each of the variables (the path flows and the link labor productivity enhancements) at two successive iterations differs by no more than this value.

Fig. 8.2 Supernetwork topology of the multiperiod supply chain network numerical examples

190

8 Multiperiod Supply Chain Network Investments in Labor Productivity

8.3.1 Examples 8.1, 8.2, and 8.3 In this set of examples, there are two time periods, and the firm has two production facilities, a single distribution center for storage, and serves two demand markets. The supernetwork topology of this multiperiod supply chain network problem is depicted in Figure 8.2. A generic product is considered for which one can draw insights based on the numerical computations and results. The purpose is to demonstrate that the model is computable and provides a breadth of information. The complete example input data as well as solutions for Examples 8.1, 8.2, and 8.3 are reported, followed by results of sensitivity analysis. In Example 8.1, the link operational costs and the demand price functions are the same for each period, and the demand price functions do not depend on the link productivity enhancements. The latter reflect a scenario in which consumers do not care about the work environment of the workers associated with the supply chain network for the product. Example 8.2 then has the identical data to the data in Example 8.1 but with the consumers now being responsive to the link labor productivity enhancements, which I consider to be such that make workers more comfortable, in terms of enhanced safety and health, so they can be more productive. Example 8.3, in turn, has the identical data to the data in Example 8.2, but now I consider the scenario in which the consumers at the two demand markets in the second period are willing to pay a much higher price for the product than they did in the first time period. Hence, the demand price functions reflect this. The scenario in Example 8.3 could reflect a holiday purchasing scenario for a product or a product that is in greater demand in the pandemic or even during a war. Example 8.1 (Same Data in Time Period 2 as in Time Period 1; Demand Price Functions Independent of Link Labor Productivity Enhancements) The operational link cost functions are 2

cˆa (fa , va1 ) = 2fa2 + .05va1 ,

2

2

cˆb (fb , vb1 ) = 2fb2 + .1vb1 ,

cˆc (fc , vc1 ) = .5fc2 + .05vc1 ,

2

2

2

2

cˆh (fh , vh1 ) = 2fh2 + .05va1 ,

cˆi (fi , vb1 ) = 2fi2 + .1vb1 ,

2

cˆk (fk , vd1 ) = .5fk2 + .05vd1 ,

2

cˆl (fl , ve1 ) = 2fl2 + .1ve1 ,

cˆd (fd , vd1 ) = .5fd2 +.05vd1 , cˆe (fe , ve1 ) = fe2 +2fe +.1ve1 , cˆf (ff , vf1 ) = .5ff2 +.1vf1 , cˆg (fg , vg1 ) = .5fg2 + .1vg1 ,

cˆj (fj , vc1 ) = .5fj2 + .05vc1 ,

2

2

cˆm (fm , vf1 ) = .5fm2 + .1vf1 , The demand price functions are

cˆn (fn , vn1 ) = 0.00,

2

2

cˆo (fo , vo1 ) = 0.00.

8.3 The Algorithm and Numerical Examples

191

ρ1,1 (d) = −5d1,1 + 800,

ρ1,2 (d) = −5d1,2 + 850.

ρ2,1 (d) = −5d2,1 + 800,

ρ2,2 (d) = −5d2,2 + 850.

The alpha link parameters are αa = 55,

αb = 50,

αc = 35,

αd = 35,

αe = 60,

αf = 38,

αg = 36,

αh = 55,

αi = 50,

αj = 35,

αk = 35,

αl = 60,

αm = 38,

αn = 10,

αo = 10, and the gamma link parameters are γa = .1, γh = .1,

γb = .1,

γc = .09,

γd = .07,

γe = .08

γf = .06,

γg = .08,

γi = .1,

γj = .09,

γk = .07,

γl = .08

γm = .06,

γn = .1,

γo = .1. The total investment cost function in productivity enhancements on the link is  2 T C = a∈Lˆ (va1 ) . 1 The bounds on the link productivity enhancements are vamax = . . . = vgmax = 5. The discount rate r = .03 (see Saberi et al. 2018 who also has used this value and provides a justification). The paths are defined as: path p1 = (n, a, c, e), path p2 = (n, b, d, e), path p3 = (n, a, c, f ), path p4 = (n, b, d, f ), path p5 = (n, a, c, g, l), path p6 = (n, b, d, g, l), path p7 = (o, h, j, l), path p8 = (o, i, k, l), path p9 = (n, a, c, g, m), path p10 = (n, b, d, g, m), path p11 = (o, i, k, m), and path p12 = (o, i, k, m). The modified projection method yields the following equilibrium product path flow pattern: xp∗1 = 22.21,

xp∗2 = 22.17,

xp∗5 = 0.00,

xp∗6 = 0.00,

xp∗7 = 22.19,

xp∗8 = 22.15,

xp∗10 = 0.00,

xp∗11 = 26.52,

xp∗12 = 26.48.

xp∗9 = 0.00,

xp∗3 = 26.54,

xp∗4 = 26.50,

The equilibrium link flows and labor values are reported in Table 8.2, whereas the equilibrium link labor productivity enhancements and hourly wages are reported in Table 8.3. The demand price at the first demand market in time period 1 is 578.09, and at the second demand market, the price is 584.79. The demand price at the first demand

192

8 Multiperiod Supply Chain Network Investments in Labor Productivity

market in time period 2 is 578.31, and at the second demand market, the price is 585.02 with the corresponding equilibrium demands of 44.38, 53.04, 44.34, and 53.00, respectively. The value of the firm’s NPV (the objective function) is 77,010.94. There is no inventorying from time period 1 to time period 2 since fg∗ = 0. The equilibrium link flows in time period 2 are lower than their respective values in time period 1. The wages also decrease on each of the operational supply chain network links in time period 2 as compared to the respective values in time period 1. All the equilibrium link labor productivity enhancements are positive except for the inventorying link one that has vg1∗ = 0, which makes sense since the inventorying link is not used/needed in this example. Example 8.2 (Same Data as in Example 8.1 But the Demand Price Functions Now Depend on the Link Productivity Enhancements) The data for Example 8.2 are identical to those in Example 8.1 but with the demand price functions at the demand markets in the two time periods expanded to include dependency on the link productivity enhancements. This is to model the possible scenario of consumers caring about the working conditions of the laborers in terms of health and safety, for example. The demand price functions are now ρ1,1 (d, v 1 ) = −5d1,1 + 10



va1 + 800,

ρ1,2 (d, v 1 ) = −5d1,2 + 10

a∈Lˆ1

ρ2,1 (d, v 1 ) = −5d2,1 + 10





va1 + 850,

a∈Lˆ1

va1 + 800,

ρ2,2 (d, v 1 ) = −5d2,2 + 10

a∈Lˆ1



va1 + 850.

a∈Lˆ1

The modified projection method yields the following equilibrium product path flow pattern: xp∗1 = 22.46,

xp∗2 = 22.42,

xp∗5 = 0.00,

xp∗6 = 0.00,

xp∗7 = 22.44,

xp∗8 = 22.40,

xp∗10 = 0.00,

xp∗11 = 26.79,

xp∗12 = 26.75.

xp∗9 = 0.00,

xp∗3 = 26.81,

xp∗4 = 26.78,

The equilibrium link flows and labor values are reported in Table 8.2, whereas the equilibrium link labor productivity enhancements and hourly wages are reported in Table 8.3. The demand price at the first demand market in time period 1 is now 584.39, and at the second demand market, the price is 617.26. The demand price at the first demand market in time period 2 is 584.61, and at the second demand market, the price is 591.10 with the corresponding equilibrium demands of 44.81, 53.51, 44.77, and 53.46, respectively.

8.3 The Algorithm and Numerical Examples Table 8.2 Equilibrium link flows and labor values for Examples 8.1, 8.2, and 8.3

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗ fg∗ fh∗ fi∗ fj∗

fk∗ fl∗ fm∗ fn∗ fo∗ la∗ lb∗ lc∗ ld∗ le∗ lf∗ lg∗ lh∗ li∗ lj∗

lk∗ ll∗ ∗ lm ln∗ lo∗

193 Equilibrium value Example 8.1 Example 8.2 48.75 49.27 48.68 49.20 48.75 49.27 48.68 49.20 44.38 44.88 53.04 53.59

Example 8.3 67.17 67.08 67.17 67.08 37.12 45.13

0.00 48.70 48.63 48.70

0.00 49.22 49.15 49.22

51.99 79.06 78.94 79.06

48.63 44.34 53.00 97.42 97.33 0.88 0.97 1.37 1.36 0.74 1.37

49.15 44.84 53.54 98.47 98.38 0.90 0.98 1.41 1.41 0.75 1.38

78.94 98.26 111.73 134.25 158.00 1.22 1.34 1.92 1.92 0.62 1.17

0.00 0.88 0.97 1.37

0.00 0.89 0.98 1.41

1.42 1.44 1.58 2.26

1.36 0.74 1.37 9.74 9.73

1.40 0.75 1.38 9.85 9.84

2.26 1.64 2.90 13.42 15.80

The equilibrium link product flows, the link labor values, as well as the link wages are all higher in Example 8.2 than in Example 8.1. The equilibrium demand market prices that consumers now pay for the product are greater in time period 2 than in time period 1 and the demands are higher. The link productivity enhancements are positive on links e and f in this example. The firm enjoys an NPV of 80,030.05, which exceeds the NPV in Example 8.1. The results for this example strongly suggest that letting consumers know that a firm cares about laborers and is willing to invest in their health and safety in the firm’s supply chain network, and having consumers be responsive to such enhancements, can have a positive impact on the firm’s value of its objective function.

194

8 Multiperiod Supply Chain Network Investments in Labor Productivity

Table 8.3 Equilibrium link productivity enhancements and hourly wages for Examples 8.1, 8.2, and 8.3

Notation va1∗ vb1∗ vc1∗ vd1∗ ve1∗ vf1∗

Equilibrium value Example 8.1 Example 8.2 0.14 0.00 0.18 0.00 0.58 0.00 0.73 0.00 0.11 0.11 0.76 0.77

Example 8.3 0.00 0.00 0.00 0.00 0.07 0.56

vg1∗ wa∗ wb∗ wc∗ wd∗ we∗ wf∗

0.00 8.84 9.70 15.22 19.46 9.23 22.81

0.00 8.96 9.84 15.64 20.08 9.33 23.04

0.66 12.21 13.42 21.32 27.38 7.72 19.51

0.00 8.83 9.69 15.21

0.00 8.95 9.83 15.63

18.26 14.37 15.79 25.10

19.44 9.22 22.79 97.42 97.33

20.06 9.32 23.02 98.47 98.38

32.22 20.45 48.30 134.25 158.00

wg∗ wh∗ wi∗ wj∗

wk∗ wl∗ ∗ wm wn∗ wo∗

Example 8.3 (Same Data as in Example 8.2 But with the Demand Price Function Intercepts Increased in the Second Time Period) The data in Example 8.3 are identical to those in Example 8.2 but with the demand price function intercepts increased in time period 2, reflecting an increase in the prices the consumers are willing to pay at the demand markets in the second time period. The demand price functions in the second time period are now ρ2,1 (d, v 1 ) = −5d2,1 + 10



va1 + 1600,

ρ2,2 (d, v 1 ) = −5d2,2 + 10

a∈Lˆ 1



va1 + 1650.

a∈Lˆ 1

The modified projection method yields the following equilibrium product path flow pattern: xp∗1 = 18.57,

xp∗2 = 18.55,

xp∗3 = 22.58,

xp∗4 = 22.56,

xp∗5 = 11.33,

xp∗6 = 11.30,

xp∗7 = 37.84,

xp∗8 = 37.79,

xp∗10 = 14.67,

xp∗11 = 41.21,

xp∗12 = 41.15.

xp∗9 = 14.69,

8.3 The Algorithm and Numerical Examples

195

The equilibrium link flows and labor values are reported in Table 8.2, and the equilibrium link labor productivity enhancements and hourly wages are reported in Table 8.3. The demand price at the first demand market in time period 1 is 627.34, and at the second demand market, the price is 676.08. The demand price at the first demand market in time period 2 is 1121.63, and at the second demand market, the price is 1104.30 with the corresponding equilibrium demands of 37.12, 45.13, 98.26, and 111.73, respectively. The firm’s value of its NPV at the optimal solution in Example 8.3 is equal to 197,428.52. In Example 8.3, the NPV is more than double the NPV in Example 8.2. Inventorying now takes place with fg∗ = 51.99. Wages now increase on all links. As in the previous two examples, those involved in the managerial links n and o earn the highest wages. Three links now have positive equilibrium productivity enhancement values at the equilibrium with the inventorying link g having the highest value with vg1∗ = 0.66. 8.3.1.1

Sensitivity Analysis

I now proceed to conduct several sensitivity analysis exercises. In the first exercise, I evaluate the effects of varying the discount rate on the NPV of the firm. In the second exercise, I explore the impact on the NPV of the firm of changing the αn and αo factors on the managerial  links. In the final sensitivity analysis exercise, I modify the factor preceding the a∈Lˆ va1 term in the demand price functions and examine 1 the impact on the NPV of the firm. Example 8.3 is used as the baseline for these sensitivity analysis exercises with the data changes for each sensitivity analysis data point as noted on the x axis of Figures 8.3, 8.4, and 8.5. The results of the first sensitivity analysis are reported in Figure 8.3.

210 205

103

200 195

Firm NPV

+

190 185 180 175 170 0.01

0.03

0.05 Discount Rate r

0.07

0.09

Fig. 8.3 Sensitivity analysis for different discount rates r and effects on the firm’s NPV

196

8 Multiperiod Supply Chain Network Investments in Labor Productivity

From Figure 8.3, it can be seen that the NPV of the firm decreases linearly as the discount rate is raised. The fact that the firm’s NPV decreases as the discount rate increases is the behavior that one would expect from corporate finance. The results of the second sensitivity analysis are reported in Figure 8.4. As has been emphasized earlier in this chapter, one of the novel contributions herein is the use of a supernetwork for representing the multiperiod supply chain network optimization problem with link labor productivity enhancements through investments. The use of a supernetwork, with the addition of links to represent managerial activities, provides a richer framework for supply chain management since the wages of management with their hourly labor values and also productivity are now included. Also, recall that for the managerial links there are no link labor productivity enhancements and, therefore, the productivity factors on the managerial links n and o in our numerical examples consist solely of the factors αo and αn . In Figure 8.4, one sees that as the values of these managerial link factors, which are the same on the managerial link in time period 1 and in time period 2, increase, then the NPV of the firm also increases, but at a decreasing rate over the range that this sensitivity analysis exercise was conducted. 204

103

202

Firm NPV

+

200

198

196

10

20 30 40 α on Managerial Links n and o

50

Fig. 8.4 Sensitivity analysis for different values of α for managerial links o and n and effects on the firm’s NPV

The final sensitivity analysis results are reported in Figure 8.5. In this exercise, all the demand price function coefficients preceding the a∈Lˆ va1 term were the same 1 and as delineated in the x axis in Figure 8.5. Note that the result for the NPV for 10 is precisely that for Example 8.3. Here, one sees that the greater the value of this coefficient, the greater the NPV of the firm. This means that the firm should make sure that consumers respond to the investments of the firm in labor productivity enhancement and are made aware of the investments.

8.4 Summary, Conclusions, and Suggestions for Future Research

197

8.4 Summary, Conclusions, and Suggestions for Future Research Supply chain networks serve as the critical infrastructure for the production, transportation, storage, and distribution of products globally. The COVID-19 pandemic, and the increase in climate-related disasters and their severity, along with global conflicts and wars, have demonstrated the need to maintain the functionality of supply chains. Essential to the functionality of supply chains is labor. Indeed, without labor, no product can be produced, transported, stored, and distributed. The productivity of labor, in turn, depends on workers being healthy and safe.

103

198

Firm NPV

+

196

194

192

0

2

4

6

10

8

Factor Preceding v1 Expression in Each Demand Price Function

Fig. 8.5 Sensitivity analysis for different values of factor preceding the demand price functions and effects on the firm’s NPV



a∈Lˆ1

va1 expression in all

In this chapter, I present a multiperiod supply chain network optimization model, using the firm’s NPV, and a supernetwork formalism, to identify both optimal product flows as well as labor productivity enhancements. A novelty of the model is that not only are operational links included but managerial links are as well since management needs to be involved in supporting the supply chain activities over the time periods. The model is formulated and solved as a variational inequality problem. The solution of the model yields optimal product path and link flows, optimal labor values associated with the various supply chain activities, wages that should be paid, as well as the optimal productivity enhancements. The numerical examples provide insights that include that both the firm as well as the laborers can gain when a firm invests in labor productivity enhancement, which here is considered to enhance health and safety. Future research can include the construction of multiperiod game theory models with the inclusion of labor that can also address the competition for labor, another vivid feature of the pandemic. It would also be very interesting to formulate service

198

8 Multiperiod Supply Chain Network Investments in Labor Productivity

supply chains, as in the healthcare sector, to include labor specifically. We address this issue, in part, in Chapter 11. It would be interesting to consider extensions of the model in this chapter (as well as related ones) to allow for the number of demand markets to change from time period to time period, perhaps, with appropriate additional investments in different time periods. It would also be interesting to allow the number of production sites and distribution centers to change from time period to time period, with an increase in number having an associated investment (but note that the type of investment would be different from the investments in the model in this chapter, which are for productivity of labor). Finally, allowing for investments in each time period in terms of labor productivity would also be a valuable extension to the model in this chapter.

8.5 Sources and Notes This chapter is based on the paper by Nagurney (2021) with updates and standardization of notation as well as linkages to other chapters in this book. This chapter showcases the first supply chain network optimization model with labor and multiple time periods, with a focus on enhancing labor productivity, which is a topic of great current relevance in the pandemic and climate change, as well. Chapter 2 constructs a supply chain network optimization model for perishable products, with a focus on food, in which there were bounds on the availability of labor on each link. This allows for the quantification of the impacts of disruptions on labor availability in different supply chain network activities on product flows, consumer prices, and the profit of the firm. Chapter 3 proposes another supply chain network optimization model with labor and three distinct sets of constraints, with the first consisting of upper bounds on labor availability on links, the second set consisting of an upper bound on labor availability on each tier of the supply chain network (production, transportation, storage, and distribution), and the third set of constraints consisting of a single labor bound in the supply chain network. Chapter 4 then extends the contributions in Chapter 3 to competing multiple firms using Nash equilibrium (cf. Nash 1950, 1951) for the model under the first set of constraints and generalized Nash equilibrium as a concept for the second two sets of constraints, where firms competed for a limited amount of labor resources. All the above noted models, however, are single period models, whereas the model in this chapter is a multiperiod one. Chapter 5 contains a single period supply chain game theory model with labor but with fixed amounts of labor on the links. Recall that, in that model, production output increases as a function of wages, and there are no investments in enhanced labor productivity. As already noted, Chapter 7 also considers a single period model, with investments in labor productivity, but the investments do not affect the demand price functions and the financial outlay has a specific form, unlike the general construct in this chapter with respect to the total cost associated with productivity enhancements, focusing on health and safety of workers. Addressing the health and safety of laborers is now of paramount

References

199

importance with the double major stressors of the COVID-19 pandemic and climate change as well as the impacts on workers of conflicts and wars. Olmos (2021) reveals how outdoor laborers involved in picking produce, who already have difficult jobs, are suffering and even dying because of the extreme heat conditions, including in the Pacific Northwest of the USA.

References Cartier, K.M.S., 2021. Global agriculture will be drastically altered by climate change. GreenBiz, February 18. D’Agostino, S., 2022. Global hunger crisis looms as war in Ukraine sends food prices soaring. Bulletin of Atomic Scientists, April 12. Day, E., Fankhauser, S., Kingsmill, N., Costa, H., Mavrogianni, A., 2019. Upholding labour productivity under climate change: An assessment of adaptation options. Climate Policy, 19(3), 367–385. de Lima, C.Z, Buzan, J.R., Moore, F.C., Lantz, U., Baldos, C., Huber, M., Hertel, T.W., 2021. Heat stress on agricultural workers exacerbates crop impacts of climate change. Environmental Research Letters, 16, 044020. Deaton, A., Muellbauer, J., 1981. Functional forms for labor supply and commodity demands with and without quantity restrictions. Econometrica, 49(6), 1521–1532. Hertel, T.W., de Lima C.Z., 2020. Viewpoint: Climate impacts on agriculture: Searching for keys under the streetlight. Food Policy, 95, 101954. International Labour Organization, 2020. In the face of a pandemic: Ensuring safety and health at work. Geneva, Switzerland. IPCC, 2021. Sixth assessment report. Released August 9, Geneva, Switzerland. Karp, E., 2021. The case for raising wages in manufacturing. Forbes, March 2. Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York. Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. Kotsireas, I.S., Nagurney, A., Pardalos, P.M., Editors, 2016. Dynamics of Disasters: Key Concepts, Models, Algorithms, and Insights. Springer International Publishing Switzerland. Kotsireas, I.S., Nagurney, A., Pardalos, P.M., Editors, 2018. Dynamics of Disasters: Algorithmic Approaches and Applications. Springer International Publishing Switzerland. Kotsireas, I.S., Nagurney, A., Pardalos, P.M., Tsokas, A., Editors, 2021. Dynamics of Disasters: Impact, Risk, Resilience, and Solutions. Springer Nature Switzerland AG. Lelieveld, J., Klingmüller, K., Pozzer, A., Burnett, R.T., Haines, A., Ramanathan, V., 2019. Effects of fossil fuel and total anthropogenic emission removal on public health and climate. PNAS, 116(15), 7192–7197. Lindsay, C., 2020. Health and safety move to the fore of workplace issues. The Wall Street Journal, October 13. Liu, Z., Cruz, J.M., 2012. Supply chain networks with corporate financial risks and trade credits under economic uncertainty. International Journal of Production Economics, 137(1), 55–67. Liu, Z., Wang, J., 2019. Supply chain network equilibrium with strategic supplier investment: A real options perspective. International Journal of Production Economics, 208, 184–198. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A., 2021. A multiperiod supply chain network optimization model with investments in labor productivity enhancements in an era of COVID-19 and climate change. Operations Research Forum, 2, 68.

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Nagurney, A., Dong, J., 2002. Supernetworks: Decision-Making for the Information Age. Edward Elgar Publishing, Cheltenham, United Kingdom. Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49. Nash, J.F., 1951. Noncooperative games. Annals of Mathematics, 54, 286–298. Olmos, S., 2021. Hard jobs turn perilous on farms. The New York Times, September 5. Pacheco, I., 2020. How much Covid-19 cost those businesses that stayed open. The Wall Street Journal, June 23. Partlow, J., 2021. ‘This is a very dangerous combination’: New study says wildfire smoke linked to increased covid cases, deaths. The Washington Post, August 13. Rosenberg, E., 2021. These businesses found a way around the worker shortage: Raising wages to $15 an hour or more. The Washington Post, June 10. Rott, N., 2021. Study finds wildfire smoke more harmful to humans than pollution from cars. NPR, March 5. Saberi, S., Cruz, J.M., Sarkis, J., Nagurney, A., 2018. A competitive multiperiod supply chain network model with freight carriers and green technology investment option. European Journal of Operational Research, 266(3), 934–949. Sainato, M., 2021. ‘We’re not animals, we’re human beings’: US farm workers labor in deadly heat with few protections. The Guardian, July 16. Samenow, J., Patel, K. 2021. Extreme weather tormenting the planet will worsen because of global warming, U.N. panel finds. The Washington Post, August 9. Shapiro, E., 2021. Farmworkers especially vulnerable as dangerous heat wave scorches wide swath of US. ABC News, June 20. Stiglitz, J.E., 1982. Alternative theories of wage determination and unemployment: The efficiency wage model. In: The Theory and Experience of Economic Development - Essays in Honor of Sir W. Arthur Lewis. M. Gersovitz, C.F. Diaz-Alejandro, G. Ranis, and M.R. Rosenzweig, Editors, George Allen & Unwin, London, England, pp. 78–106. Van Beusekom, M., 2020. Studies trace COVID-19 spread to international flights. CIDRAP, University of Minnesota, Minneapolis, Minnesota, September 21. Van Biesebroeck, J., 2015. How tight is the link between wages and productivity? A survey of the literature. International Labour Office, Geneva, Switzerland. Wolters, J., Zilinsky, J., 2015. Higher wages for low-income workers lead to higher productivity. Peterson Institute for International Economics, January 13. Yu, M., Cruz, J.M., Li, D., Masoumi, A.H., 2022. A multiperiod competitive supply chain framework with environmental policies and investments in sustainable operations. European Journal of Operational Research, 300(1), 112–123.

Part III

Advanced Supply Chain Network from Profit to Non-Profit Organizations

Chapter 9

Multitiered Supply Chain Networks with Labor

Abstract In this chapter, multitiered supply chain network equilibrium models are constructed that include labor associated with supply chain network link activities. The supply chain consists of multiple competing manufacturers, multiple competing retailers, plus consumers located at demand markets. Bounds on labor availability are included along with labor productivity factors. The formulation, analysis, and solution of the multitiered supply chain network equilibrium models make use of variational inequality theory. The multitiered supply chain network framework allows for the quantification of various disruptions on the equilibrium product flows, the product prices at the various tiers of the supply chain network, as well as the shadow prices (Lagrange multipliers) associated with the labor constraints. Numerical examples, for which complete input and output data are reported, illustrate the types of insights that can be gained.

9.1 Introduction Supply chains that link spatially separated producers through transportation with consumers around the globe depend on labor. The pandemic has resulted in both supply shocks as well as demand shocks, coupled with immense challenges to transportation provision, as workers became ill from COVID-19, many manufacturing plants were shuttered; freight services were disrupted, and shopping patterns were transformed as people worked, learned, and shopped at home (see del Rio-Chanona et al. 2020). Shortages of various products arose, including those of essential products, such as PPEs and sanitation items, as well as computer chips, and prices for many products increased (Kowalick 2021). Businesses, that could, including retailers, such as grocery stores, and restaurants, pivoted to electronic commerce for safety reasons and in order to sustain their operations (Perez 2020). Numerous trade measures were enacted by countries seeking to preserve critical product availability for their denizens (cf. Nagurney et al. 2022). Supply chains are being reenvisioned and redesigned with an eye toward greater resiliency (see Iakovou and White 2020), a topic that is discussed also in Chapters 3 and 4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_9

203

204

9 Multitiered Supply Chain Networks with Labor

What has been missing in many supply chain network models and analyses, however, and has been brought into focus in the COVID-19 pandemic, is the importance of including labor as an essential resource. Motivated by the need to be able to quantify the impact of disruptions to labor, brought about by the pandemic, and associated with various links of manufacturing, transportation, storage, and distribution, this book presents a series of optimization and game theory models that include labor, along with constraints on labor in terms of various bounds. In this chapter, the first multitiered supply chain network equilibrium models with labor are presented. Specifically, a model without bounds on labor on manufacturing, on the handling and storage of the products, and on the servicing of customers at the retailers, is first described. Then, the model is extended to include bounds on labor on such links. The models capture the profit-maximizing behavior of spatially separated producers/manufacturers, and of the retailers, plus the behavior of consumers. I utilize, as the foundation, the first supply chain equilibrium model due to Nagurney et al. (2002), but with significant extensions, and inspired by the pandemic and the realities of the world today. Manufacturers pay the workers an hourly wage and the workers have an associated productivity factor. The retailers also pay their workers an hourly wage for their activities, which can differ according to whether the tasks deal with the handling of the product or with the servicing customers. The framework allows for the quantification of disruptions due to labor shortfalls on the profits of firms at different tiers, the product flows, and the demand market prices, plus the prices that the manufacturers charge and the prices that the retailers charge. The novelty of the contributions in this chapter are: (1) This is the first set of supply chain network equilibrium models with multiple tiers of decision-makers that includes labor. (2) It is the first set of supply chain network equilibrium models that explicitly includes the behavior of retailers and also incorporates costs associated with consumer purchasing. (3) It is the first time that productivity factors associated with labor at manufacturing sites, with labor at storage/handling facilities of retailers, and the servicing of customers by retailers are included in a supply chain network model. (4) An easy to implement algorithm, the Euler Method, is proposed and utilized for the computation of the equilibrium patterns. It has not previously been applied to solve supply chain network models with labor. The Euler Method, which is induced by the general iterative scheme of Dupuis and Nagurney (1993), can be viewed as a discrete-time algorithm for the approximation of trajectories associated with the projected dynamical system, whose set of stationary points corresponds to the set of solutions to the associated variational inequality problem. (5) The solved examples reveal the impacts of disruptions in terms of decreases in labor availability as well as decreases in labor productivity, on other tiers and stakeholders, providing a general, holistic representation of actual supply chains.

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor

205

The organization of the chapter is as follows. The models are constructed in Section 9.2, with their variational inequality formulations. In Section 9.3, the Euler Method, which has nice features for easy implementation, is stated, along with its full realization for the solution of the proposed models. Numerical examples, for which the complete computed solutions are reported, are detailed in Section 9.4. Section 9.5 summarizes the results, presents conclusions, and provides suggestions for future research. Section 9.6 is the Sources and Notes Section for this chapter.

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor I now construct the multitiered supply chain network equilibrium models with labor. In Section 9.2.1, the model without link labor bounds is presented. In Section 9.2.2, I then extend the model in Section 9.2.1 to include bounds on the availability of labor. The latter model allows for the investigation of disruptions to labor, as is happening in the 2020s, and also reveals in which supply chain network economic activities investment would be beneficial. The framework uses as the basis the classical model of Nagurney et al. (2002), which introduced the concept of a multitiered supply chain network equilibrium. This work has generated much interest and further advances (cf. Nagurney 2006, Yamada et al. 2011, Nagurney et al. 2013, Qiang et al. 2013, Toyasaki et al. 2014, Nagurney and Li 2016, and the references therein). There are m spatially separated manufacturers, n retailers, also spatially separated, and consumers located at o distinct demand markets. Denote a representative manufacturer by i, a representative retailer by j , and a typical demand market by k. The manufacturers produce a homogeneous product, which they then have transported to the retailers. The retailers handle and store the products from the various manufacturers and interact with the consumers. Both manufacturers and retailers are assumed to be profit-maximizers. The consumers, take their unit transaction costs under consideration, and the product prices at demand markets, in making their purchasing decisions. With each supply chain network economic activity of manufacturing, handling and storage, and servicing consumers at the demand markets in retail outlets, is associated labor, with labor quantified in terms of labor hours. In addition, as in the preceding modeling chapters, I assume linear “production” functions with the supply chain network activities, so that the product flow associated with an activity is equal to the labor availability (in hours) times the respective positive productivity factor. Both manufacturers and retailers pay their workers an hourly wage, which is fixed. Refer to Figure 9.1 for a depiction of the topology. Vectors are all column vectors.

206

9 Multitiered Supply Chain Networks with Labor

Fig. 9.1 The supply chain network topology

9.2.1 The Multitiered Supply Chain Network Equilibrium Model with Labor and No Bounds on Labor Availability I now discuss the manufacturers’ behavior, followed by the behavior of the retailers, and, finally, that of the consumers at the demand markets. The complete supply chain network equilibrium model with labor is then built with the variational inequality formulation of the governing equilibrium conditions.

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor

9.2.1.1

207

The Behavior of the Manufacturers and Their Optimality Conditions

Let qi1 denote the production output of the homogeneous product by manufacturer m. i and group the production outputs of all manufacturers into vector q 1 ∈ R+ Manufacturer i has a production cost function fi , such that: fi = fi (q 1 ),

i = 1, . . . , m.

(9.1)

The product amount transported from manufacturer i to retailer j is qij1 with all product shipments between the manufacturers and the retailers grouped into the mn-dimensional vector Q1 , where the superscript 1 corresponds to the manufacturers’ activities in the supply chain network in Figure 9.1. Associate with each manufacturer and retailer pair (i, j ) a transportation cost, cij . Transportation is outsourced by the manufacturers and does not involve the manufacturers’ workers. The transportation cost, cij , is:  cij = cij qij1 ,

i = 1, . . . , m; j = 1, . . . , n.

(9.2)

The quantity of the product manufactured by manufacturer i must satisfy the conservation of flow equation: qi1 =

n

qij1 .

(9.3)

j =1

This means that the quantity of the product manufactured by manufacturer i is equal to the sum of the product quantities transported from the manufacturer to the retailers. Equation (9.3) must hold for each manufacturer i; i = 1, . . . , m. Noting (9.1) and (9.3), without loss of generality, one can rewrite the production cost functions as fi = fi (Q1 ); i = 1, . . . , m. Associated with the manufacturing activity of manufacturer i, i = 1, . . . , m, is li1 , which denotes the labor hours available for manufacturing at firm i. The positive parameter αi1 is the productivity parameter associated with production at manufacturer i, where qi1 = αi1 li1 ,

(9.4)

or, with the use of (9.3): n j =1

qij1 = αi1 li1 .

(9.5)

208

9 Multitiered Supply Chain Networks with Labor

According to (9.4) (or (9.5)), the quantity of the product produced at manufacturer i is equal to the labor hours available times the productivity of labor at manufacturer i. ∗ denote the price charged for the product by manufacturer i to retailer Let ρ1ij j , and let wi1 denote the hourly wage paid by manufacturer i at its production site. Then, one can state the objective function of profit maximization for manufacturer i, i = 1, . . . , m, as: Maximize

n

∗ ρ1ij qij1 − fi (Q1 ) −

j =1

n

 cij qij1 − wi1 li1 ,

(9.6)

j =1

subject to qij1 ≥ 0, for all j = 1, . . . , n, and equation (9.5) for j = 1, . . . , n. The first term in the objective function (9.6) is the revenue, whereas the subsequent two terms are the production cost and the total transportation costs, respectively, with the next term denoting the wages paid to workers engaged in manufacturing at manufacturer i. Using equation (9.5), one can rewrite the objective function (9.6) solely in terms of the product shipments, and, hence, the optimization problem of manufacturer i, i = 1, . . . , m, becomes: Maximize

n j =1

∗ ρ1ij qij1

− fi (Q ) − 1

n j =1

 cij qij1 − wi1

n

1 j =1 qij αi1

,

(9.7)

subject to qij1 ≥ 0, for all j = 1, . . . , n. Assume that the production cost functions are convex and that they also are continuously differentiable. The same assumptions are made for the transportation cost functions. The manufacturers compete in a noncooperative manner. The governing equilibrium concept underlying noncooperative behavior is that of Nash (1950, 1951). Hence, each manufacturer determines his optimal product shipments, given the optimal ones of his competitors, with the optimality conditions for all manufacturers simultaneously satisfying the variational inequality (cf. Bazaraa et al. 1993, Gabay and Moulin 1980); see also Dafermos and Nagurney mn satisfying: 1987, Nagurney 1999, and Nagurney et al. 2002): Determine Q1∗ ∈ R+   n m 1∗   ∂fi (Q1∗ ) ∂cij (qij ) wi1 ∗ 1 1∗ + + − ρ − q × q ij ij ≥ 0, 1ij ∂qij1 ∂qij1 αi1 i=1 j =1

mn ∀Q1 ∈ R+ .

(9.8) For completeness and clarity, I now identify the various link flows in the supply chain network in Figure 9.1 associated with the manufacturers. Specifically, note that associated with the production links joining node i with node Mi , i = 1, . . . , m, is the production output: qi ; i = 1, . . . , m, respectively, as well as the labor hours

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor

209

available: li1 ; i = 1, . . . , m. Also, associated with each transportation link (i, j ) joining node Mi with retailer node j , j = 1, . . . , n, is the product shipment: qij1 . 9.2.1.2

The Behavior of the Retailers and Their Optimality Conditions

The retailers transact both with the manufacturers, from whom they purchase the product for their outlets, and with the consumers, who are at the bottom tier of the network (cf. Figure 9.1) and are the “final” buyers of the products. Retailer j faces a handling cost that includes the display and storage (inventory) cost associated with the product. This cost is cj , where: cj = cj (Q1 ),

j = 1, . . . , n.

(9.9)

The retailers associate a price with the product at their retail outlet, which is ∗ , for retailer j , j = 1, . . . , n. These prices, as the prices associated denoted by ρ2j with the manufacturers, are endogenously determined in the model. Later, I show how they are recovered once the full model is solved. The retailers also employ workers, who handle the storage as well as the interactions, in terms of sales, with the consumers. Each retailer j , j = 1, . . . , n, has available lj2 hours of labor for handling the products with an associated positive productivity factor with this supply chain network activity of αj2 . In addition, the number of hours of labor availability in dealing with customers at demand market k is denoted by lj2k , with associated productivity factor of αj2k for k = 1, . . . , o. The retailer j , j = 1, . . . , n, is faced with the following labor equations: m

qij1 = αj lj2 ,

(9.10)

i=1

and qj2k = αj2k lj2k ,

k = 1, . . . , o.

(9.11)

The superscript 2 refers to the second tier of decision-makers in the supply chain in Figure 9.1, that is, the retailers. In addition, retailer j , j = 1, . . . , n, cannot sell more than he has in inventory and, therefore, the following inequality must also hold: o k=1

qj2k ≤

m i=1

qij1 ,

(9.12)

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9 Multitiered Supply Chain Networks with Labor

and the nonnegativity assumptions: qij1 ≥ 0,

i = 1, . . . , m,

(9.13)

qj2k ≥ 0,

k = 1, . . . , o.

(9.14)

and

Retailer j , j = 1, . . . , n, pays his workers engaged in handling and inventorying the product an hourly wage of wj2 . He also pays his workers engaged in servicing customers at the demand markets k, k = 1, . . . , o, an hourly wage of wj2k . Making use of expressions (9.10) and (9.11), I construct the optimization problem of retailer j , j = 1, . . . , n, thus: Maximize

∗ ρ2j

o

qj2k

− cj (Q ) − 1

k=1

m

m ∗ ρ1ij qij1

− wj2

i=1

1 i=1 qij αj2

−

o

q2 2 jk wj k 2 αj k k=1

(9.15) subject to: (9.12), (9.13), and (9.14). The first term in (9.15) is the revenue of retailer j acquired through sales. The second term is the handling/storage cost. The third term in (9.15) is the payout to the manufacturers for the product. The final two terms in (9.15) are, respectively, the wages paid to the workers by retailer j for their storage/handling activity and the wages paid for serving the customers at the demand markets and transacting with them. Referring to Figure 9.1, note that associated with each storage node link from 1 and the j to node Nj , j = 1, . . . , n, is the product inventory amount m q i=1 ij hours of available labor: lj2 , j = 1, . . . , n, respectively. Finally, associated with the customer service and transaction link joining node Nj with node k representing a demand market, for j = 1, . . . , n; k = 1, . . . , o, is the product volume transacted: qj2k and the hours of available labor: lj2k . Assume that the handling cost for each retailer is continuously differentiable and convex, and that the retailers also behave in a noncooperative manner in the sense of Nash. Then, it follows from Nagurney et al. (2002) that the optimality conditions for all the retailers coincide with the solution of the variational inequality: Determine ∗ mn+no+n (Q1 , Q2∗ , γ ∗ ) ∈ R+ satisfying:  n m ∂cj (Q1∗ ) ∂qij1  2 o n wj k

∗ + ρ1ij

i=1 j =1

+

j =1 k=1

αj2k

+

wj2

 ∗ − ρ2j



− γj∗ αj2

+ γj∗

  × qij1 − qij1∗

  × qj2k − qj2∗k

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor

+

 m n j =1

i=1

qij1∗

−

o

 qj2∗k

  × γj − γj∗ ≥ 0,

211

mn+no+n ∀(Q1 , Q2 , γ ) ∈ R+ ,

k=1

(9.16) where the term γj is the Lagrange multiplier (shadow price) associated with constraint (9.12) for retailer j ; γ is the n-dimensional vector of all multipliers, and Q2 is the no-dimensional vector of product flows between retailers and demand markets.

9.2.1.3

The Consumers at the Demand Markets and the Equilibrium Conditions

The behavior of the consumers is now discussed. They take into account in making their purchases not only the price charged for the product by the retailers but also the unit transaction cost, which includes the cost of transportation. Let cj k denote the unit transaction cost of obtaining the product by consumers at demand market k from retailer j , where recall that qj2k is the amount of the product bought from retailer j by consumers at k. Assume that the unit transaction cost is continuous and positive: cj k = cj k (Q2 ),

j = 1, . . . , n; k = 1, . . . , o.

(9.17)

Denote the price of the product at demand market k by ρ3k and the demand at demand market k by dk . The demand functions are continuous and are as follows: dk = dk (ρ3 ),

k = 1, . . . , o,

(9.18)

where ρ3 is the o-dimensional vector of demand market prices. This model can include both brick and mortar retailers as well as electronic commerce retailers with the costs (9.17) adapted accordingly. ∗ for Consumers take the price charged by retailers, which is denoted by ρ2j retailer j , plus the unit transaction cost associated with obtaining the product, in making their consumption decisions. Following the well-known spatial price equilibrium conditions (cf. Samuelson 1952, Takayama and Judge 1971, and Nagurney 1999), which have also been applied in Nagurney et al. (2002); see Nagurney (2006) and Nagurney and Li (2016) for additional references, the equilibrium conditions for consumers at demand market k take the form: For all retailers j , j = 1, . . . , n: & ∗ ρ2j

+ cj k (Q ) 2∗

∗ , if = ρ3k ∗ ≥ ρ3k , if

qj2∗k > 0 qj2∗k = 0,

(9.19)

212

9 Multitiered Supply Chain Networks with Labor

and " dk ρ3∗

⎧ n ⎪ ⎪ = qj2∗k , if ⎪ ⎪ #⎨

∗ >0 ρ3k

j =1

n ⎪ ⎪ ⎪ ≤ qj2∗k , if ⎪ ⎩

(9.20) ∗ ρ3k

= 0.

j =1

In equilibrium, conditions (9.19) and (9.20) hold for all retailers j , j = 1, . . . , m, and all demand markets k, k = 1, . . . , o. They can be expressed as a variational inequality problem, such as (9.8) and (9.16), and given by: Determine (Q2∗ , ρ3∗ ) ∈ no+o , such that R+ o  n    ∗ ∗ ρ2j × qj2k − qj2∗k + cj k (Q2∗ ) − ρ3k j =1 k=1

⎤ ⎡ o n   ∗ ⎣ + ≥ 0, qj2∗k − dk (ρ3∗ )⎦ × ρ3k − ρ3k k=1

j =1

no+o ∀(Q2 , ρ3 ) ∈ R+ .

(9.21) 9.2.1.4

The Equilibrium Conditions for the Supply Chain Network with Labor

In equilibrium, the product shipments that manufacturers transport to the retailers must be the shipments that the retailers accept from the manufacturers. In addition, the amounts of the product bought by consumers at the demand markets must be equal to the amounts sold by the retailers. The equilibrium product shipment and price pattern must satisfy the sum of the optimality conditions for all manufacturers, as expressed by inequality (9.8), the optimality conditions for all retailers, as expressed by inequality (9.16), and the equilibrium conditions for the demand markets, as expressed by inequality (9.21) in order to formalize the agreements between the tiers of decision-makers. This formalized definition is below. Definition 9.1 (Supply Chain Network Equilibrium with Labor and Multiple Tiers) The equilibrium state of the supply chain network with labor and multiple tiers is one where the product shipments/transactions between decision-makers coincide and the product shipments and prices satisfy the sum of optimality conditions (9.8), optimality conditions (9.16), and the demand market equilibrium conditions as in (9.21). I now derive the variational inequality formulation of the above equilibrium state. The variational inequality formulation is an elegant formulation that allows for the qualitative analysis of the properties of the equilibrium product flows and prices and

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor

213

also their rigorous computation, and that of the labor values. It is the fundamental methodology utilized in this book. Theorem 9.1 (Variational Inequality Formulation) The equilibrium state of the supply chain network model with labor satisfying Definition 9.1 is equivalent to the solution of the variational inequality problem: Determine the vectors of equilibrium product shipments, shadow prices, and demand market prices (Q1∗ , Q2∗ , γ ∗ , ρ3∗ ) ∈ K , satisfying:   n m 1∗   wj2 ∂fi (Q1∗ ) ∂cij (qij ) ∂cj (Q1∗ ) wi1 ∗ 1 1∗ + + + + − γ − q × q j ij ij ∂qij1 ∂qij1 ∂qij1 αi1 αj2 i=1 j =1   o n   wj2k 2∗ ∗ ∗ + cj k (Q ) + 2 + γj − ρ3k × qj2k − qj2∗k αj k j =1 k=1  m  n o   1∗ 2∗ + qij − qj k × γj − γj∗ j =1

i=1

k=1

⎤ o n   ∗ ⎣ + ≥ 0, qj2∗k − dk (ρ3∗ )⎦ × ρ3k − ρ3k ⎡

k=1

∀(Q1 , Q2 , γ , ρ3 ) ∈ K 1 ,

j =1

(9.22) where K

1

mn+no+n+o ≡ R+ .

Proof I first prove that the equilibrium state according to Definition 9.1 implies variational inequality (9.22). Adding inequalities (9.8), (9.16), and (9.21) yields, after algebraic simplifications, the variational inequality (9.22). I now show the converse, that is, that a solution to variational inequality (9.22) satisfies the sum of conditions (9.8), (9.16), and (9.21) and is, therefore, an equilibrium according to Definition 9.1. ∗ + ρ ∗ to the term in the first set of brackets To inequality (9.22) add: −ρ1ij 1ij ∗ + ρ ∗ to the term before preceding the multiplication sign and add the term: −ρ2j 2j the second multiplication sign. Such “terms” do not alter the value of the inequality since they are equal to zero, with the resulting inequality:   n m 1∗   ∂fi (Q1∗ ) ∂cij (qij ) wi1 ∗ 1 1∗ + + − ρ − q × q ij ij 1ij ∂qij1 ∂qij1 αi1 i=1 j =1   n m 2   w ∂cj (Q1∗ ) j ∗ ∗ 1 1∗ + + ρ + − γ − q × q j ij ij 1ij ∂qij1 αj2 i=1 j =1

214

9 Multitiered Supply Chain Networks with Labor

+

 2 o n wj k j =1 k=1

+

 m n j =1

+

αj2k qij1∗

i=1

 ∗ − ρ2j

−

o

+ γj∗ 

qj2∗k

  × qj2k − qj2∗k

  × γj − γj∗

k=1

o  n    ∗ ∗ ρ2j × qj2k − qj2∗k + cj k (Q2∗ ) − ρ3k j =1 k=1

⎤ ⎡ o n   ∗ ⎣ + ≥ 0, qj2∗k − dk (ρ3∗ )⎦ × ρ3k − ρ3k k=1

∀(Q1 , Q2 , γ , ρ3 ) ∈ K 1 .

j =1

(9.23) But inequality (9.23) is the sum of the optimality conditions (9.8) and (9.16) and the equilibrium conditions (9.21) and, therefore, is a supply chain network equilibrium with labor according to Definition 9.1. The proof is complete. Variational inequality (9.22) can be put into standard variational inequality form (cf. (2.14)) as follows. Define X ≡ (Q1 , Q2 , γ , ρ3 ),

F (X) ≡ (Fij , Fj k , Fj , Fk )i=1,...,m;j =1,...,n;k=1,...,o ,

with components of F given by the functional terms preceding the multiplication signs in (9.22), and K ≡ K 1 . Here, N = mn + no + n + o. In the variational inequality problem in standard form, the feasible set K is closed and convex, and the function F is continuous. The variables in the variational inequality formulation of the supply chain network equilibrium problem are: the product shipments/transactions from manufacturers to retailers, Q1 , from which one can obtain the production quantities using (9.3); the product flows from the retailers to the demand markets, Q2 ; the shadow prices associated with handling the product by the retailers, γ , and the demand market prices ρ3 . Also, labor values on various links are computed, once the problem is solved using: (9.5), (9.10), and (9.11). The solution of variational inequality (9.22) is given by (Q1∗ , Q2∗ , γ ∗ , ρ3∗ ). ∗ , for I now describe how to obtain the equilibrium manufacturers’ prices ρ1ij ∗ , for all j , from the solution of all i, j , and the retailers’ equilibrium prices, ρ2j (9.22). In Section 9.3, an algorithm is presented for obtaining the solution. The ∗ can be obtained (cf. (9.8)) by finding a q 1∗ > 0, and then top-tiered prices ρ1ij ij ( ' 1∗ ) 1 1∗ ) ∂c (q w ij ∂f (Q ij ∗ ∗ i setting ρ1ij = + ∂qij + 1i or, equivalently (see (9.16)), ρ1ij = ∂qij αi ' ( 2 w ∂c (Q1∗ ) γj∗ − j∂qij − 2j . In the case of a zero product flow between a manufacαj

turer/retailer pair, there is, in effect, no “agreed upon” price associated with the

9.2 The Multitiered Supply Chain Network Equilibrium Models with Labor

215

transaction. One can, nevertheless, analyze the corresponding optimality conditions and note that a manufacturer would, in this case, accept any price provided it was not higher than the manufacturer’s marginal production and transportation cost (associated with the product shipment) plus

wi1 . αi1

Also, if qj∗k > 0, for some k

∗ is precisely equal to γ ∗ + and a particular j , then, using (9.16), ρ2j j

o

wj2k k=1 α 2 , jk

which can the solution of (9.22); equivalently, from (9.21), then  be obtained from  ∗ = ρ ∗ − c (Q2∗ ) for q ∗ > 0. Under the above pricing mechanisms, the ρ2j jk jk 3k optimality conditions (9.8), the optimality conditions (9.16), and the equilibrium conditions (9.21) will each hold separately (and simultaneously). Consequently, in solving variational inequality (9.22), one also ensures then that the optimality conditions for the manufacturers, the optimality conditions for the retailers, and the equilibrium conditions for the consumers at the demand markets are satisfied.

9.2.2 The Multitiered Supply Chain Network Equilibrium Model with Labor and Link Bounds on Labor Availability The model in Section 9.2.1 is now extended to include bounds on labor on links. The previous notation is retained with bounds added as follows. Let l¯i1 , i = 1, . . . , m, denote the maximum hours of labor available for manufacturing at firm i. Then, one has that l¯j2 denotes the maximum hours available at retailer j for laborers to handle and inventory the products, where j = 1, . . . , n. Also, let l¯j2k denote the maximum hours of labor available at retailer j to service and transact with consumers at demand market k, where j = 1, . . . , m and k = 1, . . . , n. Making use of (9.5), (9.10), and (9.11), the additional constraints on labor now faced by manufacturer i, i = 1, . . . , m, are: n

1 j =1 qij αi1

≤ l¯i1 ,

(9.24)

and the additional constraints faced by retailer j ; j = 1, . . . , n, on labor, are: m

1 i=1 qij αj2

qj2k αj2k

≤ l¯j2k ,

≤ l¯j2 ,

k = 1, . . . , o.

(9.25)

(9.26)

The objective functions of the manufacturers and of the retailers remain as in Section 9.2.1, but, since they are now faced with the above constraints, the feasible

216

9 Multitiered Supply Chain Networks with Labor

set K in the governing variational inequality now becomes K 2 with the inclusion of (9.24) for all manufacturers i = 1, . . . , m, and the incorporation of constraints (9.25) and (9.26) for all retailers j , j = 1, . . . , n. It follows from classical variational inequality theory (cf. Kinderlehrer and Stampacchia 1980, Nagurney 1999, and Nagurney et al. 2002) that a solution exists to variational inequality (9.22) over K 1 and over K 2 under reasonable conditions, with note also that the labor bounds in the model in this section, captured in the feasible set K 2 , guarantee that the product flows between tiers are also bounded. Although one can solve variational inequality (9.22) over the feasible set K 2 and, of course, variational inequality (9.22) over K 1 , using the Euler method that is outlined in Section 9.3, I now provide an alternative variational inequality for the model with link bounds on labor availability, over a feasible set K 3 , which allows for closed form expressions at each step of the iterative algorithm, for the variables. This is already naturally the case for variational inequality (9.22) over K 1 because there the feasible set K 1 is the nonnegative orthant. Before stating the alternative variational inequality for the model with labor and bounds on labor availability, one needs to introduce additional Lagrange multipliers, which also have the interpretation of being shadow prices. Specifically, associate the nonnegative Lagrange multiplier γi1 with constraint (9.24) for each i = 1, . . . , m; the nonnegative Lagrange multiplier γj2 with constraint (9.25) for each j = 1, . . . , n, and, finally, the nonnegative Lagrange multiplier γj2k is associated with constraint (9.26) for each j = 1, . . . , m and each k = 1, . . . , o. m ; the γ 2 s for all j into Group the γi1 s for all i into the vector γ 1 ∈ R+ j n , and the γ 2 s into the vector γ 2,2 ∈ R no . Define the the vector γ 2 ∈ R+ + jk feasible set K 3 ≡ {(Q1 , Q2 , γ , ρ3 , γ 1 , γ 2 , γ 2,2 )|(Q1 , Q2 , γ , ρ3 , γ 1 , γ 2 , γ 2,2 ) ∈ mn+no+n+o+m+n+no R+ }. Using arguments as in Nagurney et al. (2017); see also Chapters 3 and 4, it follows that the solution of the following variational inequality yields an equilibrium: Determine (Q1∗ , Q2∗ , γ ∗ , ρ3∗ , γ 1∗ , γ 2∗ , γ 2,2∗ ) ∈ K 3 , such that  n m ∂fi (Q1∗ ) i=1 j =1

1 ∂qij

+

1∗ ) ∂cij (qij 1 ∂qij

+

∂cj (Q1∗ ) 1 ∂qij

 γj2∗ γi1∗ ∗ + 1 + 2 − γj + 1 + 2 αi αj αi αj wi1

wj2

  o n     w2 γ 2∗ 1 − q 1∗ + 2∗ ) + j k + γ ∗ − ρ ∗ + j k × q 2 − q 2∗ (Q c × qij jk ij j j k j k 3k αj2k αj2k j =1 k=1 ⎤ ⎡ ⎤ ⎡ n m o o n    1∗ 2∗ ∗ 2∗ ∗ ∗  ⎣ ⎣ qij − qj k ⎦ × γj − γj + qj k − dk (ρ3 )⎦ × ρ3k − ρ3k + j =1

i=1

k=1

k=1

j =1

n m   1∗   1∗   m n   j =1 qij i=1 qij 1 1 1∗ 2 2 − γ 2∗ ¯ ¯ + − γ − + li − × γ l × γ i i j j j αi1 αj2 i=1 j =1

9.3 The Algorithmic Procedure

+



qj2∗k l¯j2k − 2 αj k j =1 k=1 o n



  × γj2k − γj2∗ k ≥ 0,

217

∀(Q1 , Q2 , γ , ρ3 , γ 1 , γ 2 , γ 2,2 ) ∈ K 3 .

(9.27) Variational inequality (9.27) can be put into standard form (2.14) by defining the vector X ≡ (Q1 , Q2 , γ , ρ3 , γ 1 , γ 2 , γ 2,2 ) with  F (X) ≡ Fij , Fj k , Fj , Fk , Fi1 , Fj2 , Fj2k

i=1,...,m;j =1,...,n;k=1,...,o

,

with the specific components of F given by the functional terms preceding the multiplication signs in (9.27), and with K ≡ K 3 and N ≡ mn + no + n + o + m + n + no.

9.3 The Algorithmic Procedure In contrast to the algorithmic procedure applied in the previous modeling chapters, which is the modified projection method, in this chapter, the Euler method is used to solve the models with labor introduced in Sections 9.2.1 and 9.2.2. This algorithm is induced by the general iterative scheme of Dupuis and Nagurney (1993), where at an iteration τ , one determines: Xτ = PK (Xτ −1 − aτ −1 F (Xτ −1 )),

(9.28)

with PK being the projection on the feasible set K and with F as defined above (see also (2.14)). The Euler method can be interpreted as a discrete-time adjustment process associated with the corresponding projected dynamical system, whose set of stationary points corresponds to the set of solutions to the associated variational inequality. Specific conditions for convergence of this algorithm and applications to other network models can be found in Nagurney et al. (1994, 1995a,b), and Nagurney (2006). Additional supply chain network models that have been solved via the Euler method are constructed in Toyasaki et al. (2014), Besik and Nagurney (2017), and Saberi et al. (2018). The Euler Method for the Solution of Variational Inequality (9.27) Step 0: Initialization 0 10 20 0 10 20 2,20 ) ∈ K ≡ K 3 . Let τ = 1 and set {a } such Set (Q τ ∞, Q , γ , ρ3 , γ , γ , γ that: τ =0 aτ = ∞, aτ > 0, aτ → 0, as τ → ∞. Step 1: Computation Compute (Q1τ , Q2τ , γ τ , ρ3τ , γ 1τ , γ 2τ , γ 2,2τ ) ∈ K by solving the variational inequality

218

9 Multitiered Supply Chain Networks with Labor n m



 qij1τ

+ aτ

i=1 j =1

−γjτ −1 +

o n

∂cij (qij1τ −1 ) wj2 w11 ∂cj (Q1τ −1 ) ∂fi (Q1τ −1 ) + + + + ∂qij1 ∂qij1 ∂qij1 αi1 αj2 γi1τ −1

+

αi1 

 qj2τk

+ aτ

γj2τ −1

+



αj2

cj k (Q

2τ −1

 − qij1τ −1

)+

j =1 k=1

wj2k αj2k

  × qij1 − qij1τ

+ γjτ −1

τ −1 − ρ3k

+

γj2τk −1



αj2k

 − qj2τk −1

  m   n o     1τ −1 2τ −1 τ −1 2 2τ τ × qj k − qj k + γj + aτ − γj × γj − γjτ qij − qj k

+

o

j =1

⎡

⎛

τ ⎣ρ3k

+ aτ ⎝

+

i=1

+

n



qj2τk −1

 γi1τ

⎤    τ −1 ⎠ τ −1 ⎦ τ − dk ρ3 − ρ3k × ρ3k − ρ3k

 γj2τ + aτ

o n j =1 k=1

1τ −1 j =1 qij αi1

m

1τ −1 i=1 qij 2 αj

l¯j2 −



 γj2τk −1

+ aτ



n

l¯i1 −

+ aτ



j =1

+

k=1

⎞

j =1

k=1 m

n

i=1

l¯j2k

−

qj2τk αj2k

 − γi1τ −1



  × γi1 − γi1τ

   − γj2τ −1 × γj2 − γj2τ



 − γj2τk −1

∀(Q1 , Q2 , γ , ρ3 , γ 1 , γ 2 , γ 2,2 ) ∈ K .

  × γj2k − γj2τk ≥ 0,

(9.29)

Step 2: Convergence Verification τ − ρ τ −1 | ≤ , If |qij1τ − qij1τ −1 | ≤ , |qj2τk − qj2τk −1 | ≤ , |γjτ − γjτ −1 | ≤ , |ρ3k 3k

for all i = 1, . . . , m; j = 1, . . . , n; k = 1, . . . , o, and |γi1τ − γi1τ −1 | ≤ , for all i = 1, . . . , m, |γj2τ − γj2τ −1 | ≤ , for all j = 1, . . . , n, and |γj2τk − γj2τk −1 | ≤ , for all j = 1, . . . , n: k = 1, . . . , o, and with  > 0, a desired tolerance, then stop; otherwise, set τ := τ + 1, and proceed to Step 1. The variational inequality (9.29) can be solved explicitly and in closed form since the feasible set is the nonnegative orthant. Indeed, variational inequality decomposes τ , γ 1τ , γ 2τ , and γ 2τ variables for all into subproblems in the qij1τ , qj2τk , γjτ , ρ3k i j jk i, j, k, o. For completeness, I now provide the explicit formulae. Specifically, variational inequality (9.29) can be solved exactly and in closed form as follows.

9.3 The Algorithmic Procedure

219

Computation of the Product Shipments/Transactions At iteration τ , compute the qij1τ s, i = 1, . . . , m, j = 1, . . . , n, according to: & qij1τ



1τ −1 ∂fi (Q1τ −1 ) ∂cij (qij ) ∂cj (Q1τ −1 ) w11 − aτ + + + 1 ∂qij1 ∂qij1 ∂qij1 αi $ wj2 γj2τ −1 γ 1τ −1 + 2 − γjτ −1 + i 1 + (9.30) . αj αi αj2

0, qij1τ −1

= max

Also, at iteration τ , compute the qj2τk s, j = 1, . . . , n, k = 1, . . . , o, according to: &



qj2τk = max 0, qj2τk −1 − aτ

cj k (Q2τ −1 ) +

wj2k αj2k

τ −1 + γjτ −1 − ρ3k +

γj2τk −1 αj2k

$ . (9.31)

Computation of the Prices (Demand Market Prices and Lagrange Multipliers) The shadow prices, γjτ , j = 1, . . . , n, are computed at iteration τ according to: & γjτ = max 0, γjτ −1 − aτ

 m

qij1τ −1 −

i=1

o

$ qj2τk −1

,

(9.32)

k=1

τ , k = 1, . . . , o, are computed according to: whereas the demand market prices, ρ3k

τ ρ3k

⎞⎫ ⎛ n ⎬  τ −1 = max 0, ρ3k − aτ ⎝ qj2τk −1 − dk ρ3τ −1 ⎠ . ⎭ ⎩ ⎧ ⎨

(9.33)

j =1

Also, the shadow prices associated with the labor bounds are computed, at iteration τ , as below. The shadow prices, γi1τ , i = 1, . . . , m, are determined according to: &



γi1τ = max 0, γi1τ −1 − aτ

n

l¯i1 −

1τ −1 j =1 qij αi1

$ .

(9.34)

The shadow prices, γj2τ , j = 1, . . . , n, are computed according to: & γj2τ

= max

 0, γj2τ −1

− aτ

l¯j2

m −

1τ −1 i=1 qij αj2

$ .

(9.35)

220

9 Multitiered Supply Chain Networks with Labor

Finally, the shadow prices, γj2τk , j = 1, . . . , n; k = 1, . . . , o, are computed according to: &



γj2τk = max 0, γj2τk −1 − aτ

l¯j2k −

qj2τk −1 αj2k

$ (9.36)

.

Remark 9.1 I now show how the above statement of the Euler method is adapted to solve variational inequality (9.22) governing the multitiered supply chain network equilibrium model without labor bounds presented in Section 9.2.1. Step 0 is modified as: Set (Q10 , Q20 , γ 0 , ρ30 ) ∈ K ≡ K 1 . The first term in the variational inequality (9.29) has the term: + expression in (9.29), the term

γi1τ −1 αi1

γj2τk −1 αj2k

+

γj2τ −1 αj2

removed. In the second summation

is excised and the final three expressions with

the summation signs in (9.29) and their preceding + signs are also removed. Plus, in (9.28), ∀(Q1 , Q2 , γ , ρ3 , γ 1 , γ 2 , γ 2,2 ) ∈ K is replaced by: ∀(Q1 , Q2 , γ , ρ3 ) ∈ K . And, in terms of the explicit formulae, (9.30) through (9.36), (9.34)–(9.36) are no longer needed; in (9.30), +

γi1τ −1 αi1

+

γj2τ −1 αj2

is excised, and in (9.31),

γj2τk −1 αj2k

is removed.

9.4 Multitiered Supply Chain Numerical Examples The Euler method is coded in FORTRAN on a Linux system at the University of Massachusetts Amherst. It is initialized thus: All variables are set equal to zero. The  is 10−7 , which means that the absolute value of each variable value at two successive iterations differs by no more than this value. The series {aτ } is: {1, 12 , 12 , 13 , 13 , 13 . . .}. Example 9.1 (2 Manufacturers, 2 Retailers, 2 Demand Markets—No Bounds on Labor) The first numerical example in this section, depicted in Figure 9.2, consists of two manufacturers, two retailers, and two demand markets. The production, transportation cost, transaction cost functions, and the demand functions are as in Example 9.2 in Nagurney et al. (2002), which I report, for completeness, below, followed by the additional data for the wages at the manufacturers, the wages at the retailers, plus the productivity factors associated with manufacturing, the handling and storage of the products at the retailers, and the servicing of the customers. In this example there are no bounds on labor.

9.4 Multitiered Supply Chain Numerical Examples

221

Fig. 9.2 Supply Chain Network for Numerical Examples 9.1 and 9.2

The data for this example are constructed for easy interpretation. Production cost functions for the manufacturers (cf. (9.1)) are given by: f1 (q 1 ) = 2.5q112 + q11 q21 + 12q11 ,

f2 (q 1 ) = 2.5q212 + q11 q21 + 2q21 .

Transportation cost functions faced by the manufacturers and associated with transacting with the retailers (see (9.2)) are:   1 12 1 1 12 1 = q11 = q12 c11 q11 + 3.5q11 , c12 q12 + 3.5q12 ,   1 12 1 1 12 1 c21 q21 = 0.5q21 = 0.5q22 + 3.5q21 , c22 q22 + 3.5q22 . The handling costs of the retailers (see (9.9)) are:  c1 (Q1 ) = 0.5

2



2 1 qi1

,

i=1

c2 (Q1 ) = 0.5

2

2 1 qi2

.

i=1

The demand functions at the demand markets (cf. (9.18)) are: d1 (ρ3 ) = −2ρ31 − 1.5ρ32 + 1000,

d2 (ρ3 ) = −2ρ32 − 1.5ρ31 + 1000,

and the transaction costs (cf. (9.17)) are: 2 c11 (Q2 ) = q11 +5,

2 c12 (Q2 ) = q12 +5,

2 c21 (Q2 ) = q21 +5,

2 c22 (Q2 ) = q22 +5.

222

9 Multitiered Supply Chain Networks with Labor

The hourly wages at the manufacturers are: w11 = 10 and w21 = 10 and the wages 2 = w 2 = 10, w 2 = w 2 = 9. at the retailers are: w12 = 10 and w22 = 10 and w11 12 21 22 1 The respective productivity factors are: α1 = 2, α21 = 2, α12 = 4, α22 = 4, and 2 = α 2 = 5, α 2 = α 2 = 4. In Example 9.1 there are no bounds on the labor α11 12 21 22 availability. Hence, I use the Euler method to solve variational inequality (9.22). Alternatively, once can set a very large value for l¯11 and l¯21 , for l¯12 and l¯22 , and the l¯j2k s with j = 1, 2 and k = 1, 2. The Euler method yields the following equilibrium pattern. The product shipments/transactions between the manufacturers and the retailers are: 1∗ 1∗ Q1∗ := q11 = 13.99, q12 = 13.95,

1∗ 1∗ q21 = 16.67, q22 = 16.60.

The product shipments (consumption volumes) between the two retailers and the two demand markets are: 2∗ 2∗ Q2∗ := q11 = q12 = 15.33,

2∗ 2∗ q21 = q22 = 15.27;

the vector γ ∗ has components: γ1∗ = 254.64,

γ2∗ = 254.44,

and the demand market prices at the demand markets are: ∗ ∗ ρ31 = ρ32 = 276.97.

The equilibrium labor values are, in turn: l11∗ = 13.97, l21∗ = 16.64, l12∗ = 7.67, 2∗ = l 2∗ = 3.07, l 2∗ = l 2∗ = 3.82. and l22∗ = 7.64, and l11 12 21 22 ∗ The prices ρ1ij , i = 1, 2, j = 1, 2, are recovered according to the discussion at ∗ = 221.47, ρ ∗ = 221.39, ρ ∗ = the end of Section 9.2.1, yielding values: ρ111 112 121 ∗ = 221.39. Also, the prices charged by the retailers are: ρ ∗ = 256.64 221.47, ρ122 21 ∗ = 256.69. The profit is: 2342.56 for Manufacturer 1 and for Manufacturer and ρ22 2 the profit is: 3044.11. The profit for Retailer 1 is: 470.19 and that for Retailer 2: 466.65. The optimality/equilibrium conditions hold with excellent accuracy. Example 9.2 (2 Manufacturers, 2 Retailers, 2 Demand Markets—Bounds on Labor at Manufacturers and Reduction in Productivity) Example 9.2 has the same data as Example 9.1 except that I now consider a disruption to labor of the following form. Manufacturer 1 is impacted the most severely with its labor bound now l¯11 = 5 and with a reduction in productivity (both due to COVID-19) with now α11 = 0.1. Manufacturer 2 has a labor bound of l¯21 = 20. The Euler method now converges to the following equilibrium solution. The equilibrium product shipments/transactions between the two manufacturers and the two retailers are now:

9.4 Multitiered Supply Chain Numerical Examples 1∗ 1∗ Q1∗ := q11 = 0.27, q12 = 0.23,

223 1∗ 1∗ q21 = 20.04, q22 = 19.96.

The product shipments (consumption volumes) between the two retailers and the two demand markets are: 2∗ 2∗ Q2∗ := q11 = q12 = 10.15,

2∗ 2∗ q21 = q22 = 10.09;

the vector γ ∗ now has elements: γ1∗ = 262.78,

γ2∗ = 262.59,

and the demand market prices are: ∗ ∗ ρ31 = ρ32 = 279.93.

The equilibrium labor values are: l11∗ = 5.00, l21∗ = 20.00, l12∗ = 5.08 and 2∗ = l 2∗ = 2.03, l 2∗ = l 2∗ = 2.52. Since both l 1∗ and l 1∗ are at the l22∗ = 5.05, and l11 12 21 22 1 2 imposed labor bounds of l¯11 = 5 and l¯21 = 20, respectively, the associated Lagrange multipliers are positive and take on values of: γ11∗ = 8.14 and γ21∗ = 17.87. ∗ ; i = 1, 2; j = 1, 2, are now: ρ ∗ ∗ The prices ρ1ij 111 = 158.54, ρ112 = ∗ ∗ 158.56, ρ121 = 231.04, ρ122 = 230.96. The prices charged by the retailers are: ∗ = 264.78 and ρ ∗ = 264.84. The profit is: 0.75 for Manufacturer 1 and for ρ21 22 Manufacturer 2 the profit is: 4399.87. The profit for Retailer 1 is: 405.97 and that for Retailer 2: 399.84. Since the labor values are at the bounds at both manufacturers, the associated Lagrange multipliers (shadow prices) are positive and at values, respectively, of γ11∗ = 8.10 and γ22∗ = 15.41. These shadow prices provide valuable information and reflect by how much the profits would increase for the respective manufacturer if there was a unit increase in labor at the manufacturing facility. Observe that, with the disruption to labor availability at the production sites of Manufacturer 1 and of Manufacturer 2, with the former having a much lower labor availability than the latter and also with the former experiencing a decrease in labor productivity, due to COVID-19, Manufacturer 1 suffers a huge loss in profits, as compared to his profit in Example 9.1. On the other hand, Manufacturer 2, who has higher labor availability than Manufacturer 1, now enjoys a substantial increase in profit, as compared to that which he earned in Example 9.1. Consumers at both demand markets in this example experience a higher price for the product, and there is a lower product flow of about 33% to each demand market, as compared to that in Example 9.1. This result shows that consumers also can suffer from a labor disruption in terms of prices. Both retailers now also suffer from lower profits than they had earned in Example 9.1. This example demonstrates the propagation of impacts in a multitiered supply chain network due to disruptions to labor as well as a loss in labor productivity. Example 9.3 (3 Manufacturers, 2 Retailers, 3 Demand Markets—No Bounds on Labor) The third numerical example in this section, Example 9.3, has three manufacturers, two retailers, and three demand markets. The network structure is depicted in Figure 9.3.

224

9 Multitiered Supply Chain Networks with Labor

Fig. 9.3 Supply chain network for numerical Examples 9.3 and 9.4

The data for this example are constructed from the data for Example 9.1, with the addition of functions for the third manufacturer and the third demand market. The production cost functions for the manufacturers are given by: f1 (q 1 ) = 2.5q112 + q11 q21 + 2q11 ,

f2 (q 1 ) = 2.5q212 + q11 q21 + 2q21 ,

f3 (q 1 ) = 0.5q312 + 0.5q11 q31 + 2q31 . The transportation cost functions faced by the manufacturers and associated with transacting with the retailers are: 1 12 1 c11 (q11 ) = 0.5q11 + 3.5q11 ,

1 12 1 c12 (q12 ) = 0.5q12 + 3.5q12 ,

1 12 1 c21 (q21 ) = 0.5q21 + 3.5q21 ,

1 12 1 c22 (q22 ) = 0.5q22 + 3.5q22 ,

1 12 1 c31 (q31 ) = 0.5q31 + 2q31 ,

1 12 1 c32 (q32 ) = 0.5q32 + 2q32 .

The handling costs of the retailers, in turn, are:  c1 (Q ) = 0.5 1

3



2 1 qi1

,

c2 (Q ) = 0.5 1

i=1

3

2 1 qi2

.

i=1

The demand functions at the demand markets are: d1 (ρ3 ) = −2ρ31 − 1.5ρ32 + 1000, d3 (ρ3 ) = −2ρ33 − 1.5ρ31 + 1000,

d2 (ρ3 ) = −2ρ32 − 1.5ρ31 + 1000,

9.4 Multitiered Supply Chain Numerical Examples

225

and the transaction costs between the retailers and the consumers at the demand markets are: 2 c11 (Q2 ) = q11 + 5,

2 c12 (Q2 ) = q12 + 5,

2 c13 (Q2 ) = q13 + 5,

2 c21 (Q2 ) = q21 + 5,

2 c22 (Q2 ) = q22 + 5,

2 c23 (Q2 ) = q23 + 5.

The hourly wages at the manufacturers are: w11 = 10, w21 = 9, w31 = 9 and the 2 = 7, w 2 = 8, w 2 = 8, wages at the retailers are: w12 = 9 and w22 = 9 and w11 12 13 2 = w 2 = 7, w 2 = 8. The respective productivity factors are: α 1 = 4, α 1 = 4, w21 22 23 1 2 2 = 6, α 2 = 7, α 2 = 7, α 2 = 6, α 2 = 7, α31 = 4, α12 = 6, α22 = 5, and α11 12 13 21 22 2 = 6. In Example 9.3 there are no bounds on the labor availability. α23 The Euler method yields the following equilibrium pattern. The product shipments/transactions between the three manufacturers and the two retailers are now: 1∗ = 12.16, q 1∗ = 12.10, q 1∗ = 12.19, q 1∗ = 12.13, q 1∗ = 49.32, q 1∗ = 49.25. Q1∗ := q11 12 21 22 31 32

The product shipments between the two retailers and the three demand markets are: ∗ = 24.57, q ∗ = 24.51, q ∗ = 24.60, q ∗ = 24.52, q ∗ = 24.61, q ∗ = 24.36, Q2∗ := q11 12 13 21 22 23

whereas the vector γ ∗ , which coincides with ρ2∗ in values, is now equal to: γ1∗ = 241.00, γ2∗ = 241.04, and the demand prices at the three demand markets are: ∗ ∗ ∗ ρ31 = 271.72, ρ32 = 271.65, ρ33 = 271.73.

The equilibrium labor values are: l11∗ = 6.07, l21∗ = 6.08, l31∗ = 19.71, l12∗ = 2∗ = 4.09, l 2∗ = 3.50, l 2∗ = 3.51, l 2∗ = 4.09, 12.28 and l22∗ = 14.70, and l11 12 13 21 2∗ = 3.52, l 2∗ = 4.06. l22 23 ∗ = 165.82, The top tier prices that the manufacturers charge the retailers are: ρ111 ∗ ∗ ∗ ∗ ∗ ρ112 = 165.76, ρ121 = 165.52, ρ122 = 165.76, and ρ131 = 165.82, ρ132 = 165.76. ∗ = 242.13 and ρ ∗ = 242.20. The The prices charged by the retailers are: ρ21 22 profit is: 1619.35 for Manufacturer 1 and for Manufacturer 2 the profit is: 1626.78, whereas the profit for Manufacturer 3 is: 7.892.64. The profit for Retailer 1 is: 2713.31 and that for Retailer 2: 2712.26. Example 9.4 (3 Manufacturers, 2 Retailers, 3 Demand Markets—Bound on Labor at a Retailer and Loss in Productivity) Example 9.4 has the identical data to those in Example 9.3 except that now I consider a disruption at the first retailer of the following form. The labor bound there l¯12 = 10 and the new productivity factor

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α12 = 1, which represents a loss in productivity. Again, this example is inspired by effects of the COVID-19 pandemic on workers. Recall that the affected associated link represents the product handling and storage activity of Retailer 1. The algorithm converges to the following equilibrium pattern: the product shipments/transactions between the three manufacturers and the two retailers are now: 1∗ 1∗ 1∗ 1∗ Q1∗ := q11 = 0.00, q12 = 18.91, q21 = 0.00, q22 = 18.96,

1∗ 1∗ q31 = 10.00, q32 = 57.59.

The product shipments between the two retailers and the three demand markets are: ∗ ∗ ∗ ∗ ∗ ∗ Q2∗ := q11 = 3.34, q12 = 3.29, q13 = 3.37, q21 = 31.84, q22 = 31.93, q23 = 31.68,

whereas the vector γ ∗ , which coincides with ρ2∗ in values, is now equal to: γ1∗ = 266.19, γ2∗ = 237.69, and the demand prices at the three demand markets are: ∗ ∗ ∗ ρ31 = 275.70, ρ32 = 275.62, ρ33 = 275.70.

The equilibrium labor values are: l11∗ = 4.73, l21∗ = 4.74, l31∗ = 13.52, l12∗ = 2∗ = 0.56, l 2∗ = 0.47, l 2∗ = 0.48, l 2∗ = 5.31, 10.00 and l22∗ = 19.09, and l11 12 13 21 2∗ 2∗ l22 = 4.56, l23 = 5.28. Since l 2∗ = 10.00, which is the imposed upper bound on labor at Retailer 1 available for handling/storage of the products, the associated Lagrange multiplier is positive: γj2∗ = 154.35. Retailer 1 should do everything possible to increase labor supply at its facility since that would have a positive impact on profit. ∗ = 121.52, The top tier prices that the manufacturers charge the retailers are: ρ111 ∗ = 140.43, ρ ∗ = 121.47, ρ ∗ = 140.43, and ρ ∗ = 92.84, ρ ∗ = 140.43. ρ112 121 122 131 132 ∗ = 267.36 and ρ ∗ = 238.69. The The prices charged by the retailers are: ρ21 22 profit is: 1072.89 for Manufacturer 1 and for Manufacturer 2 the profit is: 1078.57, whereas the profit for Manufacturer 3 is: 4272.50. The profit for Retailer 1 is: 1593.04 and that for Retailer 2: 4572.07. The prices increase at all the demand markets under this disruption. Both manufacturers suffer a decrease in profits as does Retailer 1. Retailer 2, on the other hand, benefits and enjoys an increase in profit, as compared to his profit in Example 9.3. This example further illustrates the importance of having a multitiered supply chain network equilibrium framework that includes labor and can assess the impacts of labor disruptions at different tiers and the ramifications throughout the supply chain in terms of prices, product flows, as well as profits of the various firms. A generic product is considered here but the framework can be adapted accordingly to handle parameterized examples in specific applications. This work is a

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proof of concept study that shows that general multitiered supply chain network equilibrium models with labor and productivity factors, as well as with labor bounds, can be formulated and solutions to them computed.

9.5 Summary, Conclusions, and Suggestions for Future Research The pandemic has disrupted economies and societies globally. The associated pain and suffering due to illnesses and deaths have been of historic proportions. At the same time, the pandemic has pushed scholars and practitioners to assist in whatever manner that they can to address various shortfalls in order to enable the understanding of the negative impacts and the mitigation of them. In this chapter, multitiered supply chain network equilibrium models are constructed that include labor. A case without bounds is considered, as well as one with bounds on labor availability. The work builds on the existing literature but introduces significant extensions. The models with labor consist of competing profit-maximizing manufacturers and competing profit-maximizing retailers, plus consumers. The behavior of the various stakeholders is captured, along with the governing equilibrium conditions. The formulation, analysis, and solution of the models are conducted using variational inequality theory. An elegant computational procedure, the Euler method, with ease of implementation, is proposed and used to compute solutions to a series of numerical examples with the complete input and output data reported. In previous methodological chapters the modified projection method is used for computations. This chapter provides an alternative algorithm, which has a natural dynamic adjustment process interpretation in discrete-time. The solution of the models yields the equilibrium product flows along the tiers; the demand market prices, the various shadow prices associated with the labor and constraints, along with the labor volumes on the various links and the prices charged by the manufacturers and the retailers. The system-wide supply chain network framework allows for the quantification of various disruptions that we are seeing now on our planet, from decreases in labor availability as well as decreases in labor productivity, among others. This work opens up exciting, new research directions on the integration of labor into other multitiered, including multiperiod, supply chain network equilibrium problems. With the global supply chain network economy, a disruption to a supply chain network node or link in one part of the world can have “unforeseen” consequences in terms of product availability as well as product prices in other parts. Many of such disruptions have included labor issues in the pandemic and have been exacerbated by climate change, plus Russia’s war on Ukraine. I expect that specific applications to food supply chains as well as to high tech and to consumer product supply chains, with distinct supply chain network topologies and tiers of decision-makers, would be of value, recognizing also the potential of investments

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in productivity, the topic of Chapters 7 and 8. The quantification of the resilience of multitiered supply chain networks with labor is also a very timely endeavor. Results from Chapters 3 and 4 on supply chain resilience can be used as a foundation for such an extension.

9.6 Sources and Notes The results in this chapter are new and have not been previously reported elsewhere. Supply chains are networks with flows of products, prices, and information, and with multiple stakeholders, each with individual specific behavior and objective function (cf. Nagurney et al. 2002, Nagurney 2006, Nagurney et al. 2007, and citations therein). Supply chain management, as a formalism, can be traced to the 1980s (Handfield and Nichols 1999) but, as argued in Nagurney (2014), the intellectual origins are in spatial economics and regional science, with such innovations as the classical models of Samuelson (1952) and Takayama and Judge (1964, 1971) and numerous insights as to networks of production processes, transportation, and distribution in the seminal book by Beckmann et al. (1956). Importantly, such scholarly precedents recognized the need for different supply and demand price functions associated with regions, as well as with commodities, and that transportation networks provide the critical backbone. Furthermore, the work of Beckmann et al. (1956) made the concept of congestion and alternative behaviors of travelers rigorous. The impacts of congestion in the pandemic in terms of ships circling at ports in the USA and trucks seeking to unload and to obtain goods for further transport continue (Goodman and Luxen 2021). As argued by Donaghy (2007), analytical (or conceptual) as well as methodological advances are needed (see also, Donaghy 2012 and Cooper et al. 2007) to better understand globalization and the impacts, whose ramifications we are dealing with now. Indeed, challenges associated with practice have spurred major advances in methodologies, such as variational inequality theory (cf. Dafermos 1980, 1982, 1986, Nagurney 1999) and projected dynamical systems (Dupuis and Nagurney 1993; Nagurney and Zhang 1996) for network equilibria, from traffic (cf. Smith 1979, Dafermos 1980, Sheffi 1985, Ran and Boyce 1996) to spatial price and oligopolistic problems (Florian and Los 1982, Dafermos and Nagurney 1987, Miller et al. 1991) and supply chains (see, e.g., Nagurney and Dong 2002, Nagurney 2006, Nagurney et al. 2013, and Nagurney and Li 2016, and citations therein). The advances in computable general equilibrium models, including those constructed in the pandemic (cf. Rose et al. 2021, He et al. 2021, and citations therein), have also been notable; the same for input-output models (see Haddad et al. 2021) dating back to the seminal contributions of Isard (1951).

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Kowalick, C., 2021. Continued supply-chain issues may impact back-to-school shopping. Times Record News, Wichita Falls, Texas. Miller, T.C., Tobin, R.L., Friesz, T.L., 1991. Stackelberg games on a network with Cournot-Nash oligopolistic competitors. Journal of Regional Science, 31(4), 435–454. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A. 2006. Supply Chain Network Economics: Dynamics of Prices, Flows and Profits. Edward Elgar Publishing, Cheltenham, United Kingdom. Nagurney, A., 2014. Supply chains and transportation networks. In: Handbook of Regional Science. M. Fischer and P. Nijkamp Editors, Springer, Berlin, Germany, pp. 787–810. Nagurney, A., Cruz, J.M., Wakolbinger, T., 2007. The co-evolution and emergence of integrated international financial networks and social networks: Theory, analysis, and computations. In: Globalization and Regional Economic Modeling. R. Cooper, K.P. Donaghy, and G.J.D. Hewings, Editors, Springer, Berlin, Germany, pp. 183–226. Nagurney, A., Dong, J., 2002. Supernetworks: Decision-Making for the Information Age. Edward Elgar Publishing, Cheltenham, United Kingdom. Nagurney, A., Dong, J., Zhang, D., 2002. A supply chain network equilibrium model. Transportation Research E, 38(5), 281–303. Nagurney, A., Dupuis, P., Zhang, D., 1994. A dynamical systems approach for network oligopolies and variational inequalities. Annals of Regional Science, 28, 263–283. Nagurney, A., Li, D., 2016. Competing on Supply Chain Quality: A Network Economics Perspective. Springer International Publishing Switzerland. Nagurney, A., Salarpour, M., Dong, J., 2022. Modeling of Covid-19 trade measures on essential products: A multiproduct, multicountry spatial price equilibrium framework. International Transactions in Operational Research, 29(1), 226–258. Nagurney, A., Takayama, T., Zhang, D., 1995a. Massively parallel computation of spatial price equilibrium problems as dynamical systems. Journal of Economic Dynamics and Control, 18, 3–37. Nagurney, A., Takayama, T., Zhang, D., 1995b. Projected dynamical systems modeling and computation of spatial network equilibria. Networks, 26, 69–85. Nagurney, A., Yu, M., Besik, D., 2017. Supply chain network capacity competition with outsourcing: A variational equilibrium framework. Journal of Global Optimization, 69(1), 231–254. Nagurney, A., Yu, M., Masoumi, A.H., Nagurney, L.S., 2013. Networks Against Time: Supply Chain Analytics for Perishable Products. Springer Science + Business Media, New York. Nagurney, A., Zhang, D., 1996. Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic Publishers, Norwell, Massachusetts. Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49. Nash, J.F., 1951. Noncooperative games. Annals of Mathematics, 54, 286–298. Perez, S., 2020. COVID-19 pandemic accelerated shift to e-commerce by 5 years, new report says. TechCrunch, August 24. Qiang, Q., Ke, K., Anderson, T., Dong, J., 2013. The closed-loop supply chain network with competition, distribution channel investment, and uncertainties. Omega, 41(2), 186–194. Ran, B., Boyce, D.E., 1996. Modeling Dynamic Transportation Networks, second revised edition, Springer-Verlag, Berlin, Germany. Rose, A., Walmsley, T., Wei, D., 2021. Spatial transmission of the economic impacts of COVID-19 through international trade. Letters in Spatial and Resource Sciences, 14, 169–196. Saberi, S., Cruz, J.M., Sarkis, J., Nagurney, A., 2018. A competitive multiperiod supply chain network model with freight carriers and green technology investment option. European Journal of Operational Research, 266(3), 934–949. Samuelson, P.A., 1952. Spatial price equilibrium and linear programming. American Economic Review, 42, 283–303. Sheffi, Y., 1985. Urban Transportation Networks. Prentice-Hall, Englewood Cliffs, New Jersey.

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Chapter 10

International Migrant Labor and Supply Chains

Abstract The pandemic has disrupted supply chains globally with a major shortfall being that of labor shortages from production through distribution activities. In this chapter, I construct a supply chain network optimization model that includes both domestic labor and international migrant labor from multiple countries, with the latter made possible through investments in attracting labor subject to a budget constraint. The model allows for different wage settings for domestic versus migrant labor and also has the flexibility of having true information as to the wages of migrants provided or not. Variational inequality formulations of the model are derived, along with qualitative properties, and an algorithm presented that yields closed form expressions for the underlying problem variables at each iteration. The model is one of the very few variational inequality operations research models with nonlinear constraints. Three series of algorithmically solved numerical examples, motivated by a high value agricultural product—that of truffles, reveal interesting insights in terms of profits, prices, product path flows, and investments, with variations in the data including that of truthful and untruthful wages being used to attract migrant labor.

10.1 Introduction Events of the past few years, with the pandemic, climate change, and increasing global conflicts, including the war of Russia against Ukraine, have demonstrated the importance of supply chains and their effective and efficient operation, with disruptions adversely affecting product prices and deliveries, as well as the prosperity of companies and the health and well-being of consumers (Helper and Soltas 2021). The reasons for the disruptions have been multifaceted with shocks both on the demand side as well as on the supply side and challenges associated with transportation (del Rio-Chanona et al. 2020; Ivanov 2020; Chowdhoury et al. 2021; Sodhi and Tang 2021; Novoszel and Wakolbinger 2022). One of the major characteristics of the pandemic has been that of labor shortages. Workers throughout the pandemic have been falling ill; some, sadly, have lost their lives, whereas others chose to switch jobs or to leave the labor force (Lynch 2021). Furthermore, various © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_10

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countries imposed restrictions further impeding the flow of workers (Murzakulova et al. 2021). Even with vaccinations continuing, the challenges of ameliorating labor shortages throughout many supply chains remain (Cave and Schuetze 2021). Employers have had difficulties recruiting workers not only with advanced technical skills but also those with low and middle level skills. Countries are increasingly looking toward immigration policy to mitigate the labor shortage crises (see Hogarth 2021 and Hooper 2021) with the new variant Omicron adding to the complexities (Douglas et al. 2021). Attracting workers from other countries, who are known as international migrant laborers, may assist in alleviating labor shortages. According to Amo-Agyei (2020), noting the most recent estimate by the International Labour Organization, there are 164 million migrant laborers globally. In many countries, they are a major proportion of the workforce, with contributions to economies both where they work and in terms of remittances to their countries of origins. Nevertheless, many migrant workers face inequality in terms of a wage gap (being paid less than the workers from the particular nation) among other discriminatory practices (Costa and Martin 2018). For example, migrant laborers in High Income Countries (HICs) earn, on the average, approximately 12.6% less than nationals, with the wages earned by migrants widening over the past 6 years or so in many HICs (see Amo-Agyei 2020). Furthermore, migrant laborers are among the most negatively affected workers by the economic recession due to the COVID-19 pandemic, in terms of job losses and a decrease in earnings for those who have been able to stay employed. This is happening despite the fact that the United Nations’ Sustainable Development Goals (SDGs), in the framework of the UN agenda for 2030, have as their targets 8.5 and 8.8: equal pay for work of equal value and protected labor rights for all workers, including migrant workers (Smale 2017; Amo-Agyei 2020). This chapter aims to integrate and advance two streams of literature, which have received significant attention in the pandemic: that of incorporating labor into supply chain network modeling, analysis, and computations, which is a major theme of this book, and problems of human migration, which have been exacerbated under COVID-19 (cf. Nagurney et al. 2020, 2021a,b; Cappello et al. 2021). Specifically, in this chapter, a supply chain network optimization model is described that captures the profit-maximizing behavior of a firm with respect to its supply chain network activities of production at multiple sites, the transport of the product to multiple storage sites, the storage at these facilities, and, finally, the ultimate distribution of the product to multiple points of demand. Associated with each of the supply chain network activities is a bound on domestic labor availability with possible investment in labor migration from other countries to attract workers. The migrants are responsive to the wages that they are told they will be paid for the respective supply chain network activities, as well as to the investments made. Such investments can include assistance with visas, additional training to bring the laborers to the same level of productivity as the domestic labor, assistance with relocation, enhanced marketing about various job openings, etc. (see Cave and Schuetze 2021). Associated with each supply chain link, hence, is a migrant attraction investment function that is distinct for each country and each link activity

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of production, transportation, storage, or distribution. This is quite reasonable since some migrants, for example, may be more or less adverse to moving across a greater distance from their present country of origin to work. A budget constraint associated with the firm’s total possible investment outlay is also included. Here, for the sake of flexibility, and, in order to evaluate what is happening in practice in some parts of the world, the model allows for wages paid to international migrants to be different (or the same) as the wages paid to the domestic workers. Distinct wages have been noted, for example, in the agricultural sector in the pandemic (Costa 2021) as well as other sectors with migrants sometimes being paid much lower wages than nationals (Asian Development Bank Institute et al. 2021). In addition, in the international labor migration attraction investment functions, the wages told to the prospective migrants can be the true ones or not. Hence, I allow for the investigation computationally of the effects of different wage scenarios (and wage information) and investments in attracting international migrant laborers on the profits of the firm, the product flows, the labor volumes of both the nationals and the international migrant workers, and the prices that the consumers pay for the product at different demand markets. In particular, in terms of the investigation of different wage scenarios, the model allows for the quantification of the impacts of: (1) Paying international migrants the same wages for corresponding activities as the domestic workers are paid, under the scenario that the international migrant laborers are informed honestly of the wages that they will be paid. (2) Paying international migrants less than the domestic workers are paid, but they are informed of this honestly before they migrate for work. (3) Paying the international migrants less than the domestic workers but being dishonest as to what wages they will receive in order to attract them. The supply chain optimization model is sufficiently flexible to allow for “honesty” in terms of the actual wages that the international migrants will be paid coming from certain countries and not telling others their true wages. However, the above three cases can be viewed as primary ones. Note that workers, migrants, or domestic ones should be paid the prevailing wages, and at least the minimum wage, if it exists, but in many cases they are not (cf. Swanson et al. 2021). This unfairness, in terms of payment of proper wages to migrants, is unethical and can even be criminal if there are minimum wage regulations. Furthermore, it leads not only to wage gaps but can also exacerbate worker exploitation and human trafficking, which are not directly the focus of this chapter, although herein there are contributions to this literature on the periphery (cf. Konrad et al. 2017; Winterdyk 2020; Dimas et al. 2021; LeBaron 2021, and the references therein). The numerical examples that are solved are motivated by shortages of labor in the agricultural sector, an issue in many parts of the world (see Schulte and Pitt 2021; Elkin et al. 2021). The novelty of the contributions in this chapter is supported by the following: (1) This is the first optimization model to integrate a supply chain network and investments in attracting international migrant labor.

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(2) The model can handle as many countries as are of interest in attracting migrant labor from. (3) The international migrant attraction functions can differ for each supply chain network activity and country from which the migrants originate from. (4) The model can handle different wages for national/domestic labor and for international migrant labor. The model can also evaluate impacts of truthful versus untruthful wage information provided to potential international migrants, as implemented in the attraction functions. (5) The theoretical framework is that of variational inequalities and this work is one of the very few that includes nonlinear constraints in the model and these can arise due to the form that the international migrant attraction functions take. (6) The solution of a series of numerical examples, inspired by a high value agricultural product—that of truffles, having a variety of the above features, via the proposed algorithm with nice features for implementation, yields interesting insights. The framework constructed here is inspired by issues that have become more notable in the pandemic such as limited national labor in many countries for various supply chain network activities of firms, which is handled by tightening the link labor bounds in the model, as need be; restrictions due to border controls and other regulations to decrease the chances of the spread of the coronavirus that causes COVID-19, which is handled by our model through the ability to reduce the number of countries from which migrant labor can be attracted, while, at the same time, allowing for laborers from other countries; and a reduced budget for attracting labor faced by a firm in the pandemic due to financial challenges. The latter is handled, as need be, by reducing the budget parameter in the model. Importantly, and as already emphasized, the model allows for the setting of the same or different wages for national laborers and international migrants, as well as the wage information that the international migrants are given. Such wage differentials have attracted the interest of policy-makers in the pandemic. Of course, the supply chain network optimization model that is constructed here is also relevant to times of large labor resources, the free movement of workers across borders, and rich financial budgets for attracting migrant labor, through the appropriate setting of various parameters in the model. The chapter is organized as follows. In Section 10.2, the investment supply chain network optimization model in attracting migrant laborers from multiple countries is developed and the optimality conditions are shown to satisfy a variational inequality. The model, under the above-described wage settings, can computationally reveal the impacts of distinct wage payment schemes and truthfulness in terms of the revelation of actual wages to be paid to international migrants. Some qualitative properties are also presented. An alternative variational inequality is then derived. The realization of the algorithm for the solution of the alternative variational inequality is given in Section 10.3. The algorithm results in closed form expressions for the variables (the product path flows, the investments, as well as several sets of Lagrange multipliers). Section 10.4 presents the solutions to three series of numerical examples, which are inspired by a valuable agricultural product—that of truffles, which are now being

10.2 The Supply Chain Network Model with Investments in Attracting. . .

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cultivated in the UK. Section 10.5 summarizes the results, presents the conclusions and possibilities for future research. Section 10.6 is the Sources and Notes for this chapter.

10.2 The Supply Chain Network Model with Investments in Attracting Migrant Labor There are nM possible production locations, nD possible storage locations, and J demand markets. The supply chain network is represented by a graph G = [N , L ], where N is the set of nodes and L is the set of links. Links are denoted by a, b, etc. The supply chain network activities of production, transportation, storage, and distribution take place on the links. Please refer to Figure 10.1 for the problem’s supply chain network topology.

Fig. 10.1 The supply chain network topology for the model

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Table 10.1 Notation for the supply chain network model with international migrant labor— parameters Notation Parameters wa1 Hourly wage for a unit of labor on link a paid to domestic workers, ∀a ∈ L wa2 Hourly wage for a unit of labor on link a paid to international migrant workers, ∀a ∈ L

j

w˜ a αa δap l¯a1 B

Hourly wage for a unit of labor on link a that international migrant workers are told they will be paid, ∀a ∈ L , for all countries j = 1, . . . , L The link productivity on link a, ∀a ∈ L , which maps labor hours into product flow Indicator taking on the value 1 if link a is contained in path p and 0, otherwise The maximum available domestic labor hours locally for work associated with link a, a∈L The amount of financing in the budget for investments in attracting migrant labor from different countries

A path p is defined as a sequence of directed links from node 1 to a demand market node k, k = 1, . . . , J . There are nL links in the supply chain and nP paths. Denote the set of paths to demand market node k by Pk and the set of all paths by P. Different specific applications will differ in their supply chain network topologies by the number of relevant nodes and links and costs and demand price functions, etc. There are L countries that the labor can migrate from. I do not include the country the firm under consideration is located in this set of country indices. The fundamental notation for the parameters is given in Table 10.1. The notation for the variables is given in Table 10.2. All vectors are column vectors. Note that, as mentioned in the Introduction, for purposes of generality and relevance, I allow the wages for international migrant laborers to be the same (or different) from the wages of domestic laborers. In other words, the model allows for the flexibility of having wa1 = wa2 , for some a ∈ L , whereas for other links b ∈ L , one may have: wb1 = wb2 . Also, the wage used in the attraction function in attracting international migrant workers from a specific country j for a supply j chain activity a, w˜ a , can be the actual wage that will be paid (or not). I investigate the implications of different wage settings in the numerical examples, for which the data and computed solutions are reported in Section 10.4. In Table 10.3, the notation for the functions is provided. I first present the constraints and then I construct the objective function of the firm. Conservation of Flow Equations The sum of the product path flows to each demand market must be equal to the demand at the demand market: xp = dk , k = 1, . . . , J, (10.1) p∈Pk

10.2 The Supply Chain Network Model with Investments in Attracting. . .

239

Table 10.2 Notation for the supply chain network model with international migrant labor— variables Notation Variables dk The demand for the product at demand market k, k = 1, . . . , J ; group the demands J into the vector d ∈ R+ xp The product flow on path p, ∀p ∈ P ; group the product path flows into the vector nP x ∈ R+ fa The product flow on link a, ∀a ∈ L; group the product link flows into the vector n f ∈ R+L 1 la The hours of domestic labor available for link a supply chain activity, ∀a ∈ L la2 The hours of international migrant labor available for link a supply chain activity, ∀a ∈ L j va The investment in attracting migrant labor from country j , j = 1, . . . , L for link a, Ln ∀a ∈ L . The investments in attracting labor are grouped into the vector v ∈ R+ L ζ The nonnegative Lagrange multiplier associated with the budget constraint λa The nonnegative Lagrange multiplier associated with the bound on domestic labor nL hours on link a; group the Lagrange multipliers into the vector λ ∈ R+ δa1 The nonnegative Lagrange multiplier associated with the constraint guaranteeing that la1 is nonnegative on link a; group all such Lagrange multipliers into the vector n δ 1 ∈ R+L δa2 The nonnegative Lagrange multiplier associated with the constraint guaranteeing that la2 is nonnegative on link a; group all such Lagrange multipliers into the vector n δ 2 ∈ R+L Table 10.3 Notation for the supply chain network model with international migrant labor— functions Notation cˆa (f ) ρk (d) j j j ga (w˜ a , va )

Functions The operational cost function associated with link a, ∀a ∈ L The demand price function for the product at demand market k, k = 1, . . . , J The investment function associated with link a in attracting migrant workers from country j to work on link a of the firm’s supply chain network, ∀a ∈ L and j = 1, . . . , L

with all the path flows being nonnegative: xp ≥ 0,

∀p ∈ P.

(10.2)

In addition, the amount of product flow on each link must be equal to the sum of product flows on paths that contain that link: fa =

p∈P

xp δap ,

∀a ∈ L .

(10.3)

240

10 International Migrant Labor and Supply Chains

Labor Constraints Furthermore, as throughout this book, linear production functions are assumed, but they are extended to differentiate between domestic labor and international migrant labor. Hence, one has that fa = αa (la1 + la2 ),

∀a ∈ L .

(10.4)

According to (10.4), the amount of product flow on a supply chain link is equal to the hours of labor availability on the link, which represents a supply chain network activity, times the productivity factor of labor associated with the link. Here, it is assumed that, through investments, the productivity of migrant labor, for a given activity, will be the same as that of the domestic workers where the firm’s supply chain network activity is located. The domestic worker labor hours available for each link activity cannot exceed the bound on domestic labor hours of availability, that is: la1 ≤ l¯a1 ,

∀a ∈ L .

(10.5a)

Furthermore, we must have that the domestic labor hours available are nonnegative: la1 ≥ 0,

∀a ∈ L .

(10.5b)

Also, the amount of international migrant labor hours available for a link a is:

la2 =

L

j

j

j

ga (w˜ a , va ),

∀a ∈ L ,

(10.6a)

j =1

with nonnegativity of international migrant labor hours also holding: la2 ≥ 0,

∀a ∈ L .

(10.6b)

Constraints on Investments The investments by the firm in attracting international migrant labor must be nonnegative: j

va ≥ 0,

j = 1, . . . , L,

∀a ∈ L .

Finally, the firm’s budget constraint in terms of attracting migrants is:

(10.7)

10.2 The Supply Chain Network Model with Investments in Attracting. . . L

j

va ≤ B.

241

(10.8)

j =1 a∈L

Here, for simplicity, it is assumed that all the links a ∈ L of the firm’s supply chain network are located in the same country. If they are not, the above model could still be applicable/relevant, but one would have to make sure that the summations over the countries in the constraints above would exclude the country that the link a is in. The Optimization Problem The optimization problem faced by the firm in optimizing its supply chain network can now be stated. Specifically, the firm seeks to maximize its objective function, denoted by U , which represents its profits, subject to the constraints (10.1)–(10.8): U=

Maximize

J

ρk (d)dk −



cˆa (f ) −

j

va −

j =1 a∈L

a∈L

k=1

L



(wa1 la1 + wa2 la2 ).

a∈L

(10.9) J The first term after the equal sign in (10.9), ρ (d)d , is the revenue; the k k k=1  second term, a∈L cˆa (f ), represents the operational costs, whereas the third term, L   j 1 1 2 2 j =1 a∈L va , is the total investments. The last term, a∈L (wa la + wa la ), in (10.9) is the payout to labor for wages. Assume that the demand price functions are monotone decreasing and that each ρk (d)dk is concave for each k, and that the operational link cost functions are convex with both the demand price functions and the operational cost functions being continuously differentiable. Furthermore, assume that the investment functions j j j ga (w˜ a , va ) are all concave, which is reasonable, since one can expect diminishing returns to such investments and that they are bounded. These are also assumed to be continuously differentiable. Making use of the various expressions comprising the feasible set for the above problem enables one to transform the objective function in (10.9) to be in path flow and investment amount variables only. In lieu of (10.1), one can define demand price functions ρ˜k (x) ≡ ρk (d), for k = 1, . . . , J , and, in lieu of (10.3), one can define link operational total cost functions c˜a (x) ≡ cˆa (f ), for a ∈ L . Furthermore, using (10.4) and (10.6), note that  la1

=

p∈P

xp δap

αa

−

L

j

j

j

ga (w˜ a , va ),

∀a ∈ L .

(10.10)

j =1

Then, the firm’s supply chain network optimization problem (10.9) can be reexpressed as:

242

10 International Migrant Labor and Supply Chains

U˜ (x, v) =

Maximize

J

−

)

p∈P

wa1

xp δap

−

αa

a∈L



xp −

p∈Pk

k=1





ρ˜k (x)

L

c˜a (x) −

* −

j =1

j

va

j =1 a∈L

a∈L

j j j ga (w˜ a , va )

L



wa2

L

j

j

j

ga (w˜ a , va );

j =1

a∈L

(10.11a) equivalently: U˜ (x, v) =

Maximize

J

−

) wa1

p∈P

xp δap

* +

αa

a∈L



xp −

p∈Pk

k=1





ρ˜k (x)

c˜a (x) −

L

j

va

j =1 a∈L

a∈L

(wa1 − wa2 )

L

j

j

j

ga (w˜ a , va ),

(10.11b)

j =1

a∈L

subject to: 

p∈P

xp δap

αa 

p∈P

−

J

j j ga (w˜ a , va ) ≤ l¯a1 ,

∀a ∈ L ,

(10.12a)

∀a ∈ L ,

(10.12b)

j =1

xp δap

αa

−

L

j

j

ga (w˜ a , va ) ≥ 0,

j =1 L

j

j

ga (w˜ a , va ) ≥ 0,

∀a ∈ L ,

(10.13)

j =1 L

j

va ≤ B,

(10.14)

∀p ∈ P,

(10.15)

j = 1, . . . , L; ∀a ∈ L .

(10.16)

j =1 a∈L

xp ≥ 0, j

va ≥ 0,

n

+Ln

L The feasible set K 1 ≡ {(x, v) ∈ R+P and satisfying (10.12a, b)–(10.14)}. Here, it is assumed that for each link a ∈ L , wa1 ≥ wa2 , which is reasonable since one can expect the firm to pay domestic workers at least the amount that it is paying to international migrant laborers. And, if it pays migrants the same wage that it pays the domestic workers on each link, then the fifth term after the equal sign in

10.2 The Supply Chain Network Model with Investments in Attracting. . .

243

the objective function (10.11b) is equal to 0. The objective function in (10.11b), under the assumptions, is concave and the underlying functions are continuously differentiable. The underlying feasible set K 1 is convex, since constraints (10.14), (10.15), and (10.16) are linear and the investment functions are assumed to be concave, so minus each investment function is convex. Hence, it follows from the classical theory of variational inequalities (see Kinderlehrer and Stampacchia 1980 and Nagurney 1999) that the optimal solution (x ∗ , v ∗ ) ∈ K 1 satisfies the variational inequality problem: −

L ∂ U˜ (x ∗ , v ∗ ) ∂ U˜ (x ∗ , v ∗ ) j j∗ × (xp − xp∗ ) − × (va − va ) ≥ 0, j ∂xp ∂va

∀(x, v) ∈ K 1 .

j =1 a∈L

p∈P

(10.17) By expanding out the partial derivatives of the utility functions in (10.17), one obtains the equivalent variational inequality: Determine (x ∗ , v ∗ ) ∈ K 1 , such that J k=1 p∈Pk

+

⎡

⎤ J   ∗) w1 ∂ C˜ p (x ∗ ) ∂ ρ ˜ (x l a ⎣ + δap − ρ˜k (x ∗ ) − xq∗ ⎦× xp − xp∗ ∂xp αa ∂xp

L j =1 a∈L

a∈L

 −(wa1

q∈Pl

l=1

j j j∗ ∂ga (w˜ a , va ) − wa2 ) j ∂va



  j j∗ + 1 × va − va ≥ 0,

∀(x, v) ∈ K 1 , (10.18)

where ∂ cˆb (f ) ∂ C˜ p (x) ≡ δap , ∂xp ∂fa a∈L b∈L

∀p ∈ P ,

and

∂ ρ˜l (x) ∂ρl (d) ≡ , ∀p ∈ Pk , ∀k. ∂xp ∂dk (10.19)

A solution (x ∗ , v ∗ ) ∈ K 1 to both variational inequalities (10.17) and (10.18) exists since the feasible set K 1 is compact and the underlying functions, under our imposed assumptions, are continuous. I now provide an alternative variational inequality, equivalent to the one in (10.18), which is utilized for computational purposes. The alternative variational inequality below follows from analogous results in Nagurney and Daniele (2021) and Nagurney et al. (2019). According to Table 10.2, one can associate a nonnegative Lagrange multiplier λa with each link labor bound constraint (10.12a), for each a ∈ L , and the nonnegative Lagrange multiplier ζ with the budget constraint (10.14). Then, the equivalent variational inequality to the one in (10.18) is: Determine (x ∗ , λ∗ , v ∗ , ζ ∗ , δ 1∗ , δ 2∗ ) ∈ K 2 , where n

K 2 ≡ {(x, λ, v, ζ, δ 1 , δ 2 )|(x, λ, v, ζ, δ 1 , δ 2 ) ∈ R+P such that

+nL +LnL +1+2nL

},

244

10 International Migrant Labor and Supply Chains

⎡

J

⎣

k=1 p∈Pk

J w1 ∂ C˜ p (x ∗ ) ∂ ρ˜l (x ∗ ) ∗ a + δap − ρ˜k (x ∗ ) − xq ∂xp αa ∂xp a∈L

q∈Pl

l=1

⎤ ⎡   L ∗δ   1∗ λ∗ x ap δ p p∈ P j j j∗ a ⎣¯la1 − + δap − a δap × xp − xp∗ + + ga (w˜ a , va )⎦ αa αa αa a∈L

×



j =1

a∈L

λa − λ∗a



L

+

 1+ζ

∗

− (wa1

− wa2

+ λ∗a

− δa1∗

j =1 a∈L



j∗

j

× va − va



⎡ + ⎣B −

L

j j j∗ ∂ga (w˜ a , va ) + δa2∗ ) j ∂va



⎤

  j∗ va ⎦ × ζ − ζ ∗

j =1 a∈L

+



⎡ ⎣

a∈L

+

a∈L

⎡ ⎣

L

p∈P

xp∗ δap

αa

−

L

⎤

  j j j∗ ga (w˜ a , va )⎦ × δa1 − δa1∗

j =1

⎤

  j j j∗ ga (w˜ a , va )⎦ × δa2 − δa2∗ ≥ 0,

∀(x, λ, v, ζ, δ 1 , δ 2 ) ∈ K 2 .

j =1

(10.20) Variational inequality (10.20) is now put into standard form (2.14). Set K ≡ n +n +LnL +1+2nL , X ≡ (x, λ, v, ζ, δ 1 , δ 2 ), and N = nP + nL + LnL + R+P L 1 + 2nL . Define the vector F ≡ (F1 , F2 , F3 , F4 , F5 , F6 ), with components of F1 consisting of the elements, ∀p ∈ P: ⎡

⎤ J 1 1 ˜ ∂ C (x) wa ∂ρl (x) λa δ ⎣ p + δap − ρ˜k (x) − xq + δap − a δap ⎦ ; ∂xp αa ∂xp αa αa a∈L

q∈Pl

l=1

the components of F2 consisting of elements: ⎤ ⎡  L x δ p ap p∈ P j j j ⎣¯la1 − + ga (w˜ a , va )⎦ , αa

a∈L

∀a ∈ L ;

j =1

the components of F3 being:  1+ζ

− (wa1

− wa2

+ λa − δa1

j j j ∂ga (w˜ a , va ) + δa2 ) j ∂va

 ,

∀a ∈ L ,

    j and the component of F4 being: B − L v j =1 a∈L a . Finally, the components   x δ p ap j j j of F5 are the elements: p∈P − L ˜ a , va ), for a ∈ L , whereas j =1 ga (w αa

10.3 The Algorithm

245

 j j j∗ the components of F6 are: L ˜ a , va ), for a ∈ L . With such definitions, j =1 ga (w variational inequality (10.20), clearly, coincides with variational inequality (2.14).

10.3 The Algorithm The algorithm that is implemented and applied in the next section to compute solutions to numerical examples, with a goal of gaining insights into impacts of wage settings and investments in attracting international migrant labor, is the modified projection method of Korpelevich (1977), which is outlined in Chapter 2 and in the Appendix. For easy reference and completeness, I spell out the form that it takes in the solution of the model as governed by variational inequality (10.20). Notably, each iteration of the algorithm yields closed form expressions for each of the variables (the product path flows, the Lagrange multipliers associated with the domestic labor bounds, the link investments in attracting migrant labor from different countries, and the Lagrange multiplier associated with the investment budget constraint) at each of the two steps. Explicit Formulae at Iteration τ for the Product Path Flows in Step 1 The closed form expression for each path flow in Step 1 in the solution of variational inequality (10.20) is, for each p ∈ Pk , ∀k: ⎧ ⎪ L ⎨ ∂ ρ˜il (x τ −1 ) τ −1 ∂ C˜ p (x τ −1 ) τ x¯p = max 0, xpτ −1 + η(ρ˜k (x τ −1 ) + xq − ⎪ ∂xp ∂xp ⎩ l=1 q∈P i l

⎞⎫ ⎬ ) w 1 + λτ −1 − δa1(τ −1) * a a δap ⎠ . − ⎭ αa

(10.21)

a∈L

Explicit Formulae at Iteration τ the Link Domestic Labor Bound Lagrange Multipliers in Step 1 The closed form expression for each Lagrange multiplier for each link a ∈ L in Step 1 at an iteration τ is: λ¯ τa

 + ) *% L τ −1 p∈P xp δap j j j τ −1 τ −1 1 = max 0, λa + η − l¯a + − ga (w˜ a , va ) . αa j =1

(10.22)

246

10 International Migrant Labor and Supply Chains

Explicit Formulae at Iteration τ for Investments in Attracting International Migrant Labor in Step 1 The closed form expression for the investment in attracting migrant labor from different countries in Step 1 is, for each country j ; j = 1, . . . , L, and for each link a ∈ L : , jτ v¯a = max 0, vaτ −1 + η(−1 − ζ τ −1 + (wa1 − wa2 + λτa −1 − δa1(τ −1) + δa2(τ −1) ) j

j τ −1

j

∂ga (w˜ a , va j

)

$ .

(10.23)

∂va

Explicit Formula at Iteration τ for the Lagrange Multiplier Associated with the Budget Constraint in Step 1 The closed form expression for the Lagrange multiplier associated with the budget constraint in Step 1 at an iteration τ is: + ) *% L j τ −1 ¯ζ τ = max 0, ζ τ −1 + η va −B .

(10.24)

j =1 a∈L

Explicit Formulae at Iteration τ for the Lagrange Multipliers Associated with Domestic Labor Nonnegativity Constraints in Step 1 The closed form expression for the Lagrange multiplier associated with the domestic labor nonnegativity constraint in Step 1 is, for each link a ∈ L : δ¯a1τ

+ )  *% L τ −1 p∈P xp δap j j j τ −1 1(τ −1) = max 0, δa +η − + ga (w˜ a , va ) . αa

(10.25)

j =1

Explicit Formulae at Iteration τ for the Lagrange Multipliers Associated with International Migrant Labor Nonnegativity Constraints in Step 1 The closed form expression for the Lagrange multiplier associated with the international migrant labor nonnegativity constraint in Step 1 is, for each link a ∈ L : + ) *% J j j j τ −1 ¯δa2τ = max 0, δa2(τ −1) + η − ga (w˜ a , va ) .

(10.26)

j =1

The analogues of expressions (10.21) through (10.26) for Step 2 of the modified projection method follow in a straightforward manner.

10.4 High Value Food Product Numerical Examples

247

10.4 High Value Food Product Numerical Examples The modified projection method is coded in FORTRAN and a Linux system at the University of Massachusetts Amherst used for the computation of solutions to the numerical examples satisfying variational inequality (10.20). The demand for each demand market is initialized at 40 and equally distributed among the paths connecting each demand market from the origin node 1 (the Firm). The investments are initialized to 0.00 as are all the Lagrange multipliers. The algorithm is considered to have converged if the absolute difference of the path flows differed by no more than 10−7 and the same for investments and the Lagrange multipliers. The numerical examples are inspired by recent issues surrounding agricultural supply chains in the UK. The UK has been pummeled with shortfalls in labor due to COVID-19 as well as Brexit. I am particularly interested in fresh produce, which requires minimal processing, and, hence, the production links correspond also to harvesting and some basic packaging. Even prior to the pandemic and Brexit, many domestic laborers in the UK shied away from farm work (see Butler 2020) with only 1% of pickers and packers being British citizens in 2019, with the percentage rising to 11% in the summer of 2020 due to a British campaign. Obtaining sufficient truckers to transport produce and other perishables has also been a challenge in the UK and this has drawn attention in the news with the government getting involved and issuing special visas to both farmers and truckers (Segal 2021; Akintade and Essien 2021). Fresh fruit and vegetables, as well as milk, have gone to waste because of canceled or delayed deliveries (see Mayes 2021). According to Fernandez-Reino and Rienzo (2021), in 2019, an estimated 18% of workers in the UK were migrant workers, with migrants being over-represented in transport and storage at 28%. It is clear that international migrant labor is essential to supply chains, including agricultural ones, in the UK as well as beyond—in Australia (see Sherrell and Coates 2021), the USA (see Economic Research Service 2021), and the European Union (Kalantaryan 2021). The numerical examples are quite broad with the data being motivated by a very interesting agricultural product, now being grown in the UK—truffles (Davison 2018). Truffles are a high value agricultural product and are considered a delicacy, with challenges associated with production and harvesting (Floyd 2019). As noted in Cormack (2021), in the Fall of 2021, due to a shortage, white truffle prices were about $4500 a pound, whereas, in 2019, white truffle prices were in the range $1100 to $1200 a pound. Different types of truffles command different prices (see Truffle.Farm 2021) but their costliness and desirability are well-known. They are considered among the most expensive foods on the planet (Floyd 2019). Although southern Europe is a primary growth area for this agricultural delicacy, with changes in climate plus advances in science, the UK is also becoming a location for the farming of truffles. The cost and price data, as well as the wages and the profits, in the numerical examples are in British pounds. The unit for the truffle product flows is a pound of weight. According to Elison (2021), vegetable pickers in the UK, due to shortages of labor, are being paid 30 pounds an hour to pick the produce.

248

10 International Migrant Labor and Supply Chains

The numerical examples are organized into three series. Series 1 Numerical Examples In this series of examples, I explore the impacts of different wage settings, both truthful and untruthful ones. Example 10.1 (Baseline Example: Migrant Workers Earn the Same Wage as Domestic Laborers and Migrants Are Told Their Truthful Wages in the Attraction Functions) Example 10.1 has the supply chain network topology depicted in Figure 10.2. The agricultural firm has two locations requiring production/harvesting of fresh produce, which consists of truffles, which then must be transported to a single distribution center from which the produce is distributed to three demand markets.

Fig. 10.2 The supply chain network topology for the numerical examples

The operational link cost functions are: cˆa (f ) = 2.5fa2 , cˆe (f ) = fe2 + 2fe ,

cˆb (f ) = 2.5fb2 , cˆf (f ) = .5ff2 ,

The demand price functions are:

cˆc (f ) = .5fc2 , cˆg (f ) = .5fg2 ,

cˆd (f ) = .5fd2 , cˆh (f ) = .5fh2 .

10.4 High Value Food Product Numerical Examples

ρ1 (d) = −5d1 + 800,

249

ρ2 (d) = −5d2 + 850,

ρ3 (d) = −5d3 + 900.

αc = .35,

αd = .35,

αe = .60,

αg = .36,

αh = .40.

The α link parameters are: αa = .55,

αb = .50,

αf = .38,

I assume that the supply chain firm considers a single country to obtain international migrants from, but the specific country can differ from supply chain network activity to activity. Hence, one can suppress the superscript j in the international migrant attraction functions and in the investments. The international migrant attraction functions are of the form: ga (w˜ a , va ) = w˜ a va −γa va2 , for all links a ∈ L . These functions are concave. The γ parameters in these functions are: γa = .2,

γb = .2,

γc = .4,

γd = .4,

γe = .3

γf = .4,

γg = .4,

γh = .4.

The wages in Example 10.1 are: wa1 = wa2 = w˜ a = 30,

wb1 = wb2 = w˜ b = 20,

wc1 = wc2 = w˜ c = 18,

wd1 = wd2 = w˜ d = 18,

we1 = we2 = w˜ e = 17,

wf1 = wf2 = w˜ f = 19,

wg1 = wg2 = w˜ g = 19,

wh1 = wh2 = w˜ h = 19.

Hence, in Example 10.1, it is assumed that the international migrants are paid the same hourly wage for each supply chain network activity as are the domestic workers and that the international migrants are informed of the truthful wage when contacted about migrating for work. The bounds on domestic labor are: l¯a1 = 100, for all links in the supply chain from a through h and the budget B = 1000. The paths (see Figure 10.2) are defined as follows: path p1 = (a, c, e, f ), path p2 = (b, d, e, f ), path p3 = (a, c, e, g), path p4 = (b, d, e, g), path p5 = (a, c, e, h), and path p6 = (b, d, e, h). The modified projection method computes the following optimal product path flow pattern: xp∗1 = 9.60,

xp∗2 = 10.41,

xp∗3 = 11.74,

xp∗5 = 17.51,

xp∗6 = 18.32.

xp∗4 = 12.55,

250

10 International Migrant Labor and Supply Chains

The equilibrium link flows and labor values are reported in Table 10.4, whereas the optimal investments and Lagrange multipliers λ∗a , for all a ∈ L , are reported in Table 10.5. The domestic labor on supply chain links of c, d, and e is at the capacity of 100, with international migrant labor hours being positive on those links. Hence, the associated Lagrange multipliers λs on those links are positive. Note that all other Lagrange multipliers, that is, the elements of δa1∗ and δa2∗ for all a ∈ L , and ζ ∗ are equal to 0.00. The total investment outlay is: 3.69. The demand price at the first demand market is 699.99 and at the second demand market the price is: 728.52, with the demand price at the third demand market being: 756.66. The corresponding respective demands are: 20.00, 24.30, 35.84. The firm earns a profit of: 41,453.10. Example 10.2 (Domestic Workers Earn a Higher Wage at Production Sites than Migrants and Are Told Their Truthful Wages) Example 10.2 has the same data as that in Example 10.1, except that now the domestic workers earn a higher wage at the two production sites with wa1 = 40 and wb1 = 30. The modified projection method yields the following optimal product path flow pattern: xp∗1 = 9.60,

xp∗2 = 10.40,

xp∗3 = 11.74,

xp∗5 = 17.51,

xp∗6 = 18.32.

xp∗4 = 12.55,

The optimal link flows and labor values are reported in Table 10.4 and the optimal investments and Lagrange multipliers λ∗a , for all a ∈ L , are reported in Table 10.5. In Example 10.2, one sees that the firm now invests in attracting international migrants also on links a and b, the production/picking sites for truffles of the firm. It, hence, obtains international migrant laborers for links a through e, whereas in Example 10.1, it had international migrants working only on links c, d, and e. Since domestic workers are now more expensive, the firm hires exclusively migrant laborers on the production links a and b. The total investment outlay increases from 3.69 to 10.39. One now has δa1∗ and δb1∗ being positive at values of 9.97 and 9.95, respectively. All other δ ∗ values, including the δ 2∗ s as well as η∗ , are equal to 0.00. The demand price at the first demand market is now 700.00. The demand price at the second demand market is: 728.54, with the demand price at the third demand market being: 756.67, with the corresponding respective demands of: 20.00, 24.29, and 35.83. The firm earns a profit of: 41,444.64. Observe that the profit has now decreased, as compared to that in Example 10.1. Example 10.3 (Domestic Workers Earn a Higher Wage at Production Sites Than Migrants But Migrants Are Told Untruthfully That They Will Be Paid the Same Wage as the Domestic Workers)

10.4 High Value Food Product Numerical Examples

251

In Example 10.3, I investigate the impact of the firm being untruthful. Specifically, the firm now tells the international migrant laborers that it will pay them the same (higher) wage at each production site that it is paying its domestic laborers, Table 10.4 Optimal link flows and domestic and international migrant labor values for Examples 10.1, 10.2, 10.3 Optimal value Example 10.1 38.86 41.28 38.86 41.28 80.13 20.00

Example 10.2 38.85 41.27 38.85 41.27 80.12 20.00

Example 10.3 38.85 41.27 38.85 41.27 80.13 20.00

fg∗ fh∗ la1∗ lb1∗ lc1∗ ld1∗ le1∗ lf1∗

24.30 35.84 70.65 82.56 100.00 100.00 100.00 52.64

24.29 35.83 0.00 0.00 100.00 100.00 100.00 52.63

24.29 35.83 0.00 0.00 100.00 100.00 100.00 52.63

lg1∗ lh1∗ la2∗ lb2∗ lc2∗ ld2∗ le2∗ lf2∗

67.49 89.59 0.00 0.00 11.01 17.94 33.56 0.00

67.48 89.58 70.64 82.54 11.01 17.91 33.54 0.00

67.48 89.58 70.64 82.55 11.01 17.92 33.54 0.00

lg2∗ lh2∗

0.00 0.00

0.00 0.00

0.00 0.00

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗

but it actually will pay them less. The data, hence, are exactly as in Example 10.2, but now we have that w˜ a = 40 and w˜ b = 30. The modified projection method computes the product path flow pattern: xp∗1 = 9.60,

xp∗2 = 10.40,

xp∗3 = 11.74,

xp∗5 = 17.51,

xp∗6 = 18.32.

xp∗4 = 12.55,

Please refer to Table 10.4 for the computed link flows and labor values and to Table 10.5 for the computed investments and Lagrange multipliers λ∗a , for all a ∈

252

10 International Migrant Labor and Supply Chains

L . The δa2∗ = 0.00 for all links in the supply chain network and ζ ∗ = 0.00, as in the preceding examples. As in Example 10.2, the firm uses exclusively international migrant laborers at its production/picking sites on links a and b with the Lagrange multipliers δa1∗ = δb1∗ = 9.97. The total investment outlay now decreases to 8.27, since not as much is needed as in Example 10.2 due to the untruthfulness about the wages that migrants Table 10.5 Optimal link international migrant attraction investments and domestic labor bound Lagrange multipliers for Examples 10.1, 10.2, and 10.3 Notation va∗ vb∗ vc∗ vd∗ ve∗ vf∗ vg∗ vh∗ λ∗a λ∗b λ∗c λ∗d λ∗e λ∗f

λ∗g λ∗h

Optimal value Example 10.1 0.00 0.00 0.62 1.02 2.05 0.00

Example 10.2 2.39 4.31 0.62 1.02 2.05 0.00

Example 10.3 1.78 2.80 0.62 1.02 2.05 0.00

0.00 0.00 0.00 0.00 0.06 0.06 0.06 0.00

0.00 0.00 0.00 0.00 0.06 0.06 0.06 0.00

0.00 0.00 0.00 0.00 0.06 0.06 0.06 0.00

0.00 0.00

0.00 0.00

0.00 0.00

will be paid for work at the production sites for truffles. Migrant laborers, again, work on links a through e. The demand price at the first demand market remains at 700.00. The demand price at the second demand market is: 728.53, with the demand price at the third demand market being: 756.67, with the corresponding respective demands of: 20.00, 24.29, and 35.83. The demand prices and the demands are essentially unchanged for their values in Example 10.2. The firm earns a profit of: 41,447.33. Observe that the profit now increases suggesting that, without oversight, “cheating can pay.” The product path flows are essentially the same in Examples 10.1, 10.2, and 10.3 in this series due to the demand price functions, in part, with the reallocation of labor from domestic ones to migrants. Series 2 Numerical Examples In this series of examples, I investigate the impacts of increases in the price that consumers are willing to pay for the truffles at the demand markets. Specifically,

10.4 High Value Food Product Numerical Examples

253

I conduct sensitivity analysis associated with the intercept in the demand price functions. For each of the examples in this series, I report the computed link flows and the domestic and international migrant labor values in Table 10.6 and the migrant attraction function investments and domestic labor Lagrange multipliers in Table 10.7. Example 10.4 (Example 10.1 Data with All the Demand Price Function Intercepts Doubled) The data in Examples 10.4 through 10.7 are identical to the data in Example 10.1 but with the intercept for each demand function increased. In Example 10.4, I consider a doubling of the intercept so that the demand price functions are now: ρ1 (d) = −5d1 + 1600,

ρ2 (d) = −5d2 + 1700,

ρ3 (d) = −5d3 + 1800.

The modified projection method yields the following product path flow pattern: xp∗1 = 22.90,

xp∗2 = 23.70,

xp∗3 = 27.32,

xp∗5 = 39.33,

xp∗6 = 40.13.

xp∗4 = 28.12,

The demand price at the first demand market is: 1366.98 and at the second demand market the price is: 1422.80, with the demand price at the third demand market being: 1482.17. The respective demands at the demand markets are: 46.60, 55.44, and 79.46. The consumers are now willing to pay a higher price for the truffles at each demand market, the demand increases at each market, and the firm now enjoys a profit of: 171,789.94. This profit is more than four times that in Example 10.1. The domestic labor on each supply chain link is at its bound of 100, with the associated Lagrange multipliers now all positive. In addition, there is a positive investment on each link for attracting migrant labor and all values of migrant labor hours are positive on each link. The total investment outlay is: 57.96. With the demand being more than twice that in Example 10.1 at each demand market, all the domestic labor is hired and migrant labor is also hired across all the supply chain network links from production through distribution. Example 10.5 (Example 10.1 Data with All the Demand Price Function Intercepts Tripled) In Example 10.5, the intercept of each demand price function is triple that of the corresponding intercept in Example 10.1. Hence, demand price functions are now: ρ1 (d) = −5d1 + 2400,

ρ2 (d) = −5d2 + 2550,

ρ3 (d) = −5d3 + 2700.

The modified projection method computes the product path flow pattern:

254

10 International Migrant Labor and Supply Chains

xp∗1 = 24.05,

xp∗2 = 24.84,

xp∗3 = 30.74,

xp∗5 = 46.28,

xp∗6 = 47.07.

xp∗4 = 31.53,

The demand price at the first demand market is now: 2155.57. The price at the second demand market is: 2238.66, with the demand price at the third demand market being: 2326.58. The corresponding demands are: 48.89, 62.27, and 93.35. The firm now enjoys a profit of: 349,417.22, which is more than double the profit in Example 10.4. Clearly, doing marketing to enhance consumers’ value of truffles can yield economic benefits. The domestic labor on each supply chain link is, as in Example 10.4, at its bound of 100, with the associated Lagrange multipliers again all positive. In addition, there is a positive investment on each link for attracting migrant labor and all values of migrant labor hours are positive on each link. The total investment outlay is now: 86.15, which exceeds that in Example 10.4 by more than 25%. The Lagrange multipliers: δa1∗ and δa2∗ are equal to 0.00 for all links a ∈ L , and ζ ∗ remains at 0.00. Example 10.6 (Example 10.1 Data with All the Demand Price Function Intercepts Quadrupled) The demand price functions in Example 10.6 have each demand price function intercept in Example 10.1 quadrupled so that the functions are: ρ1 (d) = −5d1 + 3200,

ρ2 (d) = −5d2 + 3400,

ρ3 (d) = −5d3 + 3600.

The algorithm computes the product path flow pattern: xp∗1 = 21.62,

xp∗2 = 22.41,

xp∗3 = 30.58,

xp∗5 = 48.87,

xp∗6 = 49.66.

xp∗4 = 31.37,

The demand price at the first demand market is now: 3090.21. The price at the second demand market is: 2979.84. The demand price at the third demand market is: 3205.88. The corresponding respective demands are: 44.03, 61.96, and 98.53. The firm’s profit is: 525,725.1, which is more than 50% higher than the profit in Example 10.5. The domestic labor on each supply chain link is, as in Examples 10.4 and 10.5, at its bound of 100, with the associated Lagrange multipliers all positive. There is, again, a positive investment on each link for attracting migrant labor and all values of migrant labor hours are positive on each link. The total investment outlay is now 86.50, which is almost identical to the investment total in Example 10.5. The Lagrange multipliers: δa1∗ and δa2∗ are equal to 0.00 for all links a ∈ L , and ζ ∗ is also 0.00. Note that in all the numerical examples, to this point, and the following ones, the variational inequality (10.20) holds with excellent accuracy. Furthermore,

10.4 High Value Food Product Numerical Examples

255

Table 10.6 Optimal link flows and domestic and international migrant labor values for Examples 10.4 through 10.7 Optimal value Example 10.4 89.54 91.96 89.54 91.96 181.50 46.60

Example 10.5 101.07 103.44 101.07 103.44 204.51 48.89

Example 10.6 101.07 103.45 101.07 103.45 204.52 44.03

Example 10.7 101.08 103.45 101.08 103.45 204.54 39.18

fg∗ fh∗ la1∗ lb1∗ lc1∗ ld1∗ le1∗ lf1∗

55.44 79.46 100.00 100.00 100.00 100.00 100.00 100.00

62.27 93.35 100.00 100.00 100.00 100.00 100.00 100.00

61.96 98.53 100.00 100.00 100.00 100.00 100.00 100.00

61.65 103.71 100.00 100.00 100.00 100.00 100.00 100.00

lg1∗ lh1∗ la2∗ lb2∗ lc2∗ ld2∗ le2∗ lf2∗

100.00 100.00 62.81 83.91 155.84 162.73 202.50 22.64

100.00 100.00 83.76 106.88 188.76 195.55 240.83 28.65

100.00 100.00 83.77 106.89 188.78 195.56 240.83 15.87

100.00 100.00 83.78 106.91 188.80 195.59 240.83 3.10

lg2∗ lh2∗

54.00 98.64

72.97 133.39

72.10 146.32

71.25 159.27

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗

the prices obtained are reasonable, since truffles are such an exclusive, high value agricultural product. Example 10.7 (Example 10.1 Data with All the Demand Price Function Intercepts Quintupled) And, finally, in Example 10.7, each demand price function intercept is 5 times the value found in Example 10.1, and as below: ρ1 (d) = −5d1 + 4000,

ρ2 (d) = −5d2 + 4250,

ρ3 (d) = −5d3 + 4500.

The computed optimal product path flow pattern is: xp∗1 = 19.19,

xp∗2 = 19.99,

xp∗3 = 30.43,

xp∗5 = 51.46,

xp∗6 = 52.25.

xp∗4 = 31.22,

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10 International Migrant Labor and Supply Chains

Table 10.7 Optimal link international migrant attraction investments and domestic labor bound Lagrange multipliers for Examples 10.4 through 10.7 Notation va∗ vb∗ vc∗ vd∗ ve∗ vf∗ vg∗ vh∗ λ∗a λ∗b λ∗c λ∗d λ∗e λ∗f

λ∗g λ∗h

Optimal value Example 10.4 2.12 4.39 11.70 12.53 17.03 1.22

Example 10.5 2.85 5.67 16.64 18.33 28.33 1.56

Example 10.6 2.85 5.67 16.64 18.33 28.33 0.85

Example 10.7 2.85 5.67 16.65 18.34 28.33 0.16

3.04 5.93 0.03 0.05 0.12 0.13 0.15 0.06

4.21 8.56 0.03 0.06 0.21 0.30 395.82 0.06

4.16 9.67 0.03 0.06 0.21 0.30 907.84 0.05

4.10 10.87 0.03 0.06 0.21 0.30 1419.82 0.05

0.06 0.07

0.06 0.08

0.06 0.10

0.06 0.10

The demand price at the first demand market is: 3804.11. The price at the second demand market is: 3941.75 and that at the third demand market is: 4085.17. The respective demands are: 39.18, 61.65, and 103.71. The firm’s profit is now: 702,571.38, a sizeable increase from that in Example 10.6, with the total demand close to that in Example 10.6. The total investment outlay is almost identical to that in Example 10.6 and is at 86.97. The domestic labor on each supply chain link is, as in Examples 10.4, 10.5, and 10.6, at its bound of 100, with the associated Lagrange multipliers all positive. There is, again, a positive investment on each link for attracting migrant labor and all values of migrant labor hours are positive on each link. The Lagrange multipliers: δa1∗ and δa2∗ are equal to 0.00 for all links a ∈ L , and ζ ∗ is also 0.00. In Figure 10.3, the profits are displayed for Example 10.1, which serves as the baseline, and for Examples 10.4, 10.5, 10.6, and 10.7. It is clear from Figure 10.3 that a firm can greatly benefit by having consumers recognize the value of its product and be willing to pay a higher price for it, even in the pandemic. Recall that, in Example 10.4, the demand price function intercepts are doubled for all the demand markets, as compared to their values in Example 10.1, tripled in Example 10.5, and, so on, until they are quintupled in Example 10.7. In Figure 10.4, I display the total labor hours at optimality for Example 10.1 and Examples 10.4 through 10.7, in the supply chain network of the firm for both the domestic laborers and the international migrant laborers. One sees that, clearly, with consumers being willing to pay a higher price for the truffles, and the bounds

10.4 High Value Food Product Numerical Examples

257

700

600

103

500

Profit

+

400

300

200

100

0

Ex.1

Ex.4

Ex.5

Ex.6

Ex.7

Fig. 10.3 Effect on profit of the firm when the demand price function intercepts are doubled, tripled, and so on, with Example 10.1 being the baseline

of domestic labor being at a value of 100 for the links, international migrant labor is essential. Further, one sees that, in Example 10.5 onwards, the total amount of international migrant labor remains essentially the same, at about 1050 hours, and, from Table 10.7, one sees that the investments are essentially the same for links a through e in Examples 10.5 through 10.7, signifying a kind of stabilization. 1100 1000

Total Optimal Labor Hours

900 800 700 600 500 Domestic Labour

400

International Migrant Labor

300 200 100 0

Ex.1

Ex.4

Ex.5

Ex.6

Ex.7

Fig. 10.4 Effect on optimal total labor hours of domestic labor and of international migrant labor in the supply chain network when the demand price function intercepts are doubled, tripled, and so on, with Example 10.1 being the baseline

258

10 International Migrant Labor and Supply Chains

Series 3 Numerical Examples In the third, final series of numerical examples, I first, again, explore the impacts of being untruthful in terms of wages in recruiting international migrant laborers and then we investigate the impact of a tighter budget on investments. For each of the examples in this series, I report the computed link flows and the domestic and international migrant labor values in Table 10.8 and the migrant attraction function investments and domestic labor Lagrange multipliers in Table 10.9. Example 10.8 The data in Example 10.8 are as in Example 10.7 except that, now, I consider untruthfulness in informing migrant laborers prior to hiring about their wages. In fact, the ws ˜ are set to values even higher than those that were being paid to domestic and migrant workers, who were paid the same wages in Example 10.7. Specifically, the wage settings are now: wa1 = wa2 = 30, w˜ a = 50,

wb1 = wb2 = 20, w˜ b = 40,

wc1 = wc2 = 18, w˜ c = 38,

wd1 = wd2 = 18, w˜ d = 38,

we1 = we2 = 17, w˜ e = 37,

wf1 = wf2 = 19, w˜ f = 39,

wg1 = wg2 = 19, w˜ g = 39,

wh1 = wh2 = 19, w˜ h = 29.

The computed product path flow pattern is: xp∗1 = 62.84,

xp∗2 = 63.65,

xp∗3 = 74.08,

xp∗5 = 104.80,

xp∗6 = 105.61.

xp∗4 = 74.88,

The demand price at the first demand market is: 3367.57. The price at the second demand market is: 3505.20 and that at the third demand market is: 3658.35. The demand market prices are all lower than in Example 10.7. The demands are now: 126.49, 148.96, and 210.41. These are all higher than their respective values in Example 10.7. The firm’s profit is now: 1,085,374.25, a big increase from the profit in Example 10.7, which was 702,571.38. The total investment outlay increases to: 116.15. There is a huge increase in the hiring of international migrant laborers, which are all attracted to the work; however, under false pretenses in the form of higher wages than either the domestic laborers or the migrant laborers are being paid! The domestic labor on each supply chain link is, as in Examples 10.4, 10.5, and 10.6, at its bound of 100, with the associated Lagrange multipliers all positive. There is, again, a positive investment on each link for attracting migrant labor and all values of migrant labor hours are positive on each link and these are much higher

10.4 High Value Food Product Numerical Examples

259

than those in Example 10.7. The Lagrange multipliers δa1∗ and δa2∗ are equal to 0.00 for all links a ∈ L , and ζ ∗ is also equal to 0.00. Example 10.9 (Same Data as Example 10.8 with Budget Decrease) Example 10.9 has the identical data to the data in Example 10.8 but with a decrease in the budget B from 1000 to 100. The new computed product path flow pattern is: xp∗1 = 58.22,

xp∗2 = 58.09,

xp∗3 = 69.06,

xp∗5 = 94.19,

xp∗6 = 94.01.

xp∗4 = 68.88,

The product path flows all decrease in comparison with their values in Example 10.8. Table 10.8 Optimal link flows and domestic and international migrant labor values for Examples 10.8 and 10.9 Optimal value Example 10.8 241.72 244.14 241.72 244.14 485.86 126.49

Example 10.9 221.48 220.97 221.48 220.97 442.45 116.31

fg∗ fh∗ la1∗ lb1∗ lc1∗ ld1∗ le1∗ lf1∗

148.96 210.41 100.00 100.00 100.00 100.00 100.00 100.00

137.94 188.20 100.00 100.00 100.00 100.00 100.00 100.00

lg1∗ lh1∗ la2∗ lb2∗ lc2∗ ld2∗ le2∗ lf2∗

100.00 100.00 339.49 388.28 590.63 597.54 709.77 232.86

100.00 100.00 302.68 341.95 532.79 531.35 637.42 206.09

lg2∗ lh2∗

313.78 426.03

283.16 370.50

Notation fa∗ fb∗ fc∗ fd∗ fe∗ ff∗

260

10 International Migrant Labor and Supply Chains

The demand price at the first demand market is: 3418.44. The price at the second demand market is: 3560.32 and that at the third demand market is: 3747.19. The respective demands are now: 116.31, 137.94, and 188.20. These are all lower than their respective values in Example 10.8. The firm’s profit is now: 1,069,553.88, a decrease from the profit in Example 10.8, which is to be expected. The total investment outlay is now at the bound of 100, with ζ ∗ being positive and at a value of 1068.04. The domestic labor on each supply chain link is, as in Example 10.8, at its bound of 100, with the associated Lagrange multipliers all positive. Notably, the value of the Lagrange multipliers, the λ∗ s, is quite high now, demonstrating that the firm should try to make more domestic labor interested and willing to work in the production and harvesting of truffles. There is, again, a positive investment on each link for attracting migrant labor and all values of migrant labor hours are positive on each link and these are much higher than those in Example 10.7. The Lagrange multipliers: δa1∗ and δa2∗ are equal to 0.00 for all links a ∈ L , and ζ ∗ is also 0.00. This is another example of “cheating” in that international migrants are attracted to work through being told that they will be paid higher wages than what they will actually be paid. This example, cheating or not, also demonstrates the importance of having a sufficient budget in order to be able to attract the needed international migrant labor. Furthermore, through the use of Lagrange multipliers, one can see the value of increasing the availability of domestic labor in this endeavor, as well as the budget for investing in attracting international migrant labor. The above examples

Table 10.9 Optimal link international migrant attraction investments and domestic labor bound Lagrange multipliers for Examples 10.8 and 10.9 Notation va∗ vb∗ vc∗ vd∗ ve∗ vf∗ vg∗ vh∗ λ∗a λ∗b λ∗c λ∗d λ∗e λ∗f

λ∗g λ∗h

Optimal value Example 10.8 6.98 10.23 19.58 19.89 23.76 6.39

Example 10.9 6.21 8.95 17.10 17.04 20.70 5.61

8.85 20.47 0.02 0.03 0.04 0.05 0.04 0.03

7.90 16.56 22.50 29.35 43.95 43.87 43.49 30.97

0.03 0.08

32.71 67.86

10.5 Summary, Conclusions, and Suggestions for Future Research

261

are stylized, but do provide insights and, importantly, demonstrate both the breadth of the model and the effectiveness of the computational procedure. The numerical examples, focusing on a high value agricultural product, with input and output data reported, illustrate both the effectiveness of the computational procedure, as well as the information that the solution of the model reveals in terms of the optimal product path flows, the optimal national and international labor values needed, the optimal Lagrange multipliers associated with labor, budget, and other constraints, as well as the profit earned by the firm under different scenarios. The model can be applied to distinct products not only in agriculture but also even to high technology through appropriate parameterization. One can also construct a generalized network model to capture product perishability, as in the case of fresh produce, as is done in Chapter 2, with the inclusion of migrant labor.

10.5 Summary, Conclusions, and Suggestions for Future Research The pandemic, climate change, and global conflicts and wars are transforming our societies and economies and also demonstrating the criticality of labor resources to supply chains. Many countries have been grappling with shortages of workers from the agricultural and manufacturing sectors to various services including healthcare. Attracting international migrant labor may help to assuage shortages in domestic labor. In this chapter, the aim is to build the foundation for the integration of labor in supply chains coupled with international human migration, with the latter focused on labor. An advance to the existing literature is provided through the synthesis of a supply chain network optimization model with labor that includes both domestic as well as migrant labor with the latter requiring investments subject to a budget constraint. The model has the flexibility to handle wages that are the same or different for domestic and migrant laborers and also the use of the truthful wage in the migration attraction functions that will be paid the migrant laborers for their work or not. The theoretical framework that is utilized for the formulation, analysis, and solution of three series of numerical examples is the theory of variational inequalities. The alternative variational inequality that is constructed is in path flow and investment variables plus several sets of Lagrange multipliers, including those associated with the upper bounds on domestic labor on the various supply chain network links and the budget constraint. The alternative variational inequality allows for the implementation of an elegant computational method with closed form expressions for the underlying variables at each iteration. To date, there are very few variational inequality models for operations research problems with nonlinear constraints. Hence, this work also adds to that literature.

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10 International Migrant Labor and Supply Chains

Three series of numerical examples are detailed, with input and output data with the examples being motivated by a fairly rare, high priced, high value agricultural product—truffles, which are now even being grown in the UK. The numerical examples reveal the benefits of having Lagrange multiplier information, and show that generating additional interest in an agricultural product in terms of prices that consumers are willing to pay, can yield sizeable increases in profit. The examples also show the impacts if a firm is untruthful in reporting wages that international migrants will actually be paid in attracting them, which suggests that policy-makers need to be aware of such “cheating” since it can lead to higher profits. In addition to the possible extensions noted at the end of Section 10.4, the model in this chapter can be extended to include, for example, competition among different firms for international migrant labor and also to allow for different productivity factors associated with different levels of skilled migrants. Also, it would be interesting and valuable to include a tier of brokers, along with their behavior, who work to procure migrant laborers for firms’ supply chain networks. Of course, there are also possibilities of using tools developed in this chapter to study issues associated with human trafficking. The use of networks, game theory, and variational inequalities has been successful in the modeling of deviant networks including those in cybercrime (Nagurney 2015) as well as in the modeling of investments in cybersecurity (cf. Nagurney and Shukla 2017).

10.6 Sources and Notes This chapter is based on Nagurney (2022), with notation standardized, as well as updates to the discussion and references. The integration of international migrant networks with supply chain networks is a timely topic due to labor needs in various countries in different economic sectors, coupled with many challenges associated with the pandemic, climate change, as well as conflicts and wars around the globe. It is expected that the relevance of such modeling and analysis research will continue to grow. For example, given the number of refugees due to Russia’s 2022 war on Ukraine, to start, as well as millions displaced within Ukraine, investing in human capital and the skilling of workers, whether in their own countries, or the countries where they move to as refugees, will be critical. Some of the research in this book lays the foundation (see also Gorodnichenko et al. 2022).

References Akintade, A., Essien, H., 2021. UK launches special visas for truck drivers, farmers. Peoples Gazette, September 27. Amo-Agyei, S., 2020. The migrant pay gap: Understanding wage differences between migrants and nationals. International Labour Organization, Geneva, Switzerland.

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Asian Development Bank Institute, Organisation for Economic Co-operation and Development, and International Labour Organization, 2021. Labor migration in Asia: Impacts of the COVID19 crisis and the post-pandemic future. Tokyo, Japan. Butler, S., 2020. UK farmers to get more help from overseas workers for 2021 harvest. The Guardian, December 20. Cappello, G., Daniele, P., Nagurney, A., 2021. A system-optimization model for multiclass human migration with migration costs and regulations inspired by the Covid-19 pandemic. Minimax Theory and Applications, 6(2), 281–294. Cave, D., Schuetze, C.F., 2021. Contending with the pandemic, wealthy nations wage global battle for migrants. The New York Times, November 23. Chowdhoury, P., Kumar Paul, S., Kaisar, S., Moktadir, M.A., 2021. COVID-19 pandemic related supply chain studies: A systematic review. Transportation Research E, 148, 102271. Cormack, R., 2021. A shortage of white truffles is pushing prices up to around $4500 a pound. Robb Report, October 20. Costa, D., 2021. The farmworker wage gap continued in 2020. Working Economics Blog, Economic Policy Institute, Washington DC, July 20. Costa, D., Martin, P., 2018. Temporary labor migration programs. Economic Policy Institute, Washington DC, August 1. Davison, N., 2018. Truffle economy: how a UK scientist sniffed out a culinary opportunity. Financial Times, April 20. del Rio-Chanona, R.M., Mealy, P., Pichler, A., Lafond, F., Doyne Farmer, J., 2020. Supply and demand shocks in the COVID-19 pandemic: an industry and occupation perspective. Oxford Review of Economic Policy, 36, Supplement_1, S94-S137. Dimas, G.L., Konrad, R., Maass, K.L., Trapp, A.C., 2021. A survey of operations research and analytics literature related to anti-human trafficking. Preprint. Worcester Polytechnic Institute, Worcester, Massachusetts. Douglas, J., Lovett, I., Emont, J., 2021. Omicron disrupts government plans to lure migrant workers as labor shortages bite. The Wall Street Journal, December 10. Economic Research Service, 2021. Farm Labor, United States Department of Agriculture, August 18. Elison, A., 2021. Broccoli pickers paid £30 an hour as Britain runs short of vegetables. The Sunday Times, October 4. Elkin, E., Ngoc Chau, M., de Sousa, A., 2021. Your food prices are at risk as the world runs short of workers. Bloomberg, September 21. Fernandez-Reino, M., Rienzo, C., 2021. Migrants in the UK labour market: An overview. The Migration Observatory, University of Oxford, United Kingdom, December 1. Floyd, C., 2019. Rare Italian white truffles cost over $4000 per kilo – here’s why real truffles are so expensive. Insider, October 26. Gorodnichenko, Y., Kudlyak, M., Sahin A., 2022. The effect of the war on human capital in Ukraine and the path for rebuilding. CEPR Policy Insight No 117, London, England. Helper, S., Soltas, E., 2021. Why the pandemic has disrupted supply chains. The White House, June 17. Hogarth, T., 2021. COVID-19 and the demand for labour and skills in Europe: Early evidence and implications for migration policy. Migration Policy Institute Europe, Brussels, Belgium. February. Hooper, K., 2021. Labor shortages during the pandemic and beyond: What role can immigration policy play? Migration Policy Institute, Washington DC, October. Ivanov, D., 2020. Predicting the impacts of epidemic outbreaks on global supply chains: A simulation-based analysis on the coronavirus outbreak (COVID-19/SARS-CoV-2) case. Transportation Research E, 136, 101922. Kalantaryan, S., Scipioni, M., Natale, F., Alessandrini, A., 2021. Immigration and integration in rural areas and the agricultural sector: An EU perspective. Journal of Rural Studies, 88, 462– 472.

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Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York. Konrad, R.A., Trapp, A.C., Palmbach, T.M., Blom, J.S., 2017. Overcoming human trafficking via operations research and analytics: Opportunities for methods, models, and applications. European Journal of Operational Research, 259(2), 733–745. Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. LeBaron, G., 2021. The role of supply chains in the global business of forced labour. Journal of Supply Chain Management, 57(2), 29–42. Lynch, D.J., 2021. Inside America’s broken supply chain. The Washington Post, October 2. Mayes, J., 2021. UK has its own truck driver shortage. Transport Topics, July 21. Murzakulova, A., Dessalegn, M., Phalkey, N., 2021. Examining migration governance: evidence of rising insecurities due to COVID-19 in China, Ethiopia, Kyrgyzstan, Moldova, Morocco, Nepal and Thailand. Comparative Migration Studies 9, Article number: 44. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Kluwer Academic Publishers, Dordrecht, The Netherlands. Nagurney, A., 2015. A multiproduct network economic model of cybercrime in financial services. Service Science, 7(1), 70–81. Nagurney, A., 2022. Attracting international migrant labor: Investment optimization to alleviate supply chain labor shortages. Operations Research Perspectives, 9, 10023. Nagurney, A., Daniele, P., 2021. International human migration networks under regulations. European Journal of Operational Research, 291(3), 894–905. Nagurney, A., Daniele, P., Cappello, G., 2021a. Capacitated human migration networks and subsidization. In: Dynamics of Disasters - Impact, Risk, Resilience, and Solutions, I.S. Kotsireas, A. Nagurney, P.M. Pardalos, and A. Tsokas, Editors, Springer Nature Switzerland AG, pp. 195–217. Nagurney, A., Daniele, P., Cappello, G., 2021b. Human migration networks and policy interventions: Bringing population distributions in line with system-optimization. International Transactions in Operational Research, 28(1), 5–26. Nagurney, A., Daniele, P., Nagurney, L.S., 2020. Refugee migration networks and regulations: A multiclass, multipath variational inequality framework. Journal of Global Optimization, 78, 627–649. Nagurney, A., Salarpour, M., Daniele, P., 2019. An integrated financial and logistical game theory model for humanitarian organizations with purchasing costs, multiple freight service providers, and budget, capacity, and demand constraints. International Journal of Production Economics, 212, 212–226. Nagurney, A., Shukla, S., 2017. Multifirm models of cybersecurity investment competition vs. cooperation and network vulnerability. European Journal of Operational Research, 260(2), 588–600. Novoszel, L., Wakolbinger, T., 2022. Meta-analysis of supply chain disruption research. Operations Research Forum, 3(1), 1–25. Schulte, G., Pitt, D., 2021. There are fewer people in rural America. That’s a problem for ranch and farm labor. USA Today, August 15. Segal, D., 2021. As U.K. beckons truck drivers, many in Poland say ‘No thanks’. The New York Times, November 28. Sherrell, H., Coates, B., 2021. Australia’s new agricultural work visa could supercharge the forces of exploitation. The Conversation, November 28. Smale, A., 2017. What the SDGs mean. United Nations. Sodhi, M.S., Tang, C., 2021. Supply chain management for extreme conditions: Research opportunities. Journal of Supply Chain Management, 57, 7–16. Swanson, A., Edmondson, C., Wong, E., 2021. U.S. effort to combat forced labor targets corporate China ties. The New York Times, December 23. Truffle.Farm, 2021. Truffle price tracker. April 18. Winterdyk, J., Jones, J., Editors, 2020. The Palgrave International Handbook of Human Trafficking. Palgrave Macmillan, Cham, Switzerland.

Chapter 11

Labor and Blood Services

Abstract Blood is a unique product since it cannot be manufactured but must be donated. At the same time, blood is a perishable, life-saving product and needed in many medical procedures, accidents, and other traumatic events as well as in times of war. Blood supply chains are, hence, essential supply chains in healthcare, and each link is associated with labor, whether the procurement and collection of blood donations, the testing and processing, shipment, etc. In this chapter, a blood service organization supply chain network model is presented with labor and bounds associated with links and with frequencies of operation as well as uncertainty with respect to demand at hospitals. The formulation of the model utilizes variational inequality theory. Unlike the models in the preceding chapters, the model in this chapter is for a nonprofit organization and not for a commercial firm. In addition, the focus here is on services. The integration of labor into services and a supply chain network for a nonprofit, specifically, a healthcare, organization is another novel feature of the framework.

11.1 Introduction Blood is a precious, life-saving product that cannot be manufactured and must be donated. It is essential to many medical procedures, including surgeries, treatment for trauma, sickle-cell anemia, and certain cancers, for example. At the same time, blood is perishable, with red blood cells, typically, under appropriate conditions, lasting 42 days and with plasma about 5 days. The blood services industry in the USA has experienced economic challenges in the past several years that have resulted in, among others, the closure of testing facilities and mergers and acquisitions (Masoumi et al. 2017). The COVID-19 pandemic has now created, in the third year into the pandemic, a crisis situation with the American Red Cross reporting in the winter of 2022 the worst blood shortage in over a decade (American Red Cross 2022). In the USA, the American Red Cross, under non-crisis scenarios, provides about 40% of the nation’s blood supply used by hospitals. Blood Centers of America, in turn, is a blood supply network in the USA, consisting of over 60 independent © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Nagurney, Labor and Supply Chain Networks, Springer Optimization and Its Applications 198, https://doi.org/10.1007/978-3-031-20855-3_11

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community blood centers, that collects and distributes more than 50% of the nation’s blood supply (Blood Centers of America 2022). Some major hospitals also run their own blood collection sites. As of mid-January 2022, 68% of the blood centers had less than 3 days of supply to meet the minimum demand (Lennon 2022). In pre-pandemic times, less than 10% of those who could donate blood in the US population would do so in a given year, although 38% were eligible (Nagurney 2022). In the COVID-19 pandemic, some collection sites were closed in the USA, including many of those at colleges and businesses (cf. Romero 2020). The closure of certain mobile collection sites resulted in difficulties for some donors who had to travel greater distances to donate blood. Furthermore, extra sanitation procedures had to be implemented, resulting in increasing spacing of collections, along with the implementation of social distancing measures. Winter is also always a very challenging time for blood donations with inclement weather, the holiday season, etc. Although respiratory viruses such as SARS-CoV-2, the virus that causes COVID-19, are not spread through blood donations, donors are advised not to come to donate while they are sick. Many hospitals curtailed and postponed elective procedures to preserve both blood and other healthcare resources, including that of their workers. Patients who had their medical procedures postponed, ultimately, may have experienced greater healthcare issues that required more serious medical interventions (Dawson 2021). Supply shocks, as well as demand shocks, hence, affected blood services and, thus, the availability of blood products. It is important to note that, in the pandemic, blood shortages have not been limited to the USA but have occurred around the globe. The World Health Organization estimated that the COVID-19 pandemic resulted in a reduction of 20% to 30% in the blood supply in all its six regions (see Loua et al. 2021). Although certain measures may have mitigated shortages in the short term, the problem of maintaining the blood supply in the ongoing pandemic is a long-term one, given the consistent demand for blood for medical procedures that require transfusion (see Haw et al. 2022). Shortages of blood, as well as blood bags, and other supplies, can also be a feature in wars, with Russia’s war against Ukraine being a vivid, sad example (Tevge 2022). In addition, in the pandemic, blood service organizations have experienced labor shortages, with some staff getting ill and being unable to work and with others quarantining or leaving the workforce, as has happened in multiple industries early in the pandemic as well as, more recently, with the Omicron surge (Rosenberg 2022). For example, blood service organizations have had to deal with a shortage of blood collectors, known as phlebotomists, in certain regions of the USA and are trying to train and attract workers by raising wages (see Banse 2021). Staff shortages associated with collection have led to canceled blood drives and postponed appointments for donors (Blake 2022). The staffing shortages in blood service organizations in the pandemic have affected each link in the blood supply chain network (cf. Nagurney and Masoumi 2012; Nagurney et al. 2012), impacting also such blood supply chain network activities as: blood testing and processing, transportation, and distribution to the hospitals (Shaw-Tulloch 2022).

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267

Labor is essential to the functionality of supply chains with disruptions to labor negatively impacting the production, transportation, storage, and distribution of many products from food to lumber as well as high-technology products (Ferguson 2022). The healthcare sector has also been adversely affected with the intensity and spread of the COVID-19 coronavirus since frontline workers in hospitals and other healthcare facilities are essential workers and cannot conduct their services, in many cases, remotely (see Lagasse 2021). In this book, previous modeling chapters have focused exclusively on commercial applications, such as that of food supply chains and personal protective equipment (PPE) and medical supplies. In this chapter, in contrast, I turn to the nonprofit sector with the added unique characteristic of a product that must be donated—that of blood—and consider services, rather than a production/manufacturing setting in the commercial space. The blood service organization model that I construct in this chapter uses a supply chain network structure and has the following features. The model focuses on red blood cells being collected, tested, processed, and distributed by a regional blood service organization. It builds upon the foundations in Nagurney and Masoumi (2012) and Masoumi et al. (2017) in which demand uncertainty was introduced in the former and frequencies of operation associated with the link activities and capacities in the latter. Furthermore, a generalized network approach is used here as well since blood is a perishable product and the arc multipliers on the links capture the throughput. A related perishable product supply chain network model, but a purely deterministic one, and for food, was described in Chapter 2 of this book. Here, frequencies of operation and link capacities are included as well as a bound on the supply of available product as in the case of blood donations. Importantly, since this has been a big issue in the pandemic, the supply of available blood to be donated/collected at various sites is subject to an upper bound plus each of the blood service organization supply chain network activities has associated with it a maximum availability of labor. The model assumes that the labor availability is wage-dependent, with higher wages attracting more labor, and with labor availability affecting how much blood can be “processed” on the various links through the use of a productivity factor. In addition, in the food supply chain model in Chapter 2, the wages were fixed a priori, whereas in the model for blood service organizations in this chapter, the wages are determined endogenously. Plus, and importantly, the objective function faced by a blood service organization is distinct from that of a profit-maximizing firm. Here, I consider the minimization of total costs, which include the link operational costs, with each being also a function of the frequency of operation, the total payout of wages for labor, the discarding cost associated with wastage on links, and the expected costs associated with either a surplus or a shortage at the various hospitals, with corresponding penalties. The blood service organization supply chain network model with labor that is developed in this chapter is an optimization model as are the models in Chapters 2, 3, 7, 8, 10, and also the model in Chapter 12. Here, as in the preceding modeling chapters, the theory of variational inequalities is utilized for the formulation and analysis.

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The chapter is organized as follows. The blood service organization supply chain network optimization model is constructed in Section 11.2 and a variational inequality formulation provided. Section 11.3 describes the computational procedure that can be applied for its solution. Section 11.4 summarizes the contributions in this chapter and presents the conclusions and suggestions for future research. Section 11.5 contains the Sources and Notes.

11.2 The Supply Chain Network Model of the Blood Service Organization with Labor In this section, I construct the supply chain network model of the blood service organization (BSO) with labor. The blood service organization (also sometimes referred to as a blood bank) collects, tests, processes, and distributes the donated blood in a region to medical centers, which are referred to as “hospitals.” The topology of the supply chain network is depicted in Figure 11.1 and is denoted by the graph G = [N , L ], where N is the set of nodes and L is the set of links. Activities of the blood service organization are associated with the links. The nodes in the network consist of the organization and the facilities and include the collection sites, which can be fixed or mobile, the blood centers, from which the collected blood is shipped to the component labs, which are also nodes. At these labs, the collected blood is separated into parts, that is, red blood cells and plasma, since most recipients need only a specific component for transfusions. Each unit of donated whole blood –450 to 500 milliliters on average– can yield one unit of red blood cells (RBCs) and one unit of plasma. What is referred to as the flow of product is the amount of whole blood (WB) on the first three sets of links. Likewise, the flow on the links afterward is the number of units of RBCs processed at the component labs that are, ultimately, delivered to the hospitals. Note that the topology of the supply chain in Figure 11.1 can be adapted depending on the specific blood service organization. Furthermore, the component labs may be located geographically at the same locations as the blood centers. The testing and processing are usually done concurrently, as noted in Masoumi et al. (2017), with testing taking place at testing labs to which a small amount of each unit of blood collected is sent overnight. Such samples are thrown away after the tests. The storage is for a short term at the storage facilities, also denoted by nodes, and, typically, located in the same location as the processing labs. From the storage facilities, the RBCs are shipped to distribution centers and further on to the hospitals. Alternatively, as depicted in Figure 11.1, the RBCs can be shipped, if feasible and desirable, directly from the distribution center nodes to the hospitals. The focus here in the model is on the processes plus the resources needed, including that of labor with operational costs associated with the links as well as labor volumes and wages of the workers. To capture perishability and waste, as in the work of Nagurney and Masoumi (2012), and as in Chapter 2, arc multipliers are used.

11.2 The Supply Chain Network Model of the Blood Service Organization. . .

Fig. 11.1 Blood service organization supply chain network topology

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Uncertainty on the demand side is handled, and capacities on the links are included as well as blood donation supply bounds on the top tier links. In addition, bounds on the labor availability on the links are imposed to capture disruptions to labor in the pandemic. This feature is also relevant in times of war, in which blood donations may be sorely needed and it is challenging to obtain the donations. In the supply chain (cf. Figure 11.1), there are nC collection sites, nB blood centers, nCL component labs, nS storage facilities, nDC distribution centers, and nH hospitals. The nodes are as labeled in Figure 11.1. Tables 11.1, 11.2, and 11.3 contain the notation for the blood service organization supply chain network model. As throughout this book, vectors are column vectors. Table 11.1 Notation for the blood services supply chain network model—parameters Notation PHk P

u¯ a Sa δap βa βap

μp λ− k λ+ k l¯a αa wa πa

Parameters The set of paths, representing alternative supply chain network processes, joining the blood service organization node 1 to a specific hospital node Hk , k = 1, . . . , nH The set of all paths in Figure 11.1 The capacity of link a, a ∈ L The upper bound on the blood supply available for donations on collection link a, ∀a ∈ L 1 , where L 1 denotes the set of blood collection links δap = 1, if link a is contained in path p and δap = 0, otherwise The arc multiplier associated with link a, which represents the percentage of throughput on link a. βa ∈ (0, 1], a ∈ L The arc path multiplier, which is the product of the multipliers of the links on path p that precede link a, a ⎧ ∈ L and p ∈ P , that is,  ⎪ δap βb , if {a  < a}p = Ø, ⎨  βap ≡ b∈{a 0, there is a δ > 0, such that ψ ∈ K and ψ − X < δ imply that |f (ψ) − f (X)| < . Definition A.3 (A Differentiable Function) Consider a nonempty set K such that K ⊂ R N , with a column vector X lying in the interior of K , and let f be a function such that f : K → R. Then f is said to be differentiable, if there exists a column vector ∇f (X) ∈ R N , called the gradient of f at X, and defined as ) ∇f (X) = T

* ∂f (X) ∂f (X) ∂f (X) , , ,..., ∂X1 ∂X2 ∂XN

(A.1)

and a function β(X; y) → 0 as y → x such that: f (y) = f (X) + ∇f (X), (y − X) + y − X β(X; y),

∀y ∈ K ,

(A.2)

where ·, · denotes the inner product in the N-dimensional Euclidean space. Definition A.4 (A Twice Differentiable Function) The function f is said to be twice differentiable at X if, in addition to the gradient vector, there exists an N × N matrix H (X), called the Hessian matrix of the function f at X, defined as ⎛ ⎜ ⎜ ⎜ H =⎜ ⎜ ⎝

∂2f ∂2f ∂X1 ∂X1 ∂X1 ∂X2 ∂2f ∂2f ∂X2 ∂X1 ∂X2 ∂X2

...

.. .

..

.

... .. .

∂2f ∂XN ∂X1

...

...

∂2f ∂X1 ∂XN ∂2f ∂X2 ∂XN

.. .

∂2f ∂XN ∂XN

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

(A.3)

and a function β(X; y) → 0 as y → X such that: f (y) = f (X) + ∇f (X), (y − X) +

1 (y − X)T H (X) (y − X) + ||y − X||2 β(X; y), 2

∀y ∈ K .

(A.4)

Example A.2 An example of a twice differentiable function in two dimensions is f (X) = 20X12 + 6X22 . Definition A.5 (Convex and Concave Functions) Let K be a nonempty convex set, and consider a function f : K → R. Then the function f (X) is said to be

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307

a convex function on K if, for any two distinct points X1 , X2 ∈ K , and for all λ ∈ [0, 1], the following holds: f [λX2 + (1 − λ) X1 ] ≤ λf (X2 ) + (1 − λ) f (X1 ).

(A.5)

The function f is strictly convex on K if the above inequality holds as a strict inequality. A function f (X) is said to be concave (strictly concave) if −f (X) is convex (strictly convex). Hence, in two dimensions, a function f (X) is convex (concave) if a line segment joining any two points [X1 , f (X1 )] , [X2 , f (X2 )] on the surface of f (X) lies on or above (below) that surface. The previous definition can be used directly to determine whether a function is convex or not. For example, if one assumes that the function f (X) is continuous and that it has second-order partial derivatives over K , then another way to determine whether a function is convex or not is to evaluate whether the Hessian of the function is positive semidefinite or not. In particular, if the Hessian matrix H of second-order partial derivatives is positive semidefinite, then f (X) is convex, and if H is negative semidefinite, then f (x) is concave. Recall that a matrix is positive (negative) definite if all its eigenvalues are positive (negative) or, equivalently, if all of its principal determinants have positive (negative) values. A matrix is positive semidefinite if all its eigenvalues are nonnegative as are all of its principal determinants. Example A.3 (A Convex Function) Consider the function f (X1 , X2 ) = 5X12 +6X22 , whose Hessian matrix is ) * 10 0 H = . 0 12 This Hessian matrix is a diagonal matrix with positive elements on the diagonal and, therefore, positive-definite; hence, f is convex and, in fact, it is strictly convex. Definition A.6 (Quasiconvex and Quasiconcave Functions) Let K be a nonempty convex set, and consider a function f : K → R. The function f (X) is said to be a quasiconvex function on K , if for any two distinct points X1 , X2 ∈ K, and ∀λ ∈ [0, 1]: f [λX2 + (1 − λ) X1 ] ≤ maximum (f (X1 ), f (X2 )) . The function f (X) is said to be quasiconcave on K if −f (X) is quasiconvex. Definition A.7 (Pseudoconvex and Pseudoconcave Functions) Let K be a nonempty convex set, and consider a function f : K → R, which is differentiable on K . Then the function f (X) is said to be a pseudoconvex function on K , if, for any two distinct points X1 , X2 ∈ K , with ∇f (X1 ), X2 − X1  ≥ 0, one has f (X2 ) ≥ f (X1 ).

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The function f (X) is said to be pseudoconcave on K if −f (X) is pseudoconvex. Some fundamentals of optimization theory are now presented, along with some of the concepts and ideas that are utilized throughout this book. An optimization problem in R N is a problem in which one seeks to optimize a function f , which is said to be the objective function, subject to some constraints. An objective function may represent profits in the case of a firm to be maximized or costs or risk to be minimized (or a combination thereof, as in the case of multicriteria decision-making problems). It is important to identify the appropriate objective function and to make clear distinctions between those of a profit-maximizing firm and that of a nonprofit organization such as a humanitarian one. Constraints, in turn, can capture budget or other resource constraints, as well as nonnegativity assumptions, bounds on labor and/or wages, etc. A point X0 ∈ R N is called a feasible solution to the optimization problem if it satisfies all the constraints of the problem. Furthermore, a point X∗ ∈ R N is called an optimal solution to the optimization problem if it is a feasible solution and it provides the best possible value for the objective function. There are different classes of optimization problems, depending on the structure of the objective function and the constraints. For example, if the objective function is linear, as are the constraints, and the variables are continuous (rather than discrete), then the problem is a linear programming problem (see Bazaraa et al. 1990). On the other hand, if either the objective function or the constraints are nonlinear expressions of the variables, then the problem is a nonlinear programming problem. For further details, see Bazaraa et al. (1993). In this book, the problems are nonlinear ones. Nonlinearity of the underlying components of objective functions helps one to capture important features of supply chain network problems that are being experienced now that include congestion as well as risk. On the other hand, if one or more of the variables in a problem are constrained to be discrete, that is, to take on integer values, then one is faced with an integer programming problem. Such problems are not considered in this book.

A.2 Karush–Kuhn–Tucker Optimality Conditions I begin with some basic definitions, followed by the presentation of important conditions for optimality. Definition A.8 (Global Maximum and Minimum) The function f : K → R N is said to take its global maximum at point X∗ if f (X) ≤ f (X∗ ),

∀X ∈ K .

(A.6)

Also, the function f : K → R N is said to take its global minimum at point ψ ∗ if

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Optimization Theory, Variational Inequalities, and Game Theory

f (ψ) ≥ f (ψ ∗ ),

∀ψ ∈ K .

309

(A.7)

Definition A.9 (Local Maximum and Minimum) The function f : K → R N is said to take its local maximum at point X∗ if there exists a δ > 0 such that for every X = X∗ that belongs to K and is in a δ-neighborhood of X∗ , the following holds: f (X) ≤ f (X∗ ),

" # ∀X ∈ K ∩ B(X∗ , δ) ,

(A.8)

where B(X∗ , δ) denotes the ball with center X∗ and radius δ. The function f : K → R N is said to take its local minimum at point ψ ∗ if there exists a δ > 0 such that for every ψ = ψ ∗ that belongs to K and is in a δ-neighborhood of ψ ∗ , the following holds: f (ψ) ≥ f (ψ ∗ ),

" # ∀ψ ∈ K ∩ B(ψ ∗ , δ) .

(A.9)

Karush–Kuhn–Tucker (KKT) Conditions Karush (1939) and Kuhn and Tucker (1951) independently proposed a set of necessary and sufficient conditions for an optimal solution of a general mathematical optimization problem. Their work yielded the rigorous mathematical foundation upon which qualitative theory and algorithms for optimization problems have been based. A concise presentation of these conditions is now given for reference purposes. Karush–Kuhn–Tucker Necessary Conditions Let K be a nonempty open set such that K ⊂ R N , and let {f : R N → R}, {gi : R N → R} for i = 1, 2, . . . , m, and {hj : R N → R} for j = 1, 2, . . . , t. Furthermore, consider the general optimization problem of the following form: Minimize

f (X)

subject to: gi (X) ≤ 0,

for

i = 1, 2, . . . , m,

hj (X) = 0,

for

j = 1, 2, . . . , t,

X∈K. Let X∗ be a feasible solution, and let I = {i : gi (X∗ ) = 0}. In addition, assume that f and gi are differentiable at X∗ for i ∈ I and that the gi are continuous at X∗ for i ∈ I . Finally, assume that the hj are continuously differentiable at X∗ for all j = 1, 2, . . . , t. Suppose that ∇gi (X∗ ) for i ∈ I and ∇hj (X∗ ) for j = 1, 2, . . . , t are linearly independent. If X∗ locally solves the minimization problem, then there

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exist unique scalars vi∗ for i ∈ I and γj∗ for j = 1, 2, . . . , t, such that: ∇f (X∗ ) +

m

vi∗ ∇gi (X∗ ) +

t

γj∗ ∇hj (X∗ ) = 0,

(A.10)

j =1

i=1

vi∗ gi (X∗ ) = 0, for i = 1, 2, . . . , m,

(A.11)

vi∗ ≥ 0, for i = 1, 2, . . . , m.

(A.12)

The scalars vi∗ and γj∗ are called Lagrange multipliers. There is one Lagrange multiplier associated with each constraint. The Lagrange multipliers represent the marginal rate of change in the objective function f with respect to each per unit change in the right-hand side of the corresponding constraint. Lagrange multipliers have important economic interpretations, as detailed in a spectrum of supply chain network models with labor that are discussed in this book. Any point that satisfies the KKT conditions is called a KKT point. An optimization problem may include nonnegativity constraints for the variables involved, so that X ≥ 0. Such inequality constraints are highly relevant in supply chain network problems since one must guarantee that the product flows on the network, which are physical entities, be they products, prices, etc., are nonnegative. Clearly, the KKT conditions that were presented will still hold. At times, however, for reasons of convenience and simplicity, the Lagrange multipliers associated with the nonnegativity constraints are eliminated, and the conditions are reduced to ∇f (X∗ ) +

m

vi∗ ∇gi (X∗ ) +

⎣∇f (X∗ ) +

m i=1

γj∗ ∇hj (X∗ ) ≥ 0,

(A.13)

j =1

i=1

⎡

t

vi∗ ∇gi (X∗ ) +

t

⎤T γj∗ ∇hj (X∗ )⎦ X∗ = 0,

(A.14)

j =1

vi∗ gi (X∗ ) = 0, for i = 1, . . . , m,

(A.15)

vi∗ ≥ 0, for i = 1, . . . , m.

(A.16)

A geometric interpretation of the KKT conditions is that a vector X∗ is a KKT point if and only if −∇f (X∗ ) lies in the cone spanned by the gradients of the binding constraints, that is, those constraints that hold as equalities. Karush–Kuhn–Tucker Sufficient Conditions Let K be a nonempty open set such that K ⊂ R N , and let {f : R N → R}, {gi : R N → R} for i = 1, 2, . . . , m and {hj : R N → R} for j = 1, 2, . . . , t.

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Consider the general optimization problem of the following form: Minimize

f (X)

subject to gi (X) ≤ 0,

for

i = 1, 2, . . . , m,

hj (X) = 0,

for

j = 1, 2, . . . , t,

X∈K. Let X∗ be a feasible solution, and let I = {i : gi (X∗ ) = 0}. Assume that the KKT conditions hold at X∗ . In other words, assume that there exist scalars v i ≥ 0 with i ∈ I and γ j with j = 1, 2, . . . , t, such that ∇f (X∗ ) +



v i ∇gi (X∗ ) +

t

γ j ∇hj (X∗ ) = 0.

(A.17)

j =1

i∈I

Let J = {j : γ¯j > 0} and L = {j : γ¯j < 0}. Further, suppose that f is pseudoconvex at X∗ , the constraints gi are quasiconvex at X∗ for i ∈ I , and hj is quasiconvex for the j ∈ J and quasiconcave for the j ∈ L. Then X∗ is a global optimal solution to the general minimization problem. For a maximization problem, the KKT conditions are similar, where now the function f has to be pseudoconcave. Definition A.10 (Lagrangian Function) Consider the general minimization problem described previously. Then the function such that φ(X, v, γ ) = f (X) +

m

vi gi (X) +

t

γj hj (X)

(A.18)

j =1

i=1

is said to be the Lagrangian function of the general optimization problem. If one lets X∗ be a KKT point to the general optimization problem with v ∗ , γ ∗ being the Lagrange multipliers that correspond to the constraints of the problem, then the function L(X) ≡ φ(X, v ∗ , γ ∗ ) = f (X) +



vi∗ gi (X) +

i∈I

t j =1

is said to be the restricted Lagrangian function. Let ∇ 2 L denote the Hessian of (A.19). Then, if ∇ 2 L is:

γj∗ hj (X)

(A.19)

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• Positive semidefinite for all X in the feasible set, X∗ is a global minimum • Positive semidefinite for all X in the feasible set and in a δ-neighborhood B(X∗ , δ), for a δ > 0, X∗ is a local minimum • Positive-definite, X∗ is a strict local minimum Lagrangians are utilized for Lagrange analysis associated with supply chain network models with labor in Chapters 5 and 7 of this book.

A.3 Variational Inequalities Hartman and Stampacchia (1966) are credited with introducing variational inequalities, principally for the study of problems arising in the field of mechanics. Their research focused principally on infinite-dimensional variational inequalities, rather than on finite-dimensional variational inequalities, which are used in this book. The book by Kinderlehrer and Stampacchia (1980) provides an introduction to infinitedimensional variational inequality problems and the book by Nagurney (1999) to finite-dimensional ones. The book by Daniele (2006) gives an overview of evolutionary (time-dependent) variational inequalities, which are infinite-dimensional, with applications to dynamic networks. Smith (1979) presented a formulation of the equilibrium conditions of the transportation network equilibrium problem that were subsequently recognized as a finite-dimensional variational inequality by Dafermos (1980). From this connection, extensive research has been conducted on variational inequality problems and numerous applications, from oligopolistic market equilibrium problems to general economic and financial equilibrium problems, and, more recently, supply chain network equilibrium problems. Variational inequality theory is a powerful, unifying tool for the study of many equilibrium problems, since the variational inequality problem contains, as special cases, important problem classes that arise in economics, operations research, finance, and in engineering, such as systems of nonlinear equations, optimization problems, and complementarity problems. Moreover, the problem is related to fixed point problems. Indeed, it has been utilized to formulate equilibrium problems governed by entirely distinct equilibrium concepts, including (Wardrop 1952) equilibrium governing congested urban transportation networks (cf. Smith 1979 and Dafermos 1980), spatial price equilibrium (cf. Samuelson 1952; Takayama and Judge 1971; Florian and Los 1982; Dafermos and Nagurney 1987, and Friesz et al. 1984), and oligopolistic market equilibrium (cf. Gabay and Moulin 1980 and Dafermos and Nagurney 1987), governed by the Cournot (1838)–Nash (1950) equilibrium. In this book, I show how variational inequality theory (both finitedimensional) is utilized in the formalization of supply chain networks with labor. I first present the formal definition of a variational inequality problem and then discuss the relationship between the variational inequality problem and optimization

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Fig. A.1 Geometric Interpretation of VI(F, K )

problems. The definition of a variational inequality problem can also be found in Chapter 2. Definition A.11 (The Variational Inequality Problem) The finite-dimensional variational inequality problem, VI(F, K ), is to determine a vector X∗ ∈ K , such that

F (X∗ ), X − X∗  ≥ 0,

∀X ∈ K ,

(A.20)

where F is a given continuous function from K to R N , K is a given closed convex set, and ·, · denotes the inner product in R N . The solution to a variational inequality problem has an elegant geometric interpretation. Specifically, it states that F (X∗ ) is “orthogonal” to the feasible set K at the point X∗ . In Figure A.1, the geometric interpretation is provided. In order to emphasize the relationship between the variational inequality problem and optimization problems, I now present the following results. Proofs can be found in Kinderlehrer and Stampacchia (1980) and in Nagurney (1999) with additional results and applications to supply chains in Nagurney (2006), Nagurney et al. (2013), and Nagurney and Li (2016). Relationship Between the Variational Inequality Problem and Optimization Problems I now recall the relationship between variational inequality problems and optimization problems. Note that a variational inequality problem contains an optimization problem as a special case. Proposition A.1 Let X∗ be the solution to the following optimization problem: Minimize f (X) subject to X∈K,

(A.21)

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where f is a continuously differentiable function, and K is closed and convex. Then X∗ is a solution of the variational inequality problem:

∇f (X∗ ), X − X∗  ≥ 0,

∀X ∈ K ,

(A.22)

where ∇f (X∗ ) denotes the gradient of f with respect to X with components: ∇f (X∗ )T = (

∂f (X∗ ) ∂f (X∗ ) ∂f (X∗ ) , ,..., ). ∂X1 ∂X2 ∂XN

Proposition A.2 If f (X) is a convex function and X∗ is a solution to VI(∇f, K ) given by (A.22), then X∗ is a solution to the optimization problem (A.21). In the special case, where K = R N , then the above optimization problem (A.21) is an unconstrained problem. On the other hand, if a certain symmetry condition holds, the variational inequality problem can also be reformulated as an optimization problem. Theorem A.1 Assume that F (X) is continuously differentiable on K and that the Jacobian matrix: ⎫ ⎧ ∂F1 ∂F1 ⎪ ⎪ . . . ⎪ ∂X ∂X N ⎪ ⎬ ⎨ 1 . . . . ∇F (X) = (A.23) . . ⎪ ⎪ ⎪ ⎭ ⎩ ∂FN . . . ∂FN ⎪ ∂X1 ∂XN is symmetric and positive semidefinite, so that F is convex. Then there exists a realvalued function f : K → R satisfying: ∇f (X) = F (X)

(A.24)

with X∗ the solution of VI(F,-K ) also being -the solution of the optimization problem (A.21) where f (X) = F (X)T dx and is a line integral. Hence, one can see that the variational inequality problem encompasses the optimization problem and that the variational inequality problem can be reformulated as a convex optimization problem when the symmetry and the positive semidefiniteness conditions hold. The variational inequality problem is a more general problem in that it can also handle a function F (X) with an asymmetric Jacobian, that is, when (cf. (A.23)): ∂Fj ∂Fi ∂Xj = ∂Xi , for i, j . Historically, many equilibrium problems were reformulated as optimization problems under the symmetry and positive semidefiniteness condition, including the transportation network equilibrium problem (cf. Beckmann et al. 1956). This assumption, however, is restrictive in many application settings, specifically, in the case of supply chain network equilibrium problems, since it cannot adequately handle asymmetric cost functions, interacting multiple decision-makers

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315

realistically, as well as multiple modes of transportation on the networks. Hence, variational inequality formulation and the associated theory expand the breadth of applications. Variational Inequality Formulation of a Class of Constrained Optimization Problems I now present a result (see Bertsekas and Tsitsiklis 1989) that demonstrates how a class of constrained optimization problems over specific types of constraints can be formulated as a variational inequality problem. Consider the convex constrained optimization problem: Minimize

m

fi (Xi )

(A.25)

i=1

subject to ajT X ≤ bj , Xi ∈ Ki ,

j = 1, . . . , r,

(A.26)

i = 1, . . . , m,

where fi : R Ni → R is a convex differentiable function and ajT is a row vector of coefficients corresponding to the j -th constraint, and X is a vector consisting of the vectors: {X1 , . . . , Xm }. Then this problem is equivalent to the variational inequality problem of finding Xi∗ ∈ Ki , ∀, and u∗j ≥ 0, ∀j , such that m r r

∇fi (Xi∗ ) + ( u∗j aj i )T , Xi − Xi∗  + (bj − ajT X∗ ) × (uj − u∗j ) ≥ 0, i=1

j =1

j =1

(A.27) ∀Xi ∈ Ki , ∀i

uj ≥ 0, ∀j.

Note that u∗j is the Lagrange multiplier in the solution associated with inequality constraint j in the minimization problem. The coefficient aj i corresponds to the ith component of the vector aj .

A.3.1 Qualitative Properties Some basic qualitative properties of finite-dimensional variational inequality problems are now presented. Specifically, I discuss the existence and uniqueness of a solution. Definitions that are referred to in discussions of the convergence of algorithms are also included. All the theoretical results presented in this subsection, along with all the corresponding proofs, can be found in Kinderlehrer and Stampacchia (1980) and Nagurney (1999).

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The existence of a solution to a variational inequality problem follows from continuity of the function F entering the variational inequality, provided that the feasible set K is compact (i.e., closed and bounded, in the case of R N ), as stated in the following theorem. Theorem A.2 (Existence Under Compactness and Continuity) If K is a compact convex set and F (X) is continuous on K , then the variational inequality problem VI(F, K ) admits at least one solution X∗ . If the feasible set K is unbounded, then one might be able to make use of the following: Theorem A.3 VI(F, K ) admits a solution if and only if there exists an R > 0 and ∗ of VI(F, S), such that X ∗ < R, where S = {X : X ≤ R}. a solution XR R The existence and the uniqueness of a solution to a variational inequality problem are directly related to certain monotonicity conditions. In fact, monotonicity plays a role in variational inequality theory that is similar to the role that convexity plays in optimization theory. Definition A.12 (Monotonicity) F (X) is said to be monotone on K if

F (X1 ) − F (X2 ), X1 − X2  ≥ 0, ∀X1 , X2 ∈ K .

(A.28)

Definition A.13 (Strict Monotonicity) F (X) is strictly monotone on K if

F (X1 ) − F (X2 ), X1 − X2  > 0,

∀X1 , X2 ∈ K ,

X1 = X2 .

(A.29)

Definition A.14 (Strong Monotonicity) F (X) is strongly monotone on K , if for some α ≥ 0:

F (X1 ) − F (X2 ), X1 − X2  ≥ α X1 − X2 , ∀X1 , X2 ∈ K .

(A.30)

The first uniqueness result is presented in the following theorem. Theorem A.4 (Uniqueness Under Strict Monotonicity) If F (X) is strictly monotone on the feasible set K , then if a solution to VI(F, K ) exists, it is unique. Under the strong monotonicity assumption on the function F (X), one obtains both the existence and the uniqueness of a solution. Theorem A.5 (Existence and Uniqueness Under Strong Monotonicity) If F (X) is strongly monotone on the feasible set K , then there exists precisely one solution X∗ to VI(F, K ).

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In many cases, computational algorithms and, in particular, the modified projection method used to solve many of the supply chain network models with labor in this book require, for convergence purposes, that the function F that enters the variational inequality problem is Lipschitz continuous, that is: Definition A.15 (Lipschitz Continuity) F (X) is Lipschitz continuous on K if there exists a constant θ > 0, such that F (X1 ) − F (X2 ) ≤ θ X1 − X2 ,

∀X1 , X2 ∈ K ,

(A.31)

where θ is referred to as the Lipschitz constant. For completeness and easy reference, I recall the definition of a norm projection. Definition A.16 (Norm Projection) Let K be a closed convex set in R N . Then for each X ∈ R N , there is a unique point y ∈ K , such that X − y ≤ X − z ,

∀z ∈ K ,

(A.32)

and y is known as the orthogonal projection of X on the set K with respect to the Euclidean norm, that is, y = PK X = arg min X − z . z∈K

(A.33)

A.4 The Relationships Between Variational Inequalities and Game Theory In this section, some of the relationships between variational inequalities and game theory are briefly discussed. Nash (1950, 1951) developed noncooperative game theory, involving multiple players, each of whom acts in his/her own interest. In particular, consider a game with m players, each player i having, without loss of generality, a strategy vector Xi = {Xi1 , ..., Xin } selected from a closed, convex set Ki ⊂ R n . Each player i seeks to maximize his/her own utility function, Ui : K → R, where K = K1 × K2 × . . . × Km ⊂ R mn . The utility of player i, Ui , depends not only on his/her own strategy vector, Xi , but also on the strategy vectors of all the other players, (X1 , . . . , Xi−1 , Xi+1 , . . . , Xm ). An equilibrium is achieved if no one can increase his/her utility by unilaterally altering the value of its strategy vector. The formal definition of the Nash equilibrium is as follows. Definition A.17 (Nash Equilibrium) A Nash equilibrium is a strategy vector ∗ X∗ = (X1∗ , . . . , Xm )∈K,

(A.34)

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where Ui (Xi∗ , Xˆ i∗ ) ≥ Ui (Xi , Xˆ i∗ ),

∀Xi ∈ Ki , ∀i,

(A.35)

∗ , X ∗ , . . . , X ∗ ). and Xˆ i∗ = (X1∗ , . . . , Xi−1 m i+1

It has been shown by Hartman and Stampacchia (1966) and Gabay and Moulin (1980) that, given continuously differentiable and concave utility functions, Ui , ∀i, the Nash equilibrium problem can be formulated as a variational inequality problem defined on K . Theorem A.6 (Variational Inequality Formulation of Nash Equilibrium) Under the assumption that each utility function Ui is continuously differentiable and concave, X∗ is a Nash equilibrium if and only if X∗ ∈ K is a solution of the variational inequality

F (X∗ ), X − X∗  ≥ 0,

X∈K,

(A.36)

i (X) where F (X) ≡ (−∇X1 U1 (X), . . . , −∇Xm Um (X))T , ∇Xi Ui (X) = ( ∂U ∂Xi1 , i (X) . . . , ∂U ∂Xin ).

The conditions for existence and uniqueness of a Nash equilibrium are now introduced. As stated in the following theorem, Rosen (1965) presented existence under the assumptions that K is compact and each Ui is continuously differentiable. Theorem A.7 (Existence of a Solution Under Compactness and Continuity) Suppose that the feasible set K is compact and that each Ui is continuously differentiable ∀i. Then the existence of a Nash equilibrium is guaranteed. Gabay and Moulin (1980) relaxed the assumption of compactness of K and established the existence of a Nash equilibrium after imposing a coercivity condition on F (X). Theorem A.8 (Existence of a Solution Under Coercivity) Suppose that F (X), as given in Theorem 2.6, satisfies the coercivity condition (2.9). Then there exists a Nash equilibrium. Karamardian (1969), earlier, established the existence and uniqueness of a Nash equilibrium under the strong monotonicity assumption. Theorem A.9 (Existence and Uniqueness of a Solution Under Strong Monotonicity) Assume that F (X), as given in Theorem A.6, is strongly monotone on K . Then there exists precisely one Nash equilibrium X∗ . Based on Theorem A.4, uniqueness of a Nash equilibrium can be guaranteed under the assumptions that F (X) is strictly monotone and that an equilibrium exists.

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Theorem A.10 (Uniqueness of a Solution Under Strict Monotonicity) Suppose that F (X), as given in Theorem A.6, is strictly monotone on K . Then the Nash equilibrium, X∗ , is unique, if it exists. One can construct associated dynamic adjustment or tatonnenment processes associated with the players in a game. For background material on the theory of projected dynamical systems, see Dupuis and Nagurney (1993) and Nagurney and Zhang (1996), where additional results can be found, including conditions under which the set of stationary points of a projected dynamical system, which is a nonclassical dynamic system, coincides with the set of solutions to the related variational inequality problem. In the latter reference, stability analysis results are presented, of relevance to game theory problems as well as to a variety of equilibrium problems that have been formulated as variational inequality problems, from traffic network equilibrium problems to spatial price equilibrium problems and even financial network problems. An algorithm, the Euler method, based on the theory of projected dynamical systems, is applied to solve multitiered supply chain network problems with labor in Chapter 9 of this book. I now turn to a discussion of generalized Nash equilibrium (GNE) in which the constraints underlying the players’ strategies also depend on the strategies of their rivals. A frequently encountered class of generalized Nash games considers common coupling constraints that the players’ strategies are required to satisfy (Kulkarni and Shanbhag 2012). These games are also known as generalized Nash games with shared constraints (Rosen 1965; Facchinei and Kanzow 2010; Fischer et al. 2014). Definition A.18 (Generalized Nash Equilibrium) A strategy vector X∗ ∈ K ≡ m ∗ i=1 Ki , X ∈ S , constitutes a generalized Nash equilibrium if for each player i, i = 1, ..., m : Ui (Xi∗ , Xˆi∗ ) ≥ Ui (Xi , Xˆi∗ ),

∀Xi ∈ Ki , ∀X ∈ S ,

(A.37)

where ∗ ∗ ∗ , Xi+1 , . . . , Xm ), Xˆi∗ ≡ (X1∗ , . . . , Xi−1

Ki is the feasible set of individual players i, and S is the feasible set consisting of the shared constraints. Bensoussan (1974) formulated the GNE problem as a quasi-variational inequality. However, it is recognized that GNE problems are challenging to solve as quasivariational inequality problems since the state-of-the-art in terms of algorithmic development is not as advanced as that for variational inequality problems. Kulkarni and Shanbhag (2012) provide sufficient conditions to establish the theory of a variational equilibrium as a refinement of the GNE, which is highly relevant to applications in the COVID-19 pandemic. A GNE model is constructed in Chapter 4 of this book.

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Definition A.19 (Variational Equilibrium) A strategy vector X∗ is said to be a variational equilibrium of the above generalized Nash equilibrium game if X∗ ∈ K , where K ≡ K ∩ S , is a solution of the variational inequality: −

m

∇Xi Uˆ i (X∗ ), Xi − Xi∗  ≥ 0,

∀X ∈ K .

(A.38)

i=1

A.4.1 An Algorithm There are many algorithms for the computation of solutions to variational inequality problems, including those based on the iterative schemes of Dafermos (1983) and Dupuis and Nagurney (1993). In this book, the modified projection method of Korpelevich (1977) is used for the solution of many of the models. The algorithm requires only Lipschitz continuity and monotonicity of F (X) for convergence, provided that a solution exists. Of special interest are algorithms that resolve the variational inequality problem into subproblems that can be solved easily and exactly in closed form. This book provides a spectrum of alternative variational inequality formulations in order to take advantage of such a feature. The modified projection method, with τ denoting an iteration counter, is presented below. Step 0: Initialization Set X0 ∈ K . Let τ = 1, and let η be a scalar such that 0 < η ≤ θ1 , where θ is the Lipschitz continuity constant (cf. (A.31)). Step 1: Computation Compute X¯ τ by solving the variational inequality subproblem:

X¯ τ + ηF (Xτ −1 ) − Xτ −1 , X − X¯ τ  ≥ 0,

∀X ∈ K .

(A.39)

Step 2: Adaptation Compute Xτ by solving the variational inequality subproblem:

Xτ + ηF (X¯ τ ) − Xτ −1 , X − Xτ  ≥ 0,

∀X ∈ K .

(A.40)

Step 3: Convergence Verification If max |Xlτ − Xlτ −1 | ≤ , for all l, with  > 0, a pre-specified tolerance, then stop; else, set τ := τ + 1, and go to Step 1.

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Theorem A.11 (Convergence of the Modified Projection Method) If F (X) is monotone and Lipschitz continuous (and a solution exists), the modified projection algorithm converges to a solution of variational inequality (A.20).

References Bazaraa, M.S., Jarvis, J.J., Sherali, H.D., 1990. Linear Programming and Network Flows, second edition. John Wiley & Sons, New York. Bazaraa, M.S., Sherali, H.D., Shetty, C.M., 1993. Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York. Beckmann, M.J., McGuire, C.B., and Winsten, C. B., 1956. Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut. Bensoussan, A., 1974. Points de Nash dans le cas de fonctionelles quadratiques et jeux differentiels lineaires a N personnes. SIAM Journal on Control, 12, 460–499. Bertsekas, D.P., Tsitsiklis, J.N., 1989. Parallel and Distributed Computation. Prentice-Hall, Englewood Cliffs, New Jersey. Bradley, S.P., Hax, A., Magnanti, T.L., 1977. Applied Mathematical Programming. AddisonWesley, Reading, Massachusetts. Cournot, A., 1838. Researches into the Mathematical Principles of Wealth. English translation, MacMillan, London, 1897. Dafermos, S., 1980. Traffic equilibrium and variational inequalities. Transportation Science, 14, 43–54. Dafermos, S., 1983, An iterative scheme for variational inequalities. Mathematical Programming, 16, 40–47. Dafermos, S., Nagurney, A., 1987. Oligopolistic and competitive behavior of spatially separated markets. Regional Science and Urban Economics, 17, 245–254. Daniele, P., 2006. Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar Publishing, Cheltenham, England. Dupuis, P., Nagurney, A., 1993. Dynamical systems and variational inequalities. Annals of Operations Research, 44, 9–42. Facchinei, F., Kanzow, C., 2010. Generalized Nash equilibrium problems. Annals of Operations Research, 175, 177–211. Fischer, A., Herrich, M., Schonefeld, K., 2014. Generalized Nash equilibrium problems - Recent advances and challenges. Pesquisa Operacional, 34(3), 521–558. Florian, M., Los, M., 1982. A new look at static spatial price equilibrium models. Regional Science and Urban Economics, 12, 579–597. Friesz, T.L., Harker, P.T., Tobin, R.L., 1984. Alternative algorithms for the general network spatial price equilibrium problem. Journal of Regional Science, 24, 473–507. Gabay, D., Moulin, H., 1980. On the uniqueness and stability of Nash equilibria in noncooperative games. In: Applied Stochastic Control of Econometrics and Management Science. A. Bensoussan, P. Kleindorfer, and C.S. Tapiero, Editors, North-Holland, Amsterdam, The Netherlands, pp. 271–294. Hartman, P., Stampacchia, G., 1966. On some nonlinear elliptic differential functional equations. Acta Mathematica, 115, 271–310. Karamardian, S., 1969. The nonlinear complementarity problem with applications, part 1. Journal of Optimization Theory and Applications, 4, 87–98. Karush, W., 1939. Minima of functions of several variables with inequalities as side conditions. M.S. Thesis, Department of Mathematics, University of Chicago, Chicago, Illinois.

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Kuhn, H.W., Tucker, A.W., 1951. Nonlinear programming. In: Proceedings of Second Berkeley Symposium on Mathematical Statistics and Probability. J. Neyman, Editor, University of California Press, Berkeley, California, pp. 481–492, . Kinderlehrer, D., Stampacchia, G., 1980. An Introduction to Variational Inequalities and Their Applications. Academic Press, New York. Korpelevich, G.M., 1977. The extragradient method for finding saddle points and other problems. Matekon, 13, 35–49. Kulkarni, A.A., Shanbhag, U.V., 2012. On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica, 48, 45–55. Nagurney, A., 1999. Network Economics: A Variational Inequality Approach, second and revised edition. Boston, Massachusetts: Kluwer Academic Publishers. Nagurney, A., 2006. Supply Chain Network Economics: Dynamics of Prices, Flows, and Profits. Edward Elgar Publishing, Cheltenham, United Kingdom. Nagurney, A., Li, D., 2016. Competing on Supply Chain Quality: A Network Economics Perspective. Springer International Publishing Switzerland. Nagurney, A., Yu, M., Masoumi, A.H., Nagurney, L.S., 2013. Networks Against Time: Supply Chain Analytics for Perishable Products. Springer Science + Business Media, New York. Nagurney, A., Zhang, D., 1996. Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic Publishers, Norwell, Massachusetts. Nash, J.F., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, USA, 36, 48–49. Nash, J.F., 1951. Noncooperative games. Annals of Mathematics, 54, 286–298. Rosen, J.B., 1965. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3), 520–533. Samuelson P.A., 1952. A spatial price equilibrium and linear programming. American Economic Review, 42, 283–303. Smith, M.J., 1979. The existence, uniqueness, and stability of traffic equilibria. Transportation Research, 13B, 259–304. Takayama, T., Judge, G.G., 1971. Spatial and Temporal Price and Allocation Models. NorthHolland, Amsterdam, The Netherlands. Wardrop, J.G., 1952. Some theoretical aspects of road traffic research. Proceedings of the Institute of Civil Engineers, Part II, pp. 325–378.