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We consider a supply chain in which a supplier manages the supply of a durable product for a buyer and the buyer contracts with the supplier on the supplier\u2019s inventory service level. The SLAs are discontinuous incentive schemes with a multi-period review strategy, and the supplier\u2019s performance measure is the ready rate. \nThe first essay investigates the effectiveness of two common types of SLAs: a lump-sum penalty SLA and a linear-penalty SLA. We find that when the supplier can observe the performance history and dynamically adjust the investment in inventory to affect her review period performance, to mitigate the supplier\u2019s incentive for strategic behavior, the penalty should be dependent on the degree of the supplier\u2019s performance deviation from the target. \nThe second essay focuses on the effectiveness of performance measures in SLAs. The problem is similar to that in the first essay, but the supplier can invest in inventory and in lead time. We consider two inventory performance measures: immediate ready rate and time-window ready rate, and find that there exists a unique positive time window such that a ready rate with window induces the first-best investments. Our findings demonstrate the importance of choosing the right performance measure to align a supplier\u2019s incentive. \nThe third essay investigates the design of performance-based volume incentive schemes in the form of allocating business between suppliers when a buyer maximizes his long-run discounted payoff from repeated dual sourcing. We consider both the case where a supplier\u2019s effort cost is proportional to her business volume and the case where the cost is independent of her volume. We find that to induce and maintain suppliers\u2019 competition over time, the optimal scheme depends on each supplier\u2019s current share of business and is generally not a simple rank-order tournament; handicapping the definition of winner can do better than a simple first-past-the-post rule.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/17428?expand=metadata","@language":"en"}],"FullText":[{"@value":"Designing Performance Based Contracts in Supply Chains by Liping Liang B.Sc., South China University of Technology, 1993 M.Sc., University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Business Administration) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2009 c\r Liping Liang 2009 Abstract The three essays in this thesis address the design of performance-based contracts in decentralized supply chains when a supplier's e\u000bort is unobservable. The \frst two essays explore various issues in the design of service level agreements (SLAs), a type of performance-based incentive scheme widely used for outsourcing manufacturing and services. We consider a supply chain in which a supplier manages the supply of a durable product for a buyer and the buyer contracts with the sup- plier on the supplier's inventory service level. The SLAs are discontinuous incentive schemes with a multi-period review strategy, and the supplier's performance measure is the ready rate (1 - stockout rate). The \frst essay (Chapter 2) investigates the e\u000bectiveness of two common types of SLAs: a lump-sum penalty SLA and a linear-penalty SLA. The key \fnding is that when the supplier can observe the performance history and dynamically adjust the investment in inventory to a\u000bect her review period performance, to mitigate the supplier's incentive for strategic behavior, the penalty should be dependent on the degree of the supplier's performance deviation from the target. The second essay (Chapter 3) focuses on the e\u000bectiveness of performance measures in SLAs. The problem is similar to that in the \frst essay, but the supplier can invest both in inventory and in inventory replenishment lead time. We consider two inventory performance measures: the immediate ready rate and the time-window ready rate, and \fnd that there exists a unique positive time window such that a ready rate with window induces the \frst-best investments. Our \fndings demonstrate the importance of choosing the right performance measure to align a supplier's incentive. The third essay (Chapter 4) investigates the design of performance-based volume incentive schemes in the form of allocating business between suppliers when a buyer maximizes his long-run discounted payo\u000b from repeated dual sourcing. We consider ii Abstract both the case where a supplier's e\u000bort cost is proportional to her volume of business and the case where the cost is independent of her volume. We \fnd that to induce and maintain suppliers' competition over time, the optimal scheme depends on each sup- plier's current share of business and is generally not a simple rank-order tournament; handicapping the de\fnition of winner can do better than a simple \frst-past-the-post rule. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Inventory management . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Supply chain contracting . . . . . . . . . . . . . . . . . . . . 4 1.2.3 Non-cooperative games driven by demand allocation . . . . . 5 1.2.4 Principal-agent theory . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Research Methodology and Findings . . . . . . . . . . . . . . . . . . 8 1.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Designing Service Level Agreements For Inventory Management 12 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Performance Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Performance measure under a multi-period review strategy . . 18 2.4.2 Distribution of ready rate under a static base-stock policy . . 19 iv Table of Contents 2.5 Buyer's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.1 Choice of contract parameters . . . . . . . . . . . . . . . . . 22 2.5.2 Unimodality of the supplier's objective function . . . . . . . . 24 2.6 Supplier's Strategic Behavior . . . . . . . . . . . . . . . . . . . . . . 26 2.6.1 Single-review-phase problem . . . . . . . . . . . . . . . . . . 27 2.6.2 In\fnite-horizon problem . . . . . . . . . . . . . . . . . . . . . 29 2.7 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.7.1 Lump-sum penalty SLA . . . . . . . . . . . . . . . . . . . . . 31 2.7.2 Linear-penalty SLA . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8.1 SLA vs. traditional coordination contract . . . . . . . . . . . 36 2.8.2 Lump-sum penalty SLA vs. linear-penalty SLA . . . . . . . . 37 2.8.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Managing Supplier's Delivery PerformanceWith Service Level Agree- ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Optimal Ready-Rate Contract . . . . . . . . . . . . . . . . . . . . . 50 3.4.1 Performance measure . . . . . . . . . . . . . . . . . . . . . . 51 3.4.2 First-best solution . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.3 Ready-rate-with-window contract . . . . . . . . . . . . . . . . 54 3.4.4 Ready-rate-without-window contract . . . . . . . . . . . . . . 57 3.4.5 Incentive alignment using an inventory performance measure 58 3.5 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5.1 Linear delay cost . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.2 Convex delay cost . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5.3 E\u000bect of the length of review phase R . . . . . . . . . . . . . 64 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4 Volume Incentive Through Performance-Based Allocation Of De- mand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 v Table of Contents 4.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Buyer's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.1 Volume incentive under proportional e\u000bort cost . . . . . . . . 78 4.4.2 Volume incentive under demand-independent e\u000bort cost . . . 87 4.5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . 93 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Appendices A Proof for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B Proof for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 C Proof for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 vi List of Tables 2.1 Supplier's cost saving (lump-sum penalty,Poisson demand) . . . . . . 32 2.2 Supplier's cost saving (lump-sum penalty,normal demand) . . . . . . 32 2.3 Supplier's cost saving (linear penalty,Poisson demand) . . . . . . . . . 33 2.4 Supplier's cost saving (linear penalty,Poisson demand) . . . . . . . . . 34 2.5 Supplier's cost saving (linear penalty,normal demand) . . . . . . . . . 34 2.6 Supplier's cost saving (linear penalty,normal demand) . . . . . . . . . 35 2.7 E\u000bect of R (linear penalty,normal demand) . . . . . . . . . . . . . . . 36 2.8 Expected penalty vs. supplier's total cost . . . . . . . . . . . . . . . . 39 2.9 Actual penalty vs. supplier's total cost . . . . . . . . . . . . . . . . . 39 3.1 Cost increase from not using a window contract (linear delay cost) . . 62 3.2 Cost increase from not using a window contract (convex delay cost) . 64 4.1 Optimal allocation rule (proportional case) . . . . . . . . . . . . . . . 84 4.2 HWTA allocation rule (proportional case) . . . . . . . . . . . . . . . 85 4.3 SWTA allocation rule (proportional case) . . . . . . . . . . . . . . . . 86 4.4 Comparison between rules (% increase in buyer's payo\u000b) . . . . . . . 87 4.5 Comparison of SWTA and HWTA (independent case) . . . . . . . . . 92 4.6 Comparison of SWTA and HWTA (independent case) . . . . . . . . 92 B.1 Value of \u0012 for normal demand . . . . . . . . . . . . . . . . . . . . . . 114 vii List of Figures 2.1 Supplier's cost function (lump-sum penalty, L=0) . . . . . . . . . . . 25 2.2 Supplier's cost function (linear penalty, L=0) . . . . . . . . . . . . . . 25 3.1 First-best base-stock level (linear delay cost) . . . . . . . . . . . . . . 61 3.2 First-best lead time (linear delay cost) . . . . . . . . . . . . . . . . . 61 3.3 Optimal window (ready rate with window, linear delay) . . . . . . . . 62 3.4 Optimal window (ready rate with window, convex delay) . . . . . . . 63 4.1 Optimal allocation rule with maximum share = 1 . . . . . . . . . . . 79 4.2 Optimal allocation rule with maximum share < 1 . . . . . . . . . . . 80 B.1 EC(S,L) under convex delay cost . . . . . . . . . . . . . . . . . . . . 112 viii Acknowledgements I would like to express my sincere appreciation to my research supervisor, Professor Derek Atkins, for his continuous support and guidance at every step of my study and research, and for his care and encouragement through my years spent at UBC. I am extremely grateful to Derek for introducing me to the subject of incentive contracting in supply chains. I consider myself very lucky to have him as my advisor and to work with him. I have learned and bene\fted a lot from his insightful ideas and extensive experience in both research and applied work. His enthusiasm in work and his love and care for his students has set an example for me to follow in my academic career. My special thanks also go to Professors Hong Chen, Harish Krishnan and Ralph Winter, who have served on my thesis committee and provided me with valuable comments and guidance as well as constant support throughout my research; and to many other faculty members from the OpLog Division at the Sauder School of Business, for their help with my research and study at UBC. I would like to especially thank Harish, for his inspiring comments on my research work, which has helped to form the research question in the \frst essay of my thesis. Finally, I owe a debt of gratitude to my husband Wenhai and my daughter Yuyan, for accompanying me through my PhD study and sharing all my joys and pains. ix Dedication To my parents and my sister, my husband Wenhai, and my beloved daughter Yuyan, who always give me unconditional love and support in my life. x Chapter 1 Introduction 1.1 Motivation Outsourcing manufacturing and services is a common practice in both the public and private sectors. As supply chains become more decentralized and the suppliers perform tasks on behalf of the buyer, e\u000bectively managing these suppliers becomes vital for a buyer's success in business. Mechanism design for aligning a supplier's incentive with the buyer's interest has consequently drawn attention from operations researchers. This has been driving a stream of literature at the interface of operations management and economics, which addresses the contract design for various opera- tions management problems such as inventory management, production management, capacity investment, and quality control. Traditional contracts often focus on how the work is performed and the payment is either a \fxed price or cost\/revenue sharing. For example, a contract may dictate the inventory level or production capacity in which a supplier should invest. How- ever, implementing such a type of contract could be infeasible or expensive in many situations because a supplier's e\u000bort is often unobservable to the buyer and costly to monitor, which is known as the moral hazard problem in the principal-agent theory. Performance-based contracting has therefore gained a wider use in recent years due to its key feature of contracting on outcomes instead of dictating how the work is done. Service level agreements (SLAs) are a common type of performance-based contract for managing suppliers. In an SLA, the performance, or outcome of a task desired by the buyer is speci\fed in terms of a service level target. According to a survey (Oblicore Inc. 2007), 91% of organizations use SLAs for managing suppliers, inter- nal agreements, or external customer agreements. Performance-based incentives are also often used together with competition in the form of allocating business between multiple suppliers based on the suppliers' performance (volume incentive). Dyer and Ouchi (1993) report that Japanese \frms usually employ a `two-vendor policy' to mo- 1 1.2. Related Literature tivate suppliers to innovate and improve performance. Sun Microsystems allocates demand among multiple suppliers using a scorecard system (Farlow et al. 1996). An empirical study by Bensaou (1999) also shows that Japanese buyers typically split their purchases among multiple suppliers and then demand that the suppliers make specialized investments to obtain and keep their business. Despite the widespread use of performance-based contracts and volume incentives in practice, there is little theoretical research in the literature on their design. This thesis attempts to understand performance-based incentives as well as volume incen- tive schemes by studying issues in the design of SLAs for a single supplier and volume incentives under supplier competition. Some features of the performance-based in- centive schemes distinct from traditional contracts present new issues which haven't been fully addressed in the literature. For example, performance-based contracts are typically associated with one or more performance measure, which creates the basis for compensation. Because SLAs are very context dependent, we study a particular application to inventory management, in this case for a durable product. To mea- sure the inventory performance, o\u000b-the-shelf (immediate) ful\fllment rates are often used in theory, but time-window ful\fllment rates are more commonly used in practice (LaLonde and Zinszer 1976, LaLonde et al. 1988). We therefore also examine the e\u000bectiveness of both types of performance measures for incentive alignment. 1.2 Related Literature The research in this thesis is at the interface of operations management and eco- nomics, and relates to four areas of literature: inventory management, supply chain contracting, non-cooperative games driven by demand allocation, and principal-agent theory. The following is an overview of the relevant research work. More detailed review of the literature can be found in each essay. 1.2.1 Inventory management A number of papers in the inventory management literature investigate inventory per- formance measures. Examples are Schneider (1981), Choi et al. (2004), Boyaci and Gallego (2001), Wang et al. (2005), Thomas (2005), and Katok et al. (2008). Choi et al. (2004) investigate choosing supplier performance measures for vendor-managed- 2 1.2. Related Literature inventory when the supplier's capacity and inventory policy are private information. The performance measure in their study is mainly the ready rate. O\u000b-the-shelf (imme- diate) ful\fllment rates are often used and studied in theory to measure the inventory performance, but time-window ful\fllment rates are more commonly employed in prac- tice. Among the few papers that consider time-window performance measures, Boyaci and Gallego (2001) study the problem of minimizing average inventory costs subject to \fll-rate and \fll-rate-with-window service-level constraints in serial and assembly systems; and in an (s,S) inventory system with service level target represented by a time-window ready rate, Wang et al. (2005) \fnd a signi\fcant tradeo\u000b between the window length and the inventory costs, and suggest that a longer ful\fllment window and lower price may be used for price-sensitive but time-insensitive customers. The majority of the inventory management literature considers performance mea- sures in the long run using expected performance (see Zipkin 2000, for example, for an introduction on the inventory theory in the literature). This is because the research questions are generally operations-oriented for an integrated supply chain. Those con- sidering incentive issues tend to assume a contract can be on the long-run expected performance (see, e.g., Choi et al. 2004), which can have problems in implementation. When an SLA is used for a supplier with unobservable e\u000bort, the supplier's perfor- mance measure in a \fnite review period is a random variable, which often di\u000bers more or less from the long-run expectation. A contract based on the expected performance (service level) alone, either being unobservable or needing too long a review period gives no basis to the supplier for identifying and rectifying underperformance and thus cannot provide an adequate incentive. For an SLA to work, it is critical to understand and employ in its design the probability distribution of its performance over the (typically short) review period in order for incentive to provide the right alignment. Thomas (2005) and Katok et al. (2008) consider the \fll rate in a \fnite horizon. Thomas uses simulation to illustrate its distribution, and Katok et al. use an experimental method for analysis. Both papers examine how the length of a review period and the bonus\/penalty a\u000bect the agent's choice of the base-stock level. They only investigate a supplier's stationary inventory policy but do not consider time-window ful\fllment rates. To address the above issues, for the problem of contracting for inventory manage- ment service, we are therefore interested in the design of an SLA on the basis of a random performance measure, as well as the di\u000berence between the incentive provided 3 1.2. Related Literature by an immediate ful\fllment rate and that by a time-window one. These two issues are examined in the \frst and the second essays respectively. 1.2.2 Supply chain contracting We introduce two types of contracts which are related to the SLAs studied in the \frst two essays. Supply chain coordination contracts The coordination of inventory investment (order quantity) with contracts has been extensively studied in the supply chain literature. A comprehensive review can be found in Cachon (2003), which includes the coordination of single-period single or multiple newsvendor problem as well as the coordination of in\fnite-horizon inventory investment on a durable product. In particular, for a single-location base-stock model with linear backorder cost, the optimal coordination contract is essentially a cost sharing one, with the retailer (agent) and the seller (principal) each bears a portion of the total supply chain cost (inventory holding and backorder costs). Employing such a contract means micromanaging the agent because the inventory level and the backorders at any time need to be recorded, which will de\fnitely result in large administrative and transaction costs. Moreover, when the backorder cost is nonlinear, the coordination contract may have a complex form that is di\u000ecult to implement, which will not be an issue when using SLAs. Performance-based contracts The design of performance-based contracts has drawn operations researchers' at- tention in recent years. Examples can be seen in Plambeck and Zenios (2000), Plam- beck and Zenios (2003), Kim et al. (2007), and Kim et al. (2009). Plambeck and Zenios (2000) consider a principal delegating operational control of a production sys- tem to an agent who can exert unobservable e\u000bort to maintain the system, and inves- tigate the compensation scheme dependent on the observed system state transition. Plambeck and Zenios (2003) study a vendor-managed-inventory problem in which an agent chooses a privately known production rate to build inventory for the principal, and derive a contract based on the observed inventory level. Both papers consider a dynamic principal-agent problem with a risk-averse agent. Kim et al. (2007) study an after-sales service supply chain in which a customer operates assembled systems 4 1.2. Related Literature with each system consisting of distinct parts each provided by a di\u000berent supplier. If any of the parts fails, the system is down, and that part has to be replaced by a spare part. When there is no spare part available, a backorder occurs. Each sup- plier determines her stock level of spare parts, which is unobservable to the customer. Facing a system uptime requirement, the customer o\u000bers contracts to the suppliers. The authors propose a contract linear in the backorders of each part, and show that it induces the \frst-best solutions when all parties are risk neutral. They only con- sider suppliers' stationary policies in a static formulation, and study a coordination problem with no supplier competition. Kim et al. (2009) focus on the choice of per- formance measure in a contract for restoration service of mission-critical systems. A risk-averse supplier makes a one-time service capacity investment unobservable to the customer of a system with infrequent but costly failure. The supplier's incentives un- der the performance measures of sample-average downtime and cumulative downtime are compared. The SLAs on inventory management in our study are based on an aggregate performance measure { the demand ful\fllment rate, rather than on the individual performance outcome such as the number of backorders. This type of contract dif- fers from most of the aforementioned papers on inventory coordination contract and performance-based contracts. Thus it is interesting to \fnd out its structure and understand how it generates the incentive for investment. 1.2.3 Non-cooperative games driven by demand allocation In the operations management literature, a few papers investigate \frms' competi- tive behavior under an exogenous demand allocation mechanism, often driven by the switching behavior of customers in the market, which is dependent on the \frms' real- ized service levels. Lippman and McCardle (1997) consider a single-period multiple- competitive-newsvendor problem in which each newsvendor chooses an inventory level to meet a random demand and a rule speci\fes the allocation of initial market demand among the \frms and the allocation of excess demand among those \frms with remain- ing inventory. They derive the equilibrium inventory levels under speci\fc allocation rules. Hall and Porteus (2000) and Liu et al. (2007) consider a multi-period competi- tive newsvendor problem where two \frms make capacity (inventory) decision in each period, and the demand for each \frm is dependent on the realized level of customer 5 1.2. Related Literature service (product availability) in the prior period. Both papers look for the \frms' equilibrium behaviors in the dynamic game. Some other papers consider a buyer specifying the allocation rule. Cachon and Lariviere (1999) investigate a special allocation rule commonly used in the automobile industry by considering a single supplier allocating capacity to multiple retailers based on their past sales. A two-period game is studied. Both Benjaafar et al. (2007) and Cachon and Zhang (2007) consider a buyer outsourcing a \fxed demand at a \fxed unit price to multiple suppliers. Benjaafar et al. (2007) examine two competitive mechanisms used for outsourcing to a set of potential suppliers in a single-period setting. One mechanism is allocating the whole demand to one supplier with the probability of being selected increasing with her committed service level, the other allocating the demand to each supplier in proportion to her committed service level. Under both cases, it is assumed that the contractual promises of the suppliers regarding e\u000bort or service level are enforceable. The authors compare the service quality the buyer can achieve under both mechanisms. Cachon and Zhang (2007) study a queuing system where each supplier's service time is determined by her capacity investment, and the buyer allocates the demand among multiple suppliers based on their service times to minimize the average service time over an in\fnite horizon. Suppliers are homogeneous in terms of their capacity costs. Each supplier chooses a capacity level to maximize her own pro\ft. Several commonly used allocation rules are evaluated and an optimal rule is proposed. There exists a vast literature on sourcing policies for the problem of a buyer awarding a divisible business to one or more suppliers among multiple suppliers. The research questions are often related to the design of competitive mechanisms in the form of bidding and the suppliers' competitive behaviors under the bidding rule. Elmaghraby (2000) provides a survey on the research of this topic in the operations research and economics literature. Most of the models are for a one-time decision problem. The incentive from future business is largely not considered. Since the suppliers' competitive behaviors that are in\ruenced by future business and by the design of competitive mechanisms with no future consideration have been studied separately in the literature, the natural research question would be: How to design a competitive mechanism with the suppliers' incentive coming from the future business, i.e., a rule for allocating the buyer's future business among the suppliers based on their past performance? What is the impact of a supplier's current share of 6 1.2. Related Literature business on the form of the optimal allocation rule? How to design an allocation rule to maintain the suppliers' competition over time? All these questions are addressed in the third essay. 1.2.4 Principal-agent theory Recent years have seen the application of principal-agent theory to operations man- agement problems for incentive contract design. We also employ this theory as an analytical tool for all the research problems in this thesis. Bolton and Dewatripont (2005) provide a broad coverage of literature on principal-agent theory. All three es- says in the thesis are on the repeated moral hazard problem, with the \frst two essays on a single agent and the third on multiple agents. The SLAs studied in the \frst two essays are a blend of multiple-period review strategies and discontinuous incentive schemes. Radner (1985) investigates multiple- period review strategies for a repeated principal-agent game. The agent's performance in a period is a noisy signal of her e\u000bort level in that period. The agent's performance is reviewed every R periods. If the performance is within a margin of error of the expected output under the desired e\u000ecient e\u000bort level, the agent 'passes' the review. Otherwise they enter a penalty phase of M periods. The penalty in the SLAs under our study is monetary, not a subsequent phase of a non-cooperative game. With discontinuous incentive schemes, the payment from a principal to an agent changes only when some threshold of good or bad performance is reached (McMillan 1992). They are typically derived in single-period models. When this type of incentive scheme is used together with a multi-period review strategy, some new issues emerge. Spear and Srivastava (1987) study a repeated moral hazard problem with dis- counting between a principal and an agent, and show that history dependence can be represented by using the agent's expected utility as a state, and thus the problem of characterizing the optimal contract of such a model can be reduced to a constrained static variational problem. Monetary compensation is used for an incentive. The third essay considers the problem between a principal and two competing agents, and the compensation is in the form of future demand. The optimal allocation rule is also derived from a constrained static variational formulation, but we are also concerned about the equilibrium of the agents' dynamic game. Competitive compensation schemes studied in the economics literature can come 7 1.3. Research Methodology and Findings in the form of rank-order tournament or relative performance evaluation (RPE), gen- erally for a single-period problem. The relevant research can be found, for example, in Lazear and Rosen (1981), Green and Stokey (1983), Hart (1983), Holmstrom (1982), and Nalebu\u000b and Stiglitz (1983). RPE compensates the agents based on their output levels, and the total compensation varies with the agents' realized output levels. In our case the total demand to be split is a constant. In tournaments, rewards are based on rank order of the individuals, not on their actual output levels. Lazear and Rosen (1981) show that for risk-neutral agents rank-order tournaments work as well as independent contracts; and for agents with known heterogeneous ability, handicap- ping will improve the e\u000eciency of the tournaments. We want to \fnd out if rank-order tournaments are still optimal when incentives come from future business, and how the demand allocation varies with the agents' outputs. Both questions are examined in the third essay. 1.3 Research Methodology and Findings All three essays study the incentive mechanism design in the presence of suppliers' moral hazard problem. We therefore apply principal-agent theory to the analysis in each essay. The \frst two essays deal with a single supplier's service level agreement design and examine speci\fc forms of discontinuous incentive schemes, so the focus is mainly to \fnd out if by choosing the contract parameters carefully, the buyer's optimization problem constrained by the supplier's incentive compatibility and individual rational- ity (participation) constraints can result in the same optimal solution as the \frst best (the optimal solution to the unconstrained problem). In the \frst essay, the analysis is complicated by the existence of the supplier's strategic behavior over time due to the multi-period review structure of SLAs, so we need to use a dynamic programming technique to solve for the supplier's optimal policies under the contract o\u000bered by the buyer. The third essay studies a single-principal\/two-agent\/multi-period problem, in which the principal (buyer) designs an allocation rule to dictate how the two agents (suppliers) play a non-cooperative stochastic game. Therefore, in addition to principal- agent theory, we also apply stochastic game theory to the suppliers' problems and look for the suppliers' subgame perfect Nash equilibrium in a \fnite-horizon game as 8 1.3. Research Methodology and Findings well as their stationary Nash equilibrium in an in\fnite-horizon game. The contribution and \fndings of each essay are as follow. 1. The main contribution of the \frst essay is to argue for a methodology for studying the design of SLAs, speci\fcally applying principal-agent theory. We have investigated two frequently observed forms of SLAs: the lump-sum penalty SLA and the linear-penalty one. We have identi\fed the issue of potential supplier's strategic behavior under an SLA when the supplier can observe the performance history and dynamically adjust her e\u000bort level to a\u000bect her review period performance. We \fnd that to mitigate the supplier's incentive for strategic behavior, the penalty should be dependent on the amount of supplier's performance deviation from the target. In particular, a simple linear-penalty SLA can do well over a lump-sum penalty one for this purpose. When using a linear-penalty SLA, the performance threshold should be kept close to the target, i.e., the allowable deviation of the performance from the target should be small. 2. The second essay investigates the e\u000bectiveness of performance measures in SLAs for a multi-task problem. Speci\fcally, we examine two types of linear-penalty SLAs using either the immediate or time-window ready rate as a performance mea- sure when the supplier can make privately observed investments in inventory and supply lead time. We \fnd that there exists a unique positive time window such that a ready rate with window induces the \frst-best investment, and so an SLA using only the immediate ready rate generally cannot induce the \frst-best investment. The immediate ready rate can induce near optimal outcome when the buyer's cost for delayed demand ful\fllment is linear in the length of delay, but the e\u000eciency loss is higher when the cost is convex. Our \fndings demonstrate the importance of choos- ing the right performance measure to align a supplier's incentive, and provide some theoretical basis for the use of time-window ful\fllment rate in practice for incentive purpose. 3. The main contribution of the third essay is to examine special features of performance-based volume incentives under supplier competition over time, and tackles the topic of volume incentives for operations management which has been little studied in the literature. For both the case where a supplier's cost of e\u000bort to perform is proportional to the volume of business and the case where it's independent of the volume, we have found: to induce and maintain suppliers' competition over time, the optimal volume incentive scheme is generally not a simple rank-order tournament; 9 1.3. Research Methodology and Findings handicapping the de\fnition of winner can do well over a simple \frst-past-the-post rule even when the suppliers have an identical capability of doing the work, and the role of handicapping in volume incentive is either to enhance the suppliers' competition intensity or to provide a stronger incentive to the supplier with a larger share; and volume incentives often need to take into account each supplier's current share of business even when a supplier's e\u000bort cost is independent of the business volume. All these special features of volume incentives di\u000ber from those shown for the monetary incentives in the literature. We have also found that by limiting a supplier `s maximum share in each period, volume incentive schemes can always induce a unique stationary Nash equilibrium in the suppliers' stochastic game over an in\fnite horizon. Performance-based incentive schemes for operations management have opened an area with many issues for research. For example, because SLAs are context dependent and we have only studied the application to inventory management, future research can investigate SLAs for other types of services such as those for health care, logistics, and maintenance. The second essay has demonstrated the importance of choosing the right performance measure for incentive alignment. Future research can be investi- gating performance measures for various applications. This thesis has focused on a risk neutral buyer and suppliers. In practice, \frms especially the small ones often do not want to bear a lot of risk. A \frm's risk attitude may have a big impact on the e\u000bectiveness of a contract and its associated performance measures. Future research can therefore be on the optimal structure of performance-based incentives under risk- averse buyer or suppliers. Adverse selection is another issue that often exists in reality but has not been fully examined in the literature. It refers to the situation where an agent's capability to do the work is privately known but unobservable to the principal. In the presence of adverse selection, the performance measure and particularly the performance target need to be carefully selected and the structure and the values of the contract parameters may be a\u000bected as well. Choi et al. (2004) study the choice of performance measures for vendor-managed-inventory when the supplier's capacity and inventory policy are unknown to the buyer. The third essay has looked into volume incentives under competition. How monetary incentive interacts with com- petitive mechanism hasn't been well investigated and understood in the literature. With competing suppliers, some conclusions can di\u000ber from those for a single agent. The design of performance-based monetary incentives under competition can be an avenue for future research. 10 1.4. Structure of the Thesis 1.4 Structure of the Thesis The rest of this thesis consists of three chapters from two to four. Each chapter is a stand-alone paper with an introduction, a literature review, the main body and a con- clusion. Following them is the bibliography for all three chapters. The mathematical proofs for each chapter are in the appendices at the end of the thesis. 11 Chapter 2 Designing Service Level Agreements For Inventory Management 2.1 Introduction Service level agreements (SLAs) are a common type of performance-based contract for managing suppliers. A survey by Oblicore Inc. in 2007 found that 91% of organi- zations use SLAs for managing suppliers, internal agreements, or external customer agreements. In an SLA, the performance, or outcome of a task desired by the buyer is identi\fed and a service level target speci\fed. The buyer neither dictates nor needs to know how the work is done; the vendor can freely choose the most cost-e\u000ecient way. As described by the US O\u000ece of Federal Procurement Policy about Performance- Based Contracting: The Performance-Based Acquisition (PBA) means an acquisition structured around the results to be achieved as opposed to the manner in which the work is to be per- formed (see, e.g., Acquisition Central website). In return, the buyer pays a \fxed price over a certain time period. As a \fxed price alone is not enough to guarantee the required performance, incentives are needed. For example, a penalty might be imposed when the vendor underperforms over a period of time. Despite widespread use of SLAs for outsourcing manufacturing and services, few papers address their design. This chapter examines some fundamental issues of SLA design by studying an application to inventory management from a principal-agent perspective. Speci\fcally, consider a single supplier and a single buyer, where the supplier manages the supply of a single product for the buyer, and can invest in inventory to meet a service level target, but the investment level is unobservable to 12 2.1. Introduction the buyer | a moral hazard problem in agency theory. A problem with SLA design is that they are very context dependent. However, companies do need to answer \fve questions. First, what performance measure should be used. Since the performance is reviewed over a period of time, this measure should be an aggregate one. Second, what performance target is appropriate, as a higher performance means the buyer pays more. A performance target should consider both the buyer's valuation of performance and the vendor's cost of doing the task. The third question is how frequently performance will be reviewed. Any su\u000eciently complex task will result in some natural variation (noise) in performance, which will be a\u000bected by the review frequency. It is undesirable to penalize minor deviations from the target that are due purely to noise, so the fourth question is how much deviation is allowed from the target. Lastly, what penalty the vendor should pay when performance exceeds the allowable deviation. Here we mainly address the last two questions. Our objective is to study key issues in SLA design and the e\u000bectiveness of di\u000berent forms of penalties. We do not explicitly consider the buyer's backorder cost, but take the target service level as exogenous. In Section 2.8.1 we discuss the situations under which an SLA is preferred over a traditional coordination contract, e.g., when the backorder cost is nonlinear. We investigate two choices of a penalty that the buyer might employ to manage the supplier. The \frst is where the supplier incurs a lump-sum penalty if her review phase performance is below a performance threshold. The second is a linear-penalty SLA, where the supplier incurs a penalty linear in the amount of deviation from a performance threshold. For example, again from the Acquisition Central website (on Performance-Based Service Acquisition), we have: Example 1. \"The \frm-\fxed-price for this ... shall be reduced by 2% if the perfor- mance standard is not met.\" (A lump-sum penalty) Example 2. \"For each 5% degradation in ... performance observed ..., the \frm- \fxed-price for ... will be reduced by 1%.\" (A linear-based penalty) Under an SLA, the supplier's inventory performance is reviewed every R periods, the review phase. Unlike most inventory models we must measure performance (a random variable) over the \fnite period R. A contract based on the expected perfor- mance (service level) alone, being either unobservable or needing too long a review phase gives no basis to the supplier for identifying and rectifying underperformance and thus cannot provide an adequate incentive. Therefore we need the distribution 13 2.1. Introduction of review phase performance. The commonly used inventory performance measures in both the practice and the literature are \fll rate and stockout rate. Fill rate is the long-run fraction of demands that are \flled immediately. The steady state distribu- tion of the \fll rate is very hard to derive. The only study on its distribution is by Thomas (2005) using simulation for a static periodic-review base-stock model with zero leadtime. Our problem is more than obtaining the distribution of a performance measure because we need to \fnd out the supplier's optimal response and the buyer's optimal choice of contract parameters given a performance measure. For simplicity in exposition, most of this chapter focuses on the ready rate | the long-run fraction of periods that demands are \flled from stock, which is equal to 1 \u0000 stockout rate. But the major insights hold for the \fll rate as well. The conventional \fll rate and ready rate are performance measures of the immediate order ful\fllment. In practice, time window ful\fllment rates are more commonly used (LaLonde et al. 1988), so we also consider the ready rate with a window | the long-run fraction of periods that demands are \flled within a pre-speci\fed time window. We provide a theoretical approximation to the distributions of the immediate and time window ready rates. In our problem, the optimal inventory policy for the integrated supply chain is a static one. We \frst address the SLA design problem assuming the supplier employs a static inventory policy. However, the supplier can observe her performance during the review phase and adjust her inventory level, inducing supplier's strategic behavior. We investigate the di\u000berent incentives of the two penalty regimes in achieving the target performance and avoiding sub-optimal dynamic behavior. We \fnd that although both the lump-sum penalty and linear-penalty SLAs can induce a nonstrategic supplier to choose the \frst-best (system optimal) base-stock level, they can result in very di\u000berent inventory investments in the case of a strategic supplier. Speci\fcally, under a lump- sum penalty SLA, a strategic supplier can achieve a signi\fcant cost saving from using a dynamic inventory policy; but such cost saving is minimal under a linear-penalty SLA. The main contribution of this chapter is to argue for a methodology for studying the design of SLAs, speci\fcally applying principal-agent theory. Our results have direct managerial implications to the design of SLAs. When the supplier can observe the performance history and dynamically adjust her e\u000bort level to a\u000bect her review phase performance, to mitigate the supplier's incentive for strategic behavior, the penalty should be dependent on the degree of supplier's performance deviation from 14 2.2. Literature Review the target. In particular, a simple linear-penalty SLA can do well over a lump-sum penalty one for this purpose. When using a linear-penalty SLA, the performance threshold should be kept close to the target, i.e., the allowable deviation of the per- formance from the target should be small; as a consequence, the penalty rate in a linear-penalty SLA is small so that the supplier has a low chance to pay an `una\u000bord- able' penalty. Another drawback with the lump-sum penalty SLA is that the penalty is generally large, which may not be feasible in practice. Although the application is to SLAs in inventory management, the lessons are applicable to SLA design in other situations. The rest of this chapter is organized as follows. Section 2.2 reviews the literature. Section 2.3 describes the model, with the distribution of the ready rate derived in Section 2.4. Section 2.5 studies the buyer's problem and the supplier's long-run average cost under the two SLAs. Section 2.6 investigates the supplier's strategic behavior and Section 2.7 presents numerical results. The chapter concludes with a discussion in Section 2.8 and a summary in Section 2.9. 2.2 Literature Review Most inventory management literature considers performance measures in the long run using expected performance. In a \fnite review phase, and the fact that the buyer cannot (and does not wish to) observe (micromanage) supplier e\u000bort, it is critical to use the distribution of random performance to try and unravel whether poor performance results from weak e\u000bort by the supplier or simply noise. Punishing suppliers for mere noise means suppliers overinvest, and charge more, an ine\u000ecient outcome. Thomas (2005) and Katok et al. (2008) consider \fll rate in a \fnite horizon, using a static periodic-review base-stock model with zero lead time. The former considers a lump-sum penalty SLA; the latter a lump-sum bonus one, which gives the supplier a \fxed bonus if the actual \fll rate is above a threshold. Thomas uses simulation to investigate how the length of a review phase R and the penalty a\u000bect the optimal base-stock level. Katok et al. use an experimental method to examine how R and the bonus a\u000bect the human subjects' choice of the base-stock level. These two papers only consider the lump-sum penalty\/bonus incentive, and do not examine the supplier's strategic (dynamic) behavior during a review phase. Multiple-period review strategies are \frst studied by Radner (1985) for a repeated 15 2.2. Literature Review principal-agent game. The agent's performance in a period is a noisy signal of her e\u000bort level in that period, and the performance in each period is independently and identically distributed (i.i.d.). The agent's performance is reviewed every R periods. If performance is within a margin of error of the expected output under the desired e\u000ecient e\u000bort level, the agent 'passes' the review. Otherwise they enter a penalty phase of M periods. In the supply chain context, Ren et al. (2008) apply a modi\fed multiple-period review strategy to an information sharing game between a buyer and a supplier in a decentralized supply chain. In each period, the market demand is a function of a demand state and some normally distributed random variable, the buyer privately observes the realized demand state and sends a forecast to the supplier. A review strategy is used to evaluate whether the buyer truthfully shares the demand state information. The review strategy in their model di\u000bers from the one in Radner (1985) in two aspects. First, the review of the buyer's truthfulness of information sharing is started right at the beginning of each review phase and is conducted every period instead of only once at the end of a review phase. Second, the review phase is not \fxed, R periods is the maximum length, but the phase can be terminated earlier once a review indicates that the buyer will have no incentive to share information truthfully during the rest of the review phase. Both papers show that the two parties' payo\u000bs can be arbitrarily close to the equilibrium e\u000ecient payo\u000bs when R is su\u000eciently large. In our model, the SLA is also a multiple-period review strategy, but there are two major di\u000berences. Our review period length R is exogenous, and our penalty is monetary, not a subsequent phase of a noncooperative game. Also in our model, the performance in each period can be correlated. We and Radner (1985) study a moral hazard problem and Ren et al. (2008) a hidden information one. Choi et al. (2004) investigate choosing supplier performance measures in a vendor- managed-inventory context. The supplier's capacity and inventory policy are private information, so the buyer sets performance measures for the supplier to meet an end- customer service level target. They show that in a capacitated supply chain, the supplier's service level is in general not su\u000ecient to guarantee the target customer service level, and they propose a menu of contracts instead. Although supplier's actions are unobservable in their model, Choi et al. focus on the choice of performance measures rather than the noise in the observed measures and the penalty for failing 16 2.3. Model Description to meet targets. The performance measure in their study is mainly the ready rate. The economics literature studies discontinuous incentive schemes where the pay- ment from a principal to an agent changes only when some threshold of good or bad performance is reached (McMillan 1992). These are typically single-period mod- els. We study discontinuous incentive schemes under a multi-period review strategy, which bring about issues that do not exist in a single-period model. As we employ a principal-agent framework, a reasonable question is whether our work is simply a special case. The answer is `yes' at the most abstract level, but the implementation details di\u000berentiate them markedly. In the principal-agent literature, when both parties are risk neutral, a \fxed-fee contract can be used to induce the \frst-best e\u000bort level, under which the buyer pays the supplier a \fxed fee and the supplier bears all the system costs. Our study di\u000bers in two aspects. First, the SLAs in our study have a speci\fc payment structure in that the supplier only pays a penalty for underperformance. So an SLA gives a speci\fc form of risk-sharing rule for the two parties. Secondly, in the principal-agent literature, the distribution of the performance usually has a simple form such as an additive noise. As we focus on speci\fc applications, the speci\fc distribution of the performance is critical and typically quite complex. The application of SLAs to call center outsourcing is studied by Milner and Olsen (2008) and Hasija et al. (2008). Both papers consider SLAs based on the expected performance and do not consider a multi-period review strategy. 2.3 Model Description Consider a supply chain consisting of a single supplier and a single buyer, where the supplier manages the supply of a single product for the buyer near the buyer's site. Both parties are risk neutral. The demand is stochastic, and the demand in each period is i.i.d. with mean \u0015 and standard deviation \u001b. Assume the distribution of single-period demand is unimodal and can be either discrete or continuous. The supplier owns the inventory and incurs a constant unit inventory holding cost h per time period. The supplier uses a periodic-review base-stock policy with a constant inventory replenishment lead time L and a base-stock level S. Her only choice is the base-stock level. At the beginning of period t, the supplier determines the base-stock level and places an order. The order placed at time t arrives by the end of period 17 2.4. Performance Measure t+ L and is used to \fll demands occurred before the end of period t+ L. If a demand is not \flled immediately, it is backlogged, and the buyer incurs the backorder (delay) cost, CD(y), which is increasing and convex in the length of delay y. The buyer pays a unit transfer price p for the product to the supplier. The supplier's unit ordering cost c is constant. Under our assumptions on the inventory holding cost and backorder cost, the optimal inventory policy for the integrated supply chain is a static base-stock (order-up-to-S) policy, which has been shown in, for example, Zipkin (2000). In order to induce the supplier to invest in inventory, the buyer contracts with the supplier on the supplier's inventory service level. This SLA uses some aggregate level of performance, sets a target service level, and reviews the supplier's performance every R periods (a review phase). Assume R > maxfL; 1g. 2.4 Performance Measure Commonly used measures of inventory performance are \fll rate and ready rate. Fill rate is the long-run fraction of demands that are \flled immediately. Ready rate is the long-run fraction of periods in which demands are \flled immediately, which measures the inventory availability and is equal to 1 \u0000 stockout rate. The ready rate and \fll rate in our model are the \u000b-type and \r-type service levels as de\fned in Schneider (1981) and used by Choi et al. (2004). For simplicity in exposition, we mainly focus on the ready rate. We note that the major insights apply to \fll rate as well. We consider two types of ready rates: the conventional immediate ready rate and the time window ready rate, which measures the performance of \flling demands within a delivery time window. 2.4.1 Performance measure under a multi-period review strategy The supplier's performance is evaluated every review phase of length R demand peri- ods. Let D(t) be the demand in t periods, D(t) \u0015 0. Let D[t; \u001c) denote the demand in the interval [t; \u001c), i.e., from period t through period \u001c \u0000 1. Let W (0 \u0014 W \u0014 L) be the delivery time window. W = 0 means immediate demand ful\fllment, and W > 0 means demand ful\fllment within a time window W . Let the performance indicator 18 2.4. Performance Measure for period t (1 \u0014 t \u0014 R) be XWt , XWt = 1fD[t \u0000 L; t + 1 \u0000W ) \u0014 St\u0000Lg 2 f0; 1g, where XWt = 1 and X W t = 0 represent the situations where there is no or some de- mand delayed longer than time W at the end of period t, and St\u0000L is the supplier's base-stock level chosen in period t\u0000 L. We de\fne PrfXWt = 1g = PrfD[t\u0000L; t+1\u0000W ) \u0014 St\u0000Lg = PrfD(L+1\u0000W ) \u0014 St\u0000Lg = FL+1\u0000W (St\u0000L), where Fn(\u0001) is the cumulative distribution function (cdf) of demands in n periods. Let fn(\u0001) denote the corresponding probability density function (pdf) for continuous demand, or the corresponding probability mass function for discrete demand. For continuous demand, assume Fn(\u0001) and fn(\u0001) are continuous. Let the supplier's cumulative performance during a review phase be \u0011WR = RX t=1 XWt . \u0011WR is the number of periods without delay longer than W , and 0 \u0014 \u0011WR \u0014 R. So the ready rate in a review phase is AWR = \u0011 W R =R, and is random. A 0 R is the review phase immediate ready rate, and AWR (W > 0) the review phase time window ready rate. In each period of a review phase, the supplier receives a constant unit transfer price. Given a performance threshold \u000b for the review phase ready rate AWR , if the observed AWR \u0014 \u000b, i.e., \u0011WR \u0014 R\u000b, then the supplier incurs a penalty. 2.4.2 Distribution of ready rate under a static base-stock policy Under a static base-stock policy, the supplier uses the same base-stock level S in every period. Because the demand in each period is i.i.d, XWt has an identical distribution for each t. PrfXWt = 1g = FL+1\u0000W (S), and the ready rate (in the long run) is AW = limR!1AWR = FL+1\u0000W (S). The case when L = 0 : We only need to consider the immediate ready rate, W = 0. The probability of no stockout in a period is PrfX0t = 1g = PrfD(1) \u0014 Sg = F1(S). Because the demand in each period is i.i.d, the performance in a period Xt is i.i.d. So the cumulative performance \u00110R has a binomial distribution B(R;F1(S)), and Prf\u00110R = jg = \u0000R j \u0001 (F1(S)) j(1\u0000F1(S))R\u0000j. Moreover, the distribution of the review phase ready rate A0R is approximately normal N(F1(S); q F1(S)(1\u0000F1(S)) R ). The case when L > 0 : 19 2.4. Performance Measure We can have either W = 0 or W > 0. The performance in each period can be correlated. The performance outcomes in any two periods, XWi and X W j , are independent for ji\u0000jj \u0015 L+1\u0000W because D[i\u0000L; i+1\u0000W ) and D[j\u0000L; j+1\u0000W ) have no periods overlapped; and they are correlated for ji \u0000 jj \u0014 L \u0000 W because D[i\u0000L; i+1\u0000W ) and D[j\u0000L; j+1\u0000W ) have some periods in common. Proposition 2.1 describes the theoretical distribution for \u0011WR . Proposition 2.1 Under a static periodic-review base-stock policy with base-stock level S, \u0011WR \u0000E(\u0011WR ) \u001bWR converges in distribution to a standard normal random variable as R approaches 1, where E(\u0011WR ) = RX t=1 E(XWt ) = RFL+1\u0000W (S); and variance (\u001bWR ) 2 = RFL+1\u0000W (S)\u0000 [(L\u0000W )(2R\u0000 L+W \u0000 1) +R](FL+1\u0000W (S))2 + L\u0000WX n=1 2(R\u0000 (L+ 1) + n+W ) Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x): (2.1) Proposition 2.1 implies that for su\u000eciently large R, \u0011WR is approximately normally distributed with mean RFL+1\u0000W (S) and standard deviation \u001bWR . Therefore, in our numerical analysis, we will use normal approximation for the distribution of \u0011WR . Because the supplier's review phase ready rate AWR = \u0011 W R =R, Corollary 2.1 follows. Corollary 2.1 Under a static base-stock policy with base-stock level S, AWR \u0000E(AWR ) \u001bWR =R converges in distribution to a standard normal random variable as R approaches 1, where E(AWR ) = FL+1\u0000W (S): So for su\u000eciently large R, AWR is approximately normally distributed with mean FL+1\u0000W (S) and standard deviation \u001bWR =R. Proposition 2.2 shows the e\u000bect of the length of a review phase, R, on the variability of performance measure AWR . Proposition 2.2 V ar(AWR ) is decreasing in R. Proposition 2.2 implies that a long review phase reduces the variability of the supplier's performance outcome. The \fll rate distribution obtained by Thomas (2005) using simulation shows a similar property. 20 2.5. Buyer's Problem 2.5 Buyer's Problem In this section we consider a nonstrategic supplier, assuming the supplier uses a static inventory policy. We derive the optimal contract parameters for the buyer's prob- lem. In Section 2.6 we consider the dynamic inventory policy and study a strategic supplier's problem under an optimal contract. Consider two types of SLAs. The \frst one has a lump-sum penalty under which a supplier pays the buyer a lump-sum penalty K if the supplier's ready rate with window W during a review phase, AWR , is not above the performance threshold \u000b. The second type is a linear-penalty SLA, under which if AWR is no more than \u000b, then the supplier will pay the buyer a penalty proportional to the di\u000berence between AWR and \u000b. Speci\fcally, for the realized number of periods without stockout i, i \u0014 R\u000b, the supplier will pay a penalty K(R\u000b+1\u0000 i). Let CP (AWR ; \u000b;KjS) denote the supplier's average penalty per period when the supplier chooses a static base-stock policy with base-stock level S and the realized review phase ready rate is AWR , given the ready rate threshold \u000b and penalty parameter K. The supplier's objective is to maximize her long-run average pro\ft, which is her long-run average revenue minus cost, including the inventory holding cost, the ex- pected penalty and the ordering cost. Because of full backlogging, all demands are \flled. So the supplier's base-stock level in each period does not a\u000bect the average ordering cost. Without loss of generality we assume the unit ordering cost c = 0. Given the lead time L and the base-stock level S, the buyer's average cost is UL(S) = p\u0015+ \u0015ECD(yjS)\u0000 ECP (AWR ; \u000b;KjS); and the supplier's average pro\ft is \u0019L(S) = p\u0015\u0000 hE[S \u0000D(L+ 1\u0000W )]+ \u0000 ECP (AWR ; \u000b;KjS); where D(L+ 1\u0000W ) is the realized demand in L+ 1\u0000W periods. The buyer's problem is to choose contract parameters p, \u000b and K such that minp;\u000b;K;S UL(S) subject to (IR) : \u0019L(S) \u0015 \u0019 (IC) : S 2 argmaxbs \u0019L(bS) ; 21 2.5. Buyer's Problem where \u0019 is the supplier's reservation pro\ft per period, and the \frst constraint is called the individual rationality (IR) constraint, which guarantees the supplier to gain at least her reservation pro\ft so that she will accept the contract; and the second constraint is called the incentive compatibility (IC) constraint, which re\rects the supplier's optimal solution given the contract o\u000bered by the buyer. The above formulation with the (IR) and (IC) constraints is a standard formulation for incentive mechanism design in agency theory. The buyer can choose p to make the supplier earn only her reservation pro\ft and extract the rest of supply chain pro\ft1. So the buyer's problem can be reformulated as min\u000b;K;S hE[S \u0000D(L+ 1\u0000W )]+ + \u0015ECD(yjS) subject to S 2 argmaxbs \u0019L(bS) : (2.2) Let S\u0003 be the optimal solution without constraint (2.2). Under our assumptions on CD(yjS), S\u0003 minimizes the long-run average supply chain cost. So the target ready rate is FL+1\u0000W (S\u0003), and the buyer's problem is to choose \u000b and K to induce the \frst-best base-stock level S\u0003 and ready rate FL+1\u0000W (S\u0003). In practice, a reasonable performance threshold should not be above the performance target, so we only con- sider \u000b \u0014 FL+1\u0000W (S\u0003). 2.5.1 Choice of contract parameters Let VL(S) denote the supplier's average cost under a static base-stock S policy. Un- der the buyer's contract, the supplier's average revenue p\u0015 does not a\u000bect S, so the supplier's pro\ft-maximizing problem is equivalent to minimizing VL(S). Be- cause the review phase ready rate AWR = \u0011 W R =R, in the supplier's average penalty ECP (A W R ; \u000b;KjS), AWR can be replaced by \u0011WR =R. Under a lump-sum penalty SLA with the ready rate threshold \u000b and lump-sum penalty K, the supplier's problem is min S VL(S) = hE[S \u0000D(L+ 1\u0000W )]+ + K R R\u000bX i=0 Prf\u0011WR = ijSg: (2.3) Under a linear-penalty SLA with threshold \u000b and penalty rate K, the supplier's 1The unit transfer price p can be regarded as a \fxed fee. By choosing a higher p, the buyer can realize any desired pro\ft division between the two parties. 22 2.5. Buyer's Problem problem is min S VL(S) = hE[S \u0000D(L+ 1\u0000W )]+ + K R R\u000bX i=0 (R\u000b + 1\u0000 i) Prf\u0011WR = ijSg: (2.4) For L = 0 and W = 0, Prf\u00110R = ijSg = \u0012 R i \u0013 (F1(S)) i(1\u0000 F1(S))R\u0000i; (2.5) where F1(S) = PrfD(1) \u0014 Sg: For L > 0 and 0 \u0014 W \u0014 L, using the result in Section 2.4.2, the distribution of \u0011WR is approximately normal with mean RFL+1\u0000W (S) and standard deviation \u001b W R , where \u001bWR is given by (2.1). Using a continuity correction, Prf\u0011WR = 0jSg = \b(z0); Prf\u0011WR = ijSg = \b(zi)\u0000 \b(zi\u00001) 0 < i < R; Prf\u0011WR = RjSg = 1\u0000 \b(zR\u00001); (2.6) where \b(\u0001) is the cdf of the standard normal distribution and zi = i+0:5\u0000RFL+1\u0000W (S)\u001bWR . Now what values of parameters (\u000b;K) ensure that the supplier chooses S\u0003, the \frst best integrated solution? We approach this in two steps; \frst, the necessary condition for the supplier's problem gives us a set of (\u000b;K) candidates for each type of SLA, and then we look into the unimodality of the supplier's cost function under the (\u000b;K) candidates. Proposition 2.3 The (\u000b;K) values that ensure S\u0003 satis\fes the \frst order necessary conditions for optimality of either (2.3) and (2.4) are such that for continuous de- mand, there is a unique optimal K\u0003(\u000b) for a given \u000b; and for discrete demand, there is an interval of optimal K\u0003(\u000b), [K\u0003(\u000b); K \u0003 (\u000b)], for a given \u000b. Formulas for both can be found in Appendix A. Proposition 2.3 identi\fes multiple optimal candidates (\u000b;K) for both discrete and continuous demands and both types of penalties. Because \u0011WR only takes integer values, we pay particular attention to \u0002, the set of (R\u000b;K) values under which S\u0003 is the supplier's local optimum; thus 23 2.5. Buyer's Problem \u0002 = f(R\u000b;K)j0 < \u000b \u0014 FL+1\u0000W (S\u0003), R\u000b is integer, K 2 [K\u0003(\u000b); K\u0003(\u000b)]g. Note K\u0003(\u000b) = K \u0003 (\u000b) = K\u0003(\u000b) in the continuous case. We also note that \u0002 can be empty. 2.5.2 Unimodality of the supplier's objective function The candidate parameters (\u000b;K) above are derived from the necessary conditions for optimality, but may not be su\u000ecient. In this section we examine the unimodality of the supplier's average cost under the (\u000b;K) pairs derived above. If the supplier's cost function is unimodal then we can be assured that S\u0003 is a global optimum. Otherwise the solution might be only a local optimum and care needs to be exercised. The supplier's cost function is very complex due to the complicated distribution function of the performance measure. Even with strong assumptions on the demand distribution, it is generally di\u000ecult to prove its unimodality.2 We show that the supplier's cost function under a lump-sum penalty SLA is generally not unimodal. To show its unimodality under a linear-penalty SLA, in particular for L > 0, we have to rely on numerical results. Lump-sum penalty SLA For (R\u000b;K) 2 \u0002, VL(S), the supplier's average cost under a static base-stock S policy, may not be unimodal and S\u0003 not a global optimum. Consider the following example. Consider a normal demand distribution N(10; p 10), which can also be considered as an approximation for a Poisson demand with \u0015 = 10; h = 1, R = 30, L = W = 0, S\u0003 = 14, and hence the performance target FL+1\u0000W (S\u0003) = F1(S\u0003) = 90%. Figure 2.1 shows \fve supplier's cost functions V0(S) under di\u000berent pairs of parameters (R\u000b;K) 2 f(23; 283); (24; 146); (25; 92); (26; 80); (27; 77)g \u001a \u0002. Observe that in this continuous example the K\u0003(\u000b) are unique, not intervals. For example (27; 77) gives \u000b = 27=30 = 90% and (24; 146) gives \u000b = 80% compared to the target of F1(S \u0003) = 90%. The plots have increasing R\u000b corresponding to a decreasing intercept on the vertical cost axis. Each curve has S\u0003 = 14 as a local optimum, however none of the curves is unimodal and S\u0003 is the global optimum only for R\u000b = 23 and 24. For the remaining three pairs, S = 0 is the global optimum and S\u0003 only local. Intuitively, under a lump-sum penalty SLA, reducing the base-stock level will only 2For the special case when L = 0 and f 01(S \u0003) < 0, we can prove that S\u0003 is a local optimum for the supplier's problem with either penalty regime. Interested readers can refer to the Appendix. 24 2.5. Buyer's Problem increase the supplier's probability of being penalized, not the penalty itself. For a large performance threshold, the penalty is small; hence the supplier may prefer not to stock at all because the holding cost saved may exceed the expected penalty increase. Thus S = 0 can be the global optimum for large \u000b. Figure 2.1: Supplier's cost function (lump-sum penalty, L=0) Linear-penalty SLA For the same example but with a linear penalty, Figure 2.2 plots the \fve sup- plier's cost functions for (R\u000b;K) 2 f(23; 171); (24; 79); (25; 43); (26; 27); (27; 20)g \u001a \u0002. Again, the curve with higher R\u000b has lower intercept on the vertical cost axis, and the \frst-order condition gives S\u0003 = 14 for each curve, but now Figure 2.2 shows that they are all unimodal and S\u0003 is the global optimum. Figure 2.2: Supplier's cost function (linear penalty, L=0) Comparing \fgures 2.1 and 2.2, observe the cost of choosing too small a value of S is much more with a linear than a lump-sum penalty. This is because once the target 25 2.6. Supplier's Strategic Behavior has been missed, with a lump-sum penalty any further deterioration is costless. With the linear penalty however, the expected cost keeps rising as performance deteriorates. For L = 1, similar patterns can be found. Thus there are multiple choices of optimal (\u000b;K) for either penalty regime. The linear-penalty SLA appears more likely unimodal than the lump-sum penalty case. This is because the supplier's penalty increases with the amount of performance deviation from the target. When reducing the base-stock level, the supplier will not only increase the probability of being penalized, but also the probability of paying a higher penalty, so the expected penalty increase will o\u000bset the holding cost saved. From Section 2.5.1 and this section, we can conclude that in the case of a lump-sum penalty SLA, VL(S \u0003) is likely to be the global minimum only under relatively small thresholds. The linear-penalty SLA is more likely to have S\u0003 as the global optimum. 2.6 Supplier's Strategic Behavior The optimal contract parameters are derived above under a static inventory policy. However the supplier, aware of her performance, may prefer to dynamically adjust inventory as the review period progresses. If her performance so far is good with few periods left in that review phase, her probability of exceeding the performance threshold \u000b is large, then she has an incentive to decrease inventory to reduce cost. If the supplier's probability of exceeding \u000b is small due to poor performance, she may have an incentive to increase inventory to repair the damage or alternatively to abandon any chance of improving and at least reduce cost by not keeping any further inventory. In this section we formulate the dynamic program for computing a strategic sup- plier's optimal average cost under a dynamic inventory policy and the optimal contract obtained above. We \frst investigate the supplier's cost minimization problem in a single review phase under a contract o\u000bered, ignoring the impact of the inventory policy on the subsequent review phases. After we obtain the supplier's minimum cost from a single-review-phase problem, we then calculate the supplier's minimum long-run average cost. In this chapter, we mainly study the cases of L = 0 and 1 with W = 0. But the major \fndings also hold for the situation of L > 1 and 0 \u0014 W \u0014 L. For a positive lead time L > 0, the supplier's performance realization from her action in 26 2.6. Supplier's Strategic Behavior any period is delayed until L periods later. So when choosing the base-stock level in period t, the supplier has to anticipate possible performance outcomes in periods t; t + 1; :::; t + L \u0000 1, which depend on the base-stock levels chosen in period t \u0000 l (l = 1; :::; L). 2.6.1 Single-review-phase problem Consider a single review phase with R periods. The decision epochs = f1; 2; :::; R;R+ 1g, and there is no decision made in period R + 1. Let dt = demand in period t; St = inventory order-up-to level chosen in period t; and \u0011t = supplier's performance history | number of periods without a stockout up to the end of period t, 0 \u0014 \u0011t \u0014 t for 1 \u0014 t \u0014 R. Assume the supplier can always observe her performance history. For the dynamic programming problem we consider discrete demand, but, without loss of generality, we scale demand so that 1 unit is `small'. The purpose of this will be clear in Section 2.7 and is only to make the exposition clearer. The dynamic program for L = 0 is a straightforward derivation from L = 1, so below we will only give the L = 1 case. By Proposition 2.1 and letting L = 1 and W = 0, the distribution of \u0011R is approximately normal with mean RF2(S) and standard deviation \u001bR, where \u001b2R = RF2(S)\u0000 (3R\u0000 2)(F2(S))2 + 2(R\u0000 1) SX d=0 (F1(S \u0000 d))2f1(d): Let bIt = the supplier's inventory position (inventory plus the order made in the last period) at the beginning of period t, before an order is placed in period t; and It = maxf\u00001; bItg. So the base-stock level chosen in period t, St \u0015 It. Let \u00191t (Stji; It) denote the supplier's cost to go from period t in a single review phase given performance history i, It and St; and \u0019 1 t (i; It) = minSt\u0015It \u0019 1 t (Stji; It) denote the supplier's optimal cost to go from period t given i and It. Because the supplier does not incur backorder cost, for any negative net inventory in a period the supplier has the same immediate cost (zero inventory holding cost) and performance outcome (stockout) in that period. Thus in the state space, we can 27 2.6. Supplier's Strategic Behavior use a single state \u00001 to represent all the states of negative inventory. The state space is f(\u0011t\u00001; It) : 0 \u0014 \u0011t\u00001 \u0014 t\u00001;\u00001 \u0014 It \u0014 Sg, where 1 \u0014 t \u0014 R+1 and S is a large number such that F2(S) \u0019 1. Actions (inventory order-up-to level in a period): S 2 f0; 1; ::; Sg State transition: It+1 = maxf\u00001; St \u0000 dtg; \u0011t = ( \u0011t\u00001 + 1 if It \u0015 dt \u0011t\u00001 if It < dt : Let \u0011t\u00001 = i for 1 \u0014 t \u0014 R+1. Note that \u00110 = 0. All the expectation calculations `E' below are on dt. Rewards: rt((i; It); St) = hE[It \u0000 dt]+ 1 \u0014 t \u0014 R, lump-sum penalty: rR+1((i; IR+1)) = ( K if i \u0014 R 0 if i > R , linear penalty: rR+1((i; IR+1)) is similarly de\fned by replacing K by K(R\u000b + 1\u0000 i). (2.7) The supplier's base-stock level decision made in period R of a review phase cannot a\u000bect her performance in that review phase, but it determines the inventory holding cost and performance outcome in period 1 of the next review phase. So the optimal base-stock level SR in period R is SR = argminS\u0015IR E\u00191(0;maxf\u00001; S \u0000 dRg), where E\u00191(0;maxf\u00001; S\u0000dRg) = SX d=0 PrfD(1) = dg\u00191(0; S\u0000d)+F 1(S)\u00191(0;\u00001) is the supplier's expected cost to go from period 1 of the next review phase. We conjecture that if possible, the supplier will choose the \frst-best base-stock level S\u0003 in period R. So SR = maxfIR; S\u0003g. Transition probabilities: pt((j; I)j(i; u); S) = 8>>>><>>>>: PrfD(1) = S \u0000 Ig j = i+ 1; S \u0015 u; 0 \u0014 S \u0000 I \u0014 u; I \u0015 0 PrfD(1) = S \u0000 Ig j = i; S \u0015 u; S \u0000 I > u; I \u0015 0 PrfD(1) > Sg j = i; S \u0015 u; I = \u00001 0 otherwise : 28 2.6. Supplier's Strategic Behavior The dynamic program is \u00191R+1(i; IR+1) = rR+1((i; IR+1)); for 1 \u0014 t \u0014 R : \u00191t (i; It) = hE[It \u0000 dt]+ + min St\u0015It [Ef\u00191t+1(i+ 1; St \u0000 dt)jdt \u0014 Itg +Ef\u00191t+1(i;maxf\u00001; St \u0000 dtg)jdt > Itg]: From the dynamic program, we can obtain \u001911(0; I1) for various opening inventory level I1. The optimal base-stock policy obtained for this single review phase is not optimal in general for the in\fnite horizon problem because the terminal reward rR+1 ignores the inventory holding cost in the early periods of the next review phase resulting from the supplier's base-stock choice in period R of the current review phase. 2.6.2 In\fnite-horizon problem Instead of treating the in\fnite horizon as consisting of many time periods, we can think of each review phase as one time period in the in\fnite horizon, i.e., the in\fnite horizon consists of many review phases. So all of the supplier's base-stock policies in the nth \fnite review phase can be denoted by a vector \u0000!\r n = (\rn1 ; \rn2 ; :::; \rnR) 2 \u0000, where \rnt is the supplier's base-stock policy in period t of the n th review phase, and \u0000 is the set of possible base-stock policies in a review phase. Although the supplier's base-stock policy within a review phase depends on her performance history in that review phase and so is history dependent, \u0000!\r n and \u001911(0; I1) only depend on I1, the opening inventory of a review phase. Let the opening inventory of a review phase be the system states. Thus the system states and \u0000 do not vary with time, and the state transitions (from the opening inventory of a review phase to that of the subsequent phase) as well as the supplier's rewards (expected total cost in a review phase) are Markovian. So the supplier's problem in an in\fnite horizon consisting of review phases is a Markov decision process. Assume the supplier only uses deterministic base-stock inventory policies. Due to demand uncertainty, this MDP is unichain. To \fnd out the supplier's optimal average cost in an in\fnite horizon, we use value iteration. Let \u00191(I) denote the supplier's expected total cost in a single review phase with an opening inventory I as derived above. Let \u0019n(I) denote the supplier's expected total cost in n review phases with an opening inventory I for the \frst review 29 2.7. Numerical Analysis phase. With n review phases, the review phases are indexed reversely from 1 to n, i.e., the last review phase is indexed by 1 and the \frst by n. The algorithm is as follows. 1. Let \u00191(I) = \u001911(0; I), \" = 0:01, and set n = 1. 2. For each I 2 f\u00001; 0; ::; Sg, compute \u0019n+1(I) by applying the dynamic program as de\fned above for a single-review-phase problem but with the terminal reward rR+1((i; IR+1)) added by \u0019 n(IR+1). 3. If max I f\u0019n+1(I) \u0000 \u0019n(I)g \u0000 min I f\u0019n+1(I) \u0000 \u0019n(I)g < \", then go to step 4. Otherwise, increment n by 1 and return to step 2. 4. Let V D = 1 R max I f\u0019n+1(I) \u0000 \u0019n(I)g. Then V D is an approximation to the supplier's optimal long-run average cost. 2.7 Numerical Analysis We numerically investigate the supplier's incentive for strategic\/dynamic behavior using either the lump-sum or linear penalty SLA. We compare the supplier's optimal average cost when using a static base-stock S\u0003 policy with that under a dynamic policy. The numerical results demonstrate that the strategic supplier's gain under a lump-sum penalty SLA can be large, but that under an optimal linear-penalty SLA is minimal. It is also shown that a longer inventory replenishment lead time reduces such gain under both regimes. Consider two distributions: a Poisson demand with arrival rate \u0015 and a Normal(\u0015; \u001b) demand per period.3 With a normal demand, we can study the impact of various parameters on the supplier's gain from a dynamic inventory policy while keeping the performance target \fxed. For the dynamic policy we discretize the normal demand. PrfD(1) = dg = \b(d\u0000\u0015 \u001b ) \u0000 \b(d\u00001\u0000\u0015 \u001b ) for d > 0, and PrfD(1) = 0g = \b(\u0000\u0015 \u001b ). Now it can be seen why we scaled demand so that 1 unit was `small'. Leadtime L 2 f0; 1g. The performance threshold \u000b is chosen such that R\u000b is an integer. The range for the optimal penalty, [K\u0003(\u000b); K \u0003 (\u000b)], is obtained using the formulas in Section 2.5.1. 3The problem of negative demands with the normal distribution is not signi\fcant in our examples, as is ignored below. Only nonnegative demands are considered in the numerical analysis by using truncated normal distribution. 30 2.7. Numerical Analysis 2.7.1 Lump-sum penalty SLA Under a lump-sum penalty SLA, if the supplier's ready rate during a review phase does not exceed the performance threshold \u000b, then the supplier will pay the buyer a lump-sum penalty K. We show that in this case the supplier can bene\ft signi\fcantly from a dynamic policy. \u000f Poisson demand The following parameter values are used: h = 1, R = 30. With \u0015 = 8; 9; 10, for L = 0, we have S\u0003 = 12; 13; 14; and for L = 1, S\u0003 = 21; 24; 26. First consider the static base-stock policy. Consider L = 0. For \u0015 = 10 and R\u000b = 24 (i.e., \u000b = 80%), [K\u0003(\u000b); K \u0003 (\u000b)] = [146; 853]. It can be checked that S\u0003 = 14 is the global optimum for the static inventory policy for any K in [146; 853]. For K = 146, 200 and 300, the supplier's cost savings from using a dynamic policy are 20:3%, 20:9% and 22:4%, respectively. So even for the same threshold \u000b, the supplier's percentage cost saving from a dynamic policy varies with the penalty and is increasing in K. Although by de\fnition all (R\u000b;K) 2 \u0002 give S\u0003, this might not always be a global optimum. For example, with R\u000b = 25 and using the smallest K in the interval [92; 318], S\u0003 is not the global optimum. Let k\u0003(\u000b) = minfKjK 2 [K\u0003(\u000b); K\u0003(\u000b)] and S\u0003 is the supplier's global optimumg if it exists, then any K 2 [k\u0003(\u000b); K\u0003(\u000b)] can in- duce the supplier to choose S\u0003. If the interval is empty we say thatK(\u000b) = `Infeasible'. De\fne \u0002\u0003 = f(R\u000b;K)j0 < \u000b \u0014 FL+1\u0000W (S\u0003), R\u000b is integer, K 2 [k\u0003(\u000b); K\u0003(\u000b)] s.t. the interval is not emptyg. So \u0002\u0003 \u0012 \u0002 is the set of (R\u000b;K) values under which S\u0003 is the supplier's global optimum for a static policy. Now consider the dynamic base-stock policy. Table 2.1 shows the supplier's per- centage cost saving from using a dynamic policy under various R\u000b and k\u0003(\u000b) for L = 0 and 1. Note that for R\u000b = 27 (i.e. \u000b = 90%) the result is `Infeasible'. The results show that the supplier can bene\ft signi\fcantly from a dynamic policy. The smaller the value \u000b, the greater motivation for dynamic behavior, so the buyer should choose the performance threshold \u000b as close to the performance target as possible. Regardless of such choices however, under a lump-sum penalty SLA, the supplier has a considerable incentive to adopt a dynamic inventory policy and increase costs for the buyer. 31 2.7. Numerical Analysis Table 2.1: Supplier's cost saving (lump-sum penalty,Poisson demand) \u0015 R\u000b 8 9 10 24 26:14% 22:95% 20:28% L = 0 25 18:41% 16:27% 15:21% 26 13:62% 14:69% Infeasible 24 16:70% 20:51% 18:50% L = 1 25 14:24% 15:21% 14:41% 26 Infeasible Infeasible Infeasible Table 2.2: Supplier's cost saving (lump-sum penalty,normal demand) \u001b R\u000b 5 10 15 20 L = 0 24 20:09% 20:79% 21:09% 21:19% 25 15:20% 15:21% 15:23% 15:25% L = 1 24 16:68% 18:13% 18:50% 18:51% 25 15:18% 15:83% Infeasible Infeasible \u000f Normal demand We use normal demand to better study the impact of lead time on the supplier's cost saving from using a dynamic policy. The following parameter values are used: h = 1, R = 30, \u0015 = 50. With \u001b = 5; 10; 15; 20, for L = 0, we have S\u0003 = 57; 64; 71; 78; and for L = 1, S\u0003 = 110; 120; 130; 140. In all scenarios the target ready rate is the same, about 92%. For L = 0 and 1, and R\u000b = 26 and 27, S = 0 is the global optimum and the \frst-best S\u0003 is only a local optimum for all values of K in their intervals, that is k\u0003(\u000b) does not exist. For L = 1, R\u000b = 24 and 25, S\u0003 is the global optimum in some of the cases, thus k\u0003(\u000b) is within the interval. Table 2.2 shows the supplier's percentage cost saving from using a dynamic inventory policy. They are all very large. Comparing the results for L = 0 and 1, we can see that given the same performance threshold, the supplier's percentage cost savings from using a dynamic policy decreases as the lead time increases from 0 to 1. 2.7.2 Linear-penalty SLA We have shown that a lump-sum penalty SLA will greatly induce supplier's strate- gic behavior. Now we investigate the ability of a linear-penalty SLA to discourage 32 2.7. Numerical Analysis Table 2.3: Supplier's cost saving (linear penalty,Poisson demand) L = 0 L = 1 Target=91:7% Target=95:1% Target=92:2% Target=94:8% Deviation %saving Deviation %saving Deviation %saving Deviation %saving 1:7% 2:04% 1:8% 1:38% 2:2% 2:63% 1:4% 1:16% 5:0% 6:32% 5:1% 6:62% 5:5% 6:64% 4:8% 5:22% 8:3% 12:66% 8:5% 17:77% 8:9% 12:07% 8:1% 10:94% 11:7% 20:09% 11:8% 23:11% 12:2% 18:32% 11:4% 17:77% supplier's strategic behavior. We also investigate the e\u000bects of lead time L, demand variability \u001b \u0015 , and the length of a review phase R, on the supplier's cost saving from using a dynamic policy under a linear-penalty SLA. For all the scenarios in the numer- ical examples below, the supplier's cost functions have been checked to be unimodal using plots under the optimal penalty K given \u000b. \u000f Poisson demand We use h = 1 and R = 30. We compare the supplier's optimal average cost under a static base-stock S\u0003 policy with that under a dynamic base-stock policy, using a penalty K = k\u0003(\u000b). We note that for all the numerical examples here we have found that k\u0003(\u000b) = K\u0003(\u000b). To compare the supplier's cost saving under various performance thresholds \u000b, we use a demand rate \u0015 = 10. For L = 0, S\u0003 = 14 and 15 with the target service levels = 91:7% and 95:1%, respectively; and for L = 1, S\u0003 = 26 and 27 with the target service levels = 92:2% and 94:8%, respectively. Table 2.3 show the results for L = 0 and 1. The threshold \u000b is represented by the allowable deviation from the target, which is equal to the target ready rate \u0000 \u000b. Similar conclusion can be drawn here as that under a lump-sum penalty SLA. The greater the allowable deviation from the target (thus the smaller \u000b), the greater the supplier's percentage cost saving. Therefore, to reduce the supplier's cost saving from using a dynamic policy, the performance threshold should be chosen to be as close to the performance target as possible. We also investigate the supplier's cost saving from using a dynamic policy for di\u000berent demand rates, using the smallest possible allowable deviation from the target (the largest possible \u000b). For \u0015 = 7; 8; 9; 10, the corresponding parameter values for L = 0 are S\u0003 = 11; 12; 13; 14; and those for L = 1 are S\u0003 = 19; 21; 25; 26. Table 2.4 shows the results. \u000b is again represented by the allowable deviation from the 33 2.7. Numerical Analysis Table 2.4: Supplier's cost saving (linear penalty,Poisson demand) L = 0 L = 1 \u0015 Target Deviation %saving Target Deviation %saving 7 94:7% 1:3% 1:10% 92:3% 2:3% 2:62% 8 93:6% 3:6% 3:87% 91:1% 1:1% 1:89% 9 92:6% 2:6% 2:80% 95:5% 2:2% 1:50% 10 91:7% 1:7% 2:04% 92:2% 2:2% 2:63% Table 2.5: Supplier's cost saving (linear penalty,normal demand) R\u000b L = 0 L = 1 24 20:85% 18:39% 25 13:90% 12:58% 26 7:40% 6:57% 27 2:64% 2:63% target. The supplier's percentage cost saving is small in all the cases. So under a linear-penalty SLA with a performance threshold close to the target, the supplier will have little incentive to adopt a dynamic inventory policy. A linear-penalty SLA can greatly mitigate the supplier's strategic behavior. \u000f Normal demand The following parameter values are used: h = 1, R = 30, \u0015 = 50, \u001b 2 f5; 10; 15; 20g, L 2 f0; 1g. Note that we use the same demand process (single-period demand distri- bution) for both L = 0 and 1, so we can investigate the supplier's cost saving from a dynamic inventory policy as the lead time increases from 0 to 1. We \frst compare the supplier's cost saving under various \u000b, using \u0015 = 50 and \u001b = 15 for the single-period demand. For L = 0, S\u0003 = 71 with the target service level = 91:9%; and for L = 1, S\u0003 = 130 with the target service level = 92:1%. The results are shown in Table 2.5. The supplier's percentage cost saving decreases with \u000b, indicating that a tight allowable performance deviation from the target is required to limit the supplier's gain from strategic behavior. The results also indicate that given the same performance threshold, a longer leadtime will reduce the supplier's cost saving from strategic behavior. Next, we investigate the e\u000bects of demand variability and leadtime on the sup- plier's cost saving from using a dynamic policy. We \fx \u0015 = 50 while changing \u001b, so the coe\u000ecient of variation of demand \u001b \u0015 2 f0:1; 0:2; 0:3; 0:4g; we use the smallest 34 2.7. Numerical Analysis Table 2.6: Supplier's cost saving (linear penalty,normal demand) Target \u0019 92% Target \u0019 96:5% \u001b=\u0015 L = 0 L = 1 L = 0 L = 1 0:1 2:41% 2:06% 2:72% 2:17% 0:2 2:51% 2:07% 2:81% 2:44% 0:3 2:64% 2:63% 2:90% 2:45% 0:4 2:68% 2:67% 2:95% 2:45% possible deviation from the target (the largest \u000b < target and R\u000b is integer). For \u001b 2 f5; 10; 15; 20g, S\u0003 is chosen so that the target service level is equal for L = 0; 1 and di\u000berent \u001b. We consider two target service levels: one is about 92% and the other about 96:5%. Table 2.6 lists the numerical results. Each column shows that the sup- plier's gain from strategic behavior increases with the demand variability. Intuitively, when demand is less variable, the optimal safety stock under a static inventory pol- icy is small, and the optimal safety stock under a dynamic inventory policy will not deviate much from the static one, so the supplier's gain from a dynamic policy will be relatively small. Comparing the two columns for each target service level, we can see that the supplier's cost saving from a dynamic policy decreases as the leadtime increases from 0 to 1. So a longer leadtime will likely mitigate the supplier's incentive for strategic behavior. We will discuss the insights from this result in Section 2.8. \u000f E\u000bect of length of a review phase R We also evaluate the impact of the length of a review phase on the supplier's percentage cost saving from a dynamic policy. We use \u0015 = 50 and \u001b = 15 for single- period demand, R 2 f30; 60g, L 2 f0; 1g, S\u0003 is chosen so that the target service levels for L = 0 and 1 are both about 92%, and the same performance threshold \u000b = 90% (thus the same allowable deviation from the target). Table 2.7 compares the supplier's cost saving under di\u000berent values of R with the other parameters \fxed. It indicates that as the review phase becomes longer, if the performance threshold remains unchanged, then the supplier will bene\ft more from a dynamic inventory policy. This can be explained by Proposition 2.2, which says that the variance of the supplier's review phase ready rate decreases with R. So if \u000b is \fxed, then as R increases, the absolute allowable performance deviation from the target remains constant, but the relative allowable deviation increases, giving the supplier more \rexibility to dynamically adjust the base-stock level. This implies that as R increases, 35 2.8. Discussion Table 2.7: E\u000bect of R (linear penalty,normal demand) L = 0 L = 1 R = 30 2:64% 2:63% R = 60 4:12% 4:08% there should be less allowable performance deviation from the target (thus bigger \u000b) in order to discourage the supplier from strategic behavior. We have assumed that R is exogenous because in practice R can be determined by other factors such as transaction costs and the accounting policy. 2.8 Discussion The numerical results have shown that under a linear-penalty SLA with the per- formance threshold close to the target ready rate, the supplier's gain from using a dynamic inventory policy instead of a static one is small. In practice, implementing a dynamic inventory policy is more complicated than implementing a static one. To implement a dynamic inventory policy, the supplier has to determine the ordering quantity based on not only the inventory on hand and the performance history in the current review phase, but also every order placed in the past yet not arrived. So the implementation cost of a dynamic policy is higher than that of a static one. In this chapter, we do not explicitly model the supplier's costs of implementing an inventory policy. When the complexity and cost of implementing a policy is taken into account, a dynamic inventory policy will bring less bene\ft. So under a linear-penalty SLA with carefully chosen contract parameters, the supplier will have little incentive for strategic behavior. 2.8.1 SLA vs. traditional coordination contract To provide an incentive to the supplier for inventory investment, the buyer can use ei- ther a traditional supply chain coordination contract or an SLA. Traditional contracts coordinate the supply chain via a holding cost and backorder cost transfer payment. Cachon (2003) provides an analysis on this type of contract in a single-location base- stock model with linear backorder cost. As a special case of this contract, the supplier pays a penalty on each individual delayed delivery. To micro-manage the supplier in 36 2.8. Discussion this way, the buyer will have to make a detailed record of every single demand. When demands occur frequently, the buyer will incur large administrative and transaction costs. With an SLA, the buyer only needs to measure the supplier's aggregate per- formance over a period of time. So an SLA is preferable over a traditional contract for frequent demands. In most inventory theory, the backorder cost is generally linear in the amount of delay. But in practice, the backorder cost is often nonlinear; a short delay may not matter much, but a long delay is very costly. In this case, if a traditional coordination contract is used, then the contract will likely be quite complicated in order to correctly align the supplier's incentive with the supply chain. However, an SLA has a simple form and is easy to implement even if the backorder cost has a complex form, as in our model, the backorder cost is convex in the length of delay. Service levels for inventory performance are widely used in practice. The most frequently cited reason is that backorder costs are hard to measure because a stockout may a\u000bect not only a \frm but also external parties (e.g., customers), so \frms prefer to set a desired performance target. This makes traditional coordination contracts hard to design. But with an SLA, the performance target can be set exactly as desired. 2.8.2 Lump-sum penalty SLA vs. linear-penalty SLA A lump-sum penalty SLA is a highly discontinuous incentive scheme used with a multi-period review strategy. For a single period problem with a risk-neutral buyer and supplier, any optimal combination of the performance threshold and penalty can induce the supplier to choose the \frst-best e\u000bort level, and thus a lump-sum penalty SLA can be optimal. However, the multi-period review strategy brings in additional issues for the design of discontinuous incentive schemes. In our model, the supplier can observe her performance history throughout the review phase and can adjust her e\u000bort level at any time to a\u000bect her performance outcome at the end of the review phase. In this case, as demonstrated by the numerical results in Section 2.7, a discontinuous incentive scheme with a lump-sum penalty will cause supplier's strategic behavior. If either condition is violated, i.e., the supplier cannot observe her performance history or cannot adjust her e\u000bort level dynamically, then a lump-sum penalty SLA will be less vulnerable to supplier's strategic behavior. From the formulation of the supplier's dynamic program in Section 2.6.1, we can 37 2.8. Discussion see that under a lump-sum penalty SLA, after observing the performance history in any period of a review phase, the supplier can take the following strategy. If her past performance has already exceeded the threshold (call it an `early pass'), she will not put in any e\u000bort during the remainder of the review phase. If her past performance has been poor enough so that she will have no chance to attain the threshold in the current review phase (call it an `early failure'), then she will also not put any e\u000bort into the remaining periods of the review phase. In other situations, she will choose e\u000bort levels depending on the past performance and the number of periods left in that review phase. In practice, a service level target such as the ready rate for inventory performance is generally very high (above 90%), so the chance for an early pass is very small. On the other hand, the chance for an early failure is large. Under a lump-sum penalty SLA, if an early failure occurs, the supplier's penalty is \fxed no matter how poor her performance is. Thus a lump-sum penalty SLA will not mitigate the supplier's strategic behavior after an early failure. A linear-penalty SLA makes the supplier's penalty for poor performance linear in the amount of deviation from the threshold, and so can better mitigate the supplier's strategic behavior after an early failure. In the case of a positive inventory replenishment lead time, orders placed in a period will arrive L periods later, and the performance as a result of any e\u000bort will be revealed after a time lag of L periods. Because of the information delay, the supplier cannot respond to the performance history as e\u000bectively as in the case of no delay. Moreover, when the number of periods left in a review phase is less than L, the supplier cannot adjust the base-stock level to a\u000bect her performance at the end of the current review phase. So we can anticipate that the supplier will bene\ft less from her strategic behavior in the case of a positive lead time. This is supported by the numerical results in Section 2.7 comparing the supplier's percentage cost savings from a dynamic policy for L = 0 and L = 1. In practice, the target ready rate or \fll rate is usually high. Due to this and the positive lead time, unless a review phase is extremely long and\/or \u000b is small, there is little chance that the supplier will know she can meet the performance threshold with probability 1 before a review phase ends. At the low performance side, the linear-penalty scheme will provide the supplier with an incentive to prevent her performance from getting worse once it falls below \u000b. The linear-penalty SLA we study here does not provide incentives concerning performance beyond a threshold. So to induce a desired target service level, the 38 2.8. Discussion Table 2.8: Expected penalty vs. supplier's total cost \u0015 \u001b L Lump-sum Linear 10 3 0 8:9% 12:9% 10 4 0 9:3% 13:1% 10 3 1 5:1% 13:8% 10 4 1 5:1% 14:3% Table 2.9: Actual penalty vs. supplier's total cost L Lump-sum Linear 0 Prob 9:03% 22:01% 13:05% 5:96% 2:18% 0:66% 0:17% ratio 99:3% 16:3% 32:5% 48:7% 65:0% 81:2% 97:4% 1 Prob 3:28% 21:79% 14:73% 7:06% 2:40% 0:58% 0:10% ratio 156:8% 16:1% 32:2% 48:4% 64:5% 80:6% 96:7% allowable deviation for the supplier performance from the target should be small, i.e., \u000b is not too far from the target. This has been shown by the numerical results on the supplier's cost savings under di\u000berent values of \u000b. A linear-penalty SLA imposes a small penalty on the bad outcomes more likely due to random variability in the performance than to wrong e\u000bort levels. This can be seen by comparing the size of the supplier's penalty with her revenue under both types of penalty schemes in two ways. Note that the supplier's revenue is at least her total cost, including the inventory holding cost and expected penalty. So we use the supplier's total cost as a proxy for her revenue. The \frst way is to compare the supplier's expected penalty with her total cost, and the ratios are shown in Table 2.8 using normal demand distributions. The second way is to compare the supplier's actual penalty with her total cost, as shown in Table 2.9 using a Normal(10; 3), in which for a lump-sum penalty, the probability of incurring a penalty and the penalty to total cost ratio are provided; and for a linear penalty, both the probability of incurring a penalty level and the actual penalty to total cost ratio are provided. The results show that although the expected penalty under a lump-sum penalty SLA is small, the actual penalty could exceed the supplier's revenue, making the supplier earn nothing. On the contrary, the supplier is unlikely to pay a large penalty under a linear-penalty SLA. Therefore, a linear-penalty SLA is a mild penalty scheme compared with a lump-sum penalty one and thus induces a more stable inventory investment. 39 2.8. Discussion 2.8.3 Extensions When studying supplier's strategic behavior under an SLA, we have mainly focused on the immediate ready rate as a performance measure for inventory and considered a periodic-review base-stock policy. The main \fndings also extend to time-window ready rate and \fll rate as well as a continuous-review inventory policy. When a time- window ready rate is used with W \u0014 L, the performance indicator for period t , XWt = 1fD[t \u0000 L; t + 1 \u0000W ) \u0014 St\u0000Lg, and the supplier decides a base-stock level in each period to \fll demands in the subsequent L+ 1\u0000W periods instead of L+ 1 periods. So a time window can reduce the supplier's inventory risk and inventory cost. Let L0 = L\u0000W . Because an order placed in a period will still arrive L periods later, the supplier's decision problem is like the one with lead time L0 but the performance resulting from a base-stock level decision is realized after L0 + W periods, so the supplier cannot adjust the base-stock level under a dynamic policy as timely as in the case of lead time equal to L0. Therefore, the supplier's gain from using a dynamic inventory policy given lead time L under a time-window-W ready rate will not exceed that under an immediate ready rate given lead time L\u0000W . When the \fll rate is used as the performance measure, supplier's incentive for strategic behavior still exists, and the supplier's objective is to dynamically adjust the base-stock level to a\u000bect the proportion of demands \flled on time in a review phase instead of the proportion of periods with all demands \flled on time. This will only slightly change the calculation of the supplier's expected penalty, but the issues and insights from the ready rate as performance measure all hold here. Now suppose the supplier uses a continuous-review base-stock policy (L > 1). Because the supplier's performance is revealed at the end of each period, assume the supplier chooses the base-stock level once every period. The only di\u000berence between the two inventory policies is that the supplier's inventory cost is relatively lower under a continuous-review policy. The supplier's ability to dynamically adjust the base-stock level and the e\u000bect of the information lag due to positive lead time are similar under both policies. So the insights from a periodic-review policy also hold for a continuous-review policy. When the ready rate is used as inventory performance measure, the supplier may have other strategic behavior in addition to adopting a dynamic inventory policy. For example, if the supplier cannot \fll all demands in a particular period so that the performance in that period will be bad no matter what proportion of the demands are 40 2.9. Conclusions \flled, then the supplier may hold inventory instead of \flling partial demands in order to save the transportation cost in that period. This will increase the supplier's inven- tory holding cost. In our model, the supplier manages inventory near the buyer's site, so the transportation cost is negligible. In other situations where the transportation cost is large relative to the holding cost, then the \fll rate may be preferred because it measures the proportion of demands \flled. 2.9 Conclusions In this chapter, we have provided a methodology for studying service level agree- ments by applying the principal-agent theory to the design and choice of contract parameters of SLAs. Using a single-location uncapacitated inventory management problem, we have identi\fed issues in the design of SLAs for inventory management, where the ready rate is the performance measure. In the case of a positive inventory replenishment lead time, the supplier's performance in each period can be correlated. The ready rate in a \fnite review phase is a random variable; we have shown that its distribution is approximately normal for a long review phase. We have studied two types of SLAs: a lump-sum penalty SLA and a linear-penalty one. Due to their multi- period review structure, SLAs with a target service level provide the supplier with an incentive for strategic dynamic behavior. Speci\fcally, we have found that under a lump-sum penalty SLA, the supplier will have a signi\fcant incentive for strategic behavior. On the other hand, a simple linear-penalty SLA can greatly mitigate sup- plier's strategic behavior. This has implication for the design of SLAs in general: when the supplier can observe the performance history and dynamically adjust her e\u000bort level to a\u000bect her review phase performance, to mitigate the supplier's incentive for strategic behavior, the penalty should be dependent on the amount of supplier's performance deviation from the target. To e\u000bectively mitigate the supplier's incentive for strategic behavior using a linear-penalty SLA, the allowable deviation of the per- formance from the target service level should be small. For the application of SLAs to inventory management in particular, a positive inventory replenishment lead time and a high ful\fllment rate target can further mitigate such strategic behavior. 41 Chapter 3 Managing Supplier's Delivery Performance With Service Level Agreements 3.1 Introduction With the increased outsourcing of manufacturing and services to suppliers comes a need for better contractual agreements between suppliers and buyers. One of the most widely employed contractual instruments is a type of performance-based contracts called Service Level Agreements (SLAs). A survey by Oblicore Inc. in 2007 revealed that 91% of organizations use SLAs for managing suppliers, internal agreements, or external customer agreements. According to the O\u000ece of Federal Procurement Policy at the O\u000ece of Management and Budget, the US Federal Government expects agencies to make half their service contracts performance-based acquisitions in \fscal 2008, an increase from the goal of 45% for 2007. SLAs are typically employed when the buyer neither wants or is not able to micromanage the supplier and has no interest in how the product or service is delivered; but is interested only in the outcome. SLAs are often used when the parties involved have a long-term relationship, where the transactions are not one time. Since a \fxed price alone is not enough to guaran- tee the delivery of the required performance, positive and\/or negative performance incentives are needed. For example, a penalty might be imposed when the supplier underperforms compared to some target service level. The penalty is not based on daily transactions but performance over a period of time; reducing the administrative costs of enforcement and freeing the buyer to concentrate on their core business. Despite the widespread use of SLAs in practice, there is little theoretical research on their design. In Chapter 2 studying an application to inventory management, we identi\fed \fve fundamental issues of SLA design that need to be answered: what 42 3.1. Introduction performance measure should be used, what performance target is appropriate, how frequently the performance should be reviewed, how much deviation of the perfor- mance is allowed from the target, and what penalty the supplier should pay when the performance exceeds the allowable deviation. The performance measure should align the supplier's incentive with that of the buying \frm. The allowable deviation and the penalty together determine how strong the incentive is for the supplier. We mainly addressed within the framework of a simple inventory management problem where the supplier's decision variable is the base-stock level alone. In this chapter, we concentrate on the \frst two questions and brie\ry address the other three, within a more complex problem. The natural framework for the study of SLAs is from a principal-agent perspective, and that is the perspective we take. Speci\fcally, we study a single-item inventory system with a continuous-review base-stock policy, stochastic and stationary demand, and full backlogging. We con- sider a supply chain consisting of a single supplier and a single buyer, where the supplier can invest both in inventory and in inventory replenishment lead time to meet a service level target, and both investments are unobservable to the buyer. The supplier owns the inventory and incurs a linear inventory holding cost. For each de- layed delivery, the buyer incurs a cost which is a convex and increasing function of the amount of delay. An SLA uses a multi-period review strategy, under which the supplier's inventory performance is reviewed every R periods (called a review phase), and if it is below a pre-speci\fed performance threshold, then the supplier will pay a penalty linear in the amount of performance deviation from the threshold. Fill rate and stockout rate are commonly used in both the practice and the litera- ture for measuring delivery and inventory performance. Most inventory management literature studies performance measures in the long run using expected performance. But the performance measure in a \fnite review phase is a random variable. When the supplier's actions are unobservable, it is important to know the distribution of the performance measure in order to provide an incentive to the supplier. It is very di\u000ecult to derive the distribution of \fll rate. Interested readers can refer to Thomas (2005) for its distribution obtained using simulation in a static periodic-review base- stock model with zero lead time. For the same reason as in Chapter 2, we focus mainly on the ready rate, which is the long-run fraction of time that demands are \flled immediately from the stock. It measures inventory availability, and is equal to 1\u0000 stockout rate. The conventional ready rate and \fll rate are measures of the 43 3.1. Introduction o\u000b-the-shelf or immediate order ful\fllment performance. In practice, time-window ful\fllment rates are more commonly used than o\u000b-the-shelf performance measures (LaLonde and Zinszer 1976, LaLonde et al. 1988). In Quick Response and other forms of time-based competition, the performance measure for customer service is often the ability to meet delivery promises, where the promised time window is usu- ally small. For example, around 1995, Hewlett-Packard aimed at a 93% ful\fllment rate within 3 days, and IBM PC and Compaq 95% within 5 days (Hausman et al. 1998). When the performance measure is based on on-time delivery by a supplier to a buyer, time-window ful\fllment rates are also used. Therefore, we also study another form of ready rate, the ready rate with a window, which is the long-run fraction of time that demands are \flled within a pre-speci\fed time window. We study the ready rate for simplicity in exposition, but similar insights can be obtained when either the immediate or time-window \fll rate is used as the performance measure. Because the supplier is responsible for investments in both inventory and lead time and is evaluated by a single performance measure | the ready rate, our problem is a multi-task agency problem with a single output. Since both increasing the stock level and reducing the replenishment lead time can achieve a better performance, the supplier's two tasks are substitutes. The objective of this chapter is twofold. First, we address the design of SLAs in supply management, including choosing the performance measure, determining the performance target, allowable deviation and penalties for underperformance. Specif- ically, we examine two types of SLAs using either the immediate or time-window ready rates as performance measures. Second, we compare these two forms of SLAs in terms of the average supply chain cost. We show that when the supplier employs a static inventory policy, can invest both in inventory level and in supply lead time, with the investments unobservable to the buyer, an SLA using the time-window ready rate can induce the supplier to make the investments compatible with overall supply chain optimization. An SLA using only the immediate ready rate generally cannot induce this \frst-best investment. We also discuss the issue of using a single perfor- mance measure for aligning the supplier's incentive when the supplier has multiple ways to achieve inventory performance. The time window in the performance mea- sures plays three roles. It aligns the supplier's tradeo\u000b between inventory and lead time investments with that of the supply chain, allocates inventory risk between the buyer and a supplier, and to some extent transfers the buyer's delay cost structure 44 3.2. Literature Review to the supplier. The rest of this chapter is organized as follows. Section 3.2 reviews the literature. Section 3.3 describes the model and provides mathematical expressions for the waiting time distribution. Section 3.4 examines SLAs using the immediate or time-window ready rate as a performance measure and discusses the issue of incentive alignment using a single performance measure. Numerical analyses are in Section 3.5. We conclude in Section 3.6. 3.2 Literature Review This chapter relates to three primary literatures: agency theory, inventory manage- ment, and supply chain contracting and coordination. Multiple-period review strategies have been studied in the economics literature by Radner (1985) for a repeated principal-agent game. Ren et al. (2008) investigate a modi\fed strategy for an information-sharing game between a buyer and a supplier in a supply chain context. Both papers use trigger strategies as punishments for non- cooperation. Details of these approaches can be found in Chapter 2, which studies a multi-period review strategy in a service level agreement for inventory management. The review strategy di\u000bers from those in Radner (1985) and Ren et al. (2008) in two major aspects: the length of a review phase R is exogenous and the penalty is a monetary payment instead of a phase of noncooperative game. Moreover, the performance outcome of the agent (supplier) in each period can be correlated. The current chapter studies a multi-task moral hazard problem, whereas both Radner (1985) and Chapter 2 study single-task moral hazard problems and Ren et al. (2008) a hidden information one. The immediate ful\fllment rate is commonly employed in the inventory manage- ment literature, however in practice, time-window ful\fllment rates are more common. Boyaci and Gallego (2001) study the problem of minimizing average inventory costs subject to \fll-rate and \fll-rate-with-window service-level constraints in serial and as- sembly systems. In an (s; S) inventory system with service level target represented by a time-window ready rate, Wang et al. (2005) \fnd a signi\fcant tradeo\u000b between the window length and the inventory costs, and suggests that a longer ful\fllment window and lower price may be used for price-sensitive but time-insensitive customers. The above papers study inventory management from a single agent perspective, and do 45 3.2. Literature Review not deal with incentive issues in a decentralized system. Our concern is the use of either the immediate or time-window ready rate as a performance measure to induce investments by an independent supplier. The inventory management literature generally considers performance measures in the long run using expected performance. In practice, a supplier's delivery perfor- mance will be evaluated over a \fnite period of time. At the end of a \fnite review phase R, the buyer, unable to observe the supplier's e\u000bort, observes a single noisy performance signal. How is the buyer to unscramble poor e\u000bort from `bad' random e\u000bects? The buyer needs to know how the supplier's e\u000borts a\u000bect the random distri- bution of performance. Thomas (2008) uses simulation to investigate the distribution of \fll rate in a static periodic-review base-stock model with zero lead time and Er- lang demand. Chapter 2 provides a theoretical approximation for the distribution of the review-phase immediate\/time-window ready rate under a static periodic-review base-stock policy with general demand and discrete lead time. In this chapter, we employ a similar result under a continuous-review base-stock policy and continuous lead time. Choi et al. (2004) investigate choosing supplier performance measures in a vendor- managed-inventory context. The production of the supplier and the manufacturer are capacitated. The supplier holds inventory, and her capacity and inventory policy are private information. So the buyer chooses performance measures for the supplier. Choi et al. study both the ready rate and the \fll rate, and demonstrate that in a capacitated supply chain, the supplier's service level is in general not su\u000ecient to guarantee the manufacturer's target customer service level. They propose a menu of contracts with di\u000berent combinations of the ready rate and expected backorders. In our model, the buyer incurs a cost for each delayed delivery, and determines the service level target for the supplier. Moreover, we do not consider production by the buyer, and the supplier's supply is uncapacitated. Although the supplier's actions are unobservable in their model, Choi et al. only study immediate ful\fllment rates in the long run, and focus on the choice of performance measures, ignoring the variability in the observed performance measures and the penalty for failing to meet a target. We study both the immediate and time-window ready rates, and compare the e\u000eciency of each performance measure at aligning the supplier's incentive. In all the aforementioned studies, the target \fll rate or ready rate and the time window are assumed to be given. We allow both the time window and the target 46 3.2. Literature Review ready rate to be endogenous. Our study is also related to incentive contracting on inventory management. In this stream of literature, principal-agent theory is applied to inventory management in decentralized systems. Bolton and Dewatripont (2005) provide a broad coverage of literature on incentive contracts. Literature on moral hazard problems in agency contracting can be found therein. Corbett (2001) studies the allocation of decision rights between a single buyer and a single supplier in an order-quantity\/reorder-point (Q; r) inventory system with stochastic and stationary demand and backlogging. The delivery lead time between the two parties is constant. Consignment stock is studied, where the supplier holds inventory at the buyer's site and bears the holding cost until the goods are sold to the \fnal customer. Corbett considers two situations: one in which the buyer is the principal and the supplier has private information about her setup cost; another in which the supplier is the principal and the buyer has private information about backorder costs. In our model, the supplier carries inventory and incurs the holding costs, and there is information asymmetry on the supplier's base- stock level and lead time. Lutze and Ozer (2008) examine promised lead time contracts o\u000bered by a supplier to a buyer, under which the buyer places orders in advance and the supplier guarantees the shipment of full order on time after a promised lead time. Both the supplier and the buyer hold inventory. They investigate how a promised lead time contract can be used to share inventory risk between a buyer and a supplier. Our model demonstrates that both the service level agreement structure and the window in the inventory performance measure allow the two parties to share inventory risk. Lutze and Ozer study an adverse selection problem, where the buyer has private information about his shortage cost, but there is no uncertainty in the supplier's performance to meet the promised delivery lead time. We study a moral hazard problem, in which only the supplier holds inventory, the buyer o\u000bers the contract, and the supplier's performance is a random variable. Kim et al. (2007) study performance-based contracting between a single buyer and multiple suppliers in after-sales service supply chains. The buyer is a customer of assembled systems, where each system consists of some distinct parts. Each type of spare part is stocked by a di\u000berent supplier. If any of the parts fails, the system is down, and that part has to be replaced by a spare part. Failed parts are repaired and then returned to the spare part stock. When there is no spare part available, 47 3.3. Model and Preliminaries a backorder occurs. Each supplier determines her stock level of spare parts, which is unobservable to the customer. Facing a system uptime requirement, the customer o\u000bers contracts to the suppliers. Because the uptime requirement is equivalent to a system backorder target, the authors propose a contract linear in the backorders of each part, and show that it induces the \frst-best solutions when all parties are risk neutral. In our problem, the contract is based on the supplier's aggregate delivery performance | the ready rate, not on individual backorders. 3.3 Model and Preliminaries Consider a supply chain consisting of a single supplier (she) and a single buyer (he) with the supplier producing a single product for the buyer. The supplier makes to stock and the buyer makes to order. Without loss of generality, we assume that the supplier's unit ordering and processing costs are zero. The supplier holds inventory at a unit cost of h per period of time, and replenishes her inventory from an unlim- ited supply source at a constant lead time L using an order-up-to-S inventory policy. Assume the supplier can invest in the replenishment lead time with cost Cr(L) for lead time L. The lead time between the buyer and the supplier is taken as zero, rep- resenting a situation where the supplier holds inventory at a site near the buyer, such as a vendor-managed-inventory (VMI) program. Customer demands are stochastic and stationary. Demands in each period of time are independently and identically distributed (i.i.d.) with mean \u0015 and standard deviation \u001b. The buyer incurs a cost of waiting if a demand for the product cannot be \flled immediately. The buyer's processing time is negligible and is assumed to be zero. Both the buyer and the supplier are risk neutral. Assume the distribution of the demand and the supplier's inventory holding and lead time costs are common information. In order to induce the supplier to invest in inventory and lead time, the buyer contracts with the supplier on the supplier's inventory service level. The service level agreement uses a multi-period review strategy, under which the supplier's delivery performance is evaluated every R periods (a review phase). As the review period progresses the supplier has an incentive to dynamically (state-dependent) change her stock level S depending on her performance to date. In the earlier chapter, we investigated this issue and concluded that such strategic behavior was mitigated 48 3.3. Model and Preliminaries with the choice of penalties proportional to the deviation and that the supplier gains little from adopting them given a small allowable deviation from the target. Such dynamic inventory policies will also have higher implementation costs than a static one. Therefore, to allow us to focus on other aspects of SLA design we assume throughout this chapter that only such linear penalties are used and that the supplier uses a static inventory policy. The following notation is used throughout this chapter. Supplier's decision variables: S : base-stock level L : inventory replenishment lead time Buyer's decision variables: W : time window AW : supplier's expected ready rate (with window W ) p : unit transfer price \u000b : performance threshold for A K : penalty rate | penalty paid by the supplier to the buyer per 1% below Other: \u0015 : demand rate \u001b : standard deviation of demand per period R : length of a review phase, assumed to be large compared with likely lead times L A : supplier's realized ready rate (with window) in a review phase Cr(L) : cost of attaining lead time L for each unit of demand CD(y) : buyer's cost of delay per unit demand if the demand is \flled after y periods C(S; L) : average supply chain cost if S and L are chosen CB(S; L) : buyer's average cost if the supplier chooses S and L \u0019(S; L) : supplier's average pro\ft if she chooses S and L I(S; L) : average inventory level if S and L are chosen D(t) : demand in t periods D(t; u] : demand in the interval (t; u]: Because working with discrete-valued demand and decision variables S and L makes our analysis much more complex, and our purpose is to gain insights in in- centive contracting, we use continuous demand, S and L as an approximation in the analysis. We assume that the supplier uses a continuous-review inventory policy. 49 3.4. Optimal Ready-Rate Contract Let the cumulative distribution function (cdf) and probability density function (pdf) of D(t) be denoted by F (xjt) and f(xjt), respectively. F (xjt) = PrfD(t) \u0014 xg. The average inventory level given base-stock level S and lead time L is I(S; L) = E([S \u0000D(L)]+) = R x~~<>: 0 for y < 0 PrfD(L\u0000 y) \u0014 Sg for y 2 [0; L] 1 for y > L: (3.2) It follows from (3.2) that fw(yjS; L) = 8><>: PrfD(L) \u0014 Sg for y = 0 dFw(yjS;L) dy for y 2 (0; L) 0 othewise. Assume the delay cost function CD(\u0001) and lead time cost function Cr(\u0001) are con- tinuous and di\u000berentiable; C 0D(\u0001) > 0, C 00D(\u0001) \u0015 0, CD(0) = 0; C 0r(\u0001) < 0, C 00r (\u0001) > 0, limL!0Cr(L) =1, and limL!eLC 0r(L) = 0, where 0 < eL \u0014 1. Without loss of gener- ality, we assume that at time 0, the inventory is S, the base-stock level. We consider the situation where the supply chain optimal (\frst-best) base-stock level S\u0003 > 0 and lead time L\u0003 < eL, i.e., it is optimal for the supply chain to invest in both inventory and lead time. 3.4 Optimal Ready-Rate Contract A typical SLA will be of the following form, somewhat abbreviated for simplicity. The buyer will pay $25 for each part. The target is to ensure that parts are available 95% of the time. Every 90 days (one quarter), a review will determine the % of time that parts are available within 1 day, and if this \fgure falls beneath 93% a penalty of $200 per 1% below 93% will be deducted from the buyer's invoice. 50 3.4. Optimal Ready-Rate Contract The buyer makes a service level agreement with the supplier under a long-term relationship of the form (R;W;AW ; p; \u000b;K). The supplier's inventory performance is reviewed every R periods (90 days), which constitute a review phase. The per- formance measure is the ready rate A with window W (1 day), A 2 [0; 1]. The distribution of A will be discussed below. The target service level is AW (95%). A transfer price p ($25) is paid for each unit of demand. If the supplier's performance falls below \u000b (93%), then the supplier is charged with a linear penalty proportional to the di\u000berence between the actual performance and \u000b (K ($200) per 1% below \u000b). The buyer chooses the SLA to minimize his long-run average cost, including the payment to the supplier, the expected order delay cost, minus the penalty paid by the supplier. Given the SLA o\u000bered, the supplier chooses S and L to maximize her long-run average pro\ft. In practice, a reasonable performance threshold \u000b should be below the performance target AW . So to provide an incentive to the supplier, the candidate K and \u000b must be such that K > 0 and \u000b 2 (0; AW ). We assume R is exogenous because in practice R can be determined by other factors such as transaction costs and the accounting policy. Katok et al. (2008) use experimental methods to examine the e\u000bect of review periods in a \fnite-horizon periodic-review base-stock inventory model. The inventory replenishment lead time is zero. They \fnd that longer review periods may be more e\u000bective than shorter ones at inducing service improvements. In our model, the supplier's performance is reviewed every R periods repeatedly. The supplier makes investment to maximize her long-run average pro\ft over repeated review periods. The supplier chooses the lead time once, which is unchanged over time; there is no emergency expediting. 3.4.1 Performance measure The immediate ready rate and time-window ready rate are two common measures for inventory performance in practice. The immediate ready rate is the fraction of time that demands are \flled immediately; the time-window ready rate is the fraction of time that demands are \flled within a time window. When demand arrivals see time averages (e.g., Poisson process), the ready rate (with window) is equal to the \fll rate (with window), the fraction of demands that are \flled immediately (within a time window). We study the ready rate for ease of exposition, but the major \fndings still 51 3.4. Optimal Ready-Rate Contract hold if the \fll rate is used as performance measure. Let AW = AW (S; L) denote the supplier's expected ready rate with window W if she chooses the base-stock level S and lead time L. Note that A0 is the supplier's expected immediate ready rate. Using the waiting time distribution in (3.2), we obtain AW = Prfw \u0014 W jS; Lg = PrfD(L\u0000W ) \u0014 Sg = Fw(W jS; L) W 2 [0; L]: (3.3) We assume that the supplier's delivery performance is evaluated at the end of every period. Note that the review-phase ready rate with window W is the proportion of periods in the review phase that at the end of the period no demand is delayed longer than time W . Denote the distribution of the review-phase ready rate A 2 [0; 1] with the mean AW by \t(AjAW ). \t(AjAW ) 2 [0; 1], \t(1jAW ) = 1 and \t0(AjAW ) > 0. Let its pdf be denoted by (AjAW ). Proposition 3.1 Under a static continuous-review base-stock policy with base-stock level S and lead time L, A\u0000AW \u001bW converges in distribution to a standard normal random variable as R approaches 1, where \u001b2W = 1 R2 (RAW \u0000R2A2W + 2 X i 0 such that A0 > 0:5, @( Z A< \t(AjAW )dA)=@S @( Z A< \t(AjAW )dA)=@L = @AW =@S @AW =@L for any W 2 [0; L]. 54 3.4. Optimal Ready-Rate Contract Note that A0 is the expected immediate ready rate, and is usually greater than 50% in practice. This means that when the performance measure is the ready rate with window W , the ratio of the marginal changes in the supplier's expected penalty with respect to S and L is the same as that of the performance target. For normal distribution, this assumption is approximately satis\fed. Theoretical validation is not easy, but the validation can be done through numerical examples (see Appendix B). Assumption 3.3 The \frst-order conditions for the supplier's optimization problem given the contract o\u000bered by the buyer are su\u000ecient. This allows us to replace the (IC) constraint (3.11) by the \frst-order conditions for the supplier's problem. Even for \fxed L and no lead time cost it is di\u000ecult to prove the unimodality of the supplier's pro\ft function E\u0019(S; L) (see Chapter 2). So we have to rely on numerical results to check unimodality. Now we present the main result of this chapter. Theorem 3.1 Assume that the optimal solution for the problem of the supply chain is interior, then there exists a unique optimal time window W \u0003 2 (0; L\u0003) that induces the \frst-best e\u000bort levels. Theorem 3.1 implies that as long as the relative change in the supplier's expected penalty with respect to S and L is the same as that in the expected performance, then the buyer can always \fnd a time window W \u0003 and use the ready rate with window W \u0003 as the performance measure to coordinate the supply chain. Let A\u0003W = F (S \u0003jL\u0003 \u0000W \u0003). Corollary 3.1 follows from the proof of Theorem 3.1. Corollary 3.1 W \u0003 is such that @ECD(yjS\u0003; L\u0003)=@S @ECD(yjS\u0003; L\u0003)=@L = @A\u0003W=@S @A\u0003W=@L : (3.12) To understand the role of W \u0003 in the optimal SLA, note that @ECD(yjS;L)=@S @ECD(yjS;L)=@L is the marginal rate of technical substitution (MRTS) for the expected delay cost in economics theory, @AW @S and @AW @L are the marginal change in the expected performance AW with respect to the base-stock level S and lead time L, respectively. So in the service level agreement here, the role of the optimal window W \u0003 is to set a right performance measure (and target) so that the optimal MRTS in the integrated system 55 3.4. Optimal Ready-Rate Contract is transferred to the supplier, and the supplier's investments in inventory and lead time are perfectly balanced. Because Theorem 3.1 implies that the ready rate with window W \u0003 is the unique ready rate that induces the \frst-best e\u000bort levels and W \u0003 > 0, we can conclude that the immediate ready rate is suboptimal. Corollary 3.2 follows. Corollary 3.2 The SLA using the immediate ready rate as performance measure cannot induce the \frst-best e\u000bort levels. So in general, a service level agreement using the immediate ready rate as the performance measure is suboptimal. Proposition 3.3 The \frst-best ready-rate contract is such that: 1) W \u0003 is characterized by (3.12), where (S\u0003; L\u0003) is the \frst-best solution deter- mined by (3.9) and (3.10); 2) A\u0003W = F (S \u0003jL\u0003 \u0000W \u0003); 3) there are multiple choices of (K\u0003; \u000b\u0003) such that K\u0003 > 0, \u000b\u0003 2 (0; A\u0003W ) and satisfy K\u0003 = \u0000 RhF (S\u0003jL\u0003) 100 Z A<\u000b\u0003 @\t(AjA\u0003 W ) @S dA and Z A<\u000b\u0003 @\t(AjA\u0003W ) @S dA < 0; 4) p\u0003 = \u0019 \u0015 + 100K \u0003 \u0015R Z A<\u000b\u0003 \t(AjA\u0003W )dA+ h\u0015I(S\u0003; L\u0003) + Cr(L\u0003): Proposition 3.3 implies that in practice, managers have many choices for the threshold performance level \u000b and the penalty rate K. Note that the threshold performance level determines the allowable deviation of the supplier's performance from the target. As noted in Section 3.3, in Chapter 2 we had found it best that should be close to the performance target to mitigate the supplier's strategic behavior. So although the choice of the optimal \u000b is not unique, it should not be too far from the target. Proposition 3.4 If the demand distribution is normal, then for \fxed \u001b \u0015 , L\u0003, W \u0003 and A\u0003W are independent of \u0015, and S \u0003 is proportional to \u0015. Proposition 3.4 implies that for normal demand and the same coe\u000ecient of vari- ation, the \frst-best inventory replenishment lead time, the optimal window and the performance target in the optimal ready-rate-with-window contract are identical, and 56 3.4. Optimal Ready-Rate Contract the \frst-best base-stock level is proportional to the demand rate. This has implica- tions for implementing an SLA. If the demand rate changes, as long as the coe\u000ecient of variation of demand remains the same, the optimal window W \u0003 and the target ready rate A\u0003W in the SLA need not be changed. Next, we have a result for the \frst-best immediate ready rate A\u00030 for general de- mand distributions and linear delay costs under the optimal ready-rate-with-window contract. A linear delay cost means CD(y) = \u000ey, where \u000e > 0. Proposition 3.5 If the delay cost is linear, then A\u00030 = \u000e h+\u000e . Proposition 3.5 indicates that for a linear delay cost, even if the supplier can invest to attain a di\u000berent lead time, it will not a\u000bect the \frst-best immediate ready rate A\u00030. A \u0003 0 is only determined by the holding cost to delay cost ratio h=\u000e. 3.4.4 Ready-rate-without-window contract A ready-rate-without-window contract has the same interpretation as a ready-rate- with-window contract except that the supplier's delivery performance is measured in terms of the immediate ready rate, i.e., W = 0. So the buyer's optimization problem is similar to that under a ready-rate-with-window contract with W = 0, and the optimal solution is provided in Proposition 3.6. Proposition 3.6 Under a ready-rate-without-window contract, 1) the optimal (S\u0003\u0003; L\u0003\u0003) that the buyer can induce is the solution to the constrained optimization problem: min p;\u000b;K;S;L EC(S; L) (3.13) subject to hF (SjL)@A0=@L @A0=@S + h R x~~~~ 0, \u000b\u0003\u0003 2 (0; A\u0003\u00030 ) and satisfy 57 3.4. Optimal Ready-Rate Contract K\u0003\u0003 = \u0000 RhF (S\u0003\u0003jL\u0003\u0003) 100 Z A<\u000b\u0003\u0003 @\t(AjA\u0003\u00030 ) @S dA and Z A<\u000b\u0003\u0003 @\t(AjA\u0003\u00030 ) @S dA < 0; and p\u0003\u0003 = \u0019 \u0015 + 100K \u0003\u0003 \u0015R Z A<\u000b\u0003\u0003 \t(AjA\u0003\u00030 )dA+ h\u0015I(S\u0003\u0003; L\u0003\u0003) + Cr(L\u0003\u0003): Proposition 3.7 is a counterpart of Proposition 3.4 under a ready-rate-without- window contract. Proposition 3.7 If the demand distribution is normal, then for \fxed \u001b \u0015 , L\u0003\u0003 and A\u0003\u00030 are independent of \u0015, and S\u0003\u0003 is proportional to \u0015. Similar to the result in Proposition 3.4, Proposition 3.7 implies that when the demand rate changes, as long as the coe\u000ecient of variation is not changed, the per- formance target in the optimal ready-rate-without-window contract is still optimal. Both Propositions 3.4 and 3.7 imply that the optimal performance target in a ready- rate contract is dependent on the coe\u000ecient of variation of demand. Corollary 3.3 follows from Propositions 3.4 and 3.7 and the fact that EC(S;L) \u0015 is independent of \u0015 for \fxed \u001b \u0015 . Corollary 3.3 For normal demand, the system loss { the % increase in the aver- age supply chain cost from not using a window in the contract is identical when \u001b \u0015 is constant. The results in Propositions 3.4 & 3.7 and Corollary 3.3 will be useful for conducting numerical analysis in Section 3.5. 3.4.5 Incentive alignment using an inventory performance measure In the supplier's expected average penalty, let \u0005(S; L;W ) = Z A< \t(AjAW )dA, then @\u0005(S;L;W )=@S @\u0005(S;L;W )=@L = \u0012 @AW =@S @AW =@L , where \u0012 = 1\u0000 @\u001bW =@S @AW =@S \u001e(\u000b\u0000AW \u001bW )=\b(\u000b\u0000AW \u001bW ) 1\u0000 @\u001bW =@L @AW =@L \u001e(\u000b\u0000AW \u001bW )=\b(\u000b\u0000AW \u001bW ) : (3.15) 58 3.5. Numerical Analysis The derivation of \u0012 is from (B.4) in Appendix B. So Assumption 3.2 holds if and only if @\u001bW =@S @AW =@S = @\u001bW =@L @AW =@L for any S; L > 0 with A0 > 0:5. From Proposition 3.1, if the inventory replenishment lead time L \u0014 1, then the performance outcomes in any two periods i and j are independent, and RA, the number of periods in a review phase that have good performance (no demand is delayed longer than the window W ), has a binomial distribution with independent outcomes and b\u001b2W = AW (1\u0000AW )R , so @\u001bW =@S @AW =@S = @\u001bW =@L @AW =@L holds. For L > 1, if any two periods i and j di\u000ber by less than L periods, then the performance outcomes in these two periods are positively corre- lated. When changing the base-stock level S and lead time L, the relative change in the performance variability \u001bW due to S and L may be di\u000berent from that in the performance target AW , i.e., @\u001bW =@S @AW =@S = @\u001bW =@L @AW =@L may not hold. Other order ful\fllment rates for measuring inventory performance such as the \fll rate have similar properties for positive lead times. From the proof for Theorem 3.1, for a demand distribution with @\u001bW =@S @AW =@S 6= @\u001bW =@L @AW =@L , there may not exist a W 2 [0; L] such that a ready rate with windowW induces the supplier to make the \frst-best investment (S\u0003; L\u0003), or the opti- mal time windowW may not be unique. This implies that for inventory management in a decentralized system, to e\u000bectively align a supplier's incentive with the buyer's when the supplier has multiple ways to perform, a single aggregate performance mea- sure such as the ready rate may not be su\u000ecient. This is due to the performance variability resulting from positive inventory replenishment lead time. If this is the case, then other performance measures are needed to complement the ful\fllment rate. In Choi et al. (2004), a ready rate target for the supplier is not a su\u000ecient guarantee due to the capacity constraints of both the buyer and the supplier, and they propose average backorders as a second performance measure. But in a \fnite-horizon, this measure is also a random variable, and its variability also needs consideration when designing an SLA. 3.5 Numerical Analysis We use numerical examples to illustrate how the \frst-best S\u0003; L\u0003 and optimal contract parameters are a\u000bected by the demand rate \u0015, the variability of demand \u001b, and related costs including the inventory holding cost, lead time cost and delay cost. Moreover, because both the immediate and time-window ready rates are seen in practice, we compare the system performance (the optimal average supply chain cost) under both 59 3.5. Numerical Analysis types of SLAs. Assume the demand per period has a normal distribution. The lead time cost function is Cr(L) = r L (r > 0). To simplify the notation, let z = S\u0000\u0015L \u001b p L . Following the results in Propositions 3.4 & 3.7 and Corollary 3.3, under the SLA using either type of ready rate as performance measure, the optimal lead time is independent of \u001b \u0015 and the optimal base-stock level is proportional to \u0015; and the system loss from using the immediate ready rate instead of the time-window one is also independent of \u001b \u0015 . So we only need to examine the results for demands with di\u000berent coe\u000ecient of variation of demand by varying \u001b with \u0015 \fxed. In the inventory management literature, the delay cost is known as the backorder cost and is usually assumed to be linear in the waiting time. In reality, however, it may be nonlinear. For example, if the buyer provides after-sales services, customers may not mind waiting for one or two days to have their laptops, televisions or cars etc. \fxed, in which case the supplier provides spare parts to the buyer; but the loss of customers' goodwill goes up quickly when their waiting time is beyond their tolerance. In the automobile industry, if a customer's preferred vehicle model is not immediately available at a car dealer, the customer is often willing to wait for a few days before receiving it; but if the customer has to wait longer, then she may leave and go to another dealer. Therefore, we consider two types of delay costs: linear delay cost with CD(y) = \u000ey, and convex delay cost with CD(y) = \u000ey 2, where \u000e > 0. To be consistent with our continuous approximation of the underlying model, we report the continuous-valued optimal solutions in the numerical examples. We use the following parameter values: h 2 f1; 2g; \u000e 2 f2; 10; 20g; r 2 f2; 4g; \u0015 = 10, and \u001b 2 f1; 2; :::; 5g. All the formulas for the calculations can be found in Appendix B. 3.5.1 Linear delay cost Linear delay cost is a common assumption in the theoretical analysis. This represents the situation where the marginal cost of a delayed delivery does not vary with the amount of delay. Under linear delay cost,the optimal window W \u0003 can be computed from the simple formula below: W \u0003L = L \u0003 \u0000 S \u0003 \u0015\u0000 2\u0015C 0r(L\u0003)=h : (3.16) To examine the optimal ready-rate-with-window contract, we compute the \frst- 60 3.5. Numerical Analysis best (S\u0003; L\u0003) and W \u0003L for the parameters given above. The results are plotted in Figures 3.1-3.3. Figure 3.1: First-best base-stock level (linear delay cost) Figure 3.2: First-best lead time (linear delay cost) Figures 3.1-3.3 indicate that the optimal window increases with the coe\u000ecient of variation. Because greater demand variability poses higher inventory risk on the supplier, we can interpret the optimal window as being used for sharing the inventory risk between the buyer and the supplier. For \fxed demand rate, both the \frst-best base-stock level and lead time go down with the variability of demand (\u001b). Similar results are found for the optimal base-stock level and lead time under a ready-rate- without-window contract. Intuitively, with shorter lead time, the supply chain can respond more quickly to demand with large variability. We also examine the performance of a ready-rate-without-window contract by comparing its optimal average supply chain cost C\u0003\u0003 with the \frst-best one C\u0003 and 61 3.5. Numerical Analysis Figure 3.3: Optimal window (ready rate with window, linear delay) Table 3.1: Cost increase from not using a window contract (linear delay cost) Parameters % Cost Incre h = 1; \u000e = 10; r = 2 0:69% h = 1; \u000e = 10; r = 4 0:69% h = 1; \u000e = 20; r = 2 0:44% h = 2; \u000e = 10; r = 2 1:12% h = 2; \u000e = 2; r = 2 3:65% computing the percentage cost increase from not using a window contract. Table 3.1 demonstrates its performance for di\u000berent demand variability and related costs. It turns out that with other parameters \fxed, the system loss | the percentage increase in the average supply chain cost, from using a ready-rate-without-window contract, is independent of the coe\u000ecient of variation of demand for the \fxed demand rate. With other parameters \fxed, the system loss is independent of the lead time cost, decreases with the delay cost, and increases with the holding cost. Proposition 3.5 has shown that for linear delay cost, the \frst-best expected immediate ready rate A\u00030 = \u000e h+\u000e = 1 1+h=\u000e . The numerical results indicate that for small h=\u000e ratio (high target service level), the system e\u000eciency loss from using an immediate ready rate as performance measure is small. So when the holding cost h is small compared with the delay cost \u000e, an immediate ready rate will induce the system performance close to the optimal; if h is close to \u000e, then a time-window ready rate should be used. 62 3.5. Numerical Analysis 3.5.2 Convex delay cost In practice, the cost of delayed delivery to a buyer may not be linear in the amount of delay. The marginal cost of delay is increasing in the amount of delay, i.e., the delay cost is convex. Convex delay cost is often assumed for customers' value of service time in customer service models. Similar to the analysis for linear delay cost, we examine the optimal ready-rate- with-window contract by computing the \frst-best solution (S\u0003; L\u0003) and the optimal window W \u0003C for the parameters given above. The optimal window W \u0003 can be com- puted from the formula W \u0003C = L \u0003 \u0000 S \u0003\b(z\u0003) \u0015\b(z\u0003)\u0000 2\u0015 h C 0r(L\u0003)\u0000 \u001bpL\u0003\u001e(z\u0003) ; (3.17) where z\u0003 = S \u0003\u0000\u0015L\u0003 \u001b p L\u0003 . Figure 3.4: Optimal window (ready rate with window, convex delay) Similar patterns are found for convex delay cost about the \frst-best base-stock level S\u0003 and lead time L\u0003, the optimal window W \u0003C , as well as the optimal base-stock level S\u0003\u0003 and lead time L\u0003\u0003 under a ready-rate-without-window contract as those in the case of linear delay cost, and similar conclusions can be made. The optimal window here can be regarded as being used for sharing the inventory risk between the buyer and the supplier. Compared with linear delay cost, convex delay cost is smaller for short delay and larger for long delay. So the optimal window also plays a role of partially transferring the buyer's delay cost structure to the supplier. This can be seen by comparing Figure 3.3 with Figure 3.4, where the optimal window under 63 3.5. Numerical Analysis Table 3.2: Cost increase from not using a window contract (convex delay cost) h = 1; \u000e = 10 h = 1; \u000e = 10 h = 1; \u000e = 20 h = 2; \u000e = 10 h = 2; \u000e = 2 \u001b r = 2 r = 4 r = 2 r = 2 r = 2 1 3:78% 3:35% 2:75% 5:74% 10:46% 2 3:38% 2:95% 2:50% 5:17% 9:43% 3 3:35% 2:89% 2:52% 5:14% 9:16% 4 3:46% 2:94% 2:64% 5:30% 9:20% 5 3:63% 3:05% 2:79% 5:54% 9:39% convex delay cost is generally larger than that under linear delay cost. For the system loss from not using a window contract, Table 3.2 shows that unlike that in the case of linear delay cost, it varies with the coe\u000ecient of variation of demand and is not monotonic. With other parameters \fxed, the system loss decreases with the delay cost, but increases with the holding cost. Similar pattern has been found for linear delay cost. However, the e\u000bect of lead time cost on the system loss is di\u000berent. Here the system loss decreases with the lead time cost. In all the scenarios, the system loss is much greater than that of linear delay cost. From the above numerical results for linear and convex delay costs, we can see that a time-window ful\fllment rate is preferred to an immediate one for measuring a supplier's delivery performance in two situations. One situation is when the buyer's cost of delay is not large compared with the inventory holding cost of the product. An example is perishable products such as expensive electronics, of which the inventory holding cost also includes the depreciation of the product. Another situation is when the marginal cost of a delay increases with the length of delay, that is, the delay cost is small for short delay but very large for long delay, often the case in reality. Linear delay cost is commonly used in theoretical analysis. 3.5.3 E\u000bect of the length of review phase R We \frst consider an SLA using the optimal time-window ready rate. We have assumed that both the buyer and the supplier optimize their long-run average payo\u000b, so under a ready-rate-with-window contract, the optimal (S\u0003; L\u0003) and thus W \u0003 (and A\u0003W ) are not a\u000bected by R. This can be seen from Propositions 3.2 and 3.3. Similarly, under an SLA using the optimal immediate ready rate, the optimal (S\u0003\u0003; L\u0003\u0003) and thus A\u0003\u00030 are not a\u000bected by R, which follows from Proposition 3.6. 64 3.6. Conclusions For both types of SLAs, let m = \f\f\f\u000b\u0000AW\u001bW \f\f\f, W \u0015 0. m is the relative allowable deviation of the performance from the target. Because the variability of performance \u001bW is decreasing in R and \u000b < AW , for \fxed m, the performance threshold \u000b is increasing in R, i.e., the threshold should be closer to the target for large R. 3.6 Conclusions In this chapter we examine the design of service level agreements in decentralized supply chains when a supplier has multiple ways to do things. Speci\fcally, we study ready-rate contracts for managing a supplier's delivery performance from a principal- agent perspective, in which the supplier can invest both in inventory and in replen- ishment lead time, and the investments are unobservable to the buyer. Therefore, we study a multi-task moral hazard problem. The SLA in our study is a linear-penalty scheme under a multi-period review strategy. We investigate two common measures of inventory performance observed in practice | the immediate ready rate and the time-window ready rate, and show that under a static continuous-review base-stock policy, the distribution of the supplier's review-phase ready rate is approximately normal. Because a single performance mea- sure is used but the supplier has two choices | inventory and lead time | to a\u000bect the performance, we examine the e\u000bectiveness of immediate and time-window ready rates at aligning the supplier's incentive with that of the buyer. We also \fnd that due to positive inventory replenishment lead time, the performance outcome in each period can be correlated. As a result, the relative change in the variance of the review- phase ready rate with respect to the base-stock level and lead time may be di\u000berent from that of the performance target. In this case, a single performance measure such as the ready rate cannot align the supplier's incentive to make \frst-best investment in inventory and lead time, and additional performance measure is needed. We propose a ready-rate-with-window contract, and show that under some mild conditions the SLA with a time-window ready rate as performance measure always induces the supplier to choose the \frst-best level of investment in inventory and lead time. For normal demand, the optimal window is identical for demands with the same coe\u000ecient of variation; and the greater the coe\u000ecient of variation in demand, the larger the optimal window. We \fnd that for linear cost of delayed delivery, a simpler form of ready rate 65 3.6. Conclusions contract, ready-rate-without-window contract, is near optimal. However, in the cases where the delay cost is convex, a ready-rate-without-window contract can result in high system loss | increase in the average supply chain cost; thus a ready-rate-with- window contract is preferred. The gain from using a ready-rate-with-window contract instead of a ready-rate-without-window one increases with the convexity of the delay cost. Therefore, a simple inclusion of window in performance measures can make a big di\u000berence. The window used in the ready-rate contract plays three roles. First, it transfers the MRTS of the expected average delay cost to the supplier to balance the supplier's investment in inventory and lead time, making the supplier's tradeo\u000b between inven- tory and lead-time investments same as that of the supply chain. Second, it facilitates the sharing of inventory risk (cost from excessive inventory and inventory shortage from stochastic demand) between a buyer and a supplier by allowing a small delay of delivery when evaluating the supplier's delivery performance. Moreover, the window to some extent transfers the buyer's delay cost structure to the supplier. This \fnding provides a theoretical support for the common use of time-window ful\fllment rate in practice. 66 Chapter 4 Volume Incentive Through Performance-Based Allocation Of Demand 4.1 Introduction Buyers of products or services commonly employ multiple suppliers rather than a single one. There are a large number of reasons why this might happen. For example in the case of some accident (e.g., supply disruption) the buyer does not want to lose his single source of product or service. Even without accidents the buyer does not want to be in a `holdup' situation if switching to another supplier is not easy. In cases where the purchase amount is uncertain or the supply available is uncertain, then a backup or emergency source is needed. Finally of course there is the simple need often to keep supply price competition vigorous by maintaining a stable of suppliers. This chapter however assumes away all these reasons in order to concentrate on another widespread motivation. A great deal of innovation in the quality and performance of products and services is expected to be done by suppliers. The suppliers can make an e\u000bort (invest resources) to improve their performance, which bene\fts the buyer. Such investment is costly to the supplier and is often unobservable to the buyer. Through sourcing from multiple suppliers, the buyer can create competition between the suppliers and motivate them to invest for better performance. Dyer and Ouchi (1993) report that Japanese \frms usually employ a `two-vendor policy' to motivate suppliers to innovate and improve performance. An empirical study by Bensaou (1999) also shows that Japanese buyers typically split their purchases among multiple suppliers and then demand that the suppliers make specialized investments to obtain and keep their business. In this chapter we assume away the risk of supply disruption by con\fning un- 67 4.1. Introduction certainty only to the results of the investment, by all parties being risk neutral, by supply being \fxed to one standardized unit (one million components or ten thousand payrolls serviced or 1,000 kilometers of yellow lines down the centre of city streets maintained), and by suppliers being essentially uncapacitated. In addition the price per unit is \fxed. The only way for a supplier to gain a larger share of next period's (e.g., year's) contract for this unit is to improve quality or performance. To improve quality the buyer will announce in advance how next year's shares will depend on this year's observed performance or quality. This performance-based allocation of business provides an incentive in the form of business volume, which is commonly used in practice. For example, Toyota adjusts business volume between just two suppliers based on their performance to achieve e\u000bective competition between the suppliers (Dyer et al. 1998). Sun Microsystems allocates demand among multiple suppliers using a scorecard system (Farlow et al. 1996). When allocating business between two suppliers based on their performance, the buyer's objective is to keep suppliers competitive in terms of quality, delivery, or whatever supplier's performance characteristic the buyer deems important (Spekman 1988, Hahn, Kim & Kim 1986), and to motivate suppliers to improve by providing positive incentives (in the form of increased business volumes) or negative incentives (in the form of decreased business or competition). The buyer seeks for better supplier performance rather than the optimal supply chain e\u000eciency. The actual measurement of quality or performance will be kept as simple as possible in this chapter but the key assumption is that the buyer can observe the supplier's performance or quality only and this is a noisy signal of the supplier's e\u000bort level. Despite its widespread use, this volume incentive is seldom studied in the litera- ture. In the absence of direct monetary incentives, volume incentives in the form of delayed rewards or penalties from gaining or losing future business may be a substi- tute. The objective of this research is to study special features in the performance- based volume incentive schemes and the e\u000bectiveness of di\u000berent forms of volume incentives. We look for answers to the following questions: How can a buyer use the allocation of demand to induce competition between two suppliers to obtain better performance when the suppliers' investments are unobservable? What is the buyer's optimal volume incentive scheme that maintains the suppliers' competition over time? Speci\fcally, we consider a buyer repeatedly outsourcing a \fxed amount of divisi- ble service or product from two suppliers. The suppliers can make e\u000bort to improve 68 4.1. Introduction their performance, and in order to isolate the key structure of what drives an opti- mal allocation we make the two suppliers equal in every way. The e\u000bort levels are unobservable to the buyer { a moral hazard problem in agency theory. The buyer allocates his business between the two suppliers based on their past performance in order to maximize his long-run discounted payo\u000b from repeated dual sourcing. A key part of the modeling is the suppliers' cost function. We examine two cases; where a supplier's e\u000bort cost is proportional to her share of business (the proportional case) and the case where the cost is independent of her share (the independent case). An example of the latter might be where a new product is developed in a laboratory and then can be applied in their supplier's factory without further tooling; compared to the former case where the same product is made but the process improvement means that all the machines must be upgraded. An innovation of new software to payroll management that could then be rolled out to all accounts would be the latter, but one which needed the reformatting of accounts one by one would be the former. A new yellow paint formulation that dried quicker would be the latter but a drying process that cost per kilometer would be the former. An important example here is `learning by doing'. A \frm with a large share might in some circumstances have much more opportunity to innovate at a lower cost. The company with 999 of the 1,000 kilometers might have greater testing costs than the one with 1 kilometer, but the cost per kilometer is likely a lot less. Of course in practice this is going to be a lot more complex, but using these two cases and seeing how the results depend critically on them gives us insights into the importance of including this aspect. A natural allocation rule by the buyer that comes readily to mind is to give all the business to the one with the better performance, often termed a `winner-take-all' or WTA rule, essentially treating the competition as a rank-order tournament. Our \fnding addresses both the A, the `all', of this acronym and the W, the `winner'. In the proportional case the optimal is not `all', but in the independent case it essentially is `all'. However in neither case is the de\fnition of `winner' a simply `\frst-past-the-post'. We \fnd that winning must be relative to current shares, essentially an endowment with which a company enters this year's competition. For symmetric suppliers with an identical cost function, in both cases, the optimal rule of business allocation is what we might term a `handicapped' one. A parallel might be drawn to competitive sailing where point handicaps re\rect equipment endowments or past successes, or to the game of golf where success is handicapped. Both examples are mainly to ensure 69 4.1. Introduction a vigorous competition, the handicapped player has to work harder to win by being given a handicap at the start. Thus the competition may also look like a `fair' one. In some horse races, handicaps give extra weight into the saddles in order to make it a `fair' race, although this may be more to do with making the punting more interesting with a more evenly balanced \feld. Thus the supplier with a better performance in the current period may not get a bigger share in the next period. So in the proportional case, although the optimal allocation rule is not a WTA one, numerical results show that a handicapped-winner-take-all (HWTA) rule can perform well compared with the optimal when the variability (noise) in the performance measure is small, but worse when the variability (noise) is large, while a common rank-order tournament type of allocation rule, simple WTA (SWTA) rule, always performs far worse compared to both the HWTA rule and the optimal one. In the `independent' case, the optimal allocation rule for a \fnite horizon problem is a HWTA one. Both a share-dependent HWTA and a SWTA allocation rules are studied for an in\fnite horizon problem, and numerical results indicate that a HWTA rule can often perform much better than the SWTA one. Therefore, when the incentive comes from the allocation of business among competing suppliers, each supplier's current share of business plays an important role, and using a handicap can be very e\u000bective for incentive provision. The main contribution of this chapter is to examine these special features of performance-based volume incentive schemes. Our results have direct managerial im- plications to the design of volume incentive contracts in practice. To induce compe- tition among suppliers and maintain the competition over time, the optimal volume incentive scheme is generally not a simple rank-order tournament, which has been shown in literature to be e\u000bective under the monetary incentive scheme. Instead, handicapping the de\fnition of winner can do well over a simple \frst-past-the-post rule and the optimal rule may not be to give all the demand to one company. Performance- based volume incentives often need to take into account each supplier's current share of business. The rest of this chapter is organized as follows. Section 4.2 reviews the literature. Section 4.3 describes the model. Section 4.4 studies the buyer's problem under the two types of supplier's e\u000bort cost and presents numerical results. The chapter concludes with a summary and a discussion of future work in Section 4.5. 70 4.2. Literature Review 4.2 Literature Review Our study is at the interface of operations management and economics, and thus is related to both streams of literature. There exists a vast literature on dual (multiple) sourcing. Elmaghraby (2000) provides a survey on the research in operations research and economics literature on sourcing strategies for the problem of a buyer awarding a divisible business to one or more suppliers among multiple suppliers. The research questions are mostly one-time decision problems which are related to the design of competitive mechanisms in the form of bidding and the suppliers' competitive behavior under the bidding rule. In the operations management literature, a number of papers investigate the e\u000bect of demand allocation on the behavior of competing \frms. Lippman and McCardle (1997) study a single-period competitive newsvendor problem in which each newsvendor chooses an inventory level to meet a random demand and a rule speci\fes the allocation of initial market demand among the \frms as well as the allocation of excess demand among \frms with remaining inventory. They investigate the relationship between four speci\fc allocation rules and equilib- rium inventory levels. Both Hall and Porteus (2000) and Liu et al. (2007) consider a multi-period competitive newsvendor problem where two \frms make capacity (in- ventory) decision in each period, and the demand for each \frm is dependent on the realized level of customer service (product availability) in the prior period. The \frms' equilibrium behavior in the dynamic game is identi\fed. In all three papers, \frms' incentive for competition is governed by an exogenous demand allocation mechanism driven by the switching behavior of customers in the market, which is dependent on the \frms' realized service levels in the current (\frst paper) or prior period (the other two papers), and there is no buyer dictating the supplier competition. In our model, a buyer designs the incentive mechanism { a demand allocation rule which is based on the \frms' past performance levels. So the focus of our study is on the design of a demand allocation mechanism. Our study is closely related to two papers. Both Cachon and Zhang (2007) and Benjaafar et al. (2007) consider a buyer outsourcing a \fxed demand at a \fxed unit price to multiple suppliers. Cachon and Zhang (2007) study a queuing system where each supplier's service time is determined by the capacity she invests, and the buyer allocates the demand among multiple suppliers based on their service times to mini- 71 4.2. Literature Review mize the average service time over an in\fnite horizon. Suppliers are homogeneous in terms of their capacity costs. Each supplier chooses a capacity level to maximize her own pro\ft. The authors evaluate several allocation rules and show that performance- based allocation may not motivate suppliers to improve service times. In Benjaafar et al. (2007), a buyer outsources the demand to a set of potential suppliers. Com- petition between suppliers is created either by allocating the whole demand to one supplier with the probability of being selected increasing with her committed service level { market-seeking (MS) approach, or by allocating the demand to each supplier in proportion to her committed service level { market-augmenting (MA) approach. In the MA case, each supplier's service level is assumed to be independent of the demand allocated to her. Under both cases, it is assumed that the contractual promises of the suppliers regarding e\u000bort or service level are enforceable. The suppliers are hetero- geneous in production and service costs. Each supplier chooses a committed service level to maximize her expected pro\ft. The authors compare the service quality the buyer can achieve under the MA and MS mechanisms. Neither paper considers the hidden action problem. There is no noise in the suppliers' performance outcome, and the demand allocation is based on the suppliers' observable e\u000bort levels or expected performance in the \frst paper and on the suppliers' committed performance in the latter. Cachon and Lariviere (1999) study a special allocation rule commonly used in the automobile industry by considering a single supplier allocating capacity to multiple retailers based on their past sales. They examine a two-period game under a given allocation rule, so their focus of study is not on the design of allocation rule. In all the aforementioned papers, only Cachon and Zhang (2007) examine the optimal allocation rule. Our chapter di\u000bers from these papers by investigating the design of volume incentives which are on the basis of past performance and studying a multi-period multi-agent moral hazard problem. Our research is also related to the economics literature. Spear and Srivastava (1987) study a repeated moral hazard problem with discounting between a principal and an agent, and show that history dependence can be represented by using the agent's expected utility as a state, and thus the problem of characterizing the optimal contract of such a model can be reduced to a constrained static variational problem. Monetary compensation is used for an incentive. We study a repeated moral hazard problem between a principal and two competing agents, the compensation is in the 72 4.3. Model Description form of future demand, and the state is Supplier 1's current share of the business. Lewis and Yildirim (2002) examine the design of competitive mechanisms for dual sourcing with supplier learning by doing. Each time only one supplier is selected through bidding. The buyer faces an adverse selection problem because the suppliers' production cost is private information. Supplier's investment in performance is not in consideration. In the economics literature, incentive schemes are usually on the basis of mon- etary reward or penalty. Competitive compensation schemes can come in the form of rank-order tournament or relative performance evaluation. The relevant research can be found, for example, in Lazear and Rosen (1981), Green and Stokey (1983), Hart (1983), Holmstrom (1982), and Nalebu\u000b and Stiglitz (1983). The problems are generally for a single period. Relative performance evaluation (RPE) compensates the agents based on their output levels, and is often used when there is a common shock to the agents' performance, which is not considered in our problem. The total compensation in RPE varies with the agents' realized output levels, but in our case the total demand to be split is a constant. In tournaments, rewards are based on the rank order of the individuals, not on their actual output levels. Lazear and Rosen (1981) show that for risk-neutral agents rank-order tournaments work as well as in- dependent contracts; and for agents with known heterogeneous ability, handicapping will improve the e\u000eciency of the tournaments. We \fnd that when incentives are from future business, rank-order tournaments are generally not optimal, and handicapping signi\fcantly improves the e\u000eciency even when the agents are homogeneous in ability. 4.3 Model Description Consider a buyer outsourcing the supply of a \fxed one unit of a divisible product or service from two suppliers repeatedly over an in\fnite horizon. Both the buyer and the two suppliers are risk neutral. For tractability we make a number of simplifying assumptions. Each supplier can make e\u000bort to improve her performance (e.g., delivery, quality, cost, etc.). For instance, when contracting for inventory management, a supplier's demand ful\fllment performance can be measured by the \fll rate. In each period t (t = 1; 2; :::), Supplier i's realized performance xit = e i t + \"i (i = 1; 2), where eit is Supplier i's e\u000bort level in period t, \"1 and \"2 are independently and identically 73 4.3. Model Description distributed (i.i.d.) with the probability distribution N(0; \u001b)5. In this chapter, we use a supplier's e\u000bort level to refer to her target performance level. The suppliers have identical e\u000bort cost function, and the cost of e\u000bort takes the form C(e; \f) = g(\f) be 2 2 , where b > 0, \f is a supplier's share of demand in a period, g(\f) > 0 and g0(\f) \u0015 0. It is important that marginal increases in performance are increasingly costly to achieve. The actual quadratic nature is a matter of convenience. Other strictly convex functions are possible but the analysis would be formidable. The e\u000bort cost is common information. Only two special cases of g(\f) are considered: g(\f) = and g(\f) = 1, the proportional and independent cases respectively. The unit transfer price p of the product or service between the buyer and each supplier is identical and constant in every period, which can re\rect a dominant market price. The unit cost of supplying the product or service c is constant in every period. Consequently, the unit pro\ft of supplying the product or service m = p\u0000 c is also constant and identical for both suppliers. All parties have a common discount factor \r 2 (0; 1). Let \u000bt denote Supplier 1's share of demand in period t. So Supplier 2's share in period t is 1 \u0000 \u000bt. The state in period t is \u000bt, Supplier 1's share in that period, \u000bt 2 [0; 1]. Each supplier's feasible action set is A = [0; be], where be is a su\u000eciently large number. The buyer's allocation rule for the next period is restricted for simplicity to be based on the current share and performance. Rules such as based the average of previous year's shares or performance levels are not considered. In a repeated moral hazard problem between a principal and an agent, Spear and Srivastava (1987) have shown that history dependence in the compensation scheme can be represented by using the agent's expected utility as a state, thus the optimal compensation scheme is independent of the history of the agent's performance and compensation. In our problem, because a supplier's expected future payo\u000b is directly linked to her next period share of business, by analogy the optimal allocation rule is likely to depend only on the suppliers' immediate past performance outcomes and shares. So the restriction does not necessarily limit our \fndings. The sequence of events is as follows. At the beginning of the horizon, the buyer announces an allocation rule to be used for each period and gives each supplier an 5The case of correlated noise has been studied but essentially no added insights were available and the complexity was greatly increased. The normal assumption is just for tractability and appears reasonably benign. 74 4.3. Model Description initial share of the business, with the two suppliers' total share equal to one. In every period t (except for period 1), the buyer allocates his business between the two suppliers based on the suppliers' performance levels in the previous period and the allocation rule. The suppliers choose their e\u000bort levels simultaneously and incur the e\u000bort costs. Their performance levels are realized at the end of the period and observed by all parties. We study a moral hazard problem, where the buyer can only observe both sup- pliers' realized performance but not their e\u000bort levels in each period. Therefore, the share of demand allocated to each supplier in a period can only be based on immediate past performance realizations and demand allocations. Let vit and v B t denote Supplier i's pro\ft and the buyer's payo\u000b from period t onwards. The buyer's payo\u000b will be taken as the discounted weighted average quality (or performance) level in each period. So for t \u0015 1, the buyer's payo\u000b to go and the suppliers' pro\fts to go from period t onwards are vBt (\u000bt) = E( 1X \u001c=t \r\u001c\u00001[\u000b\u001cx1\u001c + (1\u0000 \u000b\u001c )x2\u001c ]) (4.1) = 1X \u001c=t \r\u001c\u00001[\u000b\u001ce1\u001c + (1\u0000 \u000b\u001c )e2\u001c ]; and v1t (\u000bt) = 1X \u001c=t \r\u001c\u00001[m\u000b\u001c \u0000 bg(\u000b\u001c )(e 1 \u001c ) 2 2 ]; v2t (1\u0000 \u000bt) = 1X \u001c=t \r\u001c\u00001[m(1\u0000 \u000b\u001c )\u0000 bg(1\u0000 \u000b\u001c )(e 2 \u001c ) 2 2 ]: The buyer uses a stationary allocation rule \f\u000b(x1; x2), which states that given Supplier 1's share in a period = \u000b and the suppliers' performance outcomes (x1; x2) in that period, Supplier 1's share in the next period is \f\u000b(x1; x2). Under the allocation rule \f\u000b(x1; x2), the two suppliers play a stochastic game in an in\fnite horizon. If the two suppliers' equilibrium policies are stationary, then the buyer's problem can be represented as a static variational problem (omitting the time index in the notations), 75 4.3. Model Description with the buyer's payo\u000b at period 1 being vB(\u000b) = \u000b1e1 + (1\u0000 \u000b1)e2 + Z Z vB(\f\u000b(x1; x2))f(x1je1)f(x2je2)dx1dx2; (4.2) where f(xje) = 1 \u001b \u001e(x\u0000e \u001b ) and \u001e(\u0001) is the probability density function (pdf) of the normal distribution, and the payo\u000bs of suppliers 1 and 2 at period 1 being v1(\u000b) = \u000bm\u0000 bg(\u000b)(e1) 2 2 + Z Z v1(\f\u000b(x1; x2))f(x1je1)f(x2je2)dx1dx2;(4.3) v2(1\u0000 \u000b) = (1\u0000 \u000b)m\u0000 bg(1\u0000 \u000b)(e2) 2 2 + Z Z v2(1\u0000 \f\u000b(x1; x2))f(x1je1)f(x2je2)dx1dx2: (4.4) In general we would like to keep the analysis simpler so that any allocation \f 2 [0; 1] was possible. Two circumstances prohibit this. First, given a buyer's allocation rule, the suppliers will play a stochastic game, so that we must ensure proper conditions for the Nash equilibrium to exist. Secondly, as the formulation of the buyer's problem is basically a repeated principal-agent formulation with the outcome of the suppliers' game as the agent, the participation of the suppliers needs to be ensured via the participation (individual rationality) constraints. Both of these considerations can place limits on the size of \f that the buyer can employ. The main results can be best appreciated by thinking that \f is between 0 and 1; however to do the modeling correctly we have to calculate the limits that \f can feasibly take. By the symmetry of the two suppliers, the limits are identical for the two suppliers. Let \f and \f denote a supplier's maximum and minimum shares in a period. The actual values will be addressed later. Note that \f = 1\u0000 \f. We are interested in the form of the buyer's optimal stationary allocation rules. For this purpose, we \frst derive the optimal allocation rule \f\u0003\u000b(x1; x2) from the static formulation of the buyer's in\fnite-horizon problem, under the assumption that the two suppliers use stationary policies to play the stochastic game; we then check that under this \f\u0003\u000b(x1; x2), the suppliers' in\fnite-horizon stochastic game has a unique Nash equilibrium which is stationary and is the one derived from the static formulation. Let e\u00031 and e \u0003 2 denote the optimal stationary-policy e\u000bort levels of suppliers 1 and 2 under an optimal allocation rule \f\u0003\u000b(x1; x2). As discussed above we consider two special forms of g(\f): g(\f) = \f, representing 76 4.4. Buyer's Problem a demand-dependent e\u000bort cost which is linear in the demand \f; and g(\f) = 1, a demand-independent e\u000bort cost. 4.4 Buyer's Problem The buyer designs a demand allocation rule to maximize the long-run discounted aggregate performance of the suppliers over the horizon. Given the allocation rule, the suppliers choose their e\u000bort levels in each period to maximize their respective long-run discounted pro\ft. So the suppliers play a stochastic game governed by the buyer's allocation rule. To simplify our analysis, assume the suppliers' incentive compatibility constraints can be written as \frst-order conditions. The validity of this assumption will be checked later for each speci\fc allocation rule6. To focus on the form of the opti- mal allocation rule and each party's equilibrium result, we mainly present the static formulation of the buyer's problem in the main body, leaving in the appendix the dynamic formulation of each party's problem and the veri\fcation of the existence of a unique stationary Nash equilibrium in the suppliers' stochastic game7. So the buyer's problem is to choose an allocation rule \f\u000b(x1; x2) such that max\f\u000b(x1;x2) vB(\u000b) subject to \u0000g(\u000b)be1 + Z Z v1(\f\u000b(x1; x2))f 1(x1je1)f(x2je2)dx1dx2 = 0 \u0000g(1\u0000 \u000b)be2 + Z Z v2(1\u0000 \f\u000b(x1; x2))f(x1je1)f2(x2je2)dx1dx2 = 0 v1(\u000b) \u0015 0 v2(1\u0000 \u000b) \u0015 0 1\u0000 \f \u0014 \f\u000b(x1; x2) \u0014 \f: (4.5) The \frst two constraints are the incentive compatibility constraints for suppliers 1 and 2 respectively, where f i = @f=@ei. The third and fourth constraints are the 6Generally an agent's incentive compatibility constraint can be replaced by both a \frst-order condition and a second-order condition (i.e., concave objective function - a su\u000ecient condition for the extreme point to be the global optimum). In the appendix we use the seond-order condition as the su\u000ecient condition for the Nash equilibrium. 7We provide in the appendix the proof of the existence of a unique stationary Nash equilibrium in the suppliers' stochastic game under the optimal general allocation rule for the case of g(\f) = \f. The proof for other forms of allocation rules follows similar methodology and is thus omitted. 77 4.4. Buyer's Problem suppliers' individual rationality constraints which guarantee each supplier's long-run discounted payo\u000b to be nonnegative. 4.4.1 Volume incentive under proportional e\u000bort cost As discussed above, the proportional e\u000bort cost is taken to be the case of g(\f) = \f. Allocation rules We \frst study the buyer's optimal allocation rule which maximizes his long-run dis- counted payo\u000b, and investigate some simple heuristics. We then obtain numerical results for each type of allocation rule to examine the e\u000eciency of simple heuristics compared to the optimal rule. \u000f Optimal allocation rule As the details of Theorem 4.1 make the main message a bit opaque we shall discuss the main message as-if \f = 1 and \f = 0. In Figure 4.1 the axes are the performance outcomes of the two suppliers. The point (e\u00032; e \u0003 1) is where the optimal e\u000borts should be, and would be if the signal was not noisy. The dashed straight line is the 45\u000e line. The optimal allocation for the buyer would be to allocate all demand to Supplier 1 or 2 in all quadrants based on the point (e\u00032; e \u0003 1) except for quadrant (+,+). Thus Supplier 1 gets all when x1 \u0000 e\u00031 > x2 \u0000 e\u00032 and vice versa. However in the (+,+) quadrant emanating from (e\u00032; e \u0003 1), Supplier 1 should get a share \f where S(\f) = x1\u0000e\u00031 x2\u0000e\u00032 and we have S(\f) = m+H=(1\u0000 \f)2 m+H=\f2 ; (4.6) where H > 0 is de\fned in Theorem 4.1. Because Theorem 4.1 indicates that e\u00031 and e \u0003 2 are functions of \u000b, the optimal allocation rule for the in\fnite horizon problem is not a WTA one and is a function of each supplier's share in the current period. Since the two suppliers' optimal e\u000bort levels in a period di\u000ber when they have unequal shares, the optimal allocation rule is a handicapped rule in the sense that the suppliers are not compared by their actual performance but by the deviation from their respective target performance. Although the two suppliers are symmetric in terms of their e\u000bort cost function, for unequal starting shares in a period, their marginal costs for the same e\u000bort level di\u000ber 78 4.4. Buyer's Problem Figure 4.1: Optimal allocation rule with maximum share = 1 because of the demand-dependent e\u000bort cost structure. Therefore, the optimal rule uses a handicap to cope with unequal marginal e\u000bort cost. The supplier with a larger current share has a lower optimal e\u000bort level and a lower handicap under the optimal rule. This is because a larger share results in a higher marginal e\u000bort cost. It can also be seen from (4:7) that when both suppliers overperform, the optimal allocation rule dictates that for a range of x1\u0000e\u00031 x2\u0000e\u00032 values, Supplier 1's share in the subsequent period is dependent on the two suppliers' relative deviations from their respective target performance, regardless of how good their actual performance levels are. Moreover, the optimal allocation rule is symmetric in the suppliers' performance deviation from their respective target one. Theorem 4.1 gives the result but with the correct limits to \f: Theorem 4.1 For supplier's cost function with g(\f) = \f: 1. The buyer's optimal allocation rule is characterized as below: given Supplier 1's share in the current period equal to \u000b, for x1 > e \u0003 1 and x2 > e \u0003 2: if x1\u0000e\u00031 > S(\f)(x2\u0000e\u00032), \f1\u0003\u000b (x1; x2) = \f; if x1\u0000e\u00031 < S(\f)(x2\u0000e\u00032), \f1\u0003\u000b (x1; x2) = \f; 79 4.4. Buyer's Problem otherwise, \f1\u0003\u000b (x1; x2) is determined by S(\f1\u0003\u000b (x1; x2)) = x1 \u0000 e\u00031 x2 \u0000 e\u00032 ; (4.7) for x1 < e \u0003 1 or x2 < e \u0003 2: 1\u0003 \u000b (x1; x2) = \f if x1 \u0000 e\u00031 > x2 \u0000 e\u00032 and \f1\u0003\u000b (x1; x2) = otherwise; where S(\f) is de\fned by (4:6), e\u00031 = p 2H p b ; e\u00032 = p 2H (1\u0000\u000b)pb , and H > 0 is the solution to \u001b p 2bH = Z y1(b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2; (4.8) yi \u0018 N(0; 1), and b\f\u0003(y1; y2) = \f1\u0003\u000b (e\u00031 + \u001by1; e\u00032 + \u001by2) and is independent of \u000b. 2. Under \f1\u0003\u000b (x1; x2), (e \u0003 1; e \u0003 2) is the unique stationary Nash equilibrium in the suppli- ers' stochastic game. The correct version of Figure 4.1 is shown below. The interpretation is the same. Figure 4.2: Optimal allocation rule with maximum share < 1 Corollary 4.1 provides the formulas for calculating the value functions of the buyer and the suppliers. 80 4.4. Buyer's Problem Corollary 4.1 Under the optimal allocation rule \f1\u0003\u000b (x1; x2), given Supplier 1's ini- tial share \u000b, the suppliers' value functions v\u00031(\u000b) = v \u0003 2(\u000b) = v \u0003(\u000b) for any \u000b due to symmetry, v\u0003(\u000b) = \u000bm\u0000 H + \rV; V = 1 1\u0000 Z Z (b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2; and the buyer's value function is v\u0003B = 2 p 2H (1\u0000 \r)pb: From the suppliers' value functions in Corollary 4.1, we can see that under the optimal allocation rule a supplier's expected future payo\u000b V is independent of her current share. Therefore, the optimal allocation rule uses a handicap to provide the suppliers with di\u000berent level of incentive, but it is also a `fair' rule in that it gives the suppliers equal expected future payo\u000b. Also note that the nonlinear equation (4:8) may not have a unique positive solution of H. Because v\u0003B is increasing in H, the buyer would prefer the largest H. However, the nonnegativity of v\u0003(\u000b) and V may restrict such choices because they may become negative at the largest H. In fact, in the later section of numerical analysis, we have found that in some cases there are two solutions of H to (4:8), but both v\u0003(\f) and V are always negative at the bigger H, thus only the smaller H is used. \u000f Simple heuristics Due to the complexity of the optimal allocation rule, we also investigate simpler rules. Consider the family of WTA allocation rules with the following form8 \f2\u0003\u000b (x1; x2) = ( b\f x1 \u0000 x2 > \u0012 1\u0000 b\f x1 \u0000 x2 < \u0012\u000b ; (4.9) 8We ignore the case of equality, i.e., x1 \u0000 x2 = \u0012\u000b, because the probability for this to occur is zero. Similarly for all the WTA rules we are investigating hereafter. 81 4.4. Buyer's Problem where \u0012\u000b and b\f 2 (12 ; 1) are parameters to be determined. Note that we use WTA to refer to both HWTA and SWTA allocation rules. Theorem 4.2 For supplier's cost function with g(\f) = \f, among the family of WTA allocation rules as de\fned in (4.9), the optimal rule has \u0012\u0003\u000b 6= 0 for \u000b 6= 12 . Let \u0001v(b\f) = 4\u0019b\u001b2b\f(1\u0000 b\f) \r2(2b\f \u0000 1) (1\u0000 s 1\u0000 m 2 2\u0019b\u001b2 (2b\f \u0000 1)2b\f(1\u0000 b\f) ); then we obtain Corollary 4.2 following Theorem 4.2. Corollary 4.2 Given any b\f 2 (1 2 ; 1), the optimal \u0012\u0003\u000b = e \u0003 1 \u0000 e\u00032 = \r\u0001v(b\f) 2 p \u0019b\u001b 1\u0000 2 \u000b(1\u0000 \u000b) ; the optimal \f\u0003 is a boundary solution, and is constrained by v(1\u0000 b\f) = m 1\u0000 \r \u0000 1 2 ( 1 (1\u0000 \r)(2b\f \u0000 1) + 1)\u0001v(b\f) \u0015 0; \r\u0001v(b\f) 2\u001b2 \u001e(1) \u0014 b(1\u0000 b\f); (4.10) and the value functions of the buyer and the suppliers are v\u0003B(\u000b) = 1\u0000 \u0001v(\f\u0003)p \u0019b\u001b ; v\u0003(\u000b) = \u000bm\u0000 (\r\u0001v( \u0003))2 8\u0019b\u001b2 + 2 (v\u0003(\f\u0003) + v\u0003(1\u0000 \f\u0003)): It is obvious from Theorem 4.2 result that the optimal `take all' allocation rule is also a handicapped one with the de\fnition of \u0012\u0003\u000b. The reasoning is similar to that for the optimal general allocation rule. Again, the supplier with a larger current share has a lower optimal e\u000bort level and a lower handicap under the optimal rule, and the optimal `take-all' allocation rule is a `fair' rule because both suppliers have equal expected future payo\u000b. We also note that v\u0003B(\u000b) is independent of \u000b. This is due to the special structure of each supplier's e\u000bort cost which is proportional to the demand and quadratic in the e\u000bort level. 82 4.4. Buyer's Problem The simplest form of a WTA allocation rule is the one with \u0012\u0003\u000b = 0, i.e., the SWTA rule, \f3\u0003\u000b (x1; x2) = ( b\f x1 > x2 1\u0000 b\f x1 < x2 : (4.11) A WTA scheme is usually used in a tournament. It has been extensively studied in the economics literature for a single-period monetary incentive problem, where the winner receives an extra reward in addition to a common compensation to each supplier. Such an incentive scheme has been shown to be optimal when the two suppliers are symmetric. Let e be the solution to e = \u0000 \rp 2b\u001b \u001e( ep 2\u001b )( (2b\f \u0000 1)2b\f(1\u0000 b\f)m+ b2e2)[2\r\b( \u0000ep2\u001b ) + 1\u0000 \r]\u00001; (4.12) where \b(\u0001) is the cumulative distribution function (cdf) of the normal distribution. Proposition 4.1 Under the SWTA rule de\fned in (4.11), the optimal \f\u0003 < 1 and is not necessarily a boundary solution; the optimal e\u000bort levels of a supplier with a current share b\f and 1\u0000 b\f are e\u0003b\f = (1\u0000b\f)e1\u00002b\f and e\u00031\u0000b\f = b\fe1\u00002b\f respectively. Corollary 4.3 follows from Theorem 4.1 and Theorem 4.2. Corollary 4.3 For supplier's cost function with g(\f) = \f, any WTA allocation rule is suboptimal, and a HWTA rule provides better incentive than a SWTA rule. Numerical analysis The method for numerical calculation of the optimal H from (4:8) and the calculation of e\u00031; e \u0003 2, v \u0003 B(\u000b) and v \u0003(\u000b) can be found in Subsection 6 of Appendix C. Tables 4.1, 4.2 and 4.3 show the numerical analysis results under the optimal allocation rule, the HWTA and SWTA rules respectively9. Under all three rules, a supplier's maximum (minimum) share \f (\f) goes up (down) with \u001b. The reason is that when a performance measure becomes noisier, the suppliers' Nash equilibrium can still be maintained at a larger gap in their shares. In return, the more extreme maximum\/minimum share can provide a stronger incentive to the suppliers and counter the disincentive e\u000bect of the 9We use the minimum share \f obtained for the HWTA rule as that for the optimal rule so that the comparison is on the structure of the rules only without the interference of the minimum share. 83 4.4. Buyer's Problem \r \u001b b m \f v\u0003B V v \u0003(\f) v\u0003(\f) 0:9 0:5 1 1 25:9% 6:08 3:81 3:51 4:11 0:9 1 1 1 11:3% 5:48 3:25 2:70 3:77 0:9 2 1 1 3:5% 3:65 2:90 2:18 3:56 0:9 3 1 1 1:7% 2:66 2:75 1:96 3:45 0:9 0:5 1 2 33:6% 7:83 8:28 7:56 8:00 0:9 1 1 2 18:1% 8:56 7:00 5:97 7:00 0:9 2 1 2 6:5% 6:51 6:11 4:75 6:37 0:9 3 1 2 3:2% 4:94 5:76 4:25 6:12 0:9 0:5 2 1 18:1% 4:28 3:50 3:08 3:91 0:9 1 2 1 6:5% 3:25 3:05 2:41 3:65 0:9 2 2 1 1:8% 1:97 2:77 1:98 3:47 0:9 3 2 1 0:8% 1:40 2:64 1:80 3:36 0:5 0:5 1 1 17:9% 0:84 0:85 0:48 1:22 0:5 1 1 1 6:3% 0:59 0:83 0:31 1:34 0:5 2 1 1 1:7% 0:33 0:82 0:23 1:39 0:5 3 1 1 0:8% 0:23 0:82 0:21 1:40 0:5 0:5 2 1 11:0% 0:52 0:84 0:38 1:29 0:5 1 2 1 3:4% 0:32 0:83 0:26 1:37 0:5 2 2 1 0:9% 0:17 0:83 0:21 1:40 0:5 3 2 1 0:4% 0:12 0:83 0:21 1:41 Table 4.1: Optimal allocation rule (proportional case) performance variability. On the other hand, as \f becomes smaller, the supplier with a share \f will put more e\u000bort and incur higher e\u000bort cost, which does not substantially bene\ft the buyer overall because this supplier only contributes to a small portion of the buyer's business, but does greatly reduce the supplier's pro\ft. However, the buyer would prefer this since he is maximizing his own payo\u000b and a larger \f will strengthen the suppliers' incentive. Table 4.4 compares the e\u000bectiveness of the three allocation rules by calculating the percentage increase in the buyer's long-run discounted payo\u000b from using a HWTA rule instead of a SWTA rule, and from using the optimal allocation rule instead of a HWTA rule. In all the cases, a HWTA allocation rule is much more e\u000bective than a SWTA one. The buyer's long-run discounted payo\u000b from using a HWTA rule can be more than doubled of that under a SWTA rule. The incentive can be further strengthened by using the optimal but more complex allocation rule, with the improvement generally small for less variable performance measure and large for 84 4.4. Buyer's Problem \r \u001b b m \f v\u0003B V v \u0003(\f) v\u0003(\f) 0:9 0:5 1 1 25:9% 6:05 3:81 3:51 4:11 0:9 1 1 1 11:3% 5:29 3:26 2:74 3:78 0:9 2 1 1 3:5% 3:30 3:01 2:36 3:66 0:9 3 1 1 1:7% 2:31 2:95 2:27 3:63 0:9 0:5 1 2 33:6% 7:82 8:28 7:90 8:67 0:9 1 1 2 18:1% 8:43 7:00 6:17 7:83 0:9 2 1 2 6:5% 6:09 6:20 5:00 7:40 0:9 3 1 2 3:2% 4:44 5:99 4:68 7:30 0:9 0:5 2 1 18:1% 4:21 3:50 3:09 3:92 0:9 1 2 1 6:5% 3:05 3:10 2:50 3:70 0:9 2 2 1 1:8% 1:72 2:95 2:27 3:63 0:9 3 2 1 0:8% 1:18 2:92 2:23 3:62 0:5 0:5 1 1 17:9% 0:83 0:85 0:48 1:22 0:5 1 1 1 6:3% 0:58 0:82 0:30 1:34 0:5 2 1 1 1:7% 0:32 0:81 0:23 1:38 0:5 3 1 1 0:8% 0:22 0:80 0:22 1:39 0:5 0:5 2 1 11:0% 0:51 0:83 0:38 1:29 0:5 1 2 1 3:4% 0:31 0:81 0:26 1:37 0:5 2 2 1 0:9% 0:17 0:81 0:22 1:39 0:5 3 2 1 0:4% 0:11 0:80 0:21 1:40 Table 4.2: HWTA allocation rule (proportional case) 85 4.4. Buyer's Problem \r \u001b b m \f v\u0003B v \u0003(\f) v\u0003(\f) 0:9 0:5 1 1 25:0% 3:21 4:85 4:47 0:9 1 1 1 16:4% 2:52 4:98 4:44 0:9 2 1 1 9:6% 1:71 5:14 4:44 0:9 3 1 1 6:7% 1:29 5:22 4:45 0:9 0:5 1 2 29:5% 4:88 9:59 8:98 0:9 1 1 2 20:5% 4:09 9:82 8:90 0:9 2 1 2 12:7% 2:99 10:12 8:87 0:9 3 1 2 9:1% 2:33 10:31 8:88 0:9 0:5 2 1 20:5% 2:05 4:91 4:45 0:9 1 2 1 12:7% 1:49 5:06 4:44 0:9 2 2 1 7:1% 0:95 5:21 4:45 0:9 3 2 1 4:9% 0:70 5:29 4:46 0:5 0:5 1 1 20:8% 0:47 1:21 0:71 0:5 1 1 1 12:5% 0:35 1:30 0:63 0:5 2 1 1 6:7% 0:22 1:38 0:58 0:5 3 1 1 4:5% 0:16 1:41 0:55 0:5 0:5 2 1 16:4% 0:29 1:25 0:67 0:5 1 2 1 9:3% 0:20 1:34 0:60 0:5 2 2 1 4:8% 0:12 1:41 0:56 0:5 3 2 1 3:2% 0:08 1:43 0:54 Table 4.3: SWTA allocation rule (proportional case) 86 4.4. Buyer's Problem \r \u001b b m HWTA vs SWTA Optimal vs HWTA 0:9 0:5 1 1 88:6% 0:5% 0:9 1 1 1 110:2% 3:7% 0:9 2 1 1 92:7% 10:6% 0:9 3 1 1 79:2% 15:3% 0:9 0:5 1 2 60:5% 0:1% 0:9 1 1 2 106:1% 1:6% 0:9 2 1 2 103:9% 6:8% 0:9 3 1 2 90:7% 11:2% 0:9 0:5 2 1 106:1% 1:6% 0:9 1 2 1 103:8% 6:8% 0:9 2 2 1 81:1% 14:6% 0:9 3 2 1 69:5% 19:1% 0:5 0:5 1 1 76:6% 0:4% 0:5 1 1 1 69:0% 1:1% 0:5 2 1 1 48:7% 2:9% 0:5 3 1 1 39:6% 3:7% 0:5 0:5 2 1 76:7% 0:8% 0:5 1 2 1 58:6% 1:6% 0:5 2 2 1 40:7% 3:6% 0:5 3 2 1 33:8% 4:2% Table 4.4: Comparison between rules (% increase in buyer's payo\u000b) noisier performance measure. Overall, the improvement in the buyer's payo\u000b from using the optimal rule over a HWTA one or from using a HWTA rule over a SWTA one is better for a larger discount factor. Apparently a handicap can be very e\u000bective for incentive provision. 4.4.2 Volume incentive under demand-independent e\u000bort cost In Section 4.4.1 we have studied the optimal allocation rule and two simple WTA rules for the volume incentive design in the case of proportional e\u000bort cost. We have found that the optimal allocation rule is not WTA and is handicapped. Sometimes, a supplier's cost of investment is independent of her share of demand, such as a \fxed investment in technology or innovation. Would the \fnding for the proportional e\u000bort cost still hold for the demand-independent e\u000bort cost? We will answer this question in this section by studying the buyer's problem in the case of g(\f) = 1. 87 4.4. Buyer's Problem Allocation rules We \frst study the buyer's optimal allocation rule for a \fnite-horizon problem, which maximizes his expected payo\u000b over the horizon. We then investigate some simple heuristics for the in\fnite-horizon problem, and use numerical analysis to compare the buyer's expected payo\u000b under di\u000berent heuristics. \u000f Optimal allocation rule for \fnite-horizon problem Before investigating the buyer's optimal allocation rule for an in\fnite-horizon problem, we \frst consider a \fnite-horizon problem with T periods and study the optimal rule for this problem, which is not necessarily a stationary rule. Let \u000bt denote Supplier 1's share in period t, ei\u0003t and x i t denote Supplier i's target and realized performance in period t, and \ft+1 2 (12 ; 1] be the maximum share a supplier can have in period t+ 1. Also let \u0000t = p (\u000bt)2 + (1\u0000 \u000bt)2. Theorem 4.3 provides the optimal allocation rule for the \fnite-horizon problem in the case of demand-independent e\u000bort cost. Theorem 4.3 For supplier's cost function with g(\f) = 1, the buyer's optimal allo- cation rule for a \fnite-horizon problem is a HWTA rule with the form: \f\u000bt+1(x 1 t ; x 2 t ) = ( \ft+1 \u000bt(x 1 t \u0000 e1\u0003t ) > (1\u0000 \u000bt)(x2t \u0000 e2\u0003t ) 1\u0000 \ft+1 \u000bt(x1t \u0000 e1\u0003t ) < (1\u0000 \u000bt)(x2t \u0000 e2\u0003t ) ; (4.13) where e1\u0003t = \r\u0001vt+1(\ft+1)p 2\u0019b\u001b \u000bt \u0000t ; e2\u0003t = \r\u0001vt+1(\ft+1)p 2\u0019b\u001b 1\u0000 \u000bt \u0000t ; (4.14) and \u0001vt+1(\ft+1) = v 1 t+1(\ft+1)\u0000 v1t+1(1\u0000 \ft+1); with v1T+1(\u000bT+1) = \u000bT+1m and v 2 T+1(1 \u0000 \u000bT+1) = (1 \u0000 \u000bT+1)m. Under this rule, f(e1\u0003t ; e2\u0003t )gt2f1;2;:::Tg constitutes a subgame perfect Nash equilibrium. \u000f Simple heuristic for in\fnite-horizon problem 88 4.4. Buyer's Problem Since Theorem 4.3 has demonstrated that the optimal allocation rule for each period of a \fnite horizon takes the same form of a HWTA, we would expect the buyer's optimal stationary rule for an in\fnite horizon is also a HWTA one. Nonetheless, we shouldn't immediately jump at the conclusion that the optimal rule for an in\fnite horizon takes the same form as the optimal one for a \fnite horizon. Noting that with a bang-bang type of allocation rule, in the \fnite-horizon problem, the choice of the optimal maximum share for a period does not a\u000bect those optimal maximum shares and thus the suppliers' optimal payo\u000bs in the subsequent periods; but in the in\fnite-horizon problem with a stationary allocation rule, the optimal maximum share for each period changes simultaneously. In the following analysis, we come back to the in\fnite-horizon problem. As the search for the optimal (WTA) rule is not straightforward, we shall demonstrate the key insights by examining WTA rules with the form \f4\u0003\u000b (x1; x2) = ( b\f x1 > k\u000bx2 + \u0012 1\u0000 b\f x1 < k\u000bx2 + \u0012\u000b ; (4.15) where \u0012\u000b is to be determined, k\u000b 2 (0; 1] for \u000b \u0015 0:5 and k\u000b \u0015 1 for \u000b < 0:5, both \u0012\u000b and k\u000b are functions of \u000b, and b\f 2 (12 ; 1]. Let \u0007\u000b = p1 + k2\u000b. Theorem 4.4 provides the optimal \u0012\u000b and the value functions of the buyer and the suppliers under the allocation rule (4.15). Theorem 4.4 For supplier's cost function with g(\f) = 1, among the family of WTA allocation rules as de\fned in (4.15), the optimal rule has \u0012\u0003\u000b = e \u0003 1 \u0000 k\u000be\u00032 for any \u000b, where e\u00031 and e \u0003 2 are the optimal target performance levels of suppliers 1 and 2 under this optimal allocation rule, e\u00031 = \r\u0001v\u0003(b\f)p 2\u0019b\u001b 1p k2\u000b + 1 ; (4.16) e\u00032 = \r\u0001v\u0003(b\f)p 2\u0019b\u001b k\u000bp k2\u000b + 1 ; (4.17) with \u0001v\u0003(b\f) = 8<: 2\u0019b\u001b2(1+k2b\f) \r2(1\u0000k2b\f) ( r 1 + 2m(2b\f\u00001) \u0019b\u001b2 1\u0000k2b 1+k2b\f \u0000 1) kb\f 6= 1 m(2b\f \u0000 1) kb\f = 1 : (4.18) 89 4.4. Buyer's Problem \u0012\u0003\u000b 6= 0 for k\u000b 6= 1, and \u0012\u0003\u000b = 0 for k\u000b = 1. The suppliers' value functions v\u00031(\u000b) = v \u0003 2(\u000b) = v \u0003(\u000b) for any \u000b due to symmetry, v\u0003(\u000b) = m\u000b\u0000 (\r\u0001v \u0003(b\f))2 4\u0019b\u001b2(k2\u000b + 1) + 2 (v\u0003(b\f) + v\u0003(1\u0000 b\f)); (4.19) and the buyer's value function is v\u0003B(\u000b) = p 2\u0019\u001b(1 + k2b\f) \r(1\u0000 \r)\u0007\u000b(1\u0000 k2b\f) [\u000b+ k\u000b(1\u0000 \u000b)]( vuut1 + \r2m(2b\f \u0000 1) \u0019b\u001b2 1\u0000 k2b 1 + k2b\f \u0000 1): Let yi = xi\u0000e\u0003i \u001b measure the standardized deviation of Supplier i's performance from the target. Then (4.15) with \u0012\u0003\u000b = e \u0003 1 \u0000 k\u000be\u00032 becomes b\f4\u0003\u000b (y1; y2) = ( b\f y1 > k\u000by2 1\u0000 b\f y1 < k\u000by2 : Under the allocation rule (4.15) with \u0012\u0003\u000b = e \u0003 1 \u0000 k\u000be\u00032, at the optimal e\u000bort levels, each supplier has equal chance to be the winner, but the whole (y1; y2) plane is split into two equal areas by the straight line y1 = k\u000by2 instead of the 45 \u000e line. From (4.16), (4.17) and (4.19), noting that the expected future payo\u000b of the supplier with share is 2 (v\u0003(b\f)+v\u0003(1\u0000b\f)), we can see that the allocation rule has an interesting property that it makes the two suppliers' expected future payo\u000bs identical (the two suppliers have equal chance to be the winner) but provides the suppliers with di\u000berent level of incentive if k\u000b 6= 1. When k\u000b = 1, due to the demand-independent e\u000bort cost, the suppliers' optimal e\u000bort levels are equal. The allocation rule (4.15) with k\u000b = 1 is a SWTA one. We are particularly interested in two special cases of \f4\u0003\u000b (x1; x2), where k\u000b = 1\u0000 or 1. Corollaries 4.4 and 4.5 follow from Theorem 4.4 by letting k\u000b = 1\u0000 and 1 in (4.15), respectively. The e\u000bect of k\u000b 6= 1 will be seen from the subsequent numerical analysis results. Corollary 4.4 Under the allocation rule \f4\u0003\u000b (x1; x2) with k\u000b = 1\u0000 , the optimal rule 90 4.4. Buyer's Problem has \u0012\u0003\u000b = \r\u0001v\u0003(b\f)p 2\u0019b\u001b 2\u000b\u0000 1 qb\f2 + (1\u0000 b\f)2 ; under this rule e\u0003\f = 1\u0000\fe \u0003 1\u0000\f, and the optimal \u0003 is a boundary solution. Corollary 4.5 Under the allocation rule \f4\u0003\u000b (x1; x2) with k\u000b = 1, the optimal rule has \u0012\u0003\u000b = 0 for any \u000b, and under this rule e \u0003 1 = e \u0003 2; the optimal \u0003 is a boundary solution. Numerical analysis We compare the e\u000bectiveness of two heuristics: a SWTA rule, and a HWTA one with k\u000b = 1\u0000 . For all the cases but one in the numerical analysis here, the limit to a supplier's maximum share is in fact 1 (the buyer's total business) because both the Nash equilibrium condition and the suppliers' participation constraints set very loose boundaries to a supplier's share to be mathematically more than 1. In reality, there is often a minimum order quantity speci\fed by a supplier. Therefore, in the analysis below, we impose a minimum share of 10% on each supplier. Table 4.5 compares the payo\u000bs of the buyer and the suppliers under both rules. It is noted that for the case where \r = 0:9; \u001b = 0:5; b = 1 and m = 2, a supplier's minimum share is limited to 12% due to the Nash equilibrium condition, which causes the HWTA rule to work worse than the SWTA one. In all the other cases where the same minimum share applies to both rules, the HWTA rule provides better incentive in terms of the buyer's long-run discounted payo\u000b. This is because the performance of the supplier with a larger share of business is more important to the buyer, and thus the buyer would often like to ensure that supplier to perform well in order to obtain a high aggregate performance of the two suppliers. For this purpose, the buyer can provide a stronger incentive to the supplier with a larger share by giving that supplier a higher award for performing above the expectation. Here a simple way to achieve this is to let k\u000b = 1\u0000 , so that one unit of extra e\u000bort will be worth \u000b units of overperformance for the supplier with a share of \u000b, compared to 1\u0000 \u000b units for the other supplier. However, this HWTA allocation rule may not always work better than a SWTA rule, as shown in Table 4.6. Here for the case where \r = 0:9; \u001b = 0:5; b = 1 andm = 3, under both rules, the Nash equilibrium condition sets a boundary to a supplier's minimum share, which is tighter under the HWTA rule. The \frst comparison of 91 4.4. Buyer's Problem SWTA HWTA \r \u001b b m v\u0003B V v \u0003(\f) v\u0003(\f) v\u0003B V v \u0003(\f) v\u0003(\f) %increase 0:9 0:5 1 1 4:06 4:17 4:57 3:77 4:44 4:40 4:74 4:06 9:3% 0:9 1 1 1 2:03 4:79 5:19 4:39 2:48 4:81 5:19 4:43 22:2% 0:9 2 1 1 1:02 4:95 5:35 4:55 1:28 4:95 5:34 4:55 26:5% 0:9 3 1 1 0:68 4:98 5:38 4:58 0:86 4:96 5:38 4:55 27:4% 0:9 0:5 1 2 8:12 6:70 7:50 5:90 7:45 8:23 8:82 7:65 \u00008:4% 0:9 1 1 2 4:06 9:17 9:97 8:37 4:76 9:31 10:04 8:58 17:3% 0:9 2 1 2 2:03 9:79 10:59 8:99 2:54 9:80 10:58 9:02 25:0% 0:9 3 1 2 1:35 9:91 10:71 9:11 1:72 9:91 10:70 9:12 26:7% 0:9 0:5 2 1 2:03 4:59 4:99 4:19 2:38 4:65 5:02 4:29 17:3% 0:9 1 2 1 1:02 4:90 5:30 4:50 1:27 4:90 5:29 4:51 25:0% 0:9 2 2 1 0:51 4:97 5:37 4:57 0:65 4:97 5:37 4:58 27:3% 0:9 3 2 1 0:34 4:99 5:39 4:59 0:43 4:99 5:39 4:59 27:7% 0:5 0:5 1 1 0:45 0:95 1:35 0:55 0:55 0:95 1:33 0:58 21:0% 0:5 1 1 1 0:23 0:99 1:39 0:59 0:28 0:99 1:38 0:59 26:1% 0:5 2 1 1 0:11 1:00 1:40 0:60 0:14 1:00 1:40 0:60 27:6% 0:5 3 1 1 0:08 1:00 1:40 0:60 0:10 1:00 1:40 0:60 27:4% 0:5 0:5 2 1 0:23 0:97 1:37 0:57 0:28 0:98 1:36 0:59 24:3% 0:5 1 2 1 0:11 0:99 1:39 0:59 0:14 0:99 1:39 0:60 27:1% 0:5 2 2 1 0:06 1:00 1:40 0:60 0:07 1:00 1:40 0:60 27:8% 0:5 3 2 1 0:04 1:00 1:40 0:60 0:05 1:00 1:40 0:60 28:0% Table 4.5: Comparison of SWTA and HWTA (independent case) \f v\u0003B V v \u0003(\f) v\u0003(\f) HWTM vs SWTM SWTA 11:7% 11:66 8:20 9:35 7:06 \u000038:1% 23:5% 8:09 11:73 12:53 10:93 \u000010:7% HWTA 23:5% 7:22 12:97 13:60 12:34 Table 4.6: Comparison of SWTA and HWTA (independent case) 92 4.5. Conclusions and Future Work these two rules is by using the individual minimum share, 11:7% for the SWTA rule and 23:5% for the HWTA one, which shows a 38:1% lower of the buyer's long-run discounted payo\u000b under the HWTA rule than that under the SWTA rule. The second comparison is by using the same minimum share, 23:5%, which still shows a 10:8% lower of the buyer's payo\u000b under the HWTA rule. This is because the two suppliers' incentives also come from \u0001v\u0003(b\f), the gain in a supplier's payo\u000b from a high starting share b\f instead of a low one 1 \u0000 b\f. The HWTA rule induces more e\u000bort from a supplier with a larger share, which results in higher e\u000bort cost and smaller \u0001v\u0003(b\f); while the SWTA rule always induces equal supplier e\u000bort and a \u0001v\u0003(b\f) independent of b\f. Therefore, in some situations a non-handicapped SWTA rule can perform better than the HWTA rule with k\u000b = 1\u0000 . 4.5 Conclusions and Future Work In this chapter, we have studied the design of performance-based volume incentive schemes, a type of incentive scheme widely used in practice but not well studied yet in the literature. We have considered a buyer repeatedly outsourcing a service or a product from two suppliers, under the assumptions of risk neutral parties, reliable and uncapacitated supply, no setup fee, zero switching cost, and unobservable sup- plier e\u000bort. We have focused on two types of supplier e\u000bort cost: the proportional case where the e\u000bort cost is proportional to the share of demand, and the demand- independent case where the e\u000bort cost is independent of the share. We have found that to maintain suppliers' competition over time, the optimal demand allocation rule is dependent on the suppliers' current shares, and is not a simple rank-order tourna- ment. Even when the supplier's e\u000bort cost is demand independent, each supplier's current share of business still plays an important role in the allocation rule, and using a handicap can greatly improve the e\u000eciency of volume incentive schemes. Handi- capping plays two roles in incentive provision. It can level the \feld and enhance the suppliers' competition when the `more important' supplier is at a disadvantage in the competition, as seen in the case of proportional e\u000bort cost. Alternatively, it gives the `more important' supplier an advantage and makes that supplier work harder than the other supplier, as seen in the case of independent e\u000bort cost. 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The United States. 192 and 206. Appendix A Proof for Chapter 2 Proof of Proposition 2.1 We \frst compute the variance of \u0011WR . For L > 0 and 0 \u0014 W \u0014 L, E(\u0011WR ) = RX t=1 E(XWt ) = RFL+1\u0000W (S) E((\u0011 W R ) 2) = E[( RX i=1 XWi )( RX j=1 XWj )] = RX i=1 RX j=1 E(XWi X W j ) = RX i=1 RX j=1 E(1fD[i\u0000 L; i+ 1\u0000W ) \u0014 Sg \u0002 1fD[j \u0000 L; j + 1\u0000W ) \u0014 Sg) The number of (i; i) terms is R, and the sum of these terms is M1 = RE(1fD(L+ 1\u0000W ) \u0014 Sg) = RFL+1\u0000W (S). The number of (i; j) terms with XWi and X W j independent is 2 R\u0000(L+1\u0000W )X i=1 i = R\u0000(L+1\u0000W )X i=1 i+ R\u0000(L+1\u0000W )X j=1 (R\u0000 L+W \u0000 j) = R\u0000(L+1\u0000W )X i=1 (i+R\u0000 L+W \u0000 i) = (R\u0000 L+W )(R\u0000 L+W \u0000 1). This is the number of pairs of XWi and X W j which di\u000ber by at least L + 1 \u0000W periods. The sum of these terms is M2 = (R\u0000L+W )(R\u0000L+W\u00001)E(1fD(L+1\u0000W ) \u0014 Sg)E(1fD(L+1\u0000W ) \u0014 Sg) = (R\u0000 L+W )(R\u0000 L+W \u0000 1)(FL+1\u0000W (S))2. The number of (i; j) terms with n (1 \u0014 n \u0014 L\u0000W ) periods of demands in common is 2(R\u0000 (L+ 1) + n+W ). This is the number of pairs of XWi and XWj which di\u000ber by exactly L+ 1\u0000 n\u0000W periods. For i < j, j = i+ L+ 1\u0000 n\u0000W , E(XWi X W j ) = PrfD[i\u0000 L; i+ 1\u0000W ) \u0014 S;D[j \u0000 L; j + 1\u0000W ) \u0014 Sg = PrfD[i\u0000 L; j \u0000 L) +D(n) \u0014 S;D(n) +D[i+ 1\u0000W; j + 1\u0000W ) \u0014 Sg 100 Appendix A. Proof for Chapter 2 = Z S 0 (PrfD(L+ 1\u0000 n\u0000W ) \u0014 S \u0000 xg)2dFn(x) = Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x) Cov(D[i\u0000 L; i+ 1\u0000W ); D[j \u0000 L; j + 1\u0000W )) = Cov(D[i\u0000L; j\u0000L)+D[j\u0000L; i+1\u0000W ); D[j\u0000L; i+1\u0000W )+D[i+1\u0000W; j+1\u0000W )) = V ar(D(n)) > 0 and increases with n ) D[i\u0000 L; i+ 1\u0000W ) and D[j \u0000 L; j + 1\u0000W ) are positively correlated, and Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x) increases with n ) E(XWi X W j ) = PrfD[i \u0000 L; i + 1 \u0000W ) \u0014 SgPrfD[j \u0000 L; j + 1 \u0000W ) \u0014 SjD[i \u0000 L; i+ 1\u0000W ) \u0014 Sg \u0015 FL+1\u0000W (S) PrfD[j \u0000 L; j + 1\u0000W ) \u0014 Sg = (FL+1\u0000W (S))2 So E(XWi X W j ) \u0015 (FL+1\u0000W (S))2: (A.1) The sum of the terms with each pair di\u000bering by exactly L+1\u0000n\u0000W periods is M3 = L\u0000WX n=1 2(R\u0000 (L+ 1) + n+W ) Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x). So E((\u0011WR ) 2) =M1 +M2 +M3, and (\u001bWR ) 2 = V ar(\u0011WR ) = E((\u0011 W R ) 2)\u0000 (E(\u0011WR ))2 = RFL+1\u0000W (S) + (R\u0000 L+W )(R\u0000 L+W \u0000 1)(FL+1\u0000W (S))2 + L\u0000WX n=1 2(R\u0000 (L+ 1) + n+W ) Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x)\u0000R2(FL+1\u0000W (S))2 = RFL+1\u0000W (S)\u0000 [(L\u0000W )(2R\u0000 L+W \u0000 1) +R](FL+1\u0000W (S))2 + L\u0000WX n=1 2(R\u0000 (L+ 1) + n+W ) Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x): By (A:1), (\u001bWR ) 2 \u0015 RFL+1\u0000W (S)\u0000 [(L\u0000W )(2R\u0000L+W \u0000 1) +R](FL+1\u0000W (S))2 + L\u0000WX n=1 2(R\u0000 (L+ 1) + n+W )(FL+1\u0000W (S))2 = RFL+1\u0000W (S)(1\u0000 FL+1\u0000W (S)): limR!1 (\u001bWR ) 2 R2=3 \u0015 limR!1R1=3FL+1\u0000W (S)(1\u0000 FL+1\u0000W (S)) =1: The sequence fXtg is (L+ 1\u0000W )-dependent because any subsequence fXtj ; j \u0015 1g \u001a fXtg, with tj + L+ 1\u0000W < tj+1 for every j \u0015 1, is a sequence of independent random variables. Moreover, Xt \u0014 1 for all t. 101 Appendix A. Proof for Chapter 2 Applying Theorem 7.3.1 (Chung 1974 page 214), \u0011WR \u0000E(\u0011WR ) \u001bWR converges in distribu- tion to a standard normal random variable Z, Z \u0018 N(0; 1) as R approaches 1. Proof of Proposition 2.2 V ar(AWR ) = (\u001bWR ) 2 R2 ; V ar(AWR+1)\u0000 V ar(AWR ) = (\u001b W R+1) 2 (R+1)2 \u0000 (\u001bWR )2 R2 = \u0000( 1 R \u0000 1 R+1 )FL+1\u0000W (S) \u0000 [(L \u0000 W )( 2R+1 \u0000 2R + L+1\u0000WR2 \u0000 L+1\u0000W(R+1)2 ) + 1R+1 \u0000 1 R ](FL+1\u0000W (S))2 \u00002 L\u0000WX n=1 ( 1 R \u0000 1 R+1 +L+1\u0000n\u0000W (R+1)2 \u0000L+1\u0000n\u0000W R2 ) Z S 0 (FL+1\u0000n\u0000W (S\u0000x))2dFn(x) = \u0000( 1 R \u0000 1 R+1 )FL+1\u0000W (S)(1\u0000 FL+1\u0000W (S)) \u00002 L\u0000WX n=1 ( 1 R \u0000 1 R+1 +L+1\u0000n\u0000W (R+1)2 \u0000L+1\u0000n\u0000W R2 )( Z S 0 (FL+1\u0000n\u0000W (S\u0000x))2dFn(x)\u0000(FL+1\u0000W (S))2) = \u0000( 1 R \u0000 1 R+1 )FL+1\u0000W (S)(1\u0000 FL+1\u0000W (S))\u0000\u0007 where \u0007 = 2 L\u0000WX n=1 R(R+1)\u0000(2R+1)(L+1\u0000n\u0000W ) R2(R+1)2 ( Z S 0 (FL+1\u0000n\u0000W (S\u0000x))2dFn(x)\u0000(FL+1\u0000W (S))2). Let \u0001n = R(R + 1)\u0000 (2R + 1)(L+ 1\u0000 n\u0000W ). \u0001n is increasing in n. By the assumption R > L, R \u0015 L+ 1, so \u00011 = R(R+1)\u0000 (2R+1)(L\u0000W ) can be negative when R is small, for example, R = L+ 1; \u0001L \u0015 R2 \u0000R\u0000 1 > 0 because R > 1: Let mn = Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x)\u0000 (FL+1\u0000W (S))2. From the proof of Proposition 2.1, mn > 0 and Z S 0 (FL+1\u0000n\u0000W (S \u0000 x))2dFn(x) increases with n. So if \u00011 < 0, let n = maxfnj\u0001n < 0g, \u0007 \u0015 2mn L\u0000WX n=1 R(R+1)\u0000(2R+1)(L+1\u0000n\u0000W ) R2(R+1)2 = mn(L\u0000W ) (2R+2)R\u0000(2R+1)(L+1\u0000W )R2(R+1)2 \u0015 0; if \u00011 \u0015 0, then \u0007 \u0015 0. So V ar(AWR+1)\u0000 V ar(AWR ) < 0, V ar(AWR ) is decreasing in R. Proof of Proposition 2.3 Lump-sum penalty SLA, L =W = 0 : optimal K given V0(S) = hE[S \u0000 D(1)]+ + KR R\u000bX i=0 Prf\u00110R = ijSg, where Prf\u00110R = ijSg is given by (2.5). 102 Appendix A. Proof for Chapter 2 1. If the demand and S are continuous, then dV0(S) dS = hF1(S) + K R R\u000bX i=0 \u0000 R i \u0001 [i(F1(S)) i\u00001(1 \u0000 F1(S))R\u0000i \u0000 (R \u0000 i)(F1(S))i(1 \u0000 F1(S)) R\u0000i\u00001]f1(S) = hF1(S)\u0000K \u0000 R\u00001 R \u0001 (F1(S)) R\u000b(1\u0000 F1(S))R(1\u0000\u000b)\u00001f1(S) gives dV0(S) dS = F1(S)[h\u0000K \u0012 R\u0000 1 R \u0013 (F1(S)) R\u000b\u00001(1\u0000 F1(S))R(1\u0000\u000b)\u00001f1(S)]: (A.2) Given any \u000b < F1(S \u0003), dV0(S \u0003) dS = 0)the optimal K\u0003(\u000b) = h\u0000 R\u00001 R \u0001 (F1(S\u0003))R\u000b\u00001(1\u0000 F1(S\u0003))R(1\u0000\u000b)\u00001f1(S\u0003) ; (A.3) where f1(\u0001) is the pdf of single-period demand. 2. If the demand and S are discrete, then given any \u000b, to induce the supplier to choose S\u0003, K\u0003(\u000b) should be chosen such that V0(S \u0003 + 1)\u0000 V0(S\u0003) \u0015 0 and V0(S\u0003 \u0000 1)\u0000 V0(S\u0003) \u0015 0: (A.4) Because E[S \u0000 d]+ = SX d=0 (S \u0000 d)f1(d) and f1(d) = PrfD(1) = dg, (A:4)) hF1(S \u0003) + K R R\u000bX i=0 (Prf\u00110R = ijS\u0003 + 1g \u0000 Prf\u00110R = ijS\u0003g) \u0015 0; and \u0000hF1(S\u0003 \u0000 1) + KR R\u000bX i=0 (Prf\u00110R = ijS\u0003 \u0000 1g \u0000 Prf\u00110R = ijS\u0003g) \u0015 0: It can be seen from (A:2) that R\u000bX i=0 Prf\u00110R = ijSg is decreasing in S, so R\u000bX i=0 (Prf\u00110R = ijS\u0003g \u0000 Prf\u00110R = ijS\u0003 + 1g) > 0; and R\u000bX i=0 (Prf\u00110R = ijS\u0003 \u0000 1g \u0000 Prf\u00110R = ijS\u0003g) > 0. So the interval of optimal K\u0003(\u000b) is [K\u0003(\u000b); K \u0003 (\u000b)] = [ RhF1(S \u0003\u00001) R\u000bX i=0 (Prf\u00110R=ijS\u0003\u00001g\u0000Prf\u00110R=ijS\u0003g) ; RhF1(S \u0003) R\u000bX i=0 (Prf\u00110R=ijS\u0003g\u0000Prf\u00110R=ijS\u0003+1g) ]. 103 Appendix A. Proof for Chapter 2 Lump-sum penalty SLA, L > 0;0 \u0014 W \u0014 L : optimal K given Let zi(x) = i+0:5\u0000RFL+1\u0000W (x) \u001bWR . From (2.3) and (2.6), VL(S) = hE[S \u0000D(L+ 1\u0000W )]+ + KR\b(zR\u000b(S)): 1. If the demand and S are continuous, then dVL(S) dS = hFL+1\u0000W (S) + KR\u001e(zR\u000b(S)) dzR\u000b(S) dS : Letting dVL(S \u0003) dS = 0 we can obtain K\u0003(\u000b) = \u0000 RhFL+1\u0000W (S\u0003) \u001e(zR\u000b(S)) dzR\u000b(S) dS . 2. If the demand and S are discrete, given any \u000b, to induce the supplier to choose S\u0003, K\u0003(\u000b) should be chosen such that VL(S \u0003 + 1)\u0000 VL(S\u0003) \u0015 0 and VL(S\u0003 \u0000 1)\u0000 VL(S\u0003) \u0015 0) hFL+1\u0000W (S\u0003) + KR [\b(zR\u000b(S \u0003 + 1))\u0000 \b(zR\u000b(S\u0003))] \u0015 0; and \u0000hFL+1\u0000W (S\u0003 \u0000 1) + KR [\b(zR\u000b(S\u0003 \u0000 1))\u0000 \b(zR\u000b(S\u0003))] \u0015 0: It follows that [K\u0003(\u000b); K \u0003 (\u000b)] = [ RhFL+1\u0000W (S \u0003\u00001) \b(zR\u000b(S\u0003\u00001))\u0000\b(zR\u000b(S\u0003)) ; RhFL+1\u0000W (S\u0003) \b(zR\u000b(S\u0003))\u0000\b(zR\u000b(S\u0003+1)) ]: Linear penalty SLA, L =W = 0 : optimal K given V0(S) = hE[S \u0000D(1)]+ + KR R\u000bX i=0 (R\u000b + 1\u0000 i) Prf\u00110R = ijSg, where Prf\u00110R = ijSg is given by (2.5). 1. If the demand and S are continuous, then dV0(S) dS = hF1(S)+ K R R\u000bX i=0 (R\u000b+1\u0000i)\u0000R i \u0001 [i(F1(S)) i\u00001(1\u0000F1(S))R\u0000i\u0000(R\u0000i)(F1(S))i(1\u0000 F1(S)) R\u0000i\u00001]f1(S)) dV0(S) dS = hF1(S)\u0000K R\u000bX i=0 \u0012 R\u0000 1 i \u0013 (F1(S)) i(1\u0000 F1(S))R\u0000i\u00001f1(S): (A.5) Given \u000b < F1(S \u0003), dV0(S \u0003) dS = 0) K\u0003(\u000b) = hF1(S \u0003) R\u000bX i=0 \u0000 R\u00001 i \u0001 (F1(S\u0003))i(1\u0000 F1(S\u0003))R\u0000i\u00001f1(S\u0003) : (A.6) 2. If the demand and S are discrete, then given any \u000b, to induce the supplier to choose S\u0003, K\u0003(\u000b) should be chosen such that 104 Appendix A. Proof for Chapter 2 V0(S \u0003 + 1)\u0000 V0(S\u0003) \u0015 0 and V0(S\u0003 \u0000 1)\u0000 V0(S\u0003) \u0015 0) hF1(S \u0003) + K R R\u000bX i=0 (R\u000b + 1\u0000 i)(Prf\u00110R = ijS\u0003 + 1g \u0000 Prf\u00110R = ijS\u0003g) \u0015 0; (A.7) and \u0000hF1(S\u0003\u0000 1) + K R R\u000bX i=0 (R\u000b+ 1\u0000 i)(Prf\u00110R = ijS\u0003\u0000 1g\u0000Prf\u00110R = ijS\u0003g) \u0015 0: (A.8) It can be seen from (A:5) that the second term in V0(S) is decreasing in S, so R\u000bX i=0 (R\u000b + 1\u0000 i)(Prf\u00110R = ijS\u0003g \u0000 Prf\u00110R = ijS\u0003 + 1g) > 0; and R\u000bX i=0 (R\u000b + 1\u0000 i)(Prf\u00110R = ijS\u0003 \u0000 1g \u0000 Prf\u00110R = ijS\u0003g) > 0: Then it follows from (A:7) and (A:8) that [K\u0003(\u000b); K \u0003 (\u000b)] = [ RhF1(S \u0003\u00001) R\u000bX i=0 (R\u000b+1\u0000i)(Prf\u00110R=ijS\u0003\u00001g\u0000Prf\u00110R=ijS\u0003g) ; RhF1(S \u0003) R\u000bX i=0 (R\u000b+1\u0000i)(Prf\u00110R=ijS\u0003g\u0000Prf\u00110R=ijS\u0003+1g) ]. Linear penalty SLA, L > 0;0 \u0014 W \u0014 L : optimal K given VL(S) = hE[S \u0000D(L+ 1\u0000W )]+ + KR R\u000bX i=0 (R\u000b+ 1\u0000 i) Prf\u0011WR = ijSg 1. If the demand and S are continuous, then using the results in Proposition 2.1, the approximation for VL(S) is VL(S) = hE[S \u0000D(L+ 1\u0000W )]+ + KR Z x\u0014R (R\u000b\u0000 x)d\b(x\u0000RFL+1\u0000W (S) \u001bWR ) = hE[S \u0000D(L+ 1\u0000W )]+ + K R Z x\u0014R \b(x\u0000RFL+1\u0000W (S) \u001bWR )dx ) dVL(S) dS = hFL+1\u0000W (S) + KR Z x\u0014R d dS \b(x\u0000RFL+1\u0000W (S) \u001bWR )dx = hFL+1\u0000W (S) + KR [\u0000RfL+1\u0000W (S)\b(R\u000b\u0000RFL+1\u0000W (S)\u001bWR ) + d\u001bWR dS \u001e(R\u000b\u0000RFL+1\u0000W (S) \u001bWR )]: Letting dVL(S \u0003) dS = 0 we can obtain K\u0003(\u000b) = RhFL+1\u0000W (S \u0003) RfL+1\u0000W (S)\b( R\u000b\u0000RFL+1\u0000W (S\u0003) \u001bW R )\u0000 d\u001b W R dS \u001e( R\u000b\u0000RFL+1\u0000W (S\u0003) \u001bW R ) , where \u001bWR is given by (2.1). 2. If the demand and S are discrete, given any \u000b, to induce the supplier to choose 105 Appendix A. Proof for Chapter 2 S\u0003, K\u0003(\u000b) should be chosen such that VL(S \u0003 + 1)\u0000 VL(S\u0003) \u0015 0 and VL(S\u0003 \u0000 1)\u0000 VL(S\u0003) \u0015 0 ) hFL+1\u0000W (S\u0003) + KR R\u000bX i=0 (R\u000b + 1\u0000 i)(Prf\u0011WR = ijS\u0003 + 1g \u0000 Prf\u0011WR = ijS\u0003g) \u0015 0; and \u0000hFL+1\u0000W (S\u0003\u00001)+ KR R\u000bX i=0 (R\u000b+1\u0000 i)(Prf\u0011WR = ijS\u0003\u00001g\u0000Prf\u0011WR = ijS\u0003g) \u0015 0: So [K\u0003(\u000b); K \u0003 (\u000b)] = [ RhFL+1\u0000W (S \u0003\u00001) R\u000bX i=0 (R\u000b+1\u0000i)(Prf\u0011WR =ijS\u0003\u00001g\u0000Prf\u0011WR =ijS\u0003g) ; RhFL+1\u0000W (S \u0003) R\u000bX i=0 (R\u000b+1\u0000i)(Prf\u0011WR =ijS\u0003g\u0000Prf\u0011WR =ijS\u0003+1g) ], where Prf\u0011WR = ijS\u0003g is given by (2.6). Proposition 2.3 follows from all the above derivations for (\u000b;K) candidates. Derivation for Section 2.5.2 Unimodality of supplier's objective function The derivation below is based on continuous-valued demand and base-stock level S. Lump-sum penalty SLA, L = 0: Substituting (A:3) into (A:2)) at \u000b and K\u0003(\u000b), dV0(S) dS = hF1(S)(1\u0000 (F1(S))R\u000b\u00002(1\u0000F1(S))R(1\u0000\u000b)f1(S)(F1(S\u0003))R\u000b\u00002(1\u0000F1(S\u0003))R(1\u0000\u000b)f1(S\u0003)) d[(F1(S))R\u000b\u00002(1\u0000F1(S))R(1\u0000\u000b)f1(S)] dS = (F1(S)) R\u000b\u00003(1\u0000F1(S))R(1\u0000\u000b)\u00001[(f1(S))2(R\u000b\u00002\u0000 (R\u0000 2)F1(S)) + f 01(S)F1(S)(1\u0000 F1(S))]: \u000b < F1(S \u0003); F 01(S) > 0; 0 \u0014 F1(S) \u0014 1 ) R\u000b\u0000 2\u0000 (R\u0000 2)F1(S) = R(\u000b\u0000 F1(S))\u0000 2(1\u0000 F1(S)) < 0 for S \u0015 S\u0003; moreover, R\u000b\u00002 R\u00002 < \u000b < F1(S \u0003); F 01(S) > 0) F\u000011 (R\u000b\u00002R\u00002 ) < S\u0003 ) R\u000b\u0000 2\u0000 (R\u0000 2)F1(S) < 0 for F\u000011 (R\u000b\u00002R\u00002 ) < S < S\u0003. Because it is assumed that the distribution of single-period demand is unimodal, if f 01(S \u0003) < 0, then f 01(S) < 0 for S \u0015 S\u0003. This is true for distributions in the location- scale family and Poisson distribution with S\u0003 > \u0015. Because f1(S) is continuous, for S > S\u0003, d[(F1(S)) R\u000b\u00002(1\u0000F1(S))R(1\u0000\u000b)f1(S)] dS < 0 and dV0(S) dS > 0; dV0(S \u0003) dS = 0, and for S < S\u0003 and S close to S\u0003, d[(F1(S)) R\u000b\u00002(1\u0000F1(S))R(1\u0000\u000b)f1(S)] dS < 0, and so dV0(S) dS < 0. Thus S\u0003 is a local optimum. 106 Appendix A. Proof for Chapter 2 But V0(S) may be not unimodal. For small S and S < S \u0003, d[(F1(S)) R\u000b\u00002(1\u0000F1(S))R(1\u0000\u000b)f1(S)] dS > 0. Depending on the value of S\u0003 and R\u000b, it is possible that for S < S\u0003, dV0(S) dS > 0. This can be seen from Figure 2.1. Linear penalty SLA, L = 0 : Substituting (A:6) into (A:5)) at \u000b and K\u0003(\u000b), dV0(S) dS = h[F1(S)\u0000 F1(S\u0003) R\u000b\u00001X i=0 (R\u00001i )(F1(S))i(1\u0000F1(S))R\u0000i\u00001f1(S) R\u000b\u00001X i=0 (R\u00001i )(F1(S\u0003))i(1\u0000F1(S\u0003))R\u0000i\u00001f1(S\u0003) ]. Let \u00031 = R\u000b\u00001X i=0 \u0000 R\u00001 i \u0001 (F1(S)) i(1\u0000 F1(S))R\u0000i\u00001. d\u00031 dS = \u0000(R\u0000 1) \u0012 R\u0000 2 R\u000b\u0000 1 \u0013 (F1(S)) R\u000b\u00001(1\u0000 F1(S))R(1\u0000\u000b)\u00001f1(S) < 0: (A.9) Using the same assumption for the case of lump-sum penalty SLA and L = 0, f 01(S \u0003) < 0, 1. for S > S\u0003, F1(S) > F1(S\u0003) and f1(S) < f1(S\u0003), then together with (A:9), dV0(S) dS > 0 for S > S\u0003; 2. for S < S\u0003, F1(S) < F1(S\u0003), and by (A:9), R\u000b\u00001X i=0 \u0012 R\u0000 1 i \u0013 (F1(S \u0003))i(1\u0000 F1(S\u0003))R\u0000i\u00001 < R\u000b\u00001X i=0 \u0012 R\u0000 1 i \u0013 (F1(S)) i(1\u0000 F1(S))R\u0000i\u00001; (A.10) for S close to S\u0003, because f 01(S) is continuous, f 0 1(S) < 0 and f1(S) > f1(S \u0003), so dV0(S) dS < 0 for S < S\u0003 and close to S\u0003; for S < S\u0003 and not close to S\u0003, dV0(S) dS < 0 if S satis\fes f1(S \u0003) F1(S\u0003) R\u000b\u00001X i=0 \u0012 R\u0000 1 i \u0013 (F1(S \u0003))i(1\u0000 F1(S\u0003))R\u0000i\u00001 < f1(S) F1(S) R\u000b\u00001X i=0 \u0012 R\u0000 1 i \u0013 (F1(S)) i(1\u0000 F1(S))R\u0000i\u00001: (A.11) The above inequality depends on the value of S\u0003 and R\u000b. Because 1 F1(S\u0003) < 1 F1(S) and (A:10), if S\u0003 is su\u000eciently large so that f1(S) f1(S\u0003) is not too small, then (A:11) will 107 Appendix A. Proof for Chapter 2 hold. Therefore, S\u0003 is a local optimum. If (A:11) holds for all S < S\u0003, then V0(S) is unimodal and S\u0003 is a global optimum. Dynamic program for L = 0 (Section 2.6.1) Compared to the case of L = 1, the de\fnition of It is modi\fed as It = the supplier's inventory on hand at the beginning of period t, before an order is placed in period t. Because the supplier does not incur the backorder cost, for any nonpositive net in- ventory in a period, the supplier has the same immediate cost (zero inventory holding cost). Due to a zero lead time, the inventory performance in any period is determined by the base-stock level rather than the net inventory in that period. Note that a zero base-stock level dominates any negative base-stock levels because the inventory costs in both cases are zero and the supplier's performance under the former choice may be better than that under the latter when the demand in a period is zero. Thus in the state space, we can use 0 to represent all the states of nonpositive inventory. The state space is f(\u0011t\u00001; It) : 0 \u0014 \u0011t\u00001 \u0014 t\u0000 1; 0 \u0014 It \u0014 Sg where 1 \u0014 t \u0014 R + 1 and S is a large number such that F1(S) \u0019 1. Actions (inventory order-up-to level in a period): S 2 f0; 1; 2; :::; Sg State transition: It+1 = [St \u0000 dt]+; \u0011t = ( \u0011t\u00001 + 1 if St \u0015 dt \u0011t\u00001 if St < dt : Rewards: rt((i; It); St) = hE[St \u0000 dt]+ 1 \u0014 t \u0014 R; and the de\fnition of rR+1((i; IR+1)) is the same as that in (2.7). Notice that in this single-review-phase model there is no consequence of having closing inventory IR+1. Transition probabilities: pt((j; I)j(i; u); S) = 8><>: PrfD(1) = S \u0000 Ig if j = i+ 1; S \u0015 u; I \u0015 0 PrfD(1) > Sg if j = i; S \u0015 u; I = 0 0 otherwise : In period t, for the order-up-to level St \u0015 It, \u00191t (Stji; It) = hE[St \u0000 dt]+ + Ef\u00191t+1(i+ 1; St \u0000 dt)jSt \u0015 dtg+ F 1(St)\u00191t+1(i; 0): The optimal cost to go from period t is \u00191t (i; It) = minSt\u0015It \u0019 1 t (Stji; It): 108 Appendix A. Proof for Chapter 2 The dynamic program is \u00191R+1(i; IR+1) = rR+1((i; IR+1)); for 1 \u0014 t \u0014 R: \u00191t (i; It) = minSt\u0015It \u00191t (Stji; It): 109 Appendix B Proof for Chapter 3 Derivation of formula (3.2) The waiting time distribution given S and L is derived as follows. Let wt denote the waiting time of a demand at time t. For the demand at time t to be \flled by time t + y, the inventory position at time t + y \u0000 L, denoted by IP (t + y \u0000 L), should satisfy the demands occuring in the interval (t + y \u0000 L; t]. Therefore, the distribution of wt given S and L is Prfwt \u0014 yjS; Lg = PrfD(t+ y \u0000 L; t] \u0014 IP (t+ y \u0000 L)g; where D(t+ y\u0000L; t] is the demand during (t+ y\u0000L; t], and IP (t+ y\u0000L) is the constant S. Note that D(t+ y \u0000 L; t] = D(L\u0000 y). So for 0 \u0014 y < L, Prfwt \u0014 yjS; Lg = PrfD(L\u0000 y) \u0014 Sg, which is independent of t. Therefore, Fw(yjS; L) = Prfw \u0014 yjS; Lg = limt!1 Prfwt \u0014 yjS; Lg = PrfD(L\u0000 y) \u0014 Sg. We also de\fne Fw(yjS; L) = 0 for y < 0 and Fw(yjS; L) = 1 for y \u0015 L. Proof of Proposition 3.1 Referring to the proof of Proposition 2.1 in the previous chapter, the proof for the distribution of A is similar. Note that a periodic-review base-stock policy is considered therein and the timing of demand is de\fned di\u000berently. So in the proof there, L is generally replaced by L+1. In this chapter, a continuous-review base-stock policy is studied, and L is not replaced by L+ 1. Moreover, both L and W are continuous. The de\fnition of XWt , the performance indicator for period t (1 \u0014 t \u0014 R), is modi\fed as: PrfXWt = 1g = PrfD(t\u0000 L; t\u0000W ] \u0014 Sg = PrfD(L\u0000W ) \u0014 Sg = AW . In the proof of Proposition 2.1, the derivation of the variance of the realized ready rate with window W is modi\fed as follows: \u001b2W = 1 R2 [ RX i=1 V ar(XWi ) + 2 X i 0), and Cr(L) = rL\u0000b (b > 0). Using the relationship between the average backorders and average waiting time (see page 192 of Zipkin (2000)), B(S; L) = \u0015E(yjS; L), where B(S; L) is the average backorders. So \u000eB(S; L) = \u0015ECD(yjS; L). B(S; L) = E([D(L)\u0000 S]+) = R x>S (x\u0000 S)f(xjL)dx = R x (x\u0000 S)f(xjL)dx+ R x~~~~ 0. 1) Unimodality of EC(S; L) in S for given L Given L > 0, C1(S; L) = @EC(S;L) @S = (h+ \u000e)F (SjL)\u0000 \u000e; C11(S; L) = @2EC(S;L) @S2 = (h+ \u000e)f(SjL) > 0 111 Appendix B. Proof for Chapter 3 so EC(S; L) is convex in S for given L. Let bS(L) be the solution to C1(S; L) = 0, F (bS(L)jL) = \u000eh+\u000e , @F (bS(L)jL)@L = 0 for any L > 0.bS(L) is the minimizer of EC(S; L) given L. 2) Unimodality of EC(bS(L); L) in L For normal demand, let z = S\u0000\u0015L \u001b p L , F (SjL) = \b(z), f(SjL) = \u001e(z) and I(S; L) = (S \u0000 \u0015L)\b(z) + \u001bpL\u001e(z). So C2(S; L) = @EC(S;L) @L = (h+ \u000e)[\u0000\u0015\b(z) + \u001b 2 p L \u001e(z)] + \u000e\u0015+ \u0015C 0r(L): By the Envelope Theorem, and noting that \b( bS(L)\u0000\u0015L \u001b p L ) = \u000e h+\u000e , dEC(bS(L);L) dL = C2(bS(L); L) = (h+ \u000e) \u001b2pL\u001e(z\u0003)\u0000 \u0015bLb+1 ;where z\u0003 = \b\u00001( \u000eh+\u000e ). Let eL be such that eLb+0:5 = 2\u0015b (h+\u000e)\u001b\u001e(z\u0003) . Then dEC( bS(L);L) dL < 0 for L < eL, dEC(bS(eL);eL) dL = 0 and dEC( bS(L);L) dL > 0 for L > eL. So EC(bS(L); L) is unimodal in L, and EC(S; L) is unimodal in (S; L). Convex delay cost: For convex delay cost CD(yjS; L) = \u000ey2, it is not easy to prove that EC(S; L) is unimodal. Instead, we have to use the plot of EC(S; L) for each speci\fc case to check its unimodality. For example, for the case of h = 2; \u000e = 10; r = 2; \u0015 = 4 and \u001b = 2, EC(S; L) is plotted in the Figure B.1, which indicates that it is unimodal. Figure B.1: EC(S,L) under convex delay cost Proof of Proposition 3.2 112 Appendix B. Proof for Chapter 3 At the optimum, constraint (3.7) is binding because otherwise the buyer's ob- jective function value can be reduced by decreasing p while (3.7) remains to hold. Solving for p from (3.7) with equality and subsituting it into the objective function, we obtain the buyer's unconstrained optimization problem: minS;LECB(S; L) = hI(S; L) + \u0015ECD(yjS; L) + \u0015Cr(L) + \u0019. So ECB(S; L) = EC(S; L) + \u0019. The \frst-order derivatives of the objective function with respect to S and L are @ECB(S;L) @S = @EC(S;L) @S = hF (SjL) + \u0015@ECD(yjS;L) @S and @ECB(S;L) @L = @EC(S;L) @L = \u0000hR x~~~~ 0. Using (B.3), when the supplier's choices of S and L are unobservable, the \frst- order conditions for (3.6) are @E\u0019(S; L) @S = \u0000100K R @\u0005(S; L;W ) @S \u0000 hF (SjL) = 0; (B.5) @E\u0019(S; L) @L = \u0000100K R @\u0005(S; L;W ) @L + h R x~~~~ 0, for (S\u0003; L\u0003) to be the solution to the supplier's \frst-order conditions (B.5) and (B.6), it is only required that there existsW 2 [0; L\u0003] such that (B.9) holds. Let g(W ) = R L\u0003 0 ( @Fw(yjS\u0003;L\u0003) @S @Fw(yjS\u0003;L\u0003) @L \u0000 @Fw(W jS\u0003;L\u0003) @S @Fw(W jS\u0003;L\u0003) @L ) @Fw(yjS\u0003; L\u0003) @L C 0D(y)dy; (B.10) and g0(yjW ) = ( @Fw(yjS\u0003;L\u0003) @S @Fw(yjS\u0003;L\u0003) @L \u0000 @Fw(W jS\u0003;L\u0003) @S @Fw(W jS\u0003;L\u0003) @L )@Fw(yjS \u0003;L\u0003) @L C 0D(y). @Fw(yjS\u0003;L\u0003) @L = @ Pr(D(L \u0003\u0000y)\u0014S\u0003) @L < 0: It has been assumed that C 0D(y) > 0, and @Fw(yjS\u0003;L\u0003) @S =@Fw(yjS \u0003;L\u0003) @L is strictly mono- tonic in y. Consider the case where @Fw(yjS \u0003;L\u0003) @S =@Fw(yjS \u0003;L\u0003) @L is increasing in y. For W = 0, g0(0jW ) = 0, and g0(yjW ) < 0 for y 2 (0; L\u0003]. So g(0) < 0. For W = L\u0003, g0(L\u0003jW ) = 0, and g0(yjW ) > 0 for y 2 [0; L\u0003). So g(L\u0003) > 0. Similarly when @Fw(yjS \u0003;L\u0003) @S =@Fw(yjS \u0003;L\u0003) @L is decreasing in y, we can show that g(0) > 115 Appendix B. Proof for Chapter 3 0 and g(L\u0003) < 0. Because g(W ) is continuous on [0; L\u0003], by the Intermediate Value Theorem, there exists W 2 [0; L\u0003] such that g(W ) = 0, and (B.9) holds. Because g(W ) is strictly monotonic in W , this W is unique; and g(0) 6= 0; g(L\u0003) 6= 0, so the optimal W \u0003 2 (0; L\u0003). Proof of Proposition 3.3 The \frst-best K and \u000b are determined by the supplier's \frst-order conditions at the \frst-best solution (S\u0003; L\u0003) and the optimal time window W \u0003: 100K R Z A< @\t(AjA\u0003W ) @S dA+ hF (S\u0003jL\u0003) = 0 (B.11) 100K R Z A< @\t(AjA\u0003W ) @L dA\u0000 hR x~~~~ 0, the optimal \u000b should be such that Z A< @\t(AjA\u0003W ) @S dA < 0. So any K > 0 and \u000b 2 (0; A\u0003W ) satisfyingZ A< @\t(AjA\u0003W ) @S dA < 0 and either equation, e.g. (B.11), are the optimal contract parameters for a ready-rate-with-window contract. Proof of Proposition 3.4 Let (\u0001) and '(\u0001) denote the cdf and pdf of the standardized demand, respectively. Also let s = S \u0015 , then S = s\u0015. Then the \frst-order conditions (3.9) and (3.10) can be represented as functions of s and L: h( s\u0000L (\u001b=\u0015) p L )\u0000 R L 0 C 0D(y) 1 (\u001b=\u0015) p L\u0000y'( s\u0000(L\u0000y) (\u001b=\u0015) p L\u0000y )dy = 0 and h[\u0000( s\u0000L (\u001b=\u0015) p L )+\u001b \u0015 1 2 p L '( s\u0000L (\u001b=\u0015) p L )]+ R L 0 C 0D(y) s+(L\u0000y) 2(\u001b=\u0015)(L\u0000y)3=2'( s\u0000(L\u0000y) (\u001b=\u0015) p L\u0000y )dy+C 0 r(L) = 0: Let (s\u0003; L\u0003) denote the solution to the above two equations. For \fxed \u001b \u0015 , (s\u0003; L\u0003) is independent of \u0015, and A\u0003W = ( s\u0003\u0000(L\u0003\u0000W \u0003) (\u001b=\u0015) p L\u0003\u0000W \u0003 ) is independent of \u0015; ECD(yjS\u0003; L\u0003) = R L\u0003 0 CD(y)d( s\u0003\u0000(L\u0003\u0000y) (\u001b=\u0015) p L\u0003\u0000y ) is independent of \u0015, @AW\u0003 @L and @ECD(yjS \u0003;L\u0003) @L are independent of \u0015. Because S\u0003 = \u0015s\u0003, S\u0003 is proportional to \u0015, @AW\u0003 @S = 1 \u0015 @AW\u0003 @s and @ECD(yjS\u0003;L\u0003) @S = 1 \u0015 @ECD(yjS\u0003;L\u0003) @s . So for \fxed \u001b \u0015 , (3.12) is independent of \u0015, and W \u0003, the solution to (3.12), is independent of \u0015. 116 Appendix B. Proof for Chapter 3 Proof of Proposition 3.5 If CD(y) = \u000ey, then 1 \u000e ECD(yjS; L) is the average waiting time and \u0015\u000eECD(yjS; L) the average backorders, and ECD(yjS; L) = \u000e\u0015(I(S; L)\u0000 (S \u0000 \u0015L)). The \frst-order condition for the expected average supply chain cost, (3.9), be- comes (h+ \u000e)F (SjL)\u0000 \u000e = 0. Using the result in Theorem 3.1, under the optimal ready-rate-with-window con- tract, the supplier's optimal base-stock level and lead time are the \frst-best S\u0003 and L\u0003, and thus satisfy the above \frst-order condition. So A\u00030 = F (S \u0003jL\u0003) = \u000e h+\u000e . Proof of Proposition 3.6 The results in Proposition 3.6 are similar to those in Proposition 3.3 with W = 0. (3.14) follows from (B.5), (B.6) and Assumption 3.2. Proof of Proposition 3.7 Following the de\fnition of (\u0001), '(\u0001) and s in the proof of Proposition 3.4, (3.1) becomes I(S; L) = S \u0000 R x~~~~ 0. We use \t(\f) and S(\f) instead of \t(\fjH) and S(\fjH) whenever there is no am- biguity. The following results will be used in the proofs.Z b a (y2 \u0000 1)\u001e(y)dy = a\u001e(a)\u0000 b\u001e(b); (C.1)Z b a y2\u001e(y)dy = a\u001e(a)\u0000 b\u001e(b) + \b(b)\u0000 \b(a): (C.2) 2. Formulation of period t problem Let Supplier 1's share in period t be \u000bt. Under the allocation rule t+1(x 1 t ; x 2 t ) for period t, the buyer's payo\u000b to go is vBt (\u000bt) = \u000bte 1 t +(1\u0000\u000bt)e2t + Z x1t Z x2t vBt+1( t+1(x 1 t ; x 2 t ))f(x 1 t je1t )f(x2t je2t )dx1tdx2t ; (C.3) the pro\fts to go of supplier 1 and 2 are v1t (\u000bt) = m\u000bt\u0000 bg(\u000bt)(e 1 t ) 2 2 + Z x1t Z x2t v1t+1( t+1(x 1 t ; x 2 t ))f(x 1 t je1t )f(x2t je2t )dx1tdx2t ; (C.4) and v2t (1\u0000 \u000bt) = m(1\u0000 \u000bt)\u0000 bg(1\u0000 \u000bt)(e1t )2 2 + Z x1t Z x2t v2t+1(1\u0000 \f\u000bt+1(x1t ; x2t ))f(x1t je1t )f(x2t je2t )dx1tdx2t : (C.5) 123 Appendix C. Proof for Chapter 4 First-order conditions ) @v1t (\u000bt) @e1t = \u0000bg(\u000bt)e1t + Z x1t Z x2t v1t+1( t+1(x 1 t ; x 2 t ))f 1(x1t je1t )f(x2t je2t )dx1tdx2t = 0; @v2t (1\u0000\u000bt) @e2t = \u0000bg(1\u0000 \u000bt)e2t + Z x1t Z x2t v2t+1(1\u0000 \f\u000bt+1(x1t ; x2t ))f(x1t je1t )f 2(x2t je2t )dx1tdx2t = 0) e1t = bg(\u000bt) Z x1t Z x2t v1t+1( t+1(x 1 t ; x 2 t ))f 1(x1t je1t )f(x2t je2t )dx1tdx2t ; (C.6) e2t = bg(1\u0000 \u000bt) Z x1t Z x2t v2t+1(1\u0000 \f\u000bt+1(x1t ; x2t ))f(x1t je1t )f 2(x2t je2t )dx1tdx2t : (C.7) (C:6) and (C:7) are necessary conditions for (e1t ; e 2 t ) to be a Nash equilibrium in period t. 3. Proof of Theorem 4.1 Due to the complexity of the problem and our focus on the suppliers' incentive issues, we will ignore the suppliers' individual rationality constraints in the following analysis and restrict a supplier's share to a range by letting \f and \f be the upper and lower bounds for a supplier's share in a period, where \f can be exogeneously de\fned by some constraints such as the minimum order quantity and also ensures the (IR) constraint is satis\fed. The proof of this theorem consists of two parts. In the \frst part, we derive the optimal allocation rule from the static formulation of the buyer's in\fnite-horizon problem, under the assumption that the two suppliers use stationary policies to play the stochastic game; in the second part, we check that under the derived optimal allocation rule, the suppliers' in\fnite-horizon stochastic game has a unique Nash equilibrium which is stationary and is the one derived from the static formulation. \u000f Part 1. Derive the optimal allocation rule We \frst consider the buyer's problem with the incentive compatibility constraints only. Other constraints will de\fne the upper and lower bounds for Supplier 1's share in the next period. Let yi = xi\u0000e\u0003i \u001b , then yi \u0018 N(0; 1) and from the \frst-order conditions 124 Appendix C. Proof for Chapter 4 in (4.5), the suppliers' best responses to an allocation rule \f\u000b(x1; x2) are e\u00031 = \u000bb\u001b Z Z v(b\f\u000b(y1; y2))y1\u001e(y1)\u001e(y2)dy1dy2; (C.8) e\u00032 = (1\u0000 \u000b)b\u001b Z Z v(1\u0000 b\f\u000b(y1; y2))y2\u001e(y1)\u001e(y2)dy1dy2; (C.9) where b\f\u000b(y1; y2) = \f\u000b(e\u00031 + \u001by1; e\u00032 + \u001by2), and v(\f) = v1(\f) = v2(\f) for any \f due to symmetry. Substituting the suppliers' best response functions into the buyer's objective func- tion, we obtain the buyer's unconstrained optimization problem, in which the buyer only needs to choose an allocation rule in terms of the suppliers' standardized per- formance: vB(\u000b) = maxb\f\u000bf Z Z [ 1 b\u001b y1v(b\f\u000b(y1; y2)) + 1b\u001by2v(1\u0000 b\f\u000b(y1; y2)) +vB(b\f\u000b(y1; y2))]\u001e(y1)\u001e(y2)dy1dy2g: Let G(b\f\u000b(y1; y2)) = 1b\u001by1v(b\f\u000b(y1; y2)) + 1b\u001by2v(1\u0000 b\f\u000b(y1; y2)) + vB(b\f\u000b(y1; y2)): Pointwise optimization of vB(\u000b) w.r.t. b\f\u000b(y1; y2)) G0(b\f\u000b(y1; y2)) = 1b\u001by1v0(b\f\u000b(y1; y2))\u0000 1b\u001by2v0(1\u0000 b\f\u000b(y1; y2)) + v0B(b\f\u000b(y1; y2)): So the optimal b\f\u0003\u000b(y1; y2) is independent of \u000b, and we omit the subscript \u000b inb\f\u000b(y1; y2) and b\f\u0003\u000b(y1; y2) from now on. It follows that vB(\u000b) = Z Z [ 1 b\u001b y1v(b\f\u0003(y1; y2)) + 1 b\u001b y2v(1\u0000 b\f\u0003(y1; y2)) +vB(b\f\u0003(y1; y2))]\u001e(y1)\u001e(y2)dy1dy2 = vB (C.10) is independent of \u000b. Let H = 2 2b\u001b2 ( Z Z y1v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2)2. Substituting it into (C:8) and (C:9), we get e\u00031 = p 2H p b ; e\u00032 = p 2H (1\u0000\u000b)pb , then due to symmetry of the two suppliers, v1(\u000b) = v2(\u000b) = v(\u000b), where v(\u000b) = \u000bm\u0000 H + Z Z v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2: (C.11) Note that both H and the last term in the above formula are independent of and H > 0, so 125 Appendix C. Proof for Chapter 4 v0(\u000b) = m+ H \u000b2 > 0 and v00(\u000b) = \u00002H \u000b3 < 0; G0(b\f(y1; y2)) = 1b\u001by1(m+ H(b\f(y1;y2))2 )\u0000 1b\u001by2(m+ H(1\u0000b\f(y1;y2))2 ); G00(b\f(y1; y2)) = \u0000 1b\u001by1 2H(b\f(y1;y2))3 \u0000 1b\u001by2 2H(1\u0000b\f(y1;y2))3 : So we have (1) for y1 > 0 and y2 > 0: G(b\f(y1; y2)) is concave, the optimal b\f\u0003(y1; y2) is either determined by G0(b\f(y1; y2)) = 0, or at the boundary with b\f\u0003(y1; y2) = \f ifG0(\f) > 0 and b\f\u0003(y1; y2) = \f ifG0(\f) < 0; (2) for y1 < 0 and y2 < 0: G(b\f(y1; y2)) is convex, b\f\u0003(y1; y2) = \f if G(\f) > G(1\u0000\f) and b\f\u0003(y1; y2) = \f otherwise; (3) for y1 > 0 and y2 < 0: G 0(b\f(y1; y2)) > 0, b\f\u0003(y1; y2) = \f; (4) for y1 < 0 and y2 > 0: G 0(b\f(y1; y2)) < 0, b\f\u0003(y1; y2) = \f. In the above analysis, G0(b\f(y1; y2)) > 0 is equivalent to y1 > S(b\f(y1; y2))y2; G0(b\f(y1; y2)) < 0 is equivalent to y1 < S(b\f(y1; y2))y2; G(\f) > G(1\u0000 \f) is equivalent to 1 b\u001b y1v(\f) + 1 b\u001b y2v(1\u0000 \f) > 1b\u001by1v(1\u0000 \f) + 1b\u001by2v(\f); , y1(v(\f)\u0000 v(1\u0000 \f)) > y2(v(\f)\u0000 v(1\u0000 \f)), y1 > y2: Because yi = xi\u0000e\u0003i \u001b , the above analysis on b\f\u0003(y1; y2) immediately translates to \f1\u0003\u000b (x1; x2) based on x1 and x2. So the optimal rule in Theorem 4.1 follows. Let C = Z Z y1v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2; (C.12) so H = \r2C2 2b\u001b2 : (C.13) In (C:11), let V = Z Z v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2, v(\u000b) = \u000bm\u0000 H\u000b + \rV , take \u000b = b\f\u0003(y1; y2), multiply both sides by y1, and integrate both sides over y1 and y2,Z Z y1v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = Z Z y1(b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1;y2))\u001e(y1)\u001e(y2)dy1dy2 + \rV Z Z y1\u001e(y1)\u001e(y2)dy1dy2 ) C = Z y1(b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1;y2))\u001e(y1)\u001e(y2)dy1dy2, together with (C:13) ) \u001b p 2bH = Z Z y1(b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1;y2))\u001e(y1)\u001e(y2)dy1dy2: 126 Appendix C. Proof for Chapter 4 Check the suppliers' second-order conditions (Nash equilibrium): Under the optimal rule \f1\u0003\u000b (x1; x2), @2v1(\u000b) (@e1)2 = \u0000\u000bb+ Z Z v1( 1\u0003 \u000b (x1; x2))f 11(x1je1)f(x2je2)dx1dx2 = \u0000\u000bb+ Z Z v1( 1\u0003 \u000b (x1; x2)) 1 \u001b4 [(x1\u0000e1 \u001b )2 \u0000 1]\u001e(x1\u0000e1 \u001b )\u001e(x2\u0000e2 \u001b )dx1dx2: At (e1; e2) = (e \u0003 1; e \u0003 2), @2v1(\u000b) (@e1)2 j(e\u00031;e\u00032) = \u0000\u000bb+ Z Z (b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1;y2)+\rV ) 1\u001b2 [(y1)2\u00001]\u001e(y1)\u001e(y2)dy1dy2 = \u0000\u000bb+ \u001b2 Z Z [(y1) 2 \u0000 1]\t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2: Using polar coordinates y1 = r sin \u0012; y2 = r cos \u0012; dy1dy2 = rdrd\u0012; \u0012 \u0018 U [0; 2\u0019]; tan \u0012 = y1 y2 = S(b\f\u0003): (C.14) Let \u0012min and \u0012max correspond to \f and \f.Z y1 Z y2 y21\t( b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = \u00031 +\t(\f) Z y2<0 Z 0 y2 y21\u001e(y1)\u001e(y2)dy1dy2 +\t(\f) Z y2<0 Z y10;y2<0 y21\u001e(y1)\u001e(y2)dy1dy2 +\t(\f) Z y1<0;y2>0 y21\u001e(y1)\u001e(y2)dy1dy2; where \u00031 = Z 1 r=0 Z \u0019=2 \u0012=0 r2(sin \u0012)2\t(b\f\u0003(y1; y2)) 12\u0019e\u0000r2=2rdrd\u0012 = Z \u0019=2 \u0012=0 (sin \u0012)2\t(b\f\u0003(y1; y2)) 1\u0019 [Z 1 r=0 r2 2 e\u0000r 2=2d( r 2 2 )]d\u0012 = 1 \u0019 Z \u0019=2 \u0012=0 (sin \u0012)2\t(b\f\u0003(y1; y2))d\u0012 = 1 \u0019 Z \u0012max \u0012min (sin \u0012)2\t(b\f\u0003(y1; y2))d\u0012+ 1\u0019\t(\f)(\u00194 \u0000 \u0012max2 + 14 sin(2\u0012max))+ 1\u0019\t(\f)( \u0012min2 \u0000 1 4 sin(2\u0012min)): By (C:2),Z y2<0 Z 0 y2 y21\u001e(y1)\u001e(y2)dy1dy2 = Z y2<0 [y2\u001e(y2) + 1 2 \u0000 \b(y2)]\u001e(y2)dy2 = Z y2<0 y2\u001e(y2)\u001e(y2)dy2 + 1 2 Z y2<0 \u001e(y2)dy2 \u0000 Z y2<0 \b(y2)\u001e(y2)dy2 = \u0000 1 4\u0019 + 1 4 \u0000 Z y2<0 \b(y2)\u001e(y2)dy2; 127 Appendix C. Proof for Chapter 4 Z y2<0 \b(y2)\u001e(y2)dy2 = (\b(y2)) 2j0\u00001 \u0000 Z y2<0 \b(y2)\u001e(y2)dy2 ) Z y2<0 \b(y2)\u001e(y2)dy2 = 1 8 ) Z y2<0 Z 0 y2 y21\u001e(y1)\u001e(y2)dy1dy2 = 1 8 \u0000 1 4\u0019 : Z y2<0 Z y10;y2<0 y21\u001e(y1)\u001e(y2)dy1dy2 = 1 2 Z y2<0 \u001e(y2)dy2 = 1 4 ;Z y1<0;y2>0 y21\u001e(y1)\u001e(y2)dy1dy2 = 1 2 Z y2>0 \u001e(y2)dy2 = 1 4 : So Z y1 Z y2 y21\t( b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = \u00031 +\t(\f)( 3 8 \u0000 1 4\u0019 ) + \t(\f)(3 8 + 1 4\u0019 ):Z y1 Z y2 \t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = \u00032 +\t(\f) Z y2<0 Z 0 y2 \u001e(y1)\u001e(y2)dy1dy2 +\t(\f) Z y2<0 Z y10;y2<0 \u001e(y1)\u001e(y2)dy1dy2 +\t(\f) Z y1<0;y2>0 \u001e(y1)\u001e(y2)dy1dy2 = \u00032 + 1 8 \t(\f) + 1 8 \t(\f) + 1 4 (\t(\f) + \t(\f)))Z y1 Z y2 \t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = \u00032 + 3 8 (\t(\f) + \t(\f)); (C.15) where \u00032 = Z y1>0;y2>0 \t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = Z 1 r=0 Z \u0019=2 \u0012=0 \t(b\f\u0003) 1 2\u0019 e\u0000r 2=2rdrd\u0012 = Z \u0019=2 \u0012=0 \t(b\f\u0003) 1 2\u0019 d\u0012 ) \u00032 = 1 2\u0019 Z \u0012max \u0012min \t(b\f\u0003)d\u0012 + 1 2\u0019 \t(\f)\u0012min + 1 2\u0019 \t(\f)(\u0019=2\u0000 \u0012max): (C.16) @2v1(\u000b) (@e1)2 j(e\u00031;e\u00032) = \u0000\u000bb+ \r\u001b2 [\u00031+\t(\f)(38\u0000 14\u0019 )+\t(\f)(38+ 14\u0019 )\u0000(\u00032+ 38(\t(\f)+\t(\f)))] 128 Appendix C. Proof for Chapter 4 = \u0000\u000bb+ \u0019\u001b2 [ Z \u0012max \u0012min ((sin \u0012)2\u0000 1 2 )\t(b\f\u0003(y1; y2))d\u0012+ 14(sin(2\u0012max)\u00001)(\t(\f)\u0000\t(\f))]; note that sin \u0019 4 = p 2 2 ; @2v1(\u000b) (@e1)2 j(e\u00031;e\u00032) \u0014 \u0000\u000bb+ \r\u0019\u001b2 [\t(\f) Z \u0012max \u0019=4 ((sin \u0012)2 \u0000 1 2 )d\u0012 +\t(\f) Z \u0019=4 \u0012min ((sin \u0012)2 \u0000 1 2 )d\u0012 +1 4 (sin(2\u0012max)\u0000 1)(\t(\f)\u0000\t(\f))]; because (sin \u0012)2 = 1\u0000cos 2\u0012 2 ; @2v1(\u000b) (@e1)2 j(e\u00031;e\u00032) \u0014 \u0000\u000bb+ \r\u0019\u001b2 [\u000012\t(\f) Z \u0012max \u0019=4 cos 2\u0012d\u0012 \u0000 1 2 \t(\f) Z \u0019=4 \u0012min cos 2\u0012d\u0012 +1 4 (sin(2\u0012max)\u0000 1)(\t(\f)\u0000\t(\f))] = \u0000\u000bb + \u0019\u001b2 [\u00001 4 \t(\f)(sin(2\u0012max) \u0000 1) \u0000 14\t(\f)(1 \u0000 sin(2\u0012min)) + 14(sin(2\u0012max) \u0000 1)(\t(\f)\u0000\t(\f))] = \u0000\u000bb+ 4\u0019\u001b2 \t(\f)(sin(2\u0012min)\u0000 sin(2\u0012max)) = \u0000\u000bb+ 4\u0019\u001b2 \t(\f)(sin(2\u0012min)\u0000 sin(2(\u00192 \u0000 \u0012min))) = \u0000\u000bb; similarly @ 2v2(1\u0000\u000b) (@e2)2 j(e\u00031;e\u00032) \u0014 \u0000(1 \u0000 \u000b)b, so (e\u00031; e\u00032) is a Nash equilibrium when the suppliers only use stationary policies. Uniqueness of Nash equilibrium: Let ai = ei\u0000e\u0003i \u001b , zi = yi \u0000 ai. By (C:1), @2v1(\u000b) (@e1)2 = \u0000\u000bb+ Z Z \t(\f1\u0003\u000b (x1; x2)) 1 \u001b4 [(x1\u0000e1 \u001b )2 \u0000 1]\u001e(x1\u0000e1 \u001b )\u001e(x2\u0000e2 \u001b )dx1dx2 +\rV Z Z 1 \u001b4 [(x1\u0000e1 \u001b )2 \u0000 1]\u001e(x1\u0000e1 \u001b )\u001e(x2\u0000e2 \u001b )dx1dx2 = \u0000\u000bb+ \u001b2 Z Z \t(b\f\u0003(y1; y2))[(y1 \u0000 a1)2 \u0000 1]\u001e(y1 \u0000 a1)\u001e(y2 \u0000 a2)dy1dy2 = \u0000\u000bb+ \u001b2 [\t(\f) Z 0 \u00001 ( Z 1 y2 [(y1 \u0000 a1)2 \u0000 1]\u001e(y1 \u0000 a1)dy1)\u001e(y2 \u0000 a2)dy2 +\t(\f) Z 0 \u00001 ( Z y2 \u00001 [(y1 \u0000 a1)2 \u0000 1]\u001e(y1 \u0000 a1)dy1)\u001e(y2 \u0000 a2)dy2 +\t(\f) Z 1 0 ( Z 0 \u00001 [(y1 \u0000 a1)2 \u0000 1]\u001e(y1 \u0000 a1)dy1)\u001e(y2 \u0000 a2)dy2 + Z 1 0 ( Z 1 0 \t(b\f\u0003(y1; y2))[(y1 \u0000 a1)2 \u0000 1]\u001e(y1 \u0000 a1)dy1)\u001e(y2 \u0000 a2)dy2] = \u0000\u000bb+ \u001b2 [\t(\f) Z 0 \u00001 ( Z 1 y2\u0000a1 (z21 \u0000 1)\u001e(z1)dz1)\u001e(y2 \u0000 a2)dy2 +\t(\f) Z 0 \u00001 ( Z y2\u0000a1 \u00001 (z21 \u0000 1)\u001e(z1)dz1)\u001e(y2 \u0000 a2)dy2 +\t(\f) Z 1 0 ( Z \u0000a1 \u00001 (z21 \u0000 1)\u001e(z1)dz1)\u001e(y2 \u0000 a2)dy2 129 Appendix C. Proof for Chapter 4 + Z 1 \u0000a1 ( Z 1 \u0000a2 \t(e\f\u0003\u000b(z1; z2))(z21 \u0000 1)\u001e(z1)dz1)\u001e(z2)dz2 = \u0000\u000bb+ \u001b2 [(\t(\f)\u0000\t(\f)) Z 0 \u00001 (y2 \u0000 a1)\u001e(y2 \u0000 a1)\u001e(y2 \u0000 a2)dy2 +\t(\f)a1\u001e(a1)\b(a2) + Z 1 \u0000a1 ( Z 1 \u0000a2 \t(e\f\u0003\u000b(z1; z2))(z21 \u0000 1)\u001e(z1)dz1)\u001e(z2)dz2: Z 0 \u00001 (y \u0000 a1)\u001e(y \u0000 a1)\u001e(y \u0000 a2)dy = 1 2\u0019 Z 0 \u00001 (y \u0000 a1) exp[\u000012((y \u0000 a1)2 + (y \u0000 a2)2)]dy = 1 2\u0019 exp[\u0000(a1\u0000a2 2 )2] Z 0 \u00001 (y \u0000 a1) exp[\u0000(y \u0000 a1+a22 )2]dy = 1 2\u0019 exp[\u0000(a1\u0000a2 2 )2][ Z 0 \u00001 (y \u0000 a1+a2 2 ) exp[\u0000(y \u0000 a1+a2 2 )2]dy \u0000a1\u0000a2 2 Z 0 \u00001 exp[\u0000(y \u0000 a1+a2 2 )2]dy] = 1p 2\u0019 exp[\u0000(a1\u0000a2 2 )2][\u0000 1 2 p 2\u0019 exp[\u0000(a1+a2 2 )2]\u0000a1\u0000a2 2 p 2 Z 0 \u00001 p 2p 2\u0019 exp[\u00001 2 ( p 2(y\u0000a1+a2 2 ))2]dy] = L 2 p 2\u0019 ; where L = \u0000 1p 2\u0019 exp[\u00001 2 (a21 + a 2 2)]\u0000 a1\u0000a2p2 exp[\u0000(a1\u0000a22 )2]\b(\u0000a1+a2p2 ): Let y and y be the solutions to (y1 \u0000 a1)2 \u0000 1 = 0, y \u0000 a1 = 1, y \u0000 a1 = \u00001, (y1 \u0000 a1)2 \u0000 1 \u0015 0 for y1 \u0015 y and y1 \u0014 y. Because \f \u0014 e\f\u0003\u000b(z1; z2) \u0014 \f, @2v1(\u000b) (@e1)2 \u0014 \u0000\u000bb+ \u001b2 [(\t(\f)\u0000\t(\f)) L 2 p 2\u0019 +\t(\f)a1\u001e(a1)\b(a2) + Z 1 0 f\t(\f) Z 1 y [(y1\u0000a1)2\u00001]\u001e(y1\u0000a1)dy1+\t(\f) Z y 0 [(y1\u0000a1)2\u00001]\u001e(y1\u0000a1)dy1 +\t(\f) Z y y [(y1 \u0000 a1)2 \u0000 1]\u001e(y1 \u0000 a1)dy1g\u001e(y2 \u0000 a2)dy2] = \u0000\u000bb+ \u001b2 [(\t(\f)\u0000\t(\f)) L 2 p 2\u0019 +\t(\f)a1\u001e(a1)\b(a2) + Z 1 0 f\t(\f) Z 1 y\u0000a1 (z21 \u0000 1)\u001e(z1)dz1 +\t(\f) Z y\u0000a1 \u0000a1 (z21 \u0000 1)\u001e(z1)dz1 +\t(\f) Z y\u0000a1 y\u0000a1 (z21 \u0000 1)\u001e(z1)dz1g\u001e(y2 \u0000 a2)dy2] = \u0000\u000bb+ \u001b2 [(\t(\f)\u0000\t(\f)) L 2 p 2\u0019 \u0000 (\t(\f)\u0000\t(\f))a1\u001e(a1)\b(a2) +(\t(\f)\u0000\t(\f))((y \u0000 a1)\u001e(y \u0000 a1)\u0000 (y \u0000 a1)\u001e(y \u0000 a1))\b(a2)] = \u0000\u000bb+ \u001b2 (\t(\f)\u0000\t(\f))[\u0000 1 2 p 2\u0019 \u001e( p a21 + a 2 2)\u0000a1\u0000a22 \u001e(a1\u0000a2p2 )\b(\u0000a1+a2p2 )\u0000a1\u001e(a1)\b(a2)+ 2\u001e(1)\b(a2)] 130 Appendix C. Proof for Chapter 4 = \u0000\u000bb+ \u001b2 (\t(\f)\u0000\t(\f))[\u00001 2 \u001e(a1)\u001e(a2)+ a2\u0000a1 2 \u001e(a1\u0000a2p 2 )(1\u0000\b(a1+a2p 2 ))\u0000a1\u001e(a1)\b(a2)+ 2\u001e(1)\b(a2)] because x\u001e(x) is maximized at x = 1 and \u0000x\u001e(x) is maximized at x = \u00001; relaxing each term in the [ ] individually, we obtain @2v1(\u000b) (@e1)2 \u0014 \u0000\u000bb+ \u001b2 (\t(\f)\u0000\t(\f))[\u00001 2 \u001e(a1)\u001e(a2) + a2\u0000a1 2 \u001e(a1\u0000a2p 2 )\u0000 a1\u001e(a1)\b(a2) +2\u001e(1)\b(a2)] \u0014 \u0000\u000bb+ \u001b2 (\t(\f)\u0000\t(\f))[ 1p 2 \u001e(1) + 3\u001e(1)\b(a2)] \u0014 \u0000\u000bb+ \r(2\f\u00001) \u001b2 (m+ H \f(1\u0000\f))( 1p 2 + 3)\u001e(1): Because 1\u0000\f \u0014 \u000b \u0014 \f, the su\u000ecient condition for the existence of a unique Nash equilibrium when the suppliers only use stationary policies is \r(2\f \u0000 1) \u001b2 (m+ H \f(1\u0000 \f))( 1p 2 + 3)\u001e(1) < b(1\u0000 \f): (C.17) Because (C:17) holds at \f = 1 2 with strict inequality, the set of the values of \f such that \f > 1 2 and (C:17) holds is nonempty. \u000f Part 2. Prove the existence of a unique stationary Nash equilibrium in the suppliers' in\fnite-horizon stochastic game under \f1\u0003\u000b (x1; x2) The proof consists of two steps: the \frst step is on a \fnite-horizon problem and uses backward recursion to show the existence of a unique subgame perfect Nash equilibrium in the two suppliers' \fnite-horizon stochastic game; and the second step shows when the horizon goes to in\fnity, the subgame perfect equilibrium becomes a stationary equilibrium. Step 1. Finite-horizon problem: First consider a T -period model, T < 1. Assume given Supplier 1's share in Period T + 1, \u000bT+1, the terminal values of each party take the following form: vBT+1(\u000bT+1) = V B T+1 = 2 p 2HT+1p b ; v1T+1(\u000bT+1) = \u000bT+1m\u0000 HT+1\u000bT+1 + \rVT+1; v2T+1(1\u0000 \u000bT+1) = (1\u0000 \u000bT+1)m\u0000 HT+11\u0000\u000bT+1 + \rVT+1; where \u000bT+1 2 [\f; \f], and HT+1 \u0015 0 and VT+1 are independent of \u000bT+1. Because there is little restriction on the value of HT+1 and VT+1, the above form has included the case of no future payo\u000b for the buyer beyond period T when HT+1 = 0 and the case of no future payo\u000b for the suppliers beyond period T when HT+1 = VT+1 = 0. 131 Appendix C. Proof for Chapter 4 Motivated by Part 1 of the proof, we consider an allocation rule of the following form for the \fnite horizon problem. Given Supplier 1's share in period t, \u000bt, the optimal allocation rule for period t+1 (t \u0014 T ), \f\u000b\u0003t+1(x1t ; x2t ), takes the form 1. for x1t > e 1\u0003 t and x 2 t > e 2\u0003 t : if x1t \u0000 e1\u0003t > S(\fjHt)(x2t \u0000 e2\u0003t ), \f\u000b\u0003t+1(x1t ; x2t ) = \f; if x1t \u0000 e1\u0003t < S(\fjHt)(x2t \u0000 e2\u0003t ), \f\u000b\u0003t+1(x 1 t ; x 2 t ) = \f; otherwise, \u000b\u0003 t+1(x 1 t ; x 2 t ) is determined by S( \u000b\u0003 t+1(x 1 t ; x 2 t )jHt) = x 1 t\u0000e1\u0003t x2t\u0000e2\u0003t ; 2. for x1t < e 1\u0003 t or x 2 t < e 2\u0003 t : \u000b\u0003 t+1(x 1 t ; x 2 t ) = \f if x 1 t\u0000e1\u0003t > x2t\u0000e2\u0003t and \f\u000b\u0003t+1(x1t ; x2t ) = \f otherwise, where e1\u0003t = p 2Ht \u000bt p b ; e2\u0003t = p 2Ht (1\u0000 \u000bt) p b ; (C.18) and Ht (t = T; T \u0000 1; :::1) is calculated as Ht = \r2 2b\u001b2 ( Z y1(b\f\u0003t+1(y1; y2)m\u0000 Ht+1b\f\u0003t+1(y1; y2))\u001e(y1)\u001e(y2)dy1dy2)2 > 0; (C.19) where b\f\u0003t+1(y1; y2) = \f\u000b\u0003t+1(e1\u0003t + \u001by1; e2\u0003t + \u001by2): In period T , let yi = xiT\u0000ei\u0003T \u001b (i = 1; 2), from (C:6) and (C:7), e1T = b\u000bT Z x1T Z x2T (\f\u000b\u0003T+1(x 1 T ; x 2 T )m\u0000 HT+1\f\u000b\u0003T+1(x1T ;x2T )+\rVT+1) x1T\u0000e1T \u001b2 f(x1T je1T )f(x2T je2T )dx1Tdx2T = b\u000bT \u001b Z y1 Z y2 (b\f\u0003T+1(y1; y2)m\u0000 HT+1b\f\u0003T+1(y1;y2))(y1\u0000 e1T\u0000e1\u0003T\u001b )\u001e(y1\u0000 e1T\u0000e1\u0003T\u001b )\u001e(y2\u0000 e2T\u0000e2\u0003T\u001b )dy1dy2; and e2T = b(1\u0000\u000bt)\u001b Z y1 Z y2 ((1\u0000b\f\u0003T+1(y1; y2))m\u0000 HT+11\u0000b\f\u0003T+1(y1;y2))(y2\u0000 e2T\u0000e2\u0003T\u001b )\u001e(y1\u0000 e1T\u0000e1\u0003T\u001b )\u001e(y2\u0000 e2T\u0000e2\u0003T \u001b )dy1dy2: It is obvious that b\f\u0003T+1(y1; y2) is symmetrical in y1 and y2, i.e., b\f\u0003T+1(y1; y2) = 1\u0000 b\f\u0003T+1(y2; y1) for any (y1; y2). SoZ y1(b\f\u0003T+1(y1; y2)m\u0000 HT+1b\f\u0003T+1(y1;y2))\u001e(y1)\u001e(y2)dy1dy2 = Z y2((1\u0000 b\f\u0003T+1(y1; y2))m\u0000 HT+11\u0000b\f\u0003T+1(y1;y2))\u001e(y1)\u001e(y2)dy1dy2. Because letting eiT = e i\u0003 T in the two equations above for e 1 T and e 2 T will make both equations hold simultaneously, obviously e1\u0003T = p 2HT \u000bT p b ; e2\u0003T = p 2HT (1\u0000\u000bT ) p b is a solution to (C:6) and (C:7) for period T problem. 132 Appendix C. Proof for Chapter 4 To show (e1\u0003T ; e 2\u0003 T ) is a unique Nash equilibrium in period T , consider the suppliers' second-order conditions. Following similar analysis as above for the second-order conditions in the static formuation, but with H and V replaced by HT+1 and VT+1, we can show that at (e1T ; e 2 T ) = (e 1\u0003 T ; e 2\u0003 T ), @2v1T (\u000bT ) (@e1T ) 2 j(e1\u0003T ;e2\u0003T ) \u0014 \u0000\u000bT b and @2v2T (1\u0000\u000bT ) (@e2T ) 2 j(e1\u0003T ;e2\u0003T ) \u0014 \u0000(1 \u0000 \u000bT )b, so (e 1\u0003 T ; e 2\u0003 T ) is a Nash equilibrium; moreover, the su\u000ecient condition for the existence of a unique Nash equilibrium in period T is 2\r(2\f\u00001) \u001b2 (m+ HT+1 \f(1\u0000\f))\u001e(1) < b(1\u0000 \f): At the Nash equilibrium (e1\u0003T ; e 2\u0003 T ), from (C:3) and noting that V B T+1 is independent of \u000bT+1, vBT (\u000bT ) = V B T = 2 p 2HTp b + \rV BT+1; v1T (\u000bT ) = \u000bTm\u0000 HT\u000bT + \rVT ; v2T (1\u0000 \u000bT ) = (1\u0000 \u000bT )m\u0000 HT 1\u0000\u000bT + \rVT ; where VT = Z v1T+1( b\f\u0003T+1(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = Z (b\f\u0003T+1(y1; y2)m\u0000 HT+1b\f\u0003T+1(y1;y2))\u001e(y1)\u001e(y2)dy1dy2 + VT+1: Suppose the above results hold for period t+1 (t \u0014 T\u00001), i.e., under the condition 2\r(2\f\u00001) \u001b2 (m + Ht+2 \f(1\u0000\f))\u001e(1) < b(1 \u0000 \f), (e1\u0003t+1; e2\u0003t+1) de\fned by (C:18) and (C:19) with the subscript t replaced by t + 1 is the unique Nash equilibrium of the suppliers in period t+ 1, and at this equilibrium, vBt+1(\u000bt+1) = V B t+1 = 2 p 2Ht+1p b + \rV Bt+2; v1t+1(\u000bt+1) = \u000bt+1m\u0000 Ht+1\u000bt+1 + \rVt+1; v2t+1(1\u0000\u000bt+1) = (1\u0000\u000bt+1)m\u0000 Ht+1 1\u0000\u000bt+1 + \rVt+1; where Vt+1 = Z (b\f\u0003t+2(y1; y2)m\u0000 Ht+2b\f\u0003t+2(y1;y2))\u001e(y1)\u001e(y2)dy1dy2 + Vt+2: Then using the backward recursive argument and similar analysis as that for period T problem, we can show that in period t, under the condition 2\r(2\f \u0000 1) \u001b2 (m+ Ht+1 \f(1\u0000 \f))\u001e(1) < b(1\u0000 \f); (C.20) (e1\u0003t ; e 2\u0003 t ) de\fned by (C:18) and (C:19) is the unique Nash equilibrium of the suppliers in period t, and at this equilibrium, vBt (\u000bt) = V B t ; v 1 t (\u000bt) = \u000btm\u0000 Ht\u000bt + \rVt; v2t (1\u0000 \u000bt) = (1\u0000 \u000bt)m\u0000 Ht1\u0000\u000bt + \rVt; 133 Appendix C. Proof for Chapter 4 where Vt = Z (b\f\u0003t+1(y1; y2)m\u0000 Ht+1b\f\u0003t+1(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 + Vt+1; (C.21) and V Bt = 2 p 2Htp b + \rV Bt+1: (C.22) Because (C:20) holds at \f = 1 2 , the set of the values of \f such that \f > 1 2 and (C:20) holds is nonempty. So under condition (C:20), f(e1\u0003t ; e2\u0003t )gt=1;2;:::;T constitutes a unique subgame perfect Nash equilibrium. Step 2. In\fnite-horizon problem: When T !1, the Nash equilibrium is stationary if there exists a solution to F (H1) = H1\u0000 2 2b\u001b2 ( Z y1(b\f\u0003(y1; y2)m\u0000 H1b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2)2 = 0: (C.23) Then in period t, given \u000bt, e 1\u0003 t = e \u0003 1 = p 2H1 \u000bt p b ; e2\u0003t = e \u0003 2 = p 2H1 (1\u0000\u000bt) p b ; the allocation rule is in fact the optimal allocation rule derived in the Part 1 proof; and from (C:21) and (C:22), V1 = 11\u0000 Z (b\f\u0003(y1; y2)m\u0000 H1b\f\u0003(y1;y2))\u001e(y1)\u001e(y2)dy1dy2; V B1 = 2p2H1(1\u0000\r)pb : It is obvious that F (H1) is continuous in H1. To prove the existence of a solution to (C:23), we can \fnd a bound to F (H1) using the fact that \f \u0014 b\f\u0003(y1; y2) \u0014 \f: For simplicity of notation, we omit H1 in \t(b\f\u0003(y1; y2)jH1) in the analysis. Using polar coordinates de\fned in (C:14), in F (H1), let B = Z y1 Z y2 y1\t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = \u00033 +\t(\f) Z y2<0 Z 0 y2 y1\u001e(y1)\u001e(y2)dy1dy2 +\t(\f) Z y2<0 Z y10;y2<0 y1\u001e(y1)\u001e(y2)dy1dy2 +\t(\f) Z y1<0;y2>0 y1\u001e(y1)\u001e(y2)dy1dy2] = \u00033+\t(\f) Z y2<0 \u001e(y2)(\u001e(y2)\u0000\u001e(0))dy2\u0000\t(\f) Z y2<0 (\u001e(y2)) 2dy2+ 1 2 (\t(\f)\u0000\t(\f))\u001e(0) = \u00033 +\t(\f)( 1 4 p \u0019 \u0000 \u001e(0) 2 )\u0000\t(\f) 1 4 p \u0019 + 1 2 (\t(\f)\u0000\t(\f))\u001e(0)) B = \u00033 +\t(\f) 1 4 p \u0019 \u0000\t(\f) 1 4 p \u0019 ( p 2 + 1); (C.24) 134 Appendix C. Proof for Chapter 4 where \u00033 = Z y1>0;y2>0 y1\t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = Z 1 r=0 Z \u0019=2 \u0012=0 \t(b\f\u0003)r sin \u0012 1 2\u0019 e\u0000r 2=2rdrd\u0012 = p \u0019 2 Z \u0019=2 \u0012=0 \t(b\f\u0003) sin \u0012 1 2\u0019 d\u0012 ) \u00033 = 1 2 p 2\u0019 [ Z \u0012max \u0012min \t(b\f\u0003) sin \u0012d\u0012 +\t(\f)(1\u0000 cos \u0012min) + \t(\f) cos \u0012max]: (C.25) Because \t0(\fjH1) > 0 and H1 \u0015 0, \u00033 < 1 2 p 2\u0019 \t(\f) Z \u0012max \u0012min sin \u0012d\u0012 + 1 2 p 2\u0019 \t(\f)(1\u0000 cos \u0012min) + 12p2\u0019\t(\f) cos \u0012max = 1 2 p 2\u0019 [\t(\f) + (\t(\f)\u0000\t(\f)) cos \u0012min]) B < 1 2 p 2\u0019 [ 1p 2 + cos \u0012min](\t(\f)\u0000\t(\f)); \u00033 > 1 2 p 2\u0019 \t(\f) Z \u0012max \u0012min sin \u0012d\u0012 + 1 2 p 2\u0019 \t(\f)(1\u0000 cos \u0012min) + 12p2\u0019\t(\f) cos \u0012max = 1 2 p 2\u0019 [\t(\f) + (\t(\f)\u0000\t(\f)) cos \u0012max]) B > 1 2 p 2\u0019 [ 1p 2 + cos \u0012max](\t(\f)\u0000\t(\f)) > 0: Let F1(x) = x\u0000 \r216\u0019b\u001b2 ( 1p2 + cos \u0012min)2(\t(\f)\u0000\t(\f))2 = x\u0000 J(\u0012min)(m+ x\f\f )2; and F2(x) = x\u0000 J(\u0012max)(m+ x\f\f )2; where J(\u0012) = 2 16\u0019b\u001b2 (2\f \u0000 1)2( 1p 2 + cos \u0012)2. So for any x \u0015 0, F1(x) < F (x) < F2(x): (C.26) F 01(x) = 1\u0000 2J(\u0012min)(m+ x\f\f ) 1\f\f ; F 001 (x) = \u00002J(\u0012min) 1(\f\f)2 < 0; F 02(x) = 1\u0000 2J(\u0012max)(m+ x\f\f ) 1\f\f ; F 002 (x) = \u00002J(\u0012max) 1(\f\f)2 < 0: Let x\u0003 be the solution to F 01(x) = 0; so F1(x) takes the maximum at x \u0003 = \f\f( 2J(\u0012min) \u0000m); and F1(x \u0003) = \f\f( 4J(\u0012min) \u0000m). Because F1(0) < 0 and F2(0) < 0, the su\u000ecient condition for the existence of a solution to (C:23) is F1(x \u0003) \u0015 0, which is equivalent to (2\f \u0000 1)2 \f(1\u0000 \f) \u0014 4\u0019b\u001b2 \r2m( 1p 2 + cos \u0012min)2 : (C.27) Because (C:27) holds at \f = 1 2 with strict inequality, the set of the values of 135 Appendix C. Proof for Chapter 4 such that \f > 1 2 and (C:27) holds is nonempty. (C:27) sets a lower bound to \f with \f > 0, and guarantees at least one solution to F1(x) = 0. (C:27)) F 02(0) = 1\u0000 2J(\u0012max) m\f\f > 1\u0000 2J(\u0012min) m\f\f \u0015 1\u0000 12 = 12 ; for su\u000eciently large x, F 02(x) < 0, so (C:27) guarantees two solutions to F2(x) = 0. Let the bigger solution to F2(x) = 0 be x, and the smaller one be x. Obviously x\u0003 2 [x; x]. It follows from (C:26) that F (x) < 0; F (x) < 0 and F (x\u0003) > F1(x\u0003) \u0015 0. By the Intermediate Value Theorem, there exists at least one ex 2 [x; x] such that F (ex) = 0. Because F (x) < F2(x) < 0 for x > 0 and x =2 [x; x], there is no \fxed point outside [x; x]. So under (C:27), (e\u00031; e \u0003 2) constitutes a unique stationary Nash equilibrium. It is noted that (C:27) is a su\u000ecient condition which can be relaxed because F1(x) only gives a lower bound to F (x), even if F1(x \u0003) < 0, there may still exist a solution to F (x) = 0 when the maximum value of F2(x) is positive. It is also noted that there could be multiple \fxed points in [x; x]. As will be shown in Corollary 4.1, the buyer's long-run discounted payo\u000b v\u0003B = 2 p 2H (1\u0000\r)pb . So when there are multiple \fxed points in [x; x], the largest \fxed point while making the (IR) constraint and Nash constraint hold is optimal. 4. Proof of Corollary 4.1 From the proof of Theorem 4.1, G0(b\f\u0003(y1; y2)) = 0) S(b\f\u0003(y1; y2)) = y1 y2 = R: (C.28) BecauseZ Z y1v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = Z Z y2v(1\u0000 b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2, (C:10) ) vB = 2 b\u001b Z Z y1v(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 + \rvBZ Z \u001e(y1)\u001e(y2)dy1dy2 ) vB = 2 p 2H (1\u0000\r)pb : In (C:11), using the de\fnition of V in the proof of Theorem 4.1, take \u000b = b\f\u0003(y1; y2) and integrate both sides over y1 and y2, V = Z Z (b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1;y2))\u001e(y1)\u001e(y2)dy1dy2 + \rV Z Z \u001e(y1)\u001e(y2)dy1dy2 ) V = 1 1\u0000 Z Z (b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1;y2))\u001e(y1)\u001e(y2)dy1dy2: Substituting this into (C:11) gives the formula for v\u0003(\u000b). 136 Appendix C. Proof for Chapter 4 In the following subsections 5 to 7, for simplicity of notation, we use \f for b\f. 5. Proof of Theorem 4.2 For simplicity of notation, we use \u0012 for \u0012\u000b. Under the allocation rule de\fned by (4:9), Supplier 1's long-run discounted payo\u000b is v1(\u000b) = \u000bm\u0000\u000bb(e1) 2 2 +\r[v1(\f)\b( \u0012 \u0000 (e1 \u0000 e2)p 2\u001b )+v1(1\u0000\f)\b(\u0012 \u0000 (e1 \u0000 e2)p 2\u001b )]: (C.29) FOC ) \u0000\u000bbe1 + \r(v1(\f)\u0000 v1(1\u0000 \f)) 1p 2\u001b \u001e( \u0012 \u0000 (e1 \u0000 e2)p 2\u001b ) = 0: (C.30) Supplier 2's long-run discounted payo\u000b is v2(1\u0000 \u000b) = (1\u0000 \u000b)m\u0000 (1\u0000 \u000b)b(e2) 2 2 +\r[v2(1\u0000 \f)\b(\u0012 \u0000 (e1 \u0000 e2)p 2\u001b ) + v2(\f)\b( \u0012 \u0000 (e1 \u0000 e2)p 2\u001b )]: (C.31) FOC ) \u0000(1\u0000 \u000b)be2 + \r(v2(\f)\u0000 v2(1\u0000 \f)) 1p 2\u001b \u001e( \u0012 \u0000 (e1 \u0000 e2)p 2\u001b ) = 0: (C.32) Because v1(\u000b) = v2(\u000b) for any \u000b, denote it by v(\u000b) and let \u0001v(\u000b) = v(\u000b)\u0000v(1\u0000\u000b): For an allocation rule to provide incentive to suppliers, we need \u0001v(\f) > 0. The buyer's problem is vB(\u000b) = max \f;\u0012 f\u000be1+(1\u0000\u000b)e2+\r[vB(\f)\b(\u0012 \u0000 (e1 \u0000 e2)p 2\u001b )+vB(1\u0000\f)\b(\u0012 \u0000 (e1 \u0000 e2)p 2\u001b )]g (C.33) subject to (C:30) and (C:32): Note that vB(\f) = vB(1 \u0000 \f). Let \u00111 and \u00112 be the Lagrangian multipliers of (C:30) and (C:32). @L @\u0012 = \u0000(\u00111 + \u00112)\r\u0001v(\f)\u0012\u0000(e \u0003 1\u0000e\u00032) 2\u001b2 1p 2\u001b \u001e( \u0012\u0000(e\u00031\u0000e\u00032)p 2\u001b ) = 0) the optimal \u0012\u0003 = e\u00031 \u0000 e\u00032: (C.34) 137 Appendix C. Proof for Chapter 4 It follows from (C:30), (C:32) and (C:34) that e\u00031 = \r\u0001v(\f) p 2b\u001b \u001e( \u0012\u0003 \u0000 (e\u00031 \u0000 e\u00032)p 2\u001b ) = \r\u0001v(\f) 2 p \u0019b\u001b ; (C.35) e\u00032 = \r\u0001v(\f) 2(1\u0000 \u000b)p\u0019b\u001b : (C.36) For \u000b 6= 1 2 , e\u00031 6= e\u00032. So \u0012\u0003 6= 0. 6. Proof of Corollary 4.2 The formula for \u0012\u0003\u000b follows from (C:35) and (C:36). Substituting this formula, (C:35) and (C:36) into the buyer's objective function, (C:29) and (C:31), we obtain vB(\u000b) = 1\u0000 \u0001v(\f)p \u0019b\u001b ; v(\u000b) = \u000bm\u0000 (\r\u0001v(\f))2 8\u0019\u000bb\u001b2 + 2 (v(\f) + v(1\u0000 \f))) v(\f) = \fm\u0000 (\r\u0001v(\f)) 2 8\u0019\fb\u001b2 + 2 (v(\f) + v(1\u0000 \f)); (C.37) v(1\u0000 \f) = (1\u0000 \f)m\u0000 (\r\u0001v(\f)) 2 8\u0019(1\u0000 \f)b\u001b2 + 2 (v(\f) + v(1\u0000 \f)): (C.38) ) \u0001v(\f) = (2\f \u0000 1)m+ (\r\u0001v(\f))2(2\f\u00001) 8\u0019b\u001b2\f(1\u0000\f) ) \u0001v(\f) is the solution to \r2(2\f \u0000 1) 8\u0019b\u001b2\f(1\u0000 \f)(\u0001v(\f)) 2 \u0000\u0001v(\f) + (2\f \u0000 1)m = 0: (C.39) Let W = 2 8\u0019b\u001b2 1\u00002 \f(1\u0000\f) . Then \u0001v(\f) = 1 2W (\u00001\u0006p1 + 4W (2\f \u0000 1)m): For \f > 1 2 , W < 0 and p 1 + 4W (2\f \u0000 1)m < 1; and for there existing a solution to (C:39), we need 1 + 4W (2\f \u0000 1)m \u0015 0: (C.40) This could constrain the value of the optimal \f. \u0001v(\f) = 0 at \f = 1 2 , so \u0001v(\f) = 1 \u00002W (1\u0000 p 1 + 4W (2\f \u0000 1)m): (C.41) Let \u0006v(\f) = v(\f) + v(1\u0000 \f): (C:37) + (C:38)) (1\u0000 \r)\u0006v(\f) = m\u0000 (\r\u0001v(\f)) 2 8\u0019b\u001b2\f(1\u0000 \f) : (C.42) 138 Appendix C. Proof for Chapter 4 It follows that v(\f) = 1 2 (\u0006v(\f) + \u0001v(\f)); v(1\u0000 \f) = 1 2 (\u0006v(\f)\u0000\u0001v(\f)): Let K = 2 8\u0019b\u001b2 . For \f > 1 2 , @W @ = \u0000K( 1 \f2 + 1 (1\u0000\f)2 ) < 0; @ @ ( 1\u00002W ) = 1 2W 2 @W @ < 0; @ @ ( p 1 + 4W (2\f \u0000 1)m) = 1 2 (1+ 4W (2\f\u0000 1)m)\u0000 12 [\u00004(2\f\u0000 1)mK( 1 \f2 + 1 (1\u0000\f)2 )+ 8Wm] = \u00002(1 + 4W (2\f \u0000 1)m)\u0000 12m[(2\f \u0000 1)K( 1 \f2 + 1 (1\u0000\f)2 ) + 2K(2\f\u00001) \f(1\u0000\f) ] = \u00002(1 + 4W (2\f \u0000 1)m)\u0000 12 K(2\f\u00001)m \f2(1\u0000\f)2 = 2(1 + 4W (2\f \u0000 1)m)\u0000 1 2 mW \f(1\u0000\f) : By (C:41), @\u0001v(\f) @ = 1 2W 2 @W @ (1\u0000p1 + 4W (2\f \u0000 1)m) + (1 + 4W (2\f \u0000 1)m)\u0000 12 m \f(1\u0000\f) = \u0000 1 2K(2\f\u00001)2 ( 2+(1\u0000\f)2)(1\u0000p1 + 4W (2\f \u0000 1)m)+(1+4W (2\f\u00001)m)\u0000 12 m \f(1\u0000\f) = (1\u0000 4K(2\f\u00001)2m \f(1\u0000\f) ) \u0000 1 2 [ 1 2K(2\f\u00001)2 (1\u0000 q 1\u0000 4K(2\f\u00001)2m \f(1\u0000\f) )( 2 + (1\u0000 \f)2)\u0000 m(2\f\u00001)2 \f(1\u0000\f) ]: At \f = 1 2 , (1\u0000 4K(2\f\u00001)2m \f(1\u0000\f) ) \u0000 1 2 = 1; \f2 + (1\u0000 \f)2 = 1 2 ; (2\f \u0000 1)2 m \f(1\u0000\f) = 0; lim\f! 1 2 1 2K(2\f\u00001)2 (1\u0000 q 1\u0000 4K(2\f\u00001)2m \f(1\u0000\f) ) = \u0000 lim\f! 12 1 8K(2\f\u00001)\f2(1+4W (2\f\u00001)m)\u0000 1 2 mW \f(1\u0000\f) = lim\f! 1 2 m 4\f3(1\u0000\f)2 (1 + 4W (2\f \u0000 1)m)\u0000 1 2 = 8m; so lim \f! 1 2 @\u0001v(\f) @ > 0: (C.43) @2\u0001v(\f) @\f2 = 1 2K(1+4W (2\f\u00001)m) 32 (2\f\u00001)3 [ \u0010 1\u0000p1 + 4W (2\f \u0000 1)m\u00112 \u00102p1 + 4W (2\f \u0000 1)m+ 1\u0011 +4m 2K2(2\f\u00001)6 \f3(1\u0000\f)3 ] ) @2\u0001v(\f) @\f2 > 0 for \f > 1 2 and @ 2\u0001v(\f) @\f2 < 0 for \f < 1 2 ; so \u0001v(\f) is convex in \f > 1 2 and concave in \f < 1 2 . By (C:43), @\u0001v(\f) @ > 0 for \f satisfying (C:40); because vB(\f) = 1\u0000 \u0001v(\f)p \u0019b\u001b , the optimal \f\u0003 should be as large as possible and is constrained by (C:40). \u000f Supplier's individual rationality constraint (C:39) and (C:42) ) \u0006v(\f) = 1 1\u0000\r [2m\u0000 12\f\u00001\u0001v(\f)]; so v(1\u0000 \f) = m 1\u0000\r \u0000 12(1 + 1(1\u0000\r)(2\f\u00001))\u0001v(\f)) v(1\u0000 \f) = m 1\u0000 \r + 1 4W (1+ 1 (1\u0000 \r)(2\f \u0000 1))(1\u0000 p 1 + 4W (2\f \u0000 1)m) \u0015 0: (C.44) \u000f Nash equilibrium condition (suppliers' second-order conditions) 139 Appendix C. Proof for Chapter 4 We need to check if at the optimal \f\u0003, the Nash equilibrium between the two suppliers exists and is unique. SOC for Supplier 1's and Supplier 2's problems: @2v1(\u000b) (@e1)2 = \u0000\u000bb+ \r\u0001v(\f) 1p 2\u001b \u0012\u0000(e1\u0000e2) 2\u001b2 \u001e(\u0012\u0000(e1\u0000e2)p 2\u001b ) and @ 2v2(1\u0000\u000b) (@e2)2 = \u0000(1\u0000 \u000b)b\u0000 \r\u0001v(\f) 1p 2\u001b \u0012\u0000(e1\u0000e2) 2\u001b2 \u001e( \u0012\u0000(e1\u0000e2)p 2\u001b ): Using the fact that x\u001e(x) is maximized at x = 1, the su\u000ecient condition for the existence of a unique Nash equilibrium is \r\u0001v(\f) 2\u001b2 \u001e(1) \u0014 bminf\u000b; 1\u0000 \u000bg = b(1\u0000 \f): (C.45) This condition also guarantees that using the suppliers' FOCs for the incentive com- patibility constraints is su\u000ecient. So the optimal \f\u0003 is constrained by (C:40), (C:44) and (C:45). @v(1\u0000\f) @ = 1 (1\u0000\r)(2\f\u00001)2 1 \u00002W (1\u0000 p 1 + 4W (2\f \u0000 1)m) \u00001 2 (1 + 1 (1\u0000\r)(2\f\u00001))[\u0000 12K(2\f\u00001)2 (\f2 + (1 \u0000 \f)2)(1 \u0000 p 1 + 4W (2\f \u0000 1)m) + (1 + 4W (2\f \u0000 1)m)\u0000 12 m \f(1\u0000\f) ] = (1\u0000p1 + 4W (2\f \u0000 1)m) 1 4K(2\f\u00001)2 ( 2 + (1\u0000 \f)2 + 1 (1\u0000\r)(2\f\u00001)) \u0000(1 + 4W (2\f \u0000 1)m)\u0000 12 m 2\f(1\u0000\f) \u0000 12 1(1\u0000\r)(2\f\u00001)(1 + 4W (2\f \u0000 1)m)\u0000 1 2 m \f(1\u0000\f) = (1+4W (2\f\u0000 1)m)\u0000 12 [(1+4W (2\f\u0000 1)m) 12 1 4K(2\f\u00001)2 ( 2+(1\u0000\f)2+ 1 (1\u0000\r)(2\f\u00001)) \u0000 1 4K(2\f\u00001)2 ( 2+(1\u0000\f)2+ 1 (1\u0000\r)(2\f\u00001))+ m \f(1\u0000\f)( 2+(1\u0000\f)2+ 1 (1\u0000\r)(2\f\u00001))\u0000 m2\f(1\u0000\f)\u0000 1 2 1 (1\u0000\r)(2\f\u00001) m \f(1\u0000\f) ] = (1+4W (2\f\u00001)m)\u0000 12f\u0000[(1\u0000 (1+4W (2\f\u00001)m) 12 ) 1 4K(2\f\u00001)2 \u0000 m\f(1\u0000\f) ](\f2+(1\u0000 \f)2 + 1 (1\u0000\r)(2\f\u00001))\u0000 m2\f(1\u0000\f) \u0000 1(1\u0000\r)(2\f\u00001) m2\f(1\u0000\f)g = \u0000(1+4W (2\f\u00001)m)\u0000 12 [ 1 4K(2\f\u00001)2 (1\u0000(1+4W (2\f\u00001)m) 1 2 )(1+4W (2\f\u00001)(\f2+ (1\u0000 \f)2 + 1 (1\u0000\r)(2\f\u00001)) + m 2\f(1\u0000\f)(1 + 1 (1\u0000\r)(2\f\u00001))]; so @v(1\u0000 \f) @ < 0 for \f > 1 2 : (C.46) Let e\f > 1 2 be the solution to 1 + 4W (2\f \u0000 1)m = 0. At e\f, v(1\u0000 e\f) = m 1\u0000\r + 1 4W ( 1 (1\u0000\r)(2e\f\u00001) + 1) = 1 4W (1\u0000\r)(2e\f\u00001) [4W (2e\f \u0000 1)m+ 1 + (1\u0000 \r)(2e\f \u0000 1)] = 14W < 0; by (C:46), the solution to v(1\u0000 \f) = 0 is smaller than e\f; for \f > 1 2 , @ @ (1 + 4W (2\f \u0000 1)m) = 4mW \f(1\u0000\f) < 0: So (C:40) holds at any \f \u0014 e\f, 140 Appendix C. Proof for Chapter 4 (C:44) implies (C:40). The optimal \f\u0003 is a boundary solution constrained by (C:44) and (C:45). 7. Proof of Proposition 4.1 Note that we only consider \f > 1 2 . Let e = e\u0003\f \u0000 e\u00031\u0000\f. From (C:30) and (C:32), e solves e = \r\u0001v(\f)p 2b\u001b \u001e( ep 2\u001b ) 1\u0000 2 \f(1\u0000 \f) : (C.47) So e\u0003\f = \r\u0001v(\f)p 2\fb\u001b \u001e( ep 2\u001b ) = (1\u0000\f)e 1\u00002\f ; e \u0003 1\u0000\f = \fe 1\u00002\f : (C:29) and (C:31) become v1(\f) = \fm\u0000 \f b(e\f) 2 2 + \r[v1(\f)\b( \u0000ep 2\u001b ) + v1(1\u0000 \f)\b( \u0000ep2\u001b )]; v2(1\u0000 \f) = (1\u0000 \f)m\u0000 (1\u0000 \f) b(e1\u0000\f) 2 2 + \r[v2(1\u0000 \f)\b( \u0000ep2\u001b ) + v2(\f)\b( \u0000ep2\u001b )] ) \u0006v(\f) = 1 1\u0000\r (m\u0000 b\f(1\u0000\f)2(2\f\u00001)2 (e)2); \u0001v(\f) = (2\f \u0000 1)m+ b 2 \f(1\u0000\f) 2\f\u00001 e 2 \u0000 \r\u0001v(\f)(2\b( \u0000ep 2\u001b )\u0000 1) ) \u0001v(\f) = [1 + \r(2\b( \u0000ep 2\u001b )\u0000 1)]\u00001[(2\f \u0000 1)m+ b 2 \f(1\u0000\f) 2\f\u00001 e 2]; substituting into (C:47), e = \u0000 \rp 2b\u001b \u001e( ep 2\u001b )( (2\f \u0000 1)2 \f(1\u0000 \f)m+ b 2 e2)[2\r\b( \u0000ep 2\u001b ) + 1\u0000 \r]\u00001: (C.48) So given any \f, the optimal e can be solved from (C:48). @e @ = \rp 2b\u001b e 2\u001b2 \u001e( ep 2\u001b )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2)[2\r\b( \u0000ep 2\u001b ) + 1\u0000 \r]\u00001 @e @ \u0000 \rp 2b\u001b \u001e( ep 2\u001b )( (2\f\u00001)m \f2(1\u0000\f)2 + be @e @ )[2\r\b( \u0000ep 2\u001b ) + 1\u0000 \r]\u00001 \u0000 \r2 \u001b2 (\u001e( ep 2\u001b ))2( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2)[2\r\b( \u0000ep 2\u001b ) + 1\u0000 \r]\u00002 @e @ ) @e @ = [(1 e + e 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2) + be]\u00001 (2\f\u00001)m \f2(1\u0000\f)2 ) @e @ < 0 for \f > 1 2 . Because vB(\f) = vB(1\u0000 \f), we only consider vB(\f) for \f > 12 . From (C:33), vB(\f) = \u0000 2\f(1\u0000\f)e(1\u0000\r)(2\f\u00001) : @vB(\f) @ = 2(2\f2\u00002\f+1) (1\u0000\r)(2\f\u00001)2 e\u0000 2\f(1\u0000\f) (1\u0000\r)(2\f\u00001) @e @ = 2(2\f2\u00002\f+1) (1\u0000\r)(2\f\u00001)2 e\u0000 2m(1\u0000\r)\f(1\u0000\f) [(1e + e2\u001b2 )( (2\f\u00001)2 \f(1\u0000\f)m+ b 2 e2) + be]\u00001 = 2 1\u0000\r [ 2\f2\u00002\f+1 (2\f\u00001)2 e\u0000 m\f(1\u0000\f)((1e + e2\u001b2 )( (2\f\u00001)2 \f(1\u0000\f)m+ b 2 e2) + be)\u00001] ) @2vB(\f) @\f2 = 2 (1\u0000\r)f\u0000 2(2\f\u00001)3 e+ (2\f2\u00002\f+1) (2\f\u00001)2 @e @ \u0000 (2\f\u00001)m \f2(\f\u00001)2 [( 1 e + e 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2)+be]\u00001 141 Appendix C. Proof for Chapter 4 + m \f(1\u0000\f) [( 1 e + e 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2) + be]\u00002[(\u0000 1 e2 + 1 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2) @e @ +(1 e + e 2\u001b2 )( (2\f\u00001)m \f2(\f\u00001)2 + be @e @ ) + b @e @ ]g = 2 (1\u0000\r)f\u0000 2(2\f\u00001)3 e+ (2\f2\u00002\f+1) (2\f\u00001)2 @e @ \u0000 @e @ + @e @ \f(1\u0000\f) (2\f\u00001) [( 1 e + e 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2) + be]\u00001[(\u0000 1 e2 + 1 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2) @e @ +(1 e + e 2\u001b2 )( (2\f\u00001)m \f2(\f\u00001)2 + be @e @ ) + b @e @ ]g = 2 (1\u0000\r)f\u0000 2(2\f\u00001)3 e+ 2\f(1\u0000\f) (2\f\u00001)2 @e @ + @e @ \f(1\u0000\f) (2\f\u00001) [( 1 e + e 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2)+be]\u00001[ @e @ [(\u0000 1 e2 + 1 2\u001b2 ) (2\f\u00001) 2 \f(1\u0000\f)m+ 3b 2 (1 + e 2 2\u001b2 )] + (1 e + e 2\u001b2 ) (2\f\u00001)m \f2(\f\u00001)2 ]g = 2 (1\u0000\r)f\u0000 14\u001b2 e 2 \f(1\u0000\f)(2\f\u00001)2 (\f (1\u0000 \f) (e2 + 32\u001b2 ) + 2m (2\f \u0000 1) 2) + @e @ \f(1\u0000\f) (2\f\u00001) [( 1 e + e 2\u001b2 )( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2)+be]\u00002 (2\f\u00001)m \f2(1\u0000\f)2 [ 3 2\u001b2 (2\f\u00001)2 \f(1\u0000\f)m+3b+ 7be2 4\u001b2 +( e 2\u001b2 )2( (2\f\u00001) 2 \f(1\u0000\f)m+ be2 2 )]g: For \f > 1 2 , because @e @ < 0, @ 2vB(\f) @\f2 < 0, thus vB(\f) is concave in \f. It follows from (C:48) that [2\r\b( \u0000ep 2\u001b ) + 1\u0000 \r]e=\u001e( ep 2\u001b ) = \u0000 \rp 2b\u001b ( (2\f\u00001) 2 \f(1\u0000\f)m+ b 2 e2), so e! \u00001 as \f ! 1. @vB(\f) @ = 2 1\u0000\r [ 2\f2\u00002\f+1 (2\f\u00001)2 e\u0000 m( 1 e + e 2\u001b2 )((2\f\u00001)2m+ b 2 \f(1\u0000\f)e2)+b\f(1\u0000\f)e ] m ( 1 e + e 2\u001b2 )((2\f\u00001)2m+ b 2 \f(1\u0000\f)e2)+b\f(1\u0000\f)e ! 0 as \f ! 1, so @vB(\f) @ ! \u00001 as \f ! 1. Because vB(\f) is concave in \f, the solution of \f to @vB(\f) @ = 0 is smaller than 1, so the optimal \f\u0003 < 1. It follows from the formulas for \u0006v(\f) and \u0001v(\f) that v(1\u0000\f) = 1 2 [ 1 1\u0000\r (m\u0000 b\f(1\u0000\f)2(2\f\u00001)2 (e)2)\u0000(1+\r(2\b( \u0000ep2\u001b )\u00001))\u00001((2\f\u00001)m+ b\f(1\u0000\f) 2(2\f\u00001)e 2)]: Similar to the case under the HWTA rule, the su\u000ecient condition for the ex- istence of a unique Nash equilibrium between the two suppliers is (C:44) together with (C:48). The optimal \f\u0003 obtained from @vB(\f) @ = 0 needs to be checked with the Nash equilibrium condition and the supplier's individual rationality constraint (v(1\u0000 \f) \u0015 0 together with (C:48)). If either constraint is binding, then the optimal \f\u0003 is a boundary solution. 8. Method for numerical calculation for the optimal allocation rule \f1\u0003\u000b (x1; x2) and g(\f) = \u000f Computing the right hand side of (4:8) Using polar coordinates de\fned in (C:14), tan \u0012 = y1 y2 = m+H=(1\u0000\f) 2 m+H=\f2 . By (C:24) and (C:25), the right hand side of (4:8) is 142 Appendix C. Proof for Chapter 4 B = 1 2 p 2\u0019 [ Z \u0012max \u0012min \t(\f) sin \u0012d\u0012 +\t(\f)( 1p 2 + cos \u0012max)\u0000\t(\f)( 1p2 + cos \u0012min)]: (C:28)) (R\u0000 1)m\f4 \u0000 2(R\u0000 1)m\f3 + (R\u0000 1)(m+H)\f2 \u0000 2RH\f +RH = 0; (C.49) where R = tan \u0012. So the value of \f corresponding to \u0012 is determined by (C:49), and Z \u0012max \u0012min \t(\f) sin \u0012d\u0012 can be computed numerically by adding \t(\f) sin \u0012 over \u0012 2 [\u0012min; \u0012max]. \u000f Steps used for numerical calculation of vB, v(\u000b) and e\u00031; e\u00032 Step 1. Compute H. A search method is used. For any H in a range of values, using (C:49) to compute \f corresponding to \u0012 and calculate Z \u0012max \u0012min \t(\f) sin \u0012d\u0012, then using the above numerical method to obtain the RHS of (4:8). If (4:8) holds at a value of H, then H is optimal under the optimal allocation rule b\f\u0003(y1; y2). Step 2. Using the result in Corollary 4.1 and Theorem 4.1, vB = 2 p 2H (1\u0000\r)pb ; e \u0003 1 = p 2H p b ; e\u00032 = p 2H (1\u0000\u000b)pb : Step 3. Compute v(\u000b). Using the result in Corollary 4.1, V = 1 1\u0000 Z Z (b\f\u0003(y1; y2)m\u0000 Hb\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2: (C.50) By (C:15) and (C:16),Z y1 Z y2 \t(b\f\u0003(y1; y2))\u001e(y1)\u001e(y2)dy1dy2 = 12\u0019Z \u0012max \u0012min \t(b\f\u0003(y1; y2))d\u0012+\t(\f)(38 + \u0012min2\u0019 )+ \t(\f)(5 8 \u0000 \u0012max 2\u0019 ): Similar to the calculation for the RHS of (4:8) as above, Z \u0012max \u0012min \t(b\f\u0003(y1; y2))d\u0012 can be computed numerically by using the optimal H in Step 1, and adding \t(\f) over \u0012 2 [\u0012min; \u0012max]. This gives the calculation of V , then for any \u000b, v(\u000b) = \u000bm\u0000H\u000b +\rV . 9. Proof of Theorem 4.3 We use backward induction to solve this problem. Let yit = xit\u0000ei\u0003t \u001b , where i = 1; 2, t = 1; 2; :::, ei\u0003t is Supplier i's optimal e\u000bort level in period t. 143 Appendix C. Proof for Chapter 4 Because there is no future business from the buyer beyond period T + 1, in pe- riod T + 1, both suppliers have no incentive for investment in performance, so both suppliers' optimal e\u000bort levels are 0, vBT+1(\u000bT+1) = 0; v 1 T+1(\u000bT+1) = \u000bT+1m; v 2 T+1(1\u0000 \u000bT+1) = (1\u0000 \u000bT+1)m: \u000f Period T problem: The buyer's pro\ft to go is vBT (\u000bT ) = \u000bT e 1 T + (1\u0000 \u000bT )e2T : (C.51) Supplier 1's expected pro\ft to go is v1T (\u000bT ) = m\u000bT \u0000 b(e 1 T ) 2 2 + \rm Z x1T Z x2T \f\u000bT+1(x 1 T ; x 2 T )f(x 1 T je1T )f(x2T je2T )dx1Tdx2T FOC ) @v1T (\u000bT ) @e1T = \u0000be1T + \rm Z x1T Z x2T \f\u000bT+1(x 1 T ; x 2 T )f 1(x1T je1T )f(x2T je2T )dx1Tdx2T = 0) e1T = \rm b Z x1T Z x2T \f\u000bT+1(x 1 T ; x 2 T )f 1(x1T je1T )f(x2T je2T )dx1Tdx2T : (C.52) Similarly, the FOC for Supplier 2's expected pro\ft to go is @v2T (1\u0000\u000bT ) @e2T = \u0000be2T +\rm Z x1T Z x2T (1\u0000\f\u000bT+1(x1T ; x2T ))f(x1T je1T )f2(x2T je2T )dx1Tdx2T = 0) e2T = \u0000 \rm b Z x1T Z x2T \f\u000bT+1(x 1 T ; x 2 T )f(x 1 T je1T )f2(x2T je2T )dx1Tdx2T : (C.53) (C:52) and (C:53) are necessary conditions for (e1T ; e 2 T ) to be a Nash equilibrium. Substituting (C:52) and (C:53) into (C:51), the buyer's problem becomes vBT (\u000bT ) = Z y1T Z y2T b\f\u000bT+1(y1T ; y2T )\rmb\u001b [\u000bTy1T \u0000 (1\u0000 \u000bT )y2T ]\u001e(y1T )\u001e(y2T )dy1Tdy2T : Pointwise optimization w.r.t. b\fT+1\u000b ) the optimal value is determined by the sign of \rm b\u001b [\u000bTy 1 T \u0000 (1\u0000\u000bT )y2T ]\u001e(y1T )\u001e(y2T ): So to induce (e1T ; e 2 T ) de\fned by (C:52) and (C:53), the optimal allocation rule is a HWTA one such that b\f\u000bT+1(y1T ; y2T ) = ( \fT+1 \u000bTy 1 T > (1\u0000 \u000bT )y2T 1\u0000 \fT+1 \u000bTy1T < (1\u0000 \u000bT )y2T ; (C.54) 144 Appendix C. Proof for Chapter 4 and \f\u000bT+1(x 1 T ; x 2 T ) = ( \fT+1 \u000bT (x 1 T \u0000 e1\u0003T ) > (1\u0000 \u000bT )(x2T \u0000 e2\u0003T ) 1\u0000 \fT+1 \u000bT (x1T \u0000 e1\u0003T ) < (1\u0000 \u000bT )(x2T \u0000 e2\u0003T ) : Using the fact thatZ y \u001e(y)\u001e( 1\u0000 y)dy = \u000bp 2\u0019\u0000 Z y \u0000 \u001e( \u0000 y)dy = \u000bp 2\u0019\u0000 ;Z y \u001e(y)\u001e( 1\u0000 \u000by)dy = 1\u0000 \u000bp 2\u0019\u0000 Z y \u0000 1\u0000 \u000b\u001e( \u0000 1\u0000 \u000by)dy = 1\u0000 \u000bp 2\u0019\u0000 ; (C.55) where \u0000 = p \u000b2 + (1\u0000 \u000b)2, e1T = \rm b\u001b Z y2T [\fT+1 Z 1 y2T 1\u0000\u000b2 \u000b2 y1T\u001e(y 1 T )dy 1 T + (1\u0000 \fT+1) Z y2T 1\u0000\u000b2\u000b2 \u00001 y1T\u001e(y 1 T )dy 1 T ]\u001e(y 2 T )dy 2 T = \rm b\u001b Z y2T (2\fT+1 \u0000 1)\u001e(y2T 1\u0000\u000b2\u000b2 )\u001e(y2T )dy2T = \rm(2\fT+1\u00001)p 2\u0019b\u001b \u000bT \u0000T : So (C:52), (C:53) and (C:54) ) e1\u0003T = \rm(2\fT+1\u00001)\u000bTp2\u0019b\u001b\u0000T ; e 2\u0003 T = \rm(2\fT+1\u00001)(1\u0000\u000bT )p 2\u0019b\u001b\u0000T ; at (e1\u0003T ; e 2\u0003 T ), v1T (\u000bT ) = m(\u000bT + 2 )\u0000 b 2 (e1\u0003T ) 2 = m(\u000bT + 2 )\u0000 b 4\u0019 ( \rm(2\fT+1\u00001)\u000bT b\u001b\u0000T )2; v2T (1\u0000 \u000bT ) = m(1\u0000 \u000bT + \r2 )\u0000 b2(e2\u0003T )2 = m(1\u0000 \u000bT + \r2 )\u0000 b4\u0019 ( \rm(2\fT+1\u00001)(1\u0000\u000bT ) b\u001b\u0000T )2; \u0001vT (\u000bT ) = v 1 T (\u000bT )\u0000 v1T (1\u0000 \u000bT ) = (2\u000bT \u0000 1)[m\u0000 14\u0019b( \rm(2\fT+1\u00001) \u001b\u0000T )2] vBT (\u000bT ) = \u000bT e 1\u0003 T + (1\u0000 \u000bT )e2\u0003T = \rm(2\fT+1\u00001)\u0000Tp2\u0019b\u001b : Because vBT (\u000bT ) is increasing in \fT+1, the optimal \fT+1 is a boundary solution. @\u0000T @\u000bT = 1 2\u0000T @\u00002T @\u000bT = 2\u000bT\u00001 \u0000T ; @ 2\u0000T @\u000b2T = 1 \u00002T [2\u0000T \u0000 (2\u000bT\u00001)2\u0000T ] = 1\u00003T ) @vBT (\u000bT ) @\u000bT = \rm(2\fT+1 \u0000 1)p 2\u0019b\u001b 2\u000bT \u0000 1 \u0000T ; (C.56) @vBT (\u000bT ) @\u000bT = 0 at \u000bT = 1 2 : @2vBT (\u000bT ) (@\u000bT )2 = \rm(2\fT+1 \u0000 1)p 2\u0019b\u001b 1 \u00003T > 0; (C.57) so vBT (\u000bT ) is convex and is minimized at \u000bT = 1 2 ; 145 Appendix C. Proof for Chapter 4 at \u000bT = 1 2 , \u00002T = 1 2 ; and @2vBT (\u000bT ) (@\u000bT )2 = 2\rm(2\fT+1\u00001)p \u0019b\u001b . @v1T (\u000bT ) @\u000bT = m\u0000 2m2(2\fT+1 \u0000 1)2\u000bT (1\u0000 \u000bT ) 2\u0019b\u001b2\u00004T ; (C.58) so for @v1T (\u000bT ) @\u000bT > 0, because \r2m(2\fT+1\u00001)2\u000bT (1\u0000\u000bT ) 2\u0019b\u001b2\u00004T is maximized at \u000bT = 1 2 ; it is required that \r2m(2\fT+1\u00001)2 2\u0019b\u001b2 < 1. @v2T (1\u0000 \u000bT ) @\u000bT = \u0000m(1\u0000 2m(2\fT+1 \u0000 1)2\u000bT (1\u0000 \u000bT ) 2\u0019b\u001b2\u00004T ) = \u0000@v 1 T (\u000bT ) @\u000bT ; (C.59) @\u0001vT (\u000bT ) @\u000bT = @v1T (\u000bT ) @\u000bT \u0000 @v2T (1\u0000\u000bT ) @\u000bT = 2 @v1T (\u000bT ) @\u000bT > 0; @2v1T (\u000bT ) (@\u000bT )2 = \r2m2(2\fT+1 \u0000 1)2(2\u000bT \u0000 1)(1 + 2\u000bT \u0000 2(\u000bT )2) 2\u0019b\u001b2\u00006T ; @2v2T (1\u0000 \u000bT ) (@\u000bT )2 = \u0000@ 2v1T (\u000bT ) (@\u000bT )2 : (C.60) \u000f Period T \u0000 1 problem: The buyer's payo\u000b to go and the suppliers' pro\fts to go are de\fned by (C:3) to (C:5) with t = T \u0000 1 and with g(\u000bt) = g(1\u0000 \u000bt) = 1. (C:6) and (C:7) become e1T\u00001 = b\u001b Z y1T\u00001 Z y2T\u00001 v1T ( b\f\u000bT (y1T\u00001; y2T\u00001))y1T\u00001\u001e(y1T\u00001)\u001e(y2T\u00001)dy1T\u00001dy2T\u00001; e2T\u00001 = b\u001b Z y1T\u00001 Z y2T\u00001 v2T (1\u0000 b\f\u000bT (y1T\u00001; y2T\u00001))y2T\u00001\u001e(y1T\u00001)\u001e(y2T\u00001)dy1T\u00001dy2T\u00001: Substituted into the buyer's objective function, vBT\u00001(\u000bT\u00001) = Z y1T\u00001 Z y2T\u00001 [ 1 b\u001b (\u000bT\u00001v1T (b\f\u000bT )y1T\u00001 + (1\u0000 \u000bT\u00001)v2T (1\u0000 b\f\u000bT )y2T\u00001) +vBT ( b\f\u000bT )]\u001e(y1T\u00001)\u001e(y2T\u00001)dy1T\u00001dy2T\u00001: Let G(b\f\u000bT ) = \u000bT\u00001b\u001b y1T\u00001v1T (b\f\u000bT ) + (1\u0000\u000bT\u00001)b\u001b y2T\u00001v2T (1\u0000 b\f\u000bT ) + vBT (b\f\u000bT ): Pointwise optimization w.r.t. b\f\u000bT and by (C:56), (C:58) and (C:59)) G0(b\f\u000bT ) = 1b\u001b @v1T (b\f\u000bT )@b\f\u000bT (\u000bT\u00001y1T\u00001 \u0000 (1\u0000 \u000bT\u00001)y2T\u00001) + @vBT (b\f\u000bT )@b\f\u000bT ; by (C:57) and (C:60), G00(b\f\u000bT ) = 1b\u001b @2v1T (b\f\u000bT )(@b\f\u000bT )2 (\u000bT\u00001y1T\u00001 \u0000 (1\u0000 \u000bT\u00001)y2T\u00001) + @2vBT (b\f\u000bT )(@b\f\u000bT )2 : Because the distribution of a supplier's performance given her e\u000bort level f(xje) is normal and satis\fes the MLRP, the allocation rule should be such that \f\u000bT (x 1 T\u00001; x 2 T\u00001) 146 Appendix C. Proof for Chapter 4 is increasing in x1T\u00001 and decreasing in x 2 T\u00001, and b\f\u000bT (y1T\u00001; y2T\u00001) is increasing in y1T\u00001 and decreasing in y2T\u00001, so for \u000bT\u00001y1T\u00001 > (1 \u0000 \u000bT\u00001)y2T\u00001, G0(b\f\u000bT ) > 0 and G00(b\f\u000bT ) > 0 for b\f\u000bT > 12 , the optimal b\f\u000b\u0003T (y1T\u00001; y2T\u00001) = \fT ; for \u000bT\u00001y1T\u00001 < (1 \u0000 \u000bT\u00001)y2T\u00001, G0(b\f\u000bT ) < 0 and G00(b\f\u000bT ) > 0 for b\f\u000bT < 12 , the optimal b\f\u000b\u0003T (y1T\u00001; y2T\u00001) = 1\u0000 \fT . Thus the optimal allocation rule is a HWTA one, b\f\u000bT (y1T\u00001; y2T\u00001) = ( \fT \u000bT\u00001y 1 T\u00001 > (1\u0000 \u000bT\u00001)y2T\u00001 1\u0000 \fT \u000bT\u00001y1T\u00001 < (1\u0000 \u000bT\u00001)y2T\u00001 : (C.61) Using (C:55), vBT\u00001(\u000bT\u00001) = \r\u0000T\u00001p 2\u0019b\u001b \u0001vT (\fT ) + 2 [vBT (\fT ) + v B T (1\u0000 \fT )]) @vBT\u00001(\u000bT\u00001) @\u000bT\u00001 = \r 2\u000bT\u00001\u00001p 2\u0019b\u001b\u0000T\u00001 \u0001vT (\fT ); @2vBT\u00001(\u000bT\u00001) @\u000b2T\u00001 = 2\rp 2\u0019b\u001b\u0000T\u00001 \u0001vT (\fT )\u0000 \r 2\u000bT\u00001\u00001p2\u0019b\u001b \u0001vT (\fT ) 2\u000bT\u00001\u00001 \u00003T\u00001 = \rp 2\u0019b\u001b \u0001vT (\fT ) 1 \u00003T\u00001 > 0: e1T\u00001 = b\u001b \u0001vT (\fT ) \u000bT\u00001p 2\u0019\u0000T\u00001 ; e2T\u00001 = b\u001b \u0001vT (\fT ) 1\u0000\u000bT\u00001p 2\u0019\u0000T\u00001 ; @e1T\u00001 @\u000bT\u00001 = \rp 2\u0019b\u001b \u0001vT (\fT ) 1\u0000\u000bT\u00001 \u00003T\u00001 : v1T\u00001(\u000bT\u00001) = m\u000bT\u00001 \u0000 b(e 1 T\u00001) 2 2 + 2 [v1T (\u000bT ) + v 2 T (1\u0000 \u000bT )]; @v1T\u00001(\u000bT\u00001) @\u000bT\u00001 = m\u0000 \r2 2\u0019b\u001b2 (\u0001vT (\fT )) 2 \u000bT\u00001(1\u0000\u000bT\u00001) \u00004T\u00001 ; @2v1T\u00001(\u000bT\u00001) (@\u000bT\u00001)2 = 2 2\u0019b\u001b2 (\u0001vT (\fT )) 2 (2\u000bT\u00001\u00001)(1+2\u000bT\u00001\u00002\u000b2T\u00001) \u00006T\u00001 : \u000f Period t problem: Based on the results for periods T and T \u0000 1, suppose in period t+ 1, e1t+1 = \r\u0001vt+2(\ft+2)p 2\u0019b\u001b \u000bt+1 \u0000t+1 ; e2t+1 = \r\u0001vt+2(\ft+2)p 2\u0019b\u001b 1\u0000\u000bt+1 \u0000t+1 ; @vBt+1(\u000bt+1) @\u000bt+1 = \r 2\u000bt+1\u00001p 2\u0019b\u001b\u0000t+1 \u0001vt+2(\ft+2)) @vBt+1(\u000bt+1) @\u000bt+1 > 0 for \u000bt+1 > 1 2 ; @vBt+1(\u000bt+1) @\u000bt+1 < 0 for \u000bt+1 < 1 2 ; @2vBt+1(\u000bt+1) @\u000b2t+1 = \rp 2\u0019b\u001b \u0001vt+2(\ft+2) 1 \u00003t+1 > 0; @v1t+1(\u000bt+1) @\u000bt+1 = m\u0000 \r2 2\u0019b\u001b2 (\u0001vt+2(\ft+2)) 2 \u000bt+1(1\u0000\u000bt+1) \u00004t+1 > 0; @2v1t+1(\u000bt+1) (@\u000bt+1)2 = 2(2\u000bt+1\u00001)(1+2\u000bt+1\u00002(\u000bt+1)2) 2\u0019b\u001b2\u00006t+1 (\u0001vt+2(\ft+2)) 2 147 Appendix C. Proof for Chapter 4 ) @2v1t+1(\u000bt+1) (@\u000bt+1)2 > 0 for \u000bt+1 > 1 2 ; @2v1t+1(\u000bt+1) (@\u000bt+1)2 < 0 for \u000bt+1 < 1 2 ; @v2t+1(1\u0000 \u000bt+1) @\u000bt+1 = \u0000@v 1 t+1(\u000bt+1) @\u000bt+1 ; @2v2t+1(1\u0000 \u000bt+1) (@\u000bt+1)2 = \u0000@ 2v1t+1(\u000bt+1) (@\u000bt+1)2 : (C.62) Then in period t, referring to the formulation de\fned in Subsection 2 of Appendix C with g(\u000bt) = g(1\u0000 \u000bt) = 1, e1t = b\u001b Z y1t Z y2t v1t+1( b\f\u000bt+1(y1t ; y2t ))y1t \u001e(y1t )\u001e(y2t )dy1t dy2t ; e2t = b\u001b Z y1t Z y2t v2t+1(1\u0000 b\f\u000bt+1(y1t ; y2t ))y2t \u001e(y1t )\u001e(y2t )dy1t dy2t : substituted into the buyer's objective function, vBt (\u000bt) = Z y1t Z y2t [ 1 b\u001b (\u000btv 1 t+1( b\f\u000bt+1)y1t + (1\u0000 \u000bt)v2t+1(1\u0000 b\f\u000bt+1)y2t ) +vBt+1( b\f\u000bt+1)]\u001e(y1t )\u001e(y2t )dy1t dy2t : Let G(b\f\u000bt+1) = \u000btb\u001by1t v1t+1(b\f\u000bt+1) + (1\u0000\u000bt)b\u001b y2t v2t+1(1\u0000 b\f\u000bt+1) + vBt+1(b\f\u000bt+1): Pointwise optimization w.r.t. b\u000bt+1 and by (C:62)) G0(b\f\u000bt+1) = 1b\u001b @v1t+1(b\f\u000bt+1)@b\f\u000bt+1 (\u000bty1t \u0000 (1\u0000 \u000bt)y2t ) + @vBt+1(b\f\u000bt+1)@b\f\u000bt+1 ; by (C:62), G00(b\f\u000bt+1) = 1b\u001b @2v1t+1(b\f\u000bt+1)(@b\f\u000bt+1)2 (\u000bty1t \u0000 (1\u0000 \u000bt)y2t ) + @2vBt+1(b\f\u000bt+1)(@b\f\u000bt+1)2 : Because the distribution of a supplier's performance given her e\u000bort level f(xje) is normal and satis\fes the MLRP, the allocation rule should be such that \f\u000bt+1(x 1 t ; x 2 t ) is increasing in x1t and decreasing in x 2 t , and b\f\u000bt+1(y1t ; y2t ) is increasing in y1t and decreasing in y2t , so for \u000bty 1 t > (1 \u0000 \u000bt)y2t , G0(b\f\u000bt+1) > 0 and G00(b\f\u000bt+1) > 0 for b\u000bt+1 > 12 , the optimalb\f\u000bt+1(y1t ; y2t ) = \ft+1; for \u000bty 1 t < (1 \u0000 \u000bt)y2t , G0(b\f\u000bt+1) < 0 and G00(b\f\u000bt+1) > 0 for b\u000bt+1 < 12 , the optimalb\f\u000bt+1(y1t ; y2t ) = 1\u0000 \ft+1. Thus the optimal allocation rule for period t+ 1 is again a bang-bang one,b\f\u000bt+1(y1t ; y2t ) = ( \ft+1 \u000bty 1 t > (1\u0000 \u000bt)y2t 1\u0000 \ft+1 \u000bty1t < (1\u0000 \u000bt)y2t : Using (C:55), vBt (\u000bt) = \r\u0000tp 2\u0019b\u001b \u0001vt+1(\ft+1) + 2 [vBt+1(\ft+1) + v B t+1(1\u0000 \ft+1)]; @vBt (\u000bt) @\u000bt = \r 2\u000bt\u00001p 2\u0019b\u001b\u0000t \u0001vt+1(\ft+1); @2vBt (\u000bt) @\u000b2t = \rp 2\u0019b\u001b \u0001vt+1(\ft+1) 1 \u00003t : 148 Appendix C. Proof for Chapter 4 e1\u0003t = b\u001b \u0001vt+1(\ft+1) \u000btp 2\u0019\u0000t ; e2\u0003t = b\u001b \u0001vt+1(\ft+1) 1\u0000\u000btp 2\u0019\u0000t ; @e1t @\u000bt = \rp 2\u0019b\u001b \u0001vt+1(\ft+1) 1\u0000\u000bt \u00003t : v1t (\u000bt) = m\u000bt \u0000 b(e 1 t ) 2 2 + 2 [v1t+1(\u000bt+1) + v 2 t+1(1\u0000 \u000bt+1)]; @v1t (\u000bt) @\u000bt = m\u0000 \r2 2\u0019b\u001b2 (\u0001vt+1(\ft+1)) 2 \u000bt(1\u0000\u000bt) \u00004t ; @2v1t (\u000bt) (@\u000bt)2 = 2(2\u000bt\u00001)(1+2\u000bt\u00002(\u000bt)2) 2\u0019b\u001b2\u00006t (\u0001vt+1(\ft+1)) 2: So all the value functions in period t have the same properties as those in period t+ 1. By recursion, the optimal allocation rule for every period takes the same form as in (C:61). Nash equilibrium condition: Note that Z b a ((y)2 \u0000 1)\u001e(y)dy = a\u001e(a)\u0000 b\u001e(b). Second-order conditions for the suppliers' period t problems: at (e1\u0003t ; e 2\u0003 t ), @2v1t (\u000bt) (@e1t ) 2 = \u0000b+ \r\u001b2 Z x1t Z x2t v1t+1( t+1(x 1 t ; x 2 t ))(( x1t\u0000e1\u0003t \u001b )2 \u0000 1)f(x1t je1\u0003t )f(x2t je2\u0003t )dx1tdx2t = \u0000b+ \u001b2 Z y1t Z y2t v1t+1( b\f\u000bt+1(y1t ; y2t ))((y1t )2 \u0000 1)\u001e(y1t )\u001e(y2t )dy1t dy2t = \u0000b + \u001b2 Z y2t [v1t+1(\f) Z 1 y2t 1\u0000\u000bt \u000bt ((y1t ) 2 \u0000 1)\u001e(y1t )dy1t + v1t+1(1 \u0000 \f) Z y2t 1\u0000\u000bt\u000bt \u00001 ((y1t ) 2 \u0000 1)\u001e(y1t )dy 1 t ]\u001e(y 2 t )dy 2 t = \u0000b+ \u001b2 [v1t+1(\f) Z y2t y2t 1\u0000\u000bt \u000bt \u001e(y2t 1\u0000\u000bt \u000bt )\u001e(y2t )dy 2 t\u0000v1t+1(1\u0000\f) Z y2t y2t 1\u0000\u000bt \u000bt \u001e(y2t 1\u0000\u000bt \u000bt )\u001e(y2t )dy 2 t ] = \u0000b < 0, similarly @2v2t (1\u0000\u000bt) (@e2t ) 2 = \u0000b, so (e1\u0003t ; e2\u0003t ) is a Nash equilibrium, and f(e1\u0003t ; e2\u0003t )gt=1;2:::;T constitutes a subgame perfect Nash equilibrium. Let \"i = eit\u0000ei\u0003t \u001b : @2v1t (\u000bt) (@e1t ) 2 = \u0000b+ \r\u001b2 Z x1t Z x2t v1t+1( t+1(x 1 t ; x 2 t ))(( x1t\u0000e1t \u001b )2 \u0000 1)f(x1t je1t )f(x2t je2t )dx1tdx2t = \u0000b+ \u001b2 [ Z y2t Z 1 y2t 1\u0000\u000bt \u000bt v1t+1(\ft+1)((y 1 t \u0000 \"1)2 \u0000 1)\u001e(y1t \u0000 \"1)\u001e(y2t \u0000 \"2)dy1t dy2t + Z y2t Z y2t 1\u0000\u000bt\u000bt \u00001 v1t+1(1\u0000 \ft+1)((y1t \u0000 \"1)2 \u0000 1)\u001e(y1t \u0000 \"1)\u001e(y2t \u0000 \"2)dy1t dy2t ] = \u0000b+ \u001b2 [ Z z2t Z 1 (z2t+\"2) 1\u0000\u000bt \u000bt v1t+1(\ft+1)((z 1 t ) 2 \u0000 1)\u001e(z1t )\u001e(z2t )dz1t dz2t + Z z2t Z (z2t+\"2) 1\u0000\u000bt\u000bt \u00001 v1t+1(1\u0000 \ft+1)((z1t )2 \u0000 1)\u001e(z1t )\u001e(z2t )dz1t dz2t ] 149 Appendix C. Proof for Chapter 4 = \u0000b+ \u001b2 \u0001vt+1(\ft+1) Z z2t (z2t + \"2) 1\u0000\u000bt \u000bt \u001e((z2t + \"2) 1\u0000\u000bt \u000bt )\u001e(z2t )dz 2 t = \u0000b+ \u001b2 1\u0000\u000bt \u000bt \u0001vt+1(\ft+1) Z z2t (z2t+\"2 (1\u0000\u000bt)2 \u00002t +\"2 \u000b2t \u00002t )\u001e(\u0000t \u000bt (z2t+\"2 (1\u0000\u000bt)2 \u00002t ))\u001e(\"2 (1\u0000\u000bt) \u0000t )dz2t = \u0000b+ \u001b2 \u000b2t (1\u0000\u000bt) \u00003t \u0001vt+1(\ft+1)\"2\u001e(\"2 1\u0000\u000bt \u0000t ) \u0014 \u0000b+ \u001b2 \u000b2t \u00002t \u0001vt+1(\ft+1)\u001e(1); similarly @2v2t (\u000bt) (@e2t ) 2 \u0014 \u0000b+ \r\u001b2 (1\u0000\u000bt) 2 \u00002t \u0001vt+1(\ft+1)\u001e(1). So the su\u000ecient condition for the existence of a unique subgame perfect Nash equilibrium in the T period problem is \u001b2 \f2t \u00002t \u0001vt+1(\ft+1)\u001e(1) \u0014 b; which sets a boundary to the value of \ft+1. 10. Proof of Theorem 4.4 To simplify the notation, we omit the subscripts of k\u000b, \u0012\u000b and \u0007\u000b when there is no ambiguity, and use \f for b\f. The suppliers' optimal e\u000bort levels e\u00031 = b Z x1 Z x2 v1( 4\u0003 \u000b (x1; x2))f 1(x1je\u00031)f(x2je\u00032)dx1dx2 = b\u001b Z y2 [v1(\f) Z 1 ky2+ 1 \u001b (\u0012\u0000(e\u00031\u0000ke\u00032)) y1\u001e(y1)dy1 +v1(1\u0000 \f) Z ky2+ 1\u001b (\u0012\u0000(e\u00031\u0000ke\u00032)) \u00001 y1\u001e(y1)dy1]\u001e(y2)dy2 = b\u001b Z y2 (v1(\f)\u0000 v1(1\u0000 \f))\u001e(ky2 + 1\u001b (\u0012 \u0000 (e\u00031 \u0000 ke\u00032)))\u001e(y2)dy2 ) e\u00031 = \r\u0001v \u0003(\f) b\u001b 1p 2\u0019\u0007 exp[\u0000 1 2\u00072\u001b2 (\u0012 \u0000 (e\u00031 \u0000 ke\u00032))2]; and similarly e\u00032 = \r\u0001v\u0003(\f) b\u001b kp 2\u0019\u0007 exp[\u0000 1 2\u00072\u001b2 (\u0012 \u0000 (e\u00031 \u0000 ke\u00032))2] ) \u0012\u0003 = e\u00031 \u0000 ke\u00032, and (4:16) and (4:17) follow. For e\u00031; e \u0003 2 > 0, we need k > 0 and \u0001v \u0003(\f) > 0. v\u0003B(\u000b) = \r\u0001v\u0003(\f) b\u001b p 2\u0019\u0007 +k(1\u0000\u000b)\r\u0001v\u0003(\f) b\u001b p 2\u0019\u0007 + Z y2 [v\u0003B(\f) Z 1 ky2 \u001e(y1)dy1+v \u0003 B(1\u0000\f) Z ky2 \u00001 \u001e(y1)dy1]\u001e(y2)dy2 ) v\u0003B(\u000b) = \r\u0001v\u0003(\f)p 2\u0019b\u001b\u0007 [\u000b+ k(1\u0000 \u000b)] + 2 [v\u0003B(\f) + v \u0003 B(1\u0000 \f)]; (C.63) v\u0003B(\f) = \r\u0001v\u0003(\f)p 2\u0019b\u001b\u0007 [\f + k\f(1\u0000 \f)] + \r2 [v\u0003B(\f) + v\u0003B(1\u0000 \f)]; v\u0003B(1\u0000 \f) = \r\u0001v \u0003(\f)p 2\u0019b\u001b\u00071\u0000 [1\u0000 \f + k1\u0000\f\f] + \r2 [v\u0003B(\f) + v\u0003B(1\u0000 \f)]: Due to symmetry, k1\u0000\f = 1=k\f, it follows that v\u0003B(\f) = v \u0003 B(1\u0000 \f), 150 Appendix C. Proof for Chapter 4 v\u00031(\u000b) = v \u0003 2(\u000b) = v \u0003(\u000b); v\u0003(\u000b) = m\u000b\u0000 1 4\u0019b (\r\u0001v \u0003(\f) \u001b 1 \u0007 )2 + Z y2 [v\u0003(\f) Z 1 ky2 \u001e(y1)dy1 +v\u0003(1\u0000 \f) Z ky2 \u00001 \u001e(y1)dy1]\u001e(y2)dy2 = m\u000b\u0000 1 4\u0019b (\r\u0001v \u0003(\f) \u001b 1 \u0007 )2 + 2 (v\u0003(\f) + v\u0003(1\u0000 \f)): FOC of the supplier's problem: @v1(\u000b) @e1 = \u0000be1 + Z x1 Z x2 v1( 4 \u000b(x1; x2))f 1(x1je1)f(x2je2)dx1dx2 ) @v1(\u000b) @e1 = \u0000be1 + \r\u0001v(\f)p 2\u0019\u0007\u001b exp[\u0000 1 2\u00072\u001b2 (e\u00031 \u0000 ke\u00032 \u0000 (e1 \u0000 ke2))2]; (C.64) @v2(1\u0000\u000b) @e2 = \u0000be2 + \r\u0001v(\f)\u001b kp2\u0019\u0007 exp[\u0000 12\u00072\u001b2 (e\u00031 \u0000 ke\u00032 \u0000 (e1 \u0000 ke2))2]; ) at any equilibrium (e1; e2), e2 = ke1, so at an equilibrium (e1; e2), @v1(\u000b) @e1 = \u0000be1 + \r\u0001v(\f)\u001b 1p2\u0019\u0007 exp[\u0000 1\u0000k 2 2\u001b2(1+k2) (e\u00031 \u0000 e1)2]; @v2(1\u0000\u000b) @e2 = \u0000be2 + \r\u0001v(\f)\u001b kp2\u0019\u0007 exp[\u0000 1\u0000k 2 2\u001b2(1+k2) (e\u00031 \u0000 e1)2]. @v1(\u000b) @e1 = 0) e1 = \r\u0001v(\f)p 2\u0019\u0007b\u001b exp[\u0000 1\u0000 k 2 2\u001b2(1 + k2) (e\u00031 \u0000 e1)2]; (C.65) substituting (C:65) and e2 = ke1 into the supplier's value function, we obtain \u0001v(\f) = m(2\f \u0000 1)\u0000 1 4\u0019b (\r\u0001v(\f) \u001b exp[\u0000 1\u0000k 2 2\u001b2(1+k2\f) (e\u00031 \u0000 e1)2])2 1\u0000k2 1+k2 ) for k\f 6= 1, \u0001v(\f) = 2\u0019b\u001b2(1+k2\f) \r2(1\u0000k2\f) ( s 1 + \r2 exp[\u0000 1\u0000k2 \u001b2(1+k2 ) (e\u00031\u0000e1)2]m(2\f\u00001) \u0019b\u001b2 1\u0000k2 1+k2 \u0000 1); \u0001v\u0003(\f) = 2\u0019b\u001b2(1+k2\f) \r2(1\u0000k2\f) ( r 1 + 2m(2\f\u00001) \u0019b\u001b2 1\u0000k2 1+k2 \u0000 1); and for k\f = 1, \u0001v(\f) = m(2\f \u0000 1); then from (C:63), v\u0003B(\u000b) = p 2\u0019\u001b(1+k2\f) \r(1\u0000\r)\u0007(1\u0000k2\f) [\u000b+ k(1\u0000 \u000b)]( r 1 + 2m(2\f\u00001) \u0019b\u001b2 1\u0000k2 1+k2 \u0000 1): From (C:64), @2v1(\u000b) (@e1)2 = \u0000b+ \r\u0001v(\f) \u001b p 2\u0019\u00073\u001b2 exp[\u0000 1 2\u001b2(1+k2) (e\u00031\u0000ke\u00032\u0000(e1\u0000ke2))2](e\u00031\u0000ke\u00032\u0000(e1\u0000ke2)) \u0014 \u0000b+ \r\u0001v(\f) (1+k2)\u001b2 \u001e(1); similarly @ 2v2(\u000b) (@e2)2 \u0014 \u0000b+ \r\u0001v(\f) (1+k2)\u001b2 \u001e(1). 151 Appendix C. Proof for Chapter 4 For k\f \u0014 1, \u0001v(\f) \u0014 \u0001v\u0003(\f). So the su\u000ecient condition for the existence of a unique Nash equilibrium is \r\u0001v\u0003(\f) (1 + k2)\u001b2 \u001e(1) \u0014 b: (C.66) 11. Proof of Corollary 4.4 Letting k\u000b = 1\u0000 in Theorem 4.4, we obtain the formula for \u0012\u0003\u000b, e \u0003 \f and e \u0003 1\u0000\f, then e\u0003\f = 1\u0000\fe \u0003 1\u0000\f. Also v\u0003B(\u000b) = p 2\u0019\u001b\u00003 \r(1\u0000\r)(2\f\u00001)( q 1 + 2m(2\f\u00001)2 \u0019b\u001b2\u00002 \u0000 1) is independent of \u000b. @v\u0003B(\f) @ = p 2\u0019\u001b \r(1\u0000\r) [ 3\u0000(2\f\u00001)2\u00002\u00003 (2\f\u00001)2 ( q 1 + 2m(2\f\u00001)2 \u0019b\u001b2\u00002 \u0000 1) + \u0000 3 2(2\f\u00001) r 1+ \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 \r2m \u0019b\u001b2 4(2\f\u00001)\u00002\u00002(2\f\u00001)3 \u00004 ] = p 2\u0019\u001b \r(1\u0000\r) r 1+ \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 [\u00008 2\u00008\f+1 (2\f\u00001)2 (1 + \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 \u0000 q 1 + 2m(2\f\u00001)2 \u0019b\u001b2\u00002 ) + 2m \u0019b\u0000\u001b2 ] = p 2\u0019\u001b \r(1\u0000\r)\u0000(2\f\u00001)2 r 1+ \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 [2m \u0019b ( \u001b )2(2\f \u0000 1)4 + \u00002(8\f2 \u0000 8\f + 1) \u0000\u00002(8\f2 \u0000 8\f + 1) q 1 + 2m(2\f\u00001)2 \u0019b\u001b2\u00002 ] = p 2\u0019\u001b\u0000 \r(1\u0000\r)(2\f\u00001)2 r 1+ \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 [2m \u0019b ( \u001b )2 (2\f\u00001) 4 \u00002 +(2(2\f\u00001)2\u00001)(1\u0000 q 1 + 2m(2\f\u00001)2 \u0019b\u001b2\u00002 )]: For 2(2\f \u0000 1)2 \u0014 1; i.e., \f \u0014 1 2 + p 2 4 , @vB(\f) @ > 0; for \f > 1 2 + p 2 4 , 2(2\f \u0000 1)2 \u0000 1 > 0, and noting that 1 \u0000 p1 + a > \u0000a for any a > 0, @v\u0003B(\f) @ > p 2\u0019\u001b\u0000 \r(1\u0000\r)(2\f\u00001)2 r 1+ \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 m(2\f\u00001)2 \u0019b\u00002 ( \u001b )2[2(2\f \u0000 1)2 \u0000 (2(2\f \u0000 1)2 \u0000 1)] = p 2\rm (1\u0000\r)b\u001b\u0000p\u0019 r 1+ \r2m(2\f\u00001)2 \u0019b\u001b2\u00002 > 0, so @v\u0003B(\f) @ > 0 for any \f > 1 2 , therefore, the optimal \f\u0003 is a boundary solution and is constrained by the (IR) constraint or Nash equilibrium constraint. Letting k\f = 1\u0000 in (4:18), \u0001v\u0003(\f) = 2\u0019b\u001b 2(\f2+(1\u0000\f)2) \r2(2\f\u00001) ( q 1 + 2m(2\f\u00001)2 \u0019b\u001b2(\f2+(1\u0000\f)2) \u0000 1): By (C:66), because the \r\u0001v \u0003(\f) (1+k2)\u001b2 \u001e(1) is decreasing in k, the Nash equilibrium con- dition becomesp 2\u0019b\f2 \r(2\f\u00001)pe( q 1 + 2m(2\f\u00001)2 \u0019b\u001b2(\f2+(1\u0000\f)2) \u0000 1) \u0014 b: 152 Appendix C. Proof for Chapter 4 (4:19) with k\u000b = 1\u0000 ) \u0006v\u0003(\f) = v\u0003(\f) + v\u0003(1\u0000 \f) = 1 1\u0000\r [m\u0000 (\r\u0001v \u0003(\f))2 4\u0019b\u001b2 ]. So v\u0003(\f) = 1 2 (\u0006v\u0003(\f) + \u0001v\u0003(\f)) = 1 2 ( 1 1\u0000\r [m\u0000 (\r\u0001v \u0003(\f))2 4\u0019b\u001b2 ] + \u0001v\u0003(\f)); v\u0003(1\u0000 \f) = 1 2 (\u0006v\u0003(\f)\u0000\u0001v\u0003(\f)) = 1 2 ( 1 1\u0000\r [m\u0000 (\r\u0001v \u0003(\f))2 4\u0019b\u001b2 ]\u0000\u0001v\u0003(\f)): The (IR) constraint is v(1\u0000 \f) \u0015 0: 12. Proof of Corollary 4.5 From the results in Theorem 4.4, letting k\u000b = 1, \u0001v(\f) = (2\f \u0000 1)m; \u0006v(\f) = 1 1\u0000\r [m\u0000 (\r(2\f\u00001)m) 2 4\u0019b\u001b2 ]; e\u00031 = e \u0003 2 = \r(2\f\u00001)m 2 p \u0019b\u001b ; \u0012\u0003 = 0: It follows that v(\f) = 1 2 ( 1 1\u0000\r [m\u0000 (\r(2\f\u00001)m) 2 4\u0019b\u001b2 ] + (2\f \u0000 1)m); v(1\u0000 \f) = 1 2 ( 1 1\u0000\r [m\u0000 (\r(2\f\u00001)m) 2 4\u0019b\u001b2 ]\u0000 (2\f \u0000 1)m); vB(\f) = 1\u0000 \u0001v(\f) 2 p \u0019b\u001b = 1\u0000 (2\f\u00001)m 2 p \u0019b\u001b : For v(1\u0000 \f) \u0015 0, 1 2 ( 1 1\u0000\r [m\u0000 (\r(2\f\u00001)m) 2 4\u0019b\u001b2 ]\u0000 (2\f \u0000 1)m) \u0015 0) \r2m(2\f\u00001)2 4\u0019b\u001b2 + (1\u0000 \r)(2\f \u0000 1) \u0014 1) \f \u0014 1 2 + 1 m\r2 ( p (\u0019b\u001b2(1\u0000 \r))2 + \u0019b\u001b2m\r2 \u0000 \u0019b\u001b2(1\u0000 \r)): (C.67) Second-order condition for Supplier 1's problem @2v1(\u000b) (@e1)2 = \u0000b+ \r(2\f \u0000 1) m 2\u001b2 \u0000(e1\u0000e2)p 2\u001b \u001e(\u0000(e1\u0000e2)p 2\u001b ) \u0014 \u0000b+ \rm(2\f\u00001) 2\u001b2 p 2\u0019e , so @ 2v1(\u000b) (@e1)2 \u0014 0 for \f \u0014 1 2 + b\u001b2 p 2\u0019e \rm ; (C.68) which can be obtained similarly for Supplier 2's problem. Because vB(\f) is increasing in \f for any \f, the optimal \f is the boundary solution constrained by (C:67) and (C:68). 153","@language":"en"}],"Genre":[{"@value":"Thesis\/Dissertation","@language":"en"}],"GraduationDate":[{"@value":"2010-05","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0068695","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Program":[{"@value":"Business Administration","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"University of British Columbia","@language":"en"}],"Rights":[{"@value":"Attribution-NonCommercial-NoDerivs 3.0 Unported","@language":"en"}],"RightsURI":[{"@value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/","@language":"en"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Title":[{"@value":"Designing performance based contracts in supply chains","@language":"en"}],"Type":[{"@value":"Text","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/17428","@language":"en"}],"SortDate":[{"@value":"2009-12-31 AD","@language":"en"}],"@id":"doi:10.14288\/1.0068695"}~~