Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition 9811965765, 9789811965760

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Uncertainty and Operations Research

Jianbin Li Xueyuan Cai Binbin Li

Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition

Uncertainty and Operations Research Editor-in-Chief Xiang Li, Beijing University of Chemical Technology, Beijing, China Series Editor Xiaofeng Xu, Economics and Management School, China University of Petroleum, Qingdao, Shandong, China

Decision analysis based on uncertain data is natural in many real-world applications, and sometimes such an analysis is inevitable. In the past years, researchers have proposed many efficient operations research models and methods, which have been widely applied to real-life problems, such as finance, management, manufacturing, supply chain, transportation, among others. This book series aims to provide a global forum for advancing the analysis, understanding, development, and practice of uncertainty theory and operations research for solving economic, engineering, management, and social problems.

Jianbin Li · Xueyuan Cai · Binbin Li

Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition

Jianbin Li Huazhong University of Science and Technology Wuhan, China

Xueyuan Cai Wuhan Textile University Wuhan, China

Binbin Li Huazhong University of Science and Technology Wuhan, China

ISSN 2195-996X ISSN 2195-9978 (electronic) Uncertainty and Operations Research ISBN 978-981-19-6576-0 ISBN 978-981-19-6577-7 (eBook) https://doi.org/10.1007/978-981-19-6577-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 State of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Option/Reservation Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Commitment Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 6 7

2 Managing Downstream Competition via Capacity Allocation in Symmetric Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proportional Allocation Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Retailers’ Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Optimal Wholesale Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Lexicographic Allocation Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Retailers’ Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Optimal Wholesale Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Case with Multiple Retailers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Proportional Versus Lexicographic Mechanisms Under Multiple Retailers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Nonlinear Wholesale Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 30 31 32 48

3 Capacity Allocation with Demand Competition in Asymmetric Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Uniform Allocation Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Equilibrium Analysis Under Uniform Allocation . . . . . . . . .

51 51 55 58 59

11 11 14 15 15 18 20 20 21 24 28

vii

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Contents

3.3.2 Supplier’s Decisions Under Uniform Allocation . . . . . . . . . . 3.4 Proportional and Lexicographic Allocation Mechanisms . . . . . . . . . 3.5 Comparison of Three Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 An Arbitrary Number of Retailers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 63 66 67 70 72 93

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Retailers’ Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Retailer 1’s Best Response Function . . . . . . . . . . . . . . . . . . . . 4.3.2 Retailer 2’s Best Response Function . . . . . . . . . . . . . . . . . . . . 4.3.3 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Supplier’s Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Impacts of Fixed Factor α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Comparison with Centralized Supply Chain . . . . . . . . . . . . . . . . . . . . 4.7 Comparison with Other Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Optimal Capacity Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 98 100 100 102 104 106 108 116 117 119 119 120 132

5 Fixed Allocation of Capacity for Multiple Retailers Under Demand Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Optimal Strategies Under Fixed Allocation . . . . . . . . . . . . . . . . . . . . . 5.3.1 Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplier’s Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Impacts of Fixed Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Comparison with Proportional Allocation . . . . . . . . . . . . . . . . . . . . . . 5.6 Comparison with Centralized Supply Chain . . . . . . . . . . . . . . . . . . . . 5.7 Extension: Case with Retailers’ Capacity Hoarding . . . . . . . . . . . . . . 5.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 135 138 139 142 144 149 150 151 153 154 167

6 Supply Chain Coordination Through Capacity Reservation Contract and Quantity Flexibility Contract . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Capacity Reservation Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Construction Decision of Manufacturer . . . . . . . . . . . . . . . . .

169 169 171 173 174

Contents

6.3.2 Reservation Decision of Retailer . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 The Benefit of the Capacity Reservation Contract . . . . . . . . . 6.3.4 Design Parameters for Channel Coordination . . . . . . . . . . . . 6.4 Quantity Flexibility Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Construction Decision of Manufacturer . . . . . . . . . . . . . . . . . 6.4.2 Reservation Decision of Retailer . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 The Benefit of the Quantity Flexibility Contract . . . . . . . . . . 6.4.4 Design Parameters for Channel Coordination . . . . . . . . . . . . 6.5 Comparison Between Quantity Flexibility Contact and Capacity Reservation Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Comparison Without Coordination . . . . . . . . . . . . . . . . . . . . . 6.5.2 Profit Distribution Under Coordination . . . . . . . . . . . . . . . . . . 6.6 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Negotiation for Setting Contract Parameters . . . . . . . . . . . . . 6.6.2 Correlation Between Demand and Retail Price . . . . . . . . . . . 6.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Outsourcing Decision and Order Policy with Forecast Updating Under Capacity Reservation Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Centralized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Decentralized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Retailer’s Optimal Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Optimal Strategy for the Supplier . . . . . . . . . . . . . . . . . . . . . . . 7.5 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Analysis of Players’ Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

176 177 178 179 180 182 184 186 187 187 190 192 192 194 197 198 211 213 213 215 216 219 219 223 226 226 227 228 251

Chapter 1

Introduction

1.1 Background For most global enterprises, capacity management has become an important issue to focus on. On one hand, capacity shortfall is frequently observed in various industries when retailers’ total order size exceeds a supplier’s available capacity. Besides, capacity investment is typically a long-term decision, and capacity adjustments can only be made infrequently. In the short run, mismatches between capacity and demand are inevitable. When capacity lags demand, rationing is necessary. As a result, in practice, a supplier with limited capacity often puts capacity on allocation, i.e., rationing capacity through quantity competition of retailers rather than through a pricing mechanism. Capacity allocation is a common occurrence in industries in which capacity expansion is costly and time consuming and price is given exogenously (e.g., steel and paper). A supplier can use his prior beliefs on his own and the retailers’ needs to construct a capacity allocation mechanism for allocation of his capacity among retailers. Commonly used capacity allocation mechanisms contain allocation rules that allocate capacity based on retailers’ order sizes, such as proportional allocation and lexicographic allocation. When the supplier’s capacity is insufficient to fill all the orders received, proportional allocation allocates capacity in proportion to order size, and lexicographic allocation allocates capacity in the order of a predetermined priority sequence. On the other hand, capacity reservation can be another effective policy for enterprises to consider when facing with difficulties in capacity. Innovative product such as electronics, semiconductor and computer memory chip, is often characterized by highly volatile demand, short life-cycle and long manufacturing lead time. Thus, capacity management plays an important role in these industries. Indeed, the overall market size of innovative product is growing at an enormous rate, and usually supplier’s capacity is significantly below the market demand while adopting a conservative capacity expansion policy. Due to this fact, supplier has constantly suffered from capacity shortages, resulting in lost revenue, eroding his long-term

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Li et al., Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-19-6577-7_1

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1 Introduction

market position. Consequently, his downstream retailer may not have adequate supplies to meet the market demand and thus damage the market reputation. Despite the need for higher revenues and improved service levels, the supplier may not be ready to expand capacity aggressively because of the significant financial risk due to higher capacity cost, long (capacity) lead time and high demand volatility. However, the supplier might be motivated to expand capacity more aggressively if the retailer agrees to mitigate the financial risk by assuming a certain level of liability. For instance, the retailer may reserve a portion of future capacity prior to demand realization, and in exchange, the supplier commits to have the excess capacity in addition to the reservation amount. This kind of capacity reservation policy provides a win-win situation for both the supplier and the retailer. In the past several years, we have built models for capacity allocation under different mechanisms as well as capacity reservation through different contracts. Our models include allocating with arbitrary number or retailers, fixed allocation, downstream competition in symmetric market, comparisons with different mechanisms, etc. We use mathematical methods to transform practical problems into manageable models, develop their analysis methods, and explore better capacity management. In this book, we provide a unified approach to all models, outline our main conclusions, comment on existing ones, and indicate potential research directions.

1.2 State of Research Capacity allocation mechanisms have been employed widely in many industries, including automobiles, pharmaceuticals, and toys (e.g., Hwang and Valeriano (1992); Blumenstein (1996)). For the properties of a large variety of capacity allocation rules, we refer to the survey paper by Hall and Liu (2010). Next, we review the studies on proportional and lexicographic allocation rules, which are also examined in our work. Lee et al. (1997) recognize that proportional allocation creates incentive for retailers to raise their orders above their desired allocation in multi-echelon supply chains. Cachon and Lariviere (1999a) demonstrate that proportional allocation with fixed price can lead retailers to order more than they desire to receive a favorable allocation, even when they directly order from the supplier. These authors also show that lexicographic allocation is truth-inducing in that it provides no incentive for retailers to order more than they desire. However, under proportional allocation, both the supplier and the whole supply chain can earn higher profits. These authors further demonstrate that a truth-inducing allocation with fixed price cannot maximize total retailer profits. Cachon and Lariviere (1999b) more specifically compare proportional and lexicographic allocations. These authors show that whether order inflation incentivized by proportional allocation helps or harms a supply chain depends on how profits are distributed within the supply chain. In general, encouraging order inflation increases supplier profits but decreases retailer profits. In addition, Sprumont (1991) shows that uniform allocation induces retailers to order their ideal allocations and thus eliminates any gaming effect. Lee et al. (1997)

1.2 State of Research

3

find that proportional allocation causes order inflation, leading to the bullwhip effect in a supply chain. Uniform allocation and proportional allocation are analyzed by Cachon and Lariviere (1999a, b), who show that when compared with an order inflation mechanism, a truth-inducing mechanism can reduce profits for the supplier, the supply chain and retailers. Cachon and Lariviere (1999c) examine a turn-and-earn allocation in a two-period setting, in which current allocations are based on past sales, and find that such allocation increases a supplier’s profit at the expense of the retailers. Lu and Lariviere (2012) extend the turn-and-earn mechanism to a multiperiod supply chain and find that it leads to high levels of sales and is robust to local demand variability. Cohen-Vernik et al. and Purohit (2013) examine this mechanism in a product line setting, and find that turn-and-earn not only helps the manufacturer but also benefits the retailer and increases the total supply chain profit. Hall and Liu (2010) examine and assess the benefit of joint capacity allocation and production scheduling decisions, and capacity allocation decisions are also investigated experimentally (Chen et al. (2012)). Besides, other approaches such as approximate dynamic programming and data envelopment analysis are used on capacity allocation in supply chain networks (Ang et al. (2020), Alkaabneh et al. (2021)). These various studies all assume that retailers sell in their individual markets, whilst we examine a scenario in which retailers compete for both limited supply and market demand, and engage in Cournot competition. Demand competition thus affects the retailers’ ordering decisions, which we assess by examining the effect of capacity allocation on a supply chain with two competing retailers. Besides, capacity allocation with demand competition has increasingly become a focus of research. Liu (2012) investigation of uniform allocation and individually responsive allocation in a supply chain with a common supplier and two retailers with demand competition suggests that uniform allocation is not necessarily truthinducing if there is demand competition. Cho and Tang (2014) extend this research by including multiple retailers and propose the unique truth-telling mechanism of competitive allocation to eliminate the gaming effect. These studies assume a fixed wholesale price, but Chen et al. (2013) compare lexicographic and proportional allocation under the condition of a supplier’s optimal pricing decision and find that lexicographic allocation produces greater profits for the supplier and the supply chain. Li et al. (2017) consider two competing retailers with asymmetric market powers, and analyze their performance with uniform, proportional and lexicographic allocations. Lexicographic allocation is found to be the most advantageous from the supplier’s perspective. Li et al. (2017) further show that in a duopoly setting, fixed allocation can incorporate lexicographic and proportional allocations from the perspectives of the supplier and the supply chain. Qing et al. (2017) investigate a capacity allocation problem in which a supplier can allocate her capacity to an external channel with a manufacturer, to an internal channel, or to both. The equilibrium decisions under these different channel choices are considered and the effect of bargaining power on the supplier’s decision is demonstrated. Yang et al. (2017) consider a supplier who sells via two channels: directly and through a retailer in the same market. They demonstrate the importance of capacity level selection, showing that the supplier, retailer, and consumers can simultaneously benefit

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1 Introduction

from a certain level of capacity. Horizontal capacity sharing between two asymmetric competitors is also studied by Fang and Wang (2020), who show competition between the firms will be softened by capacity sharing, which brings more benefits to the larger firm. The studies by Cho and Tang (2014) and Li et al. (2017) are closely related to our work. Cho and Tang (2014) define a class of individually unresponsive rules in an oligopoly setting, and their principle of predetermining a specific share of the capacity for each retailer is similar to our fixed allocation approach. They identify a unique allocation mechanism that satisfies specific rules and eliminates the gaming effect. Earlier literature relating to capacity reservation mainly discusses the retailer’s optimal strategy. Jain and Silver (1995) analyze the case where the retailer pays a nonrefundable premium to assure a certain level of availability, and derive the optimal policy for the retailer to determine the level of dedicated capacity to reserve and the size of each periodic replenishment. Brown and Lee (1998) particularly analyze “pay-to-delay” capacity reservation contract in semiconductor manufacturing and also propose methods to make optimal decisions for the retailer. Moreover, for a comprehensive review of capacity procurement games, see Cachon (2003). Generally speaking, research work on capacity reservation can be classified into two main categories in terms of retailer’s motivation to reserve. Literatures in the first category consider the case where the retailer reserves a certain portion of future capacity to achieve potential cost reduction. Bonser and Wu (2001) focus the analysis on the retailer’s perspective to minimize procurement cost in a multi-period setting, where the retailer has the option of committing to long-term contract in advance with the option of purchasing additional quantities through fluctuated spot market after the demand is realized. Ralf and Ulrich (2002) investigate the procurement problem to minimize the expected cost with long-term contracts and spot market purchases in a single-period setting. Moreover, Wu et al. (2002), Spinler et al. (2002), Araman et al. (2003), Martinez-de-Albeniz and Simchi-Levi (2005) study partial commitments that employ option contracts under various assumptions regrading demand and spot market prices. Serel et al. (2001) extends capacity reservation problem to a multiperiod setting with stationary demand and derive the equilibrium strategies based on full commitment. Ozer and Wei (2006) investigates the capacity reservation problem under asymmetric forecast information. Specifically, papers in the second category study the problem that the retailer is motivated to offer early commitment on the future capacity so as to ensure acertain level of availability. Eppen and Iyer (1997) study the capacity reservation problem through backup agreements: The retailer may choose to cancel the upfront reserved capacity by paying a penalty cost. Barnes-Schuster et al. (2002) propose a two-period setting, where the retailer can place firm orders and purchase options at the same time. Gan et al. (2004) study capacity coordination problem between two risk-averse agents. They define the coordinating contract that results in a pareto optimal solution and show that any such solution can be achieved only if the two parties share the risk appropriately. Moreover, Gan et al. (2005) extend their coordination model to supply chain involving a risk-neutral supplier and a risk-averse retailer. They show that through risk sharing contract which includes a return policy that limits the maximum amount of returns, supply chain coordination

1.3 Option/Reservation Contract

5

can be achieved. Cachon and Lariviere (2005) investigate capacity contracting in the context of supplier-retailer forecast coordination. Initially, the retailer provides a forecast and a contract consisting of firm commitments and capacity options. Then the supplier builds up the capacity, later the retailer places a firm order based on the up-to-date forecast. They conclude that only under forced compliance regime, the supply chain coordination can be achieved through options contracts in the context of full information. Jin and Wu (2006) propose a capacity expansion policy that the supplier will have excess capacity in addition to reservation amount from the retailer. With a deductible reservation contract, they show that supply chain coordination can be achieved and both players benefit from supply chain coordination.

1.3 Option/Reservation Contract When considering the literature related to option/reservation contracts, our work is closely related to the studies by Cachon and Lariviere (2001) and Erkoc and Wu (2005). Cachon and Lariviere (2001) focus on buyer-led capacity reservation contracts. They explore the importance of contract enforcement with forced and voluntary compliance regimes and show that channel coordination can be achieved only through capacity reservation contracts in the full information case under forced compliance. Erkoc and Wu (2005) and Jin and Wu (2007) concentrate on supplier-led capacity reservation contracts. Erkoc and Wu (2005) propose reservation contracts with deductible reservation fee under exogenous wholesale price. They study the effects of capacity cost, market size, and demand variability on the contract parameters and propose two channel coordination contracts. Nosoohi and Nookabadi (2016) study the implementation of an option contract in the supply chain consisting of a manufacturer and a retailer. They consider that case in which both the market demand and the processing cost are stochastic. The numerical analysis shows the value of option contracts. Merzifonluoglu (2017) studies the problem of maximizing the expected profit of a firm by selecting specific customer orders to satisfy, and procuring from the spot market and a supplier via an option contract. An efficient algorithm is proposed and numerical studies examine the sensitivity of the optimal strategies to key problem parameters, including the total cost and reservation cost weight of the option contract. A few scholars extend the framework with several buyers or sellers. For example, Jin and Wu (2007) study a capacity reservation contract that is slightly different from that investigated by Erkoc and Wu (2005) and extend the result from one buyer to n buyers. Park and Kim (2014) study a multi-period capacity reservation problem consisting of one retailer and multiple heterogeneous suppliers with different capacities and prices. A rolling-horizon implementation strategy is proposed and computational experiments show the effectiveness of the model and strategy. Jain and Hazra (2016) examine a sourcing problem where a buyer reserves capacity from multiple suppliers. The capacities of suppliers are limited, and the unit production cost decreases in capacity. Under the case of information asymmetry, the buyer and other suppli-

6

1 Introduction

ers only know the probability distribution of the supplier’s capacity rather than the accurate capacity. The optimal reservation quantity of the buyer is obtained in the presence of such capacity and cost correlation. Kole and Bakal (2017) consider the replenishment strategy of a buyer with two suppliers. The buyer utilises an option contract to hedge against the potential disruption of a regular supplier. They also investigate the effectiveness of an option contract under different levels of information (full information, partial information or no information) through analytical and numerical studies. A major difference between our work and aforementioned works is the greater flexibility in our proposed contracts. For example, both the final order quantity of the retailer and the capacity level constructed by the manufacturer are not constrained by the reserved quantity.

1.4 Commitment Contract When considering the literature related to commitment contracts, our work is closely related to the research conducted by Tsay (1999) and Wu et al. (2013). Tsay (1999) considers buyer-led contracts where the buyer commits a minimum purchasing quantity and the supplier commits to deliver no less than a certain quantity; this contract leads to channel coordination under certain conditions. Note that the buyer and the supplier in Tsay’s model are equivalent to the retailer and the manufacturer in our model, respectively. Wu et al. (2013) study α-contracts where the buyer commits a fixed fraction α of ex post realised demand to the supplier; their analysis offers a supporting rationale for the coexistence of α-contracts and reservation contracts in practice. Chung et al. (2013) propose a new contract that combines the quantity flexibility mechanism and price-only discount incentive; they find that the contract can coordinate the supply chain without the knowledge of the demand distribution and a win-win outcome occurs for the buyer and supplier under certain conditions. Bicer and Hagspiel (2016) study the case where one supplier and one retailer develop a quantity flexibility contract; they show that although the quantity flexibility helps the retailer to reduce supply-demand mismatch, it may also cause supply chain disintermediation problems for the retailer. Li et al. (2016) study the effect of the quantity flexibility contract under supply chain coordination, and find that both the retailer and the manufacturer can attain maximal profit. In the aforementioned studies, the quantity flexibility is reflected in the form of minimum commitment, whereas in our model, the quantity flexibility is realised through a quantity window. Besides having the named characteristics, our study differs from earlier studies in the following ways. Most studies on supply contracts only address the case with fixed retail price. Our study, on the other hand, considers the situation with stochastic retail price. Furthermore, the compliance mechanism in our contracts is more flexible than that in the existing literature. We allow both parties to deviate from the reservation quantity in both upward and downward directions, whereas contracts in the existing literature are much more restrictive. Finally, by comparing two different risk-sharing

References

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contracts, we can determine the preference of supply chain members for different contracts.

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Fang, D., & Wang, J. (2020). Horizontal capacity sharing between asymmetric competitors. Omega, 97, 102109. Fu, Q., Lee, C. Y., & Teo, C. P. (2010). Procurement risk management using options: Random spot price and the portfolio effect. IIE Transactions, 42, 793–811. Gan, X., Sethi, S. P., & Yan, H. (2004). Coordination of supply chains with risk-averse agents. Production and Operations Management, 13, 135–149. Gan, X., Sethi, S. P., & Yan, H. (2005). Channel coordination with a risk-netural supplier and a down-risk-averse retailer. Production and Operations Management, 14, 80–89. Hall, N. G., & Liu, Z. (2010). Capacity allocation and scheduling in supply chains. Operations Research, 58(6), 1711–1725. Hwang, S. L., & Valeriano, L. (1992). Marketers and consumers get the jitters over severe shortages of nicotine pathes. Wall Street Journal, May 22, B1. Jain, K., & Silver, E. A. (1995). The single period procurement problem where dedicated supplier capacity can be reserved. Naval Research Logistics, 42, 915–934. Jain, T., & Hazra, J. (2016). Sourcing under incomplete information and negative capacity-cost correlation. Journal of the Operational Research Society, 67(3), 437–449. Jin, M. Z., & Wu, S. (2006). Capacity reservation contracts for high-tech industry. European Journal of Operational Research, 176, 1659–1677. Jin, M., & Wu, S. D. (2007). Capacity reservation contracts for high-tech industry. European Journal of Operational Research, 176(3), 1659–1677. Kole, H., & Bakal, I. S. (2017). Value of information through options contract under disruption risk. Computers and Industrial Engineering, 103, 85–97. Lee, H. L., Padmanabhan, V., & Whang, S. (1997). Information distortion in a supply chain: The bullwhip effect. Management Science, 43(4), 546–558. Li, Y., Zhen, X., Qi, X., & Cai, G. G. (2016). Penalty and financial assistance in a supply chain with supply disruption. Omega, 61, 167–181. Liu, Z. (2012). Equilibrium analysis of capacity allocation with demand competition. Naval Research Logistics, 59(3–4), 254–265. Lu, L. X., & Lariviere, M. A. (2012). Capacity allocation over a long horizon: The return on turn-and-earn. Manufacturing and Service Operations Management, 14(1), 24–41. Martinez-de-Albeniz, V., & Simchi-Levi, D. (2005). A portfolio approach to procurement contracts. Production and Operations Management, 14, 90–114. Merzifonluoglu, Y. (2017). Integrated demand and procurement portfolio management with spot market volatility and option contracts. European Journal of Operational Research, 258(1), 181– 92. Nosoohi, I., & Nookabadi, A. S. (2016). Outsource planning through option contracts with demand and cost uncertainty. European Journal of Operational Research, 250(1), 131–142. Ozer, O., & Wei, W. (2006). Strategic commitments for an optimal capacity decisions under asymmetric information. Management Science, 52, 1238–1257. Park, S. I., & Kim, J. S. (2014). A mathematical model for a capacity reservation contract. Applied Math Model, 38(5), 1866–1880. Qing, Q., Deng, T., & Wang, H. (2017). Capacity allocation under downstream competition and bargaining. European Journal of Operational Research, 261(1), 97–107. Serel, D. A., Dada, M., & Moskowitz, H. (2001). Sourcing decisions with capacity reservation contracts. European Journal of Operational Research, 131, 635–648. Spinler, S., Huchzermeier, A., & Kleindorfer, P. R. (2002). The valuation of option on capacity, Working paper, Otto-Beisheim Graduate School of Managment, WHU, Germany. Sprumont, Y. (1991). The division problem with single-peaked preferences: A characterization of the uniform allocation rule. Econometrica: Journal of the Econometric Society, 509–519. Tsay, A. A. (1999). The quantity flexibility contract and supplier-customer incentives. Management Science, 45(10), 1339–1358. Wu, D. J., Kleindorfer, P. R., & Zhang, J. E. (2002). Optimal bidding and contracting strategies for capital-intensive goods. European Journal of Operational Research, 137, 657–676.

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Chapter 2

Managing Downstream Competition via Capacity Allocation in Symmetric Market

2.1 Introduction Capacity investment is typically a long-term decision, and capacity adjustments can only be made infrequently. In the short run, mismatches between capacity and demand are inevitable. When capacity lags demand, rationing is necessary. It is the purpose of this paper to study how the mechanism used by the seller to allocate her capacity influences the strategic behavior of the buyers and its impact on the supply chain members’ profits. Consider a supply chain with one supplier and multiple retailers. The supplier produces a single product and sells it to the retailers, who in turn sell the product to consumers. The supplier has limited production capacity. She sets the wholesale price and chooses a mechanism for allocating her capacity in case it is insufficient to satisfy all the retailers’ orders. The retailers determine their order quantities, and are engaged in a Cournot competition at the market level. Notice that the strategic interaction among the retailers occurs not only at the market level but also at the supply level for scare capacity. The wholesale price and the capacity allocation mechanism, both chosen by the supplier, define the game that the retailers play with their order-quantity decisions. Two allocation mechanisms are considered. The proportional mechanism allocates capacity in proportion to the retailers’ orders, whereas the lexicographic mechanism first establishes a priority sequence for the retailers independent of their orders and then satisfies the retailers’ orders from the highest-priority retailer to the lowestpriority retailer. A key finding of this paper is that for any given wholesale price, the supplier sells more to the retailers under the lexicographic mechanism than if she uses the proportional mechanism. (Here more means at least as much as and sometimes strictly more.) It then follows that the supplier prefers lexicographic allocation to proportional allocation. This result is actually quite intuitive. First of all, the supplier would be indifferent between the two allocation mechanisms if the wholesale price © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Li et al., Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-19-6577-7_2

11

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2 Managing Downstream Competition via Capacity …

is either very high or very low: if the price is very high, the retailers will order small quantities and there will not be a capacity shortage; if the price is very low, the supplier will end up selling all her capacity no matter which allocation mechanism she uses. For an intermediate range of wholesale prices, the lexicographic mechanism induces the highest-priority retailer (call it retailer 1) to order the entire capacity simply to keep the other retailers out of the game. In other words, if the low-priority retailers did not exist, retailer 1 would prefer to lower his order quantity to below the capacity. But because of the existence of the other retailers, lowering the order quantity to below the capacity would only invite the other retailers to enter the game and take all the remaining capacity. To keep the low-priority retailers out of the game, retailer 1 decides to order the entire capacity. Therefore, the lexicographic mechanism dampens retail competition. On the other hand, under the proportional mechanism, the retailers have the same power in securing supply, and no retailer can keep the others out of the game by ordering a large quantity. This reduces the incentive for the retailers to order large quantities, resulting in a smaller sales quantity for the supplier than under the lexicographic mechanism. In short, the supplier can manage the downstream competition to her own benefits by giving preferential treatment to one retailer through the lexicographic allocation of her capacity. Moreover, it can be shown that managing downstream competition in this way also benefits the supply chain (i.e., increasing the total profit of the chain). The reason for this is that because lexicographic allocation increases the total production in the supply chain (relative to if the proportional allocation is used), it is better at addressing the problem of low production due to double marginalization. (Interestingly, if the supplier does not face a capacity constraint, it can also be shown that prioritizing retailers, i.e., requiring the retailers to place their orders in a fixed priority sequence as opposed to having them order simultaneously, increases the total retailer order quantity and thus mitigates the double marginalization problem. This suggests that prioritizing retailers can increase the supply chain efficiency.) It is interesting to contrast the above model with one in which the retailers serve independent markets and therefore are not engaged in a Cournot competition. In this case, for any given wholesale price, each retailer has his own optimal order quantity. If the sum of these optimal order quantities is less than the supplier’s capacity, then no capacity allocation is necessary. On the other hand, if the sum of the retailer optimal order quantities exceeds the supplier’s capacity, then the supplier will be able to sell her entire capacity no matter what allocation mechanism is used. This suggests that the supplier would be indifferent between, e.g., the proportional allocation mechanism and the lexicographic allocation mechanism. This observation, together with the above result derived for the model with retail competition, suggests that market-level competition among the retailers can alter the supplier’s preference for capacity allocation mechanisms. Both the proportional and lexicographic mechanisms for capacity allocation are motivated by real-world practices and widely studied in the literature, see, e.g., Lee et al. (1997) and Cachon and Lariviere (1999a, b, c) and the references therein. The proportional mechanism is perhaps the most intuitive and represents a fair method for sharing the “pain,” i.e., the capacity shortfall is shared by the parties in propor-

2.1 Introduction

13

tion to their orders, whereas the lexicographic mechanism is based on a notion of fairness that is similar to the “first come, first serve” principle although the priority sequence used by the mechanism may be totally unrelated to the sequence in which the parties place their orders. Dhakar et al. (2010) in a case study describes that a large apparel manufacturing company allocates its capacity to customers based on their place in a priority sequence. And prioritization is based on the customer class to which each customer belongs, and the customer classes are defined by a number of factors such as overall business volume, location, strategic importance, etc. See Hwang and Valeriano (1992), Blumenstein (1996) and Gottwald (1997) for illuminating descriptions of industry capacity shortages and various allocation mechanisms used in practice. Various allocation mechanisms have also been used to allocate inventory in production-distribution systems, see, e.g., Eppen and Schrage (1981), Federgruen and Zipkin (1984a, b), Axsater (1993), Federgruen (1993), Chen and Zheng (1997), and Ha (1997). Furthermore, the proportional mechanism has also been used to allocate demand to suppliers whereby the demand allocated to a supplier is proportional to the supplier’s measured capability such as the customer service level provided, see, e.g., Cachon and Zhang (2007). The existing literature on the proportional and lexicographic mechanisms has provided many interesting insights. For example, Lee et al. (1997) shows that the use of the proportional mechanism for allocating scarce production capacity can lead to the so-called bullwhip effect, a phenomenon where as orders are passed from downstream to upstream in a supply chain, their volatility increases. Cachon and Lariviere (1999a) studies both the proportional and the lexicographic mechanisms for allocating capacity in a supply chain where the retailers serve independent markets and each possess private information about their own market. Therefore, for any given wholesale price, each retailer knows his own optimal order quantity but not the optimal quantities of the other retailers. Their analysis shows that the lexicographic mechanism induces the retailers to truthfully reveal their optimal order quantities, whereas the proportional mechanism makes the retailers inflate their orders. As a result, the supplier prefers proportional allocation to lexicographic allocation because proportional allocation leads to more sales under any wholesale price. The contribution of this paper to the extant literature is therefore to consider the competition-dampening aspect of the above two allocation mechanisms, and to show that the lexicographic mechanism is better at reducing the downstream competitive intensity, and to provide an example, albeit stylized, that illustrates that retail-level competition can alter the supplier’s preference over capacity allocation mechanisms. The rest of the paper is organized as follows. To make the paper easy to read, we provide a detailed analysis for the case with two retailers, with only summary results for the general setting with multiple retailers. Section 2.2 presents the two-retailer model and formally introduces the allocation mechanisms. The proportional allocation mechanism is considered in Sect. 2.3, where we characterize the retailers’ equilibrium order quantities and derive the supplier’s optimal wholesale price that maximizes her profits. Section 2.4 deals with the lexicographic mechanism. Section 2.5 compares the two allocation mechanisms. Section 2.6 extends the analysis to the case with multiple retailers, and shows that the key result that the supplier prefers the

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lexicographic mechanism to the proportional mechanism still holds when there are an arbitrary number of retailers. Moreover, it is shown that using nonlinear wholesale pricing to influence downstream competition does not provide any advantage to the supplier when compared with lexicographic allocation. Section 2.7 presents concluding remarks.

2.2 Model Formulation We consider a single-product supply chain with one supplier and two retailers. The supplier produces the product subject to a fixed capacity constraint, and production cost is linear with per-unit cost normalized to zero. The retailers place orders with the supplier. If the supplier’s capacity is insufficient to satisfy both orders, allocation takes place according to a pre-announced mechanism. The retailers sell the product in the same market and are engaged in a Cournot game. The supplier chooses and publicly announces a wholesale price and an allocation mechanism. Knowing these, the retailers independently choose their order quantities. These quantities, together with the allocation mechanism, determine the total supply brought to the market, which in turn determines the retail price. The supplier and the retailers are independent firms, seeking to maximize their own profits. Let K be the supplier’s capacity, and w the wholesale price. Both of these are publicly known. Denote by m i retailer i’s order quantity, i = 1, 2. Because the capacity is known, ordering more than the capacity by one retailer is a clear indication of gaming behavior. We do not allow this behavior, and thus assume m i ≤ K , i = 1, 2. If  2 i=1 m i ≤ K , each retailer receives his order. Otherwise, an allocation mechanism is used to determine how much each retailer gets. Let gi (m), with m = (m 1 , m 2 ), be the quantity received by retailer i’s, i = 1, 2. The retail price is given by p = M − g1 (m) − g2 (m), where M is a constant representing the market potential. To avoid trivial cases, we assume M > K . The supplier chooses an optimal wholesale price w to maximize her profits: s = max{w · (g1 (m 1 , m 2 ) + g2 (m 1 , m 2 ))}; w

while the retailers play a Cournot game by simultaneously choosing their order quantities: 1 (w, m 2 ) = max (M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w) · g1 (m 1 , m 2 ), m1

2 (w, m 1 ) = max (M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w) · g2 (m 1 , m 2 ). m2

2.3 Proportional Allocation Mechanism

15

Two allocation mechanisms are considered. Under the proportional allocation mechanism,   K mi gi (m) = min m i , 2 , i = 1, 2. j=1 m j A lexicographic mechanism first specifies a sequence in which the retailers’ orders are to be satisfied, irrespective of the order quantities. If retailer 1’s order is to be satisfied first, we have g1 (m) = min{m 1 , K } and g2 (m) = min{m 2 , K − g1 (m)}; otherwise, g2 (m) = min{m 2 , K } and g1 (m) = min{m 1 , K − g2 (m)}. In the sequel, for convenience we adopt: ⎧ if b < a or b < 0, ⎨ ∅, [a, b] = [0, b], if b > 0 and a < 0, ⎩ [a, b], if 0 ≤ a ≤ b, and for any real number x, (x)− denotes a number that is less than x but is as close to x as possible.

2.3 Proportional Allocation Mechanism We first characterize the retailers’ equilibrium order quantities given a wholesale price, and then show how an optimal wholesale price can be determined to maximize the supplier’s profits.

2.3.1 Retailers’ Game Under proportional allocation, the retailers’ optimization problems can be written as 1 (w, m 2 ) = max {(M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w) · g1 (m 1 , m 2 )} (2.1) m 1 ≤K   m1 · K m2 · K m1 · K = max M − ( ∧ m1) − ( ∧ m2) − w · ∧ m1 m 1 ≤K m1 + m2 m1 + m2 m1 + m2

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and 2 (w, m 1 ) = max {(M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w) · g2 (m 1 , m 2 )} (2.2) m 2 ≤K   m1 · K m2 · K m2 · K = max M − ( ∧ m1) − ( ∧ m2) − w · ∧ m2 . m 2 ≤K m1 + m2 m1 + m2 m1 + m2

The next theorem characterizes the retailers’ best response functions. (All proofs can be found in the Appendix.) Theorem 2.1 Let m 1 (·) be retailer 1’s best response function, i.e., if retailer 2 orders m 2 then it is optimal for retailer 1 to order m 1 (m 2 ). Similarly, let m 2 (·) be retailer 2s best response function. Because the game is symmetric, we only need to characterize one of these functions: (i) w ∈ (0, M − 2K ) : for any m 2 , m 1 (m 2 ) = K with 1 (w, m 2 ) = (M − K − 2 w) K K+m 2 . 2 (ii) w ∈ [M − 2K , M − K ] : if m 2 ≤ α then m 1 (m 2 ) = M−w−m with 1 (w, m 2 ) = 2 2 (M−w−m 2 )2 ; if m > α then m (m ) = K with  (w, m ) = (M − K − w) K K+m 2 , 2 1 2 1 2 4 where α=

 1

K + M − w − (M − w − K ) · (M − w + 7K ) . 2

2 (iii) w ∈ (M − K , M) : if m 2 ≤ M − w then m 1 (m 2 ) = M−w−m with 1 (w, m 2 ) = 2 2 (M−w−m 2 ) ; if m 2 > M − w then m 1 (m 2 ) = 0 with 1 (w, m 2 ) = 0. 4

The intuition behind Theorem 2.1 is fairly easy to see. It is obvious that if the wholesale price is very high, the retailer will not order very much and the capacity constraint becomes a nonissue. In this case, the best response function is the same as if there were no capacity limit. This explains (iii). Another observation one can make is that if the capacity limit is to be exceeded, then retailer 1 will surely order K (with retailer 2s order quantity fixed at m 2 ). This is because (1) if the sum of the orders exceeds the capacity, the total supply to the market will be fixed at K , which means the profit margin for each retailer stays at M − K − w, (2) as a result, retailer 1s profit is increasing in its stock allocation, and 3) ordering K , which is the maximum allowed, maximizes retailer 1s allocation. Consequently, retailer 1s best response is either K or (M − w − m 2 )/2, with the latter being the best response when the capacity constraint is nonbinding. There are two possible reasons for retailer 1 to order K (and thus the total order exceeds the capacity): the wholesale price is very low or retailer 2 orders too much. The first case is self evident. For the latter case, if retailer 1 decides to ensure that the sum of the orders stays within the capacity limit, then he would have to order a very small quantity (less than or equal to K − m 2 ). When m 2 is large, this option becomes unattractive compared to ordering K . This explains (i) and the second part of (ii). Finally, the first part of (ii) represents the “middle ground”, where the wholesale price is neither high nor low, and retailer 2

2.3 Proportional Allocation Mechanism

17

does not order very much. In this case, staying within the capacity limit turns out to be optimal for retailer 1. The following theorem characterizes the Nash equilibrium of the retailer game. √ Theorem 2.2 (i) If w ∈ (0, M + 3K − 3 2K ), then there √ is a unique Nash equilibrium: (m ∗1 , m ∗2 ) = (K , K ). (ii) If w ∈ [M + 3K − 3 2K , M − K ], then there , M−w ) and (m ∗1 , m ∗2 ) = (K , K ). Furare two Nash equilibria: (m ∗1 , m ∗2 ) = ( M−w 3 3 thermore, the former equilibrium dominates the latter in the sense of generating a higher profit for both retailers. (iii) If w ∈ (M − K , M), then there is a unique Nash , M−w ). equilibrium: (m ∗1 , m ∗2 ) = ( M−w 3 3 Theorem 2.2 describes a pattern for the retailers’ √ total order quantity as a function of the wholesale price. For w√< M + 3K − 3 2K , the total order quantity is 2K , while for w ≥ M + 3K − 3 2K , the√total quantity ordered is 2(M − w)/3, which achieves its maximum value of 2( 2 − 1)K (≈ 0.82K ) when w = M + √ 3K − 3 2K . Notice that the retailers’ total order quantity is a nonincreasing function of the wholesale price, which clearly is expected. What √ is interesting is that as the wholesale price crosses the critical point M + 3K − 3 2K , the total order quantity drops from 2K to a value strictly less than K . To understand the intuition behind this drop in the total order quantity, consider for a moment the case with unlimited capacity. In this case, for any given wholesale price w, the equilibrium order quantity for each retailer is (M − w)/3. Let’s take a w so that this equilibrium order quantity is 0.5K . Now suppose retailer 2 orders 0.5K in the capacitated scenario. What is retailer 1s optimal response? If he orders 0.5K , making the total equal to K , his profit would be 0.5(M − K − w)K . If he increases his order quantity by any small amount, the supply capacity will be exceeded, and the proportional mechanism starts to dictate allocation. Note that as long as the total order is more than the capacity, the total supply to the retail market will remain at K and thus the profit margin for retailer 1 stays at M − K − w. Seeing this, retailer 1 would want to increase his allocation as much as possible. And this is done by ordering the maximum quantity allowed, i.e., K . In short, if retailer 1 wants to order any amount over 0.5K , the best for him is to go all the way by ordering K . This strategy earns him an allocation of K K = 23 K , resulting in a profit of 23 K (M − K − w), which is more than his 0.5K +K profit if he “plays along” with retailer 2 by ordering 0.5K (i.e., implementing the uncapacitated equilibrium). This example shows that the capacity constraint and the proportional allocation mechanism create a situation where a retailer can increase his order while maintaining his profit margin (this is never the case when capacity is unlimited). This incentive structure destroys the uncapacitated equilibrium (0.5K , 0.5K ). What Theorem 2.2 has shown is that all the uncapacitated equilibria √ (θK , θK ) for all θ > 2 − 1 ≈ 0.41 have been replaced by the capacitated equilibrium (K , K ). This explains the drop of the total order quantity from 2K and 0.82K √ when the wholesale price crosses the value M + 3K − 3 2K .

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2.3.2 Optimal Wholesale Price Here we consider the wholesale pricing decision from the supplier’s perspective. Knowing how the retailers react to changes in the wholesale price (Theorem 2.2), determining the optimal wholesale price is straightforward. As noted above, for √ w < M + 3K − 3 2K , the total order quantity is 2K , which exceeds the capacity K , and thus the total sales to the retailers is K . In this case, the supplier’s profit is wK . (Recall that the supplier’s variable cost of √ production has been normalized to zero.) On the other hand, if w ≥ M + 3K − 3 2K , the total quantity ordered is the same as the total supplied, because the total order quantity 2(M − w)/3 is within the capacity constraint. In this case, the supplier’s profit is 2w(M − w)/3. Therefore, the supplier’s maximum profit can be written as max{π1 , π2 } where π1 =

wK √ w∈(0, M+3K −3 2K ) max

√ = (M + 3K − 3 2K )− K ,

and π2 =

max√ w∈[M+3K −3 2K , M)

2w(M − w) . 3

Note that the unconstrained optimization problem of maxw 2(M−w)w is solved at w = 3 M/2 with the maximum objective function value at M 2 /6. The optimal wholesale price can be determined by considering the following two cases. √ √ Case 1: M2 ≤ M + 3K − 3 2K or equivalently M ≥ 6( 2 − 1)K . In this case, √ √ 2 (M + 3K − 3 2K ) · (3 2 − 3)K 3√ √ √ = 2( 2 − 1)K (M + 3K − 3 2K ) < K (M + 3K − 3 2K )− = π1 .

π2 =

√ + 3K − 3 2K )− , with the supTherefore, the optimal wholesale price is w∗ = (M √ = K · (M + 3K − 3 2K )− . √ plier’s maximum profit at ∗s √ M Case 2: 2 > M + 3K − 3 2K or equivalently M < 6( 2 − 1)K . In this case, π2 = M 2 /6 as noted above. To determine the optimal wholesale price, we need 2 to compare π1 with √ M /6 crosses √ π2 . It is easily verified that as functions of M, from below exactly once, at M = 3(2 − 2)K . In other K (M + 3K − 3 2K ) √ 2 words, for M ≤ 3(2 − 2)K , π2 ≥ π1 and thus w ∗ = M2 with ∗s = M6 , and for √ √ √ M ∈ (3(2 − 2)K , 6( 2 −√1)K ), π1 > π2 and thus w ∗ = (M + 3K − 3 2K )− with ∗s = K (M + 3K − 3 2K )− . We thus have the following theorem (Table 2.1). √ Theorem 2.3 (i) If M ≤ 3(2 − 2)K , then the supplier’s optimal wholesale price is ∗ ∗ w M 2 /6. (ii) If M > √3(2 − √ = M/2 and the optimal profits for the supplier are s = ∗ 2)K then the supplier’s optimal wholesale price is w = (M +√3K − 3 2K )− and the optimal profits for the supplier are ∗s = K (M + 3K − 3 2K )− .

2.3 Proportional Allocation Mechanism

19

Table 2.1 Results under proportional allocation mechanism M Retailers’ Optimal Supplier’s sales equilibrium order wholesale price quantities

M M M M M≤ √ 6 , 6 2 3 3(2 − 2)K M> √ (K , K ) (M √ + 3K − K 3(2 − 2)K 3K 2)−

Supplier’s profits

M2 6

K (M √ + 3K − 3K 2)−

w ∗ , Q∗

◦

Q∗ = K

3 √ 2K

K

3K + =

M

√ 2)K ∗

w

=

◦

M 2

∗

Q

=

w∗

(2 −

−

√ (6−3 2)K 2

M 3

3(2 −

√ 2)K

M

Fig. 2.1 Proportional mechanism

Theorem 2.3 indicates that as M increases, the supplier’s profits increase (continuously). This is certainly expected, as M represents the market potential for the supplier’s product and the supplier reaps more profits from a product with higher market potential. What is not so intuitive is that as M increases, the optimal wholesale price √ threshold √ is not necessarily increasing. Note that when M crosses the the optimal wholesale price drops from M/2 = 3(2 − 2)K /2 to 3(2 − 2)K , √ √ (M + 3K − 3 2K )− = (9 − 6 2)K (see Fig. 2.1 for an illustration). Note further that before crossing the √ threshold market potential, the total retailer order quantity is ∗ = M3 < (2 − 2)K < K , i.e. the optimal selling quantity Q ∗ = M3 , whereas 2 M−w 3 after crossing the threshold value the total order quantity becomes 2K (but the optimal selling quantity Q ∗ = K for the limited capacity). In other words, when the market potential is relatively small, the supplier sets its wholesale price to induce a retail demand that is below capacity and thus purposely lets some of the capacity go unused, and when the market potential exceeds a threshold level, the supplier suddenly cuts the wholesale price to achieve full capacity utilization. Although the

20

2 Managing Downstream Competition via Capacity …

supplier’s profit increases continuously as M increases, there is a sudden drop in wholesale price, a sudden increase in retail order quantities, and a transition from excess capacity to 100% capacity utilization.

2.4 Lexicographic Allocation Mechanism We now take up the lexicographic allocation mechanism. As in the previous section, we first consider the order-quantity game between the retailers and then characterize the optimal wholesale price that maximizes the supplier’s profits.

2.4.1 Retailers’ Game Recall that the allocation mechanism gives priority to one retailer, satisfies this retailer’s order as much as possible, and fills the other retailer’s order with the remaining capacity, if any. Suppose priority is given to retailer 1. (Due to symmetry, the case when retailer 2 gets priority is the same.) Note that the retailers’ profits can be written as: ˆ 1 (w, m 2 ) = max {[M − m 1 − (K − m 1 ) ∧ m 2 − w] · m 1 } ,  m 1 ≤K

(2.3)

ˆ 2 (w, m 1 ) = max {[M − m 1 − (K − m 1 ) ∧ m 2 − w] · [(K − m 1 ) ∧ m 2 ]} . (2.4)  m 2 ≤K

Compared with the proportional allocation mechanism, the game of two retailers is not symmetric for the lexicographic allocation mechanism. The best response function for retailer 1 is as follows. Theorem 2.4 Let mˆ 1 (m 2 ) be retailer 1’s optimal order quantity, given retailer 2 orders m 2 . The best response function is characterized as follows: ˆ 1 (w, m 2 ) = (M − K − (i) w ∈ (0, M − 2K ) : for any m 2 , mˆ 1 (m 2 ) = K and  w)K . 2 ˆ 1 (w, m 2 ) = ˆ mˆ 1 (m 2 ) = M−w−m and  (ii) w ∈ [M − 2K , M − K ] : for m 2 ≤ α, 2 (M−w−m 2 )2 ˆ 1 (w, m 2 ) = (M − K − w)K , where ; for m 2 > α, ˆ mˆ 1 (m 2 ) = K and  4 αˆ = M − w −



4K (M − w − K ).

2 ˆ 1 (w, m 2 ) = and  (iii) w ∈ (M − K , M) : for m 2 ≤ M − w, mˆ 1 (m 2 ) = M−w−m 2 2 (M−w−m 2 ) ˆ ; for m 2 > M − w, mˆ 1 (m 2 ) = 0 and 1 (w, m 2 ) = 0. 4

Next we give retailer 2s best response function given that retailer 1 orders m 1 .

2.4 Lexicographic Allocation Mechanism

21

Theorem 2.5 Let mˆ 2 (m 1 ) be retailer 2s optimal order quantity, given that retailer 1 orders m 1 . The best response function is characterized as follows: (i) w ∈ (0, M − 2K ) : mˆ 2 (m 1 ) can be any value in the interval [K − m 1 , K ], with ˆ 2 (w, m 1 ) = (M − K − w)(K − m 1 ).  1 and (ii) w ∈ [M − 2K , M − K ] : if m 1 < 2K − M + w then mˆ 2 (m 1 ) = M−w−m 2 2 (M−w−m 1 ) ˆ , otherwise if m 1 ≥ 2K − M + w then mˆ 2 (m 1 ) can be any 2 (w, m 1 ) = 4 ˆ 2 (w, m 1 ) = (M − K − w)(K − m 1 ). value in the interval [K − m 1 , K ] and  1 ˆ 2 (w, m 1 ) = and  (iii) w ∈ (M − K , M) : if m 1 ≤ M − w then mˆ 2 (m 1 ) = M−w−m 2 (M−w−m 1 )2 ˆ 2 (w, m 1 ) = 0. , otherwise if m 1 > M − w then mˆ 2 (m 1 ) = 0 and  4 We are now ready to characterize the Nash equilibrium in the order-quantity game between the two retailers. Theorem 2.6 (i) If w ∈ (0, M − 2K ) then √any point in {K } × [0, K ] is a Nash equilibrium. (ii) If w ∈ [M − 2K , M − 9K −32 5K ) then any point in {K } × (α, ˆ K ] is √

, M−w ) is a Nash a Nash equilibrium. (iii) If w ∈ [M − 9K −32 5K , M − K ] then ( M−w 3 3 equilibrium and any point in {K } × (α, ˆ K ] is also a Nash equilibrium. Moreover, the former equilibrium dominates any equilibrium in {K } × (α, ˆ K ] in the sense of generating higher profits for both retailers. (iv) If w ∈ (M − K , M) then there exists , M−w ). a unique Nash equilibrium (mˆ ∗1 , mˆ ∗2 ) = ( M−w 3 3 As Theorem 2.6 indicates, the quantity game of the retailers may have multiple equilibria. In cases (i) and (ii) of the theorem, all equilibria predict that retailer 1 will order K and receive the entire capacity from the supplier. In these cases, it does not really matter what retailer 2 orders, and the different equilibria lead to the same profits , M−w ), since it gives for the two retailers. In case (iii), we adopt the equilibrium ( M−w 3 3 the retailers higher profits than any of the other equilibria does. Note also that as the wholesale price increases, the total order quantity from the retailers decreases, which is certainly expected. What is interesting, however, is that the decrease in the total √ order quantity is not continuous. To see this, note that for w ∈ [M − 9K −32 5K , M − K ], the total equilibrium order quantity is

M−w 3

+

√ 3− 5 K , and for w ∈ (M − 2 √ hand, for w < M − 9K −32 5K ,

M−w 3

≤

+ M−w < 2K . On the other K , M), the total is M−w 3 3 3 the total equilibrium order quantity is at least K . Therefore, as w crosses the critical √ point M − 9K −32 5K , the equilibrium total order quantity experiences a drop.

2.4.2 Optimal Wholesale Price To determine the optimal wholesale price that maximizes the supplier’s profits, recall √ 9K −3 5K , the total quantity ordered by the from Theorem 2.6 that (1) for w < M − 2 retailers (in equilibrium) is at least K , implying that the quantity sold by the supplier is √ K ; and (2) for w ≥ M − 9K −32 5K , each retailer’s equilibrium order quantity is M−w 3 and thus the total quantity sold by the supplier is 2(M−w) . Let f (w) be the supplier’s 3

22

2 Managing Downstream Competition via Capacity …

√ Fig. 2.2 M ≥ 9K − 3 5K

Π* s

Π* s =2/3w(Μ-w) Π* s =wΚ

M-

W

9K - 3 5K 2

√

profits as a function of the wholesale price. Write w0 for M − 9K −32 5K . Thus f (w) = wK for w < w0 and f (w) = 2(M−w)w for w ≥ w0 . Clearly, the supplier’s maximum 3 profits is either f (w0− ) = w0− K or maxw≥w0 f (w), whichever is larger. To find the optimal wholesale price, first note that f (w0− ) < f (w0 ) and that 2 arg maxw { 2(M−w)w } = M2 with maxw { 2(M−w)w } = M6 . Therefore, if M2 ≤ w0 or 3 3 √ equivalently M ≥ 9K − 3 5K , then f (w) is decreasing for w ≥ w0 , indicating that the optimal wholesale price w∗ = w0− (This situation is depicted in Fig. 2.2). Now √ suppose M2 > w0 or equivalently M < 9K − 3 5 K . In this case, we only need to 2 compare the values of f (w0− ) and M6 . If the former is larger, then w ∗ = w0− . Othercan be estabwise, w ∗ = M/2 (See Fig. 2.3 for an illustration.). The following fact  √ 2 lished after simple algebra: f (w0− ) ≤ M6 if and only if M ∈ [0, 3K − 3 5 − 2K ]. We thus have the following theorem. √ 5 − 2K , then the optimal wholesale price is M/2 Theorem 2.7 If M ≤ 3K − 3 ˆ ∗s = M 2 /6. Otherwise, if M > 3K − and the supplier’s maximum profits are  √ √ 3 5 − 2K , then the optimal wholesale price is (M − 9K −32 5K )− and the sup√ ˆ ∗s = K (M − 9K −3 5K )− . plier’s maximum profits are  2

√ 5 − 2K , From Theorem 2.7 (and its derivation), we know that for M ≤ 3K − 3 wˆ ∗ = M/2, which is greater than w0 . Therefore, at the optimal wholesale price, we have from Theorem 2.6  that the equilibrium order quantity for each retailer is

√ M−wˆ ∗ 3

=

M 6

≤

1−

√ 5−2 K 2

. The total order quantity is clearly strictly less than K ,

2.4 Lexicographic Allocation Mechanism Fig. 2.3 M√∈ [0, 9K − 3 5K )

23

Π* s Π* s =2/3w(Μ-w) 0 3K − 3 5 − 2K , we have from Theorem 2.7 that wˆ ∗ = w0− . At the optimal wholesale price, from Theorem 2.6, we know that retailer 1s equilibrium order quantity is K and retailer 2s equilibrium order quantity can be anywhere from αˆ to K , suggesting that the total order quantity exceeds K (i.e. the supplier’s optimal selling quantity Qˆ ∗ = K ). In short, when the market potential M is below a threshold, the supplier maintains a strict capacity surplus, whereas when the market potential exceeds the threshold, the supplier’s capacity is depleted. Clearly, it is not surprising that as the market potential increases, the supplier sells more. But it is interesting that the total order quantity jumps from strictly below capacity to strictly above as the market threshold is crossed. For an illustration of the supplier’s optimal wholesale price and selling quantity as a function of the market potential, see Fig. 2.4 (Table 2.2).

24

2 Managing Downstream Competition via Capacity … ˆ∗ w ˆ∗ , Q

ˆ∗ = K Q

◦

(1 −

√ 5 − 2)K ∗

ˆ w

=

M 2

M

◦

wˆ ∗

=

M

−

2

√ 5 − 2)K

9−

−

3 (1 2

3 √ 5

K

K

ˆ∗ = 3 Q 3(1 −

√ 5 − 2)K

M

Fig. 2.4 Lexicographic mechanism

2.5 Comparisons We proceed to compare the two allocation mechanisms from the perspective of the supplier, the retailers, and the supply chain. Underlining all the comparisons is the Stackelberg game: given an allocation mechanism, the supplier chooses an optimal wholesale price to maximize its profits, anticipating the retailers’ order quantities in equilibrium. First, consider the supplier’s profits. Table 2.3 above summarizes the supplier’s profits under both the proportional and lexicographic allocation mechanisms, assuming that the supplier chooses the optimal wholesale price in every scenario. (The optimal wholesale prices and the supplier’s profits are directly obtained from Theorems 2.3 and 2.7.) From Table 2.3, we see that the lexicographic mechanism dominates the proportional mechanism: when the market potential is relatively small (and there is excess capacity), the supplier is indifferent between the two mechanisms; whereas when the market potential is large, the supplier strictly prefers lexicographic. To understand why the supplier prefers the lexicographic mechanism to the proportional mechanism, consider how the retailers react as the wholesale price increases. When the wholesale price is very small, the retailers will order large quantities and the supplier’s capacity will be fully utilized, with the actual total quantity sold being equal to K . Now consider the other extreme where the wholesale price is very large. In this case, the retailers will not order very much, and there will be excess capacity. It is easy to see that in both of these extremes, the allocation mechanism does not really matter as far as the supplier is concerned. Between these extremes, the supplier wants to stay with the full-capacity-utilization regime as long as possible, because

2.5 Comparisons

25

Table 2.3 Comparison of profits M Proportional M∈ [0, 3K − √ 5 − 2K ] 3

Lexicographic

Comparison

Optimal choice

wˆ ∗ = ˆ ∗s = 

ˆ ∗s ∗s = 

the same

ˆ r∗ r∗ = 

the same

ˆ ∗sc ∗sc = 

the same

ˆ ∗s ∗s < 

Lexicographic

ˆ r∗ r∗ ≤ 

Lexicographic

ˆ ∗sc ∗sc < 

Lexicographic

ˆ ∗s ∗s < 

Lexicographic

ˆ r∗ r∗ > 

Proportional

ˆ ∗sc ∗sc < 

Lexicographic

ˆ ∗s ∗s < 

Lexicographic

ˆ r∗ r∗ > 

Proportional

ˆ ∗sc ∗sc = 

The same

Supplier’s profits w∗ = ∗s

M 2 , M2 6

= Retailers’ profits

M 2 , M2 6

2 ˆ r∗ = M 2  r∗ = M 18 18 Supply chain’s profits 2 ˆ ∗sc = 2M 2 ∗sc = 2M 

9

M∈ (3K − √ 3 5 − 2K , √ 3 7 − 3 5K ]

M∈ √ (3 7 −√ 3 5K , 6K − 3 2K ]

Supplier’s profits w∗ =

M 2 ,

∗s =

M2 6

wˆ ∗ =√(M − 9K −3 5K − ) , 2 ˆ ∗s = K (M −  √ 9K −3 5K − ) 2

Retailers’ profits √ 2 ˆ r∗ = 7−3 5 K 2  r∗ = M 18 2 Supply chain’s profits 2 ˆ ∗sc =  ∗sc = 2M 9 MK − K2 Supplier’s profits w∗ =

M 2 ,

∗s =

M2 6

Retailers’ profits

M√∈ (6K − 3 2K , +∞)

9

wˆ ∗ =√(M − 9K −3 5K − ) , 2 ˆ ∗s = K (M −  √ 9K −3 5K − ) 2

√ 5

2 ˆ r∗ = 7−3  r∗ = M 18 2 Supply chain’s profits 2 ˆ ∗sc =  ∗sc = 2M 9 MK − K2 Supplier’s profits

K2

w ∗ = (M wˆ ∗ =√(M − √+ 3K − 3 2K )− , 9K −3 5K )− , 2 ˆ ∗s = K (M −  ∗s = K√(M + √ 3K − 3 2K )− 9K −3 5K )− 2 Retailers’ profits √ ˆ r∗ = 7−3 5 K 2  r∗√= 2 (3 2 − 4)K 2 Supply chain’s profits ˆ ∗sc =  ∗sc = MK − K2 MK − K2

26

2 Managing Downstream Competition via Capacity …

for any given wholesale price, the supplier makes more profits under full utilization (selling more) than under excess capacity (selling less). It turns out that the lexicographic mechanism is able to induce the high-priority retailer to continue to order K , even after the retailers under the proportional mechanism have switched to the excess-capacity regime. The reason for this is that switching to the excess-capacity regime means splitting the profits equally between the two retailers. This is especially difficult to do for the high-priority retailer under lexicographic allocation, relative to the case with proportional allocation where the retailers are already making the same profits even under the full-utilization regime. In other words, the lexicographic mechanism wins by increasing the retailer “stickiness” to the full-utilization regime. Now consider the retailers’ profits. Suppose √proportional allocation is used. Recall from Theorem 2.3 that (1) if M ∈ [0, 6K − √ 3 2K ], the supplier’s optimal √ wholesale price is w ∗ = M2 , and (2) if M > 6K − 3 2K , w ∗ = (M + 3K − 3 2K )− . For case (1), it is easily verified that (iii) of Theorem 2.2 holds and therefore the retailers’ ∗ = M6 , leading to their profits equilibrium order quantities are m ∗1 = m ∗2 = M−w 3 ∗1 = ∗2 = (M −

M M M2 M − )× = . 3 2 6 36

For case (2), note that (i) of Theorem 2.2 holds and thus m ∗1 = m ∗2 = K with each retailer receiving K /2, which means ∗1

=

∗2

√ 3 2−4 2 K K . = (M − K − w ) = 2 2 ∗

Now suppose lexicographic allocation is used. Recall from Theorem2.7 that (a) √ √ 5 − 2K ] then wˆ ∗ = M2 , and b) if M > 3K − 3 if M ∈ (0, 3K − 3 5 − 2K √ √ 9−3 5 ∗ − then wˆ = (M − 2 K ) . For (a), since M ≤ 3K − 3 5 − 2K ≈ 1.54K , we have M − K < M/2, which implies that (iv) of Theorem 2.6 holds with equilibrium ∗ order quantities mˆ ∗1 = mˆ ∗2 = M−3wˆ = M6 and the corresponding profits ˆ ∗1 =  ˆ ∗2 = (M −  For (b), since wˆ ∗ = (M − mˆ ∗2 ∈ (α, ˆ K ], implying

√ 9−3 5 K )− , 2

M M M2 M − )· = . 3 2 6 36 (ii) of Theorem 2.6 holds with mˆ ∗1 = K and

√ 7−3 5 2 K , = (M − K − wˆ )K = 2 ˆ ∗2 = 0. 

ˆ ∗1 

∗

Let r∗ denote the sum of the retailers’ profits under the proportional allocation ˆ r∗ the sum of the retailers’ profits under lexicographic allocation mechanism, and 

2.5 Comparisons

27

ˆ r∗ can be found in Table 2.3. Note mechanism. The comparison between r∗ and  that the retailers’ preference over the two allocation mechanisms depends on the market potential: as the market potential increases, their preference goes from being indifferent, to lexicographic, and finally to proportional. (Here the comparison is based on the total retailer profits. Of course, if the priority scheme in the lexicographic mechanism is determined by a coin toss, then each retailer’s expected profits would be equal to half of the total. In this sense, the comparison also holds for the individual retailers.) Adding the supplier’s profits to the retailers’ profits, we have the supply chain ˆ ∗sc profits. Denote by ∗sc supply chain’s profits under proportional allocation and  supply chain’s profits under lexicographic allocation. Table 2.3 summarizes the comˆ ∗sc . Observe that from the supply chain’s perspective, the parison between ∗sc and  lexicographic mechanism dominates the proportional mechanism. ˆ ∗s , ∗sc ,  ˆ ∗sc , r∗ , and  ˆ r∗ as functions of the market potenFigure 2.5 depicts ∗s ,  tial. Recall that the lexicographic mechanism dominates the proportional mechanism in terms of the supplier’s profits and the supply chain’s profits, but there is no dominance from the retailers’ standpoint. The figure also provides c∗ sc , the maximum , let q be the total supply chain profits under centralized control. (To obtain c∗ sc

Fig. 2.5 Optimal profits of all players in centralized and decentralized systems

28

2 Managing Downstream Competition via Capacity …

quantity sold. The supply chain profits can be written as Csc = (M − q)q. Note M2 that the optimal selling quantity q ∗ = min{K , M2 }. Thus c∗ sc = 4 for M ≤ 2K and c∗ sc = (M − K )K for M > 2K .) It provides a benchmark for measuring the loss of efficiency in the decentralized supply chain.

2.6 The Case with Multiple Retailers 2.6.1 Proportional Versus Lexicographic Mechanisms Under Multiple Retailers We now generalize the proceeding analysis from a system with two retailers to one with an arbitrary number of retailers. Let n be the number of retailers in the supply chain. Retain all other notation and assumptions. In particular, m i is retailer i’s order quantity, i = 1, . . . , n, and gi (m) is the quantity retailer i receives, where m = (m 1 , . . . , m n ). The exact functional form of gi (·) depends on the allocation mechanism used when the orders from the retailers exceed the supplier’s capacity K . The supplier chooses an optimal wholesale price w to maximize her profits:    n gi (m) s = max w · w

i=1

while the retailers simultaneously choose their order quantities m 1 , m 2 , . . . , m n to maximize their profits:  i (w, m−i ) = max mi

M−

n 

gi (m) − w · gi (m).

i=1

Again, we consider the proportional and lexicographic allocation mechanisms. First, consider the proportional allocation mechanism, whereby 

K mi gi (m) = min m i , n j=1 m j

 , i = 1, . . . , n.

Let  √ √ (n + 1) n 2 + 4n − 4 − (n + 1)n n( n 2 + 4n − 4 − n) , βn = 1 − . αn = 2(n − 1) 2(n − 1) Similar to Theorems 2.2 and 2.3, we have

2.6 The Case with Multiple Retailers

29

Theorem 2.8 (i) If w ∈ (0, M − αn K ), then there is a unique Nash equilibrium: (m ∗1 , m ∗2 , . . . , m ∗n ) = (K , K , . . . , K ). (ii) If w ∈ [M − αn K , M − K ], then and , M−w , . . . , M−w there are two Nash equilibria: (m ∗1 , m ∗2 , . . . , m ∗n ) = M−w n+1 n+1 n+1 ∗ ∗ ∗ (m 1 , m 2 , . . . , m n ) = (K , K , . . . , K ). Furthermore, the former equilibrium dominates the latter in the sense of generating higher profit for each retailer. (iii) If ∗ ∗ ∗ w ∈ (M − K , M), then  there is a unique Nash equilibrium: (m 1 , m 2 , . . . , m n ) =

M−w M−w M−w , n+1 , . . . , n+1 . n+1 Theorem 2.9 (i) If M ≤ 2(n + 1)K (1 − βn )/n, then the supplier’s optimal wholesale price is w∗ = M/2 and the optimal profits for the supplier are ∗s = n M 2 /4(n + 1). (ii) If M > 2(n + 1)K (1 − βn )/n, then the supplier’s optimal wholesale price is w ∗ = (M − αn K )− and the optimal profits for the supplier are ∗s = K (M − αn K )− . Now consider the lexicographic allocation mechanism. Suppose retailer 1s order is satisfied first, then comes retailer 2s order, etc. Therefore, g1 (m) = min{m 1 , K },   +  i−1  , i = 2, . . . , n. gi (m) = min m i , K − g j (m) j=1

Define αˆ n =

(n + 1)(n + 1 −

√ 2

(n + 3)(n − 1))

, βˆ n =

 1−

 n (n + 1 − (n + 3)(n − 1)). 2

We first characterize the Nash equilibrium for the order-quantity game played by the retailers, and then provide the supplier’s optimal wholesale price and profits. Notice that in some cases, only partial characterization of the Nash equilibrium is provided. But fortunately this does not prevent us from carrying out the supplier’s optimization problem. Theorem 2.10 (i) If w ∈ [0, M − 2K ], then any point in {K } × [0, K ]n−1 is a Nash equilibrium. (ii) If w ∈ (M − 2K , M − αˆ n K ), then a Nash equilibrium exists and any equilibrium satisfies mˆ ∗1 = K . (iii) If w ∈ [M − αˆ n K , M − K ), then , . . . , M−w ) is a Nash equilibrium, but there are also other Nash equilibria. ( M−w n+1 n+1 However, all the other Nash equilibria must satisfy mˆ ∗1 = K . Moreover, the former equilibrium dominates all the other equilibria in the sense of generating higher profits for each retailer. (iv) If w ∈ [M − K , M), then there exists a unique Nash , . . . , M−w ). equilibrium ( M−w n+1 n+1 Theorem 2.11 If M ≤ 2(n + 1)K (1 − βˆn )/n, then the optimal wholesale price is ˆ ∗s = n M 2 /4(n + 1). Otherwise, if M/2 and the supplier’s maximum profits are  M > 2(n + 1)K (1 − βˆn )/n, then the optimal wholesale price is (M − αˆ n K )− and ˆ ∗s = K (M − αˆ n K )− . the supplier’s maximum profits are 

30

2 Managing Downstream Competition via Capacity …

Note that M − αn < M − αˆ n . Combining Theorems 2.8 and 2.10, we have Theorem 2.12 (i) If w ∈ / [M − αn K , M − αˆ n K ), then the total quantities sold to the n retailers under the two allocation mechanisms are the same. (ii) Otherwise, if w ∈ [M − αn K , M − αˆ n K ), then the total quantity sold under the lexicographic allocation mechanism is larger than the total quantity sold under the proportional allocation mechanism. The theorem indicates that for any given wholesale price w, the supplier’s sales under the proportional allocation mechanism is never larger than that under the lexicographic allocation mechanism. Hence the supplier prefers the lexicographic allocation mechanism. Note that since limn→∞ αn = limn→∞ αˆ n = 1, the supplier’s preference for the lexicographic mechanism diminishes as the number of retailers becomes large.

2.6.2 Nonlinear Wholesale Price So far, we have considered allocation mechanisms as the means for the supplier to influence the downstream competition at the retail level. An allocation mechanism maps what the retailers order to what they will receive and thus defines the orderquantity game played by the retailers. As we have seen in the proceeding analysis, different allocation mechanisms lead to not just different order quantities by the retailers but ultimately also the quantities sold to them. Another way to influence the retailers’ order quantities is of course through a nonlinear wholesale price. Given that the supplier has limited capacity, she can limit the retailers’ order quantities by charging higher prices for larger quantities. The purpose of this subsection is to investigate if the supplier is better off with such a nonlinear wholesale pricing scheme as compared to the previously considered allocation mechanisms. Let f (m i ) be the per unit wholesale price paid by retailer i if he orders m i , i = 1, . . . , n. Suppose f (·) is increasing with f (0) = 0. Further assume that the retailers will receive what they have ordered (this would not be a problem because as we will see shortly, the supplier can always choose a wholesale pricing scheme so that the total retailer orders do not exceed her capacity). Therefore, the n retailers are engaged in a Cournot game, each maximizing their own profits: ⎧⎛ ⎫ ⎞ n ⎨ ⎬  m j − f (m i )⎠ · m i , i (m 1 , m 2 , . . . , m n ) = max ⎝ M − mi ⎩ ⎭

(2.5)

j=1

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

(2.6)

Let (m ∗1 ( f ), m ∗2 ( f ), . . . m ∗n ( f )) be a Nash equilibrium n of∗ the above game. The probm i ( f ) ≤ K so as to maximize lem facing the supplier is to choose an f (·) with i=1 her profits.

2.7 Concluding Remarks

31

Theorem 2.13 There exists an f (·), which is increasing with f (0) = 0, such that  n ∗ i=1 m i ( f ) ≤ K . Theorem There does not exist an f (·), which is increasing with f (0) = 0 n 2.14 and i=1 m i∗ ( f ) ≤ K , so that the supplier’s profits are higher than those under the lexicographic allocation mechanism.

2.7 Concluding Remarks This paper has considered a supply chain with one supplier and multiple retailers, where the retailers compete for the supplier’s capacity as well as for customers. When the retailers’ orders exceed the supplier’s capacity, allocation occurs according to a pre-announced allocation mechanism. Two mechanisms are considered: the proportional mechanism that allocates capacity in proportion to orders, and the lexicographic mechanism that allocates capacity according to a pre-determined priority sequence. The competition for customers is modeled as a Cournot competition, whereby the total supply to the retail market determines a market-clearing price. The supplier charges a constant wholesale price. The members of the supply chain are independent profit maximizers. The main goal of the paper is to study the impact of the capacity allocation mechanism on the competitive behavior of the retailers. It is found that under the lexicographic mechanism, the highest-priority retailer sometimes orders a large quantity in order to keep the other retailers out of the market, and this ability to monopolize the market dampens the retail-level competition. On the other hand, under the proportional mechanism, no retailers enjoy the preemptive power, and as a result, competition is more intense. Furthermore, the competition-dampening effect of the lexicographic mechanism benefits the supplier as well as the supply chain. We have also considered nonlinear wholesale pricing as an alternative means to influence the retail-level competition, but shown that the supplier has no incentive to adopt such a method when she can implement the lexicographic mechanism. This paper represents a first attempt to include market-level competition in the study of capacity allocation mechanisms. For example, both Lee et al. (1997) and Cachon and Lariviere (1999a) consider supply chains where the retailers are local monopolists serving independent markets. Although our results demonstrate that the inclusion of market-level competition is important because it can alter the supplier’s preferences over capacity allocation mechanisms, much work remains to be done in order to establish a complete understanding of the impact of capacity allocation on supply chains with market-level competition. Interesting directions abound. For example, it would be interesting to study how asymmetric retailers impact the results of this paper, where retailer asymmetry can be in terms of market sizes, locations, or even different stages of retailer/market development. What if the products sold by the retailers are imperfect substitutes? How would uncertainties affect the results, such as uncertainty about the supplier’s capacity (as in Lee et al. (1997)) or about the

32

2 Managing Downstream Competition via Capacity …

retail market (as in Cachon and Lariviere (1999a))? Extension to dynamic models is another interesting research direction, where it is possible to base allocation on past sales such as the turn-and-earn mechanism used in the auto industry. Cachon and Lariviere (1999c)’s study of the turn-and-earn mechanism in a supply chain with one supplier and two monopolistic retailers represents an excellent starting point for this line of research. We hope future researchers will take up these and other interesting questions.

Appendix In order to prove Theorem 2.1, we first establish few lemmas. To the end, define β=

 1

K + M − w + (M − w − K ) · (M − w + 7K ) . 2

Then we have Lemma A.1 If w ∈ [M − 2K , M − K ] and α is given in Theorem 2.1, then α and β are real numbers with 0 ≤ α ≤ 2K − M + w,

(A-1)

β ≥ 2K − M + w.

(A-2)

Proof Take any w ∈ [M − 2K , M − K ]. That α and β are real numbers is obvious. For convenience, let M − w = y. Thus K √ ≤ y ≤ 2K . To see that α ≥ 0, sim+ K − (y − K )(y + 7K ) ≥ 0 or y + K ≥ ply note that α ≥ 0 if and only if y √ (y − K )(y + 7K ) or (y + K )2 ≥ (y − K )(y + 7K ), the validity √ of which is easily (y − K )(y + 7K ) verified. Similarly, to show that α ≤ 2K − y is to show y + K − √ ≤ 4K − 2y or (y − K )(y + 7K ) ≥ 3(y − K ) or (y − K )(y + 7K ) ≥ 9(y − K )2 , which can be shown to be equivalent to (y − K )(y − 2K ) ≤ 0, which is clearly true. The proof of (A-2) is similar. 

Lemma A.2 If w ∈ [M − 2K , M − K ] and α is given in Theorem 2.1, then 2 2 2) ; (i) if m 2 ≤ α, then (M − K − w) K K+m 2 ≤ (M−w−m 4 2

(ii) if α < m 2 ≤ 2K − M + w, then (M − K − w) K K+m 2 >

(M−w−m 2 )2 . 4

Proof Take any w ∈ [M − 2K , M − K ]. Take any m 2 with 0 ≤ m 2 ≤ α. As before, for convenience, we write y for M − w. And thus K ≤ y ≤ 2K . To show (y − 2 2 2) K ) K K+m 2 ≤ (y−m is to show 4 (y − K )(2K )2 ≤ (K + m 2 )(y − m 2 )2 ,

Appendix

33

which is equivalent to (y − K )(2K )2 − (y − K )(y − m 2 )2 ≤ (K + m 2 )(y − m 2 )2 − (y − K )(y − m 2 )2 , or (y − K )(2K + y − m 2 )(2K − y + m 2 ) ≤ (y − m 2 )2 (2K − y + m 2 ), or (2K − y + m 2 )[(y − m 2 )2 − (y − K )(2K + y − m 2 )] ≥ 0. It is easily verified that the terms inside the square brackets are equal to (m 2 − α)(m 2 − β), since α + β = y + K and αβ = 2K 2 − K y. Therefore, it suffices to show (2K − y + m 2 )(m 2 − α)(m 2 − β) ≥ 0. 

Lemma 2 (i) thus follows. (ii) can be proved following the same procedure. With Lemmas A.1 and A.2 in hands, we can give the proof of Theorem 2.1.

Proof of Theorem 2.1: Consider retailer 1’s optimization problem as in (2.1). Based on whether or not the orders exceed the capacity, we have  1 (w, m 2 ) = max

max

m 1 ∈[0,K −m 2 ]

G 11 (m 1 , m 2 );

max

m 1 ∈(K −m 2 ,K ]

 G 12 (m 1 , m 2 ) , (A-3)

where G 11 (m 1 , m 2 ) = (M − m 1 − m 2 − w) · m 1 m1 · K G 12 (m 1 , m 2 ) = (M − K − w) . m1 + m2 The two subproblems are easily solved: m ∗11 = arg max G 11 (m 1 , m 2 ) m 1 ∈[0,K −m 2 ] ⎧ if M−m22 −w < 0, ⎨ 0, M−m 2 −w = , if 0 ≤ M−m22 −w ≤ K − m 2 , 2 ⎩ K − m 2 , if M−m22 −w > K − m 2 ;

(A-4)

G ∗11 =

max G 11 (m 1 , m 2 ) m 1 ∈[0,K −m 2 ] ⎧ if M−m22 −w < 0, ⎨ 0, (M−w−m 2 )2 = , if 0 ≤ M−m22 −w ≤ K − m 2 , 4 ⎩ (M − K − w) · (K − m 2 ), if M−m22 −w > K − m 2 .

(A-5)

34

2 Managing Downstream Competition via Capacity … m ∗12 = arg G ∗12 =

 max

m 1 ∈(K −m 2 ,K ]

max

m 1 ∈(K −m 2 ,K ]

G 12 (m 1 , m 2 ) =

G 12 (m 1 , m 2 ) =



K − m 2 , if w > M − K , K, if w ≤ M − K ;

(A-6)

(M − K − w) K ·(KK−m 2 ) , if w > M − K , (A-7) 2 (M − K − w) K K+m 2 , if w ≤ M − K ;

(i) w ∈ (0, M − 2K ). Take any m 2 ∈ [0, K ]. Since w < M − 2K , we have 2K − 2 > K − m 2 . Therefore, from M + w < 0 and thus 2K − M + w < m 2 or M−w−m 2 ∗ ∗ (A-4) and (A-5), we have m 11 = K − m 2 and G 11 = (M − K − w)(K − m 2 ). On the other hand, since w < M − 2K < M − K , we have from (A-6) and (A-7) that 2 m ∗12 = K and G ∗12 = (M − K − w) K K+m 2 . Noting that (K − m 2 ) ≤ K 2 /(K + m 2 ), we have from (A-3) that m 1 (m 2 ) = m ∗12 = K , 1 (w, m 2 ) = G ∗12 = (M − K − w)

K2 . K + m2

(ii) w ∈ [M − 2K , M − K ]. First note that as w ≤ M − K , we have from (A-6) 2 and (A-7) that m ∗12 = K and G ∗12 = (M − K − w) K K+m 2 . Since w ∈ [M − 2K , M − K ] we have M − w ≥ K , which, together with the fact that m 2 ≤ K , implies that 2 ≥ 0. Consequently, based on (A-4) and (A-5), we have the M − w ≥ m 2 or M−w−m 2 following cases: 2 2 ≤ K − m 2 , then m ∗11 = M−w−m (a) if m 2 ≤ 2K − M + w, or equivalently M−w−m 2 2 2 2) and G ∗11 = (M−w−m ; 4 2 (b) if m 2 > 2K − M + w, or equivalently M−w−m > K − m 2 , then m ∗11 = K − m 2 2 ∗ and G 11 = (M − K − w)(K − m 2 ). Now it only remains to compare G ∗11 with G ∗12 . Using Lemmas A.1 and A.2, we have that if m 2 ≤ α, then m 1 (m 2 ) = m ∗11 =

M − w − m2 (M − w − m 2 )2 , 1 (w, m 2 ) = G ∗11 = ; 2 4

if m 2 ∈ (α, 2K − M + w], then m 1 (m 2 ) = m ∗12 = K , 1 (w, m 2 ) = G ∗12 = (M − K − w)

K2 ; K + m2

and if m 2 > 2K − M + w, then by again (K − m 2 ) ≤ K 2 /(K + m 2 ), m 1 (m 2 ) = m ∗12 = K , 1 (w, m 2 ) = G ∗12 = (M − K − w)

K2 . K + m2

(iii) w ∈ (M − K , M). Since w > M − K , we know from (A-7) that G ∗12 < 0. Again since w > M − K or equivalently 2K − M + w > K , which, together with 2 < K − m 2 . Consider m 2 ≤ K , implies 2K − M + w > m 2 or equivalently M−w−m 2

Appendix

35

2 two cases. (a) m 2 ≤ M − w. In this case, M−w−m ≥ 0. Therefore, from (A-3), m ∗11 = 2 2 M−w−m 2 2) 2 and G ∗11 = (M−w−m . Since G ∗12 < 0, we have m 1 (m 2 ) = m ∗11 = M−w−m 2 4 2 2 (M−w−m ) 2 2 and 1 (w, m 2 ) = G ∗11 = . (b) m 2 > M − w. In this case, M−w−m < 0. 4 2 ∗ ∗ ∗ Therefore, from (A-3), m 11 = 0 and G 11 = 0. Consequently, since G 12 < 0, we have m 1 (m 2 ) = m ∗11 = 0 and 1 (w, m 2 ) = G ∗11 = 0. The proof is complete. 

Proof of Theorem 2.2: We begin by making a simple observation about any Nash equilibrium of the game. Suppose (m ∗1 , m ∗2 ) is a Nash equilibrium. There can only be two possibilities: Case 1 with m ∗1 + m ∗2 < K and Case 2 with m ∗1 + m ∗2 ≥ K . In Case 1, there is ample capacity, and the equilibrium must be the same as the equilibrium after the capacity constraint has been removed, which can be easily determined to be . On the other hand, under Case 2, the retailers’ orders exceed the m ∗1 = m ∗2 = M−w 3 capacity constraint. As mentioned earlier, in this case, the total supply to the retail market is fixed at K , and the incentive for each retailer is such that they would order as ∗ ∗ much as possible given the other’s order. Consequently, we must have m 1 = m 2 = K .  In sum, the game’s equilibrium must be

M−w M−w , 3 3

, (K , K ), or both. Therefore,

to prove the theorem, we only need to check if any of these equilibrium points satisfies the best response relationships identified in Theorem 2.1. If w ∈ (0, M − 2K ) then from Theorem 2.1(i), it is clear that (K , K ) is the only equilibrium. Now suppose w ∈ [M −√2K , M − K ]. Simple algebra shows that M−w > α if and only if w < M + 3K − 3 2K . Therefore, for w ∈ [M − 2K , M + 3 √ 3K − 3 2K ), we know from Theorem 2.1(ii) that (K , K ) is the only equilibrium. Combining these scenarios, we √ have the first part of the theorem. ≤ α. Again based on Suppose w ∈ [M + 3K − 3 2K , M − K ]. Thus M−w 3  and (K , K ) are Nash Theorem 2.1(ii), it is easily verified that both M−w , M−w 3 3 equilibria. Note that the retailers’ profits under these equilibra are (M − w)2 M −w M −w ) = 2 (w, )= , 3 3 9 K 1 (w, K ) = 2 (w, K ) = (M − K − w) . 2 √ 2 It can be shown that for w ∈ [M + 3K − 3 2K , M − K ], (M−w) > (M − K − 9 K M−w M−w w) 2 . Consequently, both retailers prefer the equilibrium ( 3 , 3 ) to (K , K ). This proves the second part of the theorem. Finally, Theorem 2.1(iii) indicates that for w ∈ (M − K , M), the best response functions are exactly the ones found in a game with the capacity constraint removed. Theorem 2.2(iii) thus follows easily.  To establish Theorem 2.4, we need the following lemma. 1 (w,

36

2 Managing Downstream Competition via Capacity …

Lemma A.3 Take any w ∈ [M − 2K , M − K ] and αˆ is given in Theorem 2.4. Then 0 ≤ αˆ ≤ 2K − M + w. Furthermore, 2 ˆ (M−m42 −w) ≥ (M − K − w)K ; (i) for m 2 ∈ [0, α], 2 (ii) for m 2 ∈ (α, ˆ 2K − M + w], (M−m42 −w) < (M − K − w)K . Proof Take any w ∈ [M − 2K , M − K ]. Write y for M − w. Thus K ≤ y ≤ 2K . That αˆ ≤ 2K − y follows from simple algebra, and that αˆ ≥ 0 follows from the 2 2) − inequality a · b ≤ [(a + b)/2]2 for any real numbers a and b. The sign of (y−m 4 (y − K )K is of course the same as the sign of (y − m 2 )2 − 4(y − K )K , which can be re-written as (y − m 2 )2 − 4(y − K )K   = (y − m 2 − 4K (y − K )) · (y − m 2 + 4K (y − K ))  = (αˆ − m 2 ) · (y + 4K (y − K ) − m 2 ). √ w. With this Since y ≥ K , we have y + 4K (y − K ) ≥ 2K − y = 2K − M +√ fact, we can easily determine the sign of the expression (αˆ − m 2 )(y + 4K (y − K ) ˆ and m 2 ∈ (α, ˆ 2K − y]. The lemma follows.  − m 2 ) for m 2 ∈ [0, α] With the help of Lemma A.3, we can prove Theorem 2.4. Proof of Theorem 2.4: Depending on whether or not the capacity constraint is violated, we can rewrite (2.3) as ˆ 1 (w, m 2 ) = max 

 max

m 1 ∈[0,K −m 2 ]

Gˆ 11 (m 1 , m 2 );

max

m 1 ∈(K −m 2 ,K ]

 Gˆ 12 (m 1 , m 2 ) .

where Gˆ 11 (m 1 , m 2 ) = (M − m 1 − m 2 − w) · m 1 , Gˆ 12 (m 1 , m 2 ) = (M − K − w) · m 1 . Note that Gˆ 11 (m 1 , m 2 ) = G 11 (m 1 , m 2 ), which is defined in the previous section. Define mˆ ∗11 = argmax m 1 ≤K −m 2 Gˆ 11 (m 1 , m 2 ) and Gˆ ∗11 = maxm 1 ≤K −m 2 Gˆ 11 (m 1 , m 2 ). Therefore, mˆ ∗11 = m ∗11 and Gˆ ∗11 = G ∗11 . The following are straightforward as well: mˆ ∗12 = arg Gˆ ∗12 =

max

m 1 ∈(K −m 2 ,K ]

max

m 1 ∈(K −m 2 ,K ]

Gˆ 12 (m 1 , m 2 ) =

Gˆ 12 (m 1 , m 2 ) =





K − m 2 , if w > M − K K, if w ≤ M − K

(A-8)

(M − K − w) · (K − m 2 ), if w > M − K (A-9) (M − K − w)K , if w ≤ M − K

The best response function takes different forms for different intervals of the wholesale price, as we will see next. (i) w ∈ (0, M − 2K ): In this case, since w < M − 2K , we have M−m22 −w > M−m 2 −(M−2K ) = K − m22 ≥ K − m 2 . Consequently, from (A-4) and (A-5), we have 2

Appendix

37

mˆ ∗11 = K − m 2 and Gˆ ∗11 = (M − K − w)(K − m 2 ). On the other hand, from (A-8) and (A-9), we have mˆ ∗12 = K and Gˆ ∗12 = (M − K − w)K . Hence, comparing Gˆ ∗11 with Gˆ ∗12 , we obtain ˆ 1 (w, m 2 ) = Gˆ ∗12 = (M − K − w)K . mˆ 1 (m 2 ) = mˆ ∗12 = K ,  (ii) w ∈ [M − 2K , M − K ]: Note first from (A-8) and (A-9) that mˆ ∗12 = K and Gˆ ∗12 = (M − K − w)K . As to mˆ ∗11 and Gˆ ∗11 , we consider the following three cases: ˆ Because αˆ < 2K − M + w (Lemma A.3), we have m 2 < 2K − M + (a) m 2 ≤ α. w or equivalently M−m22 −w < K − m 2 . Therefore, from (A-4) and (A-5), we have 2 2) mˆ ∗11 = M−m22 −w and Gˆ ∗11 = (M−w−m . Since Gˆ ∗11 ≥ Gˆ ∗12 (Lemma A.3), we have 4 mˆ 1 (m 2 ) = mˆ ∗11 =

M − w − m2 (M − w − m 2 )2 ˆ 1 (w, m 2 ) = Gˆ ∗11 = ,  . 2 4

(b) m 2 ∈ (α, ˆ 2K − M + w]. Since m 2 ≤ 2K − M + w which is equivalent to M−m 2 −w ≤ K − m 2 , we have again from (A-4) and (A-5) that mˆ ∗11 = M−m22 −w and 2 2 2) Gˆ ∗11 = (M−w−m . However, since Gˆ ∗11 < Gˆ ∗12 (Lemma A.3), we have 4 ˆ 1 (w, m 2 ) = Gˆ ∗12 = (M − K − w)K . mˆ 1 (m 2 ) = mˆ ∗12 = K ,  (c) m 2 > 2K − M + w. In this case, we have M−m22 −w > K − m 2 . Therefore, from (A-4) and (A-5), we have mˆ ∗11 = K − m 2 and Gˆ ∗11 = (M − K − w)(K − m 2 ). Since Gˆ ∗11 ≤ Gˆ ∗12 , we have ˆ 1 (w, m 2 ) = Gˆ ∗12 = (M − K − w)K . mˆ 1 (m 2 ) = mˆ ∗12 = K ,  (iii) w ∈ (M − K , M): First note that in this case, Gˆ ∗12 is negative. Note further that since w > M − K and m 2 ≤ K , we have w > M − K − (K − m 2 ) = M − 2K + m 2 , which is equivalent to M−m22 −w < K − m 2 . We thus have from (A-4) 2 ˆ 1 (w, m 2 ) = and (A-5) that if m 2 ≤ M − w then mˆ 1 (m 2 ) = mˆ ∗11 = M−w−m with  2 2 (M−w−m 2 ) ∗ ∗ ˆ ˆ G 11 = , and if m 2 > M − w then mˆ 1 (m 2 ) = mˆ 11 = 0 with 1 (w, m 2 ) = 4 Gˆ ∗11 = 0. The proof is complete.  Proof of Theorem 2.5: Suppose retailer 1 orders m 1 . Consider retailer 2s ordering decision. Depending on whether or not the capacity constraint is violated, we can rewrite (2.4) as ˆ 2 (w, m 1 ) = max 

where

 max

m 2 ∈[0,K −m 1 ]

Gˆ 21 (m 1 , m 2 );



max

m 2 ∈(K −m 1 , K ]

Gˆ 22 (m 1 , m 2 )

(A-10)

38

2 Managing Downstream Competition via Capacity …

Gˆ 21 (m 1 , m 2 ) = (M − m 1 − m 2 − w) · m 2 , Gˆ 22 (m 1 , m 2 ) = (M − K − w)(K − m 1 ). Define mˆ ∗21 = argmax m 2 ≤K −m 1 Gˆ 21 (m 1 , m 2 ) and Gˆ ∗21 = maxm 2 ≤K −m 1 Gˆ 21 (m 1 , m 2 ). Note that Gˆ 21 (m 1 , m 2 ) = G 11 (m 2 , m 1 ), where G 11 was defined in the previous section. Therefore, mˆ ∗21 and Gˆ ∗21 can be easily obtained from the expressions for mˆ ∗11 and Gˆ ∗11 (see (A-4) and (A-5)), respectively, by replacing m 2 with m 1 . On the other hand, note that the objective function of the second optimization problem in (A-10) is constant. Therefore, mˆ ∗22 = arg Gˆ ∗22 =

max

m 2 ∈(K −m 1 ,K ]

max

m 2 ∈(K −m 1 ,K ]

Gˆ 22 (m 1 , m 2 ) = (K − m 1 , K ],

Gˆ 22 (m 1 , m 2 ) = (M − K − w)(K − m 1 ).

(A-11) (A-12)

(i) w ∈ (0, M − 2K ). In this case, it is easily verified that M−m21 −w > K − m 1 . Thus from (A-4) and (A-5), with m 1 and m 2 swapped, we have mˆ ∗21 = K − m 1 and Gˆ ∗21 = (M − K − w)(K − m 1 ). Since Gˆ ∗21 = Gˆ ∗22 (see (A-12)), we have from (A-11) ˆ 2 (w, m 1 ) = (M − K − w)(K − m 1 ). mˆ 2 (m 1 ) ∈ [K − m 1 , K ],  (ii) w ∈ [M − 2K , M − K ]. Since m 1 ≤ K and M − w ∈ [K , 2K ], we have ≥ 0. The following observations are based on (A-4) and (A-5): (a) if m 1 < 1 1 2K − M + w, which is equivalent to M−w−m < K − m 1 , then mˆ ∗21 = M−w−m 2 2 2 (M−w−m ) M−w−m ∗ 1 1 with Gˆ 21 = , and (b) if m 1 ≥ 2K − M + w or ≥ K − m 1 , then 4 2 ∗ ∗ ˆ ˆ mˆ 21 = K − m 1 with G 21 = (M − K − w)(K − m 1 ). Comparing G ∗21 with Gˆ ∗22 , and )2 for any real numbers x and y, we have using the inequality x y ≤ ( x+y 2 if m 1 < 2K − M + w then M−w−m 1 2

mˆ 2 (m 1 ) = mˆ ∗21 =

M − w − m1 (M − w − m 1 )2 ˆ 2 (w, m 2 ) = Gˆ ∗21 = ,  ; 2 4

otherwise if m 1 ≥ 2K − M + w then ˆ 2 (w, m 1 ) = (M − K − w)(K − m 1 ). mˆ 2 (m 1 ) ∈ [K − m 1 , K ],  (iii) w ∈ (M − K , M). First note that in this case Gˆ ∗22 is negative. Therefore 1 < mˆ 2 (m 1 ) = mˆ ∗21 . From (A-4) and (A-5), we have if m 1 ≤ M − w then 0 ≤ M−w−m 2 2 (M−w−m M−w−m 1 ∗ ∗ 1) ˆ 2 (w, m 2 ) = Gˆ 21 = K − m 1 , which implies that mˆ 21 = with  , 2 4 ∗ ∗ ˆ ˆ  and if m 1 > M − w then mˆ 21 = 0 with 2 (w, m 1 ) = G 21 = 0. Proof of Theorem 2.6: Let (mˆ ∗1 , mˆ ∗2 ) be a Nash equilibrium. A simple observation helps simplify the analysis significantly. Partition the region [0, K ] × [0, K ]

Appendix

39 m1

K

M −w 2

m ˆ

1 (m 2)

= M

− w − 2 m 2

m1 = m2

α ˆ

K

m2

Fig. 2.6 Plot of retailer 1’s best response function when mˆ 1 (α) ˆ > αˆ

into two: {(m 1 , m 2 ) : m 1 + m 2 ≥ K , m 1 , m 2 ∈ [0, K ]} and {(m 1 , m 2 ) : m 1 + m 2 < K , m 1 , m 2 ∈ [0, K ]}. In the first region, the retailers’ orders together exceed the supplier’s capacity and thus the total supply to the retail market is fixed at K , which implies that the market price is fixed at M − K . As a result, retailer 1, given its high priority in getting stock allocation, will order K to maximize his profits. So if a Nash eqiulibrium lies in the first region, it must be that mˆ ∗1 = K . Now consider the second region. Here the supplier’s capacity is sufficient to satisfy the retailers’ orders, and the game is the same as if there were no supplier capacity constraint. For the game without the capacity constraint, it is well known that the Nash equilibrium lies on , M−w ). In sum, in searching for the 45◦ line (i.e. m 1 = m 2 ). In particular, it is ( M−w 3 3 a Nash equilibrium for the game with limited supplier capacity, one only needs to consider either retailer 1 ordering the full capacity or the two retailers ordering the same quantity (the 45◦ line). (a) w ∈ (0, M − 2K ). In this case, from Theorems 2.4 and 2.5, it is clear that any point in {K } × [0, K ] is a Nash equilibrium. This shows Theorem 2.6(i). (b) w ∈ [M − 2K , M − K ]. First consider retailer 1s best response function. Note 2 for m 2 ≤ αˆ and mˆ 1 (m 2 ) = K for m 2 > α. ˆ from Theorem 2.4 that mˆ 1 (m 2 ) = M−w−m 2 Depending on whether or not this best response function lies above the 45◦ line, we have the following two cases. ˆ > αˆ then the best response function lies above the 45◦ line (see (b-a) If mˆ 1 (α) αˆ > α, ˆ which can be shown to be Fig. 2.6). The condition can be written as M−w− 2

40

2 Managing Downstream Competition via Capacity … m1

1(

m ˆ

M −w 2

m

2)

=

M

− w 2 −m 2

K

m1 = m2

M −w 3

α ˆ

K

m2

Fig. 2.7 Plot of retailer 1’s best response function when mˆ 1 (α) ˆ α then m (m ) = K , where α is defined in Theorem 2.1. j i −i j =i   M−w− j =i m j m ≤ M − w then m (m ) = ; if (iii) w ∈ (M − K , M) : if j i −i j =i 2  m > M − w then m (m ) = 0. j i −i j =i  Proof Similar to the proof of Theorem 2.1 with m 2 replaced by j =i m j .  Proof of Theorem 2.8: Suppose (m ∗1 , m ∗2 , . . . , m ∗n ) is a Nash equilibrium. can n There n m i∗ < K and Case 2 with i=1 m i∗ ≥ K . only be two possibilities: Case 1 with i=1 In Case 1, there is ample capacity, and the equilibrium must be the same as the equilibrium after the capacity constraint has been removed, which can be easily . On the other hand, under Case 2, the retailers’ orders determined to be m i∗ = M−w n+1 exceed the capacity constraint. As mentioned earlier, in this case, the total supply to the retail market is fixed at K , and the incentive for each retailer is such that they would order as much as possible given the other’s order.  Consequently, we must have m i∗ = K . In sum, the game’s equilibrium must be

M−w M−w , n+1 , . . . , M−w n+1 n+1

,

(K , K , . . . , K ), or both. Therefore, to prove the theorem, we only need to check if any of these equilibrium points satisfies the best response relationships identified in Lemma A.4. If w ∈ (0, M − 2K ) then from Lemma A.4(i), it is clear that (K , K . . . , K ) is the only equilibrium. Now suppose w ∈ [M − 2K , M − K ]. Simple algebra shows > α if and only if w < M − αn K . Therefore, for w ∈ [M − that (n − 1) · M−w n+1 2K , M − αn K ), we know from Lemma A.4(ii) that (K , K , . . . , K ) is the only equilibrium. Combining these scenarios, we have the first part of the theorem. ≤ α. Again based on Suppose w ∈ [M − αn K , M − K ]. Thus (n − 1) M−w  n+1 M−w M−w and Lemma A.4(ii), it is easily verified that both M−w , , . . . , n+1 n+1 n+1 (K , K , . . . , K ) are Nash equilibria. Note that the retailers’ profits under these equilibria are

M −w M −w M − w (M − w)2 , ,..., = i w, , n+1 n+1 n+1 (n + 1)2 i (w, K , K , . . . , K ) = (M − K − w)

K . n

42

2 Managing Downstream Competition via Capacity …

It can be shown that for w ∈ [M − αn K , M − K ], (M−w) > (M − K − w) Kn . Con(n+1) 2 M−w M−w to , , . . . , sequently, all retailers prefer the equilibrium M−w n+1 n+1 n+1 2

(K , K , . . . , K ). This proves the second part of the theorem. Finally, Lemma A.4(iii) indicates that for w ∈ (M − K , M), the best response functions are exactly the ones found in a game with the capacity constraint removed. Theorem 2.8(iii) thus follows easily.  Proof of Theorem 2.9: From Theorem 2.8, we know for w < M − αn K , the total order quantity is n K , which exceeds the capacity K , and thus the total sales to the retailers is K . In this case, the supplier’s profit is wK (because the supplier’s variable cost of production has been normalized to zero). On the other hand, if w ≥ M − αn K , the total quantity ordered is the same as the total supplied, because the total order quantity n(M − w)/(n + 1) is within the capacity constraint. In this case, the supplier’s profit is nw(M − w)/(n + 1). Therefore, the supplier’s maximum profit can be written as max{π1n , π2n } where π1n =

max

w∈(0, M−αn K )

wK = K (M − αn K )− ,

and π2n =

max

w∈[M−αn K , M)

nw(M − w) . n+1

Note that the unconstrained optimization problem of maxw n(M−w)w is solved at n+1 w = M2 with the maximum objective function value at n M 2 /4(n + 1). To obtain the optimal wholesale price, consider the following two cases. Case 1: M/2 ≤ M − αn K or equivalently M ≥ 2αn K . In this case, π2n =

n (M − αn K ) · αn K < K (M − αn K )− = π1n . n+1

Therefore, the optimal wholesale price is w ∗ = (M − αn K )− , with the supplier’s maximum profit at ∗s = K · (M − αn K )− . Case 2: M/2 > M − αn K or equivalently M < 2αn K . In this case, π2n = n M 2 /4(n + 1) as noted above. To determine the optimal wholesale price, we need to compare π1n with π2n . It is easily verified that as functions of M, n M 2 /4(n + 1) crosses K (M − αn K ) from below exactly once, at M = 2K (1 − βn )(n + 1)/n. In other words, for M ≤ 2K (1 − βn )(n + 1)/n, π2n ≥ π1n and thus w ∗ = M/2 with ∗s = n M 2 /4(n + 1), and for M ∈ (2K (1 − βn )(n + 1)/n, 2αn K ), π1n > π2n and  thus w ∗ = (M − αn K )− with ∗s = K (M − αn K )− . Proof of Theorem 2.10: Let mˆ i (m−i ) be the best response function of retailer i, i = 1, . . . , n. It is easy to see that mˆ i (m−i ) is nonincreasing in m−i . Hence a Nash equilibrium (mˆ ∗1 , mˆ ∗2 , . . . , mˆ ∗n ) exists (see Friedman 1986). To characterize the Nash equilibrium (or equilibria), we consider the following cases:

Appendix

43

Case 1: w ∈ [0, M − 2K ]. Consider Retailer 1s profit maximization. Recall that Retailer 1 has the highest priority for stock allocation under the lexicographic mechanism. Note that   n  ∗ ∗ ∗ ˆ ˆ −1 ) = M − ˆ ) − w · g1 (m ˆ ∗) gi (m 1 (w, mˆ 1 , m i=1



ˆ ∗ ) − w · mˆ ∗1 ≤ M − g1 (m

 = M − mˆ ∗1 − w · mˆ ∗1 ≤ max (M − x − w) · x x∈[0, ∞)   = (M − x − w) · x  . x= M−w 2

Since w ∈ [0, M − 2K ], (M − w)/2 ≥ K . Hence, by the concavity of the function x(M − w − x) we have ∗ ˆ 1 (w, mˆ ∗1 , m−1  ) ≤ (M − w − K ) · K .

Because Retailer 1 has the highest priority for stock allocation, he can guarantee himself a profit equal to the right-hand-side of the above inequality by ordering K . Consequently, mˆ ∗1 = K . Since Retailer 1 takes all the capacity, it really does not matter what the other retailers will order. In other words, Any point in {K } × [0, K ]n−1 is a Nash equilibrium.

(A-13)

n ˆ ∗ ) < K . To see this, simCase 2: w ∈ (M − K , M]. First, note n that ∗ i=1 gi (m ˆ ) = K , then because w > M − K , ply observe that if to the contrary, i=1 gi (m the retailer with the smallest index such that his equilibrium order quantity is positive profit; a fact that cannot be true in equilibrium. Since n n would∗make a negative ∗ ∗ ∗ ˆ ˆ g ( m ) < K , g ( m ) = m ˆ for all i. Note further that due to m ˆ i i=1 i i=1 i < K , i we have mˆ i∗ > 0, i = 1, . . . , n.

(A-14)

because otherwise, the retailer with zero equilibrium order quantity can get a positive profit by increasing his order from zero to some positive quantity. In view of (A-14), we can write the first-order equilibrium conditions as

 d M − w − nj=1 m j m i    dm i

which is equivalent to

m j =mˆ ∗j , j=1,...,n

= 0, i = 1, . . . , n

(A-15)

44

2 Managing Downstream Competition via Capacity …

mˆ i∗ +

n 

mˆ ∗j = M − w, i = 1, . . . , n

(A-16)

j=1

Clearly, the above set of equations has a unique solution: mˆ ∗1 = · · · = mˆ ∗n =

M −w . n+1

(A-17)

n ˆ ∗ ) < K or Case 3: w = M − K . There are only two possibilities: either i=1 gi (m  n ∗ ˆ ) = K . If the latter, then by w = M − K , all the retailers make zero i=1 gi (m profits. If the former, one can follow the proof for Case 2 above to show that (A-17) holds, whereby all the retailers make positive profits. Consequently, we adopt the Nash equilibrium in (A-17). ˆ ∗ . Clearly, Case 4: K (n + 1)/n, ). Take any Nash equilibrium m n M − K wn ∈ [M − ∗ ∗ ˆ i ) < K or i=1 gi (m ˆ i ) = K. either i=1 gi (m n ˆ ∗ ) < K , similar to Case 2, we must have (A-17). However, when gi (m If i=1 w ∈ [M − K (n + 1)/n, M − αˆ n K ), one can show that 

M −w n+1

2 < (M − K − w)K ,

which implies that Retailer 1, having the highest priority under the lexicographic mechanism, can strictly increase his profits by ordering K instead of (M − w)/(n + 1). Therefore, n when ∗w ∈ [M − K (n + 1)/n, M − αˆ n K ), any Nash equilibrium ˆ ) = K . On the other hand, if w ∈ [M − αˆ n K , M − K ), then gi (m must have i=1 note that for any m, 

M −w n+1

2 ≥ (M − K − w)K ≥ (M − K − w) · gi (m), i = 1, . . . , n

(A-18)

which implies that no retailer has any incentive to unilaterally increase their order (from (A-17)) such as to make the capacity constraint binding. Moreover, from the proof for Case 2, we also know that no retailer has any incentive to unilaterally deviate from (A-17) in a way that leads to excess capacity. Combining these scenarios, we know that when w ∈ [M − αˆ n K , M − K ), (A-17) is a Nash equilibrium. In sum, (a) nFor any ∗w ∈ [M − K (n + 1)/n, M − αˆ n K ), any Nash equilibrium must satisfy ˆ ) = K; i=1 gi (m , . . . , M−w ) is a Nash equilibrium . (b) For w ∈ [M − αˆ n K , M − K ), ( M−w n+1 n+1 n ∗ ˆ ) = K (it is straightforward to verify that (K , . . . , K ) is a Nash If i=1 gi (m equilibrium, thus we know there exists at least one Nash equilibrium such that n ∗ ˆ g ( m ) = K ), we have, by w < M − K , that i i=1

Appendix

45

 ∗ ˆ 1 (w, mˆ ∗1 , m ˆ −1  )

=

M−

n 

 ∗

ˆ ) − w · g1 (m ˆ ∗) gi (m

i=1

= (M − w − K ) · mˆ ∗1 ≤ (M − w − K ) · K

(A-19)

n ˆ ∗ ) = K , Retailer 1s order gi (m which indicates that in any equilibrium with i=1 must be K . Therefore, we have (c) nFor any ∗w ∈ [M − K (n + 1)/n, M − K ), there exists∗ a Nash equilibrium with ˆ ) = K and any such equilibrium must have mˆ 1 = K . i=1 gi (m Finally, note that by the first inequality in (A-18), for any w ∈ [M − αˆ n K , M − K ), the Nash equilibrium given by (A-17) dominates the Nash equilibrium given by  n ˆ ∗ ) = K with mˆ ∗1 = K in the sense of generating higher profits for every i=1 gi (m retailer. Case 5: w ∈ [M − 2K , M − K (n + 1)/n). For this case, wehave n(M − w)/(n + n ˆ ∗ ) < K (see gi (m 1) > K . Therefore, there does not exist an equilibrium with i=1 Case 2).Similar to Case 4, one can show that the Nash equilibrium in this case must n ˆ ∗ ) = K and mˆ ∗1 = K . gi (m satisfy i=1 Theorem 2.10(i) directly follows from Case 1; Theorem 2.10(ii) follows from Case 4(a, c) and Case 5; Theorem 2.10(iii) follows from Case 4 (b, c); and Theorem 2.10(iv) follows from Cases 2 and 3.  Proof of Theorem 2.11: To determine the optimal wholesale price that maximizes the supplier’s profits, recall from Theorem 2.10 that (1) for w < M − αˆ n K , the total quantity ordered by the retailers in equilibrium is at least K , implying that the quantity sold by the supplier is K ; and (2) for w ≥ M − αˆ n K , each retailer’s equilibrium order quantity is (M − w)/(n + 1) and thus the total quantity sold by the supplier is n(M − w)/(n + 1). Note that s (w) = wK for w < M − αˆ n K and s (w) = n(M − w)w/(n + 1) for w ≥ M − αˆ n K . Clearly, the supplier’s maximum profits is either s ((M − αˆ n K )− ) = K (M − αˆ n K )− or maxw≥M−αˆ n K s (w), whichever is larger. To find the optimal wholesale price, first note that s ((M − αˆ n K )− ) > s (M − αˆ n K ) and that 

n(M − w)w arg max w n+1



  n(M − w)w M nM2 = with max = . w 2 n+1 4(n + 1)

Therefore, if M/2 ≤ M − αˆ n K or equivalently M ≥ 2αˆ n K , then s (w) is decreasing for w ≥ M − αˆ n K , indicating that the optimal wholesale price w ∗ = (M − αˆ n K )− . Now suppose M/2 > M − αˆ n K or equivalently M < 2αˆ n K . In this case, we only need to compare the values of s ((M − αˆ n K )− ) and n M 2 /4(n + 1). If the former is larger, then w∗ = (M − αˆ n K )− . Otherwise, w ∗ = M/2. The following fact can be established after simple algebra: s ((M − αˆ n K )− ) ≤ n M 2 /4(n + 1) if ˆ (1 − βˆn )(n + 1)/n]. We thus have the theorem.  and only if M ∈ [0, 2K

46

2 Managing Downstream Competition via Capacity …

Proof of Theorem 2.13: Let f (x) = ax 2 for some positive constant a. The first-order conditions for (2.6) can be written as M−

n 

m j − m i − am i2 − 2am i2 = 0, i = 1, 2, . . . n.

(A-20)

j=1

The solution to (A-20) must be symmetric. Thus solving M − (n + 1)m − 3am 2 = 0, we get m∗ =

−(n + 1) ±



(n + 1)2 + 12Ma . 6a

Therefore the solution to (A-20) is m ∗1

= ··· =

m ∗n

=

−(n + 1) +

 (n + 1)2 + 12Ma . 6a

To prove the theorem, it suffices to see if there exists an a such that n 

m i∗

i=1

=

−(n + 1) +



(n + 1)2 + 12Ma ·n ≤ K 6a

(A-21)

which is equivalent to  36K 2 a 2 6K a 6K a + (n + 1) ⇐⇒ 12Ma ≤ (n + 1)2 + 12Ma ≤ + 2(n + 1) n n2 n ⇐⇒ 3K 2 a ≥ Mn 2 − n(n + 1)K . (A-22) Clearly, for any given M and K , there always exists an a such that (A-22) holds.  Proof of Theorem 2.14: Let f (·) be the wholesale price function. Consider the order-quantity game played by the n retailers. Since the game is symmetric, we will focus on a symmetric Nash equilibrium denoted by (m ∗ ( f ), m ∗ ( f ), . . . , m ∗ ( f )). The equilibrium order quantities must satisfy the first-order conditions (see (2.6)): M−

n  j=1

m j − m i − f (m i ) − m i ·

d f (m i ) = 0, i = 1, 2, . . . n. dm i

(A-23)

Case 1: M ≤ 2(n+1) K (1 − βˆn ). By Theorem 2.11, it suffices to prove that there does n not exist an increasing function f (·) with f (0) = 0 such that

Appendix

47

n · m∗( f ) ≤ K ,

(A-24)

n · m ∗ ( f ) · f (m ∗ ( f )) >

2

nM . 4(n + 1)

(A-25)

In view of (A-25), we know that m ∗ ( f ) = 0 and f (m ∗ ( f )) = 0. By the monotonicity of f (·), we have  d f (m)  ≥ 0. dm m=m ∗ ( f )

(A-26)

Using (A-23), we have 



d f (m)  M − (n + 1)m ∗ ( f ) − m ∗ ( f ) · dm m=m ∗ ( f )

 ≤ n · m ∗ ( f ) · M − (n + 1)m ∗ ( f ) (using (A-26))

n · m ∗ ( f ) · f (m ∗ ( f )) = n · m ∗ ( f ) ·

≤

sup

x∈(−∞, ∞)



{n · x · (M − (n + 1)x)}

  ≤ n · x · (M − (n + 1)x)

M x= 2(n+1)

=

nM2 4(n + 1)

(A-27)

contradicting (A-25). K (1 − βˆn ). From again Theorem 2.11, it suffices to prove that Case 2: M > 2(n+1) n there does not exist an increasing function f (·) with f (0) = 0 such that n · m∗( f ) ≤ K , ∗

(A-28) ∗

n · m ( f ) · f (m ( f )) ≥ K (M − αˆ n K ).

(A-29)

If M ≥ (n + 1)K , then similar to (A-27), we have 



d f (m)  M − (n + 1)m ∗ ( f ) − m ∗ ( f ) · dm m=m ∗ ( f )

 ≤ nm ∗ ( f ) · M − (n + 1)m ∗ ( f ) (using (A-2))

n · m ∗ ( f ) · f (m ∗ ( f )) = nm ∗ ( f ) ·

  ≤ nx · (M − (n + 1)x) 

(n + 1)K n < K (M − αˆ n K ) =K· M−

x= Kn





48

2 Managing Downstream Competition via Capacity …

contradicting (A-29). On the other hand, if M ∈ ( 2(n+1) K (1 − βˆn ), (n + 1)K ), then n one can show that nM2 < K (M − αˆ n K ) 4(n + 1)

(A-30)

which, together with (A-27) whose validity does not depend on the range of M, contradicts (A-29). 

References Axsater S. (1993). Continuous review policies for multi-level inventory systems with stochastic demand. In S. Graves, A. Rinnooy Kan & P. Zipkin (Eds.), Handbook in operations research and management science (Vol. 4). Logistics of production and inventory. North Holland. Blumenstein, R. (1996). Autos: How do you get a hot GMC suburban? you wait for a computer to dole one out. Wall Street Journal, April 10, B1. Chen, F., & Zheng, Y.-S. (1997). One-warehouse multiretailer systems with centralized stock information. Operations Research, 45(2), 275–287. Cachon, G. P., & Lariviere, M. A. (1999a). Capacity choice and allocation: Strategic behavior and supply chain performance. Management Science, 45, 1091–1108. Cachon, G. P., & Lariviere, M. A. (1999b). An equilibrium analysis of linear, proportional and uniform allocation of scarce capacity. IIE Transactions, 31, 835–849. Cachon, G. P., & Lariviere, M. A. (1999c). Capacity allocation using past sales: When to turn-andearn. Management Science, 45, 685–703. Cachon, G. P., & Zhang, F. (2007). Obtaining fast service in a queueing system via performancebased allocation of demand. Management Science, 53, 408–420. Dhakar, T. S., Schmidt, C. P., & Miller, D. (2010). Improving allocation of inventory for quick response to customer orders: a case study. In T. C. Edwin Cheng, & T.-M. Choi (Eds.), Innovative Quick Response Programs in Logistics and Supply Chain Management, International Handbooks on Information Systems. Berlin and Heidelberg: Springer. Eppen, G., & Schrage, L. (1981). Centralized ordering policies in a multiwarehouse system with leadtimes and random demand. In L. Schwarz (Ed.), Multi-level production/inventory control systems: Theory and practice (pp. 51–69). North Holland. Federgruen, A. (1993). Centralized planning models for multi-echelon inventory systems under uncertainty. In S. Graves, A. Rinnooy Kan, & P. Zipkin (Eds.), Handbook in operations research and management science (Vol. 4), Logistics of production and inventory. North Holland. Federgruen, A., & Zipkin, P. (1984a). Approximation of dynamic, multi-location production and inventory problems. Management Science, 30, 69–84. Federgruen, A., & Zipkin, P. (1984b). Allocation policies and cost approximation for multi-location inventory systems. Naval Research Logistics Quarterly, 31, 97–131. Friedman, J. F. (1986). Game theory with applications to economics. New York: Oxford University Press. Gottwald, D. (1997). Equitable inventory allocation for today’s auto industry. Working Paper, Fuqua School of Business, Duke University, Durham, NC. Ha, A. Y. (1997). Inventory rationing in a make-to-stock production system with several demand classes and lost sales. Management Science, 43, 1093–1103.

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Hwang, S. L., & Valeriano, L. (1992). Marketers and consumers get the jitters over severe shortages of nicotine pathes. Wall Street Journal, May 22, B1. Lee, H., Padmanabhan, V., & Whang, S. (1997). Information distortion in a supply chain: The bullwhip effect. Management Science, 43, 546–558.

Chapter 3

Capacity Allocation with Demand Competition in Asymmetric Market

3.1 Introduction Capacity shortfall is frequently observed in various industries when retailers’ total order size exceeds a supplier’s available capacity. For examples, capacity shortage often arises in the fashion goods, telecommunications, and electricity industries (Iyer et al. 2003). Also, it is a common practice for an automobile manufacturer to sell through multiple dealers in the same geographic region; they compete for both the manufacturer’s limited supply capacity and customer demand for popular vehicle models (Liu 2012). However, capacity investment/expansion is usually costly and difficult to timely achieve, such as for vehicles and seasonal products. Thus, it becomes an important issue for the supplier to price and allocate scarce capacity effectively. The objective of this paper is to study how different allocation mechanisms affect a supplier’s wholesale pricing and retailers’ ordering decisions, and to suggest how the supplier can choose an allocation mechanism together with pricing decisions to increase profit. Specifically, we investigate a two-echelon supply chain in which a monopoly supplier (he) sells through duopoly retailers (she) with demand competition. The capacity allocation mechanism considered works in the following way. First, the supplier announces his capacity size, unit wholesale price of this capacity, an allocation rule that defines how capacity will be allocated as a function of retailer order sizes, and a requirement that no order size can be more than total capacity. Second, the retailers place their orders. Third, the supplier allocates capacity to retailers using the pre-announced allocation rule. Finally, the retailers sell the allocated capacity to their customers. In our consideration, an allocation rule applies within an allocation mechanism. We henceforth refer to a mechanism with a specific rule, e.g., uniform, as uniform allocation mechanism. The problem under investigation fits within a two-dimensional noncooperative game framework. The first vertical gaming issue occurs as a Stackelberg game in which the supplier is a leader and the two retailers are followers. We assume that the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Li et al., Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-19-6577-7_3

51

52

3 Capacity Allocation with Demand Competition in Asymmetric Market

supplier’s capacity is given exogenously. The supplier’s decisions are over wholesale price and the allocation rule. The second horizontal gaming issue arises from competitions between the two retailers, who are in the same market and compete for both the supplier’s capacity and the demand from a common customer population. We consider Cournot competition where the two retailers sell at retail prices as functions of the total capacity provided in the retailing market by the two retailers. The two retailers are with asymmetric market powers, where the retailer with greater market power, high-type retailer, sells at a higher price than the other low-type retailer. The difference between the market powers of the two retailers is referred to as competitive gap. Three widely studied allocation rules are considered in our allocation mechanism: uniform, proportional, and lexicographic allocations. Under uniform allocation, if a retailer orders no more than the equal share of the total available capacity, then she receives her order and other retailers share the remaining capacity in a similar way. Uniform allocation is individually unresponsive (IU), in that allocation is not strictly increasing with order size. When retailers are local monopolists, uniform allocation guarantees equilibrium orders and is truth-inducing in that it incentivizes retailers to order their individually optimal order sizes, thus eliminating the gaming effect (Sprumont 1991; Cachon and Lariviere 1999a, b). However, when retailers compete for their own demand, even though uniform allocation still warrants equilibrium orders, this does not prevent the gaming effect (Liu 2012; Cho and Tang 2014). Proportional allocation is more intuitive: It allocates capacity in proportion to order size; it thus strictly increases with order size and is individually responsive (IR). Even for monopolistic retailers, proportional allocation does not guarantee equilibrium orders and is well-known to cause over-ordering (Lee et al. 1997; Cachon and Lariviere 1999a, b). Under lexicographic allocation, retailers are prioritized such that the retailer with the highest priority always has her order filled first. Lexicographic allocation also belongs to the class of IU allocations. Cachon and Lariviere (1999a) demonstrate that lexicographic allocation is truth-inducing and guarantees equilibrium orders when retailers face independent demands. When retailers face demand competition, equilibrium still exists under lexicographic allocation, but gaming effect may also occur (Chen et al. 2013). Hence, lexicographic allocation behaves similarly to uniform allocation with respect to the existence of equilibrium orders and gaming effect. Another popular IR allocation rule is linear allocation, which allocates each retailer her order size minus a common deduction. Linear allocation performs similarly to proportional allocation given the existence of equilibrium orders and gaming effect (Cachon and Lariviere 1999a; Liu 2012). In fact, consistent with the findings in Liu (2012) in a similar problem setting, linear allocation performs nearly identically to proportional allocation for our problem; hence we omit this allocation. The pre-specified upper bound on order size, total capacity, is practically meaningful, since the supplier’s capacity is publicly known, and ordering more than that obviously exceeds the supplier’s capability to meet that demand (Chen et al. 2013). Also, allowing unbounded ordering can induce arbitrarily large orders from competing retailers, leading to unpredictable allocations (Cachon and Lariviere 1999a, b).

3.1 Introduction

53

The upper bound can also be considered as embedded in allocation rules. That is, any order larger than the bound will be truncated to the bound and then an allocation rule is applied to the truncated order. Our study shows that, under uniform or proportional mechanism, when the wholesale price is relatively low, each retailer places an order equal to the pre-specified upper bound, and thus receives half the capacity. However, lexicographic mechanism performs differently. Although both retailers prefer greater allocation, the retailer with higher priority can receive the entire capacity. On the other hand, if the wholesale price is sufficiently high, then under all mechanisms considered, retailer with lower market power would be driven out of the market. For the supplier, we find that the wholesale pricing decision is sensitive to his capacity level and the competitive gap between the two retailers. Among the three mechanisms, lexicographic allocation is best for the supplier, and proportional allocation outperforms uniform allocation. Specifically, for any given wholesale price, the supplier can sell more from lexicographic allocation than from the other two allocations. Further, employing lexicographic allocation, the supplier can sell more by granting higher priority to the high-type retailer, especially when the competitive gap between the two retailers is large. We also extend our study to the case with multiple retailers. We analytically prove that proportional allocation dominates uniform allocation and lexicographic allocation with priority sequence of retailers in nonincreasing market powers dominates proportional allocation from the perspective of the supplier for any number of retailers. We numerically show that lexicographic allocation with any priority sequence of retailers dominates uniform allocation for any number of retailers. With three or more retailers, the relative performance of lexicographic allocation compared with proportional allocation depends on priority sequence of retailers: lexicographic allocation may outperform (underperform, respectively) proportional allocation if priority is given to retailers with higher (lower, respectively) market powers. Capacity allocation is studied extensively in the operations literature due to its practical importance. Hall and Liu (2010) survey capacity allocations and provide classification schemes based on whether: a game is cooperative or noncooperative; capacity is sufficient or deficient; single or multiple types of capacity are considered; timing issue is addressed or not; and what kinds of mechanisms, including auction, contract, pricing, and rule, are used. Our problem falls into the class of noncooperative games with a single type of deficient capacity allocated thorough rules without timing issue. We now review the literature on the same class of capacity allocation. Sprumont (1991) establishes important theories for rule-based allocations. This author shows that for a general capacity allocation problem where each retailer has a single-peaked preference of capacity size, uniform allocation is the unique rule that is truth-inducing, efficient, and anonymous, where efficiency means that if the total order size is greater (less, resp.) than the available capacity, then no retailer receives allocation greater (less, resp.) than her order, and anonymity means that retailers with equal orders receive the same allocation. Lee et al. (1997) show that, despite its intuitive attraction and popularity, proportional allocation can lead to order inflation and contributes to the well-known bullwhip effect. Cachon and Lariviere (1999a, b) analyze properties, equilibrium ordering decisions, supplier capacity choice, and

54

3 Capacity Allocation with Demand Competition in Asymmetric Market

supply chain performance under a variety of allocation mechanisms, including linear, proportional, lexicographic, and uniform. These authors find that, with asymmetric information, proportional allocation is not resistant to gaming effect and consequently may distort supplier capacity selection. Uniform and lexicographic allocations are both truth-inducing, but may result in lower profit for supply chain members. Cachon and Lariviere (1999c) investigate a two-period allocation using a turn and earn mechanism, where allocation in the current period is based on sales in the preceding period. These authors show that this mechanism can increase the supplier’s profit at the cost of the retailers and even the supply chain. Chen et al. (2012) study the gaming effect caused by proportional allocation, using laboratory experiments. These authors find that retailers often order much less than the Nash equilibrium. Thus, they propose a bounded rationality model and find that retailers learn to be more rational through repeated games, but that their orders still may not converge to Nash equilibrium. A common assumption of the above-mentioned literature is that each retailer is a local monopolist and her demand is independent from other retailers. To our knowledge, only four studies consider capacity allocation with demand competition, which are more relevant to our study. Liu (2012) considers demand competition where retail prices are linear functions of the total sales volume of two retailers. This author investigates how different allocations affect capacity sold, retailer profit, and supply chain profit. This author finds that a supplier can sell more with less capacity, and retailers may earn more when the supplier has less capacity. Cho and Tang (2014) extend Liu (2012)’s work to the case with multiple retailers, with a focus on gaming effect caused by uniform allocation. These authors determine exact conditions under which the gaming effect presents. They then propose competitive allocation, which eliminates gaming effect under demand competition. Chen et al. (2013) analyze the supplier’s wholesale pricing decision under proportional and lexicographic allocations, with retailers possessing the same market power in demand competition. These authors show that lexicographic allocation can bring higher profit to both the supplier and the supply chain. Yang et al. (2014) consider a supplier who sells both directly by himself and through a retailer, and who thus faces capacity allocation between himself and the retailer. These authors find that the supplier, the retailer, and consumers may simultaneously benefit from the supplier’s limited capacity. We now clarify our contribution relative to existing works of Cachon and Lariviere (1999b); Liu (2012); Chen et al. (2013) and Cho and Tang (2014). First, Cachon and Lariviere (1999b) study the allocation mechanisms when retailers are local monopolists, while we analyze the case when retailers engage in quantity competition so that a retailer’s profit is dependent upon the allocations for all the retailers. Besides, we introduce a new variable, wholesale price, to the supplier’s decision. Second, Liu (2012) and Cho and Tang (2014) consider demand competition among retailers, but do not consider lexicographic allocation or wholesale pricing decision as we do. Third, our work is more closely related to Chen et al. (2013), who consider identical retailers, while our consideration of asymmetric retailers is more general, and we also include analysis for uniform allocation that is not considered by Chen et al. (2013). We summarize the above differences in Table 3.1, where Cachon and Lar-

3.2 Model Formulation

55

Table 3.1 Contributions relative to existing literature Problem CL99 L12 CLZ13 Lexicographic allocation Proportional allocation Uniform allocation Demand competition Wholesale pricing Asymmetric retailers Two retailers More than two retailers

CT14

LYL16

No

No

Yes

No

Yes

Yes

Yes

Yes

No

Yes

Yes

Yes

No

Yes

Yes

No

Yes

Yes

Yes

Yes

No

No

Yes

No

Yes

Yes

Yes

No

Yes

Yes

Yes No

Yes No

Yes Yes

Yes Yes

Yes Yes

iviere (1999b); Liu (2012); Chen et al. (2013); Cho and Tang (2014) and this work are referred to as CL99, L12, CLZ13, CT14, LYL16, respectively. Overall, with a more general model setting, we regard our contribution as unifying and generalizing various results from the previous literature on capacity allocation with demand completion among retailers. The remainder of this paper proceeds as follows. In Sect. 3.2, we formally describe our model, specifically with two retailers. Sections 3.3 and 3.4 conduct equilibrium analysis for retailer ordering and supplier wholesale pricing decisions under uniform, proportional, and lexicographic mechanisms, respectively. In Sect. 3.5, we compare the three mechanisms from the perspective of the supplier. Results are developed for extended cases with more retailers in Sect. 3.6. Section 3.7 summarizes our work. All proofs are in the Appendix.

3.2 Model Formulation Consider a two-echelon supply chain consisting of a single supplier and two downstream retailers. The retailers order a single type of product from the supplier, who has a fixed capacity in the quantity of K . Each retailer will receive her ordered quantity if capacity is ample. However, when the supplier’s capacity is limited such that it cannot satisfy the sum of orders, an allocation rule should be implemented to allocate the capacity between the two retailers. Suppose retailer i, who places an order with quantity ri (i = 1, 2) receives an allocation with quantity qi (r1 , r2 ). Without loss of generality, the supplier’s production cost associated with unit capacity is normalized

56

3 Capacity Allocation with Demand Competition in Asymmetric Market

to zero for succinctness. The supplier charges a wholesale w for each unit of capacity sold. As discussed in Sect. 3.1 and as discussed in Chen et al. (2013), we do not allow a retailer’s order size to exceed total capacity, i.e., we require ri ≤ K . After receiving allocations, the retailers in turn sell the product to the same market and compete for customer demand. Following widely adopted convention in the capacity allocation literature, we assume that information is symmetric and complete for all parties, i.e., supplier capacity, unit capacity wholesale price, the allocation rule, the upper bound for each individual order, and the customer demand functions are common knowledge of the supplier and retailers. Let pi be the retail price of retailer i (i = 1, 2). The retail price is determined by a linear demand function pi = z i − q1 (r1 , r2 ) − q2 (r1 , r2 ), where z i represents retailer i’s market power, which may capture retailer i’s comparative advantage over the other retailer because of certain factors, such as ease of access, customer preference, and brand equity. Without loss of generality, we assume z 1 ≥ z 2 , and refer to retailer 1 as a high-type retailer and retailer 2 as a low-type retailer. Note that we characterize a duopoly model of two retailers, and it is necessary to have z 2 > (z 1 + w)/2 to retain both retailers in the market when the supplier has ample capacity; otherwise, retailer 2 cannot survive in competition with retailer 1. Such an assumption is adopted by Liu (2012) and Cho and Tang (2014). However, in our problem w is a decision variable of the supplier, and it would be artificial to impose a lower bound on w other than 0. Henceforth, we assume z 2 > z 1 /2. We next describe the sequence of events and decisions. First, before the selling season starts, the supplier announces his capacity level, unit wholesale price of the capacity, an allocation rule, and an upper bound equal to his capacity size for any order. Second, the retailers place their orders. Third, the supplier allocates capacity to retailers using the preannounced allocation rule. Finally, the retailers sell the allocated capacity to their customers. This is a Stackelberg game in which the supplier is the leader and the retailers are followers. For a given capacity level and allocation rule, the supplier’s problem is to choose a wholesale price w to maximize his profit: s = max{w · (q1 (r1 , r2 ) + q2 (r1 , r2 ))}. w

Given the wholesale price, capacity level, and allocation rule, the retailers’ problems are i (r j , w) = max{(z i − qi (ri , r j ) − q j (ri , r j ) − w) · qi (ri , r j )}, ri

where i, j = 1, 2 and i = j. Here, we consider three commonly used allocation rules: uniform, proportional, and lexicographic allocations. First, consider two retailers under uniform allocation.

3.2 Model Formulation

57

A retailer receives her order if she orders less than half of the capacity, and the other retailer receives the minimum of her order and the remaining capacity; if the retailer with smaller order size orders more than half the capacity, then she will share the capacity equally with the other retailer. Formally, uniform allocation with two retailers is defined as follows: ⎧ if ri + r j ≤ K , ⎨ ri if ri + r j > K , ri ≤ r j , qi (ri , r j ) = min{ri , K /2} ⎩ max{K − r j , K /2} if ri + r j > K , ri > r j , where i, j = 1, 2 and i = j. Under proportional allocation, a retailer receives the allocation size she orders if the capacity is sufficient; otherwise, she obtains allocation in proportion to her order size. Hence,  if ri + r j ≤ K , ri qi (ri , r j ) = K ri if ri + r j > K , ri +r j where i, j = 1, 2 and i = j. Under lexicographic allocation, the supplier grants priority to one of the retailers and satisfies her order first as far as possible; the supplier then allocates the remaining capacity to the other retailer. Suppose priority is given to retailer i and ri , r j ≤ K . We have qi (ri , r j ) = ri and  q j (ri , r j ) =

if ri + r j ≤ K , rj K − ri if ri + r j > K ,

where i, j = 1, 2 and i = j. Before examining the allocation mechanisms, we first derive order/allocation sizes in equilibrium when the supply of product is unlimited. In this case, retailer i’s order quantity equals her capacity allocation size. We do not allow capacity to be withheld by retailers, and the allocation size is the same as the selling quantity to the market. Such an assumption is used by Cachon and Lariviere (1999b); Liu (2012); Chen et al. (2013), and Cho and Tang (2014). From retailer i’s profit function πi (r1 , r2 ) = (z i − w − r1 − r2 )ri , i = 1, 2, we obtain the equilibrium order quantity without capacity constraint: r1∗ =

2z 1 − z 2 − w 2z 2 − z 1 − w , and r2∗ = . 3 3

In addition, qi∗ = ri∗ , and the total allocated capacity is q1∗ + q2∗ = z1 +z32 −2w . For convenience, in our analysis we adopt the following notations. First, we map real value interval to nonnegative interval as

58

3 Capacity Allocation with Demand Competition in Asymmetric Market

⎧ if b < a or b < 0; ⎨∅ [a, b] = [0, b] if b > 0 and a < 0; ⎩ [a, b] if 0 ≤ a ≤ b. Also, we denote [x1 , x2 ] × [y1 , y2 ] order pairs (a, b) for any x1 ≤ a ≤ x2 and y1 ≤ b ≤ y2 . In addition, we define x1 ∧ x2 = min{x1 , x2 } and x1 ∨ x2 = max{x1 , x2 }.

3.3 Uniform Allocation Mechanism We first examine uniform allocation mechanism. Given the supplier’s wholesale price and capacity level, we express the retailers’ profit functions as: 

  z i − q˜i (ri , r j ) − q˜ j (ri , r j ) − w · q˜i (ri , r j ) , ri ≤K 

˜ 1 = max = max  z i − w − ri − r j ri , i

˜ i (r j , w) = max 

ri ≤K



ri ≤K −r j

 K K  K ) − (K − ri ) ∨ · ri ∧ , 2 2 2 ri >K −r j ;ri ≤r j



  K K K ˜3 = − (r ) · (K − r , z max − w − (K − r ) ∨ ∧ ) ∨  i j j j i 2 2 2 ri >K −r j ,ri >r j ˜2 =  i



max

z i − w − (ri ∧

where i, j = 1, 2 and i = j. The following lemma characterizes the retailers’ best response functions and associated profits. Lemma 3.1 Given retailer j’s order quantity r j , let r˜i (r j ) be √ retailer i’s best response function (i, j = 1, 2 and i = j). Define α˜ i = z i − w − 2K (z i − w − K ), we then have: (i) w ∈ (0, z i − 2K ]: ˜ i (r j , w) = (z i − w − K )(K − r j ); if r j < K /2, then r˜i (r j ) ∈ [K − r j , K ] with  ˜ if r j ≥ K /2, then r˜i (r j ) ∈ [K /2, K ] with i (r j , w) = (z i − w − K )K /2. (ii) w ∈ (z i − 2K , z i − 3K /2]: ˜ i (ri , w) = (z i − w − if r j ≤ 2K − z i + w, then r˜i (r j ) = (z i − w − r j )/2 with  2 r j ) /4; ˜ i (r j , w) = (z i − if 2K − z i + w < r j ≤ K /2, then r˜i (r j ) ∈ [K − r j , K ] with  w − K )(K − r j ); ˜ i (r j , w) = (z i − w − K )K /2. if r j > K /2, then r˜i (r j ) ∈ [K /2, K ] with  (iii) w ∈ (z i − 3K /2, z i − K ]: ˜ i (r j , w) = (z i − w − r j )2 /4; if r j ≤ α˜ i , then r˜i (r j ) = (z i − w − r j )/2 with  ˜ i (r j , w) = (z i − w − K )K /2. if r j > α˜ i , then r˜i (r j ) ∈ [K /2, K ] with  (iv) w ∈ (z i − K , z i ): ˜ i (r j , w) = (z i − w − r j )2 /4; if r j ≤ z i − w, then r˜i (r j ) = (z i − w − r j )/2 with  ˜ if r j > z i − w, then r˜i (r j ) = 0 with i (r j , w) = 0.

3.3 Uniform Allocation Mechanism

59

The results show that a retailer’s best response order quantity directly depends on the supplier’s wholesale price and the other retailer’s order size. In line with the definition of uniform allocation, a very intuitive observation is that retailer i would always receive the same allocation K /2 by ordering from K /2 to K (the maximum allowed order) if retailer j’s order size is no less than K /2. This explains the last part of (i), (ii), and (iii). However, if the wholesale price is very high, then the retailers cannot afford it and order nothing (see the second part of (iv)). Further, given an appropriate wholesale price, if the competitor orders a reasonably small quantity such that total capacity is sufficient for both retailers, then the best response function is the same as in the case without capacity limit. The first parts of (ii), (iii), and (iv) illustrate this effect. On the other hand, from the second part of (i), we can see that if the wholesale price is very low, together with the fact that the other retailer’s order quantity is less than K /2, then retailer i will order from K − r2 to K and gain the same resulting allocation K − r2 .

3.3.1 Equilibrium Analysis Under Uniform Allocation In this section, our objective is to characterize retailer order sizes in equilibrium and the optimal wholesale pricing for the supplier. Using backward induction, under uniform allocation, we first derive the retailers’ equilibrium order pair given any wholesale price, and then examine the supplier’s wholesale pricing decisions. The following theorem characterizes Nash equilibria of the retailers’ order quantities. Theorem 3.1 Suppose the Nash equilibrium  order pair is (˜r1∗ , r˜2∗ ) under uniform mechanism. Define w˜ + = 2z 2 − z 1 − 49 K + 43 K 2 + 8K (z 1 − z 2 ), then we have (I) K ≤ z 1 − z 2 : (i) if w ∈ (0, z 2 − 3K /2], then any point in [K /2, K ] × [K /2, K ] is a Nash equilibrium; (ii) if w ∈ (z 2 − 3K /2, z 2 − K ], then any point in [α˜ 2 , K ] × [K /2, K ] is a Nash equilibrium; (iii) if w ∈ (z 2 − K , z 1 − 2K ], then there exists a unique Nash equilibrium (˜r1∗ , r˜2∗ ) = (K , 0); (iv) if w ∈ (z 1 − 2K , z 1 ), then there exists a unique Nash equilibrium (˜r1∗ , r˜2∗ ) = ((z 1 − w)/2, 0). (II) K > z 1 − z 2 : (i) if w ∈ (0, z 2 − 3K /2], then any point in [K /2, K ] × [K /2, K ] is a Nash equilibrium; (ii) if w ∈ (z 2 − 3K /2, w˜ + ], then any point in [α˜ 2 , K ] × [K /2, K ] is a Nash equilibrium; (iii) if w ∈ (w˜ + , z 1 − 3K /2], then (˜r1∗ , r˜2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a Nash equilibrium and any point in [α˜ 2 , K ] × [K /2, K ] is also a Nash equilibrium. Further, the former equilibrium dominates the latter by generating greater

60

3 Capacity Allocation with Demand Competition in Asymmetric Market

profits for both retailers; (iv) if w ∈ (z 1 − 3K /2, z 2 − K ], then (˜r1∗ , r˜2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a Nash equilibrium and any point in [α˜ 2 , K ] × [α˜ 1 , K ] is also a Nash equilibrium. Further, the former equilibrium dominates the latter one by generating greater profits for both retailers; (v) if w ∈ (z 2 − K , 2z 2 − z 1 ], then there exists a unique equilibrium (˜r1∗ , r˜2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3); (vi) if w ∈ (2z 2 − z 1 , z 1 ), then there exists a unique equilibrium (˜r1∗ , r˜2∗ ) = ((z 1 − w)/2, 0). Theorem 3.1 presents equilibrium order pairs when the two retailers compete for demand with different market powers. Equilibrium analysis is conducted for the cases K > z 1 − z 2 and K ≥ z 1 − z 2 separately. The results are quite intuitive. First, when capacity is no greater than z 1 − z 2 , if the wholesale price is fairly low, with w ∈ (0, z 2 − K ], then both retailers seek to gain higher allocations, but any one of them can receive only K /2 by ordering no less than K /2. Second, as the wholesale price increases, the low-type retailer cannot afford it and thus is driven out of the game. As a result, the problem reduces to a simple capacity allocation consisting of a single supplier and a single retailer. In this special case, the high-type retailer’s profit ˜ 1 = (z 1 − w − q˜1 )q˜1 . Taking the first-order condition with respect to function is  q˜1 , the optimal order follows that q˜1∗ = (z 1 − w)/2 without capacity constraint. Consequently, (a) if w ≤ z 1 − 2K , i.e., K ≤ (z 1 − w)/2, which implies that the capacity is limited for the retailer, then the equilibrium order pair and the allocation pair are , where the capacity is sufficient to both (K , 0); (b) if w > z 1 − 2K , i.e., K > z1 −w 2 satisfy the retailer’s order, then the equilibrium order pair and the allocation pair are both ((z 1 − w)/2, 0). Moreover, when the capacity level exceeds the value z 1 − z 2 , certain results are similar to the case with K ≤ z 1 − z 2 when the wholesale price is very low or sufficiently high. We also find several other interesting results. In particular, when w ∈ (z 2 − 3K /2, w˜ + ], since 2z 2 − z 1 − 3K /2 < (z 1 + z 2 − 3K )/2, it follows that where K (w˜ + ) = (z 2 − w + if 2q2∗ < q1∗ + q2∗ < K < K (w˜ + ),  (z 2 − w)2 − 8(2z 2 − z 1 − w)2 /9)/2, then the low-type retailer would inflate her order size to be no less than K /2 so as to gain more profit from the allocation pair (K /2, K /2) than from the ideal allocation pair ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3). This result reflects the fact that uniform mechanism has potential to favor the low-type retailer by offering an opportunity to diminish her competitive gap relative to the high-type retailer. Note that when w ∈ (w˜ + , z 2 − K ], though there exist multiple equilibrium order pairs, the equilibrium (˜r1∗ , r˜2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) dominates other equilibria by generating more profits for both retailers. Liu (2012) and Cho and Tang (2014) also study equilibrium ordering under uniform allocation. Cho and Tang (2014) mostly study whether the gaming effect presents under uniform allocation and does not provide complete equilibrium order pairs under different conditions of K and w. The analysis in Lemma 3.1 and Theorem 3.1 makes the following major distinctions comparing with Liu (2012): (1) we assume an upper bound (supplier’s capacity K ) on order size, while Liu (2012)

3.3 Uniform Allocation Mechanism

61

assumes no upper bound on order size; (2) Liu (2012) assumes z 2 > (z 1 + w)/2, while in our problem w is a decision variable of the supplier, and it would be artificial to impose a lower bound on w other than 0, thus our paper relax the assumption as z 2 > z 1 /2; (3) we provide complete and precise structures for retailers’ ordering decisions in Nash and dominant equilibrium in Theorem 3.1, while Liu (2012) focuses on allocations in stead of equilibrium order pairs and only provides a representative equilibrium when multiple equilibria result in the same allocation. In addition, Liu (2012) classifies equilibria based on capacity level K , while we classify equilibria based on wholesale price w, for the ease of subsequent analysis of the supplier’s wholesale pricing decision, which Liu (2012) does not consider. In fact, if we transform from w(K ˜ ) to K (w), ˜ it follows that K = (z 2 − w)/2 + ( −8z 12 + 32z 1 z 2 − 23z 22 − 16z 1 w + 14z 1 w + 14z 2 w + w 2 )/6, which is actually

the threshold K 2+ used in Liu (2012). Therefore, our results essentially are consistent with the results in Liu (2012), and are more complete from the perspective of equilibrium analysis.

3.3.2 Supplier’s Decisions Under Uniform Allocation Now we consider the supplier’s decisions on wholesale price. By anticipating the retailers’ best response order quantities and the allocations they will receive, the supplier chooses an optimal wholesale price to maximize his profit. First we establish the supplier’s profit function. Note that when K ≤ z 1 − z 2 , if w ∈ (0, z 1 − 2K ], then the total allocated capacity is K ; thus in this scenario the supplier’s profit is K w. Otherwise, if w ∈ (z 1 − 2K , z 1 ), then retailer 2 would be driven out of the market, and thus the supplier’s profit would be w(z 1 − w)/2. ˜ 2s }, ˜ 1s ,  Therefore, the supplier’s maximum profit can be characterized as max{ where w(z 1 − w) ˜ 1s = max K w,  ˜ 2s =  . max w∈(0,z 1 −2K ] w∈(z 1 −2K ,z 1 ) 2 ˜ 3s ,  ˜ 4s ,  ˜ 5s }, where Similarly, when K > z 1 − z 2 , the supplier’s problem is maxw { ˜ 3s = 

max

w∈(0,w˜ + ]

˜ 4s = K w, 

max

w∈(w˜ + ,2z

2 −z 1 ]

w(z 1 + z 2 − 2w) 5 w(z 1 − w) ˜s = max , . w∈(2z 2 −z 1 ,z 1 ) 3 2

˜ ∗s To determine the optimal wholesale price w˜ ∗ and the corresponding profit  1 2 3 4 5 ˜ s, ˜ s and  ˜ s,  ˜ s, ˜ s . The corresponding for the supplier, we need to compare  results are in Proposition 3.1 in the Appendix. We illustrate the supplier’s optimal wholesale pricing decisions in Table 3.2, where the first row presents different levels of the capacity K where a full range of K > 0 is covered, and the last column denotes the competitive gap between the two retailers’ market powers where a full range of z 1 /2 < z 2 < z 1 is covered. For any capacity level K and competitive gap between

K (z 1 − 2K )

˜ ∗s 

1 24

z 1 − 2K

w˜ ∗

+

K (z 1 − 2K )

˜ ∗s 

3z 2 −z 1 8

z 1 /2

z 1 − 2K

w˜ ∗

K˜ 2+ =

z 12 /8

K (z 1 − 2K )

˜ ∗s 

max{K w˜ + , z 12 /8}

max{K w˜ + , (2z 2 − z 1 )(z 1 − z 2 )}

z 1 )(z 1 − z 2 )}

argmax

w∈{w˜ + ,2z 2 −z 1 }

max{K w˜ + , (2z 2 −

max{K w˜ + , z 12 /8}

max{K w˜ + , z 12 /8}

argmax

w∈{w˜ + ,z 1 /2}

w∈{w˜ + ,z 1 /2} max{K w˜ + , z 12 /8}

argmax

(z 1 − z 2 , K˜ 2+ ]

argmax max{K w˜ + , (z 1 + w∈{w˜ + ,(z 1 +z 2 )/4,z 1 /2} z 2 )2 /24, z 12 /8} max{K w˜ + , (z 1 + z 2 )2 /24, z 12 /8} argmax max{K w˜ + , (z 1 + w∈{w˜ + ,(z 1 +z 2 )/4,2z 2 −z 1 } z 2 )2 /24, (2z 2 − z 1 )(z 1 − z 2 )} max{K w˜ + , (z 1 + z 2 )2 /24, (2z 2 − z 1 )(z 1 − z 2 )}

( K˜ 2+ , +∞)

 −191z 12 + 506z 1 z 2 − 311z 22 , w˜ + = 2z 2 − z 1 − 94 K + 43 K 2 + 8K (z 1 − z 2 )

z 12 /8

z 1 /2

z 1 − 2K

w˜ ∗

z ( 41 , z 1 − z 2 ]

z (0, 41 ]

K

Table 3.2 Supplier’s pricing decisions under uniform allocation

3z 1 4 < z2 < z1

5z 1 3z 1 7 < z2 ≤ 4

z1 5z 1 2 < z2 ≤ 7

–

62 3 Capacity Allocation with Demand Competition in Asymmetric Market

3.4 Proportional and Lexicographic Allocation Mechanisms

63

˜ ∗s are listed in the z 1 and z 2 , the optimal wholesale price w˜ ∗ and associated profit  corresponding cells of the tables. We can see that under different conditions, the supplier can strategically alter the wholesale price to maximize his profit.

3.4 Proportional and Lexicographic Allocation Mechanisms First we investigate proportional allocation. Under this mechanism, the retailers’ problem can be expressed as ˆ i (r j , w) = max{(z i − qˆi (ri , r j ) − qˆ j (ri , r j ) − w) · qˆi (ri , r j )}  ri ≤K

  Kr    Kr   Kr j i i ∧ ri − ∧ rj − w · ∧ ri , = max z i − ri ≤K ri + r j ri + r j ri + r j where i, j = 1, 2 and i = j. Due to space limit, retailers’ best response functions and equilibrium ordering decisions are provided in Lemma 3.2 and Proposition 3.2 in the Appendix. Using the same approach as for uniform allocation, we derive the supplier’s optimal wholesale price and the associated maximum profit under proportional allocation, as shown in Table 3.3, in a similar structure as Table 3.2. Table 3.3 shows that proportional allocation performs similar uniform allocation with respect to the supplier’s pricing decision. Specifically, the supplier’s capacity level and the competitive gap between the two retailers’ market powers both directly affect the supplier’s pricing decisions. Now we consider lexicographic allocation. Suppose priority is given to retailer i; then the retailers’ profit function can be rewritten as:   ˇ i (r j , w) = max (z i − w − qˇi (ri , r j ) − qˇ j (ri , r j ) · qˇi (ri , r j ) ,  ri ≤K

ˇ1 = = max  i ri ≤K

max

ri ≤K −r j

ˇ2 = (z i − ri − r j − w) · ri ,  i

max

ri >K −r j

 (z i − K − w) · ri ,

  ˇ j (ri , w) = max (z j − qˇi (ri , r j ) − qˇ j (ri , r j ) − w) · qˇ j (ri , r j ) ,  r j ≤K

= max

r j ≤K

ˇ1 =  j

max

r j ≤K −ri

ˇ2 = (z j − w − ri − r j ) · r j ,  j

max

r j >K −ri

 (z j − K − w) · (K − ri ) ,

where i, j = 1, 2 and i = j. Due to space limits, equilibrium analysis and the correspondent discussions for lexicographic allocation are provided in Proposition 3.3 in the Appendix. We only present the supplier’s wholesale pricing associated with the profits in Table 3.4.

K (z 1 − 2K )

ˆ ∗s 

1 24

z 1 − 2K

wˆ ∗

+

K (z 1 − 2K )

ˆ ∗s 

z 1 +z 2 12

z 1 /2

z 1 − 2K

wˆ ∗

Kˆ 2+ =

z 12 /8

K (z 1 − 2K )

ˆ ∗s 

−136z 12 + 304z 1 z 2 − 136z 22 , wˆ − = 3K +

z 12 /8

z 1 /2

z 1 − 2K

wˆ ∗

z ( 41 , z 1 − z 2 ]

z (0, 41 ]

K max{K wˆ − , z 12 /8}

( Kˆ 2+ , +∞)

argmax max{K wˆ − , {(z 1 + w∈{wˆ − ,(z 1 +z 2 )/4,z 1 /2} z 2 )2 /24, z 12 /8} 2 − max{K wˆ , z 1 /8} max{K wˆ − , {(z 1 + z 2 )2 /24, z 12 /8} − argmax max{K wˆ , (2z 2 − argmax max{K wˆ − , (z 1 + w∈{wˆ − ,2z 2 −z 1 } w∈{wˆ − ,(z 1 +z 2 )/4,2z 2 −z 1 } z 1 )(z 1 − z 2 )} z 2 )2 /24, (2z 2 − z 1 )(z 1 − z 2 )} max{K wˆ − , (2z 2 − z 1 )(z 1 − z 2 )} max{K wˆ − , (z 1 + z 2 )2 /24, (2z 2 − z 1 )(z 1 − z 2 )} z 1 +z 2 2

− 3 2

 8K 2 + (z 1 − z 2 )2

argmax

w∈{wˆ − ,z 1 /2}

max{K wˆ − , z 12 /8}

w∈{wˆ − ,z 1 /2} max{K wˆ − , z 12 /8}

argmax

(z 1 − z 2 , Kˆ 2+ ]

Table 3.3 Supplier’s pricing decisions under proportional allocation

3z 1 4 < z2 < z1

5z 1 3z 1 7 < z2 ≤ 4

z1 5z 1 2 < z2 ≤ 7

–

64 3 Capacity Allocation with Demand Competition in Asymmetric Market

K (z 1 − 2K )

ˇ ∗s 

1 24

z 1 − 2K

wˇ ∗

+

K (z 1 − 2K )

ˇ ∗s 

3z i −z j 8

z 1 /2

z 1 − 2K

wˇ ∗

Kˇ i+ =

z 12 /8

K (z 1 − 2K )

ˇ ∗s 

max{K wˇ − , z 12 /8}

( Kˇ i+ , +∞)

argmax max{K wˇ i+ , {(z 1 + w∈{wˇ i+ ,(z 1 +z 2 )/4,z 1 /2} z 2 )2 /24, z 12 /8} + 2 max{K wˇ i , z 1 /8} max{K wˇ i+ , {(z 1 + z 2 )2 /24, z 12 /8} + argmax max{K wˇ i , (2z 2 − argmax max{K wˇ i+ , (z 1 + w∈{wˇ i+ ,2z 2 −z 1 } w∈{wˇ i+ ,(z 1 +z 2 )/4,2z 2 −z 1 } z 1 )(z 1 − z 2 )} z 2 )2 /24, (2z 2 − z 1 )(z 1 − z 2 )} max{K wˇ i+ , (2z 2 − z 1 )(z 1 − z 2 )} max{K wˇ i+ , (z 1 + z 2 )2 /24, (2z 2 − z 1 )(z 1 − z 2 )}

argmax

w∈{wˇ i+ ,z 1 /2}

max{K wˇ i+ , z 12 /8}

w∈{wˇ − ,z 1 /2} max{K wˇ i+ , z 12 /8}

argmax

(z 1 − z 2 , Kˇ i+ ]

−91z i2 + 226z i z j − 115z 2j , wˇ i+ = 2z i − z j − 29 K + 23 5K 2 − 4K (z i − z j ), i, j = 1, 2, i = j

z 12 /8

z 1 /2

z 1 − 2K

wˇ ∗

z ( 41 , z 1 − z 2 ]

z (0, 41 ]

K

Table 3.4 Supplier’s Pricing Decisions under Lexicographic Allocation

3z 1 4 < z2 < z1

5z 1 3z 1 7 < z2 ≤ 4

z1 5z 1 2 < z2 ≤ 7

–

3.4 Proportional and Lexicographic Allocation Mechanisms 65

66

3 Capacity Allocation with Demand Competition in Asymmetric Market

3.5 Comparison of Three Mechanisms We have investigated uniform, proportional, and lexicographic allocations with regard to how they affect the supplier’s whole pricing decision in a duopoly model with demand competition. An interesting question is which of the three allocations is preferred by the supplier when the total order size exceeds his available capacity. In this section, we compare the performance of the three allocations from the perspective of the supplier. For notational convenience, for any given capacity level, we denote the maximum profit of the supplier obtained from uniform allocation, proportional allocation, and lexicographic allocation–1 (order priority to high-type retailer), and lexicographic allocation–2 (order priority to low-type retailer) by U , P, L 1 , and L 2 , respectively. To elaborate the comparisons of U , P, L 1 , and L 2 , let us first establish a benchmark by considering the case where the two retailers’ market powers are symmetric. Let z 1 = z 2 = z. Comparing the supplier’s profits between uniform and proportional allocation (Tables 3.2 and 3.3), we obtain the results as: (i) If K < z/3, then the ˜∗ /2) supplier’s profits under √ uniform and proportional allocations are s = K∗(z − 3K ∗ ˆs > ˜ ∗s , and ˆ s = K [z − 3( 2 − 1)K )], respectively. It is easy to check that  and  therefore the choice for the supplier; and (ii) If z/3 ≤ √ latter allocation rule is a better ˜ ∗s = z 2 /6 and  ˆ ∗s = K (z − 3K /2) (clearly, K < (2 + 2)z/6, then recall that  ˜ ∗s ); and ˆ ∗s >  proportional allocation still outperforms uniform allocation, since  ∗ ∗ 2 ˆ ˜ (iii) If capacity is sufficiently large, then s = s = z /6, which implies that the two allocation mechanisms are indifferent in allocating capacity from the supplier’s perspective. Hence, we conclude that proportional mechanism is better than uniform allocation, independent of the supplier’s capacity level. Chen et al. (2013) show that lexicographic allocation is better than proportional allocation for the supplier, through earning greater profit, in the symmetric case with z 1 = z 2 and L 1 = L 2 . Hence, we have that U ≤ P ≤ L 2 = L 1 , not affected by the supplier’s capacity level. Now we consider the case when the retailers’ market powers are asymmetric with z 1 > z 2 . Note that we assume z 2 > z 1 /2. Tables 3.2, 3.3 and 3.4 show that from the perspective of the supplier, it is indifferent among the three mechanisms when the capacity level is below the threshold level z 1 − z 2 . This result is intuitive. Within a limited capacity, the two retailers both order large quantities such that the supplier’s capacity is fully sold. Consequently, the supplier maximizes his profit by charging the same wholesale price under all the three mechanisms. However, when K > z 1 − z 2 , the problem becomes more complex. Take uniform and proportional mechanisms, for example. Observing the supplier’s profit functions under the two allocations, since wˆ − > w˜ + , the profits are different only when w is ˜ s = (z 1 + z 2 − 2w)w/3, and  ˆ s = K w. To in the interval [w˜ + , wˆ − ]. Recall that  ˆ ˜ show s < s , it suffices to show (z 1 + z 2 − 2w)/3 < K , which is equivalent to w > (z 1 + z 2 − 3K )/2. When w ∈ [w˜ + , wˆ − ], it is easy to verify that w˜ + > (z 1 + z 2 − 3K )/2. Hence, this implies that proportional mechanism outperforms uniform allocation by generating more profit for the supplier. Equivalently, we may conclude that the supplier can sell more under proportional mechanism than under uniform

3.6 An Arbitrary Number of Retailers

67

mechanism at a given wholesale price in this problem setting. Similarly, it is not difficult to verify that for the supplier, the performance of lexicographic mechanism is as least as good as proportional mechanism. Further, it is interesting to find that, for the supplier, lexicographic allocation that grants order priority to the high-type retailer outperforms the case that grants priority to the low-type retailer. We interpret this finding as follows. In general, as the wholesale price increases, retailers will shrink their order sizes, and consequently the supplier may have excess capacity. To earn more profit, the supplier would like his capacity to be fully sold with higher profit. When w ∈ (0, wˇ i+ ], (i = 1, 2), the retailer with order priority orders the entire capacity and there is no excess capacity. Since wˇ 1+ > wˇ 2+ , the supplier can sell more when priority is given to the high-type retailer. In summary, we have the following result. Theorem 3.2 U ≤ P ≤ L 2 ≤ L 1 , for any capacity level. We note that Lemma 3 and Remark 1 by Liu (2012) indicate the result U ≤ P in Theorem 3.2. To verify our findings and gain further insight, we conduct numerical studies, as shown in the Appendix. Upon examining the results, we have the following observations. First, when the supplier’s capacity is relatively small or sufficiently large, the supplier is indifferent among the three allocation mechanisms. Second, the supplier’s optimal wholesale price is not necessarily decreasing with the capacity level, and the associated profit is not necessarily increasing with the capacity level. Third, lexicographic allocation (especially when priority is given to the high-type retailer) can be evidently more profitable for the supplier than uniform and proportional allocations for any given capacity level and competitive gap. Fourth, the advantage of the superior allocation in each comparison becomes more obvious as downstream demand competition grows.

3.6 An Arbitrary Number of Retailers A key finding in earlier sections is the analytical characterization on the supplier’s preference ranking of capacity allocation rules with duopoly retailers, as in Theorem 3.2: (1) lexicographic with priority to high-type retailer; (2) lexicographic with priority to low-type retailer; (3) proportional; and (4) uniform. An interesting question is whether the finding still stands up with an arbitrary number n of retailers. We precede to answer this question in this section. Suppose there are n(n ≥ 3) retailers with market power vector z=(z 1 , z 2 , . . . , z n ). Without loss of generality, we assume z 1 ≥ z 2 ≥ . . . ≥ z n . We assume that retailer i with order quantity ri receives allocation qi , i = 1, 2, . . . , n. Also, for notational convenience, let r−i = (r1 , r2 , . . . , ri−1 , ri+1 , . . . , rn ), R−i =  nj=1, j=i r j , q−i = (q1 , q2 , . . . , qi−1 , qi+1 , . . . , qn ), and Q −i =  nj=1, j=i q j . Similar to the case with duopoly retailers, from retailer i’s profit function i (ri , r−i ) = (z i − w − ri − R−i )ri , i = 1, 2, . . . , n, we can obtain the equilib-

68

3 Capacity Allocation with Demand Competition in Asymmetric Market

rium order quantity without capacity constraint for each retailer ri∗ = ((n + 1)z i −  nj=1 z j − w)/(n + 1). In this case, every retailer’s order will be satisfied with ideal allocation qi∗ = ri∗ , and the total allocated capacity is Q ∗ = ( nj=1 z j − nw)/(n + 1). In Sect. 3.2, we describe uniform, proportional, and lexicographic allocation rules for the case with duopoly retailers, which can be easily extended to the case with n retailers and readers can refer to Hall and Liu (2010) for more details. Our objective is to find the supplier’s preference of these rules. We next achieve this objective through comparison of total allocated capacity: for any given capacity K and wholesale price w, the more the supplier sells under a mechanism, the better the mechanism performs from the perspective of the supplier. Under all the three allocation mechanisms considered, our previous analysis shows two intuitive results regarding the total allocated capacity with a given wholesale price: (i) if the supplier’s capacity level is sufficiently low, then the total order quantity exceeds the capacity and the total allocated capacity is K ; (ii) if the supplier’s capacity is sufficiently high, then every retailer will order her ideal allocation and the total allocated capacity is Q ∗ . That is, when the supplier’s capacity level is either too low or too high, the three allocation mechanisms perform the same for the supplier. However, when the supplier’s capacity is at a medium level, which can be even sufficient to supply the total ideal allocation, i.e., K ≥ Q ∗ , the three allocation mechanisms provide different incentives for retailers with demand competition to inflate their order quantities to obtain more than ideal allocation. Specifically, under each allocation mechanism, there is a threshold K ∗ such that, (i) if K < K ∗ , then the total allocated capacity is K ; (ii) if K ≥ K ∗ , then the total allocated capacity is Q ∗ . Because of order inflation, we have Q ∗ ≤ K ∗ . Observation that for any given capacity K and wholesale price w, the larger the threshold K ∗ under an allocation mechanism, the more the supplier can sell under the mechanism. Let the thresholds under uniform, proportional and lexicographic mechanisms be K u∗ , K ∗p , and K l∗t , respectively, where t ∈ {1, 2, . . . , n!} denotes one permutation, i.e., priory sequence, of the n retailers. Specifically, let l ∗ denote the priority sequence 1 → 2 → . . . n, i.e., the sequence of retailers in nonincreasing market powers. Theorem 3.3 For any wholesale price w, we have K u∗ ≤ K ∗p ≤ K l∗∗ . Theorem 3.3 means that the lexicographic allocation with priority sequence of retailers in nonincreasing market powers dominates proportional allocation, and proportional allocation dominates uniform allocation, from the perspective of the supplier, for any number of retailers. Analytical results for lexicographic allocation with all sequences are more difficult to obtain since there are n! priority sequences of n retailers that can be used. Next we compare the performance of the three allocation mechanisms through numerical study, in terms of threshold K ∗ . Note that it is not only hard to find closed-form for K ∗ , but also hard to compute K ∗ numerically. We provide in the Appendix a method for computing K ∗ for different mechanisms. To compare all the priority sequences used by lexicographic allocation, we consider n = 3, with (z 1 , z 2 , z 3 ) = (100, 85, 80). There are 3! = 6 priority sequences of the three retailers can be used under lexicographic allocation, denoted by l1 = (80, 85, 100), l2 =

3.6 An Arbitrary Number of Retailers

69

Table 3.5 Thresholds K ∗ under Different Allocations, n = 3 w K u∗ K ∗p K l∗1 K l∗2 K l∗3 5 10 15 20 25 30 35 40 45 50 55

69.20 65.22 61.22 57.20 53.16 49.10 45.01 40.88 36.72 32.51 28.23

75.43 70.88 66.32 61.73 57.13 52.51 47.84 43.14 38.37 33.51 28.51

75.11 70.55 65.99 61.41 56.84 52.25 47.66 43.06 38.44 33.80 29.15

79.16 74.58 70.01 65.42 60.82 56.21 51.59 46.94 42.28 37.57 32.81

75.97 71.30 66.63 61.95 57.28 52.60 47.92 43.23 38.55 33.85 29.15

K l∗4

K l∗5

K l∗6

77.36 72.71 68.03 63.33 58.60 53.81 48.96 44.00 38.86 33.85 29.15

82.15 77.38 72.61 67.82 63.00 58.16 53.27 48.33 43.32 38.17 32.81

82.15 77.38 72.61 67.82 63.00 58.16 53.27 48.33 43.32 38.17 32.81

(80, 100, 85), l3 = (85, 80, 100), l4 = (85, 100, 80), l5 = (100, 80, 85), l6 = (100, 85, 80), respectively. To insure that each retailer’s ideal allocation is positive, i.e., qi∗ = ri∗ = ((n + 1)z i −  nj=1 z j − w)/(n + 1) > 0, i = 1, 2, 3, we consider w ≤ 55. For given wholesale price w = 5, 10, . . . , 55, Table 3.5 summarizes the results of the thresholds K ∗ under different mechanisms. Consistent with Theorem 3.3, we have K ∗p > K u∗ for any w considered. Also, we see K l∗ > K u∗ for any whole sale price considered and any priority sequence of retailers, consistent with Theorem 3.2, theoretical result for the case with two retailers. In addition, we have K l∗1 < K ∗p when w ≤ 40, K l∗1 > K ∗p when w ≥ 45, and K l∗t > K ∗p for any w, for t = 2, . . . , 6. That is, with three or more retailers, the relative performance of lexicographic allocation compared with proportional allocation depends on priority sequence of retailers: lexicographic allocation may outperform (underperform, resp.) proportional allocation if priority is given to retailers with high (low, resp.) market powers. For a clearer view of the relative performance of different mechanisms, we directly compare the allocated capacities in Fig. 3.1. The total allocated capacities under uniform, proportional and lexicographic (with priority sequence l6 ) mechanisms are depicted by lines O-A-D-G (Q ∗u ), O-B-E-G (Q ∗p ), O-C-F-G (Q l∗6 ), respectively. On one hand, under all mechanisms, when the capacity is below the respective threshold K ∗ , the suppliers’ total selling quantity is equal to the available capacity K . On the other hand, when the capacity level exceeds the respective threshold K ∗ , the total selling quantity is equal to Q ∗ . Our numerical studies show that with more retailers, there exist more priority sequences with which proportional mechanism outperforms lexicographic mechanism. For example, with n = 5 with (z 1 , z 2 , z 3 , z 4 , z 5 ) = (100, 99, 98, 97, 96), results with lexicographic allocation with priority sequences l1 = (96, 97, 98, 99, 100), l2 = (97, 96, 98, 99, 100), l3 = (98, 96, 97, 99, 100), l4 = (99, 96, 97, 98, 100), and l5 = (100, 99, 98, 97, 96) are summarized in Table 3.6, where proportional

70

3 Capacity Allocation with Demand Competition in Asymmetric Market

Fig. 3.1 Total allocated capacities under different mechanisms, as capacity changes

Q C

Q∗l6

• B

Q∗p

•

A

Q∗u

•

Q∗

•

•

•

Ku∗

Kp∗

Kl∗6

D

O

E

Table 3.6 Thresholds K ∗ under different allocations, n = 5 w K u∗ K ∗p K l∗1 K l∗2 K l∗3 10 20 30 40 50 60 70 80

75.96 67.52 59.05 50.55 42.01 33.41 24.71 16.01

84.51 75.01 65.49 55.98 46.46 36.94 27.39 17.81

84.40 74.45 65.18 55.63 46.07 36,51 27.00 17.63

84.80 75.09 65.37 55.66 46.07 36.51 27.00 17.63

85.49 75.77 66.06 56.35 46.63 36.92 27.20 17.63

F

G

K

K l∗4

K l∗5

86.16 76.44 66.73 57.01 47.29 37.57 27.85 18.12

86.80 77.09 67.37 57.64 47.92 38.19 28.44 18.67

mechanism outperforms lexicographic mechanism with priority sequences l1 , l2 and l3 . Note that lexicographic mechanism performs better for the supplier when giving higher priority to retailers with higher market power. When there are more retailers, the gap between retailers’ market powers is relatively large and lexicographic mechanism becomes less efficient when giving higher priority to retailers with lower market power, which makes the mechanism outperformed by proportional mechanism with n ≥ 3.

3.7 Concluding Remarks In the practice of production and operations, capacity allocation is an important problem when retailer total order quantity exceeds supplier available capacity. This paper analyzes three capacity allocation mechanisms, uniform, proportional and lex-

3.7 Concluding Remarks

71

icographic allocations, in the presence of demand competition between retailers in a two-echelon decentralized supply chain. We consider the supplier’s wholesale pricing decision together with his choice of allocation mechanisms, for any given capacity level. The pricing decision is important, since with exogenously given scarce capacity and the supplier’s anticipation of retailers’ ordering behavior, it can be more profitable for the supplier to adjust his wholesale price. From a modelling perspective, we consider both horizontal and vertical games between the supplier and two retailers. Specifically, in the horizontal game, the duopoly retailers compete for demand from the same group of customers and the supplier’s capacity simultaneously. Their desired orders are determined by Cournot competition under complete information. In the vertical game, the supplier acts as a leader who first announces his capacity, wholesale price, and an allocation rule with upper bound restriction on order size. Then, the retailers determine their order quantities. Finally, capacity is allocated and market demand is realized. Our analysis focuses particularly on the impact of allocation mechanisms on supplier pricing decisions and retailer order behavior. Our results show that there may exist multiple Nash equilibria as the wholesale price changes under each allocation mechanism considered. Via equilibrium analysis, we identify exact conditions under which the gaming effect is present. Also, in our model, equilibrium orders are guaranteed under all mechanisms considered. This allows us to exactly compare the three mechanisms at any capacity level with regard to the supplier. In the asymmetric case, we show that when the capacity level is either very low or sufficiently high, the supplier is indifferent among the three mechanisms. The result is intuitive. When capacity is very small, retailers order large quantities and the supplier’s capacity is fully sold under all mechanisms considered. As a result, the supplier can sell the total capacity K by charging the same reasonably high wholesale price under all three mechanisms. On the other hand, when the capacity level is sufficiently high, each retailer would like to order her ideal order size as if capacity is unlimited. However, for an intermediate range in capacity level, the supplier can sell more from lexicographic allocation than from uniform or proportional allocation, especially when giving order priority to the high-type retailer. Our further numerical studies verify this finding. We find that the advantage of lexicographic allocation becomes more obvious as the low-type retailer’s market power is closers to that of the high-type retailer. For the case with three or more retailers, we analytically prove that proportional allocation dominates uniform allocation and lexicographic allocation with priority sequence of retailers in nonincreasing market powers dominates proportional allocation from the perspective of the supplier. Also, we numerically show that lexicographic allocation with any priority sequence of retailers dominates uniform allocation for any number of retailers. With three or more retailers, the relative performance of lexicographic allocation compared with proportional allocation depends on priority sequence of retailers: lexicographic allocation may outperform (underperform,

72

3 Capacity Allocation with Demand Competition in Asymmetric Market

respectively) proportional allocation if priority is given to retailers with higher (lower, respectively) market powers. We hope that these findings can provide a reference for suppliers in their selection of allocation mechanism and for their efficient pricing decisions, accordingly. In a two-echelon supply chain with a single supplier and oligopoly or more retailers, this paper examines the impact of capacity allocation mechanisms on supplier pricing decisions and retailer ordering behavior. This model provides a foundation for future studies, for example, extending the supply chain to be more general, e.g., some retailers have an alternative supplier or multiple suppliers with multiple retailers. Also, it is important to study how the supplier plans production if the down stream retailers face uncertainty of market demand. Further, incorporation of other allocation mechanisms, different kinds of market demand competition, and risk aversion for supply chain members, are interesting research questions.

Appendix Proof of Lemma 3.1 Because the ordering decisions are symmetric between the two retailers, to obtain retailer i’s best response function ri (r j ), we only need to consider retailer 1’s optimal problem: ˜ 1 (r2 , w) = max 



r1 ∈[0,K ]

max

r1 >K −r2 ;r1 ≤r2 ;r1 ≥ K2

˜ 11 , max 

r1 ≤K −r2

˜ 2,2  1 ,

max

r1 >K −r2 ;r1 ≤r2 ;r1 < K2

max

r1 >K −r2 ,r1 >r2 ;r2 < K2

˜ 3,1  1 ,

˜ 2,1  1 , max

r1 >K −r2 ,r1 >r2 ;r2 ≥ K2

 3,2 ˜ 1 ,

˜ 11 = (z 1 − w − r1 − r2 ) · r1 ;  ˜ 2,1 = (z 1 − w − K ) · r1 ;  ˜ 2,2 = (z 1 − w − K ) · where  1 1 ˜ 3,1 = (z 1 − w − K ) · (K − r2 );  ˜ 3,2 = (z 1 − w − K ) ·  1 1

K ; 2

K . 2

The five subproblems can be solved as follows: ⎧

z −w−r ⎪ if 1 2 2 < 0, ⎨ 0, ˜ 1 = z 1 −w−r2 , if 0 ≤ z 1 −w−r2 ≤ K − r ,  r˜11∗ = arg max 2 2 2 ⎪ r1 ∈[0,K −r2 ] 1 ⎩ z −w−r if 1 2 2 > K − r2 . K − r2 , ⎧ z −w−r ⎪ if 1 2 2 < 0, ⎪ ⎨ 0, 2 1∗ 1 (z −w−r ) z −w−r ˜ ˜ = 1 2 ,   max if 0 ≤ 1 2 2 ≤ K − r2 , 1 = r ∈[0,K ⎪ 4 −r2 ] 1 ⎪ 1 z 1 −w−r2 ⎩ > K − r2 . (z 1 − w − K )(K − r2 ), if 2 ⎧ ⎪ (K − r , K ], if r2 < K2 , w ≤ z 1 − K , 2  2,1 2,2 3,1 3,2  ⎨ K ˜ ˜ ˜ , ˜  = [ 2 , K ], r˜123∗ = arg max , , if r2 ≥ K2 , w ≤ z 1 − K , 1 1 1 1 ⎪ r1 ∈(K −r2 ,K ] ⎩ 0, if w > z 1 − K .

(A.1)

(A.2)

(A.3)

Appendix

73

˜ 23∗ =  1

max

r1 ∈(K −r2 ,K ]

 2,1 2,2 3,1 3,2  ˜ ˜ ˜ ˜  1 , 1 , 1 , 1

⎧ K ⎪ ⎨ (z 1 − w − K )(K − r2 ), if r2 < 2 , w ≤ z 1 − K , = (z 1 − w − K ) K2 , if r2 ≥ K2 , w ≤ z 1 − K , ⎪ ⎩ 0, if w > z 1 − K .

(A.4)

(i) Consider the case when w ∈ (0, z 1 − 2K ]. Since w ≤ z 1 − 2K , it follows that (z 1 − w − r2 )/2 > K − r2 , from Eqs. (A.1) and (A.2), we have r˜11∗ ∈ (K − r2 , K ] ˜ 1∗ and  1 = (z 1 − w − K )(K − r 2 ). Furthermore, since w ≤ z 1 − 2K < z 1 − K , from Eqs. (A.3) and (A.4), we have (a) if r2 < K /2, then r˜123∗ ∈ (K − r2 , K ] 23∗ ˜ 23∗ ∈ [K /2, K ] with with  1 = (z 1 − w − K )(K − r 2 ); (b) if r 2 ≥ K /2, then r1 23∗ ˜ 1 = (z 1 − w − K )K /2. Note that (z 1 − w − K )(K − r2 ) ≤ (z 1 − w − K )K /2  due to r2 ≥ K /2, thus Lemma 3.1(i) is easily verified. (ii) To prove the remaining results of Lemma 3.1, let us first establish some useful results in the following lemma. √ Lemma A1 Define α˜ 1 = z 1 − w − 2K (z 1 − w − K ), β = z 1 − w + √ 2K (z 1 − w − K ), if w ∈ (z 1 − 3K /2, z 1 − K ], then α˜ 1 and β are real numbers with K /2 < α˜ 1 < 2K + w − z 1 ≤ K ≤ β. Proof: Given any w ∈ (z 1 − 3K /2, z 1 − K ], we have 2K + w − z 1 − K /2 = w − (z 1 − 3K /2) < 0, thus 2K √ + w − z 1 > K /2. To show α˜ 1 > K /2 is equivalent to show (z 1 − w − K /2) − 2K (z 1 − w − K ) > 0, or [2(z 1 − w) − K ]2 > 8K (z 1 − w − K ). We can derive that [2(z 1 − w) − 3K ]2 > 0, and thus we have α˜ 1 > K /2. Similarly, √ to show α˜ 1 < 2K + w − z 1 − K /2, which√is equivalent to 2(z 1 − w − K ) − 2K √or 2(z 1 − w − K ) < 2K (z 1 − w − K ), it suffices √ (z 1 − w − K ) < 0, to show 2(z 1 − w − K ) < K . Since z 1 − w − 3K /2 > 0, then we have α˜ 1 < to verify α˜ 1 < K√is the equivalent √ to show α˜ 1 − K = 2K + w − z 1 . Furthermore, √ z 1 − w − K < 2K . Since z 1 − z 1 − w − K − 2K (z 1 − w − K ) < 0, or 3K − w α˜ 1 , then (z 1 − w − r2 )2 /4 < (z 1 − w − K )K /2. Proof: Through simple algebra, we obtain (z 1 − w − r2 )2 /4 − (z 1 − w − K )K /2 = 1 (r − α˜ 1 )(r2 − β), thus the lemma follows from Lemma A1. 4 2 Now we discuss the scenario as w ∈ (z 1 − 2K , z 1 − 3K /2]. Consider three cases: (a) if r2 ≤ 2K − z 1 + w, or equivalently (z 1 − w − r2 )/2 ≤ K − r2 , then it fol2 ˜ 1∗ lows that r11∗ = (z 1 − w − r2 )/2 and  1 = (z 1 − w − r 2 ) /4. On the other hand, 23∗ ˜ 23∗ if r2 ≤ 2K − z 1 + w ≤ K /2, then we have r1 ∈ (K − r2 , K ] and  1 = (z 1 − 1∗ 23∗ 1∗ 23∗ ˜ 1 with  ˜ 1 , we have  ˜1 − ˜ 1 = (r2 + z 1 − w − K )(K − r2 ). Comparing  w − 2K )2 ≥ 0. Hence, we can derive the results as follows: (a) if r2 ≤ 2K − z 1 + w, 2 ˜ 1 (r2 , w) =  ˜ 1∗ then r1 (r2 ) = r11∗ = (z 1 − w − r2 )/2 and  1 = (z 1 − w − r 2 ) /4; (b)

74

3 Capacity Allocation with Demand Competition in Asymmetric Market

Fig. 3.2 w ∈ (0, z 2 − 2K ]

r1 K

K 2

K 2

K

r2

if 2K − z 1 + w < r2 ≤ K /2, or equivalently (z 1 − w − r2 )/2 > K − r2 , it fol˜ 1∗ ˜ 23∗ lows that r11∗ = r123∗ ∈ (K − r2 , K ] and  1 = 1 = (z 1 − w − K )(K − r 2 ); (c) if r2 > K /2, then (z 1 − w − K )(K − r2 ) < (z 1 − w − K )K /2, and r1 (r2 ) = r123∗ ∈ ˜ 23∗ ˜ 1 (r2 , w) =  [K /2, K ] with  1 = (z 1 − w − K )K /2. Using the same procedure, we can verify Lemma 3.1(iii). ˜ 23∗ (iv) w ∈ (z 1 − K , z 1 ). Since w > z 1 − K , from Eq. A.4, we know that  1 = 0, so 1∗ ˜ 1 from Eqs. (A.1) and (A.2). Consider two cases: (a) if we only need to consider  2 ≤ K − r2 , and thus r1 (r2 ) = r11∗ = (z 1 − w − r2 )/2, r2 ≤ z 1 − w, then 0 ≤ z1 −w−r 2 z 1 −w−r2 2 ˜ 1 (r2 , w) =  ˜ 1∗ and  < 0, and 1 = (z 1 − w − r 2 ) /4; (b) if r 2 > z 1 − w, then 2 ˜ 1 (r2 , w) =  ˜ 1∗ thus r1 (r2 ) = r11∗ = 0 with  = 0. 1 Proof of Theorem 3.1 To find Nash equilibrium outcome resulting from the ordering quantity game, we draw the response curves from Lemma 3.1 for both retailers. First we compartmentalize the following feasible intervals with different capacity constraints. There are five possible scenarios: (I) if z 1 − z 2 < K /2, i.e., K > 2(z 1 − z 2 ), we have z 2 − 2K < z 1 − 2K < z 2 − 3K /2 < z 1 − 3K /2 < z 2 − K < z 1 − K < z 2 < z 1 ; (II) if K /2 ≤ z 1 − z 2 < K , i.e., z 1 − z 2 < K ≤ 2(z 1 − z 2 )], then we have z 2 − 2K ≤ z 2 − 3K /2 < z 1 − 2K < z 2 − K < z 1 − 3K /2 ≤ z 1 − K ≤ z 2 ≤ z 1 ; (III) if K ≤ z 1 − z 2 < 3K /2, i.e., 2(z 1 − z 2 )/3 < K ≤ z 1 − z 2 , then we have z 2 − 2K ≤ z 2 − 3K /2 ≤ z 2 − K ≤ z 1 − 2K ≤ z 1 − 3K /2 < z 2 ≤ z 1 − K < z 1 ; (IV) if 3K /2 ≤ z 1 − z 2 < 2K , i.e., (z 1 − z 2 )/2 < K ≤ 2(z 1 − z 2 )/3, then we have z 2 − 2K < z 2 − 3K /2 < z 2 − K < z 1 − 2K < z 2 ≤ z 1 − 3K /2 < z 1 − K < z 1 ; (V) if z 1 − z 2 ≥ 2K , i.e., K ≤ (z 1 − z 2 )/2, then we have z 2 − 2K < z 2 − 3K /2 < z 2 − K < z 2 ≤ z 1 − 2K < z 1 − 3K /2 < z 1 − K < z 1 . Next we analyze Nash equilibrium in each scenario. (I) K ∈ (2(z 1 − z 2 ), +∞): (i) w ∈ (0, z 2 − 2K ). In this case, See Fig. 3.2, it is clear that any point in [K /2, K ] × [K /2, K ] is a Nash equilibrium.

Appendix

75

Fig. 3.3 w ∈ (z 2 − 2K , z 1 − 2K ]

r1 K

K 2 2K − z2 + w

Fig. 3.4 w ∈ (z 1 − 2K , z 2 − 3K /2]

K 2

z2 −w 2 K

K 2

z2 −w 2 K

r2

r1 K

z1 −w 2

K 2 2K − z2 + w

2K − z1 + w

r2

(ii) w ∈ (z 2 − 2K , z 1 − 2K ]. Though it adds a new bound line r1 = 2K − z 2 + w as r1 ≤ 2K − z 2 + w, and the response curve for retailer 2 is r2 (r1 ) = (z 2 − w − r1 )/2, thus the equilibrium order pair is the same as (i) (Fig. 3.3). (iii) w ∈ (z 1 − 2K , z 2 − 3K /2]. See Fig. 3.4, any point in [K /2, K ] × [K /2, K ] is in equilibrium. (iv) w ∈ (z 2 − 3K /2, z 1 − 3K /2]. As showed in Fig. 3.5, any point in [α˜ 2 , K ] × [K /2, K ] is a Nash equilibrium. Next we determine whether the point ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is in equilibrium. It suffices to verify the conditions: (a) (2z 1 − z 2 − w)/3 ≤ α˜ 2 ; (b) (2z 2 − z 1 − w)/3 ≤ 2K − z 1 + w. Note from (a), it isequivalent to w ≤ w˜ − or w˜ + ≤ w ≤ 2z 2 − z 1 , where w˜ ± = 2z 2 − z 1 − 9K /4 ± 3 K 2 + 8K (z 1 − z 2 )/4. Recall that w ∈ (z 2 − 3K /2, z 1 − 3K /2], together with w˜ − < z 2 − 3K /2 and z 2 − 3K /2 < w˜ + < z 1 − 3K /2 < 2z 2 − z 1 , simple algebra shows that condition (a) always holds if w ∈ (w˜ + , z 1 − 3K /2]. On the other hand, for condition (b), it satisfies w ≥ (z 1 + z 2 − 3K )/2 if it holds. Note that

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3 Capacity Allocation with Demand Competition in Asymmetric Market

Fig. 3.5 w ∈ (z 2 − 3K /2, z 1 − 3K /2]

r1 K

z1 −w 2 2K − z2 + w

α ˜2 K 2

2K − z1 + w

Fig. 3.6 w ∈ (z 1 − 3K /2, z 2 − K ]

K 2

z2 −w 2

K

r2

r1 K α ˜2 z1 −w 2 K 2

2z1 −z2 −w 3

2z2 −z1 −w 3

K z2 −w 2 2

α ˜1 K

r2

(z 1 + z 2 − 3K )/2 < w˜ + , as a result, if w ∈ (z 2 − 3K /2, w˜ + ], then any point in [α˜ 2 , K ] × [K /2, K ] is a Nash equilibrium. Otherwise if w ∈ (w˜ + , z 1 − 3K /2], then ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a equilibrium and any point in [α˜ 2 , K ] × [K /2, K ] is also a Nash equilibrium. Moreover, when w ∈ (w˜ + , z 1 − 3K /2], in ˜ 12 ((2z 2 − z 1 − w)/3, w) = (2z 2 − z 1 − w)2 /9 if this case, retailer 2’s profit is  ˜ 22 (K /2, w) = the Nash equilibrium is ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3), and  (z 2 − w − K )K /2 if the Nash equilibrium is any point in [α˜ 2 , K ] × [K /2, K ], ˜ 22 (K /2, w), thus the former ˜ 12 ((2z 1 − z 2 − w)/3, w) ≥  it is easy to verify that  equilibrium dominates the latter one by gaining more profits for retailer 2. Similarly, we can prove that the equilibrium ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) dominates the other equilibrium by generating more profits for retailer 1. Hence, ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a dominant strategy for both retailers.

Appendix

77

Fig. 3.7 w ∈ (z 2 − K , z 1 − K ]

r1 K z2 − w

z1 −w 2 K 2

2z1 −z2 −w 3

2z2 −z1 −w 3

Fig. 3.8 w ∈ (z 1 − K , 2z 2 − z 1 ]

K z2 −w 2 2

α ˜1 K

r2

r1 K z2 − w z1 −w 2

2z1 −z2 −w 3

2z2 −z1 −w 3

z2 −w 2

z1 − w K

r2

(v) w ∈ (z 1 − 3K /2, z 2 − K ]. In this case (Fig. 3.6), it implies that K /2 ≤ (z 1 − w)/2 ≤ K and K /2 ≤ (z 2 − w)/2 ≤ K . Furthermore, if r2 = K /2, then r1 (K /2) = (z 1 − w − K /2)/2, it is easy to check that r1 (K /2) ≤ K /2. Similarly, if r1 = K /2, it is easy to check that r2 (K /2) ≤ K /2, then there must exist a crossing point ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) between the two response curves. Consequently, ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a Nash equilibrium and any point in [α˜ 2 , K ] × [α˜ 1 , K ] is also a Nash equilibrium. Similar to (iv), we can check that the former equilibrium dominates the latter one for both retailers by generating more profits if w ∈ (z 1 − 3K /2, z 2 − K ]. (vi) w ∈ (z 2 − K , z 1 − K ]. From the response curves in Fig. 3.7, there exists a unique equilibrium ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3).

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3 Capacity Allocation with Demand Competition in Asymmetric Market

Fig. 3.9 w ∈ (2z 2 − z 1 , z 2 )

r1 K

z1 −w 2

z2 − w

z2 −w 2 Fig. 3.10 w ∈ (z 2 , z 1 )

z1 − w K

r2

r1 K

z1 −w 2

z1 − w K

r2

(vii) w ∈ (z 1 − K , z 2 ]. See Fig. 3.8, if w ∈ (z 1 − K , 2z 2 − z 1 ], we have (2z 1 − z 2 − w)/3 ≤ z 2 − w and (2z 2 − z 1 − w)/3 ≤ z 1 − w, thus it is clear that there exists a unique equilibrium ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3). Otherwise, the equilibrium is ((z 1 − w)/2, 0) if w ∈ (2z 2 − z 1 , z 2 ] (Fig. 3.9). (viii) w ∈ (z 2 , z 1 ). In this special case, the wholesale price is so high that retailer 2 would be driven out of the market. Therefore, see Fig. 3.10, the unique equilibrium is ((z 1 − w)/2, 0). Moreover, we find that when z 1 − z 2 < K ≤ 2(z 1 − z 2 ), the equilibrium order pair is identical to the case as K > 2(z 1 − z 2 ). Similarly, we use the same approach to analyze other scenarios when K lies (0, z 1 − z 2 ]. Therefore, we can summarize the results that show in Theorem 3.1. Proposition 3.1 Under uniform allocation mechanism with asymmetric retailers, the supplier’s optimal wholesale price w˜ ∗ and the associated maximum profit denoted

Appendix

79

˜ ∗s , are as follows: by  (I) K ≤ z 1 − z 2 : (i) if z 2 ≤ 3z 1 /4, when K ≤ z 1 /4, the supplier’s optimal wholesale price is w˜ ∗ = ˜ ∗s = K (z 1 − 2K ); when z 1 − 2K and the maximum profit for the supplier is  z 1 /4 < K ≤ z 1 − z 2 , the supplier’s optimal wholesale price is w˜ ∗ = z 1 /2 and the ˜ ∗s = z 12 /8; maximum profit for the supplier is  (ii) if z 2 > 3z 1 /4, when K ≤ z 1 − z 2 , then the supplier’s optimal wholesale price is ˜ ∗s = K (z 1 − 2K ). w˜ ∗ = z 1 − 2K and the maximum profit for the supplier is  (II) K > z 1 − z 2 : (i) if z 2 ≤ 5z 1 /7, when K > z 1 − z 2 , the supplier’s optimal wholesale price is w˜ ∗ = ˜ ∗s and the maximum profit for the supplier is  ˜ ∗s = max{K w˜ + , z 12 /8}; arg max  w∈{w˜ + ,z 1 /2}

(ii) if 5z 1 /7 < z 2 ≤ 3z 1 /4, when z 1 − z 2 < K ≤ K˜ + , the supplier’s optimal whole˜ ∗s and the maximum profit for the supplier is  ˜ ∗s = sale price is w˜ ∗ = arg max  +

, z 12 /8}.

max{K w˜ arg max

w∈{w˜ + ,z 1 /2}

Otherwise, the supplier’s optimal wholesale price is w˜ ∗ = ˜ ∗s = max{K w˜ + , ˜ ∗s and the maximum profit for the supplier is 

w∈{w˜ + ,(z 1 +z 2 )/4,z 1 /2} (z 1 + z 2 )2 /24, z 12 /8}.

(iii) if z 2 > 3z 1 /4, when z 1 − z 2 < K ≤ K˜ + , the supplier’s optimal wholesale ˜ ∗s = ˜ ∗s and the maximum profit for the supplier is  price is w˜ ∗ = arg max  w∈{w˜ + ,2z 2 −z 1 }

max{K w˜ + , (2z 2 − z 1 )(z 1 − z 2 )}. Otherwise, the supplier’s optimal wholesale price ˜ ∗s and the maximum profit for the supplier is  ˜ ∗s =  arg max is w˜ ∗ = w∈{w˜ + ,(z 1 +z 2 )/4,2z 2 −z 1 } max{K w˜ + , (z 1 + z 2 )2 /24, (2z 2

− z 1 )(z 1 − z 2 )}.

Proof Consider the following two cases. Case (I): K ≤ z 1 − z 2 : ˜ 1s = K (z 1 − 2K ); (i) w ∈ (0, z 1 − 2K ]. It is easy to show that w˜ ∗ = z 1 − 2K with  (ii) w ∈ (z 1 − 2K , z 1 ). (a) If z 1 − 2K < z 1 /2, which is equivalent to K > z 1 /4, ˜ 2s = z 12 /8. (b) if K ≤ z 1 /4, then w˜ ∗ = z 1 − 2K with  ˜ 2s = then w˜ ∗ = z 1 /2 with  2 2 2 ˜ s = K (z 1 − 2K ). Since z 1 /8 − K (z 1 − 2K ) = (8K − z 1 ) /8 ≥ 0, it follows that  z 12 /8 ≥ K (z 1 − 2K ). Case (II): K > z 1 − z 2 : ˜ 3s = K w˜ + ; (i) w ∈ (0, w˜ + ]. It is easy to show that w˜ ∗ = w˜ + with  + (ii) w ∈ (w˜ , 2z 2 − z 1 ]. (a) Consider the scenario where 2z 2 − z 1 > (z 1 + z 2 )/4, i.e., z 2 > 5z 1 /7. If w˜ + < (z 1 + z 2 )/4, which is equivalent to K ≥ K˜ + , then w˜ ∗ = ˜ 4s = (z 1 + z 2 )2 /24. If w˜ + ≥ (z 1 + z 2 )/4, then w˜ ∗ = w˜ + with (z 1 + z 2 )/4 with  4 + ˜ s =(z 1 + z 2 − w˜ )w˜ + /3. (b) Consider the scenario where 2z 2 − z 1 ≤ (z 1 + z 2 )/4,  ˜ 4s = (2z 2 − z 1 )(z 1 − i.e., z 2 ≤ 5z 1 /7. It is easy to verify that w˜ ∗ = 2z 2 − z 1 with  z 2 ). (iii) w ∈ (2z 2 − z 1 , z 1 ). (a) If 2z 2 − z 1 ≤ z 1 /2, i.e., z 2 ≤ 3z 1 /4, then it follows that ˜ 5s = z 12 /8. (b) If 2z 2 − z 1 > z 1 /2, i.e., z 2 > 3z 1 /4, then it follows w˜ ∗ = z 1 /2 with  ˜ 5s = (2z 2 − z 1 )(z 1 − z 2 ). that w˜ ∗ = 2z 2 − z 1 with  The proposition directly follows the above discussions.

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3 Capacity Allocation with Demand Competition in Asymmetric Market

Lemma 3.2 Let rˆi (r j ) be retailer i’s best response order quantity given  retailer j’s order (i, j = 1, 2 and i = j). Define αˆ i = K + z i − w −  √ (z i − w − K )(z i − w + 7K ) /2; then we have ˆ i (r j , w) = K 2 (z i − K − w)/ (i) w ∈ (0, z i − 2K ]: for any r j , rˆi (r j ) = K , and  (K + r j ). (ii) w ∈ (z i − 2K , z i − K ]: ˆ i (r j , w) = (z i − w − r j )2 /4; if r j ≤ αˆ i , then rˆi (r j ) = (z i − w − r j )/2 with  ˆ i (r j , w) = K 2 (z i − w − K )/(K + r j ). if r j > αˆ i , then rˆi (r j ) = K with  (iii) w ∈ (z i − K , z i ): ˆ i (r j , w) = (z i − w − r j )2 /4; if r j ≤ z i − w, then rˆi (r j ) = (z i − w − r j )/2 with  ˆ i (r j , w) = 0. if r j > z i − w, then rˆi (r j ) = 0 with  The proof of Lemma 3.2 is similar to the symmetric case (retailers have the same market power) of Chen et al. (2013), and thus here we omit the details. Note that by involving asymmetric market powers, each retailer’s best response order size is not only affected by the wholesale price but also by the other retailer’s order quantity. Take retailer 1 for example. We interpret Lemma 3.2 as follows. First, consider the case where the wholesale price is very low; i.e., when w ∈ (0, z i − 2K ]. Lemma 3.2(i) suggests that no matter what quantity the other retailer orders, retailer 1 will order K to maximize her profit. This is because, when w ≤ z 1 − 2K , i.e., K ≤ (z 1 − w)/2, if the capacity level is relatively low, then retailer 1 would order as much as possible. Second, as Lemma 3.2(iii) states, when the wholesale price is very high, each retailer will not order much (≤ K ): (a) retailer 1 may not order when z 1 − w − K < 0 under the condition that retailer 2 orders more than z 2 − w; (b) if retailer 2’s order size is no more than z 1 − w, which means that the remaining capacity is ample for retailer 1, then the best response for retailer 1 is the same as the case without capacity constraint. Third, consider the case where the wholesale price is in the intermediate range, i.e., w ∈ [z 1 − 2K , z 1 ). Observe from the second part of Lemma 3.2(ii), if retailer 2 orders sufficiently high (> αˆ i ), then the remaining capacity is scarce for retailer 1. Since the profit margin for retailer 1 remains z 1 − w − K ≥ 0, it follows that her profit is increasing in her order size and thus ordering the maximum capacity K is her best response. If retailer 2 orders no more than αˆ i , then the result is consistent with the first part of Lemma 3.2(iii). Proposition 3.2 Let (ˆr1∗ , rˆ2∗ ) be an equilibrium order pair under proportional mech z 1 +z 2 3 − 2 anism. Define wˆ = 3K + 2 − 2 8K + (z 1 − z 2 )2 , then we have (I) K ≤ z 1 − z 2 : (i) if w ∈ (0, z 2 − K ], then there is a unique Nash equilibrium (ˆr1∗ , rˆ2∗ ) = (K , K ); (ii) if w ∈ (z 2 − K , z 1 − 2K ], then there is a unique Nash equilibrium (ˆr1∗ , rˆ2∗ ) = (K , 0); (iii) if w ∈ (z 1 − 2K , z 1 ), then there is a unique Nash equilibrium (ˆr1∗ , rˆ2∗ ) = ((z 1 − w)/2, 0). (II) K > z 1 − z 2 :

Appendix Fig. 3.11 0 < w ≤ z 2 − 2K

81

r1

K

0

K

r2

(i) if w ∈ (0, wˆ − ], then there is a unique Nash equilibrium (ˆr1∗ , rˆ2∗ ) = (K , K ); (ii) if w ∈ (wˆ − , z 2 − K ], then there exists two Nash equilibrium (K , K ) and ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3). Furthermore, the latter equilibrium dominates the former one by generating more profits for both retailers. (iii) if w ∈ (z 2 − K , 2z 2 − z 1 ], then there is a unique Nash equilibrium (ˆr1∗ , rˆ2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3); (iv) if w ∈ (2z 2 − z 1 , z 1 ), then there is a unique Nash equilibrium (ˆr1∗ , rˆ2∗ ) = (qˆ1∗ , qˆ2∗ ) = ((z 1 − w)/2, 0). Proof Before the equilibrium analysis, from Lemma 3.2, we also obtain some critical values of w, i.e., z 1 − 2K , z 1 − K , z 2 − 2K , z 2 − K , z 1 , z 2 . Using simple algebraic calculus, it follows that: (I) when K > z 1 − z 2 , we have z 2 − 2K < z 1 − 2K ≤ z 2 − K < z 1 − K < z 2 < z 1 ; (II) when (z 1 − z 2 )/2 < K ≤ z 1 − z 2 , we have z 2 − 2K < z 2 − K ≤ z 1 − 2K < z 2 ≤ z 1 − 2K < z 2 ≤ z 1 − K ≤ z 1 ; and (III) when K ≤ (z 1 − z 2 )/2, we have z 2 − 2K ≤ z 2 − K < z 1 − 2K ≤ z 2 < z 1 − K ≤ z 1 . We first discuss the case  when K ∈ (z 1 − z 2 , +∞). (i) w ∈ (0, z 2 − 2K ] (z 2 − 2K , z 1 − 2K ]. See the curves of response functions pictured in Figs. 3.11 and 3.12, it is easy to see that (K , K ) is the unique equilibrium. (ii) w ∈ (z 1 − 2K , z 2 − K ]. First let us introduce a critical value wˆ − . From Lemma 3.1, the sufficient and necessary conditions for the ideal equilibrium order pair ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) are (2z 1 − z 2 − w)/3 ≤ αˆ 2 and (2z 2 − z 1 − w)/3≤ αˆ 1 , which is equivalent to satisfy wˆ − ≤ w ≤ wˆ + , where wˆ ± = 3K + z 1 +z 2 − 23 8K 2 + (z 1 − z 2 )2 . Due to wˆ + > z 1 and the constraint w < z 1 , there2 fore, (2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is in equilibrium if and only if w ≥ wˆ − . Recall that (K , K ) is in equilibrium as w ∈ (z 1 − 2K , z 2 − K ], then we have the following results: (a) if w ∈ (z 1 − 2K , wˆ − ], there exists a unique Nash equilibrium (K , K ); (b) otherwise, (K , K ) and ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) are both in equilibrium. Figs. 3.13 and 3.14 illustrate the above results, respectively. Similar to the proof of Theorem 3.1, we can show that ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) dominates (K , K ) by generating more profits for both retailers.

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3 Capacity Allocation with Demand Competition in Asymmetric Market

Fig. 3.12 z 2 − 2K < w ≤ z 1 − 2K

r1 z2 − w K

α ˆ2 z2 −w 2

0 Fig. 3.13 z 1 − 2K < w ≤ wˆ −

r2

K

r1 z2 − w K z1 −w 2

α2 0 Fig. 3.14 wˆ − < w < z2 − K

α1

z2 −w 2

K

z1 − w

α ˆ 1 z2 −w 2

K

z1 − w

r2

r1 z2 − w K z1 −w 2 α ˆ2

0

r2

Appendix

83

Fig. 3.15 z 2 − K < w ≤ 2z 2 − z 1

r1 K z2 − w z1 −w 2

0 Fig. 3.16 2z 2 − z 1 < w ≤ z1 − K

z2 −w 2

α ˆ1

K z1 − w

r2

r1 K z1 −w 2

z2 − w

0

z2 −w 2

α ˆ1

K

z1 − w

r2

(iii) w ∈ (z 2 − K , z 1 − K ]. In this scenario, if the order pair ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is in equilibrium, it needs to satisfy two conditions: (2z 1 − z 2 − w) ≤ (z 1 − w)/2 and (2z 2 − z 1 − w)/3 ≤ (z 2 − w)/2. And the conditions is equivalent to w ≤ 2z 2 − z 1 . Together with the constraint w ∈ (z 2 − K , z 1 − K ], we obtain the results as follows: (a) when z 1 − z 2 < K ≤ 2(z 1 − z 2 ), if w ∈ (z 2 − K , 2z 2 − z 1 ], there exist a unique Nash equilibrium ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3); otherwise, the unique equilibrium is ((z 1 − w)/2, 0); (b) when K > 2(z 1 − z 2 ), z 1 − K < 2z 1 − z 2 , then there exists a unique Nash equilibrium ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3). See Figs. 3.15 and 3.16, respectively. Specifically, the figure of (b) is the same as the case with (a) when w ∈ (z 2 − K , 2z 2 − z 1 ]. (iv) w ∈ (z 1 − K , z 2 ]. As pictured in Figs. 3.17 and 3.18, we have: (a) when z 1 − z 2 < K ≤ 2(z 1 − z 2 ), there exists a unique Nash equilibrium ((z 1 − w)/2, 0); (b) when K > 2(z 1 − z 2 ), if w ∈ (z 1 − K , 2z 2 − z 1 ], then there exists a unique Nash equilibrium ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3), otherwise, the unique equilibrium is ((z 1 − w)/2, 0). Specifically, the figure of scenario (a) is the same as the case (b) when w ∈ (2z 2 − z 1 , z 2 ).

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3 Capacity Allocation with Demand Competition in Asymmetric Market

Fig. 3.17 w ∈ (z 1 − K , 2z 2 − z 1 ]

r1 K z2 − w z1 −w 2

2z1 −z2 −w 3

2z2 −z1 −w 3

Fig. 3.18 w ∈ (2z 2 − z 1 , z 2 ]

z2 −w 2

z1 − w K

r2

r1 K

z1 −w 2

z2 − w

z2 −w 2

z1 − w K

r2

(v) w ∈ (z 2 , z 1 ). It is the same as Fig. 3.10, in which the low type retailer is driven out of the supply chain, then there is a unique Nash equilibrium ((z 1 − w)/2, 0). Based on the above analysis, we can obtain the equilibrium outcome when K > z 1 − z 2 in Theorem 3.2. Similarly, the remaining results follow from the same approach. As the step is similar to the case we have analysed, here we omit the details. We interpret Proposition 3.2 in conjunction with the impact of wholesale price w on order pairs in equilibrium, for any given capacity level satisfying 0 < K ≤ z 1 − z 2 . When the wholesale price is very low, it is easy to check that the profit margin for each retailer is z i − w − K ≥ 0, (i = 1, 2). Hence, both retailers order as much as possible (by ordering K as allowed). Therefore, there exists a unique Nash equilibrium order pair (K , K ). On the other hand, when the wholesale price is sufficiently high, retailer 2 may be driven out of the game due to her lower market power; consequently, the equilibrium order quantity is always zero for retailer 2 in this case. If capacity exceeds z 1 − z 2 , then we drive the equilibrium order pairs in four scenarios. First, consider the scenario with w ∈ (0, wˆ − ]. Since 2z 2 − z 1 − 3K /2
K ≥ z1 −w . (i)2K , (ii)K , and (iii) z1 −w 2 2 This is consistent with the intuition that total order quantity is decreasing with wholesale price. This implies that retailers are inclined to order more when the supplier offers a lower wholesale price, given fixed capacity level. Moreover, the allocated as the wholesale price crosses the threshold z 1 − 2K . capacity drops from K to z1 −w 2 Further, it is easy to confirm that when the capacity level satisfies K > z 1 − z 2 , if the wholesale price exceeds the threshold wˆ − , then total allocated capacity drops from K to values strictly less than K , since K > (z 1 + z 2 − 2w)/3 > (z 1 − w)/2. However, it is not obvious which pricing option is most profitable for the supplier. Proposition 3.3 First, suppose order priority is given to the high-type retailer order pair under lexicographic allocation. (retailer 1). Let (ˇr1∗ , rˇ2∗ ) be an equilibrium √ wˇ 1+ = 2z 1 − z 2 − 29 K + 23 Define αˇ 1 = z 1 − w − 4K (z 1 − w − K ),  2 5K − 4K (z 1 − z 2 ). We have (I) K ≤ z 1 − z 2 : (i) if w ∈ (0, z 2 − K ], then any point in {K } × [0, K ] is a Nash equilibrium; (ii) if w ∈ (z 2 − K , z 1 − 2K ], then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = (K , 0); (iii) if w ∈ (z 1 − 2K , z 1 ), then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = ((z 1 − w)/2, 0). (II) K > z 1 − z 2 : (i) if w ∈ (0, z 1 − 2K ], then any point in {K } × [0, K ] is a Nash equilibrium; (ii) if w ∈ (z 1 − 2K , wˇ 1+ ], then any point in {K } × [αˇ 1 , K ] is a Nash equilibrium; (iii) if w ∈ (wˇ 1+ , z 2 − K ], then ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a Nash equilibrium and any point in {K } × [αˇ 1 , K ] is also a Nash equilibrium. Further, the former equilibrium dominates the latter by generating more profits for both retailers. (iv) if w ∈ (z 2 − K , 2z 2 − z 1 ], then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3);

86

3 Capacity Allocation with Demand Competition in Asymmetric Market

(v) if w ∈ (2z 2 − z 1 , z 1 ), then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = ((z 1 − w)/2, 0). Second, suppose order priority is given to the low-type retailer (retailer 2). With a slight abuse of notation, let (ˇr1∗ , rˇ2∗ ) be an √ equilibrium order pair under lexicographic allocation, and define αˇ 2 = z 2 − w − 4K (z 2 − w − K ), wˇ 2+ = 2z 2 − z 1 − 29 K +  3 5K 2 + 4K (z 1 − z 2 ). We have 2 (I) K ≤ z 1 − z 2 : (i) if w ∈ (0, z 2 − K ], then any point in [0, K ] × {K } is a Nash equilibrium; (ii) if w ∈ (z 2 − K , z 1 − 2K ], then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = (K , 0); (iii) if w ∈ (z 1 − 2K , z 1 ), then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = ((z 1 − w)/2, 0). (II) K > z 1 − z 2 : (i) if w ∈ (0, z 1 − 2K ], then any point in [0, K ] × {K } is a Nash equilibrium; (ii) if w ∈ (z 1 − 2K , wˇ 2+ ], then any point in [αˇ 2 , K ] × {K } is a Nash equilibrium; (iii) if w ∈ (wˇ 2+ , z 2 − K ], then ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3) is a Nash equilibrium and any point in [αˇ 2 , K ] × {K } is also a Nash equilibrium. Further, the former equilibrium dominates the latter by generating more profits for both retailers. (iv) if w ∈ (z 2 − K , 2z 2 − z 1 ], then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = ((2z 1 − z 2 − w)/3, (2z 2 − z 1 − w)/3); (v) if w ∈ (2z 2 − z 1 , z 1 ), then there is a unique Nash equilibrium (ˇr1∗ , rˇ2∗ ) = ((z 1 − w)/2, 0). The proof of Proposition 3.3 is similar to that of Theorems 3.1 and 3.2, and here we omit the details. Proposition 3.3 is intuitive. Observe that for any given capacity level, the retailer with order priority is allowed to obtain the entire capacity if the wholesale price is very low. Interestingly, when the capacity level is relatively high, gaming effect also occurs. Note that when K > z 1 − z 2 , if + + + ∗ ∗ w ∈ ((z i + z j − 3K )/2, wˇ i ], satisfying q1 + q2 < K ≤ K (wˇ i ), where K (wˇ i ) = z i −w 2

+ 16 −7z i2 − 2z i wˇ i + 5wˇ i2 + 16z i z j − 4z 2j − 8z j w, ˇ then retailer i with order priority will order K , which is more than her ideal order size ri∗ . Further, if the wholesale price lies in the interval (wˇ i∗ , 2z 2 − z 1 ), then the retailers will order their ideal orders (r1∗ , r2∗ ) and lexicographic allocation is truth-inducing. However, if the wholesale price is sufficiently high, then the low-type retailer will order nothing whether or not she has order priority. Proof of Theorem 3.2 The theorem follows the discussions preceding the theorem in Sect. 3.5. Numerical Examples with Duopoly Retailers Figure 3.19 graphically illustrates the above finding for symmetric retailers. It is evident that as the capacity K increases, the wholesale price in each allocation mechanism initially follows a descending trend and then remains unchanged when capacity becomes sufficiently large. Further, the supplier’s profit under proportional alloca-

Appendix

87 z =100

z =100

1

Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

90 80

z =100

1

2

2

2000 *

Supplier’s Profit Πs

Supplier’s optimal wholesale price w *

z =100 100

70 60 50 40 30

1500 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

1000

500

20 10 0

0

10

20

30

40

50

60

70

0

80

0

10

20

30

40

50

60

70

80

Capacity K

Capacity K

Fig. 3.19 Comparison of supplier’s pricing decisions with symmetric retailers z1=100

90

z2=60

Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

70

z2=60

1200 s

80

z1=100

1400

Supplier’s Profit Π*

Supplier’s optimal wholesale price w*

100

60 50 40 30

1000 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

800 600 400

20 200

10 0

0

10

20

30

40

50

Capacity K

60

70

80

0

0

10

20

30

40

50

60

70

80

Capacity K

Fig. 3.20 Comparison of supplier’s pricing decisions with asymmetric retailers

tion is no less than that under uniform allocation, but is no more than that under lexicographic allocation. We set forth the following interpretations for these results. As the supplier’s capacity increases, to maximize his profit, it is beneficial for the supplier to reduce the wholesale price so as to entice the retailers to order more. But when the capacity is sufficiently large to meet all the retailer demand, both the total order quantity and the supplier’s profit will stay at a constant level. From Tables 3.2, 3.3 and 3.4, our numerical studies are based on the following data: (I) when z21 < z 2 ≤ 5z71 , we have (z 1 , z 2 ) = (100, 60) or (100, 65); (II) when 5z 1 < z 2 ≤ 3z41 , we have (z 1 , z 2 ) = (100, 72) or (100, 74); (III) when 3z41 < z 1 < z 2 , 7 we have (z 1 , z 2 ) = (100, 80) or (100, 90). Results are depicted in Figs. 3.20, 3.21, 3.22, 3.23, 3.24, 3.25. Note that as K increases, the supplier has the potential to sell more in the sense that he could obtain more profits unless the total ordering quantity by the two retailers is not increasing any more. The supplier may change the wholesale price strategically with capacity, as illustrated in Figs. 3.20, 3.21, 3.22, 3.23, 3.24, 3.25.

88

3 Capacity Allocation with Demand Competition in Asymmetric Market z1=100

z2=65

z1=100

1400

z2=65

90

1200

Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

70

*

80

Supplier’s Profit Πs

Supplier’s optimal wholesale price w

*

100

60 50 40 30

1000 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

800 600 400

20 200

10 0

0

10

20

30

40

50

60

70

0

80

0

10

20

30

Capacity K

40

50

60

70

80

Capacity K

Fig. 3.21 Comparison of supplier’s pricing decisions with asymmetric retailers z =100

2

90

Supplier’s Profit Π*

70 60 50 40 30 20

1200 1000 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

800 600 400 200

10 0

2

1400 s

80

z =72

1

1600 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

*

Supplier’s optimal wholesale price w

z =100

z =72

1

100

0

10

20

40

30

50

60

70

0

80

0

10

20

30

50

40

60

70

80

Capacity K

Capacity K

Fig. 3.22 Comparison of supplier’s pricing decisions with asymmetric retailers z1=100

z1=100

z2=74 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

70

1400 *

80

60 50 40 30

1200 1000 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

800 600 400

20

200

10 0

z2=74

1600

90

Supplier’s Profit Πs

Supplier’s optimal wholesale price w*

100

0

10

20

30

40

50

Capacity K

60

70

80

0

0

10

20

30

40

50

Capacity K

Fig. 3.23 Comparison of supplier’s pricing decisions with asymmetric retailers

60

70

80

Appendix

89 z1=100

z2=80

z1=100

90 80

1600

70

*

60 50 40 30

1400 1200 1000

Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

800 600 400

20 10 0

z2=80

1800

Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

Supplier’s Profit Πs

Supplier’s optimal wholesale price w*

100

200 0

10

20

30

40

50

60

70

0

80

0

10

20

30

Capacity K

40

50

60

70

80

Capacity K

Fig. 3.24 Comparison of supplier’s pricing decisions with asymmetric retailers z =100 1

z =100

z =90 2

90 80

z =90 2

2000 1800 *

70 60 50 40 30 20

1600 1400 1200 Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

1000 800 600 400

10 0

1

2200

Lexicographic Allocation−1 Lexicographic Allocation−2 Proportional Allocation Uniform Allocation

Supplier’s Profit Πs

Supplier’s optimal wholesale price w*

100

200 0

10

20

30

40

50

Capacity K

60

70

80

0

0

10

20

30

40

50

Capacity K

60

70

80

Fig. 3.25 Comparison of supplier’s pricing decisions with asymmetric retailers

From Figs. 3.20 and 3.21, it is evident that lexicographic allocation performs no differently than the other two allocations. Technically, this is because, although K wˇ 1+ > K wˇ 2+ > K wˆ − > K w˜ + , the maximum of the four values is less than z 2 /8 for our numerical examples. Therefore, the supplier is indifferent among the three allocation mechanisms if the retailers’ market powers are significantly different. However, as the low-type retailer’s market power becomes closer to the high-type retailer’s power, we observe from Figs. 3.22, 3.23, 3.24, 3.25 that the supplier’s profit obtained from lexicographic allocation is much higher than from uniform or proportional allocations, especially when order priority is given to the high-type retailer. This demonstrates that the advantage of superior mechanism in each comparison becomes more evident when the competition between the two retailers is more intense. Proof of Theorem 3.3 First, we derive the values of K u∗ , K ∗p , K l∗∗ . Let q˜imax , qˆimax , qˇimax be retailer i’s largest received allocation under uniform, proportional and lexicographic (with priority

90

3 Capacity Allocation with Demand Competition in Asymmetric Market

∗ sequence l ∗ ) allocations given r−i = q−i , respectively. Specifically, we have

q˜imax =

1 (K − n˜ i + 1



q ∗j ),

(A.5)

j=n˜ i +1,...,n; j=i

K K, K + Q ∗−i ∗ = K − (q1∗ + · · · + qi−1 ),

qˆimax =

(A.6)

qˇimax

(A.7)

where n˜ i is defined as the largest integer less than i such that qn∗˜ ≥ n˜ 1+1 (K − i i  ∗ j=n˜ i +1,··· ,n; j=i q j ). To ensure a positive marginal profit for retailer i when the capacity is fully utilized, we assume that z i − w − K > 0, and thus for any retailer, an upper bound with K < z n − w is necessary for the assumption z 1 > z 2 > ... > z n . Let i (x, y) be retailer i’s profit with retailer i’s allocation x and the other retailers’ total allocation y. Under uniform allocation, it is obvious that if there exists any retailer i such that i (q˜imax , K − q˜imax ) ≥ i (qi∗ , Q ∗−i ), then the supplier’s total allocated capacity is K . Otherwise, each  retailer obtain her ideal allocation value. For notational convenience, let δ = j=n˜ i +1,...,n; j=i q ∗j . We next obtain the threshold K ui∗ for retailer i as follows. Note that Q ∗ = ( nj=1 z j − nw)/(n + 1) and qi∗ = ((n + 1)z i −  nj=1 z j − w)/(n + 1) for their definitions, we immediately have the important equation. Q ∗ = z i − w − qi∗ .

(A.8)

With Eq. (A.8) in hands, we know that i (q˜imax , K − q˜imax ) ≥ i (qi∗ , Q ∗−i ) which is equivalent to zi −w−K (K − δ) ≥ (z i − w − Q ∗ )qi∗ . Therefore, we have n˜ +1 i

K 2 − (z i − w + δ)K + (z i − w)δ + (n˜ i + 1)qi∗2 ≤ 0.

(A.9)

Solving Eq. (A.9), we have zi − w + δ −

[z i − w + δ]2 − 4[(z i − w)δ + (n˜ i + 1)qi∗2 ]

≤K ≤ z i − w + δ + [z i − w + δ]2 − 4[(z i − w)δ + (n˜ i + 1)qi∗2 ] 2

2

√

Note

√

z i −w+δ+

that [z i

−w+δ]2 −4[(z 2

˜ i +1)qi∗2 ] i −w)δ+(n

z i −w+δ−

[z i −w+δ]2 −4[(z i −w)δ+(n˜ i +1)qi∗2 ] 2

.

≤ Q∗ ≤

. We focus on the case K > Q ∗ in the theorem,

z i −w+δ+ [z i −w+δ]2 −4[(z i −w)δ+(n˜ i +1)qi∗2 ] 2

hence, we can obtain the threshold of retailer i = and K u∗ = max{K u1∗ , K u2∗ , . . . , K un∗ } for n retailers. Similar to the analysis of Fig. 3.1, the gaming effect occurs if K ∈ [Q ∗ , K u∗ ]. K ui∗

Appendix

91

2∗ Furthermore, we can use the same method to derive out K l∗∗ = max{K l1∗ ∗ , Kl∗ , . . . , √  z i −w+

q∗+

 [z i −w+

q ∗ ]2 −4q ∗2

j=1,··· ,i−1 j j=1,··· ,i−1 j i i∗ . Finally, we will get K ln∗ ∗ } where K l ∗ = 2 max max ∗ ∗ the value of K p . We know that i (qˆi , K − qˆi ) > i (qi , Q ∗−i ) which is equivK2 alent to (z i − w − K ) K +Q > qi∗2 , then we have −K 3 + (z i − w)K 2 − qi∗2 K − ∗ −i

qi∗2 Q ∗−i > 0. Let f (K ) = −K 3 + (z i − w)K 2 − qi∗2 K − qi∗2 Q ∗−i , it is obvious that the function f (K ) is monotone decreasing function in K ∈ [Q ∗ , z i − w). From Eq. (A.8), we have f (Q ∗ ) = −Q ∗3 + (z i − w)Q ∗2 − qi∗2 Q ∗ − qi∗2 Q ∗−i = −Q ∗3 + (Q ∗ + qi∗ )Q ∗2 − qi∗2 Q ∗ − qi∗2 (Q ∗ − qi∗ ) = qi∗ (Q ∗ − qi∗ )2 > 0, and f (z i − w) = −(z i − w)3 + (z i − w)3 − qi∗2 (z i − w) − qi∗2 Q ∗−i < 0. ∗ Consequently, there exists a value K i∗ p which belongs to [Q , z i − w) such that ∗ 1∗ 2∗ n∗ f (K ) = 0. Therefore, K p = max{K p , K p , ..., K p }. Next, we compare the values of K u∗ , K ∗p , K l∗∗ . (i) Firstly, we will prove that K u∗ ≤ K ∗p . Consider any retailer i whose profits under max (z i − w − K )q˜imax , proportional and uniform allocations are  (z i − w − K )qˆ∗i and 1 K K max ˆimax , respectively. Since q˜i = n˜ +1 (K − j=n˜ i +1,...,n; j=i q j ) < 2 < K +Q ∗ K = q −i

i

it follows that i (q˜imax , K − q˜imax ) ≤ i (qˆimax , K − qˆimax ). Recall that K ui∗ satisfies i (q˜imax , K − q˜imax ) ≥ i (qi∗ , Q ∗−i ), and thus K ui∗ also satisfies i (qˆimax , K − qˆimax ) ≥ i (qi∗ , Q ∗−i ) for any i. As K i∗ p is defined as upper bound of K that satisfies ∗ i (qˆimax , K − qˆimax ) ≥ i (qi∗ , Q ∗−i ), we can derive that K ui ≤ K i∗ p . Hence, we have ∗ ∗ Ku ≤ K p. (ii) Finally, we will prove that K ∗p ≤ K l∗∗ . It is known that i (qˆimax , K − qˆimax ) ≥ K2 ∗2 i (qi∗ , Q ∗−i ) is equivalent to (z i − w − K ) K +Q ≥ 0 with Eq. (A.8), and ∗ − qi −i

1 (qˇ1max , K − qˇ1max ) ≥ 1 (q1∗ , Q ∗−1 ) is equivalent to (z 1 − w − K )K − q1∗2 > 0. Note that (z 1 − w − K )K − q1∗2 ≥ (z i − w − K )

K2 − qi∗2 K + Q ∗−i

which is equivalent to [z 1 − w − Q ∗−i − (z i − w)]K 2 + [(z 1 − w)Q ∗−i − q1∗2 + qi∗2 ]K − (q1∗2 − qi∗2 )Q ∗−i > 0.

Thus, we have (Q ∗ − q1∗ )K 2 − [(z 1 − w)Q ∗−i − q1∗2 + qi∗2 ]K + (q1∗2 − qi∗2 )Q ∗−i < 0. Let g(K ) = (Q ∗ − q1∗ )K 2 − [(z 1 − w)Q ∗−i − q1∗2 + qi∗2 ]K + (q1∗2 − qi∗2 )Q ∗−i . We can get that g(Q ∗ ) < 0 and g(z n − w) < 0. Because of the continuity and monotone

92

3 Capacity Allocation with Demand Competition in Asymmetric Market

K ∗2 of g(K ) in K ∈ [Q ∗ , z n − w), (z 1 − w − K )K − q1∗2 ≥ (z i − w − K ) K +Q ∗ − qi −i always holds as K ∈ [Q ∗ , z n − w). For any retailer i, it follows that (z 1 − w − K2 ∗2 as K ∈ [Q ∗ , z n − w). Thus, we can conK )K − q1∗2 ≥ (z i − w − K ) K +Q ∗ − qi −i clude that if K satisfies i (qˆimax , K − qˆimax ) ≥ i (qi∗ , Q ∗−i ), then we have the result 1 (qˇ1max , K − qˇ1max ) ≥ 1 (q1∗ , Q ∗−1 ), i.e., {K |i (qˆimax , K − qˆimax ) ≥ i (qi∗ , Q ∗−i )} ⊆ {K |1 (qˇ1max , K − qˇ1max ) ≥ 1 (q1∗ , Q ∗−1 )}. Consequently, if the value of K induces the total order quantity to be greater than capacity under proportional allocation, then it also encourages retailer 1 to order the whole capacity under lexicographic allocation. Hence, we have K ∗p ≤ K l∗∗ . Above all, we have K u∗ ≤ K ∗p ≤ K l∗∗ . 2

Numerical Computation of Thresholds K ∗ The thresholds K u∗ , K ∗p , K l∗t under uniform, proportional, and proportional (with priority sequence lt , where retailers are indexed as 1, 2, . . . , n, without loss of generality) can be characterized by equilibrium ordering, as follows. Given any supplier’s wholesale price w and supplier’s capacity K ≥ Q ∗ , (i) under uniform mechanism, the order quantity vector (q1∗ , q2∗ , . . . , qn∗ ) is in ∗ ) for any i, where q˜imax = equilibrium only if i (q˜imax , K − q˜imax ) ≤ i (qi∗ , q−i 1 (K − q ∗j ) and n˜ i is defined as the largest integer less than i such  n˜ +1 i

that

qn∗˜ i

≥

j=n˜ i +1,...,n; j=i 1 (K −  n˜ i +1 j=n˜ i +1,...,n; j=i

q ∗j );

∗ ∗ ∗ (ii) under proportionalmechanism, the order quantity vector  (q1 , q2 ∗, . . .∗, qn ) is in ∗ ∗ 2 2 equilibrium only if i K /(K + q−i ), K − K /(K + q−i ) ≤ i (qi , q−i ) for any i; (iii) under lexicographic mechanism, the order quantity vector (q1∗ , q2∗ , . . . ..., qn∗ ) is ∗ q ∗j ,  q ∗j ) ≤ i (qi∗ , q−i ) for any i. in equilibrium only if i (K −  j=1,...,i−1

j=1,...,i−1

The results are intuitive. Under a specific allocation mechanism, when the supplier’s wholesale price w and capacity level K are given, retailer i will order the ideal allocation qi∗ if the profit generating from the inflated allocation (i.e., ordering as much as possible) is less than the profit resulting from the ideal allocation (q1∗ , q2∗ , . . . , qn∗ ). In other words, if every retailer’s profit satisfies this condition, then all retailers will order their ideal order quantities, i.e., the Nash equilibrium order vector (q1∗ , q2∗ , . . . , qn∗ ), and consequently the total allocation is Q ∗ . On the other hand, if there exists at least one retailer i, whose profit is larger when ordering as much as possible (i.e., K ) so as to receive the maximum possible allocation given that the total allocation is K . Accordingly, the total allocation will be equal to the available capacity K , and any retailer will order at least K /n for her best interest. Hence, the equilibrium allocation vector is (K /n, K /n, . . . , K /n). Under each allocation mechanism, the minimum value of K , given that K ≥ Q ∗ , satisfying the correspondent condition is the threshold K ∗ for the mechanism. Numerically, it is straightforward to use a binary search to locate the threshold K ∗ with specific precision for each mechanism.

References

93

References Cachon, G., & Lariviere, M. (1999). Capacity choice and allocation: Strategic behavior and supply chain performance. Management Science, 45(8), 1091–1108. Cachon, G., & Lariviere, M. (1999). An equilibrium analysis of linear, proportional and uniform allocation of scarce capacity. IIE Transactions, 31(9), 835–849. Cachon, G., & Lariviere, M. (1999). Capacity allocation using past sales: When to turn-and-earn. Management Science, 45(5), 685–703. Chen, F., Li, J., & Zhang, H. (2013). Managing downstream competition via capacity allocation. Production and Operations Management, 22(2), 426–446. Chen, Y., Su, X., & Zhao, X. (2012). Modeling bounded rationality in capacity allocation games with the quantal response equilibrium. Management Science, 58(10), 1952–1962. Cho, S.-H., & Tang, C. (2014). Capacity allocation under retail competition: Uniform and competitive allocations. Operations Research, 62(1), 72–80. Hall, N. G., Liu, Z. (2010). Capacity allocation and scheduling in supply chains. In J. J. Cochran, Editor-in-Chief, Wiley encyclopedia of operations research and management science (pp. 576– 584). New York City, New York: Wiley. Iyer, A. V., Deshpande, V., & Wu, Z. (2003). A postponement model for demand management. Management Science, 49(8), 983–1002. Lee, H. L., Padmanabhan, V., & Whang, S. J. (1997). Information distortion in a supply chain: The bullwhip effect. Management Science, 43(4), 546–558. Liu, Z. (2012). Equilibrium analysis of capacity allocation with demand competition. Naval Research Logistics, 59, 254–265. Sprumont, Y. (1991). The division problem with single-peaked preferences: A characterization of the uniform allocation rule. Econometrica, 59(2), 509–519. Yang, Z., Hu, X., Gurnani, H., Guan, H. (2014). Selling to a competing buyer with limited supplier capacity. Working paper, Lundquist College of Business, University of Oregon, Eugene, Oregon.

Chapter 4

Allocating Capacity with Demand Competition: Fixed Factor Allocation

4.1 Introduction In practice, a supplier with limited capacity often puts capacity on allocation, i.e., rationing capacity through quantity competition of retailers rather than through a pricing mechanism. Capacity allocation is a common occurrence in industries in which capacity expansion is costly and time consuming and price is given exogenously (e.g., steel and paper). A supplier can use his prior beliefs on his own and the retailers’ needs to construct a capacity allocation mechanism for allocation of his capacity among retailers. Commonly used capacity allocation mechanisms contain allocation rules that allocate capacity based on retailers’ order sizes, such as proportional allocation and lexicographic allocation. When the supplier’s capacity is insufficient to fill all the orders received, proportional allocation allocates capacity in proportion to order size, and lexicographic allocation allocates capacity in the order of a predetermined priority sequence. Even though lexicographic allocation dominates proportional allocation (lexicographic and proportional allocation mechanisms are widely used by companies in practice) in terms of increasing the profits for both the supplier and the supply chain when there exists downstream competition (Chen et al. 2013), lexicographic allocation treats retailers with identity discrimination and thus puts the supplier under risk of losing future business from some retailers. Specifically, retailers with low priority may turn to other suppliers for better wholesale price and order fulfillment. In addition, lexicographic allocation can easily drive retailers with low priority out of the market, and with fewer retailers retaining in the market, the supplier will suffer from reduced order size. Thus, these potential drawbacks motivate us to propose a new capacity allocation rule, namely, fixed factor allocation. Fixed factor allocation incorporates the principles underlying both proportional and lexicographic allocation: it prioritizes retailers as in lexicographic allocation, but guarantees only a fixed proportion of the total available capacity to the prioritized retailer, as in proportional allocation. The allocation corresponds with existing practice in pharmaceutical © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Li et al., Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-19-6577-7_4

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industry. For example, Zhong and Wu (2013) report that a manufacturer of Pien Tze Huang (traditional Chinese medicine) allocated restricted capacity to distributors based on fixed proportions. Due to the scarcity of the raw material, the short supply of Pien Tze Huang often occurred, in which cases the manufacturer guaranteed each distributor a fixed proportion of the total capacity, which resembles a fixed factor allocation. Specifically, we consider a supply chain consisting of a supplier and two retailers. The supplier sells a product to the two retailers, then the retailers sell the product to customers in a common market. The supplier has limited capacity. If the total order size from the two retailers does not exceed the supplier’s capacity, then each retailer receives what she orders; otherwise, the supplier allocates his capacity between the two retailers using a preannounced capacity allocation rule. We assume that the retail prices charged by the two retailers depend on the quantity of the product they each receive from the supplier, which is equal to the quantity they each sell in the market. Thus, the two retailers face order quantity competition, i.e., Cournot competition. The decision sequence of the supplier and retailers is as follows. First, the supplier determines a wholesale price and announces his capacity level and a capacity allocation rule. Second, the two retailers submit their order quantities simultaneously. This constitutes a Stackelberg game, with the supplier as leader and the two retailers as followers. We investigate how the supplier optimizes his wholesale price to maximize his value as a leader and how the two retailers compete for limited capacity in their order quantity determination to maximize their values. We specifically study how the proposed fixed factor allocation rule affects the decisions of the supplier and the retailers. We also compare the fixed factor allocation rule with proportional and lexicographic allocations. We show that the fixed factor allocation rule incorporates both proportional and lexicographic allocations from the perspectives of the supplier and the whole supply chain, but not necessarily the retailers. Under fixed factor allocation, we demonstrate that the supply chain’s profit is not affected by the allocation factor when it is greater than a threshold, and the retailers can share the supply chain’s profit with the supplier by negotiating the allocation factor value. Capacity allocation mechanisms have been employed widely in many industries, including automobiles, pharmaceuticals, and toys (e.g., Hwang and Valeriano 1992; Blumenstein 1996). For the properties of a large variety of capacity allocation rules, we refer to the survey paper by Hall and Liu (2010). Next, we review the studies on proportional and lexicographic allocation rules, which are also examined in our work. Lee et al. (1997) recognize that proportional allocation creates incentive for retailers to raise their orders above their desired allocation in multi-echelon supply chains. Cachon and Lariviere (1999a) demonstrate that proportional allocation with fixed price can lead retailers to order more than they desire to receive a favorable allocation, even when they directly order from the supplier. These authors also show that lexicographic allocation is truth-inducing in that it provides no incentive for retailers to order more than they desire. However, under proportional allocation, both the supplier and the whole supply chain can earn higher profits. These authors further demonstrate that a truth-inducing allocation with fixed price cannot maximize total retailer profits. Cachon and Lariviere (1999b) more specifically compare propor-

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tional and lexicographic allocations. These authors show that whether order inflation incentivized by proportional allocation helps or harms a supply chain depends on how profits are distributed within the supply chain. In general, encouraging order inflation increases supplier profits but decreases retailer profits. The following three papers on capacity allocation are closely related to the present paper in that they consider demand competition among retailers. Liu (2012) studies how different capacity allocation rules affect profits of different supply chain members, where retail prices linearly depend on the total sales volume of two competing retailers. An interesting finding is that the supplier can sell more with less capacity, and retailers may earn more when the supplier has less capacity, due to the demand competition between the two retailers. Cho and Tang (2014) develop an important extension of Liu (2012) by solving the case with multiple retailers. These authors specifically investigate the gaming effect caused by uniform allocation. They also propose a new allocation, namely, competitive allocation, which can eliminate the gaming effect with demand competition among retailers. Chen et al. (2013) consider capacity allocation in a supply chain consisting of a supplier and multiple retailers with the same market power in demand competition. These authors consider the supplier’s wholesale pricing decision when using proportional and lexicographic allocations. They show that lexicographic allocation can mitigate competition among retailers, and consequently generate higher profits for both the supplier and the supply chain. Models for capacity allocation in different settings either illustrate the computational difficulties of achieving a desirable allocation (e.g., Clark and Scarf 1960), or show that a desirable allocation can be obtained (e.g., Eppen and Schrage 1981; Federgruen and Zipkin 1984; Jonsson and Silver 1987; Schwarz 1989; Chen and Zheng 1994; Kumar et al. 1995). Note that the definition of desirable allocation varies from model to model. Several author groups (e.g., Topkis 1968; Ha 1997; Deshpande and Schwarz 2002; Deshpande et al. 2003) examine allocation of inventory with sequentially arriving customers of different priority classes in a centralized setting. Cachon and Lariviere 1999c investigate a turn-and-earn allocation mechanism over two periods, where capacity allocation in the second period is based on sales volume in the first period. The remainder of this paper proceeds as follows. In Sect. 4.2, we define the model and introduce the fixed factor allocation rule. Retailers’ best response decisions in Nash equilibrium are derived in Sect. 4.3. In Sect. 4.4, we characterize the supplier’s optimal decision. In Sect. 4.5, we investigate how allocation factor affects profits of different supply chain members. In Sect. 4.6, we compare the supply chain profit under fixed factor allocation with the profit of a centralized supply chain. We then compare fixed factor allocation with proportional and lexicographic allocation in Sect. 4.7, and study the optimal capacity choice of the supplier in Sect. 4.8. Finally, concluding remarks and suggestions for future research are set forth in Sect. 4.9. All proofs are in the Appendix.

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4.2 Model Description We consider a one-period setting in which a single supplier sells one product to two retailers, who in turn retail the product to customers in a common market. The quantity ordered by a retailer may not be the same as what she finally receives, which depends on how the supplier allocates his capacity when the total order quantity by the two retailers exceeds his capacity. We assume that order quantity by each retailer cannot exceed the supplier’s capacity and the retail price is market size minus the total quantity that the supplier allocates to the two retailers. Specifically, for each retailer i ∈ {1, 2}, given the other retailer’s allocated quantity, the market price is decreasing in her own allocated quantity, and consequently her profit function is concave with her allocated quantity. Prior to the decision period, we have the following assumptions. First, the supplier has a fixed capacity size. Second, the supplier preannounces publicly the allocation rule he will use to allocate his capacity. While both the capacity level and allocation rule are taken as given in our model, essentially they could be determined by the supplier, who wants to maximize his own profit. We investigate the supplier’s decision on capacity level and allocation rule. During the decision period, events occur in the following sequence. First, the supplier determines the wholesale price. Second, the two retailers determine their order quantities and simultaneously submit them to the supplier. Third, the supplier fills orders according to the preannounced allocation rule. Fourth, retailers sell the products in a common market and realize their profits. Note that retailers submit orders independently, and orders are the only communication between the retailers and the supplier. No retailer can credibly announce her information to other players, including the supplier and the other retailer, and no side contracts between the supplier and any retailer are allowed. In short, a retailer can influence her allotment and the other retailer’s allotment only through her order. The supplier charges a wholesale price that is determined after he chooses an allocation rule, and a retailer must accept the price and pay for any allocation up to her full order. The supplier cannot deliver to a retailer more than she has ordered. We use the following notation throughout the paper, K : the supplier’s capacity, w : the wholesale price, M : the market size, m i : retailers i’s order quantity,i = 1, 2, m : order vector m = (m 1 , m 2 ), g(m) : allocation rule defined by function g for order vector m, gi (m) : retailer i’s allocated quantity under allocation function g for order vector m, i = 1, 2, c : unit production cost,

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Recall that each retailer’s order quantity is no more than the supplier’s capacity, i.e., m i ≤ K , i = 1, 2. An allocation rule g(m) ≡ g(m 1 , m 2 ) is a function of the retailers’ order vector m = (m 1 , m 2 ), and defines an allocation vector g(m) = (g 12(m), g1 (m)). Note that for an allocation vector to be feasible, we require that i=1 gi (m) ≤ K . Also, the supplier can never allocate to a retailer more than her order quantity, i.e., gi (m) ≤ m i , i = 1, 2. Next we introduce three allocation rules: proportional, lexicographic, and fixed factor allocations. Proportional allocation allocates capacity in proportion to order size if capacity is insufficient to fill all orders. Specifically, we have  g1 (m 1 , m 2 ) =  g2 (m 1 , m 2 ) =

m1

if m 1 + m 2 ≤ K K if m 1 + m 2 > K

(4.1)

m2

if m 1 + m 2 ≤ K K if m 1 + m 2 > K

(4.2)

m1 m 1 +m 2 m2 m 1 +m 2

Lexicographic allocation prioritizes retailers and always tries to fill the order of the retailer with the highest priority. Without loss of generality, we assume that retailer 1 is granted a priority higher than retailer 2. We have g1 (m 1 , m 2 ) = m 1 ∧ K ,

(4.3)

g2 (m 1 , m 2 ) = m 2 ∧ (K − m 1 )

(4.4)

Now, we define our fixed factor allocation rule. The fixed factor allocation incorporates the principles of proportional and lexicographic allocations: it prioritizes retailers as in lexicographic allocation, but guarantees only a fixed proportion of the total available capacity to the prioritized retailer, as in proportional allocation. We denote 0 ≤ α ≤ 1 as the proportion of total capacity reserved for the retailer with higher priority. Without loss of generality, we assume that retailer 1 is granted a priority higher than retailer 2. For any order vector m = (m 1 , m 2 ), the fixed factor allocation rule is given by ⎧ if m 1 + m 2 ≤ K or m 1 + m 2 > K , m 1 ≤ αK ⎨ m1 if m 1 > αK , m 2 > (1 − α)K g1 (m 1 , m 2 ) = αK (4.5) ⎩ K − m 2 if m 1 + m 2 > K , m 2 ≤ (1 − α)K ⎧ if m 1 + m 2 ≤ K or m 1 + m 2 > K , m 2 ≤ (1 − α)K ⎨ m2 g2 (m 1 , m 2 ) = (1 − α)K if m 1 > αK , m 2 > (1 − α)K (4.6) ⎩ K − m 1 if m 1 + m 2 > K , m 1 ≤ αK When capacity is insufficient, the fixed factor allocation rule guarantees an allocation of a certain proportion of capacity to each retailer, i.e., αK to retailer with high priority, (1 − α)K to retailer 2 with low priority. If retailer i orders less than its guaranteed allocation, she will receive the quantity what she orders, otherwise, she will get at least the guaranteed quantity.

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Note that we consider two symmetric retailers. We regard retailer 1 as prioritized with higher priority in capacity allocation, and hence assume that α ∈ [1/2, 1]. Fixed factor allocation degenerates to lexicographic allocation when α = 1, and to uniform allocation when α = 1/2. For allocation rule g(·), the market price of the product is expressed as M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ), where M represents market size. The supplier’s problem is to choose a wholesale price w to maximize his profit, s (α) = max{w · (g1 (m 1 , m 2 ) + g2 (m 1 , m 2 ))}; w

(4.7)

while retailers play a Cournot game by simultaneously choosing their order quantities (m 1 , m 2 ) to maximize their individual profit as follows, 1 (w, m 2 ) = max (M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w)g1 (m 1 , m 2 ), (4.8) m1

2 (w, m 1 ) = max (M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w)g2 (m 1 , m 2 ). (4.9) m2

Here the supplier and two retailers constitute a Stackelberg game with the supplier as leader and the two retailers as followers. Meanwhile, the two retailers play a Cournot game in a common retail market.

4.3 Retailers’ Decisions In this section, we analyze the best response function of each retailer given wholesale price and the other retailer’s order quantity, and then characterize the two retailers’ equilibrium order quantities for a given wholesale price, under fixed factor allocation by the supplier. Because the fixed factor allocation rule is not symmetric for the two retailers, retailer decisions will differ. Thus, we investigate the decisions of the two retailers separately. In the remainder of this section, we analyze the decisions of retailer 1 (with higher priority in capacity allocation), retailer 2 (with lower priority in capacity allocation), and the two retailers in equilibrium, respectively.

4.3.1 Retailer 1’s Best Response Function From Equation (4.8), it is difficult to determine the concavity of retailer 1’s profit function. Thus, we analyze the profit function (4.8) by dividing the problem into two scenarios: Scenario 1: m 1 + m 2 ≤ K (i.e., retailers’ total order quantity is no

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greater than the supplier’s capacity); and Scenario 2: m 1 + m 2 > K (i.e., retailers’ total order quantity is greater than the supplier’s capacity). We can rewrite retailer 1’s profit function as follows under fixed factor allocation:   1 (w, m 2 ) = max M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w g1 (m 1 , m 2 ) m1

= max max G 11 (m 1 , m 2 ), max G 12 (m 1 , m 2 ) , m 1 ≤K −m 2

m 1 ∈(K −m 2 ,K ]

(4.10)

where G 11 (m 1 , m 2 ) = (M − m 1 − m 2 − w)m 1 ,  (M − K − w)(K − m 2 ) m 2 ≤ (1 − α)K G 12 (m 1 , m 2 ) = (M − K − w)(m 1 ∧ αK ) m 2 > (1 − α)K

(4.11) (4.12)

where G 11 (m 1 , m 2 ) represents retailer 1’s profit in Scenario 1, in which the total order quantity of two retailers is no more than the supplier’s capacity K , and G 12 (m 1 , m 2 ) denotes retailer 1’s profit in Scenario 2, in which the two retailers’ total order quantity is greater than K . Note that in Scenario 1, each retailer obtains her desired order quantity, while in Scenario 2, each retailer is allocated a quantity as specified by fixed factor allocation. It is straightforward to determine the optimal solution of m 1 for profit function G 11 (m 1 , m 2 ) in Scenario 1. If retailer 2 orders a small quantity (i.e., m 2 ≤ 2K − (M − w) and m 2 ≤ M − w), then retailer 1 will order her desired quantity (M − m 2 − w)/2 and gains an optimal profit (M − w − m 2 )2 /4; if retailer 2 orders a very large quantity (i.e., m 2 > M − w), then retailer 1 will order nothing and gains no profit. In addition, if retailer 2 orders a moderately large quantity (i.e., m 2 > 2K − (M − w)), then retailer 1 will order just the surplus capacity (K − m 2 ) to gain an optimal profit of (M − K − w)(K − m 2 ). Next, we maximize profit function G 12 in Scenario 2, as in the following Lemma 4.1. Lemma 4.1 Let m ∗12 and G ∗12 be the optimal order quantity and profit of the function G 12 for given w and m 2 , respectively. (i) Suppose m 2 ≤ (1 − α)K : m ∗12 ∈ (K − m 2 , K ] and G ∗12 = (M − K − w)(K − m 2 ). (ii) Suppose m 2 ∈ ((1 − α)K , K ]: if w < M − K , then m ∗12 ∈ [αK , K ] and G ∗12 = (M − K − w)αK ; if w ≥ M − K , then m ∗12 = (K − m 2 )+ and G ∗12 = (M − K − w)(K − m 2 )+ . Lemma 4.1 illustrates retailer 1’s best response function when the total order quantity is greater than the supplier’s capacity K . Comparing results in Lemma 4.1 with those obtained under Scenario 1, we can locate retailer 1’s global best response function given retailer 2’s order quantity in Theorem 4.1, as follows. Note that there are other ways to present the results of Theorem 4.1. We develop √ Theorem 4.1 in terms of the wholesale price. For simplicity, let αˆ = M − w − 2 (M − w − K )αK . Note that αˆ ≥ (1 − α)K .

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Theorem 4.1 Let m 1 (m 2 ) be retailer 1’s best response function; i.e., if retailer 2 orders m 2 , then it is optimal for retailer 1 to order m 1 (m 2 ). (i) Suppose w ∈ (M − K , M] : 2 2) 2 and 1 (w, m 2 ) = (M−w−m ; if m 2 ∈ [0, M − w), then m 1 (m 2 ) = M−w−m 2 4 if m 2 ∈ [M − w, K ], then m 1 (m 2 ) = 0 and 1 (w, m 2 ) = 0. (ii) Suppose w ∈ (M − (1 + α)K , M − K ] : 2 2) 2 ˆ then m 1 (m 2 ) = M−w−m and 1 (w, m 2 ) = (M−w−m ; if m 2 ∈ [0, α], 2 4 if m 2 ∈ (α, ˆ K ], then m 1 (m 2 ) ∈ [αK , K ] and 1 (w, m 2 ) = (M − K − w)αK . (iii) Suppose w ∈ (M − 2K , M − (1 + α)K ] : 2 and 1 (w, m 2 ) = if m 2 ∈ [0, 2K − (M − w)), then m 1 (m 2 ) = M−w−m 2 (M−w−m 2 )2 ; 4 if m 2 ∈ [2K − (M − w), (1 − α)K ), then m 1 (m 2 ) ∈ [K − m 2 , K ] and 1 (w, m 2 ) = (M − w − K )(K − m 2 ); if m 2 ∈ [(1 − α)K , K ], then m 1 (m 2 ) ∈ [αK , K ] and 1 (w, m 2 ) = (M − w − K )αK . (iv) Suppose w ≤ M − 2K : if m 2 ∈ [0, (1 − α)K ), then m 1 (m 2 ) ∈ [K − m 2 , K ] and 1 (w, m 2 ) = (M − w − K )(K − m 2 ); if m 2 ∈ [(1 − α)K , K ], then m 1 (m 2 ) ∈ [αK , K ] and 1 (w, m 2 ) = (M − w − K )αK . The intuition behind Theorem 4.1 is fairly easy to see. First, it is evident that if the wholesale price is high (i.e., w > M − 2K ) and retailer 2 orders little, then retailer 1 will not order much either (just (M − w − m 2 )/2), and the capacity constraint does not affect her order decision. In this case, the best response function is the same as if there does not exist capacity constraint. This explains the first element in (i)– (iii), above. Also, if retailer 2 orders large (m 2 ≥ M − w), then retailer 1 will order nothing because she cannot earn any positive profit, which explains the second part of (i). Second, if the capacity limit is exceeded by total order size with retailer 2’s order quantity no less than (1 − α)K , then retailer 1 will order no less than αK . This is because, if total order quantity exceeds capacity, then the total supply to the market will be K , which means the margin profit for each retailer is fixed at M − K − w, and consequently retailer 1’s profit is increasing in her allocation and thus she will order [αK , K ] to maximize her allocation at αK . This explains the last element in (ii)–(iv), above. Third, when w is smaller and retailer 2 does not order much, retailer 1 will order no less than the surplus capacity K − m 2 , as in the second part of (iii) and first part of (iv), above.

4.3.2 Retailer 2’s Best Response Function Because of the asymmetry of retailers caused by higher priority of retailer 1 in fixed factor allocation, we need to analyze retailer 2’s best response function to derive the Nash equilibrium ordering quantities of the two retailers. We proceed to analyze

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retailer 2’s best response function given the supplier’s wholesale price and retailer 1’s order quantity. Retailer 2 will choose her order quantity to maximize her profit function, defined by Eq. (4.9), which we rewrite as follows.   2 (w, m 1 ) = maxm 2 ≤K M − g1 (m 1 , m 2 ) − g2 (m 1 , m 2 ) − w g2 (m 1 , m 2 )

= max maxm 2 ≤K −m 1 G 21 (m 1 , m 2 ), maxm 2 ∈(K −m 1 ,K ] G 22 (m 1 , m 2 ) , (4.13)

where G 21 (m 1 , m 2 ) = (M − m 1 − m 2 − w)m 2 ,  m 1 ≤ αK (M − K − w)(K − m 1 ) G 22 (m 1 , m 2 ) = (M − K − w)(m 2 ∧ (1 − α)K ) m 1 > αK Note that G 21 is symmetric with G 11 , and we can maximize G 21 in a similar way as we maximize G 11 , as in Sect. 4.3.1, and hence we omit the details. Next, we maximize G 22 (m 1 , m 2 ). Lemma 4.2 Let m ∗22 and G ∗22 be the optimal order quantity and profit of the function G 22 for given w and m 1 , respectively. (i) Suppose m 1 ≤ αK : m ∗22 ∈ (K − m 1 , K ] and G ∗22 = (M − K − w)(K − m 1 ). (ii) Suppose m 1 ∈ (αK , K ]: if w < M − K , then m ∗22 ∈ ((1 − α)K , K ] and G ∗22 = (M − K − w)(1 − α)K ; if w ≥ M − K , then m ∗22 = (K − m 1 )+ and G ∗22 = (M − K − w)(K − m 1 )+ . In view of Lemma 4.2, we next characterize retailer 2’s best response function for a given wholesale price and retailer 1’s order quantity, as the wholesale price changes. Let  β = M − w − 2 (M − K − w)(1 − α)K . Note that β > αK . Theorem 4.2 Let m 2 (m 1 ) be retailer 2’s best response function, i.e., if retailer 1 orders m 1 , then it is optimal for retailer 2 to order m 2 (m 1 ). (i) Suppose w ∈ (M − K , M] : 2 1) 1 and 2 (w, m 1 ) = (M−w−m ; if m 1 ∈ [0, M − w), then m 2 (m 1 ) = M−w−m 2 4 if m 1 ∈ [M − w, K ], then m 2 (m 1 ) = 0 and 2 (w, m 1 ) = 0. (ii) Suppose w ∈ (M − (2 − α)K , M − K ] : 2 1) 1 and 2 (w, m 1 ) = (M−w−m ; if m 1 ∈ [0, β], then m 2 (m 1 ) = M−w−m 2 4 if m 1 ∈ (β, K ], then m 2 (m 1 ) ∈ [(1 − α)K , K ] and 2 (w, m 1 ) = (M − K − w)(1 − α)K . (iii) Suppose w ∈ (M − 2K , M − (2 − α)K ] : 1 and 2 (w, m 2 ) = if m 1 ∈ [0, 2K − (M − w)), then m 2 (m 1 ) = M−w−m 2 2 (M−w−m 1 ) ; 4

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4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

if m 1 ∈ [2K − (M − w), αK ), then m 2 (m 1 ) ∈ [K − m 1 , K ] and 2 (w, m 1 ) = (M − w − K )(K − m 1 ); if m 1 ∈ [αK , K ], then m 2 (m 1 ) ∈ [(1 − α)K , K ] and 2 (w, m 1 ) = (M − w − K )(1 − α)K . (iv) Suppose w ≤ M − 2K : if m 1 ∈ [0, αK ), then m 2 (m 1 ) ∈ [K − m 1 , K ] and 2 (w, m 1 ) = (M − w − K )(K − m 1 ); if m 1 ∈ [αK , K ], then m 2 (m 1 ) ∈ [(1 − α)K , K ] and 2 (w, m 1 ) = (M − w − K )(1 − α)K . We explain Theorem 4.2 as follows. First, it is evident that if the wholesale price is high (i.e., w > M − 2K ) and retailer 1 orders little, then retailer 2 will order her desired quantity (M − w − m 1 )/2, because the capacity constraint is not binding. In this case, the best response function is the same as if there were no capacity limit; each retailer will order her desired quantity. This explains the first part in (i)–(iii), above. Also, it is intuitive that if retailer 1 orders more (m 1 ≥ M − w), then retailer 2 will order nothing due to negative marginal profit, as reflected by the second part of (i), above. Second, if the wholesale price belongs to [M − (2 − α)K , M − K ], then retailer 2 will order no less than the guaranteed capacity √ (1 − α)K when she assumes retailer 1’s order quantity is more than M − w − 2 (M − K − w)(1 − α)K , as in the second part of (ii). Third, if the wholesale price is small, i.e., the marginal profit for the two retailers is large, then retailer 2 will order to receive the maximum possible allocation, ordering a quantity in (K − m 1 , K ] assuming retailer 1’s order quantity is small, and ordering a quantity in [(1 − α)K , K ] assuming retailer 1’s order is sufficiently large. The results are reflected in the last two parts of (iii) and the whole of (iv), above.

4.3.3 Nash Equilibrium With Theorems 4.1 and 4.2, we now characterize the two retailers’ order quantities, √which are in Nash equilibrium. Let w c (α) = M − γ(α), where γ(α) = 9αK −3 (9α−4)αK . As will be seen, there are two scenarios based on the order between 2 w c (α) and M − (2 − α)K . What’s more, w c (α) is a critical value of wholesale price that affects the retailers’ ordering decisions in equilibrium. Theorem 4.3 Denote (m ∗1 , m ∗2 ) as an order pair in Nash equilibrium, we have: (i) Suppose 1/2 ≤ α ≤ 4/5 (i.e., wc (α) ≥ M − (2 − α)K ) (i.a) if w ≤ M − (1 + α)K , then there exist multiple Nash equilibria m ∗1 × m ∗2 ∈ [αK , K ] × [(1 − α)K ), K ]; (i.b) if w ∈ (M − (1 + α)K , M − (2 − α)K ], then there exist multiple Nash equiˆ K ]. libria m ∗1 × m ∗2 ∈ [αK , K ] × [α, (i.c) if w ∈ (M − (2 − α)K , w c (α)), then there exist multiple Nash equilibria ˆ K ]. m ∗1 × m ∗2 ∈ [β, K ] × [α,

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105

(i.d) if w ∈ [w c (α), M − K ], then there exist multiple Nash equilibria (m ∗1 , m ∗2 ) = and m ∗1 × m ∗2 ∈ [β, K ] × [α, ˆ K ]. In the later set of equilibria, retailers 1 and 2 are allocated αK and (1 − α)K , respectively. Furthermore, the former , M−w ) dominates any equilibrium in the later set of equilibria in equilibrium ( M−w 3 3 that it brings higher profit to each retailer. (i.e.) if w > M − K , then there exists a unique Nash equilibrium (m ∗1 , m ∗2 ) = , M−w ) and the total order quantity of the two retailers is less than the sup( M−w 3 3 plier’s capacity level K . (ii) Suppose 4/5 < α ≤ 1 (i.e., wc (α) < M − (2 − α)K ) (ii.a) if w ≤ M − (1 + α)K , then there exist multiple Nash equilibria m ∗1 × m ∗2 ∈ [αK , K ] × [(1 − α)K ), K ]; (ii.b) if w ∈ (M − (1 + α)K , wc (α)), then there exist multiple Nash equilibria ˆ K ]. m ∗1 × m ∗2 ∈ [αK , K ] × [α, (ii.c) if w ∈ [w c (α), M − (2 − α)K ], then there exist multiple Nash equilibria , M−w ) and m ∗1 × m ∗2 ∈ [αK , K ] × [α, ˆ K ]. In the later set of equi(m ∗1 , m ∗2 ) = ( M−w 3 3 libria, retailers 1 and 2 are allocated αK and (1 − α)K , respectively. Furthermore, , M−w ) dominates any equilibrium in the later set of the former equilibrium ( M−w 3 3 equilibria in that it brings higher profit to each retailer. (ii.d) if w ∈ (M − (2 − α)K , M − K ], then there exist multiple Nash equilibria , M−w ) and m ∗1 × m ∗2 ∈ [β, K ] × [α, ˆ K ]. In the later set of equi(m ∗1 , m ∗2 ) = ( M−w 3 3 libria, retailers 1 and 2 are allocated αK and (1 − α)K , respectively. Furthermore, , M−w ) dominates any equilibrium in the later set of the former equilibrium ( M−w 3 3 equilibria in that it brings higher profit to each retailer. (ii.e) if w > M − K , then there exists a unique Nash equilibrium (m ∗1 , m ∗2 ) = , M−w ) and the total order quantity of the two retailers is less than the supplier’s ( M−w 3 3 capacity level K . , M−w ) ( M−w 3 3

Note that if w < wc (α), then the profit of each retailer and the supplier is the same at any equilibrium order pair in (αK , (1 − α)K ). Theorem 4.3 is in terms of the wholesale price w and conveniently describes the retailers’ total order quantity as a function of the wholesale price. For w < w c (α), total order quantity is K . That is, the supplier’s capacity will be used up when the wholesale price is low. While the wholesale price is slightly higher (w ≥ w c (α)), the total quantity ordered is 2(M − w)/3, which achieves its maximum value of 2γ(α)/3 when w = wc (α). Note that the retailers’ total order quantity is a nonincreasing function of the wholesale price, as expected. What is interesting is that as the wholesale price crosses the critical level wc (α), total order quantity drops from K to a value strictly less than K . This discontinuity is due to the concurrent presence of the capacity limit by the supplier and the competition between the two retailers. As Theorem 4.3 shows, there exist multiple Nash equilibria in the two retailers’ ordering. For case (i.a),(i.b) and (i.c) of the theorem, in all equilibria retailer 1 orders no less than αK , if w ≤ M − (2 − α)K , then retailer 1 orders more than αK ; while if w ∈ (M − (2 − α)K , wc (α)), then retailer 1 orders no less than β to reach an

106

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

equilibrium, and in both cases it receives the maximum possible allocation from the supplier, as specified by the allocation factor α. Accordingly, if w ≤ M − (1 + α)K , then retailer 2 orders more than (1 − α)K ; while if w ∈ (M − (1 + α)K , wc (α)), then retailer 2 orders at least αˆ to reach an equilibrium; and both cases bring retailer 2 an allocation of (1 − α)K . Note that all these equilibria lead to the same profits to all the supply chain members as when the two retailers order (αK , (1 − α)K ). For , M−w ), case (i.d) of Theorem 4.3, we proceed by adopting the equilibrium ( M−w 3 3 since it provides the two retailers higher profits than any other equilibrium. Note that, in this case, total order quantity is strictly less than K . The analysis of scenario (ii) is similar to (i). Theorem 4.3 also shows that both retailers may inflate their orders under fixed , M−w ) under sufficiently large capacfactor allocation, relative to the order pair ( M−w 3 3 ity. This happens when the wholesale price is medium. For example, under conditions that 1/2 ≤ α ≤ 4/5 and M − (2 − α)K < w < wc (α), an order in equilibrium . In such cases, the supplier’s capacity is [β, K ] × [α, ˆ K ] is strictly larger than M−w 3 sold out and retailer 1 maximizes her profit when she receives an allocation of αK in equilibrium ordering. Specifically, retailer 1 will inflate her order to guarantee an allocation of αK , which induces retailer 2 to also inflate her order to guarantee an allocation of (1 − α)K .

4.4 Supplier’s Decision In this section, we consider how the supplier, the Stackelberg game leader, determines his wholesale price to maximize his own profit. Theorem 4.3 illustrates how the two retailers react to the change in wholesale price, and thus it is straightforward to determine the supplier’s optimal wholesale price. As noted in Sect. 4.3.3, for w < w c (α), total order quantity in the non-dominated Nash equilibrium equals total capacity K , and thus the total sale to the retailers is also K . In this case, the supplier’s profit is wK . On the other hand, if w ≥ w c (α), then the total quantity ordered is less than the supplier’s capacity, since the total order quantity 2(M − w)/3 is strictly less than K . Consequently, the supplier’s maximum profit ∗s (α), as a function of the allocation factor α, can be expressed as ∗s (α) = max

 maxc

w∈(0,w (α))

wK ;

2(M − w)w . (α), M) 3

max c

w∈[w

(4.14)

√ For notational simplicity, let λ(α) = 3K − 3 (9α − 4)α + 1 − 3αK , which have the following properties. Lemma 4.3 (i) λ(α) is decreasing in α. (ii) If M > λ(α), then the supplier’s optimal profit ∗s (α) = w c (α)K , which is increasing in α.

4.4 Supplier’s Decision

107

α >α

1

Π*s(α)

2

Π*s(α ) 2

Π* (α ) s 1

λ(α ) 2

M

λ(α ) 1

Fig. 4.1 Supplier’s profit as function of market size M

Theorem 4.4 If 0 < M ≤ λ(α), then the optimal wholesale price is w∗ (α) = M/2 and the supplier’s maximum profit is ∗s (α) = M 2 /6; otherwise, i.e., M > λ(α), the optimal wholesale price is w∗ (α) = w c (α)− (which is greater than 0 for λ(α) > γ(α)) and the supplier’s maximum profit is ∗s (α) = w c (α)K . In Theorem 4.4, the term w c (α)− means that the supplier should choose his wholesale price less than but as close as possible to wc (α). This is due to the discontinuity of total order quantity as a function of the wholesale price when the wholesale price equals w c (α). Theorem 4.4 indicates that, for any fixed factor α, as the market size M increases, the supplier’s profit increases continuously, which is shown by Fig. 4.1. This is intuitive, as M represents the market potential of the supplier’s product. What is not so intuitive is that as M increases, the optimal wholesale price is not necessarily increasing. When the market size M is relatively small, the supplier chooses his wholesale price to induce retail demand, which is below capacity, and thus purposely allows certain capacity to go unused, and when the market size M exceeds a threshold level, i.e., λ(α), the supplier suddenly reduces his wholesale price to achieve full capacity utilization. Therefore, behind the continuously increasing supplier profit as M increases is a sudden drop in wholesale price, a sudden increase in total retail order quantity, and a transition from excess capacity to 100% capacity utilization, as shown in Fig. 4.2. From Theorems 4.3 and 4.4, we can locate supply chain members’ optimal decisions and their corresponding profits under different market sizes, as in Table 4.1. Table 4.1 illustrates that if M ≤ λ(α), then the optimal wholesale price is w∗ (α) = M/2 and the supplier’s maximum profit is ∗s (α) = M 2 /6, and the retailers order ∗ 2 m ∗1 = m ∗2 = M−w = M/6 and receive profits ∗1 (α) = ∗2 (α) = M36 . In this sce3 nario, total order quantity is strictly less than K , i.e., the supplier maintains a capacity surplus to maximize his profit. On the other hand, under condition M > λ(α), the optimal wholesale price is w∗ (α) = (M − γ(α))− and the supplier’s maximum profit is ∗s (α) = w c (α)K , and the two retailers’ order sizes satisfy m ∗1 + m ∗2 ≥ K

108

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation w*(α)

90

α >α >α 3

2

1

80 70 w*(α ) 3

60 50

w*(α ) 1

40 30 20 *

w (α2) 10 0

0

20

40

60 λ(α3) λ(α2)

80 λ(α1)

100

120 M

Fig. 4.2 Supplier’s optimal wholesale price as function of market size M Table 4.1 Supply chain members’ optimal decisions and profits Item M ≤ λ(α) M > λ(α) w∗ m ∗1 m ∗2 ∗s (α) ∗1 (α) ∗2 (α)

w ∗ (α)) = M/2 m ∗1 = M/6 m ∗2 = M/6 ∗s (α) = M 2 /6 ∗1 (α) = M 2 /36 ∗2 (α) = M 2 /36

r∗ (α) ∗sc (α)

r∗ (α) = M 2 /18 ∗sc (α) = 2M 2 /9

w ∗ (α) = (M − γ(α))− m ∗1 ≥ K − m ∗2 m ∗2 ≥ K − m ∗1 ∗s (α) = (M − γ(α))K ∗1 (α) = (γ(α) − K )αK ∗2 (α) = (γ(α) − K )(1 − α)K r∗ (α) = (γ(α) − K )K ∗sc (α) = (M − K )K

and they receive allocations g1 (m ∗1 , m ∗2 ) = αK and g2 (m ∗1 , m ∗2 ) = (1 − α)K , respectively. Correspondingly, retailers 1 and 2 obtain profits (from Eqs. 4.8 and 4.9) ∗1 (α) = (γ(α) − K )αK and ∗2 (α) = (γ(α) − K )(1 − α)K , respectively.

4.5 Impacts of Fixed Factor α In this section, we investigate how the fixed factor α affects supply chain member decisions. Recall that the fixed factor allocation offers priority to retailer 1. In case the two retailers’ total order size exceeds the supplier’s available capacity, retailer 1

4.5 Impacts of Fixed Factor α

109

7 6

u0=3−sqrt(3)

5

u1=1.5424

4

u2=3

τ

3 2 1 0 −1 −2 −3

0

2 u0u1

4 u2

6

8

10

u

Fig. 4.3 Relationship between τ and u

receives an allocation of max{αK , K − m 2 } if she sets an order quantity more than αK , otherwise, she just gets an allocated quantity as she orders. Then, the remaining capacity is allocated to retailer 2. Since the supplier is a game leader making wholesale price decision which directly decides retailers’ equilibrium order quantity. We firstly study the effects of α on supplier’s decision. Recall that in Theorem 4.4, λ(α) is a threshold of the market size that determines supplier’s optimal wholesale price, and by transformation we can get M = λ(α) is equivalent to α =

2 (M K ) 54

2 (M K −6) 2 −6−( M K )

6M K

when M ≤ 3K . For notational

u (u−6) and τ = 54 . Here, simplicity, it is useful to introduce the following: u = M K 6u−6−u 2 u is the ratio of market size to capacity. And the value of τ serves as a threshold of α, where α ≤ τ is equivalent to M ≤ λ(α) and α > τ is equivalent to M > λ(α) under condition u ≤ 3. Besides, for u > 3, M > λ(α) for any α ∈ [ 21 , 1]. Then with market size and capacity exogenously given, the supplier’s optimal decision relies on the value of α. As α belongs to interval [ 21 , 1], it’s necessary to find out how τ values with u ≤ 3. Figure 4.3 depicts the relationship between τ and u. Figure√4.3 shows that for u ∈ (u 0 , u 1 ), τ > 1; for u ∈ [u 1 , u 2 ], τ ∈ [ 21 , 1]. Here, u 0 = 3 − 3, u 1 ≈ 1.5425, u 2 = 3. Now by Theorem 4.4 and the definitions of u and τ , we can precisely express the supplier’s optimal wholesale price decision and profit in terms of fixed factor α, as in the following theorem. 2

2

Theorem 4.5 (i) M < u 1 K . We have M < λ(α) for any α. Thus, the optimal wholesale price is w ∗ (α) = M/2 and the supplier’s maximum profit is ∗s (α) = M 2 /6. (ii) M ∈ [u 1 K , u 2 K ]. If α ∈ [ 21 , τ ], then M ≤ λ(α), w ∗ (α) = M/2, and ∗s (α) =

110

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

Table 4.2 Optimal decisions and profits in small or large market Item M < u1 K M > u2 K w∗ ∗s (α) m ∗1 m ∗2 ∗1 (α) ∗2 (α)

w ∗ (α) = M/2 ∗s (α) = M 2 /6 ∗ m ∗1 = M−w = M6 3 M−w ∗ ∗ m 2 = 3 = M6 ∗1 (α) = M 2 /36 ∗2 (α) = M 2 /36

r∗ (α) ∗sc (α)

r∗ (α) = M 2 /18 ∗sc (α) = 2M 2 /9

Table 4.3 Optimal decisions and profits in medium market Item α≤τ w∗ ∗s (α) m ∗1 m ∗2 ∗1 (α) ∗2 (α)

w ∗ (α) = M/2 ∗s (α) = M 2 /6 ∗ m ∗1 = M−w = M6 3 M−w ∗ ∗ m 2 = 3 = M6 ∗1 (α) = M 2 /36 ∗2 (α) = M 2 /36

r∗ (α) ∗sc (α)

r∗ (α) = M 2 /18 ∗sc (α) = 2M 2 /9

w ∗ (α) = w c (α)− ∗s (α) = w c (α)K m ∗1 ∈ [αK , K ] m ∗2 ∈ [α, ˆ K] ∗1 (α) = (γ(α) − K )αK ∗2 (α) = (γ(α) − K )(1 − α)K r∗ (α) = (γ(α) − K )K ∗sc (α) = (M − K )K

α>τ w ∗ (α) = w c (α)− ∗s (α) = w c (α)K m ∗1 ∈ [αK , K ] m ∗2 ∈ [α, ˆ K] ∗1 (α) = (γ(α) − K )αK ∗2 (α) = (γ(α) − K )(1 − α)K r∗ (α) = (γ(α) − K )K ∗sc (α) = (M − K )K

M 2 /6; if α ∈ (τ , 1], then M > λ(α), w∗ (α) = w c (α)− , and ∗s (α) = w c (α)K . (iii) M > u 2 K . We have M > λ(α) for any α. Thus, the optimal wholesale price is w ∗ (α) = w c (α)− and the supplier’s maximum profit is ∗s (α) = w c (α)K . We define three types of market based on market size. Specifically, we refer to the market as small, medium, and large under conditions M < u 1 K , M ∈ [u 1 K , u 2 K ] and M > u 2 K , respectively. With Theorem 4.5, we characterize the optimal decisions and profits of the supply chain members in Tables 4.2 and 4.3. Specifically, Table 4.2 characterizes the scenarios that the market size is either relatively small or sufficiently large, and Table 4.3 covers the remaining scenario with medium market size. From Table 4.2, when market size is small but can still be larger than the capacity level K (M < u 1 K , u 1 ≈ 1.5), the supplier always sets his optimal wholesale price as M/2, which maintains a strict capacity surplus to maximize his profit at M 2 /6. If the market size is less than K , then this result is consistent with our intuition. It is interesting that the supplier still chooses to sell less than K even when M ≥ K (slightly). In this scenario, the two retailers both order M/6 and gain the corresponding profit M 2 /36, and thus the supply chain’s total profit is 2M 2 /9. As a result, the allocation factor α does not affect the supplier’s decisions when the market size is

4.5 Impacts of Fixed Factor α

111

small, which can be explained as follows. The fixed factor allocation provides priority to retailer 1, with α ≥ 1/2. When the market size is small, retailer 1 orders (M − w)/3 (her desired quantity), which is less than the quantity reserved for her (αK ). Consequently, the remaining capacity is greater than retailer 2’s optimal order quantity. Therefore, the fixed factor α does not affect the supplier and the retailers’ decisions. When market size is sufficiently large (M > u 2 K , u 2 = 3), the retailers will order as much as possible so long as the marginal profit is positive. Bearing this in mind, the supplier sets an optimal wholesale price that is increasing in α and sells out his full capacity, and consequently his optimal profit is increasing in α. That is, it is optimal for the supplier to set α = 1 if α is a decision variable. Note that retailer 1’s order quantity and allocated quantity are both increasing in α and retailer 2’s allocated quantity is decreasing in α. The supplier always faces beyond-capacity-demand and sells out his capacity. These results are consistent with our intuition. Retailers 1 and retailer 2 profits both are decreasing in α. As in the case M > u 2 K , the supply chain’s total profit does not vary with α. That is, varying α only reallocates the profit among these members of the supply chain. Remark 4.1 (i) When market size is sufficiently small, each supply chain member’s profit is independent of the allocation factor α. (ii) When market size is sufficiently large, the supplier prefers a large allocation factor (α = 1, i.e., lexicographic allocation) and his profit is related to market size, but the two retailers both prefer a small allocation factor (α = 1/2) and their profits are independent of market size. Thus far, we have analyzed cases where the market size is either sufficiently small or sufficiently large. Next, we investigate the scenario with medium market size, i.e., M ∈ [u 1 K , u 2 K ]. By Theorem 4.5, we immediately obtain the results shown in Table 4.3. To better understand the results in Table 4.3, we need the following lemma. Lemma 4.4 Under condition that M ∈ [u 1 K , u 2 K ] (i.e., u ∈ [u 1 , u 2 ], which implies that τ ∈ [ 21 , 1]), we provide the comparisons in Table 4.4 in terms of the supply chain members’ optimal decisions when α = τ and α = τ + (τ + means that α values more than but as close as possible to τ ). Table 4.4 in Lemma 4.4 shows changes of supplier’s optimal wholesale price, retailers’ allocated quantity and respective supply chain members’ profits when α approximates the threshold point τ . With Lemma 4.4, we can characterize the scenario with medium market size, as illustrated in Figs. 4.4, 4.5, 4.6, 4.7, 4.8, 4.9 and 4.10, where we show different quantities of interest as a function of α. We show the values for three different market sizes M1 , M2 , M3 . And the values τ1 , τ2 , τ3 indicate the τ values that correspond to the three different market sizes. Figure 4.4 shows how the supplier’s optimal wholesale price changes with α at medium market size. At the point α = τ , the optimal wholesale price has a sudden drop, which implies the selling√of more capacity, and then is increasing in α. Furthermore, if M ∈ [u 1 K , (9 − 3 5)K ) (the case of M1 ), then the optimal wholesale price when

112

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

Table 4.4 Supply chain members’ optimal decisions when α approaches τ Item

α=τ

α = τ+

Comparison

w∗

M/2

w c (τ )−

M/2 > w c (τ )−

∗s

M 2 /6

(M − γ(τ ))K

M 2 /6 = (M − γ(τ ))K

g1 (m ∗1 , m ∗2 ) g2 (m ∗1 , m ∗2 )

M 6 M 6

τK

M 6

(1 − τ )K

∗1

if√M ∈ [u 1 K , (6 − 3 2)K ), M6 > (1 − τ )K ; if √ M ∈ [(6 − 3 2)K , u 2 K ], M 6 ≤ (1 − τ )K

M 2 /36

(γ(τ ) − K )τ K

M 2 /36 < (γ(τ ) − K )τ K

∗2

M 2 /36

(γ(τ ) − K )(1 − τ )K

if M ∈ [u 1 K , 1.8351K ),M 2 /36 > (γ(τ ) − K )(1 − τ )K ; if M ∈ [1.8351K , u 2 K ], M 2 /36 ≤ (γ(τ ) − K )(1 − τ )K

sc

2M 2 /9

(M − K )K

2M 2 /9 < (M − K )K

< τK

45 40

M

35

M2

3

25

*

w (α)

30

20

M

1

15 10 5 0 0.5

τ1 τ2

0.6

0.7

α τ3

Fig. 4.4 Optimal Wholesale Price in Medium Market

0.8

0.9

1

4.5 Impacts of Fixed Factor α

113

30 M >M >M 3

2

1

25

m1*(α)

20

15 M

3

M

2

10 M

1

5

0 0.5

0.7

0.6

τ τ

1 2

α τ3

0.8

0.9

1

Fig. 4.5 Retailer 1s Equilibrium Allocated Quantity in Medium Market 15 M3

M3>M2>M1 M2

m*2(α)

10

M1

5

0 0.5

τ τ

1 2

0.6

0.7

α τ3

0.8

0.9

Fig. 4.6 Retailer 2s Equilibrium Allocated Quantity in Medium Market

1

114

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation 1400 M3>M2>M1 M

1200

3

1000 M

2

Πs*(α)

800

600

M1

400

200

0 0.5

τ τ

1 2

0.6

0.7

α τ3

0.8

0.9

1

Fig. 4.7 Supplier’s Optimal Profit in Medium Market

α > τ is less than that when α ≤ τ (M/2). Then Figs. 4.5 and 4.6 show how two retailers allocated quantities in equilibrium ordering change with α at medium market size corresponding to the optimal wholesale price. Retailer 1’s allocated quantity jumps at the point α = τ , then is increasing in proportion with α. It indicates that when α > τ , retailer 1 will always order more than αK regardless of the increase of optimal wholesale price and is also allocated that quantity. Different to retailer 1, retailer 2’s allocated quantity experiences a√drop or jump at the √ point α = τ which depends on the market size (M < (6 − 3 2)K or M > (6 − 3 2)K ), then is decreasing in proportion with (1 − α). However, in spite of a low allocated quantity, retailer 2 always orders more than (1 − α)K to reach an equilibrium. Figures 4.7, 4.8, 4.9 and 4.10 show how the supplier members’ optimal profits change with α at medium market size. The supplier’s profit is increasing in α. Although the supplier’s optimal wholesale price has a sudden drop at the point α = τ , the supplier’s optimal profit increases continuously with α. Besides, both retailers’ profits have a sudden jump at the point α = τ when the market size is relatively large (M ∈ [1.8351K , u 2 K ]), while both they are decreasing in α when α > τ because of the increasing optimal wholesale price. Intuitively, the two retailers both prefer allocation factor α = τ + because that the supplier adjusts her wholesale price according to the value of α. It is interesting to see, the supply chain profit jumps at the point α = τ and then maintains a constant value for any α ∈ (τ , 1] at the given market size. The reason is when α > τ , the supplier can always sell out all its capacity at its optimal wholesale price.

4.5 Impacts of Fixed Factor α

115

250 M3>M2>M1 200

M

3

150 * Π (α)

M

1

2

100

M1

50

0 0.5

τ1 τ2

0.6

0.7

α τ3

0.8

0.9

1

0.8

0.9

1

Fig. 4.8 Retailer 1’s Profit in Medium Market 200 180 160 M3 140

M2

2

* Π (α)

120 100 80 M1

60 40 20 0 0.5

τ1 τ2

0.6

0.7

Fig. 4.9 Retailer 2’s Profit in Medium Market

α τ3

116

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation 1400 M

3

1200

M2

* Πsc(α)

1000

800

600 M1 M >M >M 3

400

2

1

200

0 0.5

τ1 τ2

0.6

0.7

α τ3

0.8

0.9

1

Fig. 4.10 Supply Chain Profit in Medium Market

Remark 4.2 In medium market, the supplier prefers an allocation factor to be as large as possible (α = 1), but the two retailers prefer allocation factor α = τ + , which is related to market size M and capacity level K (M ∈ [1.8351K , u 2 K ]). The supply chain achieves the same maximum profit in the region α ∈ (τ , 1].

4.6 Comparison with Centralized Supply Chain In this section, we compare the supply chain profit under fixed factor allocation with the profit of a centralized supply chain. Let c∗ sc denote the maximum supply chain profit under centralized decisions. To obtain c∗ sc , let q be the total quantity sold. The supply chain profit can be written as Csc = (M − q)q. Note that the optimal selling quantity q ∗ = min{K , M2 }. Thus, M2 c∗ c∗ c∗ sc = 4 for M ≤ 2K and sc = (M − K )K for M > 2K . The value sc provides a benchmark for measuring the loss of efficiency in a decentralized supply chain using the fixed factor allocation, where the supplier chooses wholesale price to maximize his own value instead of the supply chain value. Table 4.5 illustrates the results of comparing supply chain profits obtained from decentralized and centralized supply chains. An important observation from Table 4.5 is that the supply chain profit in the decentralized supply chain with fixed factor allocation is the same as in the centralized supply chain when M >> K , i.e., the supplier chooses his optimal wholesale price that coordinates the supply chain, i.e., obtains the same supply chain profit as in

4.7 Comparison with Other Allocations

117

Table 4.5 Comparison with centralized supply chain M Decentralized system Centralized system M ∈ [0, u 1 K ) M ∈ (u 1 K , 2K ]

∗sc (α) ∗sc (α) α≤τ

=

2M 2 /9

=

2M 2 /9

for

c∗ = M 2 Pi sc 4 M2 c∗ = sc 4 2

M ∈ (2K , u 2 K ]

M ∈ (u 2 K , +∞)

M ∗sc (α) = (M − K )K c∗ sc = 4 for α > τ ∗sc (α) = 2M 2 /9 for c∗ sc = (M − K )K α≤τ ∗sc (α) = (M − K )K c∗ sc = (M − K )K for α > τ ∗sc (α) = (M − K )K c∗ sc = (M − K )K

Comparison ∗sc (α) < c∗ sc ∗sc (α) < c∗ sc ∗sc (α) < c∗ sc ∗sc (α) < c∗ sc ∗sc (α) = c∗ sc ∗sc (α) = c∗ sc

a centralized supply chain. Specifically, the optimal decision in the centralized system is to sell the whole capacity to the market when the market size is large enough (roughly, M > 2K ). In the decentralized system with two retailers’ equilibrium order quantities in consideration, the supplier’s optimal decision is to set a wholesale price to sell out the total capacity. Accordingly, under the manufacturer’s selected wholesale price, the two retailers each orders as much as possible (K ) because of quantity competition. Consequently, each retailer is allocated her guaranteed quantity, i.e., αK by retailer 1 and (1 − α)K by retailer 2. As a result, the centralized and decentralized systems sell the same quantity K and achieve the same supply chain profit (M − K )K . Remark 4.3 The fixed factor allocation coordinates the supply chain when M ∈ (2K , u 2 K ] with α > τ , and when M > u 2 K with α ≥ 1/2.

4.7 Comparison with Other Allocations Note that fixed factor allocation degenerates to lexicographic allocation when α = 1. We next compare fixed factor allocation with proportional allocation. We first cite a result from Chen et al. (2013), as in the following proposition. Proposition 4.1 √ (Chen et al. 2013) Under proportional allocation, (i) if M ≤ 3(2 − 2)K , then the supplier’s optimal wholesale price is w∗ = M/2, achieving a profit √ of ∗s = M 2 /6; is w ∗ = (M + (ii) if M√> 3(2 − 2)K , then the supplier’s optimal wholesale √ price − ∗ − 3K − 3 2K ) , achieving a profit of s = K (M + 3K − 3 2K ) . With Theorem 4.4 and Proposition 4.1, we can immediately obtain the following result from the view of the supplier and supply chain.

118

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

√ Theorem 4.6 If α = 2/2, then (i) the supplier and supply chain will obtain the same profit under fixed factor and proportional allocations; (ii) in equilibrium, under fixed factor allocation, retailer 1 gains more profit, retailer 2 gains less profit, and the total profit of the two retailers is the same, compared with under proportional allocation. Theorem 4.6 indicates that fixed factor √ allocation is a general rule that is equivalent to proportional allocation with α = 2/2 in terms of the supplier’s and the supply chain’s profits, under the supplier’s optimal wholesale price decision and the retailers’ equilibrium ordering. Next we compare fixed factor, lexicographic and proportional allocations numerically in terms of supply chain profit. There are three cases characterized by different levels of market size M in terms of comparing the supply chain profit under the three allocations. Specifically, 1) if M < u 1 K , then the supply chain profits are the √ same under the three allocations, which are 2M 2 /9; 2) if u 1 K ≤ M ≤ 3(2 − 2)K , then the supply chain obtains the same profit (M − K )K under fixed factor (with α > τ ) and lexicographic allocations, which is larger than that of 2M 2 /9 under proportional allocation, while when α < τ , the supply chain gains √ the same profit under fixed factor and proportional allocations; 3) if M > 3(2 − 2)K , then the three allocations induce the same √ supply chain profit (M − K )K . Comparisons under Case 2 (u 1 K ≤ M < 3(2 − 2)K ) are illustrated in Fig. 4.11. The figure shows that the supply chain obtains more profit under fixed factor allocation than that under proportional allocation with market size at certain levels. And for a large range of α, instead of single point α = 1, fixed factor allocation offers the same supply chain profit as lexicographic allocation does.

700

600

500

Π*

sc

400

300

200 −−Proportional allocation −−Fixed Factor allocation

100

−−Lexicographic allocation 0 0.5

0.6

0.7

α

τ

0.8

0.9

Fig. 4.11 Supply Chain Profits under Three Allocations, as Function of α

1

4.9 Concluding Remarks

119

4.8 Optimal Capacity Choice In this section, we consider how to choose optimal capacity and optimal wholesale price from the view of the supplier. From Theorem 4.4, for each capacity level K , the supplier’s optimal profit is ∗s (K )

=

⎧ ⎨ max K {w c (α)K − cK }, if K ∈ [0, 2

⎩ max K { M6 − cK },

if K ∈

√√

M ), (9α−4)α+1−3α (4.15) [ √√ M , +∞), 3−3 (9α−4)α+1−3α 3−3

where c is the cost for each capacity unit. To make the supplier’s profit positive, we assume M > c. Theorem 4.7 For the supplier, the optimal capacity is K ∗ (α) = )− , ( M+c 2

M−c √ , 9α−3 (9α−4)α

and

which is independent of the allocation factor α, optimal wholesale price is 2 (M−c) √ and the corresponding optimal profit is 2(9α−3 . (9α−4)α) From Theorem 4.7, we see that the supplier can achieve the maximum profit 2 (M−c) M−c √ by choosing capacity level K ∗ (α) = 9α−3√ and wholesale 2(9α−3 (9α−4)α) (9α−4)α

price ( M+c )− . This optimal setting for the supplier induces that total order quantity 2 M−c is from the two retailers be greater than K ∗ . That is, the capacity 9α−3√ (9α−4)α allocated in full.

4.9 Concluding Remarks This paper considers a supply chain consisting of one supplier and two competing retailers. The important components of our model are the supplier’s limited production capacity and the Cournot competition between the retailers. Because of the capacity constraint, an allocation rule is needed for capacity allocation in case the retailers’ total order size exceeds the supplier’s available capacity. The fixed factor allocation rule is proposed, which first guarantees a certain proportion of capacity to a prioritized retailer, then the surplus capacity can be allocated to the other retailer, incorporating the ideas of proportional and lexicographic allocations. Under fixed factor allocation, with exogenously given capacity and market demand, the supplier should carefully choose the allocation factor and the wholesale price to maximize his profit, keeping in mind that retailers order in equilibrium to maximize their own individual profits. We show that both in sufficiently small and large markets, the allocation factor has no effects on the supply chain’s profit. However, in medium market the supply chain’s profit is not affected by the allocation factor when the factor is greater than a particular threshold, and accordingly, the retailers share the supply chain profit with the supplier depending on the value of the allocation factor. Also, we prove

120

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

that retailers prefer an allocation factor different from that of the supplier, but the preferred allocation factor values by both the retailers and the supplier are in the range that maximizes total supply chain profit. Furthermore, we show that the fixed factor allocation coordinates the supply chain when the market size is sufficiently large. We also compare fixed factor and proportional allocations. And we show the fixed factor allocation can incorporate both lexicographic and proportional allocations from the perspective of the supplier and the supply chain. Moreover, the supply chain obtains more profit under fixed factor allocation than that under proportional allocation with market size at certain levels. And for a large range of α, instead of single point α = 1, fixed factor allocation offers the same supply chain profit as lexicographic allocation does. Finally, we demonstrate how the supplier can optimize his capacity level and wholesale price under fixed factor allocation. Our work can be extended in several ways. First, it would be interesting to consider asymmetric retailers with different market powers. Second, it would be valuable to study a case with more than two retailers. Third, it would be worthwhile to compare fixed factor allocation with other allocations, such as linear and uniform allocations. Finally, it would be helpful to study the gaming effect caused by fixed factor allocation and bounded rationality using laboratory experiments, as the analysis by Chen et al. (2012) for proportional allocation.

Appendix Proof of Lemma 4.1 Note that G 12 (m 1 , m 2 ) in Eq. (4.12) can be written as follows. G ∗12 =

max G 12 (m 1 , m 2 ) ⎧ ∗ π1 = maxm 1 ∈(K −m 2 ,K ] π1 , ⎪ ⎪ ⎪ ⎨ = ∗ ∗ ⎪ ⎪ π23 = max π2 = maxm 1 ∈(K −m 2 ,αK ] π2 ; ⎪ ⎩ m 1 ∈(K −m 2 ,K ]

if m 2 ≤ (1 − α)K ;

(A.1) π3∗ = maxm 1 ∈(αK ,K ] π3 ,

if m 2 ∈ ((1 − α)K , K ].

where π1 = (M − K − w)(K − m 2 ), π2 = (M − K − w)m 1 , π3 = (M − K − w)αK . Namely, we divide G 12 (m 1 , m 2 ) into two sub-scenarios based on retailer 2’s order quantity: sub-scenario 1: m 2 ≤ (1 − α)K , and sub-scenario 2: m 2 ∈ ((1 − α)K , K ]. Immediately, we can gain the optimal value π1∗ = (M − K − w)(K − m 2 ) and optimal solution m ∗12−1 ∈ (K − m 2 , K ] with the sub-scenario 1: m 2 ≤ (1 − α)K . Under the sub-scenario 2: m 2 ∈ ((1 − α)K , K ], we have

Appendix

121

 (K − m 2 )+ , if M − w ≤ K ; m ∗12−2 = arg max π2 = αK , if M − w > K . m 1 ∈(K −m 2 ,αK ]  + (M − K − w)(K − m 2 ) , if M − w ≤ K ; π2∗ = (M − K − w)αK , if M − w > K .

(A.2)

and m ∗12−3 = arg π3∗ =

max

m 1 ∈(K −m 2 ,αK ]

max

m 1 ∈(αK ,K ]

π3 ∈ [αK , K ],

π3 = (M − K − w)αK .

(A.3)

Comparing π2∗ in Eq. (A.2) with π3∗ in Eq. (A.3), we can directly gain the following result with condition m 2 ∈ ((1 − α)K , K ] holding:  (K − m 2 )+ if M − w ≤ K ; m ∗23 = arg max{π2∗ ; π3∗ } = ∈ [αK , K ] if M − w > K . m1  (M − K − w)(K − m 2 )+ , if M − w ≤ K ; ∗ = max{π2∗ ; π3∗ } = π23 (M − K − w)αK , if M − w > K . Consequently, by the analysis of sub-scenario 1 and sub-scenario 2, we prove the lemma.  Proof of Theorem 4.1 To prove the theorem, we first introduce Lemma 4.5. Lemma 4.5 Under condition M − w ∈ [K , (1 + α)K ], if m 2 ≤ αˆ or m 2 ≥ M − √ 2 2) ≥ (M − w − K )αK . Furthermore, we w + 2 (M − w − K )αK , then (M−w−m 4 have √ (a) M − w + 2 (M − w − K )αK > 2K − (M − w); (b) αˆ > (1 − α)K ; (c) αˆ < 2K − (M − w). Proof. Under condition M − w ∈ [K , (1 + α)K ], inequality w − K )αK is equivalent to

(M−w−m 2 )2 4

≥ (M −

m 22 − 2(M − w)m 2 + (M − w)2 − 4(M − w − K )αK ≥ 0, and is also equivalent to   m 2 ≥ M − w + 2 (M − w − K )αK or m 2 ≤ M − w − 2 (M − w − K )αK . √ is equivalent to 2(M − Note that M − √ w + 2 (M − w − K )αK > 2K − (M − w)√ w − K ) + 2 (M − w − K )αK > 0, and that M −√ w − 2 (M − w − K )αK > (1 − α)K is equivalent to M − w − (1 − α)K > 2 (M − w − K )αK . Further,

122

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

√ we have that M − w − 2 (M − w − K )αK < 2K − (M − w). Thus, the lemma holds.  Now we prove the theorem. By Eq. (4.10), we compare G ∗11 and G ∗12 according the range of w, which can be divided into two cases: Case 1: w ≥ M − K . In the case, we have G ∗11 G ∗12

 =  =

0, m ∗11 = 0 (M−w−m 2 )2 , m ∗11 = 4

M−w−m 2 2

if m 2 ∈ (M − w, K ]; if m 2 ∈ [0, M − w].

(M − K − w)(K − m 2 ), m ∗12 ∈ [K − m 2 , K ] if m 2 ∈ [0, (1 − α)K ]; (M − K − w)(K − m 2 )+ , m ∗12 = (K − m 2 )+ if m 2 ∈ ((1 − α)K , K ].

Case 1 can be divided into the following two subcases to analyze. Case 1.a. w ∈ (M − (1 − α)K, M]. In this subcase, 0 ≤ M − w < (1 − α)K < K . Thus, we have: 2 2) > G ∗12 = (M − K − w)(K − m 2 ). if 0 ≤ m 2 < M − w, then G ∗11 = (M−w−m 4 2 2) 2 Hence, 1 (w, m 2 ) = (M−w−m and m 1 (m 2 ) = m ∗11 = M−w−m ; 4 2 ∗ ∗ if m 2 ∈ [M − w, K ], then G 11 = 0 and G 12 is negative. Hence, 1 (w, m 2 ) = 0 and m 1 (m 2 ) = m ∗11 = 0. Case 1.a. is illustrated as follows. (M−w−m 2 )2 > G∗ 12 4

∗ G∗ 11 = 0 > G 12

∗ G∗ 11 = 0 > G 12

G∗ 12 = (M − K − w)(K − m 2 )

G∗ 12 = (M − K − w)(K − m 2 )

+ G∗ 12 = (M − K − w)(K − m 2 )

0

M −w

G∗ 11 =

m 2

(1 − α)K

K

Case 1.b. w ∈ [M − K, M − (1 − α)K ]. In this subcase, we have 0 < (1 − α)K < M − w < K . Thus, we have: 2 2) and G ∗12 = (M − K − w)(K − if m 2 ∈ (0, (1 − α)K ], then G ∗11 = (M−w−m 4 2 2) m 2 ). We have that G ∗11 > G ∗12 . Hence, 1 (w, m 2 ) = (M−w−m and m 1 (m 2 ) = 4 M−w−m 2 ∗ m 11 = ; 2 2 2) if m 2 ∈ ((1 − α)K , M − w], then G ∗11 = (M−w−m , and G ∗12 = (M − K − w) 4 (K − m 2 )+ is negative. Thus, we have that G ∗11 > G ∗12 . Hence, 1 (w, m 2 ) = (M−w−m 2 )2 2 and m 1 (m 2 ) = m ∗11 = M−w−m ; 4 2 ∗ if m 2 ∈ (M − w, K ], then G 11 = 0 and G ∗12 = (M − K − w)(K − m 2 )+ is negative. Thus, we have that G ∗11 > G ∗12 . Hence, 1 (w, m 2 ) = 0 and m 1 (m 2 ) = m ∗11 = 0. Case 1.b. is illustrated as follows. G∗ 12 = (M − K − w)(K − m 2 )

(M−w−m 2 )2 > G∗ 12 4 ∗ G 12 = (M − K − w)(K − m 2 )+

0

(1 − α)K

G∗ 11 =

(M−w−m 2 )2 > G∗ 12 4

Combining Cases 1.a. and 1.b., we have (i).

∗ G∗ 11 = 0 > G 12 + G∗ 12 = (M − K − w)(K − m 2 )

m 2

M −w

K

Appendix

123

Case 2: w < M − K . In this subcase, we have G ∗11 G ∗12

=

 (M−w−m 

=

2 2)

2 , m ∗11 = M−w−m if m 2 ≤ 2K − (M − w), 2 (M − K − w)(K − m 2 ), m ∗12 = K − m 2 if m 2 ∈ (2K − (M − w), K ].

4

(M − K − w)(K − m 2 ), m ∗1 ∈ [K − m 2 , K ] if m 2 ∈ [0, (1 − α)K ]; if m 2 ∈ ((1 − α)K , K ]. (M − K − w)αK , m ∗1 ∈ [αK , K ]

The case can be divided into the following three subcases to analyze. Case 2.a. w ∈ (M − (1 + α)K, M − K ). In this subcase, we know that 0 < (1 − α)K < αˆ < 2K − (M − w) < K . By Lemma 4.5, we consider the following conditions: 2 2) and G ∗12 = (M − K − w)(K − if m 2 ∈ [0, (1 − α)K ), then G ∗11 = (M−w−m 4 2 2) m 2 ), which implies G ∗11 > G ∗12 . Therefore, we have 1 (w, m 2 ) = (M−w−m and 4 2 m 1 (m 2 ) = M−w−m ; 2 2 2) if m 2 ∈ [(1 − α)K , α], ˆ then G ∗11 = (M−w−m and G ∗12 = (M − K − w)αK , 4 ∗ ∗ which implies G 11 > G 12 by Lemma 4.5. Therefore, we have 1 (w, m 2 ) = (M−w−m 2 )2 2 and m 1 (m 2 ) = M−w−m ; 4 2 2 2) if m 2 ∈ (α, ˆ 2K − (M − w)), then G ∗11 = (M−w−m and G ∗12 = (M − K − w) 4 ∗ ∗ αK , which implies G 11 ≤ G 12 . Therefore, we have 1 (w, m 2 ) = (M − K − w)αK and m 1 (m 2 ) ∈ [αK , K ]; if m 2 ∈ [2K − (M − w), K ], then G ∗11 = (M − K − w)(K − m 2 ) and G ∗12 = (M − K − w)αK , which implies G ∗11 < G ∗12 . Therefore, we have 1 (w, m 2 ) = (M − K − w)αK and m 1 (m 2 ) ∈ [αK , K ]. Case 2.a. is illustrated as follows. G∗ 11 =

(M−w−m 2 )2 > G∗ 12 4

G∗ 12 =(M − K − w)(K − m 2 ) 0

G∗ 11 =

(M−w−m 2 )2 > G∗ 12 4

G∗ 12 =(M − K − w)αK (1 − α)K

(M−w−m 2 )2 ∗ ∗ G∗ 12 4 ∗ G 12 = (M − K − w)(K − m 2 )

G∗ 11 =

∗ G∗ 11 = (M − K − w)(K − m 2 ) = G 12

∗ G∗ 11 = (M − K − w)(K − m 2 ) < G 12

G∗ 12 = (M − K − w)(K − m 2 )

G∗ 12 = (M − K − w)αK

2K − (M − w)

0

(1 − α)K

m2



K

From the analysis of Case 2.b., we have (iii). Case 2.c. w ≤ M − 2K . In this subcase, 0 < (1 − α)K < K , then we have if m 2 ∈ [0, (1 − α)K ), then we can obtain that G ∗11 = (M − K − w)(K − m 2 ) and G ∗12 = (M − K − w)(K − m 2 ), which implies that G ∗11 = G ∗12 . Therefore, we have 1 (w, m 2 ) = (M − K − w)(K − m 2 ) and m 1 (m 2 ) ∈ [K − m 2 , K ]; if m 2 ∈ [(1 − α)K , K ], then we can obtain that G ∗11 = (M − K − w)(K − m 2 ) and G ∗12 = (M − K − w)αK , which implies that G ∗11 ≤ G ∗12 . Therefore, we have 1 (w, m 2 ) = (M − K − w)αK and m 1 (m 2 ) ∈ [αK , K ]. Case 2.c. is illustrated as follows. ∗ G∗ 11 = (M − w − K )(K − m 2 ) = G 12

∗ G∗ 11 = (M − w − K )(K − m 2 ) < G 12

G∗ 12 = (M − K − w)(K − m 2 )

G∗ 12 = (M − w − K )αK (1 − α)K

0

From the analysis if Case 2.c., we have (iv).



m2 K



Proof of Lemma 4.2 Note that G 22 (m 1 , m 2 ) can be written as follows. G ∗22 =

max G 22 (m 1 , m 2 ) m 2 ∈(K −m 1 ,K ] ⎧ ∗ πˆ 1 = maxm 2 ∈(K −m 1 ,K ] πˆ 1 , ⎪ ⎪ ⎪ ⎨ if m 1 ≤ αK ;

= ∗ = max π ∗ = max π ˆ ˆ π ˆ ; πˆ 3∗ = maxm 2 ∈((1−α)K ,K ] πˆ 3 , ⎪ m 2 ∈(K −m 1 ,(1−α)K ] 2 23 2 ⎪ ⎪ ⎩ if m 1 ∈ (αK , K ].

(A.4)

where πˆ 1 = (M − K − w)(K − m 1 ), πˆ 2 = (M − K − w)m 2 , πˆ 3 = (M − K − w)(1 − α)K . It is easy to obtain that m ∗21 = arg

max

m 2 ∈(K −m 1 ,K ]

πˆ 1 ∈ (K − m 1 , K ],

πˆ 1∗ = (M − K − w)(K − m 1 ).

(A.5)

Appendix

125

With condition m 1 ∈ (αK , K ], we have πˆ 2∗ =

max

m 2 ∈(K −m 1 ,(1−α)K ]



=

πˆ 3∗ =

max

m 2 ∈((1−α)K ,K ]

πˆ 2

(M − K − w)(K − m 1 )+ , if M − w ≤ K ; (M − K − w)(1 − α)K , if M − w > K .

πˆ 3 = (M − K − w)(1 − α)K , m ∗2 ∈ ((1 − α)K , K ].

Comparing π2∗ with π3∗ , we can directly gain the following result with condition m 1 ∈ (αK , K ] holding: m ∗23 = arg max{πˆ 2∗ ; πˆ 3∗ } m2  if M − w ≤ K ; (K − m 1 )+ = ∈ [(1 − α)K , K ] if M − w > K . ∗ πˆ 23 = max{πˆ 2∗ ; πˆ 3∗ }  (M − K − w)(K − m 1 )+ , if M − w ≤ K ; = (M − K − w)(1 − α)K , if M − w > K .



Hence, the lemma holds.

Proof of Theorem 4.2 First, we would like to introduce the following result: inequal2 1) ity (M−w−m > (M − K − w)(1 − α)K is equivalent to m 1 < β or m 1 > M − 4 √ 2 1) w + 2 (M − K − w)(1 − α)K . Note that inequality (M−w−m > (M − K − w) 4 √ (1 − α)K√is equivalent to M − w − m 1 > 2 (M − K − w)(1 − α)K or M − w − m 1 < −2 (M − K − w)(1 − α)K , and thus the result holds. This result will be implicitly used in the following proof. With Eq. (4.13) and Lemma 4.2, we can divide the problem into the following two cases to compare G ∗21 and G ∗22 according the range of w Case 1: w ≥ M − K . In the case, we have G ∗21 = G ∗22 =

 

0, m ∗21 = 0 (M−w−m 1 )2 , m ∗21 = 4

M−w−m 1 2

if m 1 ∈ (M − w, K ]; if m 1 ∈ [0, M − w].

(M − K − w)(K − m 1 ), m ∗22 ∈ [K − m 1 , K ] if m 1 ∈ [0, αK ]; (M − K − w)(K − m 1 )+ , m ∗22 = (K − m 1 )+ if m 1 ∈ (αK , K ].

Case 1 can be divided into the following two subcases to analyze. Case 1.a. w ∈ (M − αK, M]. In this subcase, 0 ≤ M − w < αK < K . Thus, we have: 2 1) > G ∗22 = (M − K − w)(K − m 1 ). if 0 ≤ m 1 < M − w, then G ∗21 = (M−w−m 4 (M−w−m 1 )2 1 Hence, 2 (w, m 1 ) = and m 2 (m 1 ) = m ∗21 = M−w−m ; 4 2

126

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

if m 1 ∈ [M − w, K ], then G ∗21 = 0 and G ∗22 is negative. Hence, 1 (w, m 1 ) = 0 and m 2 (m 1 ) = m ∗21 = 0. Case 1.a. is illustrated as follows. (M−w−m 1 )2 > G∗ 22 4

∗ G∗ 21 = 0 > G 22

∗ G∗ 21 = 0 > G 22

G∗ 22 = (M − K − w)(K − m 1 )

G∗ 22 = (M − K − w)(K − m 1 )

+ G∗ 22 = (M − K − w)(K − m 1 )

0

M −w

G∗ 21 =

m 1

αK

K

Case 1.b. w ∈ [M − K, M − αK ]. In this subcase, we have 0 < αK < M − w < K . Thus, we have: 2 1) and G ∗22 = (M − K − w)(K − m 1 ). if m 1 ∈ (0, αK ], then G ∗21 = (M−w−m 4 2 1) We have that G ∗21 > G ∗22 . Hence, 2 (w, m 1 ) = (M−w−m and m 2 (m 1 ) = m ∗21 = 4 M−w−m 1 ; 2 2 1) if m 1 ∈ (αK , M − w], then G ∗21 = (M−w−m , and G ∗22 = (M − K − w)(K − 4 2 1) m 1 )+ is negative. Thus, we have that G ∗21 > G ∗22 . Hence, 2 (w, m 1 ) = (M−w−m 4 1 and m 2 (m 1 ) = m ∗21 = M−w−m ; 2 if m 1 ∈ (M − w, K ], then G ∗21 = 0 and G ∗22 = (M − K − w)(K − m 1 )+ is negative. Thus, we have that G ∗21 > G ∗22 . Hence, 2 (w, m 1 ) = 0 and m 2 (m 1 ) = m ∗21 = 0. Case 1.b. is illustrated as follows. G∗ 22 = (M − K − w)(K − m 1 )

(M−w−m 1 )2 > G∗ 22 4 + G∗ 22 = (M − K − w)(K − m 1 )

0

αK

G∗ 21 =

(M−w−m 1 )2 > G∗ 22 4

∗ G∗ 21 = 0 > G 22 + G∗ 22 = (M − K − w)(K − m 1 )

m 1

M −w

K

Combining Cases 1.a. and 1.b., we have (i). Case 2: w < M − K . In this subcase, we have G ∗21 G ∗22

=

 (M−w−m 

=

2 1)

1 , m ∗2 = M−w−m if m 1 ≤ 2K − (M − w), 2 (M − K − w)(K − m 1 ), m ∗2 = K − m 1 if m 1 ∈ (2K − (M − w), K ].

4

(M − K − w)(K − m 1 ), m ∗2 ∈ (K − m 1 , K ] if m 1 ∈ [0, αK ]; (M − K − w)(1 − α)K , m ∗2 ∈ [(1 − α)K , K ] if m 1 ∈ (αK , K ].

The case can be divided into the following three subcases to analyze. Case 2.a. M − w ∈ (M − (2 − α)K, M − K ). In this subcase, we know that 0 < αK < β < 2K − (M − w) < K . By Lemma 4.5, we consider the following conditions: 2 1) and G ∗22 = (M − K − w)(K − m 1 ), if m 1 ∈ [0, αK ), then G ∗21 = (M−w−m 4 2 1) which implies G ∗21 > G ∗22 . Therefore, we have 2 (w, m 1 ) = (M−w−m and m 2 (m 1 ) 4 1 = M−w−m ; 2 2 1) if m 1 ∈ [αK , β], then G ∗21 = (M−w−m and G ∗22 = (M − K − w)(1 − α)K , 4 ∗ ∗ which implies G 21 ≥ G 22 by Lemma 4.5. Therefore, we have 2 (w, m 1 ) = (M−w−m 1 )2 1 and m 2 (m 1 ) = M−w−m ; 4 2

Appendix

127

1) if m 1 ∈ (β, 2K − (M − w)), then G ∗21 = (M−w−m and G ∗22 = (M − K − w) 4 ∗ ∗ (1 − α)K , which implies G 21 < G 22 . Therefore, we have 2 (w, m 1 )=(M − K − w)(1 − α)K and m 2 (m 1 ) ∈ [(1 − α)K , K ]; if m 1 ∈ [2K − (M − w), K ], then G ∗21 = (M − K − w)(K − m 1 ) and G ∗22 = (M − K − w)(1 − α)K , which implies G ∗21 < G ∗22 . Therefore, we have 2 (w, m 1 ) = (M − K − w)(1 − α)K and m 2 (m 1 ) ∈ [(1 − α)K , K ]. Case 2.a. is illustrated as follows. 2

G∗ 21 =

(M−w−m 1 )2 > G∗ 22 4

G∗ 22 = (M − K − w)(K − m 1 )

G∗ 21 =

(M−w−m 1 )2 >G ∗ 22 4

G∗ 22 =(M−K −w)(1 − α)K

√ β=M−w−2 (M−w−K )(1−α)K

αK

0

(M−w−m 1 )2 ∗ ∗ G∗ 22 4

G∗ 22 = (M − K − w)(K − m 1 ) 0

∗ G∗ 21 =(M−K −w)(K −m 1 )=G 22

∗ G∗ 21 =(M−K −w)(K −m 1 ) < G 22

G∗ 22 =(M−K −w)(K − m 1 )

G∗ 22 =(M−K −w)(1−α)K

2K − (M − w)



αK

m1 K

From the analysis of Case 2.b., we have (iii). Case 2.c. w ≤ M − 2K . In this subcase, 0 < αK < K , then we have if m 1 ∈ [0, αK ), then we can obtain that G ∗21 = (M − K − w)(K − m 1 ) and ∗ G 22 = (M − K − w)(K − m 1 ), which implies that G ∗21 = G ∗22 . Therefore, we have 2 (w, m 1 ) = (M − K − w)(K − m 1 ) and m 2 (m 1 ) ∈ [K − m 1 , K ]; if m 1 ∈ [αK , K ], then we can obtain that G ∗21 = (M − K − w)(K − m 1 ) and ∗ G 22 = (M − K − w)(1 − α)K , which implies that G ∗21 ≤ G ∗22 . Therefore, we have 2 (w, m 1 ) = (M − K − w)(1 − α)K and m 2 (m 1 ) ∈ [(1 − α)K , K ]. Case 2.c. is illustrated as follows. ∗ G∗ 21 = (M − w − K )(K − m 1 ) = G 22

∗ G∗ 21 = (M − w − K )(K − m 1 ) < G 22

G∗ 22 = (M − K − w)(K − m 1 )

G∗ 22 = (M − w − K )(1 − α)K

0

From the analysis if Case 2.c., we have (iv).

αK



m1 K



128

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

Proof of Theorem 4.3 Case 1: w ∈ [0, M − 2K ) In this case, we have:  [K − m 2 , K ], if m 2 ∈ [0, (1 − α)K ); m 1 (m 2 ) ∈ [αK , K ], if m 2 ∈ [(1 − α)K , K ].  [K − m 1 , K ], if m 1 ∈ [0, αK ); m 2 (m 1 ) ∈ [(1 − α)K , K ], if m 1 ∈ [αK , K ]. In this case, we can obtain the Nash equilibria m ∗1 × m ∗2 ∈ [αK , K ] × [(1 − α)K , K ]. Case 2: w ∈ [M − 2K , M − (1 + α)K ). In this case, we have: ⎧ M−w−m 2 ⎪ , if m 2 < 2K − (M − w); ⎨= 2 m 1 (m 2 ) ∈ [K − m 2 , K ], if m 2 ∈ [2K − (M − w), (1 − α)K ); ⎪ ⎩ ∈ [αK , K ], if m 2 ∈ [(1 − α)K , K ]. ⎧ M−w−m 1 ⎪ , if m 1 < 2K − (M − w); ⎨= 2 m 2 (m 1 ) ∈ [K − m 1 , K ], if m 1 ∈ [2K − (M − w), αK ); ⎪ ⎩ ∈ [(1 − α)K , K ], if m 1 ∈ [αK , K ]. In this case, we can prove that 2K − (M − w) < M−w and obtain the Nash equilibria 3 m ∗1 × m ∗2 ∈ [αK , K ] × [(1 − α)K , K ]. Case 3: w ∈ [M − (1 + α)K , M − (2 − α)K ). In this case, we have:  m 1 (m 2 )

2 = M−w−m , if m 2 ≤ α; ˆ 2 ˆ K ]. ∈ [αK , K ], if m 2 ∈ (α,

⎧ M−w−m 1 ⎪ , if m 1 < 2K − (M − w); ⎨= 2 m 2 (m 1 ) ∈ [K − m 1 , K ], if m 1 ∈ [2K − (M − w), αK ); ⎪ ⎩ ∈ [(1 − α)K , K ], if m 1 ∈ [αK , K ]. √

Note that if w ≥ M − 9αK −3 (9α−4)αK , then there exist multiple Nash equilibria 2 M−w ∗ ∗ , ) and m × m ˆ K ], where in the later set (m ∗1 , m ∗2 ) = ( M−w 1 2 ∈ [αK , K ] × [α, 3 3 of equilibria we have that g1 (m ∗1 , m ∗2 ) = αK and g2 (m ∗1 , m ∗2 ) = (1 − α)K . Further, M−w ) dominates any equilibrium belonging more, the former equilibrium ( M−w 3 3 to [αK , K ] × [αK ˆ√ , K ] in the sense of it gains larger profit for both retailers. If w < M − 9αK −3 (9α−4)αK , then we have that in equilibrium the two retailers’ total 2 ˆ K ]. order size is no less than K , and the equilibria are m ∗1 × m ∗2 ∈ [αK , K ] × [α,

Appendix

129

Case 4: w ∈ [M − (2 − α)K , M − K ). In this case, we have:  m 1 (m 2 )  m 2 (m 1 )

2 = M−w−m , if m 2 ≤ α; ˆ 2 ˆ K ]. ∈ [αK , K ], if m 2 ∈ (α,

1 = M−w−m , if m 1 ≤ β; 2 ∈ [(1 − α)K , K ], if m 1 ∈ (β, K ];

√

Note that if w ≥ M − 9αK −3 (9α−4)αK , then there exist multiple Nash equilibria 2 M−w ∗ ∗ , ) and m × m ˆ K ], where in the later set (m ∗1 , m ∗2 ) = ( M−w 1 2 ∈ [β, K ] × [α, 3 3 of equilibria we have that g1 (m ∗1 , m ∗2 ) = αK and g2 (m ∗1 , m ∗2 ) = (1 − α)K . Fur, M−w ) dominates any equilibrium belongthermore, the former equilibrium ( M−w 3 3 ing to [β, K ] × [√α, ˆ K ] in the sense of it gains larger profit for both retailers. If , then we have that in equilibrium the two retailers’ total w < M − 9αK −3 (9α−4)αK 2 ˆ K ]. order size is no less than K , and the equilibria are m ∗1 × m ∗2 ∈ [β, K ] × [α, Case 5: w ∈ [M − K , M] In this case, we have:  m 1 (m 2 ) =

0, 

m 2 (m 1 ) =

M−w−m 2 , 2

M−w−m 1 , 2

0,

if m 2 < M − w; if m 2 ∈ [M − w, K ]. if m 1 < M − w; if m 1 ∈ [M − w, K ].

, From the equations, we can gain the Nash equilibrium (m ∗1 , m ∗2 ) = ( M−w 3 Combining Cases 1–5, the theorem holds.

M−w ). 3



Proof of Lemma 4.3 √ (i) To prove 3K − 3 (9α − 4)α + 1 − 3α is decreasing in α ∈ [ 21 , 1] is equivalent √ to prove that (9α − 4)α + 1 − 3α is increasing in α ∈ [ 21 , 1], which is further equivalent to   [9(α + ) − 4](α + ) + 1 − 3(α + ) > (9α − 4)α + 1 − 3α, where  > 0. After straightforward algebra operations, we have prove (i). Part (ii) can be proven similarly.



Proof of Theorem 4.4 In view of Eq. (4.14), the supplier will choose his optimal wholesale price to maximize her profit: 

2(M − w)w max max√ wK ; max √ 9αK −3 (9α−4)αK 9αK −3 (9α−4)αK 3 w∈(0,M− ) w∈[M− ,M) 2 2

 . (A.6)

130

4 Allocating Capacity with Demand Competition: Fixed Factor Allocation

Note that K (M −

√ √  9αK − 3 (9α − 4)αK 9αK − 3 (9α − 4)αK ) > (M − )(3α − (9α − 4)α)K , (A.7) 2 2 M 2(M − w)w }= , arg max{ (A.8) w 3 2 2 2(M − w)w M max{ (A.9) }= . w 3 6

Therefore, if

M 2

≤M−

√ 9αK −3 (9α−4)αK 2 ∗

√ , i.e., M ≥ 9αK − 3 (9α − 4)αK , then

by (A.7) and (A.8), we have w (α) = (M − −

√ 9αK −3 (9α−4)αK − ) ]. 2 √ M When 2 > M − 9αK −3 (9α−4)αK 2

√ 9αK −3 (9α−4)αK − ) 2

and ∗s (α) = K [(M

√ , i.e., M < 9αK − 3 (9α − 4)αK , we can

show that

√ M2 9αK − 3 (9α − 4)αK )−] ≤ K [(M − 2 6  (9α − 4)α + 1 − 3αK ];(A.10) for M ∈ (0, 3K − 3 √ M2 9αK − 3 (9α − 4)αK )−] > and K [(M − 2 6   (9α − 4)α + 1 − 3αK , 9αK − 3 (9α − 4)αK ).(A.11) for M ∈ (3K − 3 Hence, we have that √ (9α − 4)α + 1 − 3αK ], then by (A.8), (A.9) and (A.10), we if M ∈ (0, 3K − 3 2 have that w ∗ (α) = M2 and ∗s (α) = M6 ; √ √ if M ∈ (3K −3 (9α − 4)α+1 − 3αK , 9αK − 3 (9α − 4)αK ), then by (A.8), √ (A.9) and (A.11), we have w ∗ = M − 9αK −3 (9α−4)αK and ∗s (α) = K [(M − 2 √ 9αK −3 (9α−4)αK − ) ]. 2

Consequently, the theorem holds.



Proof of Theorem 4.5 The theorem follows Theorem 4.4 with straightforward algebra operations, and hence we omit the proof for preciseness.  Proof of Lemma 4.4 The lemma can be proven using straightforward algebra operations, and hence we omit the proof for preciseness.  Proof of Theorem 4.6 Part (i) follows Theorem 4.4 and Proposition 4.1. √ If M < 3(2 − 2)K , then retailers’ profits are ∗1 = M 2 /36, ∗2 = M 2 /36 and ∗1 + ∗2 = M 2 /18 under proportional allocation, the same as under fixed factor allo√ √ √ 2 3 2−4 2 ∗ cation with α = 2 . If M ≥ 3(2 − 2)K , then retailers’ profits are 1 = 2 K ,

Appendix

131

√ and ∗1 + ∗2 = (3 2 − 4)K 2 under proportional allocation. Under √ √ fixed factor allocation with α = 22 , retailers’ profits are ∗1 (α) = (3 − 2 2)K 2 > √ √ ∗1 , ∗2 (α) = (5 2 − 7)K 2 < ∗2 , and ∗1 (α) + ∗2 (α) = (3 2 − 4)K 2 = ∗1 +  ∗2 . Hence, part (ii) of the theorem holds. √ 3 2−4 2 K 2

∗2 =

Proof of Theorem 4.7 Following Eq. (4.15), the proof is divided into two cases. Case 1. If K ∈ [0,

√√

M ), (9α−4)α+1−3α

3−3

then the first order condition is

 ˆ ∗s (K )   ∂ = (M − c) − 9α − 3 (9α − 4)α K = 0. ∂K (M−c) √ 9α−3 (9α−4)α

which follows from 0
K (M − β K ), which is equivalent to √ √ 2 2(n+1)−2 (n+1) −n(n+1)β 2(n+1)+2 (n+1)2 −n(n+1)β K or M > K . Note that M< n n √ √ 2(n+1)−2 (n+1)2 −n(n+1)β 2(n+1)+2 (n+1)2 −n(n+1)β < 2β < is equivalent to β < n+1 , n n n we can obtain the following results: nM2 and w ∗ = M2 > M − β K . From (a) If 0 ≤ M ≤ τ (β)K , then ∗s = 2s = 4(n+1) Theorem 5.1, retailers’ equilibrium order quantities are equal to their ideal quantities ∗ ∗ M M , . . . , M−w ) = ( 2(n+1) , . . . , 2(n+1) ). ( M−w n+1 n+1 (b) If τ (β)K < M ≤ 2β K , then ∗s = 1s and w ∗ = (M − β K )− . From Theorem 5.1, each retailer orders no less than his guaranteed allocation αi K . K , we have 1s > 2s = nβ (M − β K ). Thus, (c) If M > 2β K , then since β < n+1 n n+1 ∗ 1 ∗ − s = s , w = (M − β K ) . Above all, Theorem 5.2 holds. 

Proof of Lemma 5.4

√ (n+1)[(n+1)x+y−1]−(n+1) [(n+1)x−(y−1)]2 −4x Define a function l(x, y) = , where x > 2[(n+1)y−n]  y/n and y ≥ 1. Let x = 1 − nj=θ∗ +1 α j and y = θ∗ . Note that β = l(x, y). Next, we  analyze the monotony of l(x, y) over x and y, and how β changes with nj=θ∗ +1 α j and θ∗ . We have (n + 1)[(n + 1)x − (y − 1)] − 2 ∂l =n+1−  , ∂x [(n + 1)x − (y − 1)]2 − 4x and thus  ∂l < 0 ⇔ (n + 1) [(n + 1)x − (y − 1)]2 − 4x < (n + 1)[(n + 1)x − (y − 1)] − 2 ∂x ⇔ (n + 1)2 {[(n + 1)x − (y − 1)]2 − 4x} < {(n + 1)[(n + 1)x − (y − 1)] − 2}2 ⇔ 4(n + 1)(y − 1) + 4 > 0.

Since y ≥ 1, we have ∂l(x, y)/∂x < 0. We also have

Appendix

163

1 ∂l [(n + 1)x − (y − 1)][(n + 1)y − n] = ·{  2 ∂y 2[(n + 1)y − n] [(n + 1)x − (y − 1)]2 − 4x  +(n + 1) [(n + 1)x − (y − 1)]2 − 4x − [(n + 1)2 x − 1]} and thus ∂l >0 ∂y  [(n + 1)x − (y − 1)][(n + 1)y − n] ⇔  + (n + 1) [(n + 1)x − (y − 1)]2 − 4x > (n + 1)2 x − 1 [(n + 1)x − (y − 1)]2 − 4x  [(n + 1)x − (y − 1)][(n + 1)y − n] ⇔{  + (n + 1) [(n + 1)x − (y − 1)]2 − 4x}2 > [(n + 1)2 x − 1]2 [(n + 1)x − (y − 1)]2 − 4x 4x[(n + 1)2 y − n]2 ⇔ > 0. [(n + 1)x − (y − 1)]2 − 4x

Thus, we have ∂l(x, y)/∂ y > 0.  From the monotony of l(x, y), we obtain that β increases with nj=θ∗ +1 α j and θ∗ .  Note that θ∗ ∈ Z + and θ∗ ≥ 1. Thus, when θ∗ = 1 and nj=θ∗ +1 α j = 0, we obtain the minimum value of β. Here, fixed proportions satisfy α1 ≤ 1 and α2 = · · · = αn = 0.  Plugging θ∗ = 1 and nj=θ∗ +1 α j = 0 into Eq. (5.4), we have βmin = [(n + 1)2 −  (n + 1) (n + 1)2 − 4]/2. In the following, we prove β < (n + 1)/n. Since fixed proportions  satisfy αi ≤ α∗ j , i ≤ j, and there exist i and j such that αi = α j , we have 1 − nj=θ∗ +1 α j > θn , which is equivalent to √    (n+1)[(n+1)(1− nj=θ∗ +1 α j )+θ∗ −1]−(n+1) [(n+1)(1− nj=θ∗ +1 α j )−(θ∗ −1)]2 −4(1− nj=θ∗ +1 α j ) n+1 < n . 2[(n+1)θ∗ −n] Thus, β < (n + 1)/n always holds.  Proof of Corollary 5.3 Note that v = M/K −

n(M/K )2 , 4(n+1)

and from Theorem 5.2,

 2(n + 1) 2(n + 1) − 2 (n + 1)2 − n(n + 1)β < . τ (β) = n n Further, through simple computation, we can obtain the following inequations:  M < [2(n + 1) − (n + 1) 4 − 2n[(n + 1) −  ⇔v < [(n + 1)2 − (n + 1) (n + 1)2 − 4]/2;



(n + 1)2 − 4]]/n K

M ≤ τ (β)K ⇔ β ≥ v.   (i) When M < [2(n + 1) − (n + 1) 4 − 2n[(n + 1) − (n + 1)2 − 4]]/n K ,  we have v < [(n + 1)2 − (n + 1) (n + 1)2 − 4]/2 < β, which is equivalent to

164

5 Fixed Allocation of Capacity for Multiple Retailers Under Demand Competition

M ≤ τ (β)K , following part (i) of Theorem 5.2. Thus, the supplier’s optimal wholesale price is w ∗ = M2 .   (ii) When [2(n + 1) − (n + 1) 4 − 2n[(n + 1) − (n + 1)2 − 4]]/n K ≤ M <  2(n + 1)/n K , we have [(n + 1)2 − (n + 1) (n + 1)2 − 4]/2 ≤ v < (n + 1)/n. If β < v, then it holds that M > τ (β), and from part (ii) of Theorem 5.2, the supplier’s optimal wholesale price is w∗ = (M − β K )− . If β ≥ v, then it holds that M ≤ τ (β)K , and from part (i) of Theorem 5.2, the supplier’s optimal wholesale price is w ∗ = M2 . (iii) When M ≥ 2(n + 1)/n K , since τ (β) < 2(n+1) , we have M > τ (β), foln lowing part (ii) of Theorem 5.2. Thus, the supplier’s optimal wholesale price is  w ∗ = (M − β K )− . Proof of Corollary 5.4 (i) When (α1 , . . . , αn ) = ( n1 , . . . , n1 ), from Theorem 5.1, the supplier’s maximum profit is ∗s = max{1s = w

max

0≤w 2(n + 1)/n K , ∗ ∗ then w = M − (n + 1)/n K and s = K [M − (n + 1)/n K ]. From Theorem 5.2 and Table 5.1, if there exist i = j such that αi = α j , then the supplier’s maximum profit is ∗s (β). Since β < (n + 1)/n, it holds that ∗s ≤ ∗s (β). (ii) When (α1 , α2 , . . . , αn ) = (1, 0, . . . , 0), from Theorem 5.2 and Table 5.1, we have that ∗s (β) decreases with β. Note that β reaches its minimum value at (α1 , α2 , . . . , αn ) = (1, 0, . . . , 0). Thus, for any fixed proportion, the supplier  achieves her maximum profit when (α1 , α2 , . . . , αn ) = (1, 0, . . . , 0). Proof of Theorem 5.3 √ n 2 +4n−4−n) When β = γn = (n+1)( 2(n−1) , we have τ (β) = 2(n + 1)(1 − ϕn )/n. Comparing Theorem 5.2 and Lemma 5.5, we have that the supplier’s optimal wholesale price and the corresponding profit under fixed allocation are respectively equal to that under proportional allocation. And under two allocations, the supply chain profit satisfies: if M ≤ 2(n + 1)(1 − ϕn )/n K , then ∗sc = n(n + 2)M 2 /[4(n + 1)2 ]; if  M > 2(n + 1)(1 − ϕn )/n K , then ∗sc = (M − K )K . Proof of Theorem 5.4 Given retailers’ allocated quantities (g1 , g2 , . . . , gn ), we first solve the best response function of the selling quantity qi (gi ), and then explore the equilibrium selling quantities of retailers.

Appendix

165

Retailer i’s objective function is formulated as: max {[M − (qi +

0≤qi ≤gi

n 

q j ) − w]qi − w(gi − qi )}

j=1, j=i

By solving the equation, we can obtain the best response function of selling quantity as follows:   gi  if gi ≤ [M − ( nj=1, j=i q j )]/2 qi (gi ) = M−( nj=1, j=i q j )  if gi > [M − ( nj=1, j=i q j )]/2 2 Substituting the conditions and equilibrium selling quantities of (i) (ii) and (iii) in Theorem 5.4 into the above equation, we can verify the results of Theorem 5.4 hold. Note that retailers are labeled by their allocated quantities, where gi ≥ gi+1 , i = 1, . . . , n − 1. Then the conditions in (i) (ii) and (iii) of Theorem 5.4  are perfect space of (g1 , g2 , . . . , gn ). Proof of Corollary 5.5 (i) From Theorem 5.4, we have the equilibrium selling quantities of retailers. Without capacity constraint, one retailer can freely choose his allocation, which just equals his order. Next, we explore retailer i’s optimal allocated quantity. Given the allocated quantities of other retailers as g−i = (g1 , . . . , gn−1 ), where g j ≥ g j+1 , j = 1, . . . , n − 2. From (ii) of Theorem 5.4, if there exits a k such that gk >  M− n−1 j=k+1 g j k+1

 M− n−1 j=k+1 g j k+1

and gk+1 ≤

, then in equilibrium the selling quantities satisfy  q ∗j

=

 M− n−1

 M− n−1 j=k+1 g j k+1 gj

for j = 1, . . . , k; for j = k + 1, . . . , n − 1.

gj

j=k+1 For gi , if gi > , then qi∗ = q ∗j = k+1 j = k + 1, . . . , n − 1. Retailer i’s objective function is

max{ gi

where i,1 = (

max 

gi >

 M− n−1 j=k+1 g j 2 ) k+1

M−

n−1 j=k+1 g j k+1

 M− n−1 j=k+1 g j k+1

i,1 ;

max 

gi ≤

M−

, j = 1, . . . , k, and q ∗j = g j ,

n−1 j=k+1 g j k+1

i,2 },

− w · gi . Since i,1 is decreasing with gi , then the opti M− n−1

gj

j=k+1 mal allocated quantity satisfies gi∗ ≤ , and the optimal selling quantity k+1 would be equal to the allocated quantity. If g−i satisfies the condition of (i) and (iii) in Theorem 5.4, we also have qi∗ = gi∗ by similar analysis. Above all, part (i) of Corollary 5.5 holds.

166

5 Fixed Allocation of Capacity for Multiple Retailers Under Demand Competition

(ii) Suppose there exists limited capacity K . Recall that retailers’ ideal quantity is when there’s no capacity limit. Next, we divide the proof into two cases based K and on whether the total ideal order quantity exceeds capacity, i.e., w ≤ M − n+1 n K . w > M − n+1 n K , i.e., n(M−w) ≤ K. Case 1: w ≤ M − n+1 n n+1 Step 1: First, we explore the condition when order inflation happens. In this case, capacity is sufficient to satisfy each retailer’s ideal quantity, one retailer would deviate from his ideal quantity only when he could gain more profit from inflating his order quantity by a high allocation, with the total order quantity exceeding capacity K . Next, we show retailer j’s maximum profit by inflating his order quantity. , . . . , M−w ), if retailer j Given other retailers’ order quantity as g− j = ( M−w n+1 n+1 (n−1)(M−w) with an allocation g j , denote k inflates his order quantity m j ≥ K − n+1 as the index of the retailer who has lower priority than retailer j and receives an ), then we have that when m j ≤ g j,max , allocation quantity in the interval (αk K , M−w n+1 retailers’ allocated quantity would be adjusted as M−w n+1

gi =

⎧ ⎪ ⎪ ⎨

M−w n+1 mj

K− ⎪ ⎪ ⎩ αi K

(k−2)(M−w) n+1

for i for i n − m j − ( i=k+1 αi K ) for i for i

= 1, . . . , k − 1, i = j; = j; = k; = k + 1, . . . , n.

When m j > g j,max , retailers’ allocated quantity would be adjusted as gi =

⎧ ⎨

for i = 1, . . . , j − 1; for i = j; ⎩ αi K for i = j + 1, . . . , n. M−w n+1 mj

From Theorem 5.4, we can qi∗ = gi for i = j, and the total selling nverify that ∗ quantity except retailer j is i=1,i= j qi = K − g j . From (ii) of Theorem 5.4, we have  if g j ≤ M − K ; gj q ∗j = M−(K −g j ) if g j > M − K . 2 

with  j (g j ) =

(M − K − w)g j if g j ≤ M − K ; [M−(K −g j )]2 − w · g j if g j > M − K . 4

From the monotonicity of  j , we have that ∗j = max 0,  j (g j,max ). Whether retailer j would inflate his order quantity depends on the comparison of )2 , which can be simplified to compare  j (g j,max ) max{0,  j (g j,max )} and ( M−w n+1 )2 . We have the result of (ii.a) in Corollary 5.5. and ( M−w n+1

References

167

Step 2: In the following, we analyze the equilibrium allocated quantities and optimal selling quantities with order inflation, and further investigates the condition where capacity hoarding occurs. )2 , then the total order quantity If there exists a j, such that (g j,max ) > ( M−w n+1 would be more than capacity. Anticipating the competition in capacity, each retailer would order as more quantity (e.g., K ) to obtain higher allocation. Finally, based on fixed allocation, each retailer except retailer 1 receives his guaranteed quantity α j K ,  and retailer 1 acquires the whole remaining capacity K − nj=2 α j K . From Theorem 5.4, we have that only  when the allocation quantities satisfy that there exists an i such that gi > [M − ( nj=1, j=i g j )]/2, there would be certain retailer sells part of his capacity. Note that under fixed  allocation, retailer 1 receives the most capacity, then we have if g1 > [M − ( nj=2 g j )]/2, which is equivalent to  M < (2 + nj=2 α j )K , then at least retailer 1 would hoard capacity. Otherwise, each retailer would sell all his available quantity out. K , i.e., n(M−w) > K. Case 2: w > M − n+1 n n+1 In this case, retailers’ total ideal quantity is more than capacity, then they would order as more quantity (e.g., K ) as possible to compete for limited quantity. The analysis of the equilibrium allocated quantities and optimal selling quantities are the same as Step 2 of Case 1, and we omit it here. Combining Step 2 of Case 1 and Case 2, (ii.b) of Corollary 5.5 is obtained. 

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Hall, N. G., & Liu, Z. (2010). Capacity allocation and scheduling in supply chains. Operations Research, 58(6), 1711–1725. Lee, H. L., Padmanabhan, V., & Whang, S. (1997). Information distortion in a supply chain: The bullwhip effect. Management Science, 43(4), 546–558. Li, J., Cai, X., & Liu, Z. (2017). Allocating capacity with demand competition: Fixed factor allocation. Decision Sciences, 48(3), 523–560. Li, J., Yu, N., Liu, Z., & Cai, X. (2017). Allocation with demand competition: Uniform, proportional, and lexicographic mechanisms. Naval Research Logistics, 64(2), 85–107. Liu, Z. (2012). Equilibrium analysis of capacity allocation with demand competition. Naval Research Logistics, 59(3–4), 254–265. Lu, L. X., & Lariviere, M. A. (2012). Capacity allocation over a long horizon: The return on turn-and-earn. Manufacturing & Service Operations Management, 14(1), 24–41. Qing, Q., Deng, T., & Wang, H. (2017). Capacity allocation under downstream competition and bargaining. European Journal of Operational Research, 261(1), 97–107. Sprumont, Y. (1991). The division problem with single-peaked preferences: A characterization of the uniform allocation rule. Econometrica: Journal of the Econometric Society, 509–519. Stangel, L. (2018). Facing an ‘extreme shortage’ of car carriers, Tesla decides to make its own. Retrieved September 09, 2020, from https://www.bizjournals.com/sanjose/news/2018/09/25/ tesla-car-carriers-model-3-delivery-problems-tsla.html Yang, Z., Hu, X., Gurnani, H., & Guan, H. (2017). Multichannel distribution strategy: Selling to a competing buyer with limited supplier capacity. Management Science, 64(5), 2199–2218.

Chapter 6

Supply Chain Coordination Through Capacity Reservation Contract and Quantity Flexibility Contract

6.1 Introduction Capacity management is always an important issue in high technology industries, such as semiconductors, telecommunication devices and optoelectronics, because capacity building is capital intensive and requires long lead time. In the semiconductor industry, a new semiconductor laboratory typically costs 1–4 billion dollars to build, and a single machine may require as much as 4–5 million dollars to purchase. In addition, the procurement lead time usually takes as long as 6–12 months (Wu et al. 2005). Market trends show that product variety is increasing and product life cycle is shrinking. According to Horn (2013), 50% of annual revenues across a range of industries are derived from new products launched within the past three years. This phenomenon makes the physical expansion of manufacturing capacity extremely risky. A similar situation also exists in industry sectors such as apparel, toys, sporting goods and electronics. Owing to the aforementioned facts, capacity management is challenging, especially in a decentralised supply chain. The upstream supplier (e.g. manufacturer) often adopts a conservative capacity-building policy to avoid the risk of demand uncertainty. Consequently, the downstream buyer (e.g. retailer) may not have adequate supplies to fulfill the market demand, thereby reducing the total profit of the manufacturer and the retailer. In other words, this situation induces the supply chain to perform poorly. To achieve supply chain coordination, which is a state where the decentralised supply chain yields total profit equivalent to the centralised system, a few coordination mechanisms are implemented to motivate the members to make system-optimal decisions (Cachon 2003). For example, in high technology industries, if the retailer is willing to share the risk of the manufacturer by assuming a certain level of liability, then the manufacturer may be willing to expand capacity aggressively, and thus the total profit can be enhanced. Often, contracts are adopted as risk-sharing mechanisms to align the objective of each firm with the objective

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Li et al., Capacity Allocation Mechanisms and Coordination in Supply Chain Under Demand Competition, Uncertainty and Operations Research, https://doi.org/10.1007/978-981-19-6577-7_6

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of the supply chain. Contracts permit better capacity planning and provide retailers the required flexibility to handle uncertain demand. Manufacturers would also prefer to have contractual agreements with retailers as it could reduce their cost due to effective capacity planning (Serel et al. 2001). In this study, we investigate contractual issues in high technology industries, such as semiconductors. These industries have a short life cycle, high capacity cost, long lead time and highly volatile demand. On the demand side, the selling price of the retailer is uncertain, which is also realistic. For example, for multinational firms, the selling price could be significantly affected by the exchange rate, which usually fluctuates (Carter and Vickery 1988). We firstly consider the case in which the retail price is independent of the demand. This situation is realistic when the market size of the retailer is limited and the retailer’s demand has negligible influence on the market price (Fu et al. 2010). In addition, many existing studies assume that the market demand of the retailer is volatile whereas the retail price is fixed (Erkoc and Wu 2005; Nosoohi and Nookabadi 2016; Li et al. 2016), which, to a certain extent, indicates that the demand and retail price are not necessarily correlated. Moreover, we also make an extension and study the case in which the retail price and demand are correlated. However, this case shows that the effect of correlation is negligible. In this context, we propose a new type of capacity reservation contract. The retailer pays a reservation fee upfront for each unit of capacity to be reserved for future usage. Then, the manufacturer constructs capacity according to the reservation of the retailer and her expectations for the market. When the retailer actually utilizes the reserved capacity, he pays the exercise price for each product. If the reserved capacity is not fully utilised within the specified time, the reservation fee associated with unused capacity is not refundable. Different from the traditional capacity reservation contract studied by Cachon and Lariviere (2001), Erkoc and Wu (2005) and Fu et al. (2010), we assume that the retailer can purchase more than his reserved quantity at a purchase price larger than the wholesale price given the available capacity of the manufacturer. On the other hand, we assume that the manufacturer can construct capacity as much as needed. This condition is also different from traditional capacity reservation contracts which usually assume that the manufacturer would not construct more than the reserved quantity. We also assume that the manufacturer is subject to penalty if he cannot satisfy the order of the retailer within the reserved quantity. We also propose a new type of quantity flexibility contract. The retailer initially reserves some capacity for future usage. No reservation fee is required in this contract. Then, the manufacturer constructs capacity according to the reservation of the retailer and her own market expectations. When actual demand is realised, the retailer places his final order at the wholesale price. The retailer has to pay penalty if his final order is outside an allowable range determined by the reserved quantity and flexibility coefficient. However, if the manufacturer cannot satisfy the entire order of the retailer, she has to pay for the unsatisfied part within the said range. The strategic interaction between the manufacturer and retailer in both contracts is modelled by a game-theoretic approach based on Stackelberg games where the retailer acts as the leader. We characterise the subgame perfect Nash equilibrium

6.2 Model Setting

171

in the case of perfect information. We show that two contracts are able to achieve channel coordination and establish ways to set contract parameters to coordinate the supply chain. Furthermore, we compare two contracts from the perspective of the manufacturer, the retailer and the supply chain, respectively. Subsequently, we derive the sufficient condition under which quantity flexibility contract can outperform capacity reservation contract. Note that different from the capacity reservation contract in which the retailer pays corresponding fees in two periods, the retailer only pays in one period with the quantity flexibility contract. This difference results from the intrinsic requirements of the option contract which typically consists of reservation price and exercise price (Fu et al. 2010). Therefore, the retailer is required to pay the reservation fee in the first period and the exercise fee in the second period. The rest of the paper is organised as follows. The next section provides a review of relevant literature. Section 6.2 presents our basic model. In Sect. 6.3, we derive optimal strategies of the manufacturer and retailer, and design parameters for channel coordination with the capacity reservation contract. Section 6.4 analyses optimal strategies of the supply chain members and derives the sufficient condition of supply chain coordination with the quantity flexibility contract. In Sect. 6.5, the two contracts with or without coordination are compared. In Sect. 6.6, the basic model is extended considering endogenous contract parameters and the correlation between the demand and price. The paper is concluded in Sect. 6.7.

6.2 Model Setting We consider a supply chain with one retailer and one manufacturer which produces an innovative product. The retailer orders from the manufacturer and sells to consumers. At the retailer side, the market demand for the product, D, is stochastic. Denote by f (t) its probability density function (pdf). f (t) > 0 for t ≥ 0 and f (t) is differentiable. Denote by F (t) the cumulative distribution function(cdf). F (t) is differentiable and invertible for t ≥ 0. Let F¯ (t) = 1 − F (t). The retail price, p, is stochastic with the pdf, φ(·) and the cdf, (·). Denote by μ p the expected retail price. In the base model, we consider the case in which the demand and price and independent. The correlated case is studied in the extension section. At the manufacturer side, the investment in production capacity is extremely expensive. Let K and c denote the capacity that the manufacturer decides to construct and the construction cost per unit capacity, respectively. Two different contracts are examined in this study, namely, capacity reservation contract and quantity flexibility contract. We aim to investigate the following problems: 1) How do the manufacturer and the retailer hedge against future risk by using the capacity reservation or quantity flexibility contract? 2) What is the benefit of our proposed contracts? 3) Does a unique equilibrium exist in the Stackelberg game for two contracts? 4) What condition can coordinate the supply chain in the two contracts? 5) What is the relation of two contracts?

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Table 6.1 Notations Notations Q K c D μD / σD f (·) / F(·) p μp / σp φ(·) / (·) po pe αc βc  M / R w αq βq d ¯ M / ¯R 

Explanations Quantity reserved by retailer, decision variable Capacity constructed by manufacturer, decision variable Construction cost per unit capacity Market demand, random variable Mean/standard deviation of D Probability density function/ cumulative distribution function of D Retail price, random variable Mean/standard deviation of p Probability density function/ cumulative distribution function of p Unit reserve price in capacity reservation contract Unit exercise price in capacity reservation contract Unit penalty to manufacturer in capacity reservation contract Unit purchasing price of extra order in capacity reservation contract Profit of manufacturer/retailer under capacity reservation contract Unit wholesale price in quantity flexibility contract Unit penalty to manufacturer in quantity flexibility contract Unit penalty to retailer in quantity flexibility contract Flexibility coefficient in quantity flexibility contract Profit of manufacturer/retailer under quantity flexibility contract

For ease of exposition, we firstly list notations used in this study, as shown in Table 6.1. For clarity, the specific parameters of capacity reservation quantity are marked with boldface and the parameters in quantity flexibility contract are marked with italic. Other parameters without any special style are common for two contracts. First, consider a centralised supply chain where the manufacturer and retailer are owned by one firm. In the centralised system, the capacity decision K is made to maximise the expected profit, denoted by C S (K ).   C∗ S = max C S (K ) = E D μ p (D ∧ K ) − cK , K

(6.1)

where μ p is the expected retail price and x ∧ y = min (x, y). The first term is the expected revenue, and the second term is the construction cost. We can verify in a straightforward manner that C S (K ) is concave in K and a unique optimal solution exists in the following equation:   c . K C∗ S = arg max C S (K ) = F −1 1 − K μp

(6.2)

6.3 Capacity Reservation Contract

173

Obviously, this solution is for the newsvendor model. As we know, the supply chain performs the best when the capacity is K C∗ S . Then, we consider another benchmark case that the retailer does not share any risk with the manufacturer in a decentralised supply chain. In detail, the retailer does not reserve any quantity and the manufacturer builds the capacity K based on her own belief of the market demand. After the demand D and price p are realised, the retailer orders D and obtains min(D, K ) with unit wholesale price w. Obvisously, the retailer bears neither overstock nor understock risk with the manufacturer. The profit functions of the manufacture and retailer can be written as follows: ˜ M (K ) = w[D ∧ K ] − cK ;  ˜ R (K ) = (μ p − w)[D ∧ K ].  The retailer does not make decision and the manufacturer faces a newsvendor  . type problem again. The optimal solution can be easily obtained as K˜ ∗ = F −1 w−c w Since w < u p , then K˜ ∗ < K C∗ S . More important, the total profit in the supply chain is also smaller than that in the centralised system. In other words, if the retailer shares no risk with the manufacturer, the building capacity is suboptimal, hindering the supply chain from satisfying the optimal amount of demand from the perspective of a channel. This conclusion has been found by Erkoc and Wu (2005). To motivate the manufacturer to build system optimal capacity and improve the performance of the supply chain, we propose two contracts, which are specified in the following.

6.3 Capacity Reservation Contract In this section, we assume that the manufacturer and retailer adopt the capacity reservation contract. Under this contract, the retailer acts as the leader and the manufacturer is the follower. The sequence of events in this contract is as follows: (1) A contract with parameters ( po , pe , αc , βc ) is offered. po denotes the reservation fee. pe denotes the exercise price. αc denotes the penalty that the manufacturer has to pay if she cannot satisfy the order of the retailer within the reserved quantity. βc denotes purchasing price that the retailer has to pay for the partial order beyond the reserved quantity. (2) Given the contract, the retailer places reservation quantity Q, paying po Q. If the realised quantity of the retailer is less than the reserved quantity, the reservation fee for unused capacity is not refundable. (3) After observing the reserved quantity, the manufacturer sets her capacity K . (4) The demand D and market retail price p of the retailer are realised, and he orders D units. As pointed out by Erkoc and Wu (2005) and Wu et al. (2005), in the high-tech environment where product specifications change frequently and the demand is difficult to forecast even in a stable economy, the buyer places the final order after demand realisation to obtain more demand information and reduce overstock or understock cost. Further,

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the buyer/retailer does not order more than the realised demand since the items have no value if not sold in the current period, and no inventory can be carried to the next period as the product will likely be obsolete. On the other hand, the retailer orders no less than the realised demand because the unit (expected) revenue is higher than the purchasing cost beyond the allowable range. Therefore, the retailer places the final order exactly as the realised demand D. (5) The manufacturer delivers (D ∧ K ) and the payment is settled. Let  R (Q, K ) and  M (Q, K ) denote the expected profit function of the retailer and manufacturer, respectively. Then, we obtain the following:  R (Q, K ) = μ p E [D ∧ K ] − T (Q, K ),  M (Q, K ) = T (Q, K ) − cK ,

(6.3) (6.4)

where T (Q, K ) is the transfer payment function from the retailer to the manufacturer with the following expression.  T (Q, K ) = po Q + E D pe (Q ∧ K ∧ D) + βc ((D ∧ K ) − Q)+ − αc ((D ∧ Q) − K )+

(6.5) Two main decision stages are observed in our model. In the first stage, the retailer determines the reservation quantity Q. In the second stage, the manufacturer chooses her capacity level K . We analyse the problem in a backward sequence. Thus, we firstly analyse the manufacturer’s capacity construction behaviour in the next section.

6.3.1 Construction Decision of Manufacturer For any given contract parameters ( po , pe , αc , βc ) and reserved quantity Q, the manufacturer determines the capacity level to maximise her expected profit. Theorem 6.1 Given reserved quantity Q, the optimal constructed capacity of the manufacturer K (Q) satisfies the following equation: K (Q) = arg max { M (Q, K )}, K ∈{ Kˆ , K˜ }

where Kˆ = K 1 ∧ Q , K˜ = K 2 ∨ Q, K 1 = F −1



αc + pe −c αc + pe



and K 2 = F −1



βc −c βc



Then, we try to find out what condition can induce the manufacturer to construct more or less. We present the following reasonable assumption before analysing the optimal strategies. Assumption 6.1 αc + pe ≥ μ p ≥ βc ≥ po + pe ≥ c ≥ po .

6.3 Capacity Reservation Contract

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Fig. 6.1 Best-response strategy of manufacturer in the capacity reservation contract

The assumption implies that the revenue (including the opportunity cost) to satisfy one unit of order in the case of K ≤ Q is larger than that in the case of K ≥ Q. Thus, we have K 1 ≥ K 2 . Now, we are ready to analyse the best-response strategy of the manufacturer in detail. Theorem 6.2 The best-response function of the manufacturer is as follows: (i) If Q ≤ K 2 , then K (Q) = K 2 . (ii) If Q ∈ (K 2 , K 1 ), then K (Q) = Q. (iii) If Q ≥ K 1 , then K (Q) = K 1 . The proof is presented in the Appendix. Based on Theorem 6.2, Fig. 6.1 characterises the manufacturer’s policy of capacity construction. The theorem illustrates that if the retailer reserves a small capacity (i.e. less than K 2 ), then the manufacturer believes that the reserved quantity cannot meet the demand, so she constructs more than the reserved capacity of the retailer, anticipating that the retailer will purchase more products than reserved with a higher price when the demand is realised. If the retailer reserves a large capacity (i.e. greater than K 1 ), then the manufacturer believes that an overstock will exist, so she produces less than the quantity reserved by the retailer to avoid residual capacity. In the middle range where Q ∈ (K 2 , K 1 ), the manufacturer builds capacity exactly as the retailer reserves. The reason is as follows. The manufacturer has two options when building capacity, i.e., K ≤ Q or K ≥ Q. For the two options, the manufacturer both faces a newsvendor-type problem. Specifically, under the first option, no βc exists. The manufacturer balances the overstock cost (c) and understock

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 , the understock cost (( pe + αc )[1 − F(K )]). When K ≤ Q < K 1 = F −1 ααc +c +pep−c e cost is higher and the optimal decision is to build as much as possible. Therefore, K ∗ = Q for K ≤ Q. Under the second option, no αc exists. The manufacturer balances the overstock

 cost (c) and understock cost (βc [1 − F(K )]). When −1 βc −c , the overstock cost is higher and the optimal decision K ≥ Q > K2 = F βc is to build as small an amount as possible. Therefore, K ∗ = Q for K ≥ Q. These two optimal decisions converge to K ∗ = Q. To conclude, when the reserved quantity is moderate, the manufacturer optimally sets K = Q to minimise all related costs.

6.3.2 Reservation Decision of Retailer In this we investigate how  subsection,    the retailer reserves capacity. Note that K 2 = −1 αc + pe −c and K are independent of Q. From Theorem 6.2, we = F F −1 βcβ−c 1 αc + pe c have the profit function of the retailer as follows:  R = max  R (Q) = max max  R (Q, K 1 ), max  R (Q, K 2 ), Q

Q≥K 1

Q≤K 2

max

Q∈[K 2 ,K 1 ]

 R (Q, Q) .

(6.6) In detail, we write the functions of  R (Q, K 1 ),  R (Q, K 2 ) and  R (Q, Q) as follows:  R (Q, K 1 ) = μ p E D (D ∧ K 1 ) − po Q − pe E D (K 1 ∧ D) + αc E D [(D ∧ Q) − K 1 ]+ , (6.7)  R (Q, K 2 ) = μ p E D (D ∧ K 2 ) − po Q − pe E D {Q ∧ D} − βc E D [(D ∧ K 2 ) − Q]+ ,  R (Q, Q) = μ p E D (D ∧ Q) − po Q − pe E D (Q ∧ D).

(6.8) (6.9)

Let Q 1 , Q 2 and Q 3 denote the optimal solution to  R (Q, K 1 ),  R (Q, K 2 ) and and  R(Q, K 2 )  R (Q, Q), respectively. Taking the first derivative of  R (Q, K 1 )

−1 αc − po , Q2 = as well as  R (Q, Q) with respect to Q, we have Q 1 = F αc



 μ − p − p o e β − p − p p . F −1 cβc −o pe e and Q 3 = F −1 μ p − pe Based on Assumption 6.1, Q 1 , Q 2 and Q 3 are well defined and we can easily obtain Q 1 ≥ Q 3 ≥ Q 2 . Furthermore, we obtain the equilibrium of the Stackelberg game in the following theorem: Theorem 6.3 Given contract parameters ( po , pe , αc , βc ), the equilibrium of the Stackelberg game is determined as follows: (1) If Q 2 ≥ K 1 , then (Q ∗ , K ∗ ) = (Q 1 , K 1 ). (2) If Q 2 ∈ (K 2 , K 1 ), Q 3 ≥ K 1 and Q 1 ≥ K 1 , then (Q ∗ , K ∗ ) = (Q 1 , K 1 ). (3) If Q 2 ∈ (K 2 , K 1 ), Q 3 ∈ (K 2 , K 1 ) and Q 1 ≥ K 1 , then (Q ∗ , K ∗ ) = (Q 3 , Q 3 ) or (Q 1 , K 1 ). (4) If Q 2 ≤ K 2 , Q 3 ≥ K 1 and Q 1 ≥ K 1 , then (Q ∗ , K ∗ ) = (Q 2 , K 2 ) or (Q 1 , K 1 ).

6.3 Capacity Reservation Contract

177

(5) If Q 2 ≤ K 2 , Q 3 ∈ (K 2 , K 1 ) and Q 1 ≥ K 1 , then (Q ∗ , K ∗ ) = (Q 2 , K 2 ) or (Q 3 , Q 3 ) or (Q 1 , K 1 ). (6) If Q 2 ≤ K 2 , Q 3 ≤ K 2 and Q 1 ≥ K 1 , then (Q ∗ , K ∗ ) = (Q 2 , K 2 ) or (Q 1 , K 1 ). (7) If Q 2 ≤ K 2 , Q 3 ∈ (K 2 , K 1 ) and Q 1 < K 1 , then (Q ∗ , K ∗ ) = (Q 2 , K 2 ) or (Q 3 , Q 3 ). (8) If Q 2 ≤ K 2 , Q 3 ≤ K 2 and Q 1 < K 1 , then (Q ∗ , K ∗ ) = (Q 2 , K 2 ). Theorem 6.3 is easily obtained by comparing  R (Q, K 1 ),  R (Q, K 2 ) and  R (Q, Q) under specific conditions. Theorem 6.3 shows that Q 1 , Q 2 and Q 3 are potential equilibrium where the reservation fee po plays an important role. If po is rather small, for example, po < (βαcc−+ppee)c that is the condition of first case in Theorem 6.3, the retailer reserves the largest quantity Q 1 . If po is rather large, for example, (u − p )c po > max{ p βc e , αcα+c cpe } that is the condition of eighth case in Theorem 6.3, the eighth case holds and the retailer reserves the smallest quantity Q 2 . In other cases where po is moderate, the retailer chooses reservation quantity among Q 1 , Q 2 or Q 3 by comparing corresponding profits.

6.3.3 The Benefit of the Capacity Reservation Contract In this subsection, we justify the benefit of our proposed capacity reservation contract in comparison with the contract of less flexibility. We consider the case that the manufacturer is forced to fully satisfy the retailer’s reservation, i.e., K ≥ Q. As a consequence, no αc exists. Let 0R and 0M denote the profit of the retailer and manufacturer with the contract without αc , respectively. We have the following:  0R (Q, K ) = μ p E [D ∧ K ] − po Q − E D pe (Q ∧ K ∧ D) + βc ((D ∧ K ) − Q)+ , (6.10)  0M (Q, K ) = po Q + E D pe (Q ∧ K ∧ D) + βc ((D ∧ K ) − Q)+ − cK , (6.11) s.t. K ≥ Q.

(6.12)

The sequence of events is the same to the flexibility contract. In brief, the retailer firstly reserves Q and the manufacturer then builds K , which should be no less than Q. With backward induction method, we also derive the equilibrium decisions, which are summarised in the following proposition. Due to high similarity to Theorem 6.3, we omit the proof. Proposition 6.1 In the less flexible contract without penalty to the manufacturer (αc ), let Q 0∗ and K 0∗ be the equilibrium reservation quantity and building capacity, respectively, then we have the following: (1) If Q 3 ≤ K 2 , then (Q 0∗ , K 0∗ ) = (Q 2 , K 2 ); (2) If Q 2 < K 2 < Q 3 , then (Q 0∗ , K 0∗ ) = (Q 2 , K 2 ) or (Q 3 , Q 3 ); (3) If Q 2 ≥ K 2 , then (Q 0∗ , K 0∗ ) = (Q 3 , Q 3 ).

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All the three equilibria can be found in Theorem 6.3, which indicates that larger flexibility yields more equilibrium results. More important, the following corollary reveals that under certain conditions, the total profit in our proposed capacity reservation contract is strictly larger than that in the contract without αc . Corollary 6.1 If αc
0∗ .

Note that this is one of sufficient conditions to achieve ∗ > 0∗ . The proof of Corollary 6.1 is in the Appendix. The main insight is that when setting rather small αc and granting the manufacturer the flexibility to partially satisfy the retailer’s order, the manufacturer may deviate downwards from the retailer’s reservation. Consequently, the building capacity accommodates the market demand better and the flexibility brings more total profit in the supply chain, which indicates the advantage of our proposed contract over that with less flexibility.

6.3.4 Design Parameters for Channel Coordination In the above subsection, we find that the flexible contract brings more profit to the supply chain under some conditions, which, however, may not guarantee maximal system profit. In other words, the supply chain is not coordinated. Therefore, now we proceed to design parameters of capacity reservation contract to achieve channel coordination. Theorem 6.3 has characterised the optimal strategies of manufacturer and the retailer and derived the Stackelberg game equilibrium in the decentralised system. Comparing these strategies with those in the centralised system, we can obtain the conditions which can coordinate the supply chain. The supply chain achieves coordination if and only if the total profit in the decentralised system is equal to that in the centralised system. The profit functions of the manufacturer and the retailer in Eqs. (6.3) and (6.4), respectively, show that the total profit is  =  M +  R = μ p E[D ∧ K ] − cK , which is only dependent on K . Therefore, the condition that achieves coordination is K ∗ = K C∗ S . In other words, given certain contract parameters in equilibrium of the Stackelberg game, if the manufacturer constructs the capacity right at the system optimal level, then supply chain coordination is achieved. Based on Theorem 6.3, contract parameters that result in K ∗ = K C∗ S can be obtained, as shown in following theorem: Theorem 6.4 The capacity reservation contract with the following parameters can coordinate the supply chain as follows: (1) αc + pe = μ p , and pco ∈ (0, βcβ−c pe ). (2) αc + pe = μ p , (3) βc = μ p and (4) βc = μ p , (5)

po c

=

po c

∈ [ βcβ−c pe ,

μ p − pe ] μp

> αcα+c pe . μ −p [ pμ e , αcα+c pe ] p

po c

po ∈ c μ p − pe ,  R (Q 3 , μp

and  R (Q 1 , K 1 ) ≥  R (Q 2 , K 2 ).

and  R (Q 2 , K 2 ) ≥  R (Q 1 , K 1 ).

Q 3 ) ≥  R (Q 1 , K 1 ) and  R (Q 3 , Q 3 ) ≥  R (Q 2 , K 2 ).

6.4 Quantity Flexibility Contract

179

Theorem 6.4 summarises the sufficient and necessary conditions to achieve channel coordination. In detail, Condition (1) or (2) each guarantees the equilibrium to be (Q 1 , K 1 ) where K 1 = K C∗ S . Condition (3) or (4) each guarantees the equilibrium to be (Q 2 , K 2 ) where K 2 = K C∗ S . Condition 5 guarantees the equilibrium to be (Q 3 , Q 3 ) where Q 3 = K C∗ S . The conditions in Theorem 6.4 indicate that achieving channel coordination by proposing the capacity reservation contract is easy. Within so many parameters that guarantee coordination, we focus on a special case in which the manufacturer builds capacity as the retailer reserves, as shown in the following theorem: Theorem 6.5 Under channel coordination, the manufacturer builds capacity exactly μ −p as the retailer reserves, if βc = μ p , αc + pe = μ p and pco = pμ e . In addition, the p profit can be arbitrarily allocated between the retailer and the manufacturer. Corollary 6.1, Theorems 6.4 and 6.5 together show the advantage of such a parameter-rich contract. Firstly, when channel coordination is not achieved, more flexibility in our proposed contract can bring more profit to the supply chain. Secondly, with enough levers in hand to adjust the behaviour of two agents, there are more conditions that can lead to supply chain coordination, compared with the contract with less flexibility. In other words, supply chain coordination conditions are less restrictive with our contract than others. Last but not the least, in addition to arbitrary profit allocation under channel coordination, as several supply chain contracts do, our contract has additional advantage of simpler calculation of profit split, even without the information of demand distribution. For example, under the conditions in Theorem 6.5, we have Q ∗ = K ∗ and we can tell the profit splits of the manufacturer and the retailer are μpe and pco , respectively, which are independent of demand p distribution.

6.4 Quantity Flexibility Contract In this section, we study the case in which the manufacturer and retailer adopt the quantity flexibility contract. The retailer acts as the leader and the manufacturer is the follower. The sequence of events under this contract is as follows: (1) A contract with parameters (w, αq , βq , d) is offered. w is the wholesale price after demand realization. αq represents the penalty per unit that the manufacturer has to pay if she cannot satisfy the order of the retailer within the allowable range. βq indicates the penalty that the retailer has to pay if his final order is outside of an allowable range. d ∈ [0, 1] is a parameter that measures the flexibility and determines the allowable range. (2) Given the contract, the retailer reserves quantity Q. Consequently, the allowable range is determined as [(1 − d)Q, (1 + d)Q]. Note that the retailer does not need to pay the reservation fee at present. (3) After observing the reserved quantity, the manufacturer sets her capacity K , which is no less than the lower bound of the allowable range, i.e., K ≥ (1 − d)Q. (4) The demand D and market retail price p of

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the retailer are realised, and the retailer orders D units. (5) The manufacturer delivers min(D, K ) and the payment is settled. ¯ M (Q, K ) denote the expected profit function of the retailer ¯ R (Q, K ) and  Let  and the manufacturer, respectively. Then, we have the following: ¯ R (Q, K ) = μ p E [D ∧ K ] − T¯ (Q, K ),  ¯ M (Q, K ) = T¯ (Q, K ) − cK , 

(6.13) (6.14)

where T¯ (Q, K ) is the transfer payment function from the retailer to the manufacturer, with the following specific expression:    T¯ (Q, K ) = E D w (D ∧ K ) + βq ((D ∧ K ) − (1 + d) Q)+ + ((1 − d) Q − D)+ − αq ((1 + d) Q ∧ D) − K )+ . (6.15)

Our model also has two main decision stages. In the first stage, the retailer determines the reservation quantity Q. In the second stage, the manufacturer chooses her capacity level K . We analyse the problem in a backward sequence. Thus, we firstly analyse the capacity construction behaviour of the manufacturer in the next section.

6.4.1 Construction Decision of Manufacturer To gain the best-response strategy for the manufacturer in the quantity flexibility contract, we firstly present the following assumption and then analyse the decision of the manufacturer given the reserved quantity Q of the retailer. Assumption 6.2 w + βq ≥ αq ≥ βq . The left inequality means that the purchasing cost of the capacity beyond the range ([(1 − d)Q, (1 + d)Q]) is greater than the penalty for the unfulfilled order, which is reasonable. On the right hand of the assumption, αq ≥ βq is easy to satisfy in reality. That is, the penalty for the manufacturer is larger, which can motivate the retailer to accept the contract. ¯ M, I I (Q, K ) denote the expected profit of the manufacturer when she plans Let  ¯ M, I I I (Q, K ) as to construct capacity in the region of [(1 − d)Q, (1 + d)Q] and  the expected profit of the manufacturer when she plans to construct capacity no less than (1 + d)Q. The profit functions are detailed as follows:  ¯ M,I I (Q, K ) = E D w(D ∧ K ) − αq [((1 + d)Q ∧ D) − K ]+ + βq [(1 − d)Q − D]+ − cK ,   ¯ M,I I I (Q, K ) = E D w(D ∧ K ) + βq [((D ∧ K ) − (1 + d)Q)+ + ((1 − d)Q − D)+ ] − cK . 

6.4 Quantity Flexibility Contract

181

¯ M,I I (Q, K ) and  ¯ M,I I I (Q, K ) Let K I I and K I I I be the optimal solutions to  without constraints, respectively. KI I KI I I

 w + αq − c , K w + αq   w + βq − c ¯ M,I I I (Q, K ) = F −1 = arg max  . K w + βq ¯ M,I I (Q, K ) = F −1 = arg max 



(6.16) (6.17)

Given the preceding expressions, we immediately have the following remark: Remark 6.1 K I I > K I I I . ¯ M,I I (Q, K ), the manufacturer is motivated to build In the profit function of  more capacity to avoid the penalty (αq ) when she cannot satisfy the demand of the ¯ M,I I I (Q, K ), the manufacturer retailer within the range. In the profit function of  is motivated to build more capacity to avoid the loss (βq ) when the retailer gives an extra order beyond (1 + d)Q, which can be deemed as an opportunity cost for the manufacturer. Based on Assumption 6.2, the manufacturer has a stronger motivation to build capacity in the former case, as Remark 6.1 shows. The manufacturer builds capacity K to maximise her own profit. Thus, we have  ¯ M,I I I (Q) , ¯ M (Q, K ) = max  ¯ M (Q) = max  ¯ M,I I (Q);   K

¯ M,I I (Q) = where 

max

K ∈[(1−d)Q,(1+d)Q]

¯ M,I I I (Q) = ¯ M,I I (Q, K ),  

max

K >(1+d)Q

(6.18) ¯ M,I I I 

(Q, K ). The following theorem summarises the optimal capacity-building policy of the manufacturer, for a given Q. Theorem 6.6 Given the reserved quantity Q of the retailer, the best-response function K¯ (Q) of the manufacturer is expressed as follows:

K¯ (Q) =

⎧ KI I (1 − d)Q, if Q > 1−d ; ⎪ ⎪ ⎪ ⎪ K KI I II ⎨ KI I , if Q ∈ ( 1+d , 1−d ]; KI I III ⎪ , 1+d ]; (1 + d)Q, if Q ∈ ( K1+d ⎪ ⎪ ⎪ ⎩ KI I I if Q ≤ 1+d . KI I I ,

The proof is in the Appendix. Figure 6.3 characterises the best-response strategy of the manufacturer. The figure illustrates that the manufacturer always constructs more than the reserved quantity if Q < K I I . Otherwise, the manufacturer always constructs less capacity. In detail, if the reserved quantity Q is extremely small ( III ), then the manufacturer always constructs K I I I , which is constant. If the Q < K1+d reserved quantity is larger, then the manufacturer builds capacity proportionally to the reserved quantity, with the multiplier 1 + d. If Q is becomes increasingly larger, then the manufacturer will construct K I I . Otherwise, she will decrease her capacity level proportionally with the multiplier 1 − d because the lower bound is (1 − d)Q.

182

6 Supply Chain Coordination Through …

KIII

K

(Q

)

=

(1

+

K

(Q

d) Q

KII

)

=

(1

−

d) Q

K

(Q

)

=

Q

¯ K(Q)

KIII 1+d

KII 1+d

KII 1−d

KII

Q

Fig. 6.2 Best-response function of the manufacturer in the quantity flexibility contract

Remark 6.2 Given any parameter in the quantity flexibility contract, the capacity level constructed by the manufacturer is not equal to the reserved quantity of the retailer except when Q = K I I . Remark 6.2 can be easily observed from Fig. 6.2. This condition indicates that when building capacity, the manufacturer easily deviates from what the retailer reserves, which is the consequence of the voluntary compliance regime.

6.4.2 Reservation Decision of Retailer The retailer chooses his reserved quantity which maximises his expected profit while anticipating the response of the manufacturer. We assume that the expected retail price is not extremely large; thus, we have the following assumption for convenient technical handling: Assumption 6.3 μ p < w +

βq (w−c) c

+ αq (

βq c

−

2d ). 1−d β (w−c)

β

Note that μ p ≥ w holds, and the following explanation is for q c + αq ( cq − 2d ) > 0. In practice, the deviation portion should not be extremely high, otherwise 1−d

6.4 Quantity Flexibility Contract

183

the manufacturer is not profitable. For example, we can assume that d < 13 (i.e. β (w−c) β 2d + αq ( cq − 1−d ) > 0 is the deviation ratio is less than 33% or so). Thus, q c equivalent to βq (w + αq ) > (βq + αq )c. Evidently, the condition is easy to satisfy. From Theorem 6.6, we have the profit function of the retailer as follows:  1 ¯ ∗R = max  ¯ R,  ¯ 2R ,  ¯ 3R ,  ¯ 4R ,  ¯ 1R = max  ¯ 2R = ¯ R (Q, K I I I ),  where  K Q< I I I 1+d

max

Q∈[

KI I KI I , ) 1+d 1−d

max

K K Q∈[ I I I , I I ) 1+d 1+d

(6.19)

¯ 3R = ¯ R (Q, (1 + d) Q),  

¯ 4R = max  ¯ R (Q, K I I ), and  ¯ R (Q, (1 − d) Q).  Q≥

KI I 1−d

¯ 4R can be characterised Based on Assumption 6.3, the property of the function  as follows:   ¯ R Q, (1 − d)Q is concavely decreasing in Q for Q > K I I . Lemma 6.1  1−d ¯3 ¯4 The proof is in the Appendix. As a result,  1  R2 ≤ 3 R . Therefore, the profit function ∗ ¯ ¯ ¯ ¯ of the retailer decreases to  R = max  R ,  R ,  R . ¯ R (Q, K I I I ), Let Q¯ 1 , Q¯ 2 and Q¯ 3 denote the optimal solutions to  ¯ R (Q, K I I ), respectively. Thus, we have the following: ¯ R (Q, (1 + d) Q) and  

1−d F ((1 − d)Q) = 1 Q¯ 1 = Q|F ((1 + d)Q) + 1+d

(1 − d)βq F ((1 − d)Q) = 1 Q¯ 2 = Q|F ((1 + d)Q) + (1 + d)(μ p − w)

(1 − d)βq ¯ Q 3 = Q|F ((1 + d)Q) + F ((1 − d)Q) = 1 . (1 + d)αq

(6.20) (6.21) (6.22)

Lemma 6.2 (i) If μ p ∈ [w + βq , w + αq ) , then Q¯ 3 > Q¯ 2 ≥ Q¯ 1 ; (ii)If μ p ≥ w + αq , then Q¯ 2 ≥ Q¯ 3 > Q¯ 1 . These results can be immediately obtained from Eqs. (6.20–6.22). ¯ R ). Let Q¯ ∗ denote the optimal reserved quantity of the retailer (i.e. Q¯ ∗ = arg max  Q

¯ R is a piecewise-concave Q function, with Lemma 6.2 in hand, we can obtain the As  equilibrium decisions in the Stackelberg game (i.e. ( Q¯ ∗ , K¯ ∗ )) in Tables 6.2 and 6.3. Tables 6.2 and 6.3 provide the equilibrium reserved quantity and capacity level under all conditions. We find that the manufacturer may construct capacity more than the reserved quantity of the retailer in most scenarios, but she may also construct less in certain scenarios, which is different from the case of the capacity reservation contract.

184

6 Supply Chain Coordination Through …

Table 6.2 Equilibrium decisions if μ p ≥ w + αq Optimal reserved quantity Equilibrium ( Q¯ ∗ , K¯ ∗ ) ∗ ¯ ¯ Q = Q1 ( Q¯ 1 , K I I I )

Conditions Q¯ 1 ≤ Q¯ 3 ≤ Q¯ 2 ≤

¯R Q¯ ∗ = arg max{ Q¯ 1 , Q¯ 2 } 

( Q¯ 1 , K I I I ) ver sus ( Q¯ 2 , (1 + d) Q¯ 2 )

Q¯ ∗ = arg max{ Q¯

KI I ( Q¯ 1 , K I I I ) ver sus ( 1+d , K I I ) Q¯ 1 ≤ { Q¯ 3 , { Q¯ 2 , K I I }

KI I 1 , 1+d

}

¯R 

Q¯ 1 ≤ { Q¯ 3 , KI I 1+d

KI I I 1+d

KI I I 1+d

} ≤ Q¯ 2 ≤

KI I I 1+d

}≤

KI I 1+d

≤

1−d

¯R Q¯ ∗ = arg max{ Q¯ 1 , Q¯ 3 } 

III ( Q¯ 1 , K I I I ) ver sus ( Q¯ 3 , K I I ) Q¯ 1 ≤ K1+d ≤ KI I ¯ { 1−d , Q 2 }

KI I 1+d

≤ Q¯ 3 ≤

Q¯ ∗ = arg max{ Q¯

( Q¯ 1 , K I I I ) ver sus K I I 1+d ( 1−d , 1−d K I I )

III Q¯ 1 ≤ K1+d ≤ ¯ ¯ Q3 ≤ Q2

KI I 1+d

≤

Q¯ ∗ = Q¯ 2

( Q¯ 2 , (1 + d) Q¯ 2 )

KI I I 1+d KI I 1+d

≤ Q¯ 1 ≤ Q¯ 3 ≤ Q¯ 2 ≤

Q¯ ∗ =

KI I ( 1+d , KI I )

KI I I 1+d

≤ Q¯ 1 ≤ Q¯ 3 ≤ KI I 1−d }

KI I 1 , 1−d

}

¯R 

KI I 1+d

{ Q¯ 2 ,

Q¯ ∗ = Q¯ 3

( Q¯ 3 , K I I )

KI I I ¯ KI I 1+d ≤ { Q 1 , 1+d } KI I { 1−d , Q¯ 2 }

Q¯ ∗ =

KI I ( 1−d , KI I )

KI I I 1+d

KI I 1−d

KI I ≤ { 1+d , Q¯ 3 ≤ Q¯ 2

KI I 1−d

KI I 1+d

≤

≤

≤ Q¯ 3 ≤

KI I 1−d ,

Q¯ 1 } ≤

6.4.3 The Benefit of the Quantity Flexibility Contract In this subsection, we justify the benefit of our proposed quantity flexibility contract in comparison with the contract of less flexibility. We consider the case that the retailer has no reservation flexibility (d = 0) and the manufacturer is forced to fully ¯ 0R satisfy the retailer’s reservation (K ≥ Q). As a consequence, no αq occurs. Let  0 ¯ and  M be the profit of the retailer and manufacturer in the less flexible conttract, respectively. Then, we have the following:  ¯ 0R (Q, K ) = μ p E[D ∧ K ] − w(D ∧ K ) − βq [(D ∧ K ) − Q]+ + (Q − D)+ , (6.23)   ¯ 0M (Q, K ) = w(D ∧ K ) + βq [(D ∧ K ) − Q]+ + (Q − D)+ − cK ,  (6.24) s.t. K ≥ Q.

(6.25)

The retailer reserves Q firstly and the manufacturer builds capacity K afterwards. We also use backward induction method to analyze the equilibrium decisions. We directly give the equilibrium results in the following.

¯ 2, KI I 1,Q 1−d

}

¯R 

Q¯ ∗ = Q¯ 3 KI I Q¯ ∗ = 1−d

{ Q¯ 2 , 1−d }

Q¯ ∗ = Q¯ 2 ¯R Q¯ ∗ = arg max{ Q¯ 2 , Q¯ 3 }  ¯R Q¯ ∗ = arg max KI I 

Q¯ ∗ = arg max{ Q¯

¯R Q¯ ∗ = arg max{ Q¯ 1 , Q¯ 2 }  ∗ ¯ ¯R Q = arg max{ Q¯ 1 , Q¯ 2 , Q¯ 3 } 

¯R 

( Q¯ 3 , K I I ) KI I ( 1−d , KI I )

1−d

( Q¯ 1 , K I I I ) ver sus ( Q¯ 2 , (1 + d) Q¯ 2 ) ( Q¯ 1 , K I I I ) ver sus( Q¯ 2 , (1 + d) Q¯ 2 ) ver sus ( Q¯ 3 , K I I ) ( Q¯ 1 , K I I I ) ver sus( Q¯ 2 , (1 + KI I d)) Q¯ 2 ) ver sus( 1−d , KI I ) ¯ ¯ ( Q 2 , (1 + d) Q 2 ) ( Q¯ 2 , (1 + d) Q¯ 2 ) ver sus ( Q¯ 3 , K I I ) ( Q¯ 2 , (1 + d) Q¯ 2 ) ver sus ( K I I , K I I )

KI I ( Q¯ 1 , K I I I ) ver sus ( 1−d , KI I )

KI I 1 , 1−d

}

Q¯ ∗ = arg max{ Q¯

{ Q1 , Q3 }

Equilibrium ( Q¯ ∗ , K¯ ∗ ) ( Q¯ 1 , K I I I ) ( Q¯ 1 , K I I I ) ver sus ( Q¯ 3 , K I I )

Optimal reserved quantity Q¯ ∗ = Q¯ 1 ¯R Q¯ ∗ = arg max ¯ ¯ 

Table 6.3 Equilibrium decisions if μ p ∈ [w + βq , w + αq )

1+d

1+d

1−d

KI I I 1+d KI I I 1+d KI I I 1+d KI I I 1+d KI I I 1+d

KI I I 1+d

≤ Q¯ 2 ≤

KI I 1+d

≤ { Q¯ 3 ,

KI I 1−d }

≤ { Q¯ 1 , ≤ { Q¯ 1 ,

KI I 1+d } KI I 1+d }

KI I ≤ Q¯ 2 ≤ Q¯ 3 ≤ 1−d K II ≤ { Q¯ 2 , 1−d } ≤ Q¯ 3

KI I KI I ≤ Q¯ 1 ≤ Q¯ 2 ≤ Q¯ 3 ≤ 1+d ≤ 1−d K KI I I I ≤ Q¯ 1 ≤ Q¯ 2 ≤ 1+d ≤ Q¯ 3 ≤ 1−d K K I I I I ≤ Q¯ 1 ≤ Q¯ 2 ≤ 1+d ≤ 1−d ≤ Q¯ 3

Q¯ 1 ≤

KI I KI I III Q¯ 1 ≤ Q¯ 2 ≤ K1+d ≤ 1+d ≤ 1−d ≤ Q¯ 3 K K KI I III II Q¯ 1 ≤ 1+d ≤ Q¯ 2 ≤ Q¯ 3 ≤ 1+d ≤ 1−d KI I KI I III Q¯ 1 ≤ K1+d ≤ Q¯ 2 ≤ 1+d ≤ Q¯ 3 ≤ 1−d

Conditions KI I KI I III Q¯ 1 ≤ Q¯ 2 ≤ Q¯ 3 ≤ K1+d ≤ 1+d ≤ 1−d KI I I KI I KI I ¯ ¯ ¯ Q 1 ≤ 1+d ≤ 1+d ≤ Q 2 ≤ Q 3 ≤ 1−d K K KI I I I I I I Q¯ 1 ≤ Q¯ 2 ≤ 1+d ≤ 1+d ≤ Q¯ 3 ≤ 1−d K K K Q¯ 1 ≤ I I I ≤ I I ≤ { Q¯ 2 , I I } ≤ Q¯ 3

6.4 Quantity Flexibility Contract 185

186

6 Supply Chain Coordination Through …

Proposition 6.2 In the contract of less flexibility with d = 0 and αq = 0, let Q¯ 0∗ and K¯ 0∗ be the equilibrium reservation quantity and building capacity, respectively, then we have the following: βq c (1) If w+β < μ p −w+β , then ( Q¯ 0∗ , K¯ 0∗ ) = (F −1 (0.5), K I I I ); q q βq c (2) If w+β ∈ [ μ p −w+β , 0.5], then ( Q¯ 0∗ , K¯ 0∗ ) = (F −1 (0.5), K I I I ) or q q

 μ p −w μ p −w −1 F −1 ( u p −w+β ), F ( ) ; u p −w+βq q

 c ¯ 0∗ , K¯ 0∗ ) = F −1 ( μ p −w ), F −1 ( μ p −w ) . (3) If w+β > 0.5, then ( Q u p −w+βq u p −w+βq q Further, we can also find that under certain conditions, the flexible contract brings strictly larger total profit in the supply chain. The conditions are revealed in Corollary 6.2. Corollary 6.2 Assuming the market demand follows uniform distribution, then if w+β (1+d)2 (μ −w)c ¯∗ > ¯ 0∗ . c > 2 q and αq ≥ c − w + (1−d)2pβq , then K¯ ∗ > K¯ 0∗ and  ¯∗ > ¯ 0∗ . The proof of Note that this is one of sufficient conditions that lead to  Corollary 6.2 is in the Appendix. By setting a relatively large αq , the manufacturer has stronger incentive to build more capacity, which brings more total profit to the supply chain. This result indicates the advantage of our proposed contract over that with less flexibility.

6.4.4 Design Parameters for Channel Coordination The above conditions bring channel performance improvement, but cannot necessarily guarantee system-optimal performance. In other words, supply chain coordination is not achieved. Now we proceed to design parameters that lead to channel coordination. Based on the profit functions of the manufacturer and the retailer in Eqs. ¯R= ¯ = ¯ M + (6.13) and (6.14), the total decentralised supply chain profit is  μ p E(D ∧ K ) − cK , which is independent of Q. Therefore, the sufficient and necessary condition to achieve coordination is K ∗ = K C∗ S . In Tables 6.2 and 6.3, diverse cases of equilibrium decisions and lack of closedform Q¯ 1 , Q¯ 2 and Q¯ 3 expressions make exhausting every condition that guarantees K ∗ = K C∗ S intractable, as we have done for the capacity reservation contract. However, we can provide some sufficient conditions. Theorem 6.7 The supply chain can be coordinated  1−d  by the following conditions, αq = μ p − w, βq = αq , F(K I I ) + 1−d F K = 1. In addition, the profit can I I 1+d 1+d be arbitrarily allocated between the manufacturer and the retailer. The proof is in the Appendix. The conditions in Theorem 6.7 guarantee the equiKI I librium to be ( Q¯ ∗ , K¯ ∗ ) = ( 1+d , K I I ), where K I I = K C∗S = F −1 (1 − μc ). For ease p

6.5 Comparison Between Quantity Flexibility Contact and Capacity Reservation Contract

187

of exposition, denote ∗ = μ p E(D ∧ K C∗ S ) − cK C∗ S and d = 1−d . Under the condi1+d tions in Theorem 6.7, the profit split of the manufacturer is λq = 1 −  dK ∗ αq [E(D∧K C∗ S )− 0 C S (dK C∗ S −D) f (D)dD] and the profit split of the retailer is 1 − ∗ (1−λq )∗ ∈ [0, μ p ), βq = αq , alently, setting αq =  dK ∗ E(D∧K C∗ S )− 0 C S (dK C∗ S −D) f (D)dD

λq . Equivw = μp −

αq leads to coordination with arbitrary profit split λq . Analogous to capacity reservation contract, quantity flexibility contract can also induce the manufacturer to build capacity as the retailer reserves and maintains channel coordination. The following theorem reveals the conditions that guarantee Q ∗ = K ∗ = K C∗S .

Theorem 6.8 The manufacturer builds capacity exactly as the retailer reserves

in the coordinated supply chain under the following conditions: αq = μ p − w, F (1 +  

  (1−d)β d)K C∗S + (1+d)αqq F (1 − d)K C∗S = 1 and 1−d F 1−d K ∗ ≤ μc . 1+d 1+d C S p

The proof is in the Appendix. In other words, the conditions here guarantee the equilibrium decision to be ( Q¯ 3 , K I I ), where K I I = Q¯ 3 = K C∗S . Note that when setting αq = βq , the third condition becomes redundant because it can be derived from the second one. By this theorem, we find that XXX

6.5 Comparison Between Quantity Flexibility Contact and Capacity Reservation Contract In this section, we compare the two contracts and investigate the preference of the manufacturer/retailer. In the real world, the choice of firms on different contracts may be influenced by kinds of factors. For example, one manufacturer may prefer the capacity reservation contract instead of quantity flexibility contract because the former provides a certain revenue at the first stage before demand realisation. However, in this research, we ignore the impact of payment time or so-called “time value of money” and measure the performance of two contracts with the total profit, which is common in relevant literature, including Erkoc and Wu (2005), Jin and Wu (2007), Fu et al. (2010), Lee et al. (2016) and Fu (2015).

6.5.1 Comparison Without Coordination 6.5.1.1

Comparison of Capacity

In the following, we compare the best-response strategy of the manufacturer for given Q in two contracts. The parameters in the two contracts have the following relations: (1) w = po + pe ; (2) w + βq = βc ; (3) αc − po = αq . The first equation means that

188

6 Supply Chain Coordination Through …

the normal procurement costs in two contracts are equal. This situation occurs when the demand is smaller than both the capacity of the manufacturer and the reservation region of the retailer. The second equation indicates equal procurement cost for the quantity beyond the reservation region, which happens when the demand is larger than the reservation region and smaller than the capacity of the manufacturer. The third equation means that the actual penalties for the manufacturer in two contracts are equal, which happens when the demand is larger than the capacity of the manufacturer and smaller than the reservation region of the retailer. Remark 6.3 (i) K I I I = K 2 ; (ii)K I I = K 1 . This remark is easily obtained by the definition of K I I I , K I I , K 2 and K 1 in Theorem 6.1 as well as Eqs. (6.16) and (6.17). Based on this remark, as well as Figs. 6.1 and 6.2, the following proposition is obtained: Proposition 6.3 (i) The manufacturer constructs the same quantity as the retailer reserves only when Q = K I I under the quantity flexibility contract. By contrast, under the capacity reservation contract, she may construct the same quantity as the retailer reserves in a range of Q. (ii) For given Q, the manufacturer constructs greater capacity in the quantity flexibility contract than in the capacity reservation contract. This proposition can be easily observed by combining Figs. 6.1 and 6.2. For the first part, more flexibility (under the quantity flexibility contract) indicates that the manufacturer likely deviates from what the retailer reserves. As illustrated in Fig. 6.2, the manufacturer considers the range parameter d and constructs (1 − d)Q or (1 + d)Q to respond to the reservation decision of the retailer. Thus, we cannot find a region where K = Q holds. The second part shows that the manufacturer has strong incentive to construct more capacity under quantity flexibility contract than in the capacity reservation contract. Referring to Eq. (6.15), the penalty of and the compensation to the manufacturer in the quantity flexibility contract are dependent on (1 + d)Q and K , whereas in the capacity reservation contract, the counterparts are Q and K (referring to Eq. (6.5)). Therefore, the manufacturer constructs more capacity to reduce penalty and gain compensation.

6.5.1.2

Comparison of Profit

In this subsection, we compare the profits of members in the supply chain. The main purpose is to investigate the conditions under which the retailer (or the manufacturer) prefers the capacity reservation contract or the other one. First, we obtain the following results: Theorem 6.9 The capacity reservation contract and quantity flexibility contract are identical if po = βq and d = 0 along with the premises of (1) w = po + pe ; (2) w + βq = βc ; and (3) αc − po = αq .

6.5 Comparison Between Quantity Flexibility Contact and Capacity Reservation Contract

189

The proof is in the Appendix. Theorem 6.9 shows that if the flexibility is negligible (d = 0), and the penalty for the retailer in the quantity flexibility contract is equal to the reservation fee po , then optimal decisions of the retailer are the same under two contracts. Therefore, the manufacturer does not have to choose from either one contract. The following theorem summarises how po and d influence the choices of the retailer and manufacturer between two contracts. ˆ the retailer Theorem 6.10 (1) If po < βc −2 pe , threshold dˆ exists so that when d > d, ˆ prefers the quantity flexibility contract, and when d < d, the retailer prefers the capacity reservation contract. The preference of the manufacturer is the opposite. (2) If po ≥ βc −2 pe , the retailer always prefers the quantity flexibility contract and the manufacturer always prefers the capacity reservation contract. The theorem is directly derived from Theorem 6.9 and the proof is omitted. Intuitively, a high reservation fee ( po ) in the capacity reservation contract makes it less attractive to the retailer. Therefore, when po is large, the retailer prefers the other contract. This theorem makes sense. Specifically, if po > βc −2 pe , we obtain βq < po with the premise of βq = βc − pe − po . Based on Theorem 6.9, because d > 0 and βq < po , the quantity flexibility contract brings about less reservation cost and more flexibility than the other contract. As a result, the profit of the retailer is evidently large in the former contract. By contrast, if po ≤ βc −2 pe , then the capacity reservation contract has the advantage of less reservation cost, whereas the quantity flexibility contract has the advantage of more flexibility. The retailer should make the tradeoff. ˆ the retailer selects The optimal decision is that when the flexibility is large (d > d), the quantity flexibility contract. Otherwise, he selects the other one. These results are illustrated through numerical experiments. Based on the assumptions made, the following parameters are set: c ∈ [1, 2), μ p ∈ [4.0, 4.5), αc ∈ [2.5, 3), pe ∈ [2, 2.5), po ∈ [0, 1], βc ∈ [3.5, 4), w = po + pe , αq = αc − po , βq = βc − w. The market demand follows a truncated normal distribution in the support of (0, 10) with the mean μ D = 5 and the standard deviation σ D = 1. To investigate the effect of po on the profit, we fix other parameters constant and adjust po . In the following experiments, we choose c = 1.5, μ p = 4.25, αc = 2.75, pe = 2.25, βc = 3.75 and po ∈ {0.6, 0.9}. The results are shown in Figs. 6.3 and 6.4. We repeat the numerical experiments for other parameters in the aforementioned regions and similar results occur, which indicate the robustness of the numerical experiments. If po is small, the results are shown in Fig. 6.3, indicating that when d is smaller than the threshold, the retailer gains more profit in capacity reservation contract because reservation cost saving outweighs the negligible flexibility. When d is larger than the threshold, the profit of the quantity flexibility contract is larger because the noticeable flexibility outweighs reservation cost saving. The preference of the manufacturer is the opposite. That is, when the retailer gains more in a certain contract, the manufacturer gains less. Similarly, if po is large, the results are shown in Fig. 6.4, indicating that as po becomes large, the reservation fee becomes expensive, which makes the capacity reservation contract less attractive to the retailer. As a result, the retailer always prefers the quantity flexibility contract.

190

6 Supply Chain Coordination Through … Quantity flexibility contract Capacity reservation contract

5.9 5.8 5.7 5.6 5.5

Quantity flexibility contract Capacity reservation contract

6.7

profit of the retailer

profit of the manufacturer

6

6.6 6.5 6.4 6.3 6.2

5.4 0

0.02 0.04 0.06 0.08 0.1 0.12 0.14

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14

d

d

7.35

Quantity flexibility contract Capacity reservation contract

7.3 7.25 7.2 7.15 7.1 7.05

Quantity flexibility contract Capacity reservation contract

5.15

profit of the retailer

profit of the manufacturer

Fig. 6.3 Manufacturer and retailer profit when po = 0.6

5.1 5.05 5 4.95 4.9 4.85

7

4.8 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

d

d

Fig. 6.4 Manufacturer and retailer profit when po = 0.9

6.5.2 Profit Distribution Under Coordination We compare the profit split between two firms in two contracts. We focus on one special coordinated case where the manufacturer builds capacity exactly as the retailer reserves (i.e. Q ∗ = K ∗ = K C∗ S ). In such a case, by maintaining the same decision variables and total profit between the two contracts, we can isolate other factors and focus on profit split in coordinated supply chains. One prime question is whether two contracts can both achieve coordination with Q ∗ = K ∗ with the following links or not: (1) w = po + pe , (2) w + βq = βc and (3) αc − po = αq . The answer is yes. Let ( po∗ , pe∗ , αc∗ , βc∗ ) represent coordinated parameters in the capacity reservation contract, and we denote by λc the profit split of the manufacturer. Analogously, let (w∗ , αq∗ , βq∗ , d ∗ ) represent coordinated parameters in the quantity flexibility contract, and we denote by λq the profit split

6.5 Comparison Between Quantity Flexibility Contact and Capacity Reservation Contract

191

of the manufacturer. By Theorem 6.5, we have po∗ = c(1 − λc ), pe∗ = μ p λc , αc∗ = μ p (1 − λc ), βc∗ = μ p . In addition, w = po∗ + pe∗ = μ p λc + c(1 − λc ), βq = βc∗ − w = (μ p − c)(1 − λc ) and αq = αc∗ − po∗ = (1 − λc )(μ p − c). By choosing a delicate d ∗ that satisfies the conditions in Theorem 6.8, Q¯ ∗ = K¯ ∗ = K C∗ S can be achieved for the quantity flexibility contract. Another purpose is to compare λc and λq . Due to the complexity to obtain d ∗ , we resort to numerical experiments. Note that in the capacity reservation contract, the profit split is determined by po (or pe ). Therefore, without loss of generality, po is adjusted and then we elaborately select other parameters to achieve channel coordination. As a representative example, the parameters values are as follows. μ D = 5, σ D = 1, c = 1.5, μ p ∈ {2.5, 3.0, 4.25} to represent low, medium and high expected price, respectively. Other parameters are elaborately set to guarantee channel coordination. As shown in Fig. 6.5, λ = λc − λq > 0 always holds, which means that the profit split of the manufacturer in the capacity reservation contract is higher, whereas the profit split of the retailer is lower. The reason is as follows. Recall the results in Theorem 6.10 and note that the condition po ≥ βc −2 pe is equivalent to μc ≥ 21 for p coordinated parameters. Therefore, when μ p = 2.5 or 3.0, the retailer always prefers the quantity flexibility contract, whereas the manufacturer prefers the other, as shown in Theorem 6.10. When μ p = 4.25, then we obtain d ∗ = 0.67, which is high enough and brings much flexibility and therefore more profit to the retailer.

Fig. 6.5 Profit ratio gaps under supply chain coordination

=2.5

0.22

=3.0

0.2

=4.25

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.2

0.4

0.6

0.8

po

1

1.2

1.4

192

6 Supply Chain Coordination Through …

6.6 Extension 6.6.1 Negotiation for Setting Contract Parameters In the basic model, we study the case in which the contract parameters are exogenous/given. In this section, we make an extension and answer the research question of how to determine these parameters? In practice, the common phenomenon is that the manufacturer and the retailer negotiate to determine corresponding parameters in the first stage before any quantity decision is made. The relative power of two parties significantly affects the negotiation outcome and we utilise generalised Nash bargaining (GNB) scheme to model the negotiation process. Initiated by Nash in 1950, the GNB scheme is now popular in pricing negotiation. For example, Nagarajan and Bassok (2008), Nagarajan and Sosic (2008), Wang et al. (2013), Nagurney and Shukla (2017) and Qing et al. (2017) study endogenous pricing issues with the GNB scheme. Note that, the Stackelberg game in which one acts the leader and the other acts as the follower is a special case of GNB model. The objective of the GNB scheme is to maximise the following function:  M )x ( R −   R )1−x . max  = ( M − 

(6.26)

Here,  is Nash product. x and 1 − x represent the bargaining power of the man M is the reserved profit of the manufacufacturer and the retailer, respectively.  turer, which is also called disagreement payoff. As the name suggests, this payoff represents the amount that the manufacturer can obtain if the negotiation with the  R is the reserved profit of the retailer that represents retailer fails. Analogously,  the payoff that he can obtain if the negotiation fails. In our model, without loss of  R = 0 are assumed. The objective function then decreases  M = 0 and  generality,  to  = ( M )x ( R )1−x . For the quantity flexibility contract, notations are highly ¯ R )1−x . ¯ M ) x ( analogous, and the objective function is  = ( In  M and  R , we incorporate contract parameters into the equilibrium decisions to highlight the dependence of the equilibrium results on them. Subscripts (c and q) are also used in  to differentiate two contracts. Therefore, the negotiation process aims to solve the following problem for the capacity reservation contract. max

c { po , pe ,αc ,βc }

 x  1−x =  M (Q ∗ ( po , pe , αc , βc ), K ∗ ( po , pe , αc , βc ))  R (Q ∗ ( po , pe , αc , βc ), K ∗ ( po , pe , αc , βc )) .

(6.27) Analogously, the negotiation process aims to solve the following problem for the quantity flexibility contract. max

q {w,αq ,βq ,d}

    ¯ R ( Q¯ ∗ (w, αq , βq , d), K¯ ∗ (w, αq , βq , d) 1−x . ¯ M ( Q¯ ∗ (w, αq , βq , d), K¯ ∗ (w, αq , βq , d)) x  = 

(6.28)

6.6 Extension

193

Q ∗ , K ∗ , Q¯ ∗ and K¯ ∗ functions were analysed in the preceding sections (see Theorem 6.3 and Tables 6.2 and 6.3). Since equilibrium decisions are in various forms with multiple parameters, obtaining an optimal solution to Eqs. (6.27) and (6.28) in closed form is intractable. Alternatively, through sufficient numerical experiments, we can find the parameters that maximise c when the bargaining powers vary. In detail, po increases with a step of 0.1 from 0.1 to 1.0, pe increases with a step of 0.05 from 2.0 to 2.5, αc increases with a step of 0.05 from 2.5 to 3.0 and βc increases with a step of 0.05 from 3.5 to 4.0. Among these 13,310 combinations, we select the optimal combination that emaximises c . In Table 6.4, we find that as the manufacturer becomes more powerful (and the retailer becomes less powerful), they negotiate and set a higher reservation price ( po ), exercise price ( pe ) and purchasing price for the retailer (βc ), whereas penalty is lower to the manufacturer(αc ). As a result, the buyer (the retailer) has a weak motivation to reserve and the constructed capacity level also decreases. In addition, the profit split of the manufacturer increases in her bargaining power, which is consistent with the intuition. We also discuss how to determine these parameters in quantity flexibility contract under generalised Nash bargaining framework. Thus, we implement comprehensive numerical experiments by varying parameters in certain ranges. In detail, w increases with a step of 0.05 from 2.5 to 3. αq increases with a step of 0.05 from 2.0 to 2.5. βq increases with a step of 0.1 from 0.5 to 1.5. d increases with a step of 0.02 from 0.02 to 0.20. Among these 13,310 combinations, the specific one that maximises q with different x is selected. The results are shown in Table 6.5. In general, as the manufacturer becomes more powerful (and at the same time, the retailer becomes less powerful), they negotiate and set a higher wholesale price (w) and penalty to the retailer (βq ), whereas penalty to the manufacturer (αq ) and flexibility (d) are lower. Again, as x increases, the profit of the manufacturer increases, whereas the profit of the retailer decreases. Further, we compare the performance of two contracts based on Tables 6.4 and 6.5. One finding is that the profits are more stable in quantity flexibility contract. That is,

Table 6.4 Negotiated parameters in capacity reservation contract x ( po∗ , pe∗ , αc∗ , βc∗ ) Q∗ K∗ 0.1 0.3 0.5 0.7 0.9

(0.10, 2.00, 3.00, 3.50) (0.10, 2.00, 2.80, 3.50) (0.50, 2.00, 2.50, 4.00) (0.80, 2.35, 2.50, 4.00) (1.00, 2.50, 2.50, 4.00)

∗M

∗R

6.83

5.99

5.51

10.59

6.80

5.97

5.52

10.58

5.67

5.84

8.02

8.09

5.04

5.84

11.29

4.81

4.57

5.84

12.93

3.18

194

6 Supply Chain Coordination Through …

Table 6.5 Negotiated parameters in quantity flexibility contract Q¯ ∗ K¯ ∗ x (w ∗ , αq∗ , βq∗ , d ∗ ) 0.1 0.3 0.5 0.7 0.9

(2.50, 2.50, 0.50, 0.20) (2.50, 2.50, 0.50, 0.20) (2.55, 2.45, 1.50, 0.14) (3.00, 2.00, 1.50, 0.02) (3.00, 2.00, 1.50, 0.02)

¯∗  M

¯∗  R

5.62

5.99

7.42

8.67

5.62

5.99

7.42

8.67

5.16

5.89

8.05

8.05

5.02

5.92

10.88

5.22

5.02

5.92

10.88

5.22

Table 6.6 The contract preference of the manufacturer and retailer The bargaining power of the The manufacturer’s preference The retailer’s preference manufacturer High Low

Capacity reservation contract Quantity flexibility contract

Quantity flexibility contract Capacity reservation contract

the profits of both the manufacturer and retailer change less in the quantity flexibility contract as the relative bargaining power varies. Specifically, the numbers show that as the manufacturer’s power increases from 0.1 to 0.9, his profit under capacity reservation contract increases about 135% (see Table 6.4), whereas his profit under quantity flexibility contract increases about 47% (see Table 6.5). From Tables 6.4 and 6.5, we also find that when x is relatively smaller than a threshold (x ≤ 0.5), the capacity reservation contract results in less profit to the manufacturer and larger profit to the retailer than quantity flexibility contract. This result implies that the manufacturer prefers the capacity reservation contract when he has stronger power when bargaining, whereas the retailer prefers the quantity flexibility contract when she has stronger power. We summarize the preference of the manufacture and retailer to two contracts in Table 6.6.

6.6.2 Correlation Between Demand and Retail Price In this subsection, we extend the basic model and study the effect of correlation between the demand and retail price. For correlated random variables, let g( p, D) denote the joint probability density function of the price and demand. In the capacity reservation contract, profit functions of the retailer and the manufacturer are as follows:

6.6 Extension

195

 R (Q, K ) = E{ p,D} p [D ∧ K ] − T (Q, K ),

(6.29)

 M (Q, K ) = T (Q, K ) − cK ,

(6.30)

∞∞ where E{ p,D} p [D ∧ K ] = 0 0 p(D ∧ K )g( p, D)dDd p. Note that the transfer payment, T (Q, K ) is independent of p. Therefore, the transfer payment function in Eq. (6.5) continues to work in this case. For the manufacturer, her best-response function remains. In Eqs. (6.5) and (6.30), we can see that the profit function of the manufacturer is independent of p and the correlation. Therefore, the best-response function in Theorem 6.2 continues to work in the correlated case. Based on the best-response function of the manufacturer, the profit function of the retailer can be expressed as follows:   R = max  R (Q) = max Q

 max  R (Q, K 1 ), max  R (Q, K 2 ),

Q≥K 1

Q≤K 2

max  R (Q, Q) . Q∈[ K 2 ,K 1 ]

(6.31) In detail,  R (Q, K 1 ),  R (Q, K 2 ) and  R (Q, Q) functions are written as follows:  R (Q, K 1 ) = E{ p,D} p(D ∧ K 1 ) − po Q − pe E D (K 1 ∧ D) + αc E D [(D ∧ Q) − K 1 ]+ ;  R (Q, K 2 ) = E{ p,D} p(D ∧ K 2 ) − po Q − pe E D (Q ∧ D) − βc E D [(D ∧ K 2 ) − Q]+ ;  R (Q, Q) = E{ p,D} p(D ∧ Q) − po Q − pe E D (Q ∧ D).

(6.32) (6.33) (6.34)

Equations (6.32), (6.33) and (6.34) are highly similar to their counterparts in the uncorrelated case, as shown in Eqs. (6.7), (6.8) and (6.9), except for the first term. Interestingly, in Eqs. (6.32) and (6.33), we can conclude that the optimal decision does not change because the decision variable Q of the retailer is not incorporated in the first term. In detail, the optimal solution to Eqs. (6.32) and (6.33) remain as Q 1 = F −1 ( αcα−c po ) and Q 2 = F −1 ( βcβ−c p−o p−e pe ), respectively, if the Q constraint is ignored. However, the optimal solution to  R (Q, Q) in Eq. (6.34) may deviate from that in Eq. (6.9) because the correlation in the first term influences the optimal decision, denoted by Q 3 . To study the effect of the correlation on Q ∗ , we implement numerical experiments for the case in which the demand and retail price follow bivariate normal distribution (i.e. ( p, D) ∼ N(μ p , σ p , μ D , σ D , ρ)). μ p and σ p are the mean and standard deviation of the retail price, respectively. μ D and σ D are the mean and standard deviation of market demand, respectively. ρ represents the correlation between these two variables. Other parameters are set as follows: μ p = 4.25, σ p = 1, μ D = 5, σ D = 1, c = 1.5, po ∈ {0.5, 1.2}, pe = 2.25, αc = 2.75, βc = 3.75. ρ increases with a step of 0.2 from −0.8 to 0.8. The results are reported in Table 6.7. When po is small ( po = 0.5), the retailer tends to reserve a large quantity. As a result, the global optimum, Q ∗ = Q 1 = F −1 ( αcα−c po ), which is independent of ρ. As po increases ( po = 1.2), the expensive

196

6 Supply Chain Coordination Through …

Table 6.7 Effect of correlation in capacity reservation contract ρ po = 0.5 po = 1.2 Q3 Q∗ K∗ ∗M ∗R Q3 Q∗ −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

5.27 5.38 5.48 5.52 5.52 5.52 5.52 5.52 5.52

5.89 5.89 5.89 5.89 5.89 5.89 5.89 5.89 5.89

5.52 5.52 5.52 5.52 5.52 5.52 5.52 5.52 5.52

5.24 5.24 5.24 5.24 5.24 5.24 5.24 5.24 5.24

6.49 6.59 6.69 6.80 6.91 7.04 7.17 7.32 7.47

5.25 5.25 5.25 5.25 5.25 5.25 5.25 5.25 5.25

4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18 4.18

K∗

∗M

∗R

5.25 5.25 5.25 5.25 5.25 5.25 5.25 5.25 5.25

8.72 8.72 8.72 8.72 8.72 8.72 8.72 8.72 8.72

3.09 3.17 3.25 3.34 3.44 3.54 3.66 3.79 3.92

reservation fee induces the retailer to reserve a small quantity. Therefore, the optimal solution to Eq. (6.34) is Q 3 = K 2 when considering the constraint that Q ∈ [K 2 , K 1 ]. In this case, Q 1 , Q 2 and Q 3 are all independent of ρ. This result shows that quantity decisions are robust when correlation is considered. In the quantity flexibility contract, the results are analogous. On the one hand, for the manufacturer, the transfer payment and her profit function are independent of ρ. As a result, her best-response function, which is shown in Theorem 6.6, continues to work. On the other hand, for the retailer, his profit function is a piecewise function of ¯ 2R (Q, (1 + d)Q),  ¯ 3R (Q, K I I ) . Similar to the ¯ ∗R = max  ¯ 1R (Q, K I I I ),  Q and  capacity reservation contract, the correlation ρ does not influence the optimal solution ¯ 3R (Q, K I I ). Therefore, Q¯ 1 and Q¯ 3 continue to satisfy Eqs. ¯ 1R (Q, K I I I ) and  to  (6.20) and (6.22), respectively. We investigate the effect of ρ on Q¯ 2 and equilibrium decisions through numerical experiments. Related parameters are set as follow: w ∈ {2.75, 3.45}, αq = 2.25, βq = 1.0, d = 0.1. Other parameters remain the same as those in the capacity reservation contract. Results are shown in Table 6.8. Although the local optimum, Q¯ 2 increases with ρ, the equilibrium decisions, Q¯ ∗ and K¯ ∗ are rather robust. More importantly, the difference of Q¯ 2 , Q¯ ∗ , K¯ ∗ between ρ = 0 and ρ = 0.8 is rather small. This condition implies that when the retail price and the demand are positively correlated, which is often the case in practice, the effect of correlation is negligible. In summary, the profit function and the best-response function of the manufacturer are independent of the correlation. For the retailer, the correlation takes effect under restrictive conditions. As a result, the equilibrium decisions are robust, especially when the retail price and demand are positively correlated. Therefore, we can conclude that in the two contracts, the correlation between the demand and retail price shows a negligible effect.

6.7 Concluding Remarks

197

Table 6.8 Effect of correlation in quantity flexibility contract ρ w = 2.75 w = 3.45 ¯∗ ¯∗ Q¯ 2 Q¯ ∗ K¯ ∗ Q¯ 2 Q¯ ∗   M

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

4.91 5.00 5.02 5.02 5.02 5.02 5.02 5.02 5.02

5.43 5.43 5.43 5.43 5.43 5.43 5.43 5.43 5.43

5.52 5.52 5.52 5.52 5.52 5.52 5.52 5.52 5.52

5.05 5.05 5.05 5.05 5.05 5.05 5.05 5.05 5.05

R

6.67 6.77 6.88 6.99 7.10 7.22 7.36 7.50 7.65

4.93 4.93 4.93 4.93 5.00 5.07 5.10 5.12 5.12

4.93 5.43 5.43 5.43 5.43 5.43 5.43 5.43 5.43

K¯ ∗

¯∗  M

¯∗  R

5.42 5.63 5.63 5.63 5.63 5.63 5.63 5.63 5.63

8.53 8.43 8.43 8.43 8.43 8.43 8.43 8.43 8.43

3.25 3.34 3.45 3.57 3.69 3.82 3.96 4.11 4.26

6.7 Concluding Remarks We study the problem in which one manufacturer and one retailer develop innovative products. The manufacturer does not have much incentive to build capacity if the retailer does not share the risk from stochastic demand and retail price, which induces a low performance of the supply chain. This hinders the supply chain from satisfying the optimal amount of demand from the perspective of a channel. We describe and analyse capacity reservation and quantity flexibility contract which are both piecewise linear as mechanisms to improve the overall performance of the supply chain. Both contracts induce a dynamic game of perfect information, and we characterise the subgame perfect Nash equilibria. We find that flexibility in our contracts can improve the supply chain profit when channel coordination is not achieved. For both contracts, we establish ways to set contract parameters to coordinate the supply chain with arbitrary profit split between the retailer and manufacturer. We also compare the two contracts when coordination is not achieved. One finding is that for given reservation quantity, the manufacturer constructs greater capacity in the quantity flexibility contract than in the capacity reservation contract. Another finding is about the conditions under which two contracts are identical or one dominates the other from the perspective of the retailer or manufacturer. In addition, we find that when coordination is achieved, the retailer prefers the quantity flexibility contract because his profit split is higher. We also extend the basic models to study the cases of endogenous parameters and the correlation between the price and demand. For the endogenous parameters, the manufacturer prefers the capacity reservation contract rather than the quantity flexibility contract when his bargaining power is high. The preference of retailer is the opposite. Numerical experiments show that the analytic results in equilibrium are robust and the correlation between the price and demand shows a negligible effect. In summary,

198

6 Supply Chain Coordination Through …

our work investigate the dynamic of game between the retailer and manufacturer under two novel contracts and the findings provide meaningful guidance to firms that make selection between different contracts. Our model framework can be extended in several ways. The first direction includes relaxing certain assumptions made in these models. One can also extend the analysis by including nonlinear cost (reflecting production economies of scale) or other pricing policies, such as quantity discount. As the study examines the scenario of one manufacturer/supplier and one retailer, an evident extension is to study the scenario with multiple suppliers and retailers where contracts are used to reduce the disruption risk from supply. Another natural extension is to study the case of one manufacturer and n retailers where all retailers compete for the capacity. In addition, future work incorporating a dynamic decision-making environment is useful, especially in multi-echelon supply chains.

Appendix Proof of Theorem 6.1. Define  1M (Q, K ) = po Q + E D pe (K ∧ D) − αc ((D ∧ Q) − K )+ − cK ;  2M (Q, K ) = po Q + E D pe (Q ∧ D) + βc ((D ∧ K ) − Q)+ − cK . Here, 1M (Q, K ) is the expected profit of the manufacturer when she plans to construct capacity less than the quantity reserved by the retailer. 2M (Q, K ) is the expected profit of the manufacturer when she plans to construct capacity no less than the quantity reserved by the retailer. Thus, we have:

 M (Q) = max  M (Q, K ) = max max 1M (Q, K ), max 2M (Q, K ) . K ≤Q

K

K ≥Q

Taking the first derivative of 1M (Q, K ) with respect to K , we can ∂1M (Q,K ) = ( pe + αc )[1 − F(K )] − c. ∂K Let K 1 be the optimal solution to 1M (Q, K ), which is concave in K . We K 1 = F −1



 αc + pe − c . αc + pe

Hence, Kˆ = arg max 1M (Q, K ) = K 1 ∧ Q. K Q

Case 1. Q ≤ K 2 . In this case, Kˆ = arg max 1M (Q, K ) = K 1 ∧ Q = Q and K  R (K 2 , K 2 ). Therefore, by Eq. (6.6),

the equilibrium decision is determined depending on the order of  R (Q 3 , Q 3 ) and  R (Q 1 , K 1 ). That is, (Q ∗ , K ∗ ) ∈ {(Q 3 , Q 3 ), (Q 1 , K 1 )}. (4) Q 2 ∈ (K 2 , K 1 ), Q 3 ∈ (K 2 , K 1 ) and Q 1 ∈ (K 2 , K 1 ). We can prove that this case does not exist. By Q 2 > K 2 , we obtain pco < βcβ−c pe . By Q 1 < K 1 , we obtain po > αcα+c pe . However, these two inequalities are in contradiction to each other with c Assumption 6.1 in hand. Therefore, this impossible case is ignored. (5) Q 2 ≤ K 2 , Q 3 ≥ K 1 and Q 1 ≥ K 1 . By the concavity of profit functions, we have max  R (Q, K 1 ) =  R (Q 1 , K 1 ) >  R (K 1 , K 1 ), max  R (Q, K 2 ) = Q≥K 1

 R (Q 2 , K 2 ) and

Q≤K 2

max

Q∈[K 2 ,K 1 ]

 R (Q, Q) =  R (K 1 , K 1 ). Therefore, by Eq. (6.6), the

equilibrium decision is determined depending on the order of  R (Q 2 , K 2 ) and  R (Q 1 , K 1 ). That is, (Q ∗ , K ∗ ) ∈ {(Q 2 , K 2 ), (Q 1 , K 1 )}. (6) Q 2 ≤ K 2 , Q 3 ∈ (K 2 , K 1 ) and Q 1 ≥ K 1 . By the concavity of profit functions, we have max  R (Q, K 1 ) =  R (Q 1 , K 1 ), max  R (Q, K 2 ) =  R (Q 2 , K 2 ) Q≥K 1

and

max

Q∈[K 2 ,K 1 ]

Q≤K 2

 R (Q, Q) =  R (Q 3 , Q 3 ). Therefore, by Eq. (6.6), the equilibrium

decision is determined depending on the order of  R (Q 2 , K 2 ),  R (Q 1 , K 1 ) and  R (Q 3 , Q 3 ). That is, (Q ∗ , K ∗ ) ∈ {(Q 1 , K 1 ), (Q 2 , K 2 ), (Q 3 , Q 3 )}. (7) Q 2 ≤ K 2 , Q 3 ≤ K 2 and Q 1 ≥ K 1 . By the concavity of profit functions, we have max  R (Q, K 1 ) =  R (Q 1 , K 1 ), max  R (Q, K 2 ) =  R (Q 2 , K 2 ) > Q≥K 1

 R (K 2 , K 2 ) and

Q≤K 2

max

Q∈[K 2 ,K 1 ]

 R (Q, Q) =  R (K 2 , K 2 ). Therefore, by Eq. (6.6), the

equilibrium decision is is determined depending on the order of  R (Q 2 , K 2 ) and  R (Q 1 , K 1 ). That is, (Q ∗ , K ∗ ) ∈ {(Q 1 , K 1 ), (Q 2 , K 2 )}. (8) Q 2 ≤ K 2 , Q 3 ∈ (K 2 , K 1 ) and Q 1 < K 1 . By the concavity of profit functions, we have max  R (Q, K 1 ) =  R (K 1 , K 1 ), max  R (Q, K 2 ) =  R (Q 2 , K 2 ) Q≥K 1

and

max

Q∈[K 2 ,K 1 ]

Q≤K 2

 R (Q, Q) =  R (Q 3 , Q 3 ) >  R (K 1 , K 1 ). Therefore, by Eq. (6.6),

the equilibrium decision is determined depending on the order of  R (Q 2 , K 2 ) and  R (Q 3 , Q 3 ). That is, (Q ∗ , K ∗ ) ∈ {(Q 2 , K 2 ), (Q 3 , Q 3 )}. (9) Q 2 ≤ K 2 , Q 3 ≤ K 2 and Q 1 < K 1 . By the concavity of profit functions, we have max  R (Q, K 1 ) =  R (K 1 , K 1 ), max  R (Q, K 2 ) =  R (Q 2 , K 2 ) > Q≥K 1

 R (K 2 , K 2 ) and

Q≤K 2

max

Q∈[K 2 ,K 1 ]

 R (Q, Q) =  R (K 2 , K 2 ) >  R (K 1 , K 1 ). Therefore, by

Eq. (6.6), we conclude Q ∗ = Q 2 and K ∗ = K 2 . Omitting the fourth impossible case, we obtain the results of Theorem 6.3.



Appendix

201

Proof of Corollary 6.1. We prove Corollary 6.1 in comparison with the Case 3 of Proposition 6.1 and Case 1 of Theorem 6.3. po c ≤ , βc − pe βc po c Q2 ≥ K1 ⇔ ≤ . βc − pe αc + pe

Q2 ≥ K2 ⇔

(A1.1) (A1.2)

po By Assumption 6.1, we have βc − ≤ αc +c pe , which is equivalent to αc ≤ (βc −pope )c − pe pe . We further compare the capacity. In Case 3 of Proposition 6.1, K 0∗ = > Q 3 = F −1 (1 − μ pp−o pe ). In Case 1 of Theorem 6.3, K ∗ = K 1 = F −1 ααc +c +pep−c e

 μ −c p F −1 μ p = K C∗ S , where the last inequality is derived from Assumption 6.1. By

αc
K ∗ . αc + pe  (μ − p )c (βc − pe )c − pe , p po e − pe = (βc −pope )c po

<





− pe ,

the equilibrium capacity decisions are K 0∗ = F −1 1 − μ pp−e pe and K ∗ =

 . In addition, K 0∗ > K ∗ > K C∗S . By the concavity of the system profit F −1 ααc +c +pep−c e

function, it is easy to prove C∗ S > ∗ > 0∗ .



Proof of Theorem 6.4. We check the cases in Theorem 6.3 one by one to find the equilibrium that can lead to supply chain coordination. (1) In this case, (Q 1 , K 1 ) is the equilibrium and the corresponding condition pe . To achieve coordination, K 1 = K C∗S is Q 2 ≥ K 1 , which is equivalent to pco ≤ αβcc − + pe yields αc + pe = μ p . Therefore, the following conditions leads to supply chain coordination: αc + pe = μ p and pco ≤ βcμ− pe . p (2) In this case, (Q 1 , K 1 ) is the equilibrium. The conditions to achieve this equilibrium are listed as follows: po c po Q2 < K1 ⇔ c po Q3 ≥ K1 ⇔ c po Q1 ≥ K1 ⇔ c

Q2 > K2 ⇔

βc − pe , βc βc − pe > , αc + pe μ p − pe ≤ , αc + pe αc ≤ . αc + pe
K2 ⇔ Q2 < K1 ⇔ Q3 > K2 ⇔ Q3 < K1 ⇔ Q1 ≥ K1 ⇔

po c po c po c po c po c

< > < > ≤

βc − pe , βc βc − pe , αc + pe μ p − pe , βc μ p − pe , αc + pe αc . αc + pe

(A3.1) (A3.2) (A3.3) (A3.4) (A3.5) μ −p

If (Q 3 , Q 3 ) achieves coordination, then Q 3 = K C∗S yields pco = pμ e . However, it p is in contradiction to (A3.1) because μ p ≥ βc . If (Q 1 , K 1 ) achieves coordination, then K 1 = K C∗S yields αc + pe = μ p . However, (A3.4) and (A3.5) are in contradiction to each other. Therefore, this case can not lead to supply chain coordination. (4) In this case, two potential equilibria exist, namely, (Q 2 , K 2 ) and (Q 1 , K 1 ), depending on which one brings the retailer more profit. The conditions to achieve these equilibria are listed as follows: βc − pe po ≥ , c βc μ p − pe po ≤ Q3 ≥ K1 ⇔ , c αc + pe αc po ≤ Q1 ≥ K1 ⇔ . c αc + pe

Q2 ≤ K2 ⇔

(A4.1) (A4.2) (A4.3)

If (Q 2 , K 2 ) achieves channel coordination, K 2 = K C∗S yields βc = μ p . Therefore, μ −p (A4.1) becomes pco ≥ pμ e . Since αc + pe ≥ μ p , (A4.2) holds only if αc + pe = p

μ −p

μ −p

μ p . In addition, (A4.3) becomes pco ≤ pμ e . Consequently, pco = pμ e . Note that p p by these equations, we can obtain  R (Q 2 , K 2 ) =  R (Q 1 , K 1 ). To summarize, the following conditions lead to supply chain coordination: βc = μ p , αc + pe = μ p and μ −p po = pμ e . c p

Appendix

203

If (Q 1 , K 1 ) achieves condition, K 1 = K C∗S yields αc + pe = μ p . Combining μ −p (A4.1), (A4.2) and (A4.3), we obtain pco ∈ [ βcβ−c pe , pμ e ]. In addition, the retailer’s p equilibrium decision is Q 1 instead of Q 2 only if  R (Q 1 , K 1 ) ≥  R (Q 2 , K 2 ). To summarize, the following conditions lead to supply chain coordination: μ −p  R (Q 1 , K 1 ) ≥  R (Q 2 , K 2 ), αc + pe = μ p and pco ∈ [ βcβ−c pe , pμ e ]. p (5) In this case, three potential equilibria exist, namely, (Q 2 , K 2 ), (Q 3 , Q 3 ) and (Q 1 , K 1 ), depending on which one brings the retailer the largest profit. The conditions to achieve these equilibria are listed as follows: po c po Q3 > K2 ⇔ c po Q3 < K1 ⇔ c po Q1 ≥ K1 ⇔ c Q2 ≤ K2 ⇔

βc − pe , βc μ p − pe < , βc μ p − pe > , αc + pe αc . ≤ αc + pe ≥

(A5.1) (A5.2) (A5.3) (A5.4)

If (Q 2 , K 2 ) achieves supply chain coordination, K 2 = K C∗S yields βc = μ p , which makes (A5.1) and (A5.2) are in contradiction to each other. Therefore, this case can not lead to supply chain coordination. μ −p If (Q 3 , Q 3 ) achieves supply chain coordination, Q 3 = K C∗S yields pco = pμ e . All p the inequalities in (A5.1), (A5.2), (A5.3) and (A5.4) automatically hold. In addition, the retailer’s equilibrium decision is Q 3 only if  R (Q 3 , Q 3 ) ≥  R (Q 1 , K 1 ) and  R (Q 3 , Q 3 ) ≥  R (Q 2 , K 2 ). To summarize, the following conditions lead to supply chain coordination:  R (Q 3 , Q 3 ) ≥  R (Q 1 , K 1 ),  R (Q 3 , Q 3 ) ≥  R (Q 2 , K 2 ) and μ −p po = pμ e . c p If (Q 1 , K 1 ) achieves supply chain coordination, K 1 = K C∗S yields αc + pe = μ p . μ −p μ −p Therefore, (A5.3) becomes pco > pμ e and (A5.4) becomes pco ≤ pμ e . However, p p these two inequalities are in contradiction to each other. Therefore, this case can not lead to supply chain coordination. (6) In this case, two potential equilibria exist, namely, (Q 2 , K 2 ) and (Q 1 , K 1 ), depending on which one brings the retailer more profit. The conditions to achieve these two equilibria are listed as follows: βc − pe po ≥ , c βc μ p − pe po ≥ Q3 ≤ K2 ⇔ , c βc αc po ≤ Q1 ≥ K1 ⇔ . c αc + pe

Q2 ≤ K2 ⇔

(A6.1) (A6.2) (A6.3)

204

6 Supply Chain Coordination Through …

If (Q 2 , K 2 ) achieves supply chain coordination, then K 2 = K C∗S yields βc = μ p . μ −p Therefore, by (A6.1),(A6.2) and (A6.3), we obtain pco ∈ [ pμ e , αcα+c pe ]. Furthermore, p the retailer’s equilibrium decision is Q 2 only if  R (Q 2 , K 2 ) ≥  R (Q 1 , K 1 ). To summarize, the following conditions guarantee supply chain coordination: μ −p  R (Q 2 , K 2 ) ≥  R (Q 1 , K 1 ), βc = μ p and pco ∈ [ pμ e , αcα+c pe ]. p If (Q 1 , K 1 ) achieves supply chain coordination, then K 1 = K C∗S yields αc + μ −p pe = μ p . Therefore, (A6.3) becomes pco ≤ pμ e . Combining with (A6.2), the only p

μ −p

feasible case is βc = μ p and pco = pμ e . Note that under these two equalities, p  R (Q 1 , K 1 ) =  R (Q 2 , K 2 ). To summarize, the following conditions guarantee supμ −p ply chain coordination: αc + pe = μ p , βc = μ p and pco = pμ e . p (7) In this case, two potential equilibria exist, namely, (Q 2 , K 2 ) and (Q 3 , Q 3 ), depending on which one brings the retailer more profit. The conditions to achieve these two equilibria are listed as follows: po c po Q3 > K2 ⇔ c po Q3 < K1 ⇔ c po Q1 < K1 ⇔ c Q2 ≤ K2 ⇔

βc − pe , βc μ p − pe < , βc μ p − pe > , αc + pe αc > . αc + pe ≥

(A7.1) (A7.2) (A7.3) (A7.4)

If (Q 2 , K 2 ) achieves supply chain coordination, then K 2 = K C∗S yields βc = μ p , which makes (A7.1) and (A7.2) in contradiction to each other. Therefore, (Q 2 , K 2 ) does not lead to supply chain coordination. If (Q 3 , Q 3 ) achieves supply chain coordination, then Q 3 = K C∗S yields pco = μ p − pe , which is in contradiction to (A7.4) because αc + pe ≥ μ p . Therefore, μp (Q 3 , Q 3 ) does not lead to supply chain coordination. To summarize, this case does not lead to supply chain coordination. (8) In this case, the equilibrium decision is (Q 2 , K 2 ). The conditions to achieve this equilibrium are listed as follows: βc − pe po ≥ , c βc μ p − pe po ≥ Q3 ≤ K2 ⇔ , c βc αc po > . Q1 < K1 ⇔ c αc + pe Q2 ≤ K2 ⇔

(A8.1) (A8.2) (A8.3)

Appendix

205

To achieve supply chain coordination, K 2 = K C∗S yields βc = μ p . Therefore, (A8.1), (A8.2) and (A8.3) reduce to pco > αcα+c pe . To summarize, the following conditions guarantee supply chain coordination: βc = μ p and pco > αcα+c pe . We have checked all equilibrium decisions that potentially lead to supply chain coordination and their corresponding conditions. The results are shown in Theoμ −p rem 6.4. Note that, if βc = μ p , αc + pe = μ p and pco = pμ e , then  R (Q 1 , K 1 ) = p  R (Q 2 , K 2 ) =  R (Q 3 , Q 3 ). Therefore, the case that βc = μ p , αc + pe = μ p and μ −p po = pμ e can be incorporated into the fifth condition of Theorem 6.4. To avoid c p overlapping, this special case is not listed in Theorem 6.4. Proof of Theorem 6.5 It is easy to verify that K 1 = K 2 = K C∗S and Q 1 = Q 2 = Q 3 = K C∗S under the folμ −p lowing conditions: (1)βc = μ p ; (2)αc + pe = μ p ; (3) pco = pμ e . Therefore, in equip librium, Q ∗ = K C∗ S and K ∗ = K C∗ S . In the decentralised system, profit functions of the manufacturer and retailer can be written as follows:      R (Q, K )  Q=K C∗ S ,K =K C∗ S = E D μ p − pe (D ∧ K C∗ S ) − po K C∗ S ,     M (Q, K )  Q=K C∗ S ,K =K C∗ S = E D pe (D ∧ K C∗ S ) − (c − po ) K C∗ S . Note also that in the centralised supply chain, the system profit is as follows:   C∗ S = E D μ p (D ∧ K C∗S ) − cK C∗S . Denote λc = μpe ∈ [0, 1] and then pco = 1 − λc . It is easy to verify that  M = p λc C∗ S and  R = (1 − λc )C∗ S . By adjusting pe , the profit can be arbitrarily allocated between the manufacturer and retailer.  Proof of Theorem 6.6. From the objective function of the manufacturer in Eq. (6.11), we need to analyse the regions of K as follows: Region I

0

Region II

(1 − d)Q

Region III

(1 + d)Q

K

As shown in the above figure, in Region I, K ≤ (1 − d)Q, which is in contradiction to the assumption that the manufacturer never constructs capacity smaller than the lower bound, namely, (1 − d)Q. Hence, the scenario does not exist. In Region II, K ∈ ((1 − d)Q, (1 + d)Q) and the objective function of the manufacturer is written as follows:

206

6 Supply Chain Coordination Through …

¯ M,I I (Q) = w 





K

x f (x)dx +

0

−αq

(1+d)Q

K (1−d)Q

+βq

K f (x)dx

K

 

+∞





(x − K ) f (x)dx +

+∞

(1+d)Q

   (1 + d)Q − K f (x)dx

((1 − d)Q − x) f (x)dx − cK .

0

¯ M,I I (Q) with respect to K , respectively, Taking the first and second derivatives of  we have:  ∞  ∞ ¯ M,I I (Q) ∂ =w f (x)dx + αq f (x)dx − c, ∂K K K ¯ M,I I (Q) ∂2 = −w f (K ) − αq f (K ) < 0. ∂K2 ¯ M,I I (Q) is concave in K and has a unique optimal solution without conThus,  ¯ M,I I (Q). Then, straints. Let K I I satisfy the first order condition of  ¯ M,I I (Q) = F −1 K I I = arg max  K

w + α − c q . w + αq

In Region III, K ≥ (1 + d)Q and we have: ¯ M,I I I (Q) = w 

 K 0

+βq +βq

x f (x)dx +

 K (1+d)Q

 (1−d)Q 0

 ∞

¯ M,I I I (Q) ∂ =w ∂K K



 +∞ K

K f (x)dx



 x − (1 + d)Q f (x)dx +

 +∞  K

  K − (1 + d)Q f (x)dx

((1 − d)Q − x) f (x)dx − cK ,

f (x)dx + βq

 ∞ K

f (x)dx − c.

¯ M,I I I (Q) is the following: Analogously, the optimal solution to  ¯ M,I I I (Q) = F −1 ( K I I I = arg max  K

w + βq − c ). w + βq

Consequently, we use the following figure to characterise the optimal decision of the manufacturer without constraints.

Appendix

207 w+α −c

q KII = F −1 ( w+α q

0

(1 − d)Q

w+β −c

q KIII = F −1 ( w+β q

)

(1 + d)Q

)

K

In order to obtain the optimal solution, we need to compare the objective function values at K I I , K I I I , (1 − d)Q and (1 + d)Q points. The expected profit function of ¯ M,I I I , ¯ M,I I and  the manufacturer is composed of two concave functions, namely,      ¯ M,I I I (1 + d)Q . ¯ M,I I (1 + d)Q =  which are equal at the boundary. That is,  Hence, we investigate the optimal decisions of the manufacturer according to the following six cases: Case 1. If K I I I < K I I < (1 − d)Q < (1 + d)Q, then K¯ (Q) = (1 − d)Q. Case 2. If K I I I < (1 − d)Q < K I I < (1 + d)Q, then K¯ (Q) = K I I . Case 3. If K I I I < (1 − d)Q < (1 + d)Q < K I I , then K¯ (Q) = (1 + d)Q. Case 4. If (1 − d)Q < K I I I < K I I < (1 + d)Q, then K¯ (Q) = K I I . Case 5. If (1 − d)Q < K I I I < (1 + d)Q < K I I , then K¯ (Q) = (1 + d)Q. Case 6. If (1 − d)Q < (1 + d)Q < K I I I < K I I , then K¯ (Q) = K I I I . Combining Cases 1–6, we can gain the best-response function of the manufacturer as follows: ⎧ KI I ⎪ ⎪ (1 − d)Q, if Q > 1−d ; ⎪ ⎪ KI I KI I ⎪ III ⎪ if Q ∈ ( K1−d , 1−d ) ∩ Q > 1+d ; KI I , ⎪ ⎪ ⎪ ⎪ ⎨ (1 + d)Q, if Q ∈ ( K I I I , K I I ); 1−d 1+d K¯ (Q) = KI I KI I I ⎪ , if Q ∈ ( , ); K ⎪ II 1+d 1−d ⎪ ⎪ ⎪ K KI I ⎪ III III ⎪ ∩ Q < 1+d ∩ Q > K1+d ; (1 + d)Q, if Q < 1−d ⎪ ⎪ ⎪ ⎩ KI I I if Q < 1+d . KI I I , KI I III Note that K1−d > 1+d ⇐⇒ d > rewritten as follows.

K¯ (Q) =

If d ≤

K I I −K I I I K I I +K I I I

K I I −K I I I K I I +K I I I

. Thus, if d >

K I I −K I I I K I I +K I I I

, then K¯ (Q) can be

⎧ KI I (1 − d)Q, if Q > 1−d ; ⎪ ⎪ ⎪ ⎪ KI I KI I ⎨ KI I , if Q ∈ ( 1+d , 1−d ]; KI I III ⎪ , 1+d ]; (1 + d)Q, if Q ∈ ( K1+d ⎪ ⎪ ⎪ ⎩ KI I I if Q ≤ 1+d . KI I I ,

, then

K¯ (Q) =

⎧ KI I (1 − d)Q, if Q > 1−d ; ⎪ ⎪ ⎪ ⎪ KI I KI I ⎨ KI I , if Q ∈ ( 1+d , 1−d ]; KI I III ⎪ , 1+d ]; (1 + d)Q, if Q ∈ ( K1+d ⎪ ⎪ ⎪ ⎩ KI I I if Q ≤ 1+d . KI I I ,

Both cases have the same results because the impact of parameter d disappears. Consequently, we have Theorem 6.6. 

208

6 Supply Chain Coordination Through …

Proof  of Lemma  6.1. ¯ R Q, (1 − d)Q can be written as follows     ¯ R Q, (1 − d)Q = (μ p − w)  +αq

 (1−d)Q 0

  (1+d)Q  (1−d)Q

x f (x)dx +

 ∞ (1−d)Q



(1 − d)Q f (x)dx

x − (1 − d)Q f (x)dx +

 ∞

 (1−d)Q   −βq (1 − d)Q − x f (x)dx.

(1+d)Q



2d Q f (x)dx



0

  ¯ R Q, (1 − d)Q with respect to Q, we have Taking the first derivative of     ∞ ¯ R Q, (1 − d)Q d f (x)dx = (μ p − w)(1 − d) dQ (1−d)Q  ∞  (1+d)Q  f (x)dx + 2d +αq − (1 − d) = (μ p − w)(1 − d)  +αq − (1 − d)

(1−d)Q  ∞

(1−d)Q

 ∞

(1−d)Q

(1+d)Q

 (1−d)Q  f (x)dx − βq (1 − d) f (x)dx 0

f (x)dx f (x)dx + (1 + d)

 ∞ (1+d)Q

 (1−d)Q  f (x)dx − βq (1 − d) f (x)dx. 0

  ¯ 4R Q, (1 − d)Q with respect to Q, we Further, taking the second derivative of  have   ¯ R Q, (1 − d)Q d2  dQ 2

  = −(μ p − w − αq )(1 − d)2 f (1 − d)Q     −αq (1 + d)2 f (1 + d)Q − βq (1 − d)2 f (1 − d)Q < 0.

  ¯ 4R Q,(1−d)Q   d 4 ¯  R Q, (1 − d)Q is concave in Q and is decreasing in Q. We just dQ   ¯ 4R Q,(1−d)Q   d ¯ 4R Q, (1 − d)Q is decreasneed to prove | Q= K I I < 0 to prove that  dQ 1−d

 ¯ M,I I (Q) = F −1 w+αq −c and thus, we have ing in Q. Note that K I I = arg max  w+α q K ∞ c f (x)dx = . Based on Assumption 6.3, we obtain the following: KI I w+αq

Appendix

209

  ¯ R Q, (1 − d)Q d | Q= K I I dQ 1−d  ∞     c c c = (μ p − w)(1 − d) + αq − (1 − d) + (1 + d) f (x)dx − βq (1 − d) 1 − 1+d w + αq w + αq w + α q K 1−d I I  ∞   w + αq 1 + d w + αq c (μ p − w) − αq − βq f (x)dx = (1 − d) + βq + αq 1+d w + αq c 1−d c K 1−d I I  w + αq 1+d  c (μ p − w) − αq − βq + βq + αq w + αq c 1−d  w + αq − c  c 2d − βq < 0. μ − w + αq = (1 − d) w + αq p 1−d c ≤ (1 − d)

  ¯ R Q, (1 − d)Q is decreasing in Q. Hence, we can conclude that 



Proof of Corollary 6.2. We prove Corollary 6.2 by comparison of Case 3 of Proposition 6.2 and the Case in the sixth row of Table 6.2, i.e., ( Q¯ 2 , (1 + d) Q¯ 2 ). We first analyze the conditions to guarantee the above equilibria. For ease of comparison, we assume the market demand follows uniform distribution, i.e., D ∼ U (0, Dh ) where Dh is the maximal demand. Now we can give closed-form solution (μ −w)D c to equilibrium decisions. In Proposition 6.2, if w+β > 21 , then K¯ 0∗ = μ pp−w+βqh . In q Table 6.2, we focus on the equilibrium case of ( Q¯ 2 , (1 + d) Q¯ 2 ). The conditions KI I III ≤ Q¯ 1 and Q¯ 2 < 1+d . With the uniform to guarantee such an equilibrium are K1+d

(w+αq −c)Dh (w+βq −c)Dh III and K I I I = . Therefore, K1+d ≤ w+αq w+βq 2 2 (1+d) (μ −w)c (1−d) KI I p c ¯ Q¯ 1 yields w+β > 2(1+d . 2 ) , and Q 2 ≤ 1+d yields αq ≥ c − w + (1−d)2 βq q (μ p −w)Dh 0∗ ¯ We further compare the capacity. We have K = μ p −w+βq , K¯ ∗ = (1+d)2 (μ p −w)Dh (μ −c)D and K C∗ S = p μ p h . It is easy to verify K¯ ∗ > K¯ 0∗ . By the con(1+d)2 (μ p −w)+(1−d)2 βq KI I dition of Q¯ 2 < 1+d , we also obtain K¯ ∗ ≤ K I I < K C∗ S . Therefore, we have K¯ 0∗ < K¯ ∗ < K C∗ S . By the concavity of the system profit function (in Eq. 6.1), it is easy to ¯ ∗ < C∗ S . ¯ 0∗ <   prove 

distribution, we have K I I =

Proof of Theorem 6.7. Firstly, we show that αq = μ p − w, βq = αq and F(K I I ) +

1−d 1+d

F

(1−d)K I I 1+d ¯∗



=1

¯∗

can coordinate the supply chain. Specifically, the equilibrium is ( Q , K ) = KI I , K I I ) where K I I = K C∗ S . ( 1+d KI I , K I I ) are as follows: In Table 6.2, the conditions that guarantee ( Q¯ ∗ , K¯ ∗ ) = ( 1+d 

(1−d)K I I KI I I KI I KI I 1−d ¯ ¯ ¯ = 1 and the ≤ Q1, Q3 ≤ and ≤ Q 2 . By F(K I I ) + F 1+d

1+d

1+d

1+d

1+d

KI I definition of Q¯ 1 , we obtain 1+d = Q¯ 1 . By βq = αq and the definition of Q¯ 3 , we obtain KI I = Q¯ 3 . By αq = μ p − w, we obtain Q¯ 3 = Q¯ 2 . Therefore, all conditions that 1+d guarantee such an equilibrium are satisfied. By αq = μ p − w, it is straightforward that K I I = K C∗ S . To conclude, the proof of supply chain coordination is completed. Secondly, we prove that the profit can be arbitrarily allocated between the manufacturer and retailer.

210

6 Supply Chain Coordination Through …

KI I Since the equilibrium decision is ( Q¯ ∗ , K¯ ∗ ) = ( 1+d , K I I ), we have K ∗ = (1 + ∗ d)Q and the profit function of the manufacturer can be simplified as follows:

¯ M (Q, K ) = wE(D ∧ K ) + βq E[(1 − d)Q − D]+ − cK   (1−d)Q [(1 − d)Q − D] f (D)dD − cK = (μ p − αq )E(D ∧ K ) + βq 0

= (μ p − αq )E(D ∧ K C∗ S ) + αq



dK C∗ S 0

[dK C∗ S − D] f (D)dD − cK C∗ S ,

. where the last equality is derived from βq = αq and d = 1−d 1+d ¯ R = C∗ S = ¯ M + Since supply chain coordination is achieved, we have  ∗ ∗ μ p E(D ∧ K C S ) − cK C S . Denote by λq the profit split of the manufacturer, then  dK ∗ (μ p − αq )E(D ∧ K C∗ S ) + αq 0 C S [dK C∗ S − D] f (D)dD − cK C∗ S λq = μ p E(D ∧ K C∗ S ) − cK C∗ S  dK ∗ αq [E(D ∧ K C∗ S ) − 0 C S (dK C∗ S − D) f (D)dD] = 1− . μ p E(D ∧ K C∗S ) − cK C∗S We first prove λq decreases in αq by showing E(D ∧ K C∗ S ) − D) f (D)dD > 0, which is shown as follows: E(D ∧ K C∗ S ) − =

K C∗ S

 −

> K C∗ S −

K C∗ S



dK C∗ S

0

 −

0

(dK C∗ S −

(dK C∗ S − D) f (D)dD

F(D)dD −

0 K C∗ S F(K C∗ S )

 dK C∗ S

dK C∗ S

F(D)dD

0 dK C∗ S F(dK C∗ S )

= 0, where the last equality is derived from F(K C∗ S ) = 1 − μc and dF(dK C∗ S ) = μc , p p 

(1−d)K I I 1−d ∗ which is derived from F(K I I ) + 1+d F = 1 and K I I = K C S . 1+d Next, we show how to achieve extreme profit allocation by adjusting αq , namely, αq ≡ λq = 0 or λq = 1. Evidently, setting αq = 0 leads to λq = 1. Setting αq =  μ p E(D∧K C∗ S )−cK C∗ S  dK ∗ ∗ E(D∧K C S )− 0 C S (dK C∗ S −D)

f (D)dD

leads to λq = 1. In addition, it is easy to verify  αq
w > w0 + we which ensures that the retailer has incentive to reserve capacity; we ≥ v which ensures that it is profitable for the retailer to exercise reserve capacity contract;

216

7 Outsourcing Decision and Order Policy with Forecast …

Fig. 7.1 Sequence of events

v < c1 < c2 which ensures the supplier has incentive to construct more than retailer reserved capacity at the beginning of the planning horizon; and the information is additive about the market which drives the demand.

7.3 Centralized System In this section, we consider the central system where the supplier and the retailer are owned by one firm. The firm’s owner maximizes his expected profit by choosing the constructing capacity K 1 and the outsourcing capacity K 2 which solve the following two-stage optimization problem. The supply chain profit at t1 denoted by t1 (K 1 , K 2 |x). t1 (K 1 , K 2 |x) = E[−c2 K 2 + r (D ∧ (K 1 + K 2 )) + v(K 1 + K 2 − D)+ − p(D − K 1 − K 2 )+ |x].

(7.1) Given the supplier’s initial constructing capacity K 1 , the supplier will make her outsourcing capacity decision K 2 to maximize t1 (K 1 , K 2 |x) at the start of selling season. (7.2) ∗t1 (K 1 |x) = max t1 (K 1 , K 2 |x) K 2 ≥0

With Eq. (7.2) on hand, we can write the supply chain profit at the beginning of planning horizon t0 which is denoted by t0 (K 1 |x). max t0 (K 1 ) = −c1 K 1 + E X [∗t1 (K 1 |X )]. K 1 ≥0

(7.3)

In the following paragraph, we will characterize the optimal solution in centralized system. Let K = K 1 + K 2 , using Eq. (7.3), the decision’s problem can be stated in centralized system as follows.  max t0 (K 1 ) = (c2 − c1 )K 1 + K 1 ≥0

where G(K 1 , x) = max K ≥K 1 g(K , x), and

0

+∞

G(K 1 , x) f X (x)dx.

(7.4)

7.3 Centralized System

217



K

g(K , x) = −c2 K +

 [r ξ + v(K − ξ )] f D|x (ξ )dξ +

0

+∞

[r K − p(ξ − K )] f D|x (ξ )dξ.

K

(7.5) Let (K 1∗ , K 2∗ ) denote the optimal solution to the centralized system problem, the following theorem can characterize their optimal solution structures. Theorem 7.1 t0 (K 1 ) is concave on the initial constructing capacity K 1 , and the optimal outsourcing capacity K 2∗ satisfies K 2∗

 =

K ∗ (x) − K 1 , if K ∗ (x) ≥ K 1 ; 0, otherwise,

where K ∗ (x) satisfies the equation FD|x (K ) =

r + p−c2 . r + p−v

Proof From Eq. (7.5), by taking the first derivative with respect to K on g(K , x), we obtain ∂g(K , x) = −c2 + v FD|x (K ) + (r + p) − (r + p)FD|x (K ). ∂K By taking twice derivative with respect to K on g(K , x), we obtain ∂ 2 g(K , x) = −(r + p − v) f D|x (K ) ≤ 0. ∂K2 Therefore, g(K , x) is concave on K , so G(K 1 , x) is concave on K 1 (see, Porteus (2002)), hence, t0 (K 1 ) is concave on K 1 . Let K ∗ (x) satisfy its first order condition of g(K , x), we have r + p − c2 . FD|x (K ∗ (x)) = r + p−v Consequently, we complete the proof.



In the following paragraph, we consider the demand distribution that is in the shifted family to facilitate our analysis. In the paper by Erkoc Murat and David Wu (2005), they also used the distribution to get some perfectly results. Two examples to shifted family distribution is the Normal distribution and Uniform distribution which are widely used to capture uncertainty in real life practice. We investigate the supplier’s constructing capacity K 1 at the beginning of planning horizon t0 . To obtain the expected function of ∗t1 (K 1 |X ) i.e. E X [∗t1 (K 1 |X )], with respect to the random variable X , we make mathematical manipulations as follows. From −1 r + p−c2 ( r + p−v ). Geometrically shifting f D|x Theorem 7.1, we know that K ∗ (x) = FD|x curve to the left by x, we may get a new curve where the location parameter X = 0. We denote the corresponding cumulative density function by F D|0 (·) where F D|0 (a) = a ∗ f −x D|0 (ξ )dξ (a ≥ −x). In the new curve, we can identify a point, K (0), by using −1 p+r −c2 ( r + p−v ). It is obvious that K ∗ (x) can be expressed as the equation K ∗ (0) = F D|0 ∗ sum of K (0) and x, i.e. K ∗ (x) = K ∗ (0) + x.

218

7 Outsourcing Decision and Order Policy with Forecast …

Therefore, using Eq. (7.4) and Theorem 7.1, we can rewrite the supply chain’s objective function at the beginning of planning horizon t0 as follows  max t0 (K 1 ) = (c2 − c1 )K 1 +

K 1 ≥0



K 1 −K ∗ (0)

g(K 1 , x) f X (x)dx +

0

+∞

g(K ∗ (x), x) f X (x)dx.

K 1 −K ∗ (0)

(7.6) Theorem 7.2 In centralized system, the supply chain chooses his optimal constructing capacity K 1∗ which satisfies the following equation. 

K 1 −K ∗ (0)

(r + p − v)

FD|x (K 1 ) · f X (x)dx = (r + p − c2 ) · FX (K 1 − K ∗ (0)) + (c2 − c1 ).

0

Proof Taking the first derivative of Eq. (7.6) with respect to K 1 , we have ∂t0 (K 1 ) ∂ K1

 K −K ∗ (0) ∂g(K 1 ,x) = (c2 − c1 ) + g(K 1 , K 1 − K ∗ (0)) · f X (K 1 − K ∗ (0)) + 0 1 · f X (x)dx ∂ K1 ∗ ∗ ∗ ∗ −g(K (K 1 − K (0)), K 1 − K (0)) · f X (K 1 − K (0))  ∂g(K ∗ (x),x) + K+∞ · f X (x)dx. (7.7) ∗ ∂K 1 −K (0) 1

From Eq. (7.5), we have ∂g(K 1 , x) = (r + p − c2 ) − (r + p − v)FD|x (K 1 ); and ∂ K1

∂g(K ∗ (x), x) = 0. ∂ K1 (7.8)

Let K 1∗ satisfy the first order condition of t0 (K 1 ), then we can obtain K 1∗ which satisfies 

K 1 −K ∗ (0)

(r + p − v)

FD|x (K 1 ) · f X (x)dx = (r + p − c2 ) · FX (K 1 − K ∗ (0)) + (c2 − c1 )

0

+[g(K 1 , K 1 − K ∗ (0)) − g(K ∗ (K 1 − K ∗ (0)), K 1 − K ∗ (0))] · f X (K 1 − K ∗ (0)).

(7.9) Note that K ∗ (K 1 − K ∗ (0)) = K ∗ (0) + [K 1 − K ∗ (0)] = K 1 ,

(7.10)

g(K 1 , K 1 − K ∗ (0)) − g(K ∗ (K 1 − K ∗ (0)), K 1 − K ∗ (0)) = 0.

(7.11)

then we have

7.4 Decentralized System

219

Therefore, substituting Eqs. (7.10) and (7.11) into Eq. (7.9), the theorem can be directly obtained. 

7.4 Decentralized System In this section, we model the decentralized system as two-stage optimization problem, where the retailer determines the initial reserved amount and order quantity, while the supplier in turn sets constructing capacity and outsourcing capacity based on the retailer’s orders. We assume the supplier is obligated to fill the retailer’s total order quantity (i.e., the contract stipulates forced compliance, see Cachon and Lariviere (2001), In Erkoc Murat and David Wu (2005) page 23, they also have similar assumption that the supplier always builds sufficient capacity to cover the reservation amount.) and that both decision makers have common knowledge of the consumer demand forecast distribution (i.e., FX (·) and FD|X (·)).

7.4.1 Retailer’s Optimal Decision In this subsection, we want to find the optimal order strategy for the retailer to maximize his expected profits by choosing reserving capacity K 0 at the beginning of planning horizon, and the actual order quantity q at the beginning of selling season. We formula it as a two-stage optimization problem. At the beginning of selling season t1 , the retailer has the knowledge to reserve K 0 . With X = x, the retailer’s problem at time t1 is to find optimal order quantity q ¯ t1 (K 0 , q|x). to maximize his profit function  ¯ t1 (K 0 , q|x) = E[−we (q ∧ K 0 ) − w(q − K 0 )+ + r (q ∧ D) + v(q − D)+ − p(D − q)+ |x]. 

(7.12)

¯ t1 (K 0 , q|x) can be rewritten as follows The function  ¯ t1 (K 0 , q|x) = 



¯ t1 ,1 (K 0 , q|x), if q ≤ K 0 ;  ¯ t1 ,2 (K 0 , q|x), if q > K 0 , 

(7.13)

where ¯ t1 ,1 (K 0 , q|x) = −we q +  ¯ t1 ,2 (K 0 , q|x) 

q 0

[r ξ + v(q − ξ )] f D|x (ξ )dξ +

 +∞ q

[rq − p(ξ − q)] f D|x (ξ )dξ ;

q = −we K 0 − w(q − K 0 ) + 0 [r ξ + v(q − ξ )] f D|x (ξ )dξ  +∞ + q [rq − p(ξ − q)] f D|x (ξ )dξ.

(7.14) (7.15)

220

7 Outsourcing Decision and Order Policy with Forecast …

Therefore, the optimal profit function at the beginning of selling season t1 is ¯ ∗t (K 0 |x) which satisfies  1 ¯ ∗t (K 0 |x) = max  ¯ t1 (K 0 , q|x) = max  1 q≥0



¯ t ,1 (K 0 , q|x); max  1

q∈[0,K 0 ]

 ¯ t ,2 (K 0 , q|x) . max  1

q>K 0

(7.16) Theorem 7.3 Given the initial reserving capacity K 0 and known the information X = x, the optimal order quantity q ∗ at the beginning of selling season in the decentralized system, has the following structure. ⎧ ⎨ q2 (x), if K 0 ≤ q2 (x), q ∗ = K 0 , if q2 (x) < K 0 ≤ q1 (x), ⎩ q1 (x), if K 0 > q1 (x), −1 p+r −we −1 p+r −w where q1 (x) = FD|x ( p+r −v ) and q2 (x) = FD|x ( p+r −v ).

Proof From Eqs. (7.14) and (7.15), by taking the first derivative with respect to q on ¯ t1 ,1 (K 0 , q|x) and  ¯ t1 ,2 (K 0 , q|x), respectively, we can obtain  ¯ t1 ,1 (K 0 , q|x) ∂ = p + r − we − ( p + r − v)FD|x (q); ∂q ¯ t1 ,2 (K 0 , q|x) ∂ = p + r − w − ( p + r − v)FD|x (q). ∂q

(7.17) (7.18)

¯ t1 ,1 (K 0 , q|x) and  ¯ t1 ,2 (K 0 , q|x) with respect to Thus taking twice derivatives of  q, we have ¯ t1 ,1 (K 0 , q|x) ∂ 2 = −( p + r − v) f D|x (q) < 0; ∂q 2 ¯ t1 ,2 (K 0 , q|x) ∂ 2 = −( p + r − v) f D|x (q) < 0. ∂q 2 ¯ t1 ,1 (K 0 , q|x) and  ¯ t1 ,2 (K 0 , q|x) are both concave functions on q. Hence,  Let q1 (x) and q2 (x) satisfy the fist order conditions of Eqs. (7.17) and (7.18), respectively. Therefore, we have −1 p+r −we ( p+r −v ); q1 (x) = FD|x

(7.19)

−1 p+r −w ( p+r −v ). q2 (x) = FD|x

(7.20)

It is obviously that q2 (x) is less than q1 (x) by assumption. In order to complete the proof, the problem can be divided into three cases. Case 1. K 0 ≤ q2 (x). If K 0 ≤ q2 (x), then from (7.16), (7.19) and (7.20), we have

7.4 Decentralized System

221

  ¯ ∗t (K 0 |x) = max  ¯ t1 ,1 (K 0 , K 0 |x);  ¯ t1 ,2 (K 0 , q2 (x)|x) .  1

(7.21)

¯ t1 ,2 (K 0 , q2 (x)|x) ≥  ¯ t1 ,2 (K 0 , K 0 |x) =  ¯ t1 ,1 (K 0 , K 0 |x). It is easy to know that  Hence, from (7.21), we have ¯ ∗t (K 0 |x) =  ¯ t1 ,2 (K 0 , q2 (x)|x).  1

(7.22)

Then,   ¯ t1 (K 0 , q|x) = arg max  ¯ t1 ,1 (K 0 , K 0 |x);  ¯ t1 ,2 (K 0 , q2 (x)|x) = q2 (x). q ∗ = arg max  q≥0

(7.23) Case 2. q2 (x) < K 0 ≤ q1 (x). If q2 (x) < K 0 ≤ q1 (x), then again from (7.16), (7.19) and (7.20), we have   ¯ ∗t (K 0 |x) = max  ¯ t1 ,1 (K 0 , K 0 |x);  ¯ t1 ,2 (K 0 , K 0 |x) =  ¯ t1 ,1 (K 0 , K 0 |x).  1 (7.24) Therefore   ¯ t1 (K 0 , q|x) = arg max  ¯ t ,1 (K 0 , K 0 |x);  ¯ t ,2 (K 0 , K 0 |x) = K 0 . q ∗ = arg max  1 1 q≥0

(7.25) Case 3. K 0 > q1 (x). If K 0 > q1 (x), then again from (7.16), (7.19) and (7.20), we have   ¯ ∗t1 (K 0 |x) = max  ¯ t1 ,1 (K 0 , q1 (x)|x);  ¯ t1 ,2 (K 0 , K 0 |x) =  ¯ t1 ,1 (K 0 , q1 (x)|x).  (7.26) Therefore   ¯ t1 (K 0 , q|x) = arg max  ¯ t1 ,1 (K 0 , q1 (x)|x);  ¯ t1 ,2 (K 0 , K 0 )|x) = q1 (x). q ∗ = arg max  q≥0

(7.27) Consequently, combining Cases 1–3 and from Eqs. (7.19), (7.20), (7.23), (7.25) and (7.27), we completes the proof of this theorem.  Next, we will investigate the optimal policy for the retailer at the beginning of planning horizon t0 . At time t0 , the retailer wants to maximize his total expected profit during the whole planning horizon by choosing the optimal reserving capacity K 0∗ . The objective function is

222

7 Outsourcing Decision and Order Policy with Forecast …

¯ t0 (K 0 ) = −w0 K 0 + E X [ ¯ ∗t (K 0 |X )]. max  1 K 0 ≥0

(7.28)

From Eq. (7.16) and Theorem 7.3, we can rewrite ∗t1 (K 0 |x) as following ⎧ ¯ t1 ,2 (K 0 , q2 (x)|x), if K 0 ≤ q2 (x); ⎨ ¯ (K , K |x), if q2 (x) < K 0 ≤ q1 (x); ∗t1 (K 0 |x) =  ⎩ ¯ t1 ,1 0 0 t1 ,1 (K 0 , q1 (x)|x), if K 0 > q1 (x).

(7.29)

To obtain the expected function of ∗t1 (K 0 |X ), i.e., E X [∗t1 (K 0 |X )], with respect to the random variable X , we make geometrical shifting. q1 (x) can then be expressed as the sum of q1 (0) and x, i.e., q1 (x) = q1 (0) + x. The same argument that q2 (x) = q2 (0) + x. Therefore, Eq. (7.29) can be written as ⎧ ¯ t1 ,2 (K 0 , q2 (x)|x), if x ≥ K 0 − q2 (0); ⎨ ¯ (K , K |x), if K 0 − q1 (0) ≤ x < K 0 − q2 (0); (7.30) ∗t1 (K 0 |x) =  ⎩ ¯ t1 ,1 0 0 t1 ,1 (K 0 , q1 (x)|x), if x < K 0 − q1 (0). Hence, from Eqs. (7.28) and (7.30), we have

=

 K 0 −q1 (0) 0

¯ t0 (K 0 ) max K 0 ≥0   ¯ t1 ,1 (K 0 , q1 (x)|x)] f X (x)dx + K 0 −q2 (0) [ ¯ t1 ,1 (K 0 , K 0 |x)] f X (x)dx [ K 0 −q1 (0)  +∞ ¯ t1 ,2 (K 0 , q2 (x)|x)] f X (x)dx − w0 K 0 . + K 0 −q2 (0) [ (7.31)

Theorem 7.4 Given the reservation capacity contract (w0 , we , w), the retailer’ optimal reservation capacity K 0∗ satisfies the following equation.  K 0 −q2 (0)

=

K 0 −q1 (0) FD|x (K 0 ) f X (x)dx (r + p−w)FX (K 0 −q2 (0))−(r + p−we )FX (K 0 −q1 (0)) r + p−v

+

w−w0 −we . r + p−v

¯ t0 (K 0 ), we Proof From Eq. (7.31), taking the first derivative with respect to K 0 on  obtain ¯ t0 (K 0 ) ∂ ¯ t1 ,1 (K 0 , q1 (K 0 − q1 (0))|K 0 − q1 (0)) f X (K 0 − q1 (0)) = −w0 +  ∂ K0  K −q (0) ∂ ¯ t1 ,1 (K 0 ,q1 (x)|x) + 00 1 f X (x)dx ∂ K0 ¯ t1 ,1 (K 0 , K 0 |K 0 − q2 (0)) f X (K 0 − q2 (0)) + ¯ t1 ,1 (K 0 , K 0 |K 0 − q1 (0)) f X (K 0 − q1 (0)) −  K −q (0) ∂ ¯ (K ,K |x) + K 00−q12(0) t1 ,1∂ K00 0 f X (x)dx

¯ t1 ,2 (K 0 , q2 (K 0 − q2 (0))|K 0 − q2 (0)) f X (K 0 − q2 (0)) −  +∞ ¯ (K ,q (x)|x) ∂ + K 0 −q2 (0) t1 ,2 ∂ K0 0 2 f X (x)dx. (7.32)

7.4 Decentralized System

223

Using Eqs. (7.14) and (7.15), we know that ¯ t1 ,1 (K 0 , q1 (x)|x) ∂ = 0; ∂ K0 ¯ t1 ,1 (K 0 , K 0 |x) ∂ = r + p − we − (r + p − v)FD|x (K 0 ); ∂ K0 ¯ t1 ,2 (K 0 , q2 (x)|x) ∂ = w − we . ∂ K0

Note that q1 (K 0 − q1 (0)) = K 0 and

q2 (K 0 − q2 (0)) = K 0 .

(7.33) (7.34) (7.35)

(7.36)

Hence, we have ¯ t1 ,1 (K 0 , K 0 |K 0 − q2 (0)) f X (K 0 − q2 (0)) =  ¯ t1 ,2 (K 0 , q2 (K 0 − q2 (0))|K 0 − q2 (0)) f X (K 0 − q2 (0)). 

Consequently, Eq. (7.32) can be rewritten as follows ¯ t (K 0 ) ∂ 0 ∂ K0

= −w0 + (r + p − we )FX (K 0 − q2 (0)) − (r + p − we )FX (K 0 − q1 (0))  K −q (0) −(r + p − v) K 00−q12(0) FD|x (K 0 ) f X (x)dx + (w − we )[1 − FX (K 0 − q2 (0))].

(7.37)

¯ t0 (K 0 ), then we have Let K 0∗ satisfies the fist-order condition of   K 0 −q2 (0)

=

K 0 −q1 (0) FD|x (K 0 ) f X (x)dx (r + p−w)FX (K 0 −q2 (0))−(r + p−we )FX (K 0 −q1 (0)) r + p−v

Consequently, we have the theorem.

+

w−w0 −we . r + p−v



7.4.2 Optimal Strategy for the Supplier In this section, we analyze the supplier’s behavior. The supplier’s problem is to choose initial constructing capacity K 1 and outsourcing capacity K 2 that maximize her own expected profit, subject to the retailer’s ordering behavior. We assume the supplier can’t sell directly to the final customer nor supply the retailer with inventory in excess of his order quantity q. Furthermore, the supplier must commit to deliver the quantity by the retailer ordered. If the supplier’s initial constructing capacity is not enough to satisfy the demand, she will outsource the capacity to meet the demand. As a result, given initial constructing capacity K 1 , the supplier will outsource K˜ 2∗ to meet the retailer’s actual order quantity.

224

7 Outsourcing Decision and Order Policy with Forecast …

K˜ 2∗ =



q ∗ − K 1 , if q ∗ ≥ K 1 ; 0, otherwise.

(7.38)

The supplier’s decision concerning her initial construct capacity K 1 is more complex. She should obviously construct at least K 0 , because she must deliver this quantity to the retailer by the terms of the contract and postponing this capacity to the next period simply increases cost. The question is, should she construct more? The answer depends, in part, on how likely the retailer is to actual order quantity at the beginning of selling season. Because the supplier has knowledge of the demand forecast, she can derive the p.d.f. of the retailer’s order quantity after observing information X = x. Therefore, the supplier’s problem of initial constructing capacity can now be stated as follows ˜ t0 (K 1 ) = w0 K 0∗ − c1 K 1 + E X [ ˜ t1 (K 1 , q ∗ |X )]. max 

K 1 ≥K 0∗

(7.39)

where, if X = x, we have ˜ t1 (K 1 , q ∗ |x) = [we (q ∗ ∧ K ∗ ) + w(q ∗ − K ∗ )+ − c2 (q ∗ − K 1 )+ + v(K 1 − q ∗ )+ |x].  0 0

(7.40)

˜ t1 (K 1 , q ∗ |x) can be rewritten as Hence,  ⎧ ˜ t1 ,1 (K 1 , q ∗ |x), if q ∗ ≤ K 0∗ ; ⎨ ∗ ˜ t1 (K 1 , q |x)  ˜ (K , q ∗ |x), if K 0∗ < q ∗ ≤ K 1 ;  ⎩ ˜ t1 ,2 1 ∗ t1 ,3 (K 1 , q |x), if q ∗ > K 1 .

(7.41)

where ˜ t1 ,1 (K 1 , q ∗ |x) = [we q ∗ + v(K 1 − q ∗ )|x];  ˜ t1 ,2 (K 1 , q ∗ |x) = [we K 0∗ + w(q ∗ − K 0∗ ) + v(K 1 − q ∗ )|x];  ˜ t1 ,3 (K 1 , q ∗ |x) = [we K 0∗ + w(q ∗ − K 0∗ ) − c2 (q ∗ − K 1 )|x]. 

(7.42) (7.43) (7.44)

Therefore, from Theorem 7.3 and Eqs. (7.41)–(7.44), the supplier’s objective function (7.39) can be rewritten as follows ˜ t0 (K 1 ) = w0 K 0∗ − c1 K 1 + max∗ 

K 1 ≥K 0

 K 0∗ −q1 (0) 0

˜ t1 ,1 (K 1 , q1 (x)|x) f X (x)dx 

 K ∗ −q (0) ˜ t1 ,1 (K 1 , K 0 |x) f X (x)dx + K ∗0−q12(0)  0  K 1 −q2 (0) ˜ t1 ,2 (K 1 , q2 (x)|x) f X (x)dx + K ∗ −q2 (0)  0  +∞ ˜ t1 ,3 (K 1 , q2 (x)|x) f X (x)dx. + K 1 −q2 (0) 

(7.45)

7.4 Decentralized System

225

Theorem 7.5 The supplier’s optimal constructing capacity K˜ 1∗ satisfies K˜ 1∗ =



Kˆ 1∗ , if Kˆ 1∗ ≥ K 0∗ ; K 0∗ , otherwise.

−1 r + p−w 1 where Kˆ 1∗ = q2 (0) + FX−1 ( cc22−c ), and q2 (0) = FD|0 ( r + p−v ). −v

Proof Using (7.42), (7.43) and (7.44) to substitute into (7.45), we have ˜ t0 (K 1 ) max K 1 ≥K 0∗   ∗ K −q (0) = w0 K 0∗ − c1 K 1 + 0 0 1 [we q1 (x) + v(K 1 − q1 (x))] f X (x)dx  K ∗ −q (0) + K ∗0−q12(0) [we K 0∗ + v(K 1 − K 0∗ )] f X (x)dx 0  K 1 −q2 (0) + K ∗ −q2 (0) [we K 0∗ + w(q2 (x) − K 0∗ ) + v(K 1 − q2 (x))] f X (x)dx 0  +∞ + K 1 −q2 (0) [we K 0∗ + w(q2 (x) − K 0∗ ) − c2 (q2 (x) − K 1 )] f X (x)dx.

(7.46)

˜ t0 (K 1 ) with respect to K 1 , we can obtain that Taking the derivative of  ˜ t (K 1 ) ∂ 0 ∂ K1 ∗ ∗  K −q1 (0)  K −q2 (0)  K −q (0) = −c1 + 0 0 v · f X (x)dx + K ∗0−q (0) v · f X (x)dx + K ∗1−q 2(0) v · f X (x)dx 1 2 0 0 +[we K 0∗ + w(q2 (K 1 − q2 (0)) − K 0∗ ) + v(K 1 − q2 (K 1 − q2 (0)))] · f X (K 1 − q2 (0)) −[we K 0∗ + w(q2 (K 1 − q2 (0)) − K 0∗ ) − c2 (q2 (K 1 − q2 (0)) − K 1 )] · f X (K 1 − q2 (0))  c · f X (x)dx + K+∞ 1 −q2 (0) 2

= c2 − c1 + (v − c2 ) · FX (K 1 − q2 (0))

+(v − c2 ) · [K 1 − q2 (K 1 − q2 (0))] · f X (K 1 − q2 (0)).

(7.47)

Note that K 1 − q2 (K 1 − q2 (0)) = 0, and let Kˆ 1∗ satisfies the firs-order condition of ˜ t0 (K 1 ). From (7.47), we can obtain  c2 − c1 ). Kˆ 1∗ = q2 (0) + FX−1 ( c2 − v So we completed the proof of this theorem.



In the following paragraph, we will demonstrate the optimal strategies for all players in centralized and decentralized systems, and derive the necessary and sufficient condition of outsourcing under a uniformly distributed demand case.

226

7 Outsourcing Decision and Order Policy with Forecast …

7.5 Numerical Studies In this section, we analyze a particular case where the demand is uniformly distributed. At the beginning of planning horizon t0 , the retailer knows that the demand D follows uniform distribution over the interval [X − m, X + m], where the location parameter X is unknown but uniformly distributed over [γ − n, γ + n]. During the lead-time (from t0 to t1 ), the retailer updates the demand forecast. At the beginning of selling season t1 , the retailer specifies the X value, X = x, based on the forecast update. We assume that γ ≥ n + m to ensure D ≥ 0. The pdf and cdf of X as well as those of D given X = x are as follows: f X (x) FX (x) f D|x (ξ ) FD|x (ξ )

=

1 , 2n

1 = 2n (x − 1 = 2m , 1 = 2m (ξ −

x ∈ [γ − n, γ + n]. γ + n),

(7.48)

x ∈ [γ − n, γ + n].

(7.49)

ξ ∈ [x − m, x + m]. x + m),

(7.50)

ξ ∈ [x − m, x + m].

(7.51)

For simplicity, in the following paragraph, we will analyze only the case c2 + v > 2c1 and w − we > 2w0 in detail. Other cases can be analyzed similarly.

7.5.1 Analysis of Players’ Strategies In this section, we analyze the optimal strategies for all players in centralized system and decentralized system with uniform distribution case specified by Eqs. (7.48)– (7.51). Firstly, from Theorem 7.1, we can characterize the optimal policies of supply chain in centralized system as following. Proposition 7.1 In centralized system, the optimal constructing capacity K 1∗ satisfies ⎧ ∗ if ⎪ ⎨ γ + K (0) + n,  8mn·(c1 −v) ∗ K1 = γ + m + n − , if r + p−v ⎪ ⎩ 2m(c2 −c1 ) ∗ γ + K (0) + r + p−v , if where K ∗ (0) =

m n m n m n

≤ ∈ ∈

r + p−v 2(c1 −v) · c2 −v ; c2 −v r + p−v 2(c1 −v) r + p−v ( c2 −v · c2 −v , c2 −v p−v (c2 −v) ( r+ · 2(c , +∞). c2 −v 1 −v)

·

(c2 −v) ]; 2(c1 −v)

(r + p+v−2c2 )·m . r + p−v

Proposition 7.2 In centralized system, the necessary and sufficient condition of outsourcing capacity is the updating information x ≥ K 1∗ − K ∗ (0), where ⎧ r + p−v 2(c1 −v) ⎪ γ + n, if m ⎪ n ≤ c2 −v · c2 −v ; ⎨  8mn·(c −v) r + p−v 2(c1 −v) r + p−v (c2 −v) ∗ ∗ m 1 K 1 − K (0) = γ + m + n − K ∗ (0) − r + p−v , if n ∈ ( c2 −v · c2 −v , c2 −v · 2(c1 −v) ]; ⎪ ⎪ r + p−v (c2 −v) ⎩ γ + 2m(c2 −c1 ) , if m r + p−v n ∈ ( c2 −v · 2(c1 −v) , +∞).

7.6 Concluding Remarks

227

Proof The proposition is directly from Theorem 7.1 and Proposition 7.1.



Next, we investigate the behaviors of the retailer and the supplier in decentralized system. Firstly, we characterize the optimal strategy of the retailer using Theorem 7.3. Proposition 7.3 In decentralized system, the retailer’s optimal reservation capacity ⎧ 1 ⎪ ⎪ ⎨ γ + n + 2 · [q1 (0)+ q2 (0)] − 0 K 0∗ = γ + q1 (0) + n − r8mnw , + p−v ⎪ ⎪ ⎩ γ + q2 (0) + 2m(w−w0 −we ) , r + p−v

2nw0 , w−we

if if if

m n m n m n

2w0 (r + p−v) ; (w−we )2 2w0 (r + p−v) r + p−v ∈ ( (w−we )2 , w−we + p−v w−we > rw−w · 2w0 . e


0; ∂K

∂g2 (K , x) = v − c2 < 0; ∂K

(A.4) ∂g3 (K , x) 1 1 and = −c2 − (r + p − v)K + [(r + p)(x + m) − v(x − m)]. ∂K 2m 2m

(A.5) Then, analyzing (7.5) carefully, we can find the proof can be divided three cases. Case 1. If K 1 ≤ x − m, then ⎧ ⎨ g1 (K , x), if K 1 ≤ K ≤ x − m; g(K , x) = g3 (K , x), if K 1 ≤ x − m < K < x + m; (A.6) ⎩ g2 (K , x), if K 1 ≤ x − m < x + m ≤ K . Therefore, from (A.1)–(A.5) and Theorem (7.1), G(K 1 , x) can be written as follows G(K 1 , x) = max g(K , x) = max



g1 (K , x); max g3 (K , x); K 1 ≤x−m 0 for K 1 ∈ (γ + K ∗ (0) − n, γ + K ∗ (0) + n]. ∂ K1

(A.57)

Hence, one has that max

K 1 ∈(γ +K ∗ (0)−n,γ +K ∗ (0)+n]

π2 (K 1 ) = π2 (γ + K ∗ (0) + n)

(A.58)

238

7 Outsourcing Decision and Order Policy with Forecast …

From (A.26) and (A.27), we have max

K 1 ∈(γ +K ∗ (0)+n,γ +m−n]

π3 (K 1 ) =

⎧ ⎨ π3 (γ + m − n),

if ⎩ π (γ + K ∗ (0) + 2m(c2 −c1 ) ), if 3 r + p−v

m ∈ ( r + p−v , r + p−v · (c2 −v) ]; n c2 −v c2 −v 2(c1 −v) m ∈ ( r + p−v · (c2 −v) , +∞). n c2 −v 2(c1 −v)

(A.59)

and  max

K 1 ∈(γ +m−n,γ +m+n]

π5 (K 1 ) =

π5 (γ + m + n −



8mn·(c1 −v) r+ p−v ),

π5 (γ + m − n),

if if

m n m n

r+ p−v (c2 −v) c2 −v · 2(c1 −v) ]; (c2 −v) · 2(c1 −v) , +∞).

p−v ∈ ( r+ c2 −v ,

∈

p−v ( r+ c2 −v

(A.60)

Therefore, from (A.55),  ∗t0 ,2 =

π5 (γ + m + n − π3 (γ + K ∗ (0) +



8mn·(c1 −v) ), r + p−v 2m(c2 −c1 ) ), r + p−v

if if

p−v r + p−v ∈ ( r+ , c2 −v · c2 −v

m n m n

p−v ∈ ( r+ · c2 −v

(c2 −v) ]; 2(c1 −v)

(c2 −v) , +∞). 2(c1 −v)

(A.61) Combining scenarios 1 and 2, we can obtain the optimal constructing capacity K 1∗ at the beginning of planning horizon in centralized system, where ⎧ ∗ if ⎪ ⎨ γ + K (0) + n,  8mn·(c1 −v) ∗ K1 = γ + m + n − , if r + p−v ⎪ ⎩ 2m(c2 −c1 ) ∗ γ + K (0) + r + p−v , if

m n m n m n

≤ ∈ ∈

r + p−v 2(c1 −v) · c2 −v ; c2 −v r + p−v 2(c1 −v) r + p−v ( c2 −v · c2 −v , c2 −v p−v (c2 −v) ( r+ · 2(c , +∞). c2 −v 1 −v)

·

(c2 −v) ]; 2(c1 −v)

(A.62) 

Hence, the proof of this proposition is completed.

Proof of Proposition 7.3 In order to prove the proposition, we firstly must derive the retailer’s optimal strategy for given information X = x at the beginning of selling season t1 . Secondly, we optimize the retailer’s strategy at the beginning of planning horizon t0 . We denote several function expression in order to obtain the optimal strategy for given X = x at time t1 clearly. From (7.12), we let  H1 (q, x) = −we q + H2 (q, x) = −we q +

x+m

x−m  q



1 [rq − p(ξ − q)]dξ ; 2m

1 [r ξ + v(q − ξ )]dξ + 2m

x−m x+m

(A.63)  q

x+m

1 [rq − p(ξ − q)]dξ ; (A.64) 2m

1 H3 (q, x) = −we q + [r ξ + v(q − ξ )]dξ ; x−m 2m  x+m 1 H4 (q, x) = −we K 0 − w(q − K 0 ) + [rq − p(ξ − q)]dξ ; x−m 2m

(A.65) (A.66)

Appendix

239

H5 (q, x) = −we K 0 − w(q − K 0 ) +

 q

 x+m 1 1 [r ξ + v(q − ξ )]dξ + [rq − p(ξ − q)]dξ ; 2m x−m 2m q

 x+m 1 H6 (q, x) = −we K 0 − w(q − K 0 ) + [r ξ + v(q − ξ )]dξ ; x−m 2m

(A.67) (A.68)

By taking the derivative of (A.63)–(A.68) with respect to q, we have ∂ H1 (q,x) ∂ H2 (q,x) p+v−2we ) = r + p − we > 0; = 0 =⇒ q ∗ (x) = x + m(r +r + ; p−v ∂q ∂q ∂ H3 (q,x) ∂ H4 (q,x) (A.69) = v − we < 0; = r + p − w > 0; ∂q ∂q ∂ H5 (q,x) p+v−2w) ∂ H6 (q,x) = 0 =⇒ q ∗ (x) = x + m(r + ; = v − w < 0. r + p−v ∂q ∂q

Using (7.12) again, we know the proof can be divided three cases at time t1 . Case 1. If K 0 ≤ x − m, then from (7.13), (7.14) and (7.15), we have ⎧ H1 (q, x), if q ≤ K 0 ≤ x − m; ⎪ ⎪ ⎨ ¯ t1 (K 0 , q|x) = H4 (q, x), if K 0 < q ≤ x − m;  (A.70) ⎪ H5 (q, x), if K 0 ≤ x − m < q < x + m; ⎪ ⎩ H6 (q, x), if K 0 ≤ x − m < x + m ≤ q. Hence, using (A.69), we can obtain that ¯ ∗t (K 0 |x) = max  ¯ t1 (K 0 , q|x) = max  1 q≥0

max

K 0 ≤x−m