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Essays in supply chain management
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Essays in supply chain management
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ESSAYS IN SUPPLY CHAIN MANAGEMENT by Mahesh Nagarajan A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) December 2003 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3133313 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3133313 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089-1695 This dissertation, written by M W W aaM aT A -A /_____________ under the direction o f h dissertation committee, and approved by all its members, has been presented to and accepted by the Director o f Graduate and Professional Programs, in partial fulfillment o f the requirements for the degree o f DOCTOR OF PHILOSOPHY Director Date December 17, 2003 Dissertation Committee - 0 — - ............................— Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Abstract iii 1. Introduction 1 2. A Bargaining Framework In Supply Chains 6 3. Coordination issues in vendor managed inventory systems 61 4. Issues in stocking substitutable products 92 Alphabetized Bibliography 136 Appendix 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT This dissertation examines three distinct topics in the area of supply chain management. First, a bargaining framework in which agents of a supply chain, negotiate over the terms of trade is proposed. Adopting this framework, using techniques from cooperative and non-cooperative game theory, the structure of the supply chain is predicted. Second, certain contracting issues in what are commonly known as “vendor managed inventory systems” are analyzed. Finally, certain issues arising in the stocking of substitutable products are analyzed. Characterizing optimal policies, certain risk pooling anomalies and the equihbrium of a dynamic game are the contributions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Introduction. This dissertation studies and seeks to contribute to the better understanding of certain tactical and strategic issues that arise in supply chain management. Different entities in a typical supply chain include suppliers, producers, manufacturers, assemblers, distributors, retailers and finally the customers. Some or all of these entities make decisions regarding production, purchasing, pricing, scheduling etc. that determine their respective utilities. This decision making process is often quite complex. The reasons for this complexity are manifold. First, the parameters that drive the decision process and the nature of the problem often present an analytical challenge. Factors like randomness o f certain parameters (demand, lead times, yield etc.), the dynamics of decisions process etc. contribute to the difficulty of these problems. Further, since each entity may only be interested in optimizing his decisions to maximize his/her individual utility, strategic issues arise. Thus, the interaction between these entities needs to be analyzed using techniques borrowed from game theory. Thus several issues in supply chain management are analyzed using tools from the mathematical theory of optimization and the theory of games and economic behavior. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation comprises of three essays (organized in chapters 2,3 & 4 respectively), each of which address issues in supply chain management that are unrelated to each other. The first essay (Chapter 2), titled: “A bargaining framework: The Assembly problem”, examines a supply chain with a single assembler who buys complementary components from N suppliers, assembles the final product and sells it to his customers. Players take actions in the following sequence. First, (Stage 3) the suppliers form coalitions between themselves. Second, (Stage 2) the coalitions compete for a position in the negotiation sequence. Finally, (Stage 1) the coalitions negotiate with the assembler on the wholesale price and quantity o f goods to be sold to the assembler. We assume: (a) The negotiation process is sequential; that is the assembler negotiates with one coalition at a time and (b) The assembler has the power to determine the negotiation sequence, while the suppliers can freely form alliances. This essay provides a framework, using the theory o f cooperative games, to analyze how the different agents will negotiate with each other and pursue strategies (in particular form strategic alliances among themselves) to enhance their relative bargaining power. Thus, applying this framework to the specific instance of a supply chain as described above, the structure of the supply chain is predicted. The essay demonstrates analytically the following: (i) the profit o f each player is a function of his negotiating power relative to the negotiating power o f the other players, the negotiation sequence and the coalitional structure (ii) in general when the assembler is weak (low negotiating power) it is in the best interest of the suppliers 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to join forces in one big alliance. In this case the assembler prefers to negotiate with as many suppliers as possible. But when the assembler is powerful it is the interest of the suppliers to stay independent, while the assembler would like to negotiate with a small number of suppliers. Supply chain coordination issues, which have received substantial attention in the operations literature in recent years, arise due to several factors, an important one being the difference in cost impact of inventory and replenishment decisions on the manufacturer and retailer. The second essay titled: “Coordination issues is a vendor managed inventory system” addresses this asymmetry in cost impact that can lead to sub-optimal channel performance. This essay considers a system where a manufacturer supplies a single product to a retailer who faces random demand. The retailer incurs a fixed cost per order, inventory holding cost and a penalty cost for a stockout (demand is back-ordered). Further, the manufacturer incurs a penalty when there is a stockout at the retailer and a fixed replenishment cost. It is assumed that the players are rational and act non-cooperatively. In traditional retailer-managed inventory systems, the retailer places orders and makes replenishment decisions that have differential impact on the costs of the retailer and manufacturer. This essay proposes alternative systems based on the emerging trend towards vendor-managed inventory systems, wherein the vendor or manufacturer makes inventory and replenishment decisions. The retailer 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. proposes a contract to the manufacturer that influences the manufacturer’s replenishment decisions and thus improves system performance. We consider a system where the decision-maker reviews inventory “continuously” as well as one where inventory is reviewed “periodically” Other than showing that some simple contracts in a vendor-managed framework can improve system performance under certain conditions, in this essay, certain other contracts are proposed that result in channel coordination and are first- best. The final essay (chapter 4) titled: “Issues in stocking substitutable products” extends the existing literature in operations management that deal with the problem of stocking substitutable products. In particular, this essay examines the nature of inventory policies in a system where a decision maker stocks two substitutable products, say A and B, whose demands are negatively correlated. Substitution is modeled by allowing a fixed proportion of the unsatisfied customers for A to purchase item B, if it is available in inventory. First, the essay characterizes the optimal inventory policy for items A and B in the single-period case. In fact, under certain conditions, the optimal inventory levels of the two items can be computed easily and follow what is referred to as "decoupled" policies, i.e. the optimal base stock levels are independent. Furthermore, the essay demonstrates that such a decoupled base stock policy is optimal even in a multi period version of this problem for a wide range o f parameter values. Second, the 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. essay analytically identifies several anomalies in the behavior of optimal inventory levels. For instance, under certain conditions, optimal inventory levels may decline when the proportion of customers who are willing to substitute decreases, negating the typical benefits of risk pooling. Also, optimal inventory levels may decline with an increase in demand variance, primarily due to the effects of negative correlation. Finally, the essay examines a competitive and dynamic version of the substitution problem with and without correlated demands in a multi-period setting. Here, a model in which two decision makers compete, each of whom face a demand over multiple periods with a fixed proportion of unsatisfied customers willing to substitute, by setting inventory levels in each period. The essay concludes by characterizing some properties of the Nash equilibrium for this problem. Since each chapter containing the essays is self-contained, all references are at the end of each essay, and relevant proofs are either in the main body of the essay or enclosed in a technical appendix at the end of the respective chapter. Since the thesis does not have one central theme on which an elaboration is performed, there are no general conclusive remarks, except for those in each essay. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. A Bargaining Framework In Supply Chains 2 . 1. Introduction Consider a supply chain in which N suppliers who each sells a component to an assembler facing random demand for the assembled (final) product. The final product requires a single unit from each of the suppliers (the components are complementary). The assembler must purchase the required components from the suppliers, assemble them in expectation of the demand and sell the final product to meet demand at the end of a single period. The parameters of the purchase contracts are negotiated between the assembler and each o f the suppliers. In this paper we study this decentralized supply chain in which players make decisions that maximize their individual utilities without taking into account the performance of the entire supply chain. Our motivation to develop a negotiation framework is manifold. First, determining the parameters of a contract and as a result the allocation of the supply chain’s profit among its players is a very important issue in supply chain management. Indeed, much of the recent work on supply chain contracting has focused on addressing this very topic. Negotiation, auctions, take-it-or-leave-it 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. offers (which can be thought of as an extreme example o f negotiation) are some mechanisms that precisely achieve this objective. Second, is the important and interesting issue of the negotiation power o f individual firms and the strategies they pursue to enhance their relative clout in a supply chain. For instance, it is generally believed that when several players join forces by forming strategic alliances, their negotiation power increases. This can affect the prices (for instance) and the profit allocations. Thus the negotiation power of agents may have several crucial ramifications, notably on (i) the profit allocations; (ii) the structure o f the supply chain (the kinds of coalitions that players may form) (ii) the contracting mechanism. Any contracting mechanism that determines the prices (e.g. wholesale, buyback, revenue sharing etc.) achieves an allocation o f the total system profit (pie) among the players. We are interested in designing a negotiation framework through which we can determine this exact allocation o f the total system profit that each player receives. This immediately allows us to determine the exact parameters of any contracting mechanism that the players might choose. Our use of a negotiation approach to understand certain contracting and profit sharing issues is not farfetched. Indeed, anecdotal evidence and articles in the academic literature have overwhelmingly indicated that contractual relationships between agents in a supply chain are characterized by negotiating over terms of the trade. Sellers and buyers often negotiate price, quantity, delivery schedules etc. Bajary et. al. [2002] argue that in the construction industry 43% of all projects are 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. awarded using negotiation while the rest use some form o f auction. They find that supplier reputation, complexity of the project and the absence of a large supplier base, are positive determinants for negotiations to be preferred over other mechanisms. Bonaccorsi et. al. [2000] claim that in medical procurement, negotiation is often the norm when quality is uncertain. The Taiwanese semi conductor industry association (TSIA, www.tsia.org) reports that over a third of all contractual terms between its members and OEM’s are negotiated. Helmberger and Hoos [1965] provide a seminal analysis of negotiations in fruit and vegetable markets. Iskow and Sexton [1992], Worley etal. [2000] etc. are other examples of recent research that have largely described existing bargaining institutions and contracting in commodity sectors. Other examples include timber procurement (Elyakime etal.[2000]), automobile industry and retail (****). Thus adopting a negotiation framework to examine profit allocations seems very natural. In the above paragraph, we have argued using several examples that negotiation is commonplace in supply chains. We now illustrate using several examples, the link between negotiations and the structure of the supply chain. Indeed, there has been a profound increase in strategic alliances, formations of coalitions, joint ventures etc. among different entities in various supply chains. Greene [2002] indicates several instances of alliance formation in the semiconductor industry. Reasons for such alliances are manifold. Some o f these include capacity sharing, savings due to economies o f scales, increased competitiveness and other strategic reasons. In addition, another important reason 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for firms to form strategic alliances is to improve their relative negotiating position. Examples include alliances between SMIC and IMEC in the chip manufacturing sector, Asyst and Shinko in equipment manufacturing etc. The number of alliance partners, vary from as small as two, to as large as eleven. Hueth and Marcout [2003] discuss the issue of cooperatives as bargaining institutions in several U.S. agricultural markets. Stallkamp [2001] discusses alliances formations among auto part suppliers with the view of increasing supplier power. In his article, he mentions a recent move, in which, Delphi and Lear (interior trim manufacturers) have independently displayed interest in forming strategic alliances with suppliers of wiring, carpets and molded plastic with the aim of being major cockpit suppliers to the OEMs (in this case, one of the big three auto manufacturers). Zettelmeyer etal. [ 2003] explain the emergence of internet auto referral platforms as a proxy for effective collective negotiations by customers. Oftentimes, these alliances are dynamic and witness splits and reformations. Adopting a negotiation framework allows us to make important observations on the structure o f the supply chain viz. the alliances and their dynamics as a function of the negotiating power of its’ players. Thus, using this framework, we are interested in addressing the following two issues: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. How are the parameters o f a contract (and thus the profit allocations o f the players) determined and how are they affected by the relative negotiating power of the different players? 2. If players can pursue strategies to enhance their negotiating positions (we focus on the strategy that players can freely form alliances), how would these strategies affect the profit allocations and the very structure of the supply chain? We provide a brief review of topics that bear relevance to our paper and its objectives. In this process we wish to highlight our relative contribution to the existing supply chain management literature. The literature on supply chain contracting and profit sharing is vast. It is not our intention to provide a comprehensive review of these studies. For an excellent review, please refer to Cachon [2002]. Papers by Pasternack [1985], Cachon and Lariviere [2001], Bames-Schuster et. al. [2002], Gerchak and Wang [2000], Padmanabhan and Png [1997], Deneckere and Peck [1997] are some examples. In almost all o f these papers, though the mechanisms and structure of the supply chains differ, there are certain commonalities. Typically, the agents function non-cooperatively and maximize their own expected profits. These studies identify various mechanisms that coordinate the channel. In addition, some of these papers have made the observation that the channel can be coordinated in many different ways. That is, the mechanism is flexible in that it allows the parameters to be set to allow arbitrary allocations of the first best profit among the players. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Indeed, it was neither the intent nor was it within the scope of the above models to elaborate on the question of how the channel profits are allocated among the agents. In many of these studies, a Stackelberg game is used to model the supply chain. The Stackelberg game is a simple form of negotiation in which an agent (leader) makes a credible take-it-or-leave-it offer to the follower. The follower can either accept or reject the offer. The agent who is the leader (in some cases the follower) usually enjoys the first mover advantage, thereby having control over extracting shares of profits. The Stackelberg game may be an appropriate model in many situations. Such situations implicitly assume that the follower has no negotiating power except to veto a proposal offered by the leader. At the same time, it is natural to expect environments in which the Stackelberg game is not very realistic. In such environments, there may not be any player who is all-powerful or powerless. Indeed, one could argue that negotiating power is not as discrete or extreme as dictated by a Stackelberg game. It may be more desirable to have negotiating power as a continuous property. This would imply that shares that players receive are not “all or nothing” but is a function of the players’ negotiating power. We now review some papers that examine the “structure” of a supply chain as pertaining to formation of strategic alliances. In the operations literature, Corbett and Karmarkar [2001] study the effects of competition in serial supply chains. They provide a framework for studying a variety of supply chain structures and 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the effect of cost structures and entrants at different levels o f the chain. In a more recent work Carr and Karmarkar [2003] look at competition in a multi-echelon supply chain with an assembly structure. Their focus is on establishing equilibrium prices, given an assembly structure. Granot and Sosic [2001] predict the formation of stable alliances in an electronic market place under price competition. They use the idea of “farsightedness”, a dynamic concept of coalition formation. In a recent work, Majumder and Srinivasan [2003] study a serial supply chain in which different players adopt the role of a Stackelberg leader. They focus on determining the equilibrium prices and on the optimal location of the Stackelberg leader(s). All these papers focus on price competition in a deterministic environment. Negotiation between the agents, or their relative negotiation powers and its impact on the parameters of the contracts is not the main objective. Indeed as we mentioned, when the Stackelberg game is used (as in some of the above papers), the leader is exogenously endowed with complete control. With the exception of Granot and Sosic [2001] these papers do not examine the issue of coalitional stability. Our approach is completely different. In our model, agents form coalitions to enhance their negotiating positions. Thus we predict stable supply chain structure and the profit allocation o f each player as a function of the negotiating powers of the individual players. In addition, we allow for players to be farsighted in forming alliances, thus capturing the dynamics of alliance formations that may be seen in practice. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The operations management literature has seen very little work that uses economic bargaining theory to model relations between agents in a supply chain. Notable among these are Gumani and Shi [2000], Ertogral and Wu [2001]. Plambeck and Taylor [2002], consider the effects of re-negotiation on contracts. Consequently, one of our motivations is to develop a general negotiation framework and to examine profit/cost allocations as a function of negotiating power in various supply chains. Thus, to summarize, our main contributions are as follows: We develop a framework to model negotiations between agents. This framework allows us to explicitly compute the resulting prices and the profit allocations of each player as a function o f his relative negotiating power. We assume that information is symmetric and that all players are rational. We allow the players to pursue strategies that may possibly enhance their negotiating power. In particular we assume that players may form coalitions among themselves to enhance their negotiating positions. In the process of forming alliances, we assume that the players are “farsighted”. We predict the stable alliances that players will form. We show the effect of negotiation power and transaction costs on these stable alliances. Thus, using our negotiation framework, we predict the final structure of the supply chain as a function o f the negotiating power o f the various players. The paper is organized as follows: In the first part of section 2 we motivate the basic negotiation process and describe the Nash bargaining (referred to in the future as NB) solution. In the second part we describe the Nash bargaining game 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as it applies to our problem. In section 3, we describe how we model negotiating power in this paper using commitment tactics. In section 4 we describe the basic A-B-P and provide some basic results. In section 5, we extend the A-B-P to a situation in which the players use commitment tactics to enhance their negotiation power. In section 6, we first discuss the concept of farsighted coalitional stability and then characterize the stable coalitional structures that suppliers will form. At the end of section 6, we examine the issue of costs incurred by the agents due to multiple negotiations and its impact on the stable outcomes. Future research and conclusions are in section 7. All relevant proofs are attached in the appendix. 2 .2 The Negotiation Process To put things into perspective, we will now look at the issues that arise when we try to establish a negotiation framework for the assembly problem. Let us assume that the shares of the “profit” that the suppliers and the assembler receive are calculated through some sort of negotiations between the parties. We then have the first issue: How do we model negotiations between agents? To do so, we first need to describe an “atomic” negotiation process in which two players negotiate to determine their allocation of the profit (pie). Once we have specified this atomic model (note that it may be independent o f the specific supply chain in question) to capture the negotiation process, we turn our attention to the issues that relate specifically to the assembly problem. There are at least two ways in which the assembler can negotiate with the N suppliers. The assembler and all of the N suppliers could sit together and simultaneously negotiate on the profit 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. allocations or alternatively, the assembler could negotiate with the suppliers sequentially. Either style of negotiation is plausible. In this paper, we assume that the assembler negotiates sequentially with the N suppliers. In section 4, we describe these negotiations and the issues that arise thereof. To model the atomic negotiation process, we use the cooperative bargaining process initiated by Nash [1951]. In the next section, for the sake of completeness, we detail the structure of the Nash bargaining problem where its theoretical elegance will become clear. Moreover, at this point, it is pertinent to mention that experimental bargaining theory indicates stronger empirical evidence of Nash’s bargaining theory than any other. For an excellent review, we refer the reader to Roth [1995]. 2.2.1 Nash Bargaining Solution Negotiations between agents in a supply chain can be quite complex. We do not presume to model these exact processes or their specific outcomes. Often, we take a diluted view of reality by making several simplifying assumptions. We borrow and modify existing economic bargaining theory, mainly utilizing the Nash bargaining (NB) concept as the building block o f a general negotiation framework We now turn our attention to the formulation of the model. Nash engaged in an axiomatic derivation of the bargaining solution. The solution refers to the resulting payoff allocation that each of the participants unanimously agrees upon. The axiomatic approach requires that the resulting solution should possess a list of 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. properties. The axioms do not reflect the rationale of the agents or the process in which an agreement is reached but only attempts to put restrictions on the resulting solution. Further, the axioms do not influence the properties of the feasible set. Before listing the axioms, we will now describe the construction of the feasible set o f outcomes. Formally, Nash defined a two-person bargaining problem as consisting of a pair (F,d) where F is a closed convex subset of R 2 an dd = {dx,d2) is a vector in R 2. □ is the pie to be allocated between the players and it may represent profit or revenues. The set F is constructed from the set A = { ( a j ^ ) ^ +&2 < FI} in the following natural way: F = {(ul(al),u2(a2))>d;(al,a2)e A}, where u, • is the utility of player i. F is convex, closed, non-empty and bounded. Here, F , the feasible set, represents the set of all feasible utility allocations of the fixed pie I I , and d represents the disagreement payoff allocation or the disagreement point. The disagreement point may capture the utility o f the opportunity profit. Thus, we are looking for an allocation of the fixed pie in which the two players together do not get more than the entire pie. Further, each player must get at least his/her utility of the opportunity or actual costs. Hence, the problem o f developing a theory of negotiation may be considered to be a problem of finding an appropriate solution function < f > from the set of all two person bargaining problems into R 2, the set of payoff allocations. At this stage we introduce some more notations. Let any 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x e R 2 be x = (x^xj) . Thus ® (F, d) = (d>, (F , d), 0 2(F, d)) . Also let, for any x ,y e R 2,x > y < = > x ,- > y ;;z = 1,2. Nash looked for a bargaining solution d>(F, d) that satisfied a set of axioms. The axioms ensure that the solution is symmetric, i.e. identical players receive identical utility allocations, feasible i.e. the sum of the allocations do not exceed the total pie, Pareto optimal, it is impossible for both players to improve their utilities over the bargaining solutions, the solution be preserved under linear transformations and be independent of “irrelevant” alternatives (i.e. if F2 c i 7 , andd)(Fl5x) € F2 then < X > (F j,x) = <D(F2,x )). To understand this axiom, let the bargaining solution of <Fx,d > be (u(ax),u(a2)) e F2. Then, the bargaining solution of <F2,d >is also (u(ax),u(a2)) . Thus the points in Fx \F2 are irrelevant to the bargaining process. Due to constraints on space, we refer the reader to Roth [1979], for a very good description of the solution approach and a more detailed explanation of the axioms. The remarkable result due to Nash is that there is a bargaining solution that satisfies the above axioms and it is unique! Theorem (J. Nash, 1951): There is a unique solution function <D(.,.) that satisfies all the “ Axioms This solution function satisfies, fo r every two person bargaining game (F,d), 0(F,d) e argmax (x, - dx )(x2 -d f) x e F ,x ^ d 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.2 It is useful to note that the solution maximizes the geometric mean of the gains that the players obtain by reaching an agreement instead o f disagreeing. In the above discussion, we assumed that the players bargain on the profits. In doing so, we have tacitly assumed that the disagreement points reflect the opportunity costs of the players. Other interpretations are possible. This issue will be discussed shortly. The axiomatic approach, though simple, can be used as a building block to discuss more complex bargaining problems. However, this approach fails to capture the negotiation process of players making offers and counter-offers. This leads one to wonder whether the Nash bargaining solution, although mathematically elegant and simple, fails to accurately model real life negotiation scenarios. Fortunately, Rubinstein [1982] has proved that non-cooperative models in which players make alternating offers, when parameters are assigned appropriate values yields results identical to the Nash bargaining solution concept. This seminal result justifies our approach as we resort to the NB concept and greatly benefit from its inherent simplicity. Once again, in our discussion we have only provided a description of the bargaining problem and its’ solution between two players. However, this result can easily be generalized to any number of players simultaneously negotiating for their share of the fixed pie. The Nash bargaining problem in our setting. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We start by building the basic negotiation model in which a risk neutral supplier and a risk neutral assembler negotiate the profit allocations. The assembler faces a random demand the density of which is / (s ). He has to place an order of Q units in anticipation of the demand. He incurs purchasing, holding (interpreted as cost o f disposing unsold inventory), and shortage costs, and he gains revenue proportional to the quantity he sells. We will denote his revenue per unit as s , the other costs per unit as w, h and v respectively. The supplier faces a production cost per unit, denoted as c and sells to the assembler at a wholesale price w. Denote the profit of the assembler and the supplier as p A (w, Q, e) and ps(w,Q,s) respectively and E(pA(w ,Q ,s)) = 7tA (w,Q) and E(pg(w,Q,s)) = 7ts (w,Q) where s is the demand realization. Further, denote the expected profit obtained by assembling Q units, of the corresponding centralized system as T ^ Q ). Recall that the NB game requires us to identify a feasible set of payoffs and a disagreement point. The economic theory of bargaining intuits from the problem of dividing a fixed pie, with the players having fixed disagreements. The payoffs to the players are the utilities of their pie allocation. The size of the pie and the disagreement payoffs are pre-determined and are independent of the negotiations. To make sure that this is the case in our problem we assume that the players negotiate on the share of the expected profits. Their disagreement points can be 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. interpreted as their expected opportunity profits. This ensures that the size of the pie is fixed and is independent of demand realizations. Let us first suppose that the supplier and assembler negotiate on (w, Q ). The feasible set can then be written as: Q = {(7ts(w,Q)),7tA(w ,Q )): (w,Q) e T} for some suitable compact set T cz R 2 . The set T is assumed to be a rectangular set. This will be the case when both the wholesale price and the purchasing quantity are allowed to take values in a closed interval in the positive real axis. Note that Q is a convex set. Note also that in the above formulation, the feasible set is the set of feasible pairs of expected profit allocations of the total pie. Thus, any pair (w, Q) gives a unique allocation of T'(Q ). Thus, negotiating on (w, Q) can be thought of negotiating on shares o f a pie of size T ^ Q ). Risk neutrality o f the players implies that the shares obtained as fractions of the pie are independent of the size of the actual pie. This follows directly from the NB solution. Thus, we can assume that negotiations are conducted to determine the fraction of a pie, the fraction being independent of the actual pie itself. Note that one of the axioms o f the NB solution was the requirement of Pareto optimality. This ensures that when the players are risk neutral the negotiated quantity is always Q c (where Q c is the channel coordinating quantity), thus maximizing the size of X P (Q ). It is important to note that this is not an assumption but an outcome of using the NB concept together with the assumption that the players are risk neutral. Indeed, this result is extremely 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. intuitive since the players cooperatively negotiate on the share (which for risk neutral players are proportional to the size) of the pie and thus it is mutually beneficial to maximize the expected profits o f the channel. Thus, from hereon, we assume that the channel is coordinated and need only consider Q = Qc . Thus far, we have only examined wholesale price contracts. From the preceding discussion, o it is clear that players are negotiating on fractions of T (Q ). The NB solution determines the resulting fractions. Once the fractions are determined, it possible to determine the parameters o f any contracting mechanism. Thus, with a slight abuse of notation, we can rewrite the feasible set as Q = { ( 7 i A , 7 i s ) : + T tjg = X F(QC)} • This is the approach we take in the rest of this paper. As the situation demands, we will carefully define the pie in question. This approach, echoes the sentiments of Brandenburg and Nalebuff, who in their popular book Co-opetition, introduce the concept of “Value Net” in which supply chain players are partners in creating value for the supply chain, while the same players compete for their shares o f he supply chain profit. They stress the value of a cooperative symbiosis between the various agents in a supply chain. Before concluding this section, we point out a couple of assumptions that we follow in the rest of the paper. First, we assume that the parameters o f our problems are such that that the channel makes positive expected profit. Further, we assume that the expected profit is larger than the sum of the disagreement points (opportunity profits) of the players. This ensures that the feasible set of the 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bargaining problem is not empty. Thus as the players are rational, they will find it advantageous to agree on some allocation of the pie and will never choose the disagreement points. Second, we assume that players are risk neutral. Characterizing the Nash bargaining solution for general risk averse utilities is an interesting but a difficult problem. The bargaining solution, in such cases bears a dependence on the contracting mechanism. This important issue is not an objective of this paper. Further, observations to the effect that Q = Q c , are not generally true. For more on these related matters see Nagarajan and Bassok [2003]. Thus, as in the above discussion, in this paper, we will focus on the case of risk neutral players. 2.3. Negotiation Power In this paper, we allude to negotiating power or bargaining power several times. There are two aspects to negotiating power. The first is specific to the economic model that is used to describe the negotiation process. This could translate to specific characteristics that the players may possess (utility of the players, risk preferences, credibility of threats etc.) The second aspect is the negotiating power that arises due to the structure of the supply chain. Examples would be the number of suppliers in the supply chain the timing of their negotiation etc. This aspect is much more complex and we will defer elaboration 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on this second aspect to the very end, when we would have put forth a complete description of the assembly problem and the results thereof. When two players negotiate over a fixed pie (fl(say), with disagreement points (di,d 2)) it is reasonable to expect that the player with the higher negotiation power (the more “powerful” player) receives a larger share of the pie than his weaker counterpart. With this simple underlying idea we can now speak about a situation in which a player is more powerful than his counterpart. First, it is clear that two “similar” players must get equal shares of the pie. This is due to the NB solution requiring symmetry as a necessary condition. Nash intiutes the idea of similarity based on the risk preferences o f the players. In other words, two players who are risk neutral must obtain the same effective profit. Assuming that the two players are risk neutral and applying the Nash bargaining concept results in the two player maximizing the following expression: max (x, - d, )(II - x l - d 2). Taking the derivative with respect to (*i , n _* 3 )>d jqand equating to zero we get: x, = ——^ + — , where xx is the share of the pie that player one obtains. Notice that * is increasing with dx and decreasing with d2. We can also calculate the “effective profit/share” (defined as the profit over . n-d, + d, , n - d 7 — d, the disagreement point) of player one as nx ---------^--- - ~ d x = — —^ Similarly we can calculate the effective profit o f the second player. We can immediately see that the effective profits of the two players are equal. Thus we 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. observe that risk neutral players, despite having unequal disagreement points, have the same effective profit. Thus, risk-neutral players in a Nash bargaining game are equally powerful. Indeed, this result is strongly supported by empirical evidence. See for instance Raiffa [****] and Stone [1958]. Risk Aversion and Negotiating Power: The behavior of the solution changes, however, when the players exhibit different preferences (or equivalently, when they have different utilities). Using measures of comparative risk aversion as suggested by Arrow [1965] [1971], Pratt [1964] and Yaari [1969], it can be shown (Khilstrom, Roth, Schmeidler [1979]) that in a two person bargaining problem, the Nash bargaining solution assigns a player increasing utilities as his opponent becomes more risk averse. Thus a player’s negotiation power increases, as his opponent becomes more risk averse. Commitments and Negotiating Power: Risk preference is not the only way to capture negotiating power. Risk aversion makes it very hard, if not impossible, to obtain closed form expressions for allocations of profits. Further, as mentioned before, the use of risk aversion forces the discussion to be contract specific. This distracts us from the objective o f getting tractable expressions and structural results, which makes it imperative to preserve the linear structure offered by risk neutrality. Thus we adopt the idea of commitment tactics. Though both players are risk-neutral, additional considerations ensure that one player can be viewed as being more powerful than the other. The main idea in such models is that before embarking on some 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. negotiation process (Nash bargaining, in our case), players take actions that partially commit them to certain bargaining positions. The meaning of the commitment is that a player is unwilling to accept a share smaller than the announced commitment. It is important to note that the commitment is not directly related to the disagreement point. However, it is evident that each of the players will commit to a share that is not smaller than their disagreement points. Actually, it seems natural to expect the players to inflate their commitments and thus increase their share of the pie. But, in doing so the players need to exercise some caution, because, if the sum o f the commitments is larger than the pie, and if the negotiation process is to have a non-trivial solution, then, at least one o f the players must revoke his commitment. Revoking such a partial commitment is costly. It could, for instance be attributed to or construed as a loss of credibility. After such partial commitments have been made, the players engage in the Nash bargaining process to strike a deal and arrive at their allocations. Simultaneously they try to minimize the extent to which they must revoke their commitments. Such commitment tactics are not unusual and often signal a player's negotiation power. Bacharach and Lawler [1981] claim that commitment tactics are an oft-used tool in real life negotiation situations. Schelling [1960] provides a detailed discussion on commitments and their role in negotiations. Cutcher- Gershenfeld et. al. [1995] point out that in negotiations involving industrial relations, commitments are very common. Indeed, readers familiar with negotiations in a bazaar would immediately recognize that threats of walkouts if a 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. certain agreement is not reached are examples of commitments made, and later, possibly revoked. In the political arena, we find politicians hinting that accepting certain solutions would spell the end of their political careers (and indeed sometimes their lives!). Such statements aim to signal the high credibility of their commitments. Muthoo [1996], models the partial commitments as players simultaneously choosing an allocation o f the pie. Thus the players take actions that partially commit them to accepting no less a share than the previously chosen allocation. Such partial commitments can be revoked at a cost proportional to the extent revoked. Thus this game comprises o f two sub-stages. The first sub-stage involves an independent announcement of the commitments (modeled as a competitive game). The second sub-stage is a Nash bargaining game. Let n be the pie, the allocation of which the players are negotiating. Let zi z ' = l,2be the independent choices of the two players. Thus, players commit to receiving no less than these choices. However, since these commitments are partial, let Ui(xi,zi) = xi - C i{xi,zi) be the actual utility to player-i from receiving a share xt of the pie and making a commitment to receive no less than z; . This utility, as seen from the above expression, is equal to the actual allocation the player receives discounted by the cost of revoking his commitment. Muthoo, assumes a linear cost, given by: 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 + k 1 + k Muthoo proves that(z,,z?) = (--------- 1 — H ,- -----— — FI)is the unique 2 + kx+k2 2 + kx + k2 Nash equilibrium of players’ commitments. Thus a player’s partial commitment, which is identical to the equilibrium share of the pie that he obtains, is increasing in his cost o f revoking a commitment and decreasing in the corresponding cost of his opponent. In other words, a player whose revoking costs are high is considered more credible when he makes a commitment and consequently is more powerful and is awarded a larger share of the pie. It is interesting to note the relationship that this commitment game bears to the generalized Nash bargaining solution. If we ignore the axiom that requires the Nash bargaining solution to be symmetric, the other axioms determine a whole set o f solutions o f the form: <&(F,d) e argmax (x, - d 1)a(x2 - d 2)p with or + /? = l. X€.F,X2Ld The indices are loosely representative of the individual powers of the agents. However, given commitment-revoking costs as described above, it can be proved that the Nash equilibrium pair of partial commitments is exactly the solution of the generalized bargaining game with a = ---------! — . This result will be useful in 2 * ” i — kx - I - k2 understanding the source of negotiating power. Further, it will play an important role in the discussion of coalitions among suppliers. 2.4. The Single assembler N suppliers Assembly Bargaining Problem (A-B-P): 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Having described the two-person bargaining game, we are now in a position to describe the general assembly problem in which there are n suppliers. First, we denote the centralized optimal purchasing quantity by Qc and the centralized revenue by FT. These can be obtained by solving a suitable newsboy problem. We begin by considering a single-period problem in which all players are risk neutral. In other words, we are not exogenously empowering any of the players through their risk preferences. As mentioned, we assume that all information is symmetric and the players are rational. We assume that each of the suppliers faces an individual per-unit production cost c; . Hence, his profit function is simply 7t§(w,Q) = Q(w - q ) = Rj(w ) - q Q , where Rt represents the supplier’s revenue. Since the selling quantity is Qc , each supplier faces a total cost o f q(9C ~ ci■ The assembler faces a unit assembly cost cA and a holding (cost associated with disposal or salvage) and shortages cost L(QC) and sells the final products for s . More precisely, the expected profit of the assembler is: Q XA (w> Q) = ~W -Q-caQ+ j W - KQ - £)]f(e)de + j[s.Q - v(Q - s)~\f{s)ds o Q Q so = -w.Q - c a Q + Js.ef(e)ds + js.Qf (s)ds + L(Q) 0 Q 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Q = QC. For the sake o f simplicity, we assume that the salvage values of the final product and the components are all equal to zero. It is trivial to extend the results for any salvage value. We show the relationship between the negotiated solution and the production and assembly costs. Recall that we assume that the players negotiate on the share of the expected profit and their disagreement points reflect their opportunity profits. Alternatively, we assume that the suppliers’ disagreement points are equal to ct and that the assembler’s disagreement point is equal to ca Qc +L(Qc) denoted by cA . This interpretation of the disagreement points forces us to assume that the players negotiate on the total expected revenues rather than expected profits. In the spirit o f our discussion in section 2.2, the discussion can be extended to include contracting mechanisms other than a pure wholesale price contract. In such cases, the “revenue” needs to carefully defined. Note that any contract achieves an allocation of the pie that is negotiated. The actual contracting mechanism is irrelevant. The only reason we include w in the above formulation is to make clear the revenue terms and the costs. The disagreement points ensure that the players’ allocated revenues are larger than their costs. Since the players are risk neutral, the two representations are identical and result in exactly the same effective profit allocations. In the original formulation of the bargaining problem, Nash looks at the problem o f allocating an 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. abstract pie, without worrying about its exact interpretation (i.e profit pie = revenue - costs etc.). Because o f risk neutrality, our interpretations are consistent. The following observation will clarify this point: Observation: (51): Let the two players bargain over a profit pie I I , with disagreement points being (d1 ,d 2) . The profit allocations are given by: argmax (x, - dx )(TI -Xj - d 2). (52): Let FI = Yl+Cx + C2, where (^represent the cost incurred by player i. Let the two players bargain over the revenue pie II • The disagreement points now include the costs incurred as well as d,-.Let C = (C ,,C 2). Their revenue allocations are given by: argmax (x, - d x- C x)(IT -xx- d 2 - C 2) . The ( x l , f [ - x i )3.d+C effective profit allocations of the players across (SI) and (S2) are the same. The only difficulty of interpreting the disagreement points as costs is that if the players choose not to play, they are still allocated their costs and not a zero revenue. But, as we mentioned before, the expected channel profits are positive. This means that the channel revenues are larger than the cost and the feasible set is not empty. Thus, rational players will always choose to participate in the game. Our model is a three-stage game. The indices o f the stages in the game are reversed. Thus in stage 1 (the last stage), the assembler negotiates with the different suppliers or coalitions of suppliers. These coalitions were formed in an 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. earlier stage (to be discussed shortly). In stage 2, suppliers (or the coalitions of suppliers) compete for positions in the negotiation sequence. In stage 3 (the first stage), the suppliers form coalitions. Since all information is symmetric and perfect, the suppliers know the outcome of each stage and will hence first form coalitions, compete for positions in the negotiation sequence and then negotiate with the assembler. For the sake of clarity, in this section, we discuss the exact negotiation process between the assembler and the individual suppliers. The reader, without any loss of generality, may substitute a supplier by a coalition of suppliers. To extend the two-person game to model negotiations across N suppliers and a single assembler, we adopt a sequential framework in which the assembler negotiates with each of the suppliers separately. Despite the apparent length and cumbersomeness of this process, several examples indicate that sequential negotiation is quite prevalent. A recent article (WSJ, FEB. 28, 2003) describes how State health officials meet with drug companies (GlaxoSmithKline, Merck & Co, Pfizer etc.) sequentially to negotiate prices of drugs. Another article (WSJ, Mar, 10, 2003) describes the sequential negotiation approach used by Iberia Airlines, in negotiating with Airbus and Boeing when shopping for jetliners. Unions in the automobile industry and the construction industry routinely engage in pattern bargaining, which is essentially a sequential approach. These examples may not exactly resonate with the model in our paper. However, they do demonstrate that sequential negotiations is quite common place. Furthermore, in 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the A-B-P, it is unlikely that a simultaneous negotiation between all players is a natural choice, as their roles are complementary. Thus, the assembler, first negotiates with supplier 1, and then with supplier 2 etc., and finally with supplier N. A sequential framework raises several interesting issues. If the profit allocations obtained by the bargaining solution is a function o f the negotiation sequence (as one might expect), then it is necessary to decide how the sequence is determined. Although all players have equal negotiation power because of their similar risk preferences (at this point, we will not bring in the issue of commitments, or equivalently we assume all players have an infinite revoking cost), we assume that the assembler has the power to determine the negotiation sequence. In what follows we describe the sequential negotiations (Stage 1) and the issue of determining the negotiation sequence (Stage 2). We formulate Stage 1 of the Assembly-Bargaining-Problem (A-B-P) as an N-sub-stage problem. At the first sub-stage, the assembler negotiates with the first supplier, and at the final sub-stage he negotiates with the N-th supplier. We assume that at the first sub-stage the first supplier gets his share of the profits. The assembler gets the share of the pie that will be further allocated between him and the remaining suppliers. Similarly, the assembler negotiates with the N -l-th supplier for the portion of the pie (the pie itself is carried over from negotiations with the N-2-th supplier) that he will carry over for himself and the N-th supplier 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the portion that he will leave for the N -l-th supplier. Finally at the end of the negotiations with the N-th supplier, the assembler will get his share of the pie. To describe the feasible set in this game, we look at the simple case in which there are two suppliers, indexed 1 and 2, respectively. Given the assumptions and the description of the negotiations, it is natural to define the feasible set for the first sub-stage of the game in the following way: F - {Ra,2’R\} where RA 2 is the revenue to be allocated between the assembler and supplier 2, and Rx is the revenue allocated to supplier 1. Since at the first sub stage the assembler negotiates for the share of the revenues to be allocated between himself and the second supplier, it is natural to assume that the assembler’s disagreement point is (cA + c 2). Having motivated the problem with two suppliers, we will now tackle the N supplier A-B-P. We look at a supply chain in which the assembler creates the product using a component from each of the N suppliers. As mentioned, we now have a game with N-sub-stages. . Let RA be the assembler’s share of the channel revenue. Let R a j be the revenue that the assembler and suppliers i,i + l,...N divide between themselves. Let Rj denote the share of the channel revenue that the i-th supplier receives. Since the NB solution is Pareto optimal, R aj+ i + Ri = Ra,1 • Thus, if Rn is the share of the channel revenue that the N-th supplier receives, we must have Ran =Rn +Ra . Before we write the bargaining problems we will define the 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. feasible sets. At the first sub-stage, define Fj = {(Ra,2»Ri) • ' R a,2 + Ri = FIC}to be the feasible set. Note that the feasible sets are defined recursively with Fj = {(RA,i+i,R i): RA,i+i +R i = RA,i) • At the i-th sub-stage, the i-th bargaining game is constrained by the i-th feasible set. Thus, at the N-th sub-stage, the bargaining problem reduces to: max (Ra - c a))(Rn - c n). ( Ka ,tiN )€ /,w At the N -l-th sub-stage, similarly, we have: max (Ra n - c a - cA ,))(i?A ,_1 - cN_x) . (,R A ,N ’R f l - l And in general at the i-th sub-stage, max (R A,i+l - C A - (ci+l +•■• + cN))(Ri - Ci) (RA ,+ 1 ,Ri)6Fi Solving this recurrence and obtaining the shares of the assembler and supplier, we have the following: Solving this recurrence and obtaining the shares of the assembler and supplier, we have the following: Theorem: (2.4,1) The solution o f the A-B-P with N suppliers has the following properties: 1. The channel is coordinated. The revenue of the assembler and the i-th supplier are respectively: 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. The assembler is indifferent to the sequence of negotiation. 3. The suppliers prefer to bargain earlier in the sequence. 4. A supplier’s profit is completely determined by his position in the sequence and is independent of the other suppliers’ positions. The revenues that the players receive are obtained by solving the above recurrence. To calculate the effective profit, we need to subtract the costs cA and c( . respectively from the above revenues. nc N n - c t - Z c t Indeed, k a =R a - ca = --------p— and n C N n -ct-Ticj , ’ T = R. - c, = ------------: — --— 2' Note that the assembler's revenue and hence profit is unaffected by the sequence. Moreover, it is also clear that suppliers’ prefer to negotiate earlier in the sequence because their profits decrease with i. Further, in the absence of a sequencing mechanism, the assembler prefers (perhaps counter intuitively) having fewer suppliers. This is in contrast with the outcome of the second stage of the game, which will be described shortly. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since a supplier will achieve a higher profit by negotiating earlier in the sequence, we can think of the suppliers competing with each other for earlier positions by transferring some of their profits to the assembler through payments. Equivalently, one can think of the suppliers competing for positions by possibly offering discounts on the wholesale price. Call this game, the Assembly Bargaining Problem with competition (A-B-P-C). This is the second stage of our game. We now carefully describe the A-B-P-C game. 2.4.1 The A-B-P-C Game: In the second stage of the game the suppliers compete for a position in the negotiation sequence. At this stage, every supplier knows the relationship between his profit and his position in the negotiation sequence. Since suppliers would prefer to bargain earlier and the assembler is indifferent to the sequence, it is reasonable to expect that the suppliers would compete for positions in the negotiation sequence. We have assumed that the assembler chooses the sequence. Hence to resolve this competition, the suppliers will be willing to make payments to the assembler to obtain a “favorable” position. Payments made for favorable positions could be through discounts on wholesale prices etc. To avoid any confusion between indices, we denote suppliers by i, i = 1,2,...A and the positions in the sequence by j, j = 1,2,...A. The strategy of the i-th supplier is denoted by STt. We define STt = {(p{,p’ 2,...pl N) e R N} , 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where p'j is the payment supplier i makes in order to be in position j. Thus, we assume that every supplier announces a payment vector, whose components are the payments he is willing to pay for each position in the negotiation sequence. Further let R l j be the revenue that the i - th supplier will make by occupying the j - th position in the sequence, prior to making the payment to the assembler. „ N ne - C /1-£ e i +2-'e, Indeed, Rj = -------------- M -------------. Hence, his net profit, after making the 2J payment, will be R l j - p ‘ j - c(. Note that, the supplier who is the last member of N n Q - C A - E c j the negotiation sequence, makes a profit o f -— —J~1 This is the lowest profit that any supplier can possibly incur. Thus, p ^ = 0, Vi. In fact, extending N nQ - ca- Z c j this line of reasoning, if R *; - c; - p j = -— —~ — , it sufficient to restrict J J ^ JN ourselves to the strategy space ST* = {(pl,p^,— Pn) e R N};0 < Pj^Pj- To summarize, in the A-B-P-C game, the N suppliers, by offering the assembler the payment vector ST*, compete with each other for a position in the negotiation sequence. The payment scheme that maximizes the assembler’s profits determines the final position of each supplier in the sequence. We assume perfect and complete information among the players. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem: (2.4.2) The A-B-P-C game has at least one Nash equilibrium. Every Nash equilibrium satisfies the following properties: 1. The channel is coordinated. 2. The supplier in position j makes a payment p j . 3. The suppliers make equal profits irrespective of their position and costs. 4. The assembler’s profit increases with the number o f suppliers. The expressions for the net profits of the suppliers and the assembler at this unique equilibrium are as follows: Profit of the assembler n ‘‘ - c t - t c , =(IIe - c A - ^ T c k) ( \ - N / 2 N). Profit of each supplier = ----------N...— . k= \ 2 Notice that the ability of the assembler to determine the negotiation sequence forces all the suppliers to accept profits equal to the profits of the last supplier in the negotiation sequence. Thus, the assembler’s ability to determine the negotiation sequence, endows upon him a high level of negotiating power. Do the suppliers have any mechanism to retaliate and improve their profits? The suppliers could form alliances (coalitions) hoping to improve their profits. This issue will be discussed shortly. Would the assembler prefer to negotiate with a small number or a large number of suppliers? 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To answer the last question we assume that the components are divisible. That is, it is possible to substitute a single supplier who produces a kit with M components with M suppliers each producing only a sub-part, without an increase in the total production cost. The above result indicates that at equilibrium, the assembler would prefer to have a large number of suppliers. In fact, if we assume that the components are infinitely divisible, the assembler would then choose to have an infinite number of suppliers, thereby capturing the entire channel profit. Note that this is in contrast to Theorem 4.1, where the assembler preferred fewer suppliers. The difference is due to the fact that in the second stage of the game, the suppliers compete for positions and thereby lose some of their negotiation power, thereby allowing the assembler to capture a larger share of the profits. This result is also driven by the fact that we have not accounted for any kind of negotiation and managing costs that are related to the number o f the suppliers. This is an important issue. We defer our analysis with negotiation costs to section 7, at which time the impact of such costs on the resulting coalitional structures can be fully appreciated. Note that we have not exogenously empowered the assembler with a greater negotiation power than any supplier (we assume that all players are risk neutral and hence have the same negotiation power). It is quite interesting to note that (as will be seen in section 6) when we relax this assumption, the assembler may no longer prefer to have a large number of suppliers. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5. A-B-P with N-Suppliers and Commitment Tactics (A-B-P- COMMT): We now turn our attention to the A-B-P with the additional flexibility of allowing the players to make partial commitments on their allocations. Such commitments, as discussed earlier, are not unusual and often signal the players’ negotiation power. Recall that in the last stage (stage one) the negotiation outcome is determined. We will now describe this stage, with the additional provision for players to make revocable commitments (note that in our discussions in the previous section, the players could have been thought of as having an infinite cost to revoke), for the case of a single supplier and an assembler with per unit revoking costs k s and kA respectively. This stage, as before has N sub-stages. However, each of the N sub-stages is divided into two further sub-steps. In the first sub-step in any sub-stage of this game, the players (the assembler and a supplier (or a coalition)) independently announce commitments z^and z s , respectively. However, these commitments are partial and can be revoked at a certain cost. The second sub-step determines the profits of each player as a function of their partial commitments. Following Muthoo [1996], if NB is used to obtain the result of the second sub-stage, then (working backward), there is a unique Nash equilibrium of the commitments and consequently the payoffs. This unique equilibrium is given by (ZA,Zs) = m ^ A+ds))*a + dA,(nHdA+dsW - a ) + ds), 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where a is the negotiating power of the assembler and is defined as 1 +k a = ---------A — . Thus, at the end of the negotiations, for the general case of N 2 + ks + kA suppliers with revoking costs k*, we can calculate the profit o f the assembler and each supplier, similar to the previous section. We have: Theorem 2.5.1: 1. The assembler is indifferent to the sequence. 2. The suppliers prefer an earlier position. 3. Every supplier prefers to follow weaker suppliers. 4. The channel is coordinated. Note that in the game without commitments, though the results were somewhat similar (Theorem 4.1), the differences are important. As in the previous game, the assembler is indifferent to the negotiation sequence. But unlike before, the profits of the suppliers are determined not only by their position in the negotiation sequence, but also by the set of suppliers negotiating ahead of them in the sequence. Thus an equilibrium analysis, similar to that o f the previous section, where suppliers pay for favorable positions becomes complex. We conjecture that with a more complex strategy space, the resulting Nash equilibrium can be characterized. While developing a strategy space to understand the properties of the Nash equilibria is important, it is beyond the scope of this study. Due to afore stated difficulties, we proceed with the assumption that all suppliers have equal revoking costs. Under this assumption, the negotiation sequence at equilibrium can be obtained using the same analysis as Theorem 4.2. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.6. Analysis of Supplier Coalitions. Finally, we reach the third stage o f the game. Here the suppliers may form coalitions to increase their profits. To structurally validate a coalition of say M suppliers, assume that such a coalition is selling the assembler a kit composed of the M parts. The coalition’s production cost is equal to the sum of the production cost o f its own M members. Examples o f coalitions formed by suppliers/manufacturers of complementary products are not unusual. As we mentioned (in the introduction), this is an emerging trend in the automobile industry, where suppliers seek to increase their negotiation power. In yet another example, Symbol technologies, a leading scanner manufacturer has formed a strategic partnership with Paxar/Monarch, a leader in bar code labeling. Together, they have become a major supplier of kits (a bundle that includes a bar code labeler and a scanner) that are purchased by retailer. Thus in this stage of the game, suppliers producing complementary components may form coalitions among themselves. Consequently, each coalition competes for its position in the negotiation sequence (stage 2) and then negotiates with the assembler (stage 1). We make a few simplifying assumptions. As before, we denote the set of suppliers by {1,2,3...«} = N . We assume (for purposes stated earlier) that every supplier has the same cost of revoking a partial commitment and thus equal negotiation power. We make no assumption on the assembler’s revoking cost. Thus we let the assembler’s negotiation power to be arbitrary (0 < a < 1) and 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. examine cases when he is more as well as less powerful than any one supplier. We assume that the revoking cost of a coalition is the average revoking cost of the coalition’s members. In this case, the revoking cost o f every coalition is exactly the same and is equal to the revoking cost of each of its members. Finally, we assume that the every coalition member gets an equal share of the coalition’s profit. Since the suppliers have identical revoking costs, they thus have the same negotiating power. Thus obtaining equal shares o f the coalition’s profit is consistent with the idea that the members negotiate (modeled using NB) and split the coalition’s profits. Notice that while each coalition has the same negotiation power as any one of the suppliers, forming coalitions alter the structure of supply chain, thus possibly increasing the profits of the individual coalitions and the suppliers thereof. A coalition formation game (in our context) is defined by G = (N,Z,{<;}fe J V ,{->s}S s J V s^ ) , where N is the set o f suppliers, and Z = P(N) is the set o f outcomes. An outcome is a partition of N. For example, the grand coalition g = {l,...,n} is an outcome, u = {!},{2},...{n} is another outcome in which each supplier acts on his own. {<,} are the players’ strong preferences defined on Z . a <i b implies that the profit of player i in outcome b is higher than his profit in outcome a . Symbolically; > na t . Similarly, a <s b implies a <; b for every i e S. In this case we say that a is directly dominated by b . Finally, {-*s} are relations defined on Z that describe possible defections and 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coalition formations. For example, a = {l,2...,n}-»{ 1 2 3 } {1,2,3}, {4,...,«} = b means that the coalition {1,2,3} defects from outcome a, resulting in outcome b. Another example could be a = {1,2,3}, {4,5,6} {7,8...n} -» { 2 j4 j7 } {1,3}, {5,6}, {8,...nj {2,4,7} in which players 2,4 and 7 defect from several existing coalitions to form a new coalition by themselves. The game is played in the following manner. When the game begins, there is a status quo outcome, say a e Z.Thus, the outcome describes the set of coalitions that the suppliers form, and consequently the profit allocation to each supplier at equilibrium. (Note that the profit allocations are determined by the unique Nash equilibrium, resulting from the payments made by each coalition to the assembler.) Now, if a certain subset of suppliers (S), decide to break away and form a coalition, the new status quo becomes b e Z , where (a — > s b ). Thus, outcome b is obtained from outcome a , by a group of suppliers leaving their earlier coalitions and forming the new coalition S , in the outcome b . It is important to note that the other suppliers remain in the same position as they were in outcome a and in this step do not themselves regroup. Thus at each step, we look at the formation of exactly one new coalition. Note that, from the new status quo b , other coalitions may move and so forth, without limit. The only restriction is that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for the game to move from an outcome x to y , it is necessary that there exists J c N, such that (x ~>r y ) . We have assumed that every supplier is equally powerful. The assembler may be more or less powerful than the suppliers. Assuming equal power among the suppliers (and hence equal shares) allows us to borrow results from Section 4, where each supplier pays for a position in the negotiation sequence. There we proved that there is a unique Nash equilibrium in which the profits o f equally powerful suppliers are equal to the profits o f the last supplier in the negotiation sequence. As mentioned, we denote by a , the share of the assembler when he negotiates with any supplier (or a single coalition). Thus if a > 0.5, the assembler is more powerful than any one supplier. Note from the A-B-P-C game described in section 4.2, it is clear that each supplier makes a profit that is equal to the profit o f the last supplier in the negotiation sequence. From here on, we normalize the profits to one. If there is only one coalition (supplier), then the assembler makes a and the coalition makes 1 - a to be allocated equally among the n suppliers. Assume for a moment that the coalitions don’t compete for position. Now, if there are two coalitions, the coalition that is second in the negotiation sequence makes a (l-a ) and the assembler makes l- ( l- a ) - a ( l- a ) = a 2. When the coalitions compete for position, each coalition makes the profit of the coalition that is in the last position in the negotiation sequence. Thus, if there are t coalitions, each coalition makes a ‘~l( l - a ) o f the profit. Now, in this t coalitional structure, if in 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. some coalition there are m suppliers, then each supplier in that coalition makes a M (l-«) m In general, a game in coalitional form requires a payoff function that describes the utility level that each player gets (by belonging to a coalition) in any outcome. In our scenario, from our previous discussions, the payoff structure should be quite clear. Hence, we will not explicitly associate a payoff function, thus simplifying the notation. Having formally defined the game and the payoff structure for any outcome, we now seek to answer the following question. Assume that the suppliers are rational, they can freely form coalitions and all actions are public. In this case, what would be the resulting “STABLE” coalitions? How are these coalitional structures affected by the negotiation power of the players? In answering this question, one needs to examine the various stability concepts available within the framework o f games in coalitional form. In this paper, we will not do this analysis. For a detailed survey o f this topic, please refer Greenberg [1994], We will now perform a detailed analysis on the outcomes of the third stage, using the farsighted coalitional stability proposed by Chwe [1993]. The farsighted solution concept is WEAKER and thus richer than almost all other stability concepts. Thus far, most of the existing concepts do not satisfactorily consider situations in which a coalition contemplates the possibility that once it acts, a 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. second coalition may react, then a third coalition may react and so on, eventually benefiting all the involved players. Due to this reason, in our problem, most other existing concepts (the Bargaining set, the coalitional set core etc) predict supplier coalition outcomes that are not interesting. Oftentimes, using these concepts yields the set o f stable outcomes as empty or uniquely the grand coalition. Indeed in reality, coalitional structures are much richer. Thus we analyze stage 3, using the farsighted concept, which considers these types of dynamics. Thus from here on, when we speak of stability, we imply farsighted stability. In the ensuing discussion, instead of motivating Chwe’s concept through an elaborate discussion, we will illustrate it by way of adopting it to our problem. We begin by considering the game G as described above, with a few additional notations and definitions. An outcome a is indirectly dominated by b(a <b), if there exists sequences a = a0,ax,a2,...an = b and Sv S2,...Snsuch that,(aj -»g. ^ aj+i) and a^ <j b,Vj e Sj+i , where i = 0,...n - 1 . Note that the final outcome b is better than its status quo position, for every deviating coalition. Suppose the status quo is a0 and Sx (i.e. every member) prefers an outcome a 2 over a0. Further suppose that Sx cannot move to a2. However, say 5) can move from a0 to an outcome ax and another coalition S2 can move from ax to a2. Further suppose that the members of the coalition S2 actually prefer a2 to ax. Then 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. coalition S x may actually move from a0 to ax (even though its members do not directly benefit from this move) anticipating that the final outcome is a2. Chwe motivates the stability o f an outcome as follows: Consider an outcome a. Suppose that a coalition S deviates, and the new status quo outcome is d . Two things can happen. There may be no further deviations thus the ending outcome e = d , or there could be several further deviations and which end up at e where d < e. In either case, the outcome e should be stable. Now if some member of the coalition S does not prefer the outcome e to the original outcome a , then the above deviation by 5 is deterred. An outcome is stable if every deviation is deterred. Since the stability of the outcome is dependent on the stability of other outcomes, the following consistency condition is satisfied by any set of stable outcomes. DEFINITION. A set Y c Z is consistent if a e Y if and only if W , S such that a — d , 3 eeY , where d = e or d < e, such that NOT a <s e. Thus if an outcome is in a consistent set it is possible that it will be stable. On the other hand, if an outcome is in no consistent set, it cannot possibly be stable. Thus a sufficient condition for an outcome a to be stable can be easily obtained by letting Y = {a} in the previous definition. Chwe proves that when the set of all possible outcomes are finite, a stable outcome is guaranteed to exist. Thus, for our problem, we are guaranteed the existence of certain “farsighted” stable outcomes. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The operations literature has predominantly analyzed coalition structures using the Core, ignoring the issue of farsighted stability. A notable exception is Granot and Sosic (2001) who use the “farsighted idea” to analyze stable structures in electronic market place with three players. Before we state the results on stable coalitions, note that farsightedness does not imply that the dynamics illustrated in the previous examples are meant to reflect and explain the exact dynamics o f splits and mergers o f firms in the real world. The spirit of the farsighted stable concept lies in the fact that decision makers may account for long term trade-offs when forming coalitions. In the following results, we characterize how the stmcture o f stable coalitions changes as the assembler’s power a increases. We begin by characterizing a situation when the grand coalition g is uniquely stable. Lemma 2.6.1: When □< 1/n the grand coalition g is the only stable outcome. The above lemma indicates that the grand coalition of suppliers is uniquely stable when the assembler is weak. This fits well with commonly held intuition that joining forces results in increased negotiation power. The next two results characterize some stable coalitions for a < 0.5. In particular, we observe that g is always stable in this range o f a . Notice the evolution o f other stable outcomes with two unequal coalitions. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 2.6.2: Let n be even and n> 3. Further, let ~ < a < — . Then, the n n grand coalition g and the outcome ax = {1}, {2,...«} are stable outcomes. This immediately implies that when facing a more powerful assembler than in the previous instance, the suppliers may form stable outcomes in which there is more than one coalition. Thus, suppliers in this case may actually benefit by leaving the grand coalition. Further, the notion that joining forces makes for greater negotiation power is no longer completely tenable! 1 l One can show that in the interval a e (—,------ ) , no coalitional structures n n - 1 with three or more coalitions can be stable. For a < 0.5 we can make additional assumptions on the range o f a and demonstrate the stability of certain other coalitional structures. However, for large values of n, the assumptions are far too restrictive and the resulting values of a are not interesting. Moreover, this does not contribute to the overall picture of stable coalitions that we wish to establish. Thus, we state our final result for the case o f a< 0.5 in the following lemma and subsequently focus on the case when a > 0.5. Lemma 2.6.3: For a < 0.5 and n even, the grand coalition g is stable but not necessarily unique. The previous two results demonstrate that g is not necessarily uniquely stable. However we observe that when the assembler is weak, i.e. □ < 0.5, then the grand coalition is always stable. Thus, so far, we have established that when the assembler is “very weak” the grand coalition is the only stable outcome. As 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the assembler becomes stronger other outcomes with more coalitions also become stable. We will now, using a series of lemmas, demonstrate the structure of certain stable outcomes for a > 0.5. For some o f these results, we require that n be large and even. From the proofs, it will be clear that extensions to odd n are straightforward and require very similar techniques. We begin with a very useful result. Proposition 2 .6 .1: Let a > 0.5. For every outcome a and for any i e N if a — > . b then a<i b. This proposition implies that when a > 0.5 it is always myopically beneficial for a single player to defect from a coalition. This simple observation is very useful in characterizing stable outcomes for a > 0.5. Lem m a 2.6.4: Assume «is even and that 0.5 < a <2/3. Then, the outcome x = {l,2,...w/2},{«/2 + l,...n} is stable. Recall that we observed stable outcomes with two coalitions (of unequal size) when a <0.5. When the assembler becomes stronger, we observe that outcomes with two coalitions are still stable, but unlike before, the size o f the coalitions become close to being equal. Indeed, lemma 6.4 demonstrates the stability of an outcome with two coalitions o f equal size. But as will be seen in the next result, outcomes with coalitions with sizes being close to equal are also stable. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lem m a 2.6.5: Let n large and even. Let a n A = {1,2,...—-l} ;i? = There exists {bx,b2) a ( 0.5,2/3)such that V a e {bx,b2) , the outcome A is stable. We will now demonstrate that this structure of the stable coalitions can be further extended. To do so, define 3 3 as the outcome with three “equal” coalitions. As before, we will focus on n large enough to justify the following lemmas. W hennmod3 = 0, 3 3contains three equal coalitions. Whenn = k + \, where £m od3 = 0, 3 3has 2 coalitions o f size k / 3. Finally when« = k + 2, &mod3 = 0, 3 3has 2 coalitions o f sizek /3 + l . Thus every coalition in 3 3has less than or equal to (n + 2) / 3 members. Lem m a 2.6.6: For n large, there exists 2 /3 < a x < a 2 <3/4 such that for a e (a r, a 2) , 3 3 is stable. This result extends on lemmas 6.4 & 6.5 and demonstrates the stability of outcomes with three coalitions (of equal or close to equal sizes) as the assembler becomes even more powerful. Using similar proof techniques as in the above lemma, we can demonstrate the following theorem, which is quite general in its prediction of stable outcomes. Theorem 2.6.1: 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For a > 0.5 and n large, there exists 0.5 = 6X < 62 < ... < 6n = 1, such that 3 k (defined in a similar fashion as above) is stable for a e { 0 ^ , 6 ^ . Theorem 6.1 demonstrates the evolution of stable structures with increasing number of coalitions as the assembler becomes increasingly powerful. One may hypothesize that a natural culmination would be that when facing an extremely strong assembler, the suppliers would not form any coalitions (or one could alternatively say that they form n coalitions) among themselves. Indeed, we will now show that for very large a , the outcome u = (l},(2),...{«}is uniquely stable. To achieve this objective, we need the following proposition. 1 Proposition 2.6.2: Let n>2 and a > (0.5)n-2 . Assume that a b, b bx -> S 2 b2 -» ... -» s bm and i g S j V y = 1 . Then bj >t a Vy = 1 . The above proposition implies that if supplier i defects from outcome a to b and subsequently there is a sequence of defections that do not involve supplier i , then, the profit of supplier i in the each one of the resulting outcomes is higher than his profit in the status quo outcome a . We are now in a position to predict the structure of the supply chain when the assembler is very powerful. i Lem m a 2.6.7: Let n>2 and a>(Q .5)n ~2, then u = {1},{2},...{«} is the only stable outcome. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This result is quite interesting. Assume that there are 5 suppliers. Also, assume that the assembler is powerful (□ = 0.8). In this case it is in the best interest o f the suppliers to act independently and not to form any coalition. By staying independent each supplier gets 8.1% of the total profit. This is in contrast to the 4% that each member of the grand coalition obtains. By staying independent the suppliers are able to double their profits as compared to the profits of members of the grand coalition. This is somewhat surprising. One would expect that the suppliers will always benefit by joining forces. While this is true when the assembler is weak, it is not true when the assembler is powerful. We have thus exhibited stable outcomes for different values of a . hi some cases we can show uniqueness. In certain cases, although we cannot show uniqueness, we can rule out certain possibilities. For instance, when a e (1/2 ,2/3), we know that 3 2is stable. Note that at this point, we can eliminate g as a possible stable outcome using the following rationale. Suppose g is farsighted stable. But we have, in the given range o f a , 3 2 >,. g for every i . Thus,3 2 y g , independent of S. It stands to reason that, in this case, g would never be a stable outcome. We have not explicitly described the assembler’s preferences on the number of suppliers with respect to a . Since the assembler preferences are exactly the opposite of the suppliers and since the suppliers’ preferences are demonstrated in the stable coalitional outcomes, it is trivial to now make statements about the assembler’s preferences. Thus, when the assembler is weak (low values of a ), he 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. prefers to negotiate with a large number o f suppliers. On the other hand, when the assembler is very powerful (large a ), he prefers to deal with very few suppliers. An aspect of negotiation that we have ignored is the associated cost. Consider the case when the assembler suffers a cost b per supplier (or coalition) that he negotiates with. Indeed, the assembler is burdened with a certain cost that is proportional to the number of negotiations he conducts. Thus if he has to deal with n supplier coalitions, his total costs due to the negotiation process is b n . In general, the cost b can be allowed to reflect any kind of transaction costs. Solving the A-B-P game with this additional cost we get very similar outcomes for all the three stages of the games. Indeed, the structure of the stable outcomes are quite similar, except that the corresponding ranges of a are now shifted. Notice, that applying the cost of negotiation, the normalized profits shrinks from 1 to (1-tb), where t is the number of coalitions. Notice that even though the we assume that the bargaining cost is incurred only by the assembler, it also affects the profits of the suppliers. Thus, when all suppliers form a single coalition the profit of each player is: — —— (1 - b ) . The maximum profit o f any supplier in the outcome n {l},{2,3,...,n} is a (l~ a ) (\ - 2b). As a result, when — —— (1 - 6) > a il - a)(l - 2b), the grand coalition directly dominates every n other outcome for every player. Thus we have, the observation corresponding to Lemma 6.1: 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /I _ Observation: For a < —---------- the grand coalition g is uniquely stable. n (l- 2 b ) As expected, we observe that due to the introduction of the cost of negotiation, it is now more likely that the grand coalition will be the only stable outcome. Similarly, assuming that nb < 1, we have u = {1},...{«} as the unique stable outcome for a > [——^ - ] 1 /(”~ 2). Notice that when nb is approaching 1 the 2(1 - nb) outcome u = { 1 } is stable only when a > 1, which is never the case. Again, this result is not surprising. To reduce the negotiation cost (thereby increasing the size of the pie) the suppliers are likely to form fewer coalitions. 2.7. Conclusions and Future Research In this paper, we have developed a general framework for negotiations in a supply chain. We are able to extend the Nash Bargaining solution to a sequential negotiation process, and are able to model negotiating power as a function of the cost o f revoking a partial commitment. We apply this framework to the assembly bargaining problem. The assembler is given the power to choose the sequence in which he negotiates with the suppliers for allocations of the profit pie. In our paper, we use a wholesale price mechanism to transfer these profit allocations. Clearly, the results are unaffected if we choose more complex contracts. In a sequential setting, we show that the assembler’s profit is not affected by the sequence of negotiation, but is a function of the number of suppliers and their 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relative negotiation powers. The profit of the suppliers is a function of their position in the negotiation process and their negotiation power. With regard to their position (and the sequence), we explicitly show the preference of the suppliers in two different settings (identical powers and different negotiation powers). Assuming suppliers with identical negotiating power, we model the competition among the suppliers for a position in the negotiation sequence. We show that there is a unique Nash equilibrium in payments, and at equilibrium the effective profit o f the suppliers are identical. The assembler would ideally prefer to have as many suppliers as possible to extract the entire profit. This is an artifact of ignoring costs incurred by the assembler associated with having a supplier. However we empower the suppliers to form coalitions among themselves. We borrow the concept of “farsighted stability” to analyze the suppliers’ response. Doing so, we are also able to determine the stable coalition structure of the supply chain. The coalition structure is a function of the number of suppliers and their negotiating power. Surprisingly, we find that it is not always in the best interest o f the suppliers to join forces and form the grand coalition. In fact the structure seems to have a certain interesting dynamics with respect to the assembler’s negotiation power. When the assembler is very weak, the suppliers unite as one grand coalition and negotiate for their share. However, at the other extreme, when the assembler is immensely powerful, the suppliers organize themselves individually and do not form any coalitions. In situations not at either 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of these extremes, we see stable outcomes that are neither o f the above two outcomes. Further, we show that as the assembler’s negotiation power increases the suppliers tend to form smaller coalitions. We now pause and reflect on the comments made in section 3. When speaking o f negotiation power, we mentioned two aspects. The first is an artifact of the economic model that is used to capture the bargaining process. Our choice of the Nash Bargaining implied that risk aversion and the ability to make credible commitments (as demonstrated) impacted the individual negotiation power of the players. The second aspect that we alluded to was extraneous to the choice of the model. Having completed our analysis o f the supply chain, we are now in a position to expound on this feature. Recall that we assumed that all players are risk neutral and the suppliers having identical revoking costs. The assembler at the first stage, chooses a sequential negotiation process. The outcome dictated that he received a rather small portion of the profits. Indeed recall that at this stage, the assembler preferred to have as few suppliers as possible. However, this immediately gave rise to a situation where the suppliers competed for positions in the negotiation sequence. Competition among the suppliers immediately increased the bargaining position of the assembler. This though an artifact of the model, is quite natural. Indeed, the fact that the suppliers were in competition, reversed the assembler’s preference on the number of suppliers. Determinants of negotiation power in a supply chain are manifold. Clearly, competition is one of them and we demonstrated its impact on our supply chain. The ability to form alliances is 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. another. The third stage of our game precisely addresses this aspect. The suppliers retaliate by forming coalitions. Note that we did not empower the negotiation power o f the coalition extraneously. We assumed that the effective revoking cost of the coalition is exactly the average and thus equal to the revoking cost of any individual supplier. However, by forming coalitions, the suppliers impacted the structure o f the supply chain. They traded off the benefits of competing for a position in the negotiation sequence with the number of times they would negotiate with the assembler. Once again, this changed the negotiation power of the suppliers as well as the assembler. One can ask if such coalitional structures are seen in practice. We have illustrated a few examples in various parts of the paper (bar code labelers Paxar/Monarch , automobile suppliers etc.). Companies like Cisco systems do not purchase components from individual suppliers but rather trade with sellers of kits o f components (ex. Solectron). A supplier of a kit o f components can be thought of as a virtual coalition. In another example TSIA reports that it provides a platform for smaller suppliers to come together and negotiate trade terms with manufacturers from overseas. These platforms usually involve a coalition of semi conductor component manufacturers who trade with a buyer. We are in no way suggesting that out-sourcing or the existence of virtual coalitions can be fully explained through our model. The reasons may be manifold. Another aspect of our results addresses the trend in several supply chains, towards manufacturers and assemblers preferring to have fewer suppliers. Well, 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. how few is fewer? Our model suggests that due to the effects of transaction costs and negotiation power, it may not always be advisable to just have one supplier or at the other extreme, a whole lot of them. We are not in any way suggesting that the model and the results thereof are complete or general. Indeed, the issues at hand are much more complex, and our model pertains to a supply chain with a very specific structure. We believe that future research should focus in modeling negotiating power as a function of risk aversion, competition among suppliers manufacturing similar components etc. It will also be interesting to model and understand negotiation situations in which the different parties posses asymmetric information. An issue of interest is to model negotiation power exogenously as a function of the position in the sequence. Indeed, one could argue that a supplier who goes last in the sequence may have more negotiation power, as his ability to veto may be quite costly to the assembler. This can be accommodated in our framework by letting the cost of revoking commitments be a function of the position. Indeed, the question of an equilibrium set of payments (position) is to be reckoned with. This is an important and an interesting extension. Our results are sensitive to the definition o f stability that we use. A broader notion o f stability and the examination o f the results thereof are left for future research. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Coordination issues in vendor managed inventory systems 3.1. Introduction Supply chain coordination issues have received significant attention in the operations literature in recent years. While there are myriad reasons for lack of coordination in supply chains, an important one is the differential impact of inventory-related decisions on the agents in the supply chain. For instance, the cost of a stockout o f say a Hewlett Packard (HP) printer at a HP reseller is quite different for HP and the reseller depending on the substitute products carried by the reseller, brand loyalty for HP products and other factors. Hence, the optimal inventory level of H P’s printer that the reseller might carry may be quite different from the optimal inventory that HP might carry if it was allowed to determine inventory levels at the reseller. The latter scenario is becoming more plausible with the growth o f vendor-managed inventory (VMI) systems where the supplier is responsible for maintaining the “correct” inventory level at its customers ('www.vendormanagedinventory.com). For example, Hewlett-Packard practices such VMI relationships with its resellers and many consumer product manufacturers have VMI relationships with Wal-Mart and other retailers. Specifically, this paper addresses a scenario where a manufacturer is selling a product through a retailer and the manufacturer replenishes the stock at the retailer on a regular basis. In traditional retailer-managed inventory (RMI) systems, the retailer places orders periodically with the manufacturer who ships 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the items to the retailer. The retailer’s decisions determine both the cycle and safety stock levels. As the retailer ignores the implicit cost of any stockout for the manufacturer and replenishment costs incurred by the manufacturer, the retailer may set cycle and safety stock levels that are different than the optimal levels from the manufacturer’s or the supply chain’s standpoint. Such scenarios are common in many industries. The well-known HBS case study on Barilla (HBS Publishing (1994)) illustrates a situation where the manufacturer o f pasta is unhappy with the fill rates achieved by its distributors and proposes to manage the distributors’ inventory levels. One of the authors was involved in a consulting project where a large printer manufacturer was having difficulty ensuring that its resellers, who carried competitors’ products, maintain adequate inventory levels to achieve a desired fill rate. In another project, a Fortune 50 food manufacturer was initiating a vendor-managed inventory system with a major grocery chain to achieve better service levels. Allowing the manufacturer to make replenishment decisions may appear to address the above concerns, but the manufacturer may not act necessarily in the best interest of the retailer and the supply chain. So, the retailer can specify contracts that appropriately influence the manufacturer’s optimal inventory and replenishment decisions. In this paper, we propose contracts that a retailer can specify in such a vendor-managed system so as to improve system performance relative to a retailer-managed system. In particular, we consider a simple contract where the retailer charges the manufacturer an amount R per unit of average 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. inventory in the store and lets the manufacturer make the inventory decisions. R is essentially a holding cost subsidy (HCS) paid by the manufacturer to the retailer. Such holding cost subsidies are prevalent in practice. Moreover consignment sales and credit terms frequently offered by manufacturers to retailers are conceptually similar to holding cost subsidies. We show that such contracts, which are simple to implement, do not generally coordinate the channel (i.e. achieve the minimum system cost that a single decision-maker managing the entire system can obtain) but can improve performance relative to a RMI system. We also propose other contracts that can coordinate the channel and allow for arbitrary allocations of channel costs — referred to as first best contracts. We analyze contracts in a system with a single manufacturer supplying a single product to a retailer or distributor. The retailer incurs a fixed replenishment cost per order, an inventory holding cost and a penalty cost Pr when there is a stock out; we assume that unsatisfied demand is back-ordered. The manufacturer also incurs a penalty cost Pm when there is a stock out at the retailer (more on this later) and a fixed replenishment cost per order. We assume that the players are rational and act non-cooperatively. We look at both, (i) a continuous review inventory system where the decision maker continuously reviews his inventory level and places an order Q with the manufacturer when the inventory is equal to a re-order point r and (ii) a periodic review system where the decision maker orders once every T periods and orders are placed so as to raise the inventory position to a level Q. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This paper makes the following contributions. First, the supply chain coordination literature can be broadly classified into two streams — deterministic demand models with fixed replenishment costs using incentive schemes such as quantity discounts to achieve channel coordination or, the second and larger body of work using newsboy type single-period, stochastic demand models to study channel coordination issues. This paper attempts to extend both streams of literature by studying channel coordination issues in a model with stochastic demand and fixed replenishment costs. To our knowledge, ours is the first paper that considers contracts in a multi-period continuous/periodic review setting. Second, we propose and analyze novel retailer-initiated contracts to improve system performance and coordinate the channel. Furthermore, we show that VMI systems wherein the manufacturer manages inventory and makes replenishment decisions may be superior to retailer-managed systems under certain conditions. An interesting aspect of our model is that we assume both the manufacturer and the retailer incur a penalty cost when there is a stockout at the retailer. Typically, stockouts impact both the retailer and manufacturer negatively - both incur loss of goodwill and potential future sales and profits (Emmelhainz, Stock & Emmelhainz (1991) and Straughn (1991)). We do not account for an immediate loss of revenue as we assume that demand is back ordered. Our approach is similar to that taken by Cachon & Zipkin (1999) and Raman & Narayanan (1999). Cachon and Zipkin (1999) consider similar stockout costs at the retailer and the supplier when there is a stockout at the retailer. Raman & 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Narayanan (1999) also discuss the possibility that a manufacturer’s stockout cost may be different from that of the retailer. In fact, one might expect the manufacturer’s stockout cost to be greater than that o f the retailer for several reasons. For instance, a customer may be more loyal to the retailer than the manufacturer if the retailer carries a wide variety o f items, provides other services or has an attractive location. Also, the retailer may carry substitutes and the customer may switch to one o f these over time if there are repeated stockouts of the manufacturer’s item. While we do not model these aspects, one can capture them implicitly through the relative values of Pr and Pm . In particular, we assume in some of our results that Pr < Pm . We now briefly review the related literature; in particular, we only focus on papers very closely related to our work to illustrate our relative contribution. There are several comprehensive reviews of this literature (Tayur, Ganeshan & Magazine (1998), Cachon (2002)) which are recommended to the interested reader. Most of the works reviewed in these surveys consider single-period models with stochastic demand. There are also several papers that use deterministic demand, EOQ type formulations (Weng (1995), Corbett and de Groote (2000)) and show how various types of quantity discounts can be used to coordinate channel decisions. Cachon (2002) reviews and extends the literature on models for the management of incentive issues in a supply chain. A majority of the models discussed in this paper are single-period newsboy models and variations that 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. allow for retail price and effort to be endogenous, that feature multiple competing retailers, and at most two replenishment opportunities. An infinite horizon model is also studied where the retailer follows a base stock policy and the supplier and/or the retailer can hold inventory. These models do not consider fixed replenishment costs. Raman and Narayanan (1999) consider a scenario similar to ours where a manufacturer and retailer may have different stockout costs resulting in incentive misalignments and sub-optimal outcomes. Their work is different from ours in several respects. First, they consider manufacturer effort in their model, which we ignore. Second, they consider a single-period newsboy model whereas we consider a multi-period continuous/periodic review system with fixed replenishment costs. Third, they assume that the manufacturer cannot observe retailer inventory levels. We do not make this assumption, based on our observations and experience in industry of VMI systems. Finally they do not consider holding cost subsidy type contracts; also, we show that there exist contracts a retailer can offer that can coordinate the channel. Our broad focus will be on variations of HCS type contracts. Such contracts encompass traditional holding cost subsidies, rent charged on average inventory (due to this reason we use HCS and Rent interchangeably), etc. Our motivation for considering such contracts stem from anecdotal evidence and conversations with corporate managers at Nestle and Universal Studio. We propose HCS type contracts, evaluate their performance and compare them with 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. other contracts. When designing a contract, one has to make tradeoffs between important factors such as flexibility, simplicity in implementation and efficiency. Indeed, a very complex contract that allows for the best performance of the channel (and the players) may not be implemented due to its inherent complexity. So, we consider some simple contracts that are easy to implement as well as more complex contracts that are efficient and flexible. This paper is organized as follows. In section 2, we first analyze the deterministic case. In section 3 we establish the model with random demand where decision makers follow a continuous review policy and evaluate single parameter rent contracts. In section 4, we analyze the model with alternate two parameter contracts. In section 5 we examine a model where inventory is reviewed periodically. We conclude in section 6 with a summary and future research ideas. Proofs of all propositions are deferred to the appendix. 3.2. The Deterministic Demand M odel We first motivate the discussion with an analysis of the deterministic case and then present the analysis for the case with random demand. In particular, we consider a standard EOQ-like setting where the decision-maker only has to determine the order quantity. Consider a single manufacturer supplying one product to a single retailer who faces a deterministic demand D per period. In all the systems considered in the paper, inventory is held only at the retailer, incurring a holding cost H per unit per period. We do not model the manufacturer’s production and inventory 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. decisions. While inventory is held only at the retailer, the manufacturer or the retailer may make stocking and replenishment decisions. The retailer and manufacturer respectively incur replenishment costs Am and A r in each order cycle. The values of A m and Ar depend on the specific context, but typically consist of fixed order processing, material handling and delivery costs. In many real-world scenarios, the manufacturer bears the fixed transportation costs and order processing/material handling costs incurred at the manufacturer’s location. The retailer bears the order processing/handling costs incurred at the retailer’s location. Traditionally, the retailer makes replenishment decisions and we refer to this system as a “Retailer Managed Inventory" or RMI system. In such a system with deterministic demand D, the retailer determines the well-known economic order quantity or EOQ, QR = ^ 2 ArD / H by minimizing the sum of his replenishment cost A r D/Q and inventory holding cost HQI2. But the retailer ignores the impact of his decisions on the replenishment cost borne by the manufacturer. (Note: The superscript R in indicates that this Q corresponds to a RMI system.) Therefore, this order quantity need not minimize the total supply chain costs, comprised o f the manufacturer’s and the retailer’s total costs. Consider a “centralized” system in which a single agent is responsible for the entire supply chain and minimizes the total costs of the supply chain, comprising of the retailer’s costs, Ar D!Q and HQ!2, and the manufacturer’s costs AmP/Q. This is a hypothetical system that is useful in evaluating whether a 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. specific contract coordinates the channel. Also, in comparing the RMI and VMI systems, we use the total supply chain cost as a benchmark. If we denote the c * * economic order quantity for this centralized system by Q , then it is straightforward to show that Qc = ^ 2 (Ar +Am)D /H minimizes the total cost of the supply chain. Now consider a VMI system wherein the manufacturer determines the order quantity. Clearly, the retailer would not want the manufacturer to replenish quantities that ignore the retailer’s costs. Suppose the retailer charges the manufacturer, say, a rent R per unit inventory held, to appropriately influence the manufacturer’s replenishment quantity. This amount also can be seen as a holding cost subsidy paid by the manufacturer to the retailer. In this case, the manufacturer minimizes (AJDIQ + RQI2) to determine the optimal replenishment quantity, Qv = ^2 A m D / R . Hence, by setting R = Am H/(Am + A r) which implies that Qv = Qc, the retailer can ensure that the total supply chain cost of the VMI system is the same as that of the centralized system, i.e. the channel is coordinated. Note that R determines the share of the total costs borne by the retailer and manufacturer. Though the rent R may coordinate the channel, the retailer’s share of the total costs may be greater than his costs in the RMI system. Hence, the retailer may not have an incentive to participate in such a system. More generally, while such a contract may coordinate the channel, it cannot ensure that both agents are better off in a VMI system when compared to a RMI system. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, one can ask the following question: Is there a rent contract with a value R such that the manufacturer (vendor) will pick a Q that optimizes his costs, and simultaneously ensure that both the retailer and the vendor can have lower costs than in a RMI system? Indeed if there is such a R , a simple rent contract is attractive. We have the following result: Observation 3.2.1: There exists a rent R such that the manufacturer will choose Q so that the manufacturer and retailer have lower costs in the VMI system as compared to the RM I system, if and only i f A r < A m . But this contract may not coordinate the channel. The condition Ar < A m , seen in the quantity discount based contracting literature (Corbett and DeGroote (2000)), may be reasonable. As mentioned earlier, the supplier’s replenishment cost Am typically includes transportation cost unlike the retailer’s cost Ar and this may often dominate the other costs. For instance, Nestle typically assumes the delivery costs to its customers and these costs dominate their order processing and material handling costs. Indeed, one can ask what additional conditions on the parameters will ensure that the channel is coordinated and the retailer and manufacturer will both be better off. We have: Observation 3.2.2: Let A r < A J3. Then there is a unique rent R ' such that the channel is coordinated and both players have lower costs than in a RMI system. We now extend the above discussion to a situation where the retailer would use two parameters to coordinate the channel as well as to determine his share of the total supply chain costs. In particular, consider a contract where the 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. retailer announces the rent (as before) and in addition decides to subsidize the replenishment costs borne by the manufacturer. Thus he announces R and (3 , where the retailer pays the manufacturer a proportion p of his fixed replenishment cost. Indeed one can demonstrate that this is a first best contract. We have: Observation 3.2.3: The retailer can announce a contract (R, P) such that: 1. The manufacturer will pick the channel coordinating quantity and 2. Arbitrary fractions o f the total cost can be assigned to the two players. Hence, using such a two-parameter contract, both parties can do better than in a RMI system by suitably allocating the channel coordinated cost. So we observe that several two parameter contracts are first best and can allocate costs so that both players are better off. Indeed, existing literature on quantity discount models deals with a similar problem in which the supplier offers the buyer a menu of contracts from which the buyer picks the contract that optimizes his costs. We will not review this literature but refer the reader to Corbett and de Groote (2000) for a brief summary of models and results thereof. This literature presents a quantity discount contract offered by the supplier, which coordinates the channel and improves the performance of both the buyer and seller. Suppose a seller offers a buyer a quantity discount schedule, say, {(v,w ,K )}. This means that if the buyer purchases a quantity greater than K, the wholesale price is w and if the purchase quantity is less than K, the price is v. This quantity discount contract can be thought of as equivalent to a “two-parameter” contract. The issue of equivalence of contracts is important but it is not our intent to discuss this in great 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. detail. However, in section 4, we present a discussion on equivalence o f contracts for the case of stochastic demand. 3.3. The continuous review model. In this section, we assume that the relevant decision-maker follows a continuous review (r,Q) policy, wherein they continuously monitor inventory levels and replenish inventory with a fixed order quantity Q when inventory level reaches the reorder point, r. Since the demand is random, the retailer holds safety stock to hedge against potential stock outs. When there is a stockout at the retailer, we assume that all excess demand is backordered and both the retailer and the manufacturer respectively incur penalty costs, Pr and Pm . We assume a fixed lead time for delivery. The (r,Q) policy has been well studied. Zipkin (1986) showed that the long run average cost function is jointly convex in the decision variables. As our main objective is to analyze contracts in this system, rather than an analysis of exact inventory policies, we use an approximate cost formulation suggested by Hadley & Whitin (1963). As Hadley & Whitin (1963) note, this approximate model is useful due to its inherent simplicity. Also, special cases for which exact equations are available do not represent the real world situations more accurately than the approximate models, i.e. the additional assumptions needed to reduce the exact model to the simple model are frequently warranted in practice (Hopp, Spearman and Zhang (1997)). We consider two possible cases that are typical in the literature: (1) a penalty is incurred for every stockout occasion and (2) a penalty is incurred for 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. every unit stocked out. We assume that at any time no more than a single order is outstanding. Our objective is to evaluate a few contracts to see which ones are simple and flexible, can coordinate the channel, and to provide some general observations. 3.31. O ne-param eter contract (The base case) We start by considering a simple one-parameter contract offered by the retailer that is similar to the one considered in the deterministic case. In the stochastic demand case, the average inventory held by the retailer is (Q/2+r). The expected on hand inventory is actually Q /2 + r - ju, where ju is the mean lead time demand, but fi is ignored as it does not affect the results. The retailer charges the manufacturer an amount R(Q/2+r). As before, this amount can be seen as a holding cost subsidy paid by the manufacturer to the retailer. We assume that demand is represented by an exponential distribution. (Some o f our analysis and results do not require this assumption.) For a demand generated by a Poisson process, the use o f exponential density is justified. Further, it has been shown (Bitran and Dasu (1994)) that several demand distributions are exponential in the tail, which is usually the region of interest. For further justification, see for instance Glasserman (1997) and Harrison (1998). In particular, we assume that lead time demand is represented by the density function / (< f;) = e~a?. We have omitted the constant multiplier in this density function but there is no loss of generality in using this representation as will be clear from our analysis. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that F(r) = ^f{f)d^ is the probability o f a stockout in a r replenishment cycle, given that the reorder point is r and / (.) is the probability density function of the lead time demand. Let n(r) represent the expected number oo of stockouts per replenishment cycle, where n(r)= J(£ ~r)f{^)dE, . Then we r have the following result. Lem m a 3.3.1, I f lead time demand distribution is f (g) = , then n(r) = F ( r ) /a . Lemma 1 will be useful in generalizing any results for the case with penalty for each stockout occasion to the case with penalty for each unit stocked out. We first consider the case with penalty for each stockout occasion. Then, the total cost o f the supply chain for the centralized system comprising of holding cost, fixed replenishment costs and stockout penalty costs for both the manufacturer and retailer is: U c sc (Q, r) = (Ar + A J D /Q + H(Qi2 + r) + (Pm + Pr )F(r)D / Q . (3.1) where Fir ) and /( .) have been defined earlier. We use the following convention in our notation: superscripts C, R and V respectively denote centralized, RMI and VMI systems and the subscripts R, M and SC represent retailer, manufacturer and the total supply chain (manufacturer + retailer) or system cost respectively. First, we have: 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lem m a 3.3.2. The functions Y\c sc{Q,r), Tlr M(Q,r), and YlR R{Q,r) are jointly convex in (Q, r) . Differentiating Y{c sc{Q,r) with respect to r and Q and rearranging terms, c we get the following two equations to solve for the optimal re-order point r and order quantity Qc in the centralized system QC=(Pm+Pr)Df(rC)/H (3.2) (Q ) = 2D[(Ar +Am) + (Pm+ Pr)F(r )]/H . (3.3) For the RMI system, the retailer makes the re-order point and order quantity decisions. The retailer’s cost expression is: U R R(Q,r) = ArD/Q + HQ/2 + PrF(r)D / Q (3.4) From first-order conditions, his optimal response (QR , rR) satisfies the following equations: Qr =PrDf(rs )/H (3.5) (QR)2 =2D[Ar +PrF(rR)]/H (3.6) Finally, for the VMI system, the retailer charges the manufacturer an amount R{Q/2+r) and so the manufacturer determines (Q,r) that minimizes his total cost: n v M(Q ,r) = Am DIQ + R(Q/2 + r) + Pm F(r)D/Q (3.7) 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The optimal response of the manufacturer (QV (R), rv(R)), given a rent R, satisfies the following: Qr =Pm Df(rv)IR (3.8) (Qv)2 =2D[Am+PmF(rv)]/R. (3.9) In the future, we denote the manufacturer's response as (Qv, r v ) with the implicit understanding that (Qv, rv ) is a function of the rent R. Dividing both sides o f the expression in (9) by the corresponding expression in (8), we have: 2 A, , 2 F(rv) ' Pmf ( r V) f { r v) Using the fact that f(t) = e * and F(x) = jf(t)dt =f(x)/a , we have 9 A 9 Qv = ^ - p f ^ +~ ( 3 - 1 0 ) P J (r ) « Similarly, using (6) and (7), we get Qr = 2\ + - . (3.11) PJ(r*) a We can immediately make the following observation: Observation 3.3.1: By offering a HCS contract to the manufacturer, the retailer in general cannot coordinate the channel. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Though the channel is not coordinated, in the next theorem we show that the VMI system can result in lower total supply chain costs than the RMI system under certain conditions. Theorem 3.3.1: Assume that A r < 5A m and Pr < Pm . With a HCS contract o f the type R(Q/2+r), the VMI system achieves lower system costs than an RM I system in both cases: (I) a penalty is incurred fo r every stockout occasion and (2) a penalty is incurred fo r every unit stocked out. Theorem 3.1 tells us that the holding cost subsidy contract, under VMI, can result in lower system wide costs as compared to the RMI system under certain conditions that are not restrictive. We only require that A r not be very large as compared to A m . Since fixed delivery costs, which are a substantial part of replenishment costs, are typically borne by the manufacturer, this appears to be reasonable. As discussed briefly earlier, the condition Pr < Pm implies that the cost impact of a stockout is greater for the manufacturer relative to the retailer. This is reasonable when the customer is more loyal to the retailer than the manufacturer’s brand which may be due to various reasons - the retailer’s location, large variety of items including close substitutes carried by the retailer, retailer’s low prices, etc. We do not model these factors but simply identify them to explain why the condition Pr < Pm may hold. While a VMI system may do better than an RMI system using a holding cost subsidy contract, a VMI system need not always coordinate the channel even under the conditions specified. More important, the manufacturer and retailer 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. need not always do better under VMI as compared to RMI under such a contract. We illustrate this with an example. 3.3.2. An Illustrative Example Consider an example with Ar - 5, D = 200, a = 0.05, and H = 0.1. Table 1 provides details on the costs incurred by the retailer and manufacturer in the optimal solution for the RMI, VMI and centralized system for different sets of values o f Pr, Pm , Am and R. Observe in Table 1 that all the parameters are the same in Scenarios 1 and 2. The total cost of the VMI system is lower in Scenario 1 (with R = 0.057) than in scenario 2 (with R - 0.025). When R = 0.057, while the manufacturer’s cost is higher in the VMI system as compared to the RMI system, the retailer benefits significantly with VMI. However, for the same set o f parameters, the second column shows that when R = 0.025, both manufacturer and retailer have lower costs in a VMI system relative to a RMI system. Unlike Scenarios 1 and 2 where Pr < Pm , in Scenario 3, we set Pr = Pm . We then find that when R = 0.057, the total costs for VMI are lower than that for RMI. But the manufacturer has to incur significantly higher costs in VMI as compared to RMI and so the manufacturer would have no incentive to participate in a VMI system. Further analysis shows that in this case there is no R value at which both manufacturer and retailer have lower costs in a VMI system relative to a RMI system. While there are R values at which the manufacturer may have lower costs under VMI, the total system costs at these R values are higher than the cost under RMI. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1 Scenario 1 Scenario 2 Scenario 3 Scenario 4 Pr = 40, Pm = Pr = 40, Pm = Pr — 40, Pm = Pr ~ 40, Pm = 120 120 40 40 System Cost > I I > 3 II Ar — 5, Am — Ar — 5, Am — Ar — 5, Am 5 5 5 15 R = 0.057 R = 0.025 R = 0.057 R = 0.057 Centralized SC 36.66 36.66 35.27 42.89 s c 40.82 40.82 36.82 49.10 RMI RC 28.68 28.68 28.68 28.68 MC 12.14 12.14 8.14 20.42 SC 36.76 38.55 35.32 43.02 VMI RC 16.31 26.93 16.12 13.72 MC 20.45 11.62 19.20 29.30 Note: SC, MC and RC stand respectively for system cost, manufacturer’s cost and retailer’s cost. Note that Pm is equal to 40 in Scenario 3 while it is equal to 120 in Scenarios 1 and 2. We observe that an increase in Pm does not have a significant impact on the total costs of the centralized system and VMI system but does have an impact on the total cost of the RMI system. In general, we find that the VMI system proves to be truly superior to RMI when Pm values are high relative to Pr. As Pm increases relative to Pr, the RMI system’s cost gets progressively higher 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relative to VMI because the RMI system ignores the manufacturer’s stockout cost in determining order quantities and safety stocks. A similar reasoning holds when we compare Scenarios 3 and 4, where the only difference is the value of Am . Observe that an increase in Am results in a substantial increase in the total cost of RMI relative to VMI, again because the RMI system ignores the change in the manufacturer’s replenishment cost. This is clear from the fact that the retailer’s cost (RC-28.68) under RMI is the same in all four scenarios. Thus, when Pm is high relative to Pr, VMI is better than RMI for both players. This is interesting because manufacturers with significant brand power such as Procter and Gamble, Nestle and Universal Studios who are likely to have high Pm values have implemented VMI systems with their customers. In fact, some of them have been willing to incur many of the fixed and recurring costs of VMI systems, presumably because they expect VMI to provide them greater benefits. However, our examples show that while VMI can result in lower system costs than RMI, the manufacturer may sometimes incur higher costs. This is not unlike what has been observed in practice - see for instance the discussion at www.vendormanagedinventory.com. Some firms have found that they ended up incurring higher costs when they implemented VMI systems with some of their major customers. Thus, while a VMI system results in a reduction of total system costs in many situations, the benefits may not be shared appropriately. This may result in one of the parties being reluctant to participate in the VMI system, as they do not benefit relative to the status quo, which is typically the RMI system. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nevertheless, one can argue that the savings obtained by adopting a VMI system can be allocated between the manufacturer and the retailer thereby ensuring that their respective costs are lower than their costs in a retailer managed system. Moreover, using the same techniques as in the proof of the above theorem, we can show that when Pm » Pn both players are better off in the VMI system using the above contract as compared to the RMI system. This observation is interesting because, even though the condition implies that the cost of a stockout is much greater for the manufacturer, both the retailer and the manufacturer are better off in the VMI system. Moreover, a holding cost subsidy is equivalent to a penalty imposed on the manufacturer by consignment sales contracts (see for instance http://www.consi gnment-shop-store.com and for the legal aspects of consignment sales contracts, please refer to http://secure.uslegalforms.com/cgi- bin/forms/query.pl?S-T-B-B-consign). When a manufacturer sells at a retailer (or a distributor) on a consignment basis, the inventory is held at the retailer but the manufacturer owns the inventory and can take it back it at any time. Thus, the manufacturer incurs the cost of capital invested in the inventory and the retailer incurs the cost o f space, security, etc. Hence, the manufacturer is incurring part of the holding cost and so this is equivalent to a holding cost subsidy. Note that in our models, due to the multi-period setting, the retailer eventually sells everything that he holds. This allows us to ignore the issue of revenues and commissions 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtained by the retailer through sales. One can make a similar argument for credit terms such as “30 days net”. A contract based on a holding cost subsidy or a variation of it is simple and has intuitive appeal. Its merit as compared to buy back policies or revenue sharing contracts lies in the fact that minimal monitoring costs are incurred. It may not always be the “best” contracting mechanism, however, as it is neither efficient (achieves channel coordination) nor flexible (can allocate arbitrary fractions of the realized cost). In the next section, we investigate other related contracts that may be more complex but are efficient and flexible. In our discussion, we have compared a VMI system where the retailer offers the manufacturer a contract and the manufacturer decides (Q, r) with a RMI system where the retailer manages the inventory without any explicit contracting. Though comparing a system with a contract to one without any explicit contracting is not uncommon in the literature (the deterministic quantity discount based contracting literature, for instance), it will be useful to compare the VMI system in this section with an RMI system where the manufacturer offers the retailer a HCS and the retailer decides (Q, r). Next, we provide a brief discussion o f this system. Consider a “modified” retail managed inventory system in which the manufacturer offers to pay the retailer a holding cost subsidy (1- y)H(Q/2+r), y > 0 and in return lets the retailer pick (Q, r). The retailer’s cost expression is now Yl^R{Q,r) = ArD ! Q + yH{Q 12 + r) + PrF {r)D I Q . His optimal response is 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. similar to equations (5) and (6). For the modified retailer managed inventory system, denote the total supply chain cost as ]Q ^ (y ), parameterized by y that the manufacturer offers the retailer. Denote, in the case of the VMI system, the total supply chain cost by ris C(f?), parameterized by R, the rent offered by the retailer to the manufacturer. In Theorem 3.1, we essentially showed the existence of R*, such that II £ c (^*) - (0). Using a very similar proof technique (which is omitted), we can show the following stronger result. Observation 3.3.2: Let Ar <5Am andPm >Pr. Given y>0, there exists R*(y) such that risC(i?*(x)) < n "c O ) • 3.4. Two Param eter contracts. We now examine other contracts for the continuous review model in section 3, which are efficient and more flexible. We consider the case where the players incur a penalty for every stock out occasion at the retailer. The retailer is the Stackelberg leader and announces contracts to which the manufacturer responds. Let us denote a contract offered by the retailer to the manufacturer as S(Q,r). This implies that the manufacturer's cost function is as follows: n v u (Q,r) = Am DIQ + Pm DF(r)/Q + S(Q,r) The function S(Q,r) can be quite general but should be such that the manufacturer's cost function is convex. The holding cost subsidy contract discussed in the previous section is one such example. Our results from the 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. previous section imply the following: There exists a linear contract S*(Q, r) = R(Q/2+r) such that by adopting this contract, under certain conditions, the channel lowers its costs as compared to a RMI system. By "adoption", we imply that the retailer offers a contract S(Q, r) and the manufacturer chooses an optimal response (Q, r). Now, consider a contract o f the form Si(Q, r) ~ aQ + pr, which can be thought of as a generalized version of a HCS contract. When the retailer offers such a contract, the manufacturer responds by picking a pair (Q, r) that represents his optimal response. Clearly, this response depends on Sj(Q, r) which is completely characterized by (a, P). Lemma 3.4.1: There exists a contract S'(Q,r) such that the manufacturer's response selects the channel coordinating solution. Consider now an alternative contract S2(Q, r) wherein the manufacturer pays a rent R on average inventory but the retailer subsidizes the manufacturer’s replenishment cost. In particular, the manufacturer pays only a fraction a of his replenishment cost. The manufacturer’s cost is: n:,(Q ,r) = A,D /Q + P „D F (r)/Q + S(Q,r) = aA„D/Q + PmD F (r)/Q + R(Q /2 + r) Lemma 3.4.2: There exists a contract S * 2{Q,r)such that the manufacturer's response selects the channel coordinating solution. While the contracts SfQ , r) and S2(Q, r) can coordinate the channel, they cannot allocate arbitrary fractions of the total supply chain cost between the 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. players. However, we can get stronger results than in section 3. Let [dM, dR) represent the maximum cost that the two players would accept. It is natural that these costs are equal to their costs in the RMI system, i.e. (dM ,d R) ^ (I lf,,FI*). Indeed one should expect any new contract to improve the positions of the players. When we examined a rent contract of the form R(Q/2+r) , we could show that a VMI system performs better than an RMI system under certain conditions. Moreover, we needed additional conditions to ensure that both players can lower their costs in a VMI system. We will now show that such conditions are not required with a two parameter contract of the form Si(Q, r). A similar result is true for S2(Q, r) but is omitted. So, we now have the following: Theorem 3.4.1: With a contract o f the type Sj(Q, r), the VMI system can achieve lower costs than an RMI system and furthermore, n R M(Sl)< d M a n d U R R(Sl) < d R. As we had remarked earlier, in a single period newsboy type problem, results on equivalence of certain two parameter contracts have been proposed. See for instance Cachon & Lariviere (2001) who show that a revenue sharing contract is essentially equivalent to a buy back contract. We will now see that equivalence of such two parameter contracts is not generally true in a continuous review model. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Two contracts Sa and Sb are equivalent (Sa = Sb) if and only if (U M(Sa) , n R(S J )= (flM(Sb),IlR{Sb)) where we have omitted the superscript to signify that this equivalence may be applicable to any set of systems. Lem m a 3.4.3: Si(Q, r) is not equivalent to S2(Q, r). Proceeding along the same lines, we can examine contracts with more than two parameters. Indeed it can be shown that contracts with more than two parameters are powerful in that they achieve arbitrary allocations of the first best profit. But they are clearly neither simple nor easy to implement. 3.5. The periodic review model. In this section, we model the same problem analyzed in the previous section using a periodic review cost formulation and show that a “rent” contract offered by the retailer improves system performance under certain conditions. By “rent”, we again mean a holding cost subsidy specified by the retailer and in return the manufacturer makes the inventory decisions (quantity and replenishment schedule). As before we have a single manufacturer supplying to a single retailer. Before we embark on an analysis of the contract, we briefly describe the model formulation. There are several models that are used to analyze periodic review systems (Silver, Pyke and Peterson (1998), Zipkin (2000)). We use the (Q,T) policy, where T represents the time between replenishment orders and Q is the order-up to level. Thus at each instance of review, a sufficient amount is ordered to bring 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the inventory position to a level Q . We assume that there are no crossovers of orders. Analysis of the (Q,T) system has been sparse in the literature as compared to continuous review models. Some of these papers have assumed that Thas been pre-specified; see for example Hopp and Kuo (1998). Atkins and Iyogun (1998) provide a heuristic approach and computational results for this problem. In a recent paper Rao (2003) compares the (Q, T) policy with the continuous review policy and provides several analytical results. To our knowledge, there has been no work that analyzes contracting mechanisms in a (Q, I) system. Our interest is in exploring this issue and not in computing exact inventory policies. The structure of the supply chain, the players and their motivations, and the cost terms are essentially the same as in section 3. The average demand during the horizon is D and the lead-time is constant. The demand during lead-time is random with a mean //. We will now present the analysis for an arbitrary decision maker. The decision maker is the retailer in an RMI system and the manufacturer in a VMI system. We will assume that excess demand is back ordered. As before, the model can be extended to cover situations with lost sales. In the continuous review model, we extended the result to the case with penalty per unit stocked out; a similar extension applies here too. The formulation bears some resemblance to the continuous review model, but there is an important difference. If the lead time is L, an order placed at time t will arrive at time (t + L). Observe that as soon as the order is placed at t, the inventory position reverts to Q. Also, the next order arrives at time (t + L + T). So, 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a back order occurs only if the demand during the time period (L + T) is greater than Q. Thus, unlike the continuous review case the probability that a backorder occurs in any period depends on the choice o f both the order up to level and the time between orders. This makes the cost formulations different and complicates the analysis. Consistent with our analysis of the continuous review model, we propose a demand density function f(y,T) = aTe*®. We then have the following result. G O Lem m a 3.5.1: 1. ^f{y,T)dy = 1; o A 2. For any A > 0, \f(y ,T )d y is non-increasing in T . 0 1 0 0 3. — \f(y,T )d y is jointly convex in (Q ,T). t q The above lemma establishes the following: First, it establishes the legitimacy of the demand function as a probability density function. Second, the lemma ensures that as the time between orders increases, the probability o f stock out increases as well. Finally, it helps to establish that the cost function, Yl(Q, T), to be described shortly, is jointly convex in (Q,T). Before we analyze the FICS contract, we make a key observation. The order up to level Q can be restated as the sum of the safety stock level, say, s and the demand between successive orders DT. Flence, the decision maker now determines (s, T) rather than (Q, T). 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As before, we denote a RMI system as one where the retailer chooses (s, T). We compare the performance of this system with a VMI system in which the retailer charges the manufacturer R per unit average inventory and in return allows the manufacturer to choose the values o f (s, T). As before, we compare the two systems by looking at the system-wide or total supply chain cost. The question that interests us is: Is there a value R that the retailer can charge the manufacturer so that the system wide costs are lowered and in addition, each player can benefit relative to the RMI system? We first state the cost functions faced by the different players in the different systems. While the cost formulation we present is an approximate one, it compares well with the exact cost formulation (Hadley & Whitin (1963)). First, the total supply chain cost that is minimized in the centralized system is: n c x ^ J ) = ^ L^ ml + H(s + ^ f - M) + ^ S A ) f ( y , T ) d y , 1 s+DT In a retailer managed system (RMI), the retailer makes decisions by solving M in n R R(s,T) = M in ^ - + H (S + ^ - M) + ^ ) f ( y ,T ) d y ; (s,n (s,T) T 2 T s+J D T When the retailer charges the manufacturer a HCS and in return allows him to make the replenishment decisions in a VMI system, the manufacturer follows a policy that solves 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Minn V M(s,T) = M i n ^ - + R(s + — -ju) + ^ ff(y ,T )d y . (,,r) M V (s,T) T I T Jn^ s+DT We now show that a VMI system has lower system costs than a RMI system under certain conditions. A P Theorem 3.5.1: Assume that — — > — ^ . Then there exists R such that the total Ar Pr supply chain cost in the VMI system is lower than the equivalent cost in the RMI system, i.e. F Isc C ^ ) ^ n * c Cs,r). Thus we have shown that using a HCS, the retailer can lower system costs. Indeed one can further argue that by designing side payments, both players can be better off in a VMI system. However, using only a HCS contract without any side payments, one can show under additional conditions that both players are better off than in a RMI system, as in the continuous review case. We note that this contract does not coordinate the channel for reasons similar to those identified in the continuous review case. Despite this limitation, it is attractive due to its inherent simplicity. One can, as before, propose variants of this contract and check when the channel is coordinated and what type of contracts are first best, etc. We have presented such an analysis for the continuous review model (Section 3.4) and to be concise, do not repeat it here. 3.6. Conclusions and Future Research This paper addresses the issue of misalignment in incentives of players in a supply chain that can lead to sub-optimal channel performance. We consider a 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. system where a manufacturer supplies a single product to a retailer who faces random demand. We consider retailer-managed inventory systems, where the retailer places orders and makes replenishment decisions as well as vendor- managed inventory systems, wherein the vendor or manufacturer makes these decisions. In the VMI system, the retailer proposes a contract to the manufacturer that influences the manufacturer’s replenishment decisions and thus improves system performance. We showed that there exist simple holding-cost subsidy based contracts that can improve system performance as well as more complex contracts that are first best. We were able to prove these results both when the decision-maker reviews inventory continuously and follows a (Q, r) policy as well as when inventory is reviewed periodically using a (Q, T) policy. In future research, we plan to consider the following extensions. First, we would like to model a scenario wherein there are two manufacturers selling partially substitutable goods at the retailer and analyze the performance of the type of contracts discussed in the paper. Second, we would like to consider situations where there is a constraint on the amount o f inventory that can be held at the retailer and explore whether the types of contracts discussed here would be effective in such situations. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. Issues in stocking substitutable products. 4.1. Introduction This paper attempts to explore the impact of demand substitution effects on the optimal inventory decisions of a retailer. In particular, we consider a scenario with two products whose demands are negatively correlated and that are partial substitutes. The substitutes may be items purchased from competing manufacturers, for example Duracell and Energizer batteries, or from the same manufacturer, say Dreyer’s vanilla and chocolate ice creams. The retailer knows the total demand and the average demand proportions for the two items, say A and B, but the proportion of customers who come in for either item on any particular day is random and so the retailer may stock out of one or both products. A fixed proportion y of unsatisfied customers for a product will purchase the other one if it is available in stock. This scenario, though stylized, captures two important elements in stocking substitutable products: when items are substitutes, their demand is negatively correlated and out-of-stock situations result in partial substitution o f one item for the other. The revenue and cost tradeoffs used to determine stocking levels are similar to those found in the inventory literature. We analyze this model to answer questions such as: What is the optimal inventory policy for the two products in both single-period “newsboy” type situations as well as multi-period (finite and infinite horizon) scenarios that would maximize the retailer’s profits? One would expect the optimal inventory level for 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. an item to be a function o f the inventory level of the other item. However, we find that for a reasonable set of parameter values, the optimal inventory level of a product is independent of the other product’s inventory in both single and multi period situations (in fact, this is more likely in multi-period situations) - we call these “decoupled” policies. Further, the optimal inventory levels can be computed easily using closed-form formulas. The intuition behind our results may also offer insights into developing simple but effective inventory policies for more complex scenarios with multiple products, etc. Moreover, the decoupled and closed-form nature of the results offers the potential to build upon these results and analyze more complex problems. We also demonstrate analytically some of the anomalies that arise in the behavior o f optimal inventory levels in the single-period model: specifically, we show that inventory levels may increase (i) with an increase in the substitution fraction y and (ii) with a decline in the variance in demand under certain conditions. We also consider the case where two competing decision makers, say the manufacturers of A and B, determine the optimal inventory levels of the products. We study the dynamic competitive version of this game in two settings: (i) when the demands for the two products are uncorrelated and (ii) when the demands are negatively correlated (as in the single decision-maker model discussed earlier). We characterize the unique infinite horizon Nash equilibrium in such games, demonstrate that it is myopic and compute it as a limit of the equilibrium of finite horizon games. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There is a substantial literature in inventory management and there are several papers that have investigated inventory policies when there is demand substitution under various scenarios (see Mahajan and Van Ryzin (1998) for a recent survey). However, models with substitution effects tend to be inherently hard and so only a few analytical results have been obtained in scenarios that allow partial substitution. As Mahajan and Van Ryzin (1998) point out, dynamic substitution models, where the substitution behavior depends on the inventory status at the time a customer makes his choice (as in our case), are more realistic but are typically less tractable and the profit functions are in general quite complex and not easily amenable to analysis. Since the inventory literature is vast, we only review the papers that address demand substitution and are directly related to this work. One of the earliest papers to address substitution effects in a multi-product scenario is Ignall and Veinott (1969) where they study a periodic review multi period inventory problem with full substitution when there are stockouts. Bassok, Anupindi and Akella (1999) consider a model with full downward substitution and obtain optimal inventory levels. McGillivary and Silver (1978) examine the impact of partial substitution when there are stockouts in a two-product problem and develop some heuristics. Parlar and Goyal (1984) also study the partial substitution problem and show that the expected profit function is concave. Ernst and Kouvelis (1999) study a problem of packaged goods with three substitutable products and conduct an extensive numerical analysis of the optimal policy. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pasternack and Drezner (1991) show some interesting characteristics of optimal inventory levels in a two-product problem with full substitution. Gerchak and Mossman (1992) showed that, contrary to intuition, substitution might result in an increase in optimal inventory levels under certain cost and demand conditions. Netessine and Rudi (2003) study the optimal inventory system with substitution with and without retail competition and show that optimal inventory levels are higher in the competitive scenario. Anupindi and Bassok (1999) compare inventory levels and profits in centralized versus decentralized systems and find that optimal profits may be lower in a centralized system. In the above papers, while substitution effects are modeled in some detail, demand for the products are not correlated. There is another recent stream of literature, representative works being Smith and Agarwal (2000) and Mahajan and Van Ryzin (2001a) which study inventory systems where consumer choice and substitution effects are modeled in great detail and substitution in out-of-stock situations is probabilistic. The focus of this work is on determining both optimal product assortments as well as inventory levels as a function of consumer purchase behavior and they consider single-period scenarios. To our knowledge, there are only a few recent papers that study substitution effects when demand is correlated. One of them is Rajaram and Tang (2001), where the impact of partial substitution and correlation in demand is studied in a single-period model. They indicate that characterizing exact optimal 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. policies analytically is complex and so they develop heuristics to compute inventory levels when demand is a bivariate normal distribution. More recently, Yang and Schrage (2002) show that increased substitution may result in higher inventory levels under certain conditions and discuss other interesting anomalies related to risk pooling. While they provide analytical results in the full substitution case, they provide numerical results for the partial substitution case when demand is correlated. Models where two independent players manage the inventory levels of the two substitutable products and thus compete have received considerable attention, beginning with Parlar (1988) and followed by later papers including Anupindi & Bassok (1999). The analysis has mainly been for the case of a static game (single period) with the strategy space o f the players being the stocking decisions (purchasing quantities). In a more recent paper, Netissine, Rudi and Wang (2003) analyze a dynamic version of this problem with full substitution and i.i.d. demands and provide several interesting results. The organization of this paper is as follows. In section 2, we present the single period model in great detail. We establish some properties of the profit function and calculate the optimal inventory levels. We provide closed form expressions. In section 3, we present our analysis of the finite and infinite horizon problems and characterize the optimal policy for a wide range of parameter values. In section 4, we analyze the sensitivity of the optimal inventory levels to (i) the substitution parameter y , (ii) the variance of the demand distribution. We 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. demonstrate certain anomalies that the optimal inventory levels exhibit. In section 5, we present a competitive version o f the model and demonstrate properties of the Nash equilibrium. Finally in section 6, we present some of our ongoing and future research. Several of the proofs (and sometimes the outlines) are in the main body of the text. However, due to paucity of space, more detailed proofs are deferred to a technical appendix. 4.2. The Single Period Model Consider a retailer stocking two products, say 1 and 2, which are partial substitutes for sale in a single period. To focus attention on the demand substitution effects, we study a model where the total demand for both products is constant but the demand for each product is uncertain. Let the total demand be denoted by D; demand for product A is p*d while it is (1- p)*D for product B where p is a random variable taking values in the range [0,1]. We assume that p has a continuous distribution function denoted by F. A fixed proportion y of the customers who come in for a product and do not find it in stock will switch to the other product and buy it if it is available, while the proportion (1- y) will walk away. Thus, the products are substitutes in two respects: their demands are negatively correlated and unsatisfied customers for one product may switch to the other. A typical example of such a scenario is an airline that often knows the total number of passengers on a flight ahead o f time and offers two types of entrees for lunch but does not know the proportion that will choose each entree. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We assume that all the parameters are the same for both products. The retailer purchases the items at a unit cost c and obtains a revenue s by selling one unit. The retailer incurs a holding cost h per unit of unsold inventory, which is equal to the starting inventory (the purchased quantity) less the inherent demand for the product and the demand from customers who switch. Further, a penalty cost k is incurfed per instance of an unsatisfied customer. Unsatisfied customers include customers who when faced with a stock out of their first preference were not willing to substitute and those who were willing to switch but could not find the other product. We assume that c, n, h, s > 0 and s > c. These simple assumptions on the cost parameters ensure that it is profitable to sell to the original demand rather than deny a sale and possibly substituting. The assumptions satisfy all requirements of the usual single period newsvendor problem and problems with substitutable products. See for instance Silver and Peterson [1985] and Pasternack and Drezner [1991]. Salvage value for excess inventory can be incorporated easily. The retailer has to decide how much of each product to stock to maximize expected profits. We first demonstrate that the problem is indeed concave. The demand for the two products are pD and (l-p)D respectively. Let the order quantities of the two products be Q\ = aD and Qi = f3D. We specify two different symbols, a and P, for the two order quantities even though we assume identical parameters for both items for ease o f exposition. We represent the actual demand realization by di. Naturally, for any realization of the random variable p, the demands are 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d\ = pD and d j = (1 - p)D. Let r / be the number of demand units of product i satisfied by product j (i, j = 1,2). Note that given demand realizations dt and initial inventory levels Qj, r / is deterministic. Let o; - and «z - represent the excess inventory and shortage respectively of the two products, after allowing for demand substitutions. Thus, these represent the actual inventory excesses and shortages at the end of the period. Also, W.L.O.G. we assume that D=1. We now write the profit function FI(Q)- To do so, we use p{Q ,p) to express the profit conditioned on a given demand realizationp, for any Q = (Q\,Q 2 )- hi fact, 2 n<Q) = + \p(Q ,p)dF (p) (4.1) 1= 1 where, 2 2 2 2 p (Q ,p ) = Max { n it v-J 7 = 1 r‘ = 1 7 = 1 (4.2) (io.,u.,r,) 1 = 17=1 1=1 =1 1=1 subject to: 2 7=1 (4.3) (4.4) °l ’r/ > ui > o ; i,j = 1,2. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem 4.2.1: The expected profit function ]Q(Q) is concave and continuously differentiable. Proof: Clearly it suffices to show that Jp(Q, p)dF(p) is concave. The allocation problem p(Q ,p) = Max I(o, u, f) is concave in Q because I is linear in the (o.,u.,r/ ) 'iii' decision variables. Thus from proposition 7 in Van Slyke and Wets [1966], FI(Q)is concave in Q. This approach is used in Robinson [1990] and Bassok, Anupindi and Akella [1999]. Further, using Prop. 5 in Robinson [1990], it follows that ri(Q) is continuously differentiable. ♦ The above formulation is solely to demonstrate concavity. We now characterize the optimal policy and to that end, we first write E[(Q) explicitly and then arrive at the policy using first order conditions. The explicit formulation of II(Q) is typically quite messy and involves taking several cases. We adopt the following approach, which simplifies the formulation and the first order conditions thereof. n,(a,y?) a + J3> 1 Let n(Q)= In what follows, we will explicitly n 2(a, J3) a + /3 <1 derive fl2(a,(3), i.e. the case a + j3< 1. The derivation of fIi(a ,P ) is similar, though much simpler, and is relegated to the appendix. We consider all possible cases o f the realization of the random variable p to identify the exact formulation. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Case 1: a > p . Since a + p < 1, this immediately implies that 1 - p > J3. Hence (1 - p - P ) D is the excess demand of product 2 and y(l - p - /?) customers search the store for product 1 before leaving the store. Note that some or all of these searching customers can be satisfied by the excess inventory (a - p) o f Product 1. Thus we have to analyze further sub cases. To facilitate this analysis, let a * = a .@ 1 and p* = —P 1°^ . Note that a* < a and /?* > 1 - /?. (1 -r) (1 -r) Case l.A: p < a All the searching customers are satisfied sincep < a* = — —— —— < = > a - p > y(l - p - f i ) . Thus in this case, total sales 0 - 7 ) equals p + J3 + y ( l - p - ft) from sale of product 1, sale of product 2 and all the customers who substituted product 1 for product 2. The excess inventory (of product 1) after all substitution has occurred is (a - p) - y(l - p - J3). Further, the lost sales (of product 2) from the customers who were not willing to substitute is equal to (1 - y)( 1 - p - P ) . Case l.B: p > a In this case a - p < y { \ - p - (3), which implies that not all searching customers are satisfied. So, the total sales is equal to p + j3 + ( a - p ) = a + j3. Clearly since all inventory is used up by the originating demands and the searching customers, 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. there is no excess inventory at the end of the period. The lost sales equals (1 -y )(l - p - 0 ) + y(l - p - P ) - ( a - p ) = \-(a + (3). Case 2: p > a and (1 - p)> (3 = > p < I - . In this case, both products have excess demand and so there is no excess inventory at the end of the period. The total sales equals a + J3 and the lost sales equals 1 - (a + /3). C ase3: p > a and ( l - p ) < j 3 = > p > \ - / 3 . Here, we have excess demand for product 1 and excess inventory of Product 2 and some unsatisfied customers of product 1 are willing to substitute Product 2. As in Case 1, we have to consider sub cases: Case3.A: ft < p < (3*. P < P* < = > y (p - « ) > ( / ? - ( 1 - p)) which implies that not all searching customers are satisfied. Thus the total sales is a + /?, the lost sales is y{p - a ) - (/? - (l - /?)) + (l - y \ p - a) = 1 - a - ( 3 and there is no excess inventory at the end of the period. Case 3.B: < p <1. In this case, y{p - a ) < { p - { l - p ) ) and so all searching customers are satisfied. The total sales is a + (l - p)+ y(p - y ) , the lost sales is (l - y \ p - a ) and the inventory left over at the end of the period is (/? - (l - p ) ) - y ( p - a ) . 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Observe that the total sales, lost sales and excess inventory at the end of the period are identical in cases (IB), (2) and (3A) and these three cases can be combined into one scenario where a *< p< P*. So, n 2(a,P ) = J{s[p + j3 + y { l - p - p ) ] - h [ ( a - p ) - y { \ - p - p ) ] - r c { 1 - y ^ - p - P ) } d F { p ) 0 P * + \[s(a + p)-7t({-{(x +p))]dF(p) + J{s[a + ( l - p)+ y{p - a )]-h {p - (l - p ) - y(p - a ) ) - x ( l - y j p -a)}dF(p) p ’ - c (a + p ) ; A similar but somewhat simpler derivation, which is omitted, follows for the case a + p > 1. f\-p a 1 U i(cc,p) = s\ ^ \p + p + y { i-p -p )\p F { p )+ J dF(p) + J[(l - p ) + a + y(p -a)]dF (p) j [ 0 1 - p a - / z j ^ [ a - p - y { \ - p - p ) \ l F { p ) + j [ a - p + P - (I-p )] d F (p )+ ^ [ p - ( I - p ) - y ( p - a ) } l F ( p ) i-p ■ j [(1 - r ) 0 - - p -P)W(P) + J(1 - y )(p - a ) ^ ( ^ ) | - c(a + P) The function FI(Q)was shown to be concave in Q. It is easy to see that IIj(a ,p ) is strictly concave in (a,P). Thus the global maximum of ]~I(Q) is uniquely 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. attained. The optimal values o f (a, ft) can be obtained by solving the corresponding first order conditions of I I j( a ,P ) . We have the following: 2(fi - f - % Theorem 4.2.2: Let y* = 1---- . If y < y , the optimal policy is given by: s + 7t + h F(a) - s + 7 t ~c~y(s + 7l + ^) - i h + c s + ft + h - y ( s + 7T + h) ( 1 - y )(s + rc + h) (2.1) If y > y*, the optimal policy is given by: 1 h + c s + x - c F(a*) = l + y (1 + y)(s + 7r + h) (1 + y)(s + 7t + h) (2.2) where a* is a function of both a and p. Proof: Since the global maxima is uniquely attained and the function II(Q) is concave and continuously differentiable, we check the first order conditions forIT j(a,P ), i=l,2. Note that we have shown the explicit formulation o f ITj (a, P ). Taking derivatives, we have: ^ = ~h[\dF{p) + )ydF(p)] - * J(1 - y)dF(p) + s J(1 - y)d F (p )- c 0 a a a = ~h[jdF(p) + jyd F (p )]-n J ( l - y)dF(p) + s |(1 - y)dF (p)- c 0 I-/? 0 0 ATI a f t 1 - ~ L = -h jdF (p) + J(s + x)dF(p) + J[(l - y)(s + / r ) - yH]dF(p) - c iiiX n 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d n 2 da u y i = -h \dF (p) + \(s + 7t)dF(p)+ J[(l -y ){s + x ) -y H ] d F { p )-c Note the following: The maxima is unique and the first order conditions are satisfied uniquely. The function FI(Q) is actually a family of functions, parametrized by y. Thus, we need only determine, for what values o f y , the respective first order conditions hold. It is easy to check that = 0 and L = 0 are feasible only da dp when y < y*. Thus, setting the above first derivatives equal to zero in the respective feasible regions of y, we obtain the result. ♦ Salvage value for excess inventory can be incorporated easily - if v represents the salvage value, then we replace h with (h-v) in the above formulas. Observe that when y < y*, we have a “decoupled” inventory policy wherein the optimal base-stock level of a product is independent of the base stock level of the other product. This is quite striking given that the demand for the two products are negatively correlated and we have partial substitution. The condition y < y* is not very restrictive and higher the newsboy ratio, higher the value of y*. For instance, suppose the parameters are such that c = 0.2s, 0.15s and h - 0.12c and v = 0.5c. In this case, y* = 0.79 where the newsboy ratio for each product without substitution is 0.88. So, we find that for a wide range of y values between 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 and 0.79, the decoupled policy is optimal. We will find in the next section that in the multi-period problem, a decoupled policy is optimal for an even wider range of y values. We also discuss the intuition behind this result in the next section and speculate on its implications in more complex scenarios. Further, the closed-form nature o f the optimal inventory expression (2.1) allows for an analysis o f the sensitivity of the inventory levels to the substitution fraction and demand variance in section 4. For instance, it is transparent from (2.1) that optimal inventory levels will decline as the substitution fraction y increases when y < y * . 4.3 The M ulti-period Problem. In this section, we study the multi-period version of the substitution problem with correlated demand. Our assumptions on costs, demand distributions, etc. remain the same as in the single period problem. Our analysis proceeds as follows: We first examine the discounted (discount factor 8) infinite horizon problem. We then examine the corresponding problem over a finite time horizon using an inductive analysis. We restrict our attention in this section to values o f y, to be defined shortly, so that the total inventory level at the beginning of each period is at least D. i.e. a + ( 3 > 1. Finally, in all our analysis, we assume that unfulfilled demand (after all possible substitution takes place) at the end of a period is lost and is not backlogged. For ease of exposition, we also assume a zero lead time for replenishment. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Denote by G (Ii,I2) the total discounted expected profit obtained by starting with an initial inventory 7 , of product i and using an optimal policy. Henceforth, we use dF instead of dF(p) to be concise in our notation. Then, G satisfies a functional equation of the form: i-p G ( /,,/2) = ( M ax^JIl1(a ,j3 )-c (a + / 3 - I 1- I 2) + S jG (a - p - y ( \ - p - J3),0)dF a 1 + £ jG (a - p , p - { 1 - p))dF + 8 |G (0 ,/? -(1 -p )-y{p -cc ))d F ] 1-/3 a ... (4.3.1) where defined earlier represents the expected single period profit, excluding purchase cost. Define, 1-p T (a,p ,II,I 2 , G ) = n 1( a ,P ) - c ( a + p - I 1 - I 2) + 5 } G ( a - p - y ( l- p - p ) ,0 ) d F 0 a 1 + 5 }G(a - p, P - (1 - P))dF + 5 Jg(0, p - (1 - p) - y(p - a))dF l-P a Then (4.3.1) can be re-written as G(Ij ,I 2) = Max T (a ,p ,I1,I 2,G ). (a,P)>(I,,I2) Now let G o (Ii,I2)be a bounded and continuous function. Define inductively, for n=l,2...oo, G n (l!,I2) = Max T (a ,p ,I1,I 2,G n_1), for every (a,P)>(Ij,I2) 7, < oo, i = 1,2. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then, following Theorem 1 in Bellman, et al. (1955), we have that Lim G n (Il 5I 2) = G (l!,I2) • This essentially demonstrates the existence and n — >co uniqueness of the solution to (3.1). We now characterize the solution of the infinite horizon problem. To do so, we rewrite (3.1) as G (Ii ,I 2) = Max [V (a,p) + c ^ + 12) ] and solve the equations: Va = 0 and (<x,p)>(I,,I2) Vj3 = 0 simultaneously, where Va denotes a derivative with respect to a. Note that l-p Va = n la ( a ,P ) - c + 8 jG a ( a - p - y (l- p - p),0)dF O a 1 J«G „ (a - p, p - (1 - p))dF + j 7G a (0, p - (1 - p) - r(p - a))dF 1-P a The expression for Vp is similar. Further, note that when y < y and the order quantities are such that a + P > 1, we have that G a = Gp = c . This immediately implies that ra(a-) = 0=>F(a') = {s + ’t - c)- r{s + ,I + h - Sc) (4.3.2) ° (s + 7 u + h -S c )(l-y) , 2(h + (1 - 5)c) _ (s + 7 r - c) - y(s + 7 r + h - 5c) ^ _ _ Note that y < 1— -----—— — =» F (a') = — ---------— Li— — 1 > 0.5 . s + 7 t + h — 8c (s + 7 t + h - 5c)(l - y) Further, Vaa =[c-(s + /r + h)](l-y) < 0 and thus V is concave with respect to each variable. An identical expression can be derived for /3'. Thus the optimal policy for the infinite horizon problem is to follow a stationary policy of ordering 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. up to a' and/? '(= « ') in every period in which the initial inventory is below this order up to level. Thus if we let y ® = min(l - ^ + — —^ ^ - ,1 ) and noting that y * < y ® , we have s + k + h - Sc essentially established the following: Theorem 4.3.1: Let y <y®. Then the optimal policy for the infinite horizon problem is to follow a stationary base stock policy with order up to levels a' and /?'. Thus, we find that our earlier result for the single period problem extends to the infinite horizon problem too. That is, for certain y values, we have a decoupled policy where the optimal inventory levels of the two products are determined independently. In fact, the range of y values is wider in this case and can even extend to a value close to 1. For instance, if c = 0.2s, n= 0.15s and h ~ 0.12c and 8 - 0.95c, then y ® = 0.93. The decoupling effect is sharpened in an infinite horizon setting, where the optimal inventory level is typically higher than in a single period setting for the same set of cost parameters. Thus, the retailer often can determine the optimal inventory levels for each product, ignoring the inventory level of the other product. The intuition behind this rather striking result is as follows. Note that when the sum of the inventory levels of the two products is at least D, i.e. a + f}> 1, then all the customers who are willing to substitute are satisfied, i.e. they will not leave the system unsatisfied. Thus, in this case, the marginal benefit of stocking an extra unit of 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. either product does not arise from the opportunity to meet the demand arising from a substituting customer. So, a retailer can effectively ignore the other product in making inventory decisions for one product even in a multi-period setting for a wide range of substitution rates. Based on our results and this intuition, there is perhaps a more general conjecture one can make. We observed that if the cost parameters are such that the inventory level o f an item is sufficiently high, then all substitution demand is fully satisfied and so the marginal benefit of additional inventory does not come from substitution demand and in turn the optimal inventory level of the item is not impacted by the inventory level of a substitute. One may then expect that a similar phenomenon might occur even in more general inventory models with multiple products, if the newsboy ratio is reasonably high for an item (in turn implying high values of y ®), the substitution fractions are not too high and the demands for the substitute items are negatively correlated (even if weakly so). Hence, one might be able to compute inventory levels of an item using an equation such as (2.1) or (3.2) which ignores the inventory levels of substitute items and they may be near optimal in the original problem. This conjecture, if validated, can be very valuable since substitution problems are in general very complex to analyze. Finite horizon case Having examined the infinite horizon problem, we briefly discuss the finite horizon version of the same problem. If G n (Ij, 12 ) represents the expected profit 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. when there are n periods until the end of the horizon and an optimal policy is used, then the corresponding dynamic program is as follows: G i(Ii,I2) = Max [fI1( a ,P ) - c ( a + p - I i - I 2 )] and (a,P)>(I,,I2) 1-P G n+i( I i,I 2) = Max [ n 1( a ,P ) - c ( a + p - I 1 - I 2) + § }Gn ( a - p - y ( l - p - P ) , 0)dF (a,p)>(I„I2) q a 1 + 5 |G n ( a - p , p - ( l - p ) ) d F + 5 jG n ( 0 ,p - ( l- p ) - y ( p - a ) ) d F ] 1~P a From the analysis of the single period problem in the previous section, we know that the optimal policy when there is exactly one period until the end of the horizon is to follow a base stock policy for each product with an order up to level , . (s + 7 t - c ) - y { s + 7i + h) given by: F (a , ) = --------- — ----------- . Rewntmg (1 - y)(s + 7r + h) G n+](I1,I 2) = Max [Vn (a,P) + c(I] + I 2)], we have from our previous (a,p)>(I„I2) analysis that is concave. Using standard induction arguments, that are similar in spirit to those used in the proofs of theorem 5.1 and 5.3 (which we provide later), the following result is obtained. Lemma 4.3.1: Vn is concave in both variables. The optimal policy for every n is as follows: 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. c(Il+ I 2) + Vn(a:,p:); Ix < an,I2 <p: c(I1+I2) + Vn(a:,I2y, /, < a nJ 2 >p: Gn( h A ) = + I x> a n,I2 <p: c{Ix+I2) + Vn{Ix,I2y /, > a nJ 2 > pi % & where, Vn ( a n , (3 ) = 0 and Vn (a, (3 n ) = 0. a fi * * 1 r» * r»* Moreover, a n+l > a n and pn+l > pn . Thus, we are able to extend the results for the single product problem to our model. 4.4. Sensitivity analysis. Intuition suggests that when the substitution fraction y increases, the optimal inventory held in expectation of the demand should decrease. This is due to the increasing significance of risk pooling benefits as products become closer substitutes in stockout situations. However, the existing substitution literature suggests that this need not always be the case and has pointed out numerous anomalies in the behavior of inventory levels with increasing substitution. The focus of this section is not to perform a detailed analysis of this issue, but rather to report certain anomalies in the behavior of inventory levels with increasing substitution and demand variance and demonstrate them analytically in the presence of correlation and demand substitution. Gerchak and Mossman ( l 992) and Pasternack and Drezner (1991) demonstrate that inventory levels may actually increase when the level of 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. substitution is higher and in particular demonstrate this anomaly for the case of full substitution (y = 1) for specific demand distributions. In a more recent paper Yang and Schrage (2003) demonstrate that inventory anomalies exist even when there is partial substitution for certain ranges of costs and for specific types of demand distribution. Though most of their results are obtained for the i.i.d case, they demonstrate using numerical simulations that anomalies exist and are sharper when there is correlation between the demand for the two products. A general difficulty in analyzing the optimal policy for the case of partial substitution arises due to the complexity of the first order optimality conditions. As pointed out by Yang and Schrage (2003), this limits their analysis to specific distributions. However, it is interesting to note that a common structure that several of these problems (including ours) possess is that they are concave, continuously differentiable with respect to the decision variables and the first order conditions are continuous (both with respect to the decision variables and with respect to y) and have a unique solution. This immediately implies the following result that the optimal ordering quantity is continuous with respect to y, at least when 0 < y < 1. Since the following result is true in general, for a wide range of substitution problems that satisfy the above mentioned conditions, we use slightly different notation than the one used in earlier sections. Denote the unique first order conditions of the concave and continuously differentiable function as J(Q, y) = 0 and let Q*(y) be its solution. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lemma 4.4.1: Q*(y) is continuous in y for 0 < y < 1. Proof: We need to show that Lim Q*(y) = Q*(y0), 0 < y0 < 1. It is sufficient to r-*r<s show that LimQ*(a0 +e) = Q*(a0) . Suppose the contrary. Then there exists s-> 0 sequence sn -> 0 such that Q*(a0 +sn) does not -> Q*(a0). But this implies that there further exists a subsequence sn -> 0 such that Q* (a0 + sn.) -> j3 (say) and t(j) := J (Q* (a 0 + snj ),aQ + s n,) = 0 V j. Since J(Q, y) is continuous with respect to its parameters, t(j) -> J(J3,a0)=> J(/3,a0) = 0. But J(Q*(a0),a0) = 0. Uniqueness o f the first order condition leaves us with a contradiction. ♦ We now examine our model where the individual demand for the two products are negatively correlated and study the effects o f the substitution parameter y on the optimal inventory levels. First of all, note that when y = 1, the optimal decision is to hold a total of D units of inventory, with any arbitrary allocation of inventory to each product. On the other hand, when y = 0, if the holding cost h is very large, the total inventory levels of the two products may be less than D. Thus what is commonly referred to as an inventory anomaly for the case of full substitution is trivially observed here. Since we are interested in exploring the effect of y on the optimal inventory level, in the spirit of the above lemma, we represent the optimal inventory for any y as cc(y). From the above discussion, we have that a(y) is 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. continuous with respect to y. The aforementioned difficulty in analyzing the optimal policy is mitigated to some extent in our analysis in the case where y < y*, as we have closed-form expressions for computing a that are fairly easy to analyze. Lemma 4.4.2: If y < y*, then a(y) is non-increasing and convex. The result is easily demonstrated by differentiating the expression (2.1) for the optimal inventory level with respect to y . Note that no assumption on the demand distribution is required. This result is consistent with the commonly held intuition that inventory levels decrease when the substitution parameter increases. However, the analysis of y > y* presents some difficulties that are overcome if specific distributions are assumed. We now present some results assuming that p is distributed as U niform [0,1]. Lemma 4.4.3: Let p be uniformly distributed in the range [0, 1]. There exists 1 > y° > y * such that when y < y°, a(y) is non-increasing and when y > y°, a(y) is non-decreasing. Moreover a(y) is convex. Proofs of the above two lemmas: When y < y*, we have that a (y) = -------------------— -------- . This demonstrates the (s + re + h)(l - y )f(a ) claim in Lemma 4.2. To prove the remaining claims, we henceforth assume thatp 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. follows a uniform distribution between [0 1]. We observe that when y > y*, i L(yl a (y ) = —----------- — -------—, where ( s + 7 i + h)(l + y ) L(y) = -2(s + n - c)(l + y) + (1 - 2y2 )h + 2y(l + y)(s + 7 r + h ) . Assuming that s + 7 i > h + 2c implies that L(1)>0. Further, note that L is continuous and L(0.5)<0. This proves the existence of y °. Assuming p is uniform, it is straightforward to check the convexity of a (y ). ♦ Note that lemma 4.3 indicates that inventory levels actually increase when the level of substitution increases. It is interesting to note that when the demands are perfectly negatively correlated, for a wide range of very reasonable cost structures (indeed, the only assumption on the costs is that s + T t > h+2c), there is always a region of substitution values where an inventory anomaly exists. This is contrary to the anomalies reported in the literature for the i.i.d. cases where the results are restricted to certain narrow ranges o f the newsboy ratio. For instance, Yang and Schrage show the anomaly analytically when demand is i.i.d, has a normal distribution and the newsboy ratio or fractile is between 0.5 and 0.68. Observation: A simple observation from the above discussion is that if we assume a uniform distribution for demand, the optimal inventory level is a non monotonic convex function of y. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Note that we have used the fact that the demand is uniformly distributed. In fact, the structure of the above result holds for several other distributions with compact [0, 1 ] support. Impact of variance Another aspect of interest is certain singularities that the optimal inventory levels exhibit when the variance o f the demand process changes. If all else is kept constant, one may expect that, for reasonable values of the newsboy ratio, as variance o f the demand increases, the optimal inventory levels are likely to increase. However, consider the following analysis. 4x; x < 0.5 Let 0 < x < 1 and consider the distributions f x (x) = - 4x + 4; x > 0.5 and f 2 (x) = 1 / x. Note that f\{x) represents the density o f a triangular demand process while ^(x ) represents the density of the uniform demand process. We alternatively let p be distributed in the above two demand modes and observe their respective impact on the inventory levels. Denote by ( X j ( y ) the optimal inventory levels as a function of y, in the two scenarios, i=T,2. Further assume that s + n> h. A simple analysis yields the following observation: Observation: There exists 0 e {0,1} such that when 0 < y < 0, oi2 (y) ^ otj(y) and when 0 < y < 1, we have 0C 2 (y) - ai(y) • Clearly, the uniform demand process has a higher variance than the triangular distribution. There is a range of values of the substitution fraction y 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the optimal inventory level is greater if the variance is higher. Note that this seems intuitively clear if y is small, since the products are then essentially non-substitutes. However, for values of y > 0, this no longer holds. In fact, we conjecture that given any 0 < k < 1, we can find two demand processes, one with a higher variance than the other, such that the optimal order quantity for the higher variance process is lower than that of the lower variance process when 1 > y > k . Thus, in this section, we note certain anomalies in optimal inventory levels that arise in the presence of risk pooling and variance of the demand process, when the demand is negatively correlated. Also, unlike some of the results in prior literature, our results are not restricted to a narrow range o f newsboy ratios that tend to be lower than typical newsboy ratios (service levels) in industry. In future work, we plan to explore some of these anomalies when the products are asymmetric and w henp has a support that is strictly contained in [0, 1]. 4.5. The multi-period competitive version In this section, we examine a competitive version o f the problem. To do so, we consider a situation where two manufacturers (denoted as Player 1 & 2 respectively) compete in a market by selling substitutable goods. The demand and substitution process remain exactly the same as before. The essential difference here is that the inventory level chosen by the two players arises as an equilibrium 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solution to a game with competitive strategies employed by the players. We assume that the two players are symmetric. There is an important difference in the lost sales penalty cost structure here. In the previous sections, we assumed that a penalty is incurred per unit of lost sale where a lost sale is counted only when an unsatisfied customer walks out of the retail outlet. In the present scenario, when the two players compete, a player incurs a lost sale (and consequently a penalty) when he is unable to satisfy a customer originating from his demand stream. Thus he incurs a penalty, even though the unsatisfied customer may switch to the second player’s product and eventually be satisfied. Thus, even though the customer exits the system satisfied, a player may incur a penalty. We are interested in examining two situations: (1) The competitive version of the problem discussed in the previous sections which we call the CCID (Competitive-correlated identical demand) problem. (2) The competitive version o f the classical two-product substitution problem with i.i.d demands. We call this the CIID (Competitive-independent identical demand) problem. In this section, we focus on the latter problem. The analysis for the case o f correlated demands is quite similar in spirit, but more involved. The single period substitutable problem with multiple players, for the case of uncorrelated demands, has been well analyzed. Parlar [1988] showed the existence of a unique Nash equilibrium in a two-player game. Anupindi & Bassok [1999] look at the effects of the of the substitution parameter y on equilibrium 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. base stock levels. Lippman & McCardle [1997] analyze inventory levels as a function of the substitution parameter, using a competitive version of the newsvendor problem. Mahajan and Van Ryzin (2001b) analyze a model where n firms provide substitutable goods and consumers choose among them based on their availability and characterize the Nash equilibrium in a single-period game and provide many interesting properties. We are interested in studying the properties of this equilibrium in the context of a multi-period problem, in particular the infinite horizon problem. In what follows, we outline the steps followed in this analysis and examine each step in greater detail. To examine the infinite horizon substitutable product problem, we make a few restrictive assumptions. First, we only look at static policies where players make all decisions in the first period. Thus we are not concerned about the various interesting issues that arise when strategies include making decisions dynamically over time. Further, we restrict ourselves to the strategy space of stationary base stock policies. Finally, we assume that unsatisfied demand is lost and not backlogged. We characterize the Nash equilibrium for this problem using the following steps. First, we consider a finite horizon problem in periods). Assume that player 1 follows a stationary base stock strategy (whereby, he orders up to a in every period). The optimal response o f Player 2 to this strategy of Player 1 is then determined. Player 2 ’s optimal n-period strategy o f order up to levels, though non-stationary, converges. In fact, if we allow n to approach infinity, Player 2 ’s 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strategy converges to his optimal stationary response in the infinite horizon problem. Further, if Player 1 is restricted to stationary base stock policies, the optimal response o f Player 2 for the infinite horizon problem is also stationary. We then prove the existence of a unique Nash equilibrium for the infinite horizon problem (restricting ourselves to stationary base stock policies). Then using Sobel [1981], the Nash equilibrium is myopic and thus it can be derived as the Nash equilibrium of a suitably defined single period problem. Denote now by flj(a,(3)the single period, expected profit o f player i, when inventory levels after ordering, but before demand realization are (a,j3).The holding costs and the revenue obtained through sale are exactly as in the previous problem. Denote the penalty due to a lost sale as n m . If we momentarily ignore the purchasing cost c, we have: a n , (a, J3)= J j W , - h(a - dx)}dFx dF2 0 0 a y + J \[sd, + sr (d2 - f ! ) - h ( a - d , - r(d 2 - P)]dF,dF, 0 p a o o d o + J | sadFldF2 + J[^a - nm {dx - a)]dFx dF2 0 p i a ~d i « r Let - G y n (In ,(3) be player l ’s maximum expected payoff when he faces n periods until the end of the horizon, where In is player l ’s initial inventory in the 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. current period and ( 3 is player 2’s stationary stock level after replenishment. Thus we have: (DP) Gu (In,fi) = Min £*>/„ c(a-I„)+L (a,P)+ j\SGx _,_x( a -</, - r(d 2 -/?)*, 0 0 where, L(oc,P) = - n ^ o i jP ) . Let Vn (a, P) be such that: a> I„ Vn (a, P) is important for our analysis. Note that, if Vn (a, P) is convex in a , for a given P , then it is easy to show that the optimal response o f the first player is an order up to policy. In this case, the order up to level can be obtained as the point a n that minimizes Vn (a, P ). Due to the substitution and gaming effects inherent to this problem, a n is a function o f P in every period n. Convexity of Vn (oc, p) is obtained using an inductive argument. Note that in the case of (CCID), by exploiting the structure of the problem demonstrated in our earlier analysis, convexity of a corresponding function in a suitably written dynamic programming formulation can be shown using an inductive argument. However, as mentioned, in this section, we focus on the CUD problem. We now make the following observation. Theorem 4.5.1: Vn (a,P) is convex in a for n e Z +. Further, Vn e Z +, there exists a n > 0 such that Vn ( a n ,p) = 0 . 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The proof of convexity uses an induction on n, the number of periods until the end of the horizon. The existence of a n follows from convexity and from the i i fact that Vn, Vn (0,P) < 0 and Lim Vn (a,p ) > 0, which is once again established a — using an inductive argument. Proof: For n= l, (a,p ) = ca + L (a,P ). » t * It is routine to check that (a,P) > 0. Note that Lim V j(a, p) = c - s - n m <0 a — >0 and » Lim Vi (a, p) = c + H > 0, This implies the existence o f cq > 0 such that a - * oo Vi(oq,P) = 0 . We will assume that the induction hypothesis is true for 1,2...n. Thus, if we denote the initial inventory before ordering in period n by In , we have: ~ d n +Vn(an,/3), a n > In - c ln + V n(In,p), I„> an (4.5.1) 0000 Since Vn+1 (a,p) = ca + L(a,P) + J J b G ^ C a - d j- y ( d 2 - p)+,p)dF2dF1, in 00 writing Vn+j(a ,P ), we need to consider cases when I n is greater than a n and vice versa. First o f all, note that whenever dj > a , the initial inventory In in period n is zero and the optimal policy in period n is to order up to a n . So in this case, 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. there is no term with the purchasing cost i.e. c l . When dj < a , the decision in period n will be not to order only when In = a - dj - y(d2 - P)+ > a n . Clearly, for this to happen, a necessary condition is that a - dj > a n . This is precisely the last term in what follows. Thus other than this particular scenario, Gi n (In ,P) = - c l n + Vn ( a n ,P ), which represent the third and fourth terms below. We can thus write: vn + 1 («> P)-ca + L(a, P) + SVn (a„, p) - ) (sc(a-dl -y{d2- P))dF2(d2)dFx{dx) o p (a-<xn)+ y + J \8{Vn(a -d x-y(d2 - f3)l[d2> j}]),P)-Vn{an,f})]dF2{d2)dFx(dx) o o (4.5.2) Taking derivatives with respect to a , we write V 'n« {a,P ) = K n(a,p) a p+ ^ i= 3l + { J Sl[a>aA^-[Vn{ a -d ,-r{d2-P)),P)W2{d2)dFl(dd, 0 0 d a (4.5.3) where, after some simplifications, 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p- a y Kn(a,P) = c + [s + h - S c \\ jdF2(d2)dFl(dl) - nm F\(a) - 1 o o (4.5.4) represents the derivative of the first four terms in (5.2). Note that, from the above expression, it is clear that we can drop the subscript n from K n (a , p ). It is now a routine exercise to prove convexity using the induction hypothesis. This is done by taking two cases, namely a > a n and a < a n respectively. Note that, when a < a n , V;+1(a,p) = K ’(a,j3) = [s + H - Sc] F; (a)F2 (fi) + S + H - S c: ) F; ( f - 4 _ + P)dF, {d, ) + n X (a) > 0 Y o Y and when a > a n , V;+l(a,P) = K'(a,P) a-d\-a„ + ( f ' J S - ^ T[V„(a-d,-r(d2-P )+ ),P)]dF2(d2)dFl(.dP 0 0 + K X P ) j" ^ {p + a ~di - a ».)dF] {di) + sv; ( a , , P)F2 (P)F; (a -a „ ) 7 o Y which is again greater than or equal to zero since the first three terms are positive, the fourth term is positive by induction and the last term is zero if ocn is finite. Note also that according to the induction hypothesis otn > 0 . Thus in the above t expression (5.3) for Vn+1(a ,p ), taking limits, we have: » Lim Vn+1 (a, P) = c - nm - s < 0. This implies that a n+i > 0. oc— >0 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This finishes the proof. ♦ Further, it is straightforward to infer that ocn+i is bounded using an inductive argument. In fact, we have an upper bound that is naturally inferred as: Observation 4.5.1: The minimizer a n of Vn (a,P ) is uniformly bounded by a* < oo, where K(oc*,p) = 0. Proof: First we have to demonstrate that for a given P , K (a, P) = 0 at some finite value o f a . This is done by noting the value of K(0, P) which is negative and K(oo, p) which is positive. We denote this by a* . The fact that a n is bounded by a* is demonstrated by induction. When n= l, note that P , we have that 04 < a * . Assume that for a given p , Vn (a,P ) is minimized at a all feasible values of the demands in the double integrals, we clearly have a - dj - y(d2 - P) > a n . This implies that the integral term is non-negative. But K (a,P )is by itself the derivative of a convex function and thus, K(oc,P) > Ofor x y J JdF1(d1)dF2 (d2) = K (a,P).T hus, 5c J |dF1(dj)dF2(d2) > 0 . Since V i(a,P) is convex for a given value a n < a * . Thus Vn (a n ,P) = 0. Now from equation (5.3), when a > a n , for 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a > a * . Thus we have that Vn+i(a,P ) > Ofor a > a* and by the convexity of V, the proof concludes. ♦ The functions Vn (a,P) and K (a, p) are very useful in establishing several properties of the order up to levels. An immediate consequence of analyzing these two functions yields a useful monotonicity property that is akin to the property exhibited by the order up to levels in the single product, multi-period problem. Theorem 4.5.2: Let n e Z + and as before, let a n be the optimal order up to inventory level with n periods to go. Then a n+j > a n . » The proof of the theorem essentially analyzes Vn+i(a ,P ) and continues the analysis employed in proving observation 4.1. We know that Vn+j(a,P ) is » convex and thatVn+i ( a n+i,P) = 0. Thus to prove the theorem, it is sufficient to i t demonstrate that Vn+i ( a n ,P) < Oor equivalently, Vn+j(a ,P ) < 0 whenever f 0 < a < a n . But in this interval of a , note that Vn+1(a,P) = K (a,p). Since K (a,p) is the derivative of a convex function and a n < a * , we have the required. An immediate consequence o f Theorem 5.2 is the convergence of the order up to levels. This follows, since the sequence of order up to levels is monotonic increasing and uniformly bounded. Thus we have demonstrated that over a finite horizon o f length n, if player 2 employs a stationary base stock policy of ordering up to p (say) in every period, then player 1 follows an optimal policy 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of ordering up to {ock}£_i, whose properties we have established in the previous theorems. Thus the optimal response in a finite horizon problem is non-stationary. In order to analyze the competitive infinite horizon problem, we proceed as follows. We first show that the finite horizon cost function ^ (I, P) converges to a function G ij00(I,p) which satisfies (DP). Once we establish this convergence, we show that for the infinite horizon problem, when player 2 uses a stationary base stock policy of ordering up to p in every period, player 1 optimally responds by ordering up to a level a , where Lim a n = a . Thus the optimal response of n —»oo player 1 also is to follow a stationary base stock policy that can be obtained as the limit of finite horizon base stock levels. Using this, we then establish the Nash equilibrium o f the infinite horizon problem. In analyzing the above program (DP), we have from observation 5.1 that ocn is bounded above by the maximal root a* of K (a, P) = 0 for any positive p . Thus, assuming that the demand has a compact support (or if the mean demands are finite), whenever I < a* we have G i n (I,P)| < oo. Now, for any fixed P , define the mapping: Min Tp(G(,),I) = K a > I c ( a - I ) + L ( a ,P ) + { j5 G ( a - d 1 - y ( d 2 - p)+ ,P)dF2 (d 2 )dF1(d1) 0 0 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, for every n we have from the (DP) th atG in+i(I,(3) = T p(G in ,I ) . Note that if the minimizer in (DP) is a*(I,n + l) = a i n+i, we have shown that a ;,n + 1={a ” I+ T 1' I < a n + 1 . We write = We thus I’ I - a n+l have, P )_ C(oti n+i,G ijn,I) < C (aj n ,G ^ n ,I) and G ljn(I,(3) = C (ai n ,G l n_1,I) < C (ai n+1, G l n_1,1). This immediately implies the following: Gl,n+l(I ,P )- G l,n (I,P)|< M ax{C(aj n+i jG ^ jj,1)— C(cxj>n+j,G j n_ j,I ) , C(oc^n ,G j n ,I) — C(otj n , G ^n_ j,I)} Thus, we have [Gi;„+i( I ,P ) - G 1 )n(I,P)|< O O O O Max { f f8 G u ( a y - d j - y ( d 2 - P ) ,P ) - G u _ i(a £ i - d ! - y ( d 2 -p),P)jdF2 (d2)dFi(d,)} i=n,n+l „ „ 1 00 As observed before, we know that the order up to levels are bounded by a Thus, Sup G 1 )n+1( I ,P ) - G l n (I,P)| < 8 Sup G u ( I ,P ) - G u _1a,P ) le[0,a") le[0,a ') 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Assuming WLOG that Gj 0 ( I 5 P ) = Oand iteratively applying the above inequality for w=l,2,3..., we get Sup |G1 >n+1(I ,P )-G l n (I,p)| < 5n Sup ^ ( L P ) . I e [ 0 , a ’ ) l e [ 0 , a ' ) I * Since Gi n (I, P) < ao whenever I takes values less than a , we have that Gi n (I,P) converges uniformly for every le [ 0 ,a * ) . Since G in (I,p) is continuous (see equation (5.1)), the limit G^ ^ (I,P ) is also continuous. Now, taking limits in (DP), we have: Lim Tp(Gi>n,I) = Lim G l n+i(I,P) = Lim M in C (a,G l n ,I ) . Since, C (a,G ,I)is n — >oo ’ n — >c» ’ n— a>I * continuous in a for any finite I and since we need only consider a < a , we can rewrite the above as Lim G i> n + 1 (I,P ) = Min Lim C ( a ,G l n ,1) = G 1 ;O 0 (I,P ) • n - » c o a > I n - » o o * Note that, since we need only consider I e [0, a ) , C (a,G ,I) is bounded. Thus by the bounded convergence principle, we can interchange the limit and the integral. Thus we have, Min Lim C (a,G j n ,I) = M inC(a, Lim Gi n ,I) = M in C (a,G i)00,I) = G i;00(I,p) a > I n -> oo ’ a > I n -» o o ’ a > I Thus, Lim Tp(G1 )n,I) = Tp(Lim G ljn,I) = Tp(G 1 ;00,I) = G 1 ;00(I,P). Thus we n -» c o n->oo have shown that Gi;00(I,P) satisfies (DP) and is unique since Tpis a contraction. 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lem m a 4.5.1: (1) Lim a n = a* . n->ao (2) a* is the solution of the infinite horizon problem. Proof: We have shown that the sequence {an } is monotonically increasing and that it is bounded above by a * . To establish the above lemma, it is sufficient to show that if y < a * , y is not an upper bound for the sequence. To show this, we use the fact that Gij00(I,P) satisfies (DP). Define, V® as: G O 00 V oo (a, p) = ca + L(a, P)+ J |5 G 1 ;0 0 (a - dj - y(d2 - P)l[d2 >p] > P)dF(d2 )dF(dj). 00 Thus, since Lim Vn (a, P) = (a, P ), (a, P) is convex in a . Note that, since n— » a o the functions are continuous and uniformly bounded, we can interchange the limit differentiation operators and thus we have, for any upper bound y < a * , v ;(a ,p ) = K(a,P) p+Tzizz a - y y + f J 51[a> ] -^[V 00( a - d 1-y(d2 -P)),P)]dF2(d2)dF1(d1) o 0 da (4.5.5) But note that from our analysis of the (DP), we know that the above equation is solved at some a > y. Thus, the least upper bound for the sequence {an } is a* . This proves (1). To show (2), replace y in equation (5.5) by a* and observe that Wo(a*,P) = K(a*,P) =0. Since (a,P )is convex, we are done. ♦ 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now consider a situation when the two players compete against each other. Assume that players follow stationary base stock strategies. In all of the above discussion, we assumed that player 2 follows a base stock strategy of ordering up to P in any period when his inventory level is lower than p and not ordering otherwise. We showed that the optimum strategy o f player 1 involves * . . him following a base stock strategy of ordering up to a (in the infinite horizon case) whose properties we have demonstrated. It is easy to see that a similar discussion for the symmetric case could have been accomplished (wherein we would have stated the properties of P ). We now state the main result of this section. To do so, we use several notations developed above. Denote by G ^ V ^ K ^ , i € { 1,2}, exactly the same functions used in the above discussion, except that we now allow the indices to vary across both players. Thus, for instance, V 2 (a , /?) is the corresponding version of Vx (a, /?) = Vl (a, f3) , where we assume that player 2 responds to player 1 who follows a strategy o f ordering up to a in every period. We have: Theorem 4.5.3: The Infinite horizon (CCID) game has a unique Nash equilibrium in pure strategies in the class o f stationary base stock policies. The equilibrium is obtained as the intersection o f the curves K l(a,/3) and K 2(a,fi). Moreover, this Nash equilibrium is myopic i.e. it can be obtained as the equilibrium of a single period problem. 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proof : To show the existence and uniqueness of the equilibrium, we will first show that the two curves K l (a, fi) and K 2(a,/3) intersect exactly once. This is done by using the mean value theorem on the function J{a,P) = K 2(a ,(3 )-K l{a,P) . We need to show that J(a,(3) = 0 has exactly one root. Let /?,'(«) be the implicit derivative obtained from differentiating K l{a,P )implicitly with respect to f3 . In fact, P\(a) = — and (s + h -S c ) J>2 ' (0 + a - d J / y)dFy (dx) (s + h -Sc) J>; (/3 + a - d ,) /y ) d F 2 (d2) p' (a ) =-----------------------------------------2------------------------------------------------------------ , (s + h - 8 c ) a r , (s + h — 5c)Fx (a)F2 (fi) + i “ j Jf, {fi + a - dx) / y)dF2 (d2) + kF2 (0) r o Then, evaluating /? (a) = (3 ' 2 (a) - /?,' (a) implicitly from J(a,P ) = 0 , it is not t hard to see that P (a) >0. This establishes the unique intersection. To demonstrate myopicity, consider a single period game with profit functions of two symmetric manufacturers obtained by stocking (a,P) as: 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a y Ul(a,j3)=ca- J J & t a - r f j -y(d2 - /J)] dFx (dx )d F 2 (d2 )+L(a, p) 0 0 and I l 2 (a,P ) obtained symmetrically. Using these objective functions, the Nash equilibrium that is obtained is exactly the same as in the infinite horizon problem that was discussed. Verification of myopicity of this NE is exactly as in Sobel (1981) and follows from the following observations: • The payoff is a sum of two functions: one purely depends on the current state (the initial inventories) and the other only depends on the action (order upto levels). • Feasible equilibrium strategies (this is because of a* being a finite and uniform bound) • The transition probabilities clearly depend on the actions, i.e. the order up to levels and not on the initial inventories (state). 4.6. Conclusions and future research We have derived the optimal inventory policy for a single decision-maker in a model with two symmetric products whose demands are negatively correlated and that are partial substitutes in stockout situations in both single-period and multi period scenarios. We show that the inventory levels for the two products may be determined independently in many scenarios. We also analytically examine some anomalies in the behavior of inventory levels as substitution levels and demand 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. variance are increased. Finally, we are able to extend the analysis to situations where different players manage the inventories of the two products. There are natural extensions of this work. First, we would like to explore situations where the cost and other parameters of the two products, for instance their demand proportions, are different. We could explore how the inventory levels and fill rates of the two products would vary in this case. Second, we plan to extend the model to allow for total demand D to be random. Third, we are curious as to whether the optimality o f decoupled inventory policies extends to other multiple product scenarios where demand is either not correlated or weakly correlated (unlike our model where the correlation is -I) and the parameters are not identical for all the products. Even if the decoupled policies are not optimal, we conjecture that they may do quite well in such situations but this is to be investigated. Finally, we would like to investigate the contractual aspects of a single decision-maker managing the inventories o f the two products versus two independent players managing them. The closed-form nature o f inventory policies we derived for certain y values may facilitate this analysis. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. References Anupindi, R., Y. Bassok (1999). Centralization of stocks: Retailers vs. Manufacturer. Management Science. 45 (2) p .178-191. Anupindi, R., Y. Bassok, E.Zemel. 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(1964), Risk Aversion in the small and large. Econometrica: Vol.32 .pp. 122-136 Rajaram, K., C.S. Tang. (2001). The impact o f product substitution on retail merchandising. EJOR. 135. p.582-601. Raman, A., V.G. Narayanan. 1999."Contracting for Inventory in a Distribution Channel with Stochastic Demand and Substitute Products." Working Paper. Harvard Business School. Rao, U. 2003. Properties of the periodic review (R,T) inventory control policy. Manufacturing & Service Operations Management. 5 (1) p.37-53. Robinson, L.W. (1990). Optimal and approximate policies in multi-period, multi location inventory models with transshipments. Operations Research. 38 (2) p.178-295. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Roth, A. (1979), Axiomatic Models in Bargaining. Springer-Verlag. Roth, A. (1995), Handbook of Experimental Economics. Princeton Univ. Press. Rubinstein, A. (1982). Perfect Equilibrium in a bargaining model. Econometrica: Vol.50. pp.97-110. Schelling, T. (1960), The Strategy of conflict. Harvard University Press. Silver, E., R. Peterson. (1985). Decision systems for inventory management and production planning. John Wiley & Sons. 2n d edition. Simchi-Levi, D., E. Simchi-Levi, P. Kaminsky. 1999. Designing and managing the Supply chain. Concepts, Strategies and Cases. McGraw Hill/Irwin. Smith, S.A., N. Agrawal. (2000). Management of multi-item retail inventory System with demand substitution. Operations Research. 48 (1) p.50-64. Sobel, M.J. (1981). Myopic solutions of Markov decision processes and stochastic games. Operations Research. 29 (5) p.995-1009. Stallkamp., T.T. (2001). Fixing a broken economic model: A case for supplier alliances. Management briefing seminars, MSX international. Traverse city, Michigan. Stone, J. (1958). An experiment in bargaining games. Econometrica, Vol. 26, No. 2. pp. 286-296. Straughn, K. 1991. The relationship between stockouts and brand share. Unpublished doctoral dissertation: Florida State University. Tayur, S., R. Ganeshan, M. Magazine. 1998. Quantitative Models fo r Supply Chain Management (International Series in Operations Research & Management Science, 17). Kluwer Academic Press. 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Van Slyke, R., R. Wets. (1966). Programming under uncertainty and stochastic optimal control. SIAM J. Control. 4. p.179-193. Weng, Z.K. 1995. Channel coordination and quantity discounts. Management Science. 41 (9) p .1509-1522. Worley, T., R. Folwell, J. Foltz, A. Jacqua. (2000). Management of a cooperative bargaining association: A case in the pacific northwest asparagus industry. Review of Agricultural Economics. Vol. 22(2), 548-565. Yaari, M.E. (1969), Some remarks o f risk aversion and their uses. Journal of economic theory. Vol.l pp. 315-329. Yang, H., L. Schrage. (2002). An inventory anomaly: Risk pooling may increase inventory. Working Paper. Graduate School of Business, University of Chicago. Zipkin, P. 1986. Inventory service level measures. Convexity and approximations. Management Science. 32 (8) p.975-981. Zipkin, P. 2000. Foundations o f inventory management. McGraw Hill.Boston,MA. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix Theorem 2.4.1: We have to solve the following recurrence, which at each stage is a Nash Bargaining problem. At the N-th stage is: Max(RA - ca))(Ran - Ra - cn ) subject to Ra n = Rn + Ra _|_ Q --- Q Solving we get, RA = — : ——— --------. Again, solving the N-l th stage problem etc., we get, RA,M+CA+ 'Z cj -Ci_ Ra . = ------------- - ^ ■ ■ — -------- . Noting that Rj = j - R a j+j and R a ,1 = 11^, we can calculate the revenues of the suppliers and the assembler. We get: U Q+(2N -I)cA -XCj n e - c A - Z cj+ T c; R a = -----------------------------------------— a n d R i = ---------------------Y --------------------- The above equations hold for any fixed Q. Note from the above expressions that the profit of the suppliers and the assembler are increasing with Y[Q ■ Pareto optimality of the bargaining solution at each stage will ensure that the players choose Q = Qc. This proves that he channel is coordinated. Again, from the above expressions, we see that the assembler's revenues (and hence profit) depend on the sum of the costs of the suppliers. Hence the assembler’s profit is sequence independent. The i-th supplier's revenue (and hence profit) decreases with i. Hence every supplier would prefer to negotiate earlier in the sequence. The statements in the theorem follow. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Theorem 2.4.2: k-\ V Before we begin, recall that Rj = ----- N n Q - c A - ^ c k j * * - and p^f = 0. Let AC be his net profit attained by the z'-th supplier if he is allotted the y-th position in the negotiation sequence. Thus Nj = Rj - p l j - ct. Note that given S; = {(p[,pl 2,...pl N) e R n} Vz, the assembler allocates positions by solving the following: MaxZ 2 X j P j , such that = 1, = 1 ; b y e { 0,1}, V i,j (ALP) i=lj=l '= 1 The proof proceeds as follows. First we establish (2) i.e. that a necessary condition satisfied by any Nash equilibrium is that the supplier allotted to position y will pay exactly p*. (1) (3) & (4) of the theorem follow directly from (2). To finish the proof, we demonstrate existence by explicitly constructing a Nash equilibrium. Proof of (2): Suppose that, at equilibrium, suppliers are allotted positions z= 1,2...N and the payment made by the supplier occupying position iis ppAs mentioned, the positions were determined by the assembler who solved (ALP), faced with a set of N payment vectors. WLOG, we call the supplier occupying position z as supplier-i.. Further, suppose that for some j , pj < P j. Let us now focus on supplier-N who was allotted the last position and paid p ^ = 0. We know the following: (a) Supplier-j makes a higher net N ~ c a ~ y . C k profit than supplier-N. (b) Supplier-N makes a net profit of — ----- 2 n Consider the revised strategy of supplier N, in which he offers to pay the assembler (pj +Pj)/2 for position j, and zero for every other position. Assume that all other 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. suppliers do not change their payment strategies. Note that, if we show that by this unilateral deviation, supplier N increases his net profit from the status quo, we have essentially proved (2) in the theorem. Denote by ALP-O, the problem solved by the assembler with the status quo payment vectors (before, the N-th assembler revised his strategy) and by ALP-N, the problem solved by the assembler to arrive at the supplier positions based on the current strategies of the players (i.e. all players except N remain the same and player N offers the revised strategy mentioned above). Let OP = {(i, j) : 8jj = 1 in the solution of ALP-O } and NP = {(i, j) : 5 y = 1 in the solution of ALP-N}. By assumption (j, j) and (N, N) are elements of OP. Note that, by solving ALP-N, the assembler strictly increases his profit as compared with the status quo profit obtained by solving ALP-O. This is because, switching the positions of the j-th and N-th suppliers and keeping all other positions as the same as in OP results in an element of the feasible set for A LP-N where the assemblers profit strictly dominates the status quo. We claim that (N, N) g NP. Suppose this were the case, note that since the only difference in the payment vectors in the two problems is assembler-N, we can use the elements of NP to generate a feasible solution to ALP-O, which strictly dominates the solution generated by OP. This is a contradiction to the definition of OP. Thus (N,N) cannot be an element in NP. Thus, we have shown that by revising his strategy, the N-th supplier has forced the assembler to change his position in the sequence. Now, his revised payment strategy implies that he makes a strictly higher net profit in any position other than N. This implies that the N-th supplier has an incentive to deviate and this proves (2). Computing the net profit of the players at any equilibrium is now easy. Indeed, we have that the N Net profit of the assembler = (TI^ — c a - ^ C ] j )(1 - N / 2 N) and the net profit of k=l N each supplier equals (IT^ C A ^jC k )/2 . (3) & (4) are immediate from the above k=l 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. expressions. Since the net profit of every player increases with , Q = Q° and this proves (1). Finally, we establish that the (A-B-P-C) game has a Nash equilibrium. Consider pj ;1 < i, j < N as follows: Pj = p*;i = j;l < i, j < N . For i * j , define pj = 0;1 < i < N - 1 ;1 < j < N . Finally, N T j j e for i = N and for every j, define pj = p j . It is easy to verify that (A LP) implies that N given this strategy, the assembler makes (JI^ -C a ~ X lck )( l~ N /2 ) and each k=l N supplier makes (JI^ - C a — ^ ) / 2 ^ . Note that this is the net profit that each k = l supplier would make if he were to occupy the last position in the negotiation sequence. Indeed, as mentioned, this s lowest possible net profit that a supplier can possibly make. Now, note that if a supplier unilaterally deviates, (ALP) will ensure that the assembler makes no less than before by suitably interchanging (if required) the position of the deviating supplier with another. Thus the assembler’s profit will not decrease. The net profit of the non-deviating suppliers cannot decrease as they were making the lowest possible net profit before the deviation. Thus, the deviating cannot supplier cannot increse his profits. Thus the above strategy represents a weakly dominant strategy for each supplier and is thus a Nash equilibrium. Theorem 2.5.1: Let the n suppliers and the assembler have revoking cost kx,k 2,...kN and kA respectively and let their disagreement points be their respective production costs. Let 1 + k (X f = ----------— . At the end of N stages, where each stage results in an equilibrium 2 - f- k A - t ~ k^ 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N i-1 Rf =ci +[TI S c / — ](1 — ) IT ^ / -From the above expressions, we see that the y = i j= i revenue (and consequently the profit) of the assembler and the supplier increases with n 2 . Since at each stage, the bargaining solution is Pareto optimal we see that Q - Q c. The assembler's profit depends on the product of and the sum of the disagreement points of the suppliers. Hence he is indifferent to the sequence as both these quantities are preserved by permutations. However, the preference of the suppliers needs to be analyzed more carefully. Sincekt < kj < = > «■ > we can say the following. Given that a supplier takes the i-th position in the negotiation sequence, he would prefer to have weaker predecessors than successors. More precisely, if for s, t such that 1 < s <i <t < N , and if a t < a s then the i-th supplier would prefer to switch the positions of the s-th and t-th suppliers. Thus he would prefer to have weaker suppliers ahead of him, as having a lower cost of revoking can be thought of as being weaker in a negotiation situation. However, note that if a supplier cannot choose his predecessors but can only choose his position, he would prefer to go as early as possible. Lem m a 2.6.1: To prove that the grand coalition, g = {l,...,n} is stable it is sufficient to prove that for every possible coalition S and outcome a there is a stable outcome e so that g - » 5 a — >e = g, g does not directly dominate g and g > a . To show g > a , it is sufficient to show that g>{ a i - Notice that 1 — o c each player in g gets ------ and the single player in the outcome a, = {1},{2 ,..., n n } gets a (l - a ) . Thus the only outcome in which a player may get more than 1 1 — CL {1} in outcome ax is g . But, a < — =>------- > a ( l - a ) , implies n n that g >j x i = 1 x e Z . This proves that g is stable. We now prove that g is uniquely stable by contradiction. Let a e Z, a ^ g be stable. Clearly, it is 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. possible to move from every outcome I to the grand coalition g, where S ={l,...,n}. To prove that g is the only stable outcome, it is sufficient to prove that for every a there is no e so that a — ^ g-»s,ux — > Sl ... — > S n an = e and e> g . But we have seen that g >, x i = l,...n, x e Z . Thus, e does not indirectly dominate g , and a can’t be stable. This completes the proof. ■ 1 2 Lemma 2.6.2: Notice that for n> 4, when — < a < —, we have g >j a i = l,...n , n n for every a e Z , & * a\. Thus From lemma 6.1, when d ^ a x, we know that g ->§ a is deterred. To prove that g is stable it is sufficient to show that for every outcome a, there exists an outcome d^ and a coalition Sj such that g — > S o a -» S i dx — > S g g and g > a . Thus, we need only verify this sufficiency condition when a = av Let outcome d x = {l},{2,...,k},{k+l,... n }, k>2. Clearly, a <{ 2 ^ k) g and d x <i g i = l,...,n . This proves that outcome g is stable. To prove that at is stable it is sufficient to prove that for every outcome a , such that ax->S o a. — > Sj dx — > S g e , e is stable, e > a and the following does not hold e >S o ax. Let ax->S o a and outcome a has exactly 3 coalitions, then { 1 } < £ Sx. Thus, the outcome a is of the following structure a = {l},{2,...,k},{k + l,...,n} and we can substitute di by ah showing that a! is stable.. (Further, It is easy to verify that all conditions for stability hold.) If d has only 2 coalitions then di = g. Clearly, d <x g i = l,...,n , and outcome g does not dominate outcome a{. This completes the proof. ■ Lemma 2.6.3: The fact that g is not a unique stable outcome was established in lemma 6.2. We now prove that it is stable. It is sufficient to show that for every outcome d there is a sequence g -» S o d — > Si a3 — > ... ->S m am = g and g > d . Notice that since outcome a is generated by a defection from outcome g , it must have exactly 2 coalitions. Clearly, the number of players in one of these coalitions is no smaller than n/2. 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus outcome a has the following structure {l,...A:,},{& + l,...,n}, k <n/2. It is possible for a coalition of size no smaller than n/3, to defect from outcome a to a3 . In general it is always possible for a coalition of size no smaller than m to defect from a m _jto am, till the coalition {l},...,{n} is reached. Since { 1 ± n n / 2 n /3 This completes the proof. I - a a ( l - a ) a (1 - a ) . a < 0.5 =>------- > — ----- > ....... .... >.... > a (1 — a) we get that g > a Proposition 2.6.1: Consider a status quo structure a with exactly t coalitions. Each coalition makes a*~l (1 - a ) . Let b be a structure with exactly t + \ coalitions. Let a — » ,• b . Then, by definition, {/} € b. In the structure a , i makes no more than od ^ (1 — crl ----- -. In the structure b , he makes exactly a ‘(l-a ). Thus when a > 0.5, a <t b . ■ C C { 1 (X ) 1 { jC Lemma 2.6.4: Note that, a > 0.5 implies — > -------- . Further, n/2 n . .. a ( l - a ) a 2(I-a ) n a <2/3 =>----------- > --------------. Indeed for a m this range, n /2 n /3 a ( \ - a ) a k( l- a ) ^ ^ „ „ , , -----------> --------------- Consider x — > s a . The following are the only possibilities: n /2 n/k Case 1: d = g. Consider x — g x . In this sequence the members moving out of the status quo are not described explicitly as they are obvious. Since x > { ( .} g \fi G N , x >v g = > x > g . Clearly xNOT >g x . Case 2: d has 3 coalitions. Consider the chain x — > s d — > S } y 4 ... - ^ s ( n -> x -± c x, where y k + l is obtained by an arbitrary split of the largest coalition in y k , ties being arbitrarily broken. Note that this describes Sk. Further, x = {1,2,...n/2},{n/2 +1},...{n} and C — {n/2 + l,...n} . This completely 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. describes the above chain. By our choice of a , x >S k y k + l where y n = u . Note that xhas n/2+1 coalitions. Further, ----- — > a n/2( l- a ) implies a < (— ) 2/"~2. Indeed, n /2 n for large values of n, this falls into the range of a that we assume. Thus, x >c x. This implies x > d. Clearly, xNOT >s x. Case 3: d has 2 coalitions. Consider the chain, x -> s d — > S 2 y 3 — » ... -+ y n = n -» x — > c x where y 3 is an outcome with 3 coalitions obtained by splitting arbitrarily the largest coalition in d . Since the largest coalition in d has more than n/2 members x d. Thus using arguments as before, x O d and xNOT >s x . The three cases prove that x is stable. ■ Lemma 2.6.5: The technique is very similar to the previous lemmas. Let ^ ^ 1 t 2» , . a (l - a ) (1 - a) h =0.5 + — ,b7 —----------- . Note that a>b, =>----------- > --------- and 1 n 2 3(n + 2) 1 n /2 + 1 n a ( l - a ) a k~l( l - a ) , - . ^ , b2 > a => -------— > -------------- , k = 3,4,...n . Consider x — d , for every n /2 + 1 n/k legitimate (S, d). There are only three possible cases. Case 1: d = g.It can be seen that by choosing x — > g -» e = x , this deviation is deterred. Case 2: d — y 2 ^ x, has two coalitions. Note that, in this case, 3/ e A, such that i e S. Consider the chain x — y 2 — > ■ y 3 - > ...- + n - + g - » e = {1,2,...n/2},{n/2 + l,...n}, where as before, y M is obtained from y k , by an arbitrary split of the largest coalition. Ensure that S 2 the coalition that splits from y 2 is such that S 2 c {l,2,...n/l} . Clearly, by the choice of a , e > d and trivially NOT e> x . 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Case 3: d = y % . Now there are two possibilities. If S c A , choose the same chain as above. On the other hand if 3i e B fl S , choose e = x in the previous chain. These three cases conclusively show that all possible deviations are deterred. ■ 2 2 Lemma 6.6: Choose, for n large, a x < a 2 such that a x > — [1 + — ] and 3 n 3 r n , . a 2( I -a ) a w ( l - a ) a < — [-------]. Note that when a e (al,a 2) , --------------> — — -----—, for 4 n + 2 (n + 2)/3 n!k k = l,3,4,...w. To show that 3 3 is stable, we need to look at 3 3 — d , for every legitimate S, d. The following are the possibilities. Case 1: g — d . Consider the chain 3 3 — >{ ? ) g ~^s y 2 — 33. y 2 is chosen such that it has two coalitions and we can get 3 3 fromy2 by breaking the largest coalition in y 2 , namely S3. S is arbitrary and this completely describes the chain. By choice, 2s3 > g and 2s3NO T > { ? } 3 3. Case 2: d = y A has 4 coalitions. Consider the chain T4 ->s4 Ts - -* Ti -> S t - -> tt -> g -> y 2 3 3. From an outcome y k , obtain y k + l by breaking S k the largest coalition in y k arbitrarily. Thus, y n =u and the segment of the chain u — > g — > y 2 — >S j 3 3 is arrived at exactly as in Case 1. Now, 3 3 t> y 4 = d and 3 3NOT > 5 3 3. Case 3: d - y 3 has three coalitions. WLOG, assume that y 3 ^ 3 3. It is easy to see that there is a coalition in 3 3 that has increased in size in the outcome y 3. That is, if the partitions corresponding to 3 3 and 3 3 y 3 are represented by 3 3 = [ J and y 3 = [ J Yj , then X t a Yj for some i,j. WLOG I- 1 j ~ l 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. assume that X x c Yx. Choose s e X t . Consider the chain 3 3 -> s y 3 y 4- - * y k ~>sk - -» u -> 8 -> T2 ->s, 3 3. where y* -» 5t ^ +1for k>3 and w — > g — > y 2 — 3 3 are exactly as before with the additional requirement on S3 that { 5} is in the smallest coalition in 3 3. Thus 3 3 > y 3 and 3 3 NOT >s 3 3. Case 4: d — y 2 has two coalitions. 3 2 Note that if 3 3 = [ J X t and y 2 = (^J Vj represent partitions corresponding to the (= i j — \ respective outcomes, then, we can assume that WLOG ( X x U X 2) c Vx. Thus it is always possible to split the largest coalition in y 2 to get an outcome y 3 with three coalitions, such that y 3has a coalition with greater than (n+2)/3 members. We will denote such a deviation by y 2 — y 3. Consider the chain, 3 3 y 2 -> / y 3 ~^S3 J V - y k ~»sk ...-> u -* g ^ > y 2 -> 5 j 3 3. where, as before, from y k the largest coalition ( Sk), splits to form y k + x for k>2 etc. Thus 3 3 > y 2 and 3 3 NOT >s 3 3. This proves the stability of 3 3. ■ Proposition. 2.6.2: Consider any outcome z , {/} e z with k + \ coalitions. In this outcome, i makes a k(\ — a). Thus, when a > (0.5)1//* ~ 2 => a n ~l (1 - a ) > - - - - - - . This implies, that in any outcome in which i goes alone, he makes more than any outcome in which he is a part of a coalition. Thus proposition 6.2 trivially follows. g i . . . 1- ctr(l — cc) Lemma 2.6.7: Note that a > (0.5)"-2 implies that a n~ 1(l — a ) > ------------ and k a""‘(l - a ) > ------ -, for every t=1,2,.. .n-2 andA: =2,3,.. .n-1 and for (t,k)=(0,n). Thus n in outcome u , every player makes more than in any other outcome in which he will be a part of a coalition with more than or equal to two members, this ensures that u is stable. 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To see this point notice that for every a the sequence u -» S o a — bx — » ... — bm = z in which outcome b; is obtained from outcome b ;_ i by a defection of a single supplier satisfies all the conditions for farsighted stability. We need to show that u is a unique stable outcome! Proof by contradiction. Suppose that a ¥ = u,a e Z is stable. Consider,a d , for some i e N.To show stability, we need to produce e e Z such that, e > d and e a . First of all, note that, since a > 0.5, by proposition 6. 1, d > { i} a . Consider the chain, a d — >S l ax a2 e . Let at be the first outcome where {i }is not alone (i < £ Sk,\f\< k < t ). Thus, by proposition (6.2), cij >(;| a, Vj < t. Since i € S ,,a t_x — > S { at , we must have e > ( l- } at_x But if i is a member of a coalition with more than one assembler in outcome e , this can’t hold. Thus, supplier i is the only member of a coalition in e . But this ensures that e > { ;, a . This is a contradiction. Thus a cannot be stable, and the only stable outcome is u. Observation 3.2.1: In the VMI system, when the retailer announces a rent R , the manufacturer picks the order quantity Qv = ^ 2 Am D/ R and his total cost is TIm = A /2Am ILR . The corresponding cost of the retailer is FIr = S(R), where S(R) = . On the other hand, in the RMI system, the -12R retailer’s optimal order quantity is QR = ■ s jlA rD IH and the corresponding costs HD of the retailer and the manufacturer are FI^ = J l A rD H and = Am V 2 A, 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. respectively. We need to find a rent R such that f l r M ^ 11^ and ^ II* • Note A H * that the former can be true if and only if R < — = R . Now, if A r < Am , 4 A s(r')=(4 - +P a d h < n ;, Thus, if A r < Am , we have shown that there exists a rent parameter R* that ensures that the VMI system is better for both players. To finish the proof, let Ar > A m . Observe that S(R) achieves its minimum ~ HA ~ at R = ------ — and R > R*. This implies that both players can simultaneously A, - A r m never be better off. A H Observation 3.2.2: First, R’ = — is the unique rent that coordinates the A + A m channel. If A r < Am /3, notice that R* < R*. This ensures that the manufacturer’s costs are lower in the VMI system. Now consider the following expression: T = S ( R ') - U r s = S (R ')-^2 A rDH . / A, Ar Am+Ar ]j Am +Ar j r p We need to show that T <0. But = 4 lD H 4 Am + Ar A Since -—— — <1, it follows that T< 0. + A Observation 3.2.3: The optimal quantity selected by the manufacturer in the VMI system is now Q v - ^ /(l- fi)AmDI R . Suppose the total supply chain cost o f the 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. centralized system is risC and the retailer's cost in the VMI system is r is = ^ , then one can ask if there are values (R, f3) such 0 0 2 2 that the retailer can coordinate the channel and bear any proportion of the system's costs. We need, for a e [0, 1], 11* (R,j3) = o c Y lc s c and Q v = Qc where a is the V c proportion o f the total costs that the retailer wants to bear. Observe that 0 = 0 is a straight line in the variables (R, (3) as is H* (R, 13) = a FIjc • The equations of these lines are given by: U R R(R,J3) = a U c sc < = > ^ ^ + ^ R + ^ - ^ - = a p ( A r + A JD H , where 0 = 0 C and 0 0 2 2 Qv = 0 C « j3(Am DH) + R(Ar + A m )D = Indeed these lines are not parallel and thus for any a , there is a unique (R, /3) which enables the retailer to coordinate the channel and bear an arbitrary share of the system costs. oo oo Lemma 3.3.1: n(r) = J(£ - r ) f (^)d^ = f(£ - r)e~a^di; r r Integrating by parts and canceling common terms, we get: 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -a r e a F(r)= J/(£)rff= jV “srff = r r So, n(r) = F(r)/a. Lem m a 3.3.2: Since the cost functions I \ c sc(Q,r), 17^(Q,r), and Ug(Q,r) are quite similar, we will look at a general cost function of the following form: 0 0 n (Q,r) = AD!Q + (aQ + br) + [ Jf(x)dx]PD!Q r Since the sum o f convex functions is convex, to show that Y\(Q,r) is jointly 00 convex, it suffices to show that b(Q,r) = [ jf(x)dx]PD/Q is jointly convex in r (Q,r). Let H(Q, r) be the Hessian matrix of b(Q, r). To show convexity, we need to show positive semi-defmiteness of H. Since we have a 2x2 matrix, it is enough to show that the determinant and the diagonal entries are positive. From the above expressions it is clear that the latter is true. Now, WLOG, set PD = 1. The lead time demand is f(x) = e ax, a>0. We then have the following: d 2b 2 w d 2b - I rl/ N d 2b 1 „ , w - _ J /( x ) * - - / <r); — The diagonal terms of this Hessian H(Q, r j are again positive. Also, note that Det(H (Q , r ) ) > 0 o - 2 / '( r ) ( l - F(r)) > [f(r)f « 2e2^ > e2aL This implies convexity. 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Observation 3.3.1: Using equations (8) and (9) in section 3, we have a l + yjl + 2 DAma 2/R , V 1 and r = — ' / In a V P D a / R l + Vl + 2DAm a 21R V C V c It is immediate that in general, there is no R such that Q - Q and r =r simultaneously. Theorem 3.3.1: We will start with the case where a penalty is incurred for every stockout occasion. We need to show that there exists a rent R such that n (QV F r) - (QR’rR)' We will demonstrate this by taking two cases. Case 1: Let A J P m >AJPr. Consider a rent R such that rv - / = r. From (5) and (8), we have Q v = Q*HPm RPr This implies that Qv > ( f since Am /Pm > A /P r. We then have: U v sc(Qv,r) = [(Ar +Am) + (Pm +Pr)F{r)]D!Qv +HQV12 = [CAr + A J + (Pm+Pr)F(r)}D/Qv + DH[Am +Pm F(r)]/RQv n J c ( e V ) = l(Ar +Am) + (Pm +Pl.)F (r)]D / QR +HQR/2 D f j p _____________________ = [ ( 4 + Am) + (Pm + Pr) F { r ) } ~ ^ + DPr[Am + Pm F (r)\/P m Qv Q RPr Taking the difference and canceling common terms, we have, K c ( Q " A ) - n l c( Q \ r ) = ~ { Q ~ I~r)\-(Ar + Am)H P m+Pr)F<,r)-(AmPTIPm+F(r)Pm)}} Q RP„ 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Rearranging terms and canceling common ones, we have, r\ zrp_____________________________________ n U Q V>r)-nlc(Q\r) = ^ { ( \ - ~ ^ ) [ A r+Am(l-Pr/Pm ) + PrF(r)]} Q R1r Note that the expression in [.] is positive because Pr < Pm . Since our choice of rent HP implies that Qv > Q R = > ^ 1, we are done. Case 2: Let A J P m < A r!Pr. Consider a rent R such that Qv = QR = Q. As before, taking the difference and canceling common terms, we have: n U Q ,’-'")- n jr t& r * ) = ~ (P , + P ,h F ( r ') ~ V m +P,)F(r*)-H(rr - r " ) H P D PD P +P R H = ~ [ I n ( ^ ) - l n ( ^ ) ] + a RQ HQ a Pm Pr IT P A A P A P = _ /jn( v ie. \ + + zHl _ Lsl _ n a 1 yPrA j ArPm 4 Pr Set Pm /Pr = y and Am /Ar = x. Let G(x) = ln(— ) + — + x - y - l . We then have x y n u e . ' - v n ^ e . ' - ' H — g m - a G(x) is a convex function and has a unique minimum at x = y/(y+1) < 1. Further, G(y) = 0 and Pm /Pr = y> 1. G(x) has another zero at some x* < y. Hence, it is easy to see that G(x) < 0 when x € [x* y] and is positive everywhere else. By our assumption x <y. To finish the proof, we need to ensure that x > x*. As y 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. increases, x* decreases. W heny - 1, we getx*= 0.203 and this corresponds to Ar - 4.9Am . Hence, if Ar < 5Am , we can ensure that G(x) < 0. Now consider the case where a penalty cost for a stockout is incurred for every unit stocked out. In this case, the total cost of the supply chain is: n s c c (Q, r) = (Ar + Am )D / 0 + H (Q / 2 + r) + (Pm + Pr)n(r)D I Q . The only difference in the cost function is that we have n(r) instead of F(r). From Lemma 1, n(r) = F(r)/a. Also, F{r) = e ar/a -f(r)/a . So, it can be shown easily that the only difference in the proof for this case, relative to the earlier one, is that we replace F(r) with n(r) and f(r) with F(r). □ Lemma 3.4.1: Equivalently, we need only produce a pair (a, (A ) that results in channel coordination. For a given pair (a, fA), the manufacturer's optimal response (Q, r) satisfies the following pair of equations: z A R -E m D _ F { r ) + a = 0 Q2 < 2 (A l) - P D ~~~Q ~ f ( r) + P ~ 0 f" 1 C ' Let the centralized system or first best solution be (Q , r ). It is clear that there is a unique pair (a*, fA*) that satisfies the pair of equations (A l), when we set (Q, f) = (Qc, rc). But we do not claim that this solution can allocate fractions of the savings such that the participants do better than in a RMI system. 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lem m a 3.4.2: We now need to produce a pair (a, R) such that the manufacturer’s optimal response will coordinate the channel. Once again, the manufacturer’s response will satisfy the following equations: ~ aA m D _ + pm D F (r ) + R 12 = o (A2) ■ Pm Df(r) + f? = 0 0 If (Qc ,rc) is the first best solution, it is once again clear that there is a unique pair (a*, R*) that satisfies (A2) when (Q, r) - (Qc, rc). Theorem 3.4.1: We will represent the manufacturer’s optimal response in the VMI system by (Qv,rv), which satisfies equation (Al). We will denote by (QR , rR ) the decisions made by the retailer in an RMI system. The player’s costs in the VMI and RMI systems are denoted, as before, by TIm , Y O 'r, and 11^ • We need to show that there exists a contract Sj or equivalently a pair (a, J3) such that FIm r m < 0 and YVr -Elfl < 0. First choose (a, p) > 0 such that Qv = pP. Using (Qv J = 2D^Am + -P J (ry^ and p = , we get a aQ v [fl + J{l2IP>+SDA„a 2a So, for any p > 0, there exists a > 0 such that QV - Q1 * = Q - Now, r = n r M- m = ^ - [ F ( r y)-F (r')] + aQ + /)r’ 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Denote f(x) = e~tx. Using Qr = Q = > F(rR) = and writing Y as a function o f : Y = Y(/3) = - - (r )L> { ^ - - P m) + aQ v + j3rR - ^ \ n [ — ] Q H t Pm H HP But F(oo) -» -oo and 7(— — ) > 0, and Y continuous for p > 0 implies that there Pr exists a J 3 * > H such that Y(f?) < 0. But notice that if Y < 0 and a, j3 > 0 then F (rv) - F ( r R)< 0. Thus when (3 = (3*, we have X < 0. This finishes the proof. □ Lemma 3.4.3: We need only show that S ' * S * 2. Note that these contracts coordinate the channel and determine uniquely the allocations between the players. Further, the coordinated solution is unique since the problem is convex. It is then easy to show that oil (sD , n jK s,* ))* oiios*), n u sl))- Lemma 3.5.1: We will prove part (3) of the lemma. To show joint convexity, we need the positive definiteness o f the Hessian. Let H(Q,T) = (hy); i,j = 1,2. 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. GO hlx = a 3 Jy 2e_Q y7e<iy ; h2 2 — cc2Te~a® T ; hl2 = h2l = cc2Qe a@ T ; Q Clearly, the diagonal elements are positive. The determinant o f the Hessian is given by the following expression: D(H) = a 5 Te~aQ T t y e ' ^ d y - a \Q e ~ aQT)2 Q > a 5Q2 Te-aQ T je ~ ^ Td y - a 4(Qe~aQT)2 = 0. Q Thus the Hessian is positive definite. Theorem 3.5.1: The total supply chain cost of the system as a function of (5, 7) is given by: nic (* ,r) = (A + A*>+ H ( s + ^ ) d Pr+ T^ - ]f(y,T)dy * s+ D T This total system cost function will be used to compare the two systems, VMI and RMI, though the values of (s, T) would be different in the two cases. From Lemma 5.1, we have convexity of the cost function. Convexity o f the cost function implies that the optimal decisions can be obtained by solving simultaneously the two first order conditions. So, we have: = 0 o exp[-«r<> + DT)] = — ; (A3) ds Pa = 0 « H(— - 1)T2 + H( 1 - - ln[— ])T - A - — = 0 (A4) dT 2 a H a 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To analyze the VMI system, we need only set A = Am , H = R and P = Pm in (A3) and (A4). We will call the resulting optimal values as (sv, 1r) and n sc(-s'F,T F)as n^c • The corresponding optimal values (sR , Is) for the RMI system are obtained by setting A = Ar and P = Pr in (A3) and (A4). We let risC = n ^ c C ^ T * ) . Choose R = HPm /Pr . This choice of R implies that (sy + DTy)Ty = (sr + DTr )Tr . Writing H(s+ p ~ ) = ~ [(jr + DT)T] - ™ T , we have: n l -F& = [(4 + 4 ) + (P, + p. ) expt-ar-- ( / + DT’ )]((4, _ J_) + H [ ( / + DTr )Tr ](-— - V ) + S j - (T‘ - Tr ) Now notice that solving (A4) yields 1 Pa ( - ] n [ ^ ] - l ) + a H y 1 , Pa 2 1 x (1------ ln[— ] r + 4(— - 1)(— + y) a a L H J/ v 2 ~'v77 ~2 ' Furthermore, by setting R = HPm /Pr and comparing 7^ and T* using the above equation, we obtain Tv > 7^ if A J A r > P J P r. This immediately implies that nv sc -n«c< o . □ 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Asset Metadata
Creator
Nagarajan, Mahesh
(author)
Core Title
Essays in supply chain management
School
Graduate School
Degree
Doctor of Philosophy
Degree Program
Business Administration
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
business administration, management,Economics, Commerce-Business,OAI-PMH Harvest
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
[illegible] (
committee chair
), Bassok, Yehuda (
committee chair
), [illegible] (
committee member
), McBride, Richard (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c16-490574
Unique identifier
UC11340132
Identifier
3133313.pdf (filename),usctheses-c16-490574 (legacy record id)
Legacy Identifier
3133313.pdf
Dmrecord
490574
Document Type
Dissertation
Rights
Nagarajan, Mahesh
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA
Tags
business administration, management
Economics, Commerce-Business