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Essays in operations management Liao, Sha 2015

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Essays in Operations ManagementbySha LiaoB.Sc., Zhejiang University, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Business Administration)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2015c© Sha Liao 2015AbstractThis dissertation addresses two topics in the domain of operations manage-ment. First we study a single utility’s optimal policies under the RenewablePortfolio Standard, which requires it to supply a certain percentage of itsenergy from renewable resources. The utility demonstrates its complianceby holding a sufficient amount of Renewable Energy Certificates (RECs) atthe end of each year. The utility’s problem is formulated as a stochastic dy-namic program. The problem of determining the optimal purchasing policiesunder stochastic demand is examined when two energy options, renewableor regular, are available, with different prices. Meanwhile, the utility canbuy or sell RECs in any period before the end of the horizon in an outsideREC market. Both the electricity prices and REC prices are stochastic. Wefind that the optimal trading policy in the REC market is a target inter-val policy. Sufficient conditions are obtained to show when it is optimal topurchase only one kind of renewable energy and regular energy, and othersto show when it is optimal to purchase both of them. Explicit formulas arederived for the optimal purchasing quantities in each case.In the second essay, we examine the interaction between a buyer (Orig-inal Equipment Manufacturer, OEM) and his supplier during new productdevelopment. A “white box” relationship is assumed: the OEM designs thespecification of the product and outsources the production to his supplier.The supplier may suggest potential specification problems. Our research ismotivated by the fact that the supplier may detect potential specificationproblems, and one cannot take for granted that the supplier would informthe OEM. We solve an optimization problem from the perspective of theOEM. We first prove that it is strictly better for the OEM to design thecontract so that the supplier will inform the OEM should she detects anyflaws. Then we characterize the optimal solutions for the OEM. We alsoperform some sensitivity analysis at the end.iiPrefaceChapter 2 is co-authored by Tim Huh and Mahesh Nagarajan. Chapter3 is co-authored by Hao Zhang and Yimin Wang. In both chapters I wasthe main contributor. I was responsible for developing the models, carryingout analysis and reporting the results, as presented in this dissertation. Atthe same time, co-authors for both chapters have devoted tremendous timediscussing with me and contributed invaluable advice. The background storyof Chapter 3 was inspired by Yimin Wang. Both chapters will be reformattedand submitted for publication in academic peer reviewed journals.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Coping with the Renewable Portfolio Standard: A Utility’sPerspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 REC market . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Wholesale electricity market . . . . . . . . . . . . . . 132.3.3 Retail electricity market . . . . . . . . . . . . . . . . 172.3.4 Objective of the utility . . . . . . . . . . . . . . . . . 172.3.5 Dynamic programming formulation . . . . . . . . . . 172.4 Optimal Policies . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Leveraging Suppliers to Calibrate Product Specification . 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 353.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.1 A descriptive overview . . . . . . . . . . . . . . . . . 363.4 The Model Setup . . . . . . . . . . . . . . . . . . . . . . . . 38ivTable of Contents3.5 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 433.5.1 The supplier’s decision problem . . . . . . . . . . . . 433.5.2 The OEM’s decision problem . . . . . . . . . . . . . . 443.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6.1 Optimization problem N . . . . . . . . . . . . . . . . 473.6.2 Optimization problem I . . . . . . . . . . . . . . . . . 483.6.3 Overall optimal solutions for the OEM . . . . . . . . 563.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66AppendicesA Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 71B Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 84B.1 Analysis for optimization problem N . . . . . . . . . . . . . . 84B.2 Analysis for optimization problem I . . . . . . . . . . . . . . 89B.3 Analysis for overall optimal solutions for the OEM . . . . . . 105vList of Figures2.1 State RPS policies (Wiser and Barbose (2008)). . . . . . . . 52.2 Optimal energy choice in the forward market. . . . . . . . . . 212.3 Optimal strategies in the forward market when single sourcing. 282.4 Optimal strategies in the forward market . . . . . . . . . . . . 293.1 Sequence of events and cases . . . . . . . . . . . . . . . . . . 393.2 Probability of detecting flaws . . . . . . . . . . . . . . . . . . 403.3 The optimal solutions for optimization problem N . . . . . . 483.4 The optimal solutions for optimization problem I when q1 ∈[0, d1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 The optimal solutions for optimization problem I when q1 ∈(1, d1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 The global optimal solutions for the OEM in optimizationproblem I when d2 ≤1− α(1 + 2β)1− α d1. . . . . . . . . . . . . . 553.7 The global optimal solutions for optimization problem I. . . . 563.8 The optimal solutions versus θ when demands are comparable(with d1 = d2 = 0.5, β = 1, p = 4, w = 2, c = 1, h = 0.1,s = 0.2, and α = 0.3). . . . . . . . . . . . . . . . . . . . . . . 583.9 The optimal solutions versus θ when d2 is large (with d1 = 0.8,d2 = 30, β = 20, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, andα = 0.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.10 The optimal solutions versus θ when d2 is small (d1 = 0.8,d2 = 0.2, β = 0.01, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, andα = 0.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.11 The optimal solutions versus θ with intermediate q∗1 presented(with d1 = 0.46, d2 = 0.38, β = 0.8, p = 3, w = 2, c = 1,h = 0.3, s = 0.3, and α = 0.3). . . . . . . . . . . . . . . . . . 593.12 The optimal solutions versus α when demands are comparable(with d1 = d2 = 0.5, β = 1, p = 4, w = 2, c = 1, h = 0.1,s = 0.2, and θ = 0.5). . . . . . . . . . . . . . . . . . . . . . . . 61viList of Figures3.13 The optimal solutions versus α when d2 is large (with d1 =0.8, d2 = 30, β = 20, p = 4, w = 2, c = 1, h = 0.1, s = 0.2,and θ = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.14 The optimal solutions versus α when d2 is small (with d1 =0.8, d2 = 0.2, β = 0.01, p = 4, w = 2, c = 1, h = 0.1, s = 0.2,and θ = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.15 The optimal solutions versus α with intermediate q∗1 presented(d1 = 0.46, d2 = 0.38, β = 0.8, p = 3, w = 2, c = 1, h = 0.3,s = 0.3, and θ = 0.5). . . . . . . . . . . . . . . . . . . . . . . . 623.16 The optimal solutions versus β when demands are comparable(with d1 = d2 = 0.5, p = 4, w = 2, c = 1, h = 0.1, s = 0.2,θ = 0.5, and α = 0.5). . . . . . . . . . . . . . . . . . . . . . . 633.17 The optimal solutions versus β when d2 is large (with d1 =0.2, d2 = 20, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, θ = 0.5,and α = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.18 The optimal solutions versus β when d2 is small (with d1 =0.8, d2 = 0.2, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, θ = 0.5,and α = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.19 The optimal solutions versus β with intermediate q∗1 presented(with d1 = 0.46, d2 = 0.38, p = 3, w = 2, c = 1, h = 0.3,s = 0.3, θ = 0.5, and α = 0.5). . . . . . . . . . . . . . . . . . . 64viiAcknowledgementsMy years of pursuing Ph.D. at the University of British Columbia has beenan immensely rewarding and satisfying journey. I am deeply indebted tomy research advisers, Tim Huh, and Mahesh Nagarajan for their continuoussupport through these years, academically, mentally, and financially. Theyprovided me with every bit of guidance and assistance that I needed duringmy first few semesters. When I was ready to venture into research, they gaveme the freedom to choose my own topic, at the same time spent tremendouseffort discussing with me, and contributed valuable feedback, advice andencouragement.I own a great debt of gratitude to professor Hao Zhang, who collaboratedwith me on the second essay and served on my thesis committee.I am very thankful to professor Yimin Wang at Arizona State University,who collaborated with me on the second essay.I want to thank professor George Zhang, who served on my thesis com-mittee and provided invaluable comments on my research works.My thanks also go to other faculty members of the OPLOG division. Iwould particularly like to acknowledge professor Maurice Queyranne, whowas my instructor for most of the fundamental courses in optimization andoperations management.Finally, I would like to thank Guido Alexander De Maeseneer Junior, forall his love and support.viiiDedicationTo my parents, Qionggao Liao and Shanlian Wang.ixChapter 1IntroductionThis dissertation presents two essays, each contributing to the fields of oper-ations management by attempting to mitigate a gap identified in academicliterature.The first essay examines a newly initiated mechanism on renewableenergy in the United States, known as the Renewable Portfolio Standard(RPS). The mechanism requires each obligated utility to supply a certainpercentage of their energy from renewable resources. Different states mayhave different percentage levels, and the percentage levels will increase overtime. For example, in California, the percentage level was 20% annuallysince 2012, and will be raised up to 25% starting from 2016, and eventually33% starting from 2020 (Wiser and Barbose, 2008).The RPS mechanism is similar to the cap-and trade program, whichhas been widely studied in operations management literature. The wayfor utilities to demonstrate their compliance is by holding Renewable En-ergy Certificates (REC). An REC is given to energy generators for everymegawatt hour of renewable energy they generate. Utilities have the optionof purchasing these RECs with or without the renewable energy that theycame from. At the end of each year, the utilities with shortage in RECsare charged a penalty cost of $50 -$55 per unit of RECs they are short on(Wiser and Barbose, 2008).Since RECs can be traded separately from the underlying energy, anREC market has been formed, where RECs can be sold or purchased. Thetrading is often private, and the trading price of RECs can vary widely overtime and region. Utilities may trade in the REC market to speculate orhedge against the price volatility.We study a single utility’s optimal policies under the RPS. The utility’sproblem is formulated as a stochastic dynamic program. The problem ofdetermining the optimal purchasing policies under stochastic demand is ex-1Chapter 1. Introductionamined when two energy options, renewable or regular, are available, withdifferent prices. Meanwhile, the utility can buy or sell RECs in any periodbefore the end of the horizon in an outside REC market. Both the electric-ity prices and REC prices are stochastic. We find that the optimal tradingpolicy in the REC market is a target interval policy. Some sufficient condi-tions are obtained to show when it is optimal to purchase only one kind ofrenewable energy and regular energy, and others to show when it is optimalto purchase both of them. Explicit formulas are derived for the optimalpurchasing quantities in each case.This paper will be the first one investigating the RPS in OM literature.In general, this paper will be the first one to examine the policy from theperspective of a utility. This may have to do with the fact that the RPS is anewly initiated program, and many states did not start their initial compli-ance year until 2010. Most of the existing literature is from economics andfocuses on the efficacy of the RPS.Our interest is not to discuss the efficacy of the RPS. The RPS has beenrapidly expanding with increasing percentage requirements, and the veryurgent question for those obligated utilities is how to deal with it. We feel itis needed to untangle the trade-offs in choosing energy sources and provideguidelines for utilities to comply with the RPS with minimum cost.The second essay studies the interaction between a buyer (Original Equip-ment Manufacturer, OEM) and his supplier during new product develop-ment. We consider a “white box” buyer-supplier relationship: the OEMowns product specification and outsources the production to a supplier.The supplier may suggest potential specification problems. Our researchis motivated by the fact that the supplier may perceive potential specifica-tion problems based on manufacturer alternatives, local market tastes, anddifferent regulatory mandates, which the OEM firm may not be able to an-ticipate a priori.One cannot take for granted that suppliers will always be willing to pointout specification problems to OEM. Often the supplier’s objective is not per-fectly aligned with that of the OEM’s. For example, specification flaws areoften observed only after the volume production has begun, and hence thesupplier may lose significant business if it points out the specification flawbut no immediate resolutions are available.2Chapter 1. IntroductionGiven that suppliers may not always be willing to point out specifica-tion flaws, one possible solution could be integrating the supplier in projectteams. There has been extant literature discussing the pros and cons of thissolution. The discussions mainly focus on how the OEM should involve thesuppliers, e.g., the timing and depth of supplier involvement, in new productdevelopment. An implicit assumption in these papers is that the supplierwill share process knowledge with the OEM as long as they are “included”.This implicit assumption, however, is not necessarily true. We feel the needto fill the gap in literature by examining whether or not the suppliers arewilling to share their knowledge even when they are included in the projectteam.We started by identifying potential levers that the OEM may use to mo-tivate the supplier to voluntarily point out potential specification flaws. Wefocus on two levers in this essay, the ordering quantities and the contingentcancellation payment. We then solve the optimization problem from theperspective of the OEM. First we solve the optimal strategy of the OEM onthe condition that he will design the contract so that the supplier will notinform even if she detects the flaws. Then we solve the optimal strategy ofthe OEM on the condition that he will design the contract to motivate thesupplier to inform. We compare the optimal profit of the OEM in these twocases and prove that it is strictly better for the OEM to design the contractto motivate the supplier to inform. Having this principle in mind, we thenprovide full description of the optimal solutions of the OEM, and conductsensitivity analysis on some parameters. In specific, we find that the can-cellation payment provided by the OEM should decrease in his capability tofix the flaws in time, and also decrease in the spillover effect from the firstperiod sales.3Chapter 2Coping with the RenewablePortfolio Standard: AUtility’s Perspective2.1 IntroductionIn the past decade, many countries have made enthusiastic efforts in the har-nessing of renewable energy such as hydro, wind, solar and biomass power.In 2004, global investment in renewable energy was $40 billion, and the shareof renewable energy in power capacity expansion was 10%. By 2012, theserose to $244 billion and 42% (McCrone (2013)), respectively. Ambitioustargets were announced, followed by various mechanisms. China aims toprovide 15% of its annual national power from renewable resources by 2020,EU 20% by 2020, and the U.S. 25% by 2025.One new initiative in the U.S. to promote renewable energy productionis the “Renewable Portfolio Standard” (RPS). This policy breaks down theresponsibility for achieving the national target to state level, with manystates in turn requiring utility companies to supply a specified percentage oftheir energy from renewable resources by a given date. There is no federalpolicy. The target for each state is set on an individual basis and is expectedto increase over time. For example, California’s requirement was 20% in2012, but will rise to 25% by 2016 and 33% by 2020. New York’s percentagelevel was 24% in 2013 and will rise to 30% by 2015 (Wiser and Barbose,2008). The RPS has been adopted in 33 out of 50 states of the U.S. (Figure2.1). Similar mechanisms have been adopted in many other countries.Since the restructuring of the U.S. electricity market, most utilities nolonger own power generators themselves, so purchase electricity from awholesale electricity market and then supply their end-users in a retail elec-tricity market. The RPS further stipulates a certain amount of the supply42.1. IntroductionFigure 2.1: State RPS policies (Wiser and Barbose (2008)).be renewable. Given that the RPS is unlikely to be abandoned, and giventhe highly competitive nature of the electricity market, the imperative forutility companies must be to comply in the most cost effective manner. Thispaper builds a theoretical model from the perspective of a utility companyillustrating the tradeoffs of compliance options and suggest strategies en-abling the utility company to comply with the RPS but at minimum cost.How much will it cost companies to comply with their obligations underthe RPS? Consider the case of a California utility in 2011 which suppliedelectricity at an average rate of $132.2 per megawatt hour (Institute For En-ergy Research, 2011). The company was obliged to supply 20% of its energyfrom renewable resources, or face a penalty of $50 per megawatt hour short ofits target (Wiser and Barbose, 2008). Suppose the utility did not supply anyrenewable energy, the RPS would penalize the utility 8% ($50 · 20%/$132.2)of its revenue. One may consider this overestimation. After all, If a utilitycompany could easily meet its obligation, then the RPS would not incursuch a high additional cost. In reality, many utilities in California in 2011failed to meet the required percentage level of the RPS and paid penalties.What’s more, the California Public Utilities Commission placed a cap onRPS charges, so that no utility would pay more than $25 million per yearin penalties (Institute For Energy Research, 2011). The RPS has undoubt-edly resulted in challenges to companies seeking to avoid incurring increased52.1. Introductioncosts but the RPS does provide some flexibility in terms of compliance. Itis the opportunity that this flexibility offers that is the subject of this paper.The RPS works as follows. Renewable energy producers receive from thegovernment one renewable energy certificate (REC) for each megawatt hourof renewable energy they generate. Utilities who do not generate renewableenergy purchase RECs from renewable energy producers to demonstratetheir compliance of the RPS. Utilities can purchase RECs with the underly-ing energy, termed “REC-bundled” energy, or purchase RECs without theunderlying energy, termed “unbundled” RECs. At the beginning of eachyear, the Public Utility Commission (PUC) checks whether utilities havesufficient RECs to cover the required percentage of the output from the pre-vious year, and charges those who do not a penalty. Most states set thepenalty at between $50 and $60 per megawatt hour of renewable energythe utility is deficient or, equivalently, per unit of REC the utility is short(Wiser and Barbose, 2008).Since RECs can be traded separately from the underlying energy, a sec-ondary market has formed, where RECs are bought and sold. We refer thissecondary market as “REC market”. Trading is often private and can bedone bilaterally or assisted by an REC agent (U.S. Department of Energy,2014). The REC trading price varies widely over time, from less than $2to $50-$55 per unit (essentially the RPS penalty cost) (Heeter and Bird,2011). Since the PUC checks utilities’ REC amount only once a year, andsince RECs can be banked without costs, utilities have incentive to trade inthe REC market to speculate or hedge against price volatility.Aside from acquiring RECs, utility companies need to purchase electric-ity from the wholesale electricity market in order to supply their customers.Because electricity is nonstorable, and because utilities are obliged to sat-isfy the customer demands instantly with exactitude though the customerdemands are volatile and difficult to predict, utilities seek instantaneous sup-plies. This is provided by an Independent System Operator (ISO) througha“spot” market. In the spot market, parties bid on or offer electricity ona real-time basis. The ISO matches the aggregate supply and demand,announces the market-clearing price known as the “spot” price, and coor-dinates the transmission of electricity. Utilities are thus able to purchaseelectricity and supply their customers instantaneously. Inevitably, the spotprice is highly volatile, as indicated by an empirical study of the spot pricefrom 2000 to 2002 at the PJM market, the largest ISO in the United States62.1. Introduction(Longstaff and Wang, 2004). During peak hours, from 1 pm to 6 p.m., thespot price rocketed to above $1000 per megawatt hour, nearly 21 times theaverage value during these hours. At the other extreme, during the night,the spot price could drop to zero.In an attempt to mitigate the volatility of the spot price, ISOs run a“day-ahead” market before the spot market, where utilities can purchaseelectricity to be delivered during the subsequent day. The electricity priceat the day-ahead market tends to be less volatile than the spot price, thoughnot significantly so (Longstaff and Wang, 2004). To achieve greater pricestability, utilities purchase most of their power directly from power gener-ators in a “bilateral” market, not through ISOs. In Texas, for instance,95-98% of the energy is traded bilaterally (Hortacsu and Puller, 2008).Power trading occurs first in the bilateral market, then in the day-aheadmarket, and lastly on the spot market. The bilateral market and the day-ahead market are “forward” market, as the delivery times specified in thecontracts are in the future. Spot markets, on the other hand, are instanta-neous. Utilities purchase power from the forward market, and then adjusttheir output in the spot market so that they can meet the customer demandsinstantly. Specifically, if utilities electricity bought from the forward mar-ket is insufficient to meet the demand, they will purchase additional powerfrom the spot market; otherwise they will sell redundant power to the spotmarket. ISOs only facilitates the trading of regular energy (non-renewableenergy or renewable energy with the REC removed). Therefore if utilitieswant to acquire REC-bundled energy, they can only do so through bilateralcontracts.We formulate a utility company’s problem as a stochastic dynamic pro-gram. We divide an RPS compliance year into multiple periods. Duringeach period, the company purchases electricity from a wholesale electricitymarket to meet a random demand in a retail electricity market. At the sametime, the company buys or sells unbundled RECs in an REC market. TheRECs can be carried over. The wholesale electricity market is modeled astwo settlements, first a forward market and then a spot market. In the for-ward market, the utility chooses between REC-bundled and regular energy.Then the demand occurs and the utility balances its output through the spotmarket in order to meet the demand. Periodic decisions made by the utilityrelate to trading actions in the REC market and the purchasing quantitiesfrom each source in the forward market. The utility seeks to minimize the72.1. Introductiontotal discounted cost during the planning horizon.Intuitively, a utility should sell RECs when its REC amount is in ex-cess of need and should purchase RECs otherwise. How exactly should acompany reach its decision that its REC level is in excess of need? In agiven period, even if a utility has decided that its RECs are in excess, and itencounters low priced RECs, it might still have an incentive to hoard RECs.In this way, if the REC price increases later, the utility has secured someRECs purchased at a lower price, and it can even sell RECs to gain profit.Similarly, how exactly should a company decide that the REC price is lowenough? Our analysis shows that a utility company should follow a targetinterval policy in the REC market: it should purchase or sell RECs in orderto adjust its REC level between two thresholds. These two thresholds de-pend on a series of state variables, including the REC price, the electricityprice, the mount of RECs the utility has on hand and he amount of demandthe utility has met. We show that the thresholds are monotonic in some ofthe state variables.We assume there are only two energy options in the forward electricitymarket, REC-bundled energy and regular energy. One unit of REC-bundledenergy is, essentially, one unit of regular energy plus one unit of unbundledREC. Comparing these two options is more subtle than simply comparingtheir market prices. This is because from the perspective of the utility,RECs hold different values depending on the situation. For instance, RECsare more valuable to a utility with an insufficient numbers of RECs and atthe end of the horizon. We describe some conditions under which a util-ity company should purchase only one kind of renewable energy or regularenergy, and present others that show when it is optimal to purchase bothof them. In each case, we describe optimal purchasing quantities. We alsoanalyze the monotonicity of these optimal purchasing quantities and explainthe intuition behind a utility’s optimal policies.The remainder of the paper is structured as follows: We start with aliterature review in Section 2.2. Then we present the model formulationin Section 2.3 and describe the optimal strategy of the utility company inSection 2.4. Finally we conclude with a discussion in Section 2.5.82.2. Literature Review2.2 Literature ReviewAs far as we know, this paper is the first one investigating the RPS fromthe perspective of a utility. This may have to do with the fact that theRPS is a newly initiated program, and many states did not start their ini-tial compliance year until 2010. Most of the existing literature focus on theefficacy of the RPS. Wiser and Barbose (2008) conducted empirical studiesto show that approximately 76% of new renewable capacity was contributedby the states with the RPS in 2007. They therefore concluded that the RPSwas effective in promoting renewable energy. Other supporters of RPS in-clude Menz and Vachon (2006) and Hailu and Adelaja (2008). Some critics,include Bushnell et al. (2007) and Michaels (2007), pointed out that themechanism of RPS will cause unbalance development of different renewableresources. Yin and Powers (2009) asserted that the cross border trade ofRECs can “significantly” weaken the renewable energy development in cer-tain states with scarce renewable resources, since the utilities in these statescan purchase RECs from other states, essentially they are “paying the freshair in other states”.Our interest is not to discuss the efficacy of the RPS. The RPS has beenrapidly expanding with increasing percentage requirements, and the very ur-gent question for those obligated utilities is how to deal with it. This paperuntangles the trade-offs in choosing energy sources and provides guidelinesfor utilities to act both in the electricity market and REC market, and soto comply with the RPS with minimum cost.One important feature of our model is that the utility can trade un-bundled RECs in an outside market. The information of the trading priceis very limited. One reason might be that the REC market emerges as abyproduct of the RPS and thus has a short history. Another reason is thatmost of the REC tradings are private. Heeter and Bird (2011) provides astatus report of the REC market in 2010, from which we know that the RECprice is quite volatile.The REC trading scheme resembles the allowances trading scheme underthe cap-and-trade program. There is scant literature in Operations Researchstudying the REC trading scheme, but there is extant literature on the cap-and-trade program. Factories, who emit pollutants during production, aregranted with an initial allocation of allowances from the government. Theamount of the allowances represents the total amount of pollutants the fac-92.2. Literature Reviewtories can emit in the following year. The factories may purchase or sellallowances in an outside market. The factories who fail to hold sufficientallowances at the end of the year will face a fine.There are some similaritiesif we compare the problems faced by utilities under the RPS and factoriesunder the cap-and-trade. Factories under the cap-and-trade program oftenface options involving technology choice, capacity planning, investment inpollution abatement equipment, pricing decisions, etc. In addition, they canbank and trade allowances in a secondary market.In many papers in the cap-and-trade literature, the entities of interestare electricity generators. For instance, Subramanian et al. (2007) mod-eled a three-stage game in an oligopoly setting. Electricity generators makeinvestment decisions, bid for allowances and then decides production quan-tities. The selling price of the products is represented as an inverse demandfunction. Drake et al. (2010) studied a single electricity generator’s technol-ogy choice and capacity decisions in a two-period stochastic setting. Theprice is fixed. The demand is realized once with penalty for unmet demand.Zhao et al. (2010) discussed the long-term impacts of different emissions al-location schemes on electricity generator’s investment and pricing decisions,with a demand function decreasing in price. Our paper differs from thesepapers in three ways. First, they did not include the electricity generators’option of trading allowances in an outside market. Second, In these papers,electricity generators face options with fixed and known costs, or they caninternalize the costs as decision variables. They make independent decisionson the prices and quantities they produce. In our paper, utilities purchaseelectricity at exogenous and random prices, and then sell electricity at fixedprices. The demands are random, and need to be satisfied. Third, these pa-pers suggest long-term decisions of electricity generators, such as technologyor capacity choice. While in our paper, a utility makes periodic observationsand decisions.Gong and Zhou (2013) presented a muti-period model studying a fac-tory’s technology choice and production plan under the cap-and-trade pro-gram. The factory was not a electricity generator. They also captured thefactory’s option of trading allowances in a secondary market with stochas-tic prices. Two factors contribute to the differences between our modeland theirs, the difference between the cap-and-trade program and the RPSmechanism, and the specialty of electricity market. Under the cap-and-tradeprogram, a fixed amount of allowance is allocated to the factory at the be-ginning, and the factory can decide independently the producing quantity,102.2. Literature Reviewand therefore the amount of allowances it needs. The amount of allowancesthe factories need is fixed and endogenous. Under the RPS, there is no ini-tial allocation. The required amount of RECs is a percentage of the totaldemand, which is both random and exogenous. The amount of RECs theutility needs is random and exogenous. Furthermore, for factories under thecap-and-trade program, the technologies are with known and fixed emissionlevels and costs. For utilities, the electricity prices in the wholesale marketare random. As a result, the utility needs to make periodic observations ofthe electricity prices, and keeps updating the cumulative demand it has sup-plied, and then makes periodic decisions. Most importantly, the products offactories under the cap-and-trade are regular products and can be carried onfrom period to period. On the other hand, electricity is prohibitively expen-sive to be stored. As a result, utilities face a unique multi-leveled wholesaleelectricity market. In general, a utility under the RPS and a factory underthe cap-and-trade are facing different problems. The only resemblance isthe trading scheme of certificates.Another important feature of our model is the multiple settlement struc-ture of the wholesale electricity market. The wholesale electricity market indifferent regions can be at different status and under different regulations.Many technical reports discuss the deregulation or the potential design ofthe wholesale electricity market, to name a few, see Stoft (2002), Trebil-cock and Hrab (2004), Chao and Wilson (1999), and Boucher and Smeers(2001). Another stream of research on electricity market is from financeand economics. They focus on how electricity should be priced and howthe price actually behaves in those centralized markets hosted by ISOs, in-cluding day-ahead market, the hour-ahead market and the real-time market.For example, Longstaff and Wang (2004) collected data from the PJM mar-ket, and did empirical work to study the relationship between the electricityprices in the day-ahead market and the real-time market.Our paper contributes to the literature in both applied and method-ological sides. We investigated the impact of the RPS on a single utilityand have provided practical guidelines for the utility to purchase electricityand to trade RECs. On the methodological side, although some results ofour paper look similar to those in Gong and Zhou (2013), the models aredifferent.112.3. Model Formulation2.3 Model FormulationWe model an RPS compliance year as a finite horizon of T periods, indexedas 1, · · · , T . At the beginning of each period t, a utility first observes theREC price and the electricity prices. The utility then buys or sells unbun-dled RECs in an REC market, and purchases electricity from a wholesaleelectricity market in order to supply a random demand in a retail electricitymarket. At the end of the horizon, the utility needs to hold sufficient RECsto cover a percentage, say α, of its total supply during the entire horizon.If the utility does not have sufficient RECs, it pays a penalty cost pi perunit it is short on. In practice, the tradings in electricity markets and theREC market can be continuous and simultaneous, but we think of them asperiodic decisions, and assume that in each period the utility trades in theREC market first, then the electricity markets.The utility acts as a price taker in the REC market, the wholesale elec-tricity market and the retail electricity market. Therefore, we assume thesemarkets as exogenous, stochastic, and independent from each other. In thissection, we start by describing the set-ups of these three markets in subsec-tions 2.3.1, 2.3.2 and 2.3.3. We then discuss the objective of the utility insubsection 2.3.4. After that, we state the sequence of events in subsection2.3.5. Lastly, we write the dynamic programming in subsection 2.3.6.2.3.1 REC marketAt the beginning of each period t, the utility observes the REC price in anREC market. We use two random variables, bt and st, to represent the trad-ing prices of RECs, with bt being the per unit cost of buying RECs and stbeing the per unit revenue of selling RECs. Later we refer bt as the buyingprice of REC and st as the selling price of REC. We allow bt and st to bedifferent and assume bt is greater or equal to st. The gap between them canbe resulted from the transaction cost and the bid-ask spread in REC trading(Holt et al., 2011; Gillenwater, 2008). We denote Rt = (bt, st) and assume{Rt = (bt, st), 1 ≤ t ≤ T} forms a Markov chain for trackability.Theoretically, the trading prices of REC should not exceed the penaltycost of the RPS, as otherwise, the utility will have no incentive to buy anyRECs, and will wait until the end of the horizon and pays the penalty cost.This is also observed in practice (Heeter and Bird, 2011). Therefore we as-122.3. Model Formulationsume that Pr(st ≤ bt ≤ γT−t+1pi) = 1, with γ being the one-period discountfactor, 0 < γ ≤ 1.We assume the utility can sell RECs even when it has zero or negativeamount of RECs on hand. This assumption is made since the utility’s RECamount will be checked only once at the end of the horizon.Let xt and x¯t be the utility’s REC level before and after the REC tradingin period t, respectively. If the utility buys RECs, then x¯t > xt, and theutility generates a cost of bt(x¯t − xt). Otherwise x¯t < xt, and the utilityearns a revenue of st(xt − x¯t).2.3.2 Wholesale electricity marketAfter trading RECs, the utility needs to purchase electricity from a wholesaleelectricity market in order to satisfy its end-users. We capture two flavorsof the wholesale electricity market, the multiple settlements and the highvolatility of the spot price. To incorporate the first flavor, we assume theutility purchases electricity from a forward market first and then balances itsoutput (either buy or sell electricity) in a spot market. In the forward mar-ket, the utility purchases power directly from power producers. In the spotmarket, tradings are centralized through an ISO. To incorporate the secondflavor, we impose some properties on the utility’s expected cost functionin the spot market to reflect the utility’s tendency to avoid trading in thespot market. This tendency is driven by the high volatility of the spot price.Forward MarketIn the forward market, the utility company can purchase power from a va-riety of sellers through a variety of power purchase contracts. We simplifythe pool of sellers as two power producers, one sells REC-bundled renewableenergy and the other sells regular energy. In addition, we assume that thepower purchase contracts are unit-price forward contracts and the specifiedprices are for only one period. Specifically, in period t, REC-bundled re-newable energy can be purchased at p1t per unit, and regular energy canbe purchased at p2t per unit. These prices are are only for period t. Theprices for next period may be different. Since the electricity prices are ex-ogenous for the utility, we assume the prices are random variables, and132.3. Model Formulation{(p1t, p2t), 1 ≤ t ≤ T} is a Markov chain for trackability. It is also reason-able to assume REC-bundled renewable energy should be valued more thanregular energy, since one unit of REC-bundled renewable energy includesboth one unit of regular energy and one unit of REC. Nevertheless, electric-ity prices can be unpredictable, and all the results in this paper hold withor without this assumption. The prices (p1t, p2t) in the forward market arereferred to as forward prices. We denote Pt = (p1t, p2t).We assume there is no capacity constraint on the two power producers.They can provide as much electricity the utility requests. We make thisassumption because most of the contracts in the forward market are only“financially binding” (Stoft et al., 1998). Financially binding contracts en-sure the utility will, in the end, receive the exact amount of electricity atthe exact price specified in the contracts. If the power producer is not ableto provide sufficient amount of electricity specified in the contract in anyperiod, the utility will buy electricity from the spot market, and the powerproducer will compensate the expense.Let y1t be the amount of REC-bundled energy the utility purchases andy2t be the total amount of electricity the utility purchases from the forwardmarket, then y2t− y1t is the amount of regular energy the utility purchases.Spot MarketAfter the utility purchased electricity from the forward market, a randomdemand Dt is realized. If the demand is more (less) than the amount theutility had purchased from the forward market, the utility will need to buyadditional (sell redundant) electricity in the spot market, incurring expenses(revenue). The buying or selling will be conducted at the spot price withno bid-ask spread.The utility does not make decisions in the spot market. The demandhas to be satisfied, the spot price is exogenous, and the expense or revenuewill take place. However, the utility had to take into account the expenseor revenue to be incurred in the spot market when it made decisions in theforward market. At that time, neither the demand nor the spot price wasrevealed, and the utility needed to make decisions in the forward marketbased on an expectation on the expense or revenue to be incurred in thespot market. The utility’s expectation is two fold including the demand142.3. Model Formulationand the spot price. It is appealing to write the utility’s expectation as∫∞y2tet(z − y2t)fDt(z) dz −∫ y2t−∞ et(y2t − z)fDt(z) dz, where et is utility’s ex-pectation on the spot price, y2t is the amount of electricity the utility hadpurchased in the forward market, and fDt(·) is the probability density func-tion of the customer demand Dt. However, we do not use this function asutility’s expectation because this function relies on an accurate prediction ofthe spot price, which is unrealistic, and leads to some far-fetched strategiesof utilities. Under this function, if the utility expects the spot price et tobe greater than the forward price of regular energy p2t, it will purchase asmuch as it can in the forward market, sell the excessive amount into thespot market, and make profit on the price difference. On the other hand, ifthe utility expects et < p2t, it will purchase nothing in the forward market,and relies entirely on the spot market to satisfy the customer demand. Bothof these cases are very different from what we observe in practice, wherethe utility purchase most of the energy needed in the forward market andtrading in the spot market only when necessary.We define Gt(y2t) as the utility’s expectation on its expense or revenue tobe incurred in the spot market, where y2t is the amount of electricity it haspurchased from the forward market. In the following, we use one exampleof Gt(y2t) to present some desirable properties of Gt(y2t). The specific formof Gt(y2t) is not restricted to this example.Gt(y2t) =∫ ∞y2tG+t (z − y2t)fDt(z) dz −∫ y2t−∞G−t (y2t − z)fDt(z) dz. (2.1)In equation (2.1), the first term represents the case when the demand isgreater than the utility’s energy on hand. Here z represents the realizeddemand. If z > y2t, the utility would need to purchase z − y2t amountof additional energy. We assume the expense is a function of z − y2t, andwrite it as G+t (z − y2t). Therefore∫∞y2tG+t (z − y2t)fDt(z) dz is the utility’sexpectation on its expense of purchasing additional energy when the demandis greater than its energy on hand. Note that G+t (z − y2t) is nonnegative.We make the following assumptions:• G+t (z − y2t) decreases in y2t. This assumption simply means that theless electricity the utility has on hand, the more it needs to purchase,and thus the more it spends in the spot market.• G+t (z−y2t) is convex in y2t. This assumption implies that the marginalcost of buying electricity is increasingly more expensive when the util-152.3. Model Formulationity purchases more. Hence the utility wants to purchase at little aspossible in the spot market.• G+t (z − y2t) is diffrentiable, and |dG+t (z − y2t)/dy2t| > p2t. Underthis assumption, the per unit cost of buying electricity is always moreexpensive than the unit price of regular energy in the forward market.Hence the utility strictly prefers purchasing in the forward market.The second term represents the case when the demand z is less thanthe utility’s energy on hand y2t. In this case, the utility would need to selly2t − z amount of excessive energy. We assume the revenue is a function ofy2t − z, and write it as G−t (y2t − z). Therefore∫ y2t−∞G−t (y2t − z)fDt(z) dzis the utility’s expectation on its revenue of selling excessive energy whenthe demand is less than its energy on hand. Note that G−t (y2t − z) is alsononnegative. We make the following assumptions:• G−t (y2t − z) increases in y2t. This assumption simply means that themore electricity the utility sells to the spot market, the more revenueit earns.• G−t (y2t − z) is concave in y2t.• G−t (y2t−z) is diffrentiable, and dG−t (y2t−z)/d y2t approaches 0 as y2tapproaches infinity. These two assumptions suggest that the marginalrevenue of selling electricity to the spot market decreases as the utilitysells more, and will approach zero eventually as the selling amountapproaches infinity.• dG−t (y2t − z)/d y2t < p2t. This assumption implies that the per unitrevenue of selling electricity to the spot market is strictly less than theunit price of regular energy in the forward market. Hence there is noeconomic incentive for the utility to over purchase electricity from theforward market so as to sell into the spot market.In aggregate Gt(y2t) (2.1) is the utility’s expectation on its expense orrevenue in the spot market, with a positive value implying expense and anegative value implying revenue. Later we refer Gt(y2t) as the “expectedbalancing cost” in the spot market. There are four properties of Gt(y2t):• Gt(y2t) decreases in y2t;• Gt(y2t) is convex in y2t;162.3. Model Formulation• |dGt(y2t)/dy2t| > p2t when Gt(y2t) ≥ 0, and |dGt(y2t)/dy2t| < p2twhen Gt(y2t) ≤ 0;• dGt(y2t)dy2t → 0 as y2t → +∞.The specific form of Gt(y2t) should not be restricted to (2.1), but thefour properties should preserve. As we will see, the first and second prop-erties are important for deriving optimal strategies. The third and fourthproperties exclude some unreasonable boundary solutions.2.3.3 Retail electricity marketWe assume the customer demands in different periods are independent ran-dom variables, and the demand in period t is distributed with a continuousand strictly positive probability density function fDt(·). In addition, weassume the customer demands are independent from the REC prices andthe electricity prices. In practice, these random variables can be correlated.From the perspective of a utility, however, REC market and electricity mar-kets are exogenous, and the correlation between them is not the focus of thispaper.2.3.4 Objective of the utilityWe assume the utility’s objective is to minimize the expected total dis-counted cost over the planning horizon. We ignore the utility’s revenue fromselling electricity to its customers since the utility has little control over itsoperations in the retail electricity market. The retail electricity rates arecapped by the state government, and the demand elasticity is fairly small(Borenstein, 2009). We thus focus on utility’s strategy in the wholesale elec-tricity market and the REC market, not the retail electricity market.2.3.5 Dynamic programming formulationAt the beginning of each period t, utility observes its REC amount on handxt as well as its cumulative demand ut, cumulative demand being the de-mand the utility has supplied from from period 1 to period t−1. In addition,the utility also observes the buying and selling prices of unbundled RECs(Rt = (bt, st)), as well as the prices of REC-bundled energy and regular172.3. Model Formulationenergy (Pt = (p1t, p2t)). The utility then begins its decision making processby trading in the REC market, adjusting its REC level from xt to x¯t. Afterthat, the utility decides how much energy to purchase from the forward mar-ket. The utility purchases y1t amount of REC-bundled energy and y2t − y1tamount of regular energy. Finally, the demand Dt realizes, the utility bal-ances its output in the spot market, generating cost or revenue Gt(y2t). Atthe end of period t, the utility’s REC level updates to xt+1 = x¯t + y1t,and its cumulative demand level updates to ut+1 = ut + Dt. At the end ofthe horizon, if the utility does not have enough RECs, it will be charged apenalty cost.Let Vt(xt, ut, Rt, Pt) be the minimal expected cost of utility from periodt to the end of the horizon given the utility’s REC level xt, the cumulativedemand ut, the REC prices Rt and the energy prices Pt. Then the utilitysolves the following dynamic program.Vt(xt, ut, Rt, Pt)= minx¯ty2t≥y1t≥0{bt(x¯t − xt)+ − st(xt − x¯t)+ + p1ty1t + p2t(y2t − y1t)+Gt(y2t) + γEt[Vt+1(x¯t + y1t, ut +Dt, Rt+1, Pt+1)]}. (2.2)In the optimization equation (2.2), the decision variables are x¯t, y1t andy2t. These decision variables represent the three decisions the utility makesin each period, i.e., how many RECs to buy or sell, how much REC-bundledenergy to purchase, and how much regular energy to purchase. There is noconstraint on x¯t, because he utility is allowed to buy or sell RECs irrespon-sible of the amount of RECs it has on hand. The constraints on y1t and y2tmake sure the utility purchases non-negative amounts of energy from powerproducers. On the right hand side of (2.2), bt(x¯t − xt)+ − st(xt − x¯t)+ isthe utility’s expense or revenue from REC trading, p1ty1t + p2t(y2t − y1t) isthe utility’s expense to purchase electricity in the forward market, Gt(y2t)is utility’s expectation on its expense or revenue in the spot market, andfinally is the minimal expected cost from period t + 1 till the end of thehorizon. We use Et[·] to denote EDt [E(Rt+1,Pt+1)[·|(Rt, Pt)]].The value function at the end of the horizon isVT+1(zT+1, uT+1) = pi(αuT+1 − zT+1)+. (2.3)If the utility does not have sufficient RECs to cover α percent of its totalsupply during the horizon (uT+1), it will be charged pi dollars per unit ofRECs it is short. The salvage value of excessive RECs is set to be zero.182.4. Optimal Policies2.4 Optimal PoliciesIn this section, we give a complete characterization of utility’s optimal poli-cies in both the REC market and the electricity market. We divide theutility’s decision making process in each period into two stages in time se-quence. At stage one, the utility buys or sells RECs in the REC market. Atstage two, the utility purchases electricity in the forward market.The next theorem establishes the intuition that the utility should sellRECs when it has more RECs on hand, and should be buy RECs when ithas less. Specifically, the utility should follow a target interval policy withtwo thresholds Lt(ut, Rt, Pt) and Ht(ut, Rt, Pt). If the utility’s REC levelis higher than Ht(ut, Rt, Pt), it should sell RECs; if the utility’s REC levelis lower than Lt(ut, Rt, Pt), it should purchase RECs; if the utility’s REClevel is between these two thresholds, it should not trade in the REC market.We will give more detailed description of these two thresholds later whenwe move on to the utility’s optimal policies in the electricity market.Theorem 2.4.1 In each period t, t = 1, . . . , T , given state (xt, ut, Rt, Pt),the utility’s optimal REC trading policy is a target intervel policy with twostate-dependent thresholds Lt(ut, Rt, Pt) and Ht(ut, Rt, Pt), withLt(ut, Rt, Pt) ≤ Ht(ut, Rt, Pt). The optimal REC level after REC tradingcan be characterized asx¯∗t =Lt(ut, Rt, Pt) if xt ≤ Lt(ut, Rt, Pt);xt if Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt);Ht(ut, Rt, Pt) if xt ≥ Ht(ut, Rt, Pt).The threshold structure comes from the convexity of the cost-to-go function(Lemma A.0.1). There are two thresholds since the buying and selling priceof RECs may be different. Because the buying price of RECs is more thanthe selling price of RECs, if the utility purchases RECs, it should purchaseup to a lower target level than if it sells them.Both of the thresholds depend on other state variables, including thecumulative demand, the REC prices and the electricity prices. In order todevelop the monotonic property of the thresholds, we need the followinglemma.Lemma 2.4.2 The value function Vt(x, u, R, P ) is submodular on (x, u).192.4. Optimal PoliciesThe submodularity of the value function implies that REC is a economiccomplement of the cumulative demand and leads to the following propositionthat both of the thresholds increase in the cumulative demand. At the endof the horizon, the compulsory REC amount the utility needs to hold is apercentage of the cumulative demand it has supplied. Therefore if the utilityhas supplied more demand, it should be more conservative to sell RECs andmore willing to buy RECs, which is reflected as higher target interval.Proposition 2.4.3 Lt(ut, Rt, Pt) and Ht(ut, Rt, Pt) increase in ut.After the utility traded RECs in the REC market, it will purchase electric-ity from the forward electricity pmarket. There are two products to choose,REC-bundled energy and regular energy. REC-bundled energy will be sep-arated as two parts upon purchase. One part is energy, which feeds intothe power grids equivalently as regular energy. The other part is RECs,which can be stored for the RPS obligation till the end of horizon or soldfor revenue before that. In some sense, these two products are substitutes.The next proposition describes certain conditions under which it is op-timal for the utility to purchase only one of the two products. We identify4t = p1t − p2t, the price difference between REC-bundled energy and regu-lar energy, to be a critical value when compare these two products. In theremainder of this paper we refer 4t as the “intrinsic” REC price, for it isessentially the price of the RECs from REC-bundled energy. We define theoptimal purchasing quantities of REC-bundled energy and regular energy inperiod t as y∗1t and y∗2t, respectively.Proposition 2.4.4 In each period t, t = 1, . . . , T , given state (xt, ut, Rt, Pt),the utility’s optimal energy choice in the forward market can be characterizedas(a) If 4t ≥ bt, it is optimal to purchase only regular energy, i.e., y∗1t = 0;(b) If 4t ≤ st, it is optimal to purchase only REC-bundled energy, i.e.,y∗2t = y∗1t;(c) If st < 4t < bt,• when xt ≤ Lt(ut, Rt, Pt), it is optimal to purchase only REC-bundled energy, i.e., y∗2t = y∗1t;202.4. Optimal Policies• when xt ≥ Ht(u, Rt, Pt), it is optimal to purchase only regularenergy, i.e., y∗1t = 0.Figure 2.2: Optimal energy choice in the forward market.Proposition 2.4.4 is presented with Figure 2.2. If 4t ≥ bt, i.e., the in-trinsic REC price is greater than the buying price of RECs, we claim thatREC-bundled energy is dominated by regular energy in terms of price, sothat the utility has no economic incentive to purchase REC-bundled energy.To explain this, write 4t ≥ bt as p1t ≥ p2t + bt. In this case, if the utilitypurchases one unit of regular energy and one unit of REC, and combinethem together, it can get essentially the same product as one unit of REC-bundled energy, but at a cheaper price.If 4t ≤ st, i.e., the intrinsic REC price is less than the selling price ofRECs, we claim that regular energy is dominated by REC-bundled energyin terms of price, so that the utility has no economic incentive to purchaseregular energy. To explain this, write 4t ≤ st as p1t − st ≤ p2t. In thiscase, by purchasing one unit of REC-bundled energy and selling the REC,the utility gets essentially the same product as one unit of regular energy,212.4. Optimal Policiesbut at a cheaper price.Proposition 2.4.4 (c) consider the case where st < 4t < bt, i.e, the in-trinsic REC price is between the selling price and the buying price of RECs.In this case, we consider REC-bundled energy and regular energy to be com-petitive in price, and the optimal strategy of the utility is more subtle thanone might expect. The utility should make its energy choice based on itsREC level at the beginning of period t.We explain the intuition of Proposition 2.4.4 (c) by contradiction. Writest < 4t < bt as p1t < p2t+bt and p2t < p1t−st. When xt ≤ Lt(u, Rt, Pt), byTheorem 2.4.1, a utility who follows the optimal policy in the REC marketshould have bought some unbundled RECs at stage one. If the utility pur-chases some regular energy on top of that, for one unit of regular energy itpurchases together with a unit of REC it has already bought, it could havegotten them by purchasing one unit of REC-bundled energy at a cheaperprice as p1 < p2 + bt. Thus there is no incentive for the utility to purchaseregular energy. Similarly, when xt ≥ Ht(ut, Rt, Pt), the utility should havesold some RECs at stage one. If the utility purchases some REC-bundledenergy on top of that, then in aggregate it is purchasing regular energy,which is actually available at a cheaper price since p2t < p1t − st. Thus inthis case, it is always better for the utility to purchase only regular energy.There is one circumstance that is not included in Proposition 2.4.4, whichis if st < 4t < bt and Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt). In this case, sofar by Theorem 2.4.1, we know that the utility should not trade in the RECmarket. We haven’t talked about utility’s optimal energy choice in the for-ward market. For that, additional analysis is required. However, before wecome to this case, we use next two theorems to give a detailed descriptionof utility’s optimal purchasing quantities under the scenarios correspondingto Proposition 2.4.4 (a) and (b), when the utility is single sourcing.Theorem 2.4.5 In each period t, t = 1, . . . , T , given state (xt, ut, Rt, Pt),if 4t ≥ bt, then the optimal purchasing quantities of the utility in the forwardmarket, (y∗1, y∗2), can be characterized as y∗1t = 0, y∗2t = S2t(p2t), whereS2t(p2t) = arg miny≥0{p2ty +Gt(y)}. (2.4)Moreover, S2t(p2t) decreases in p2t.222.4. Optimal PoliciesTheorem 2.4.5 specifies the optimal purchasing quantities for the casepresented in Proposition 2.4.4 (a), and is presented in the fourth column ofFigure 2.3. In period t, if the intrinsic REC price 4t is above the buyingprice of unbundled REC bt, it is optimal for the utility to use the followingstrategy to comply with the RPS: purchasing regular energy to supply theend-users and buying unbundled RECs separately. These two activities doesnot interact with each other. As results, the optimal purchasing quantitydepends only on the price of regular energy, and it decreases with the price.From the definition of S2t(p2t) (2.4), we can see that the third assump-tion we made about Gt(y) ensure a positive and finite value for S2t(p2t).When y = 0, the utility had purchased nothing from the forward market,thus Gt(y) is positive, representing the utility’s cost to purchase additionalenergy from the spot market. From the third assumption we made aboutGt(y), we know that the derivative of Gt(y) at y = 0 is less than −p2t,thus p2ty + Gt(y) is strictly decreasing at y = 0, thus S2t(p2t) > 0. Onthe other hand, assume that the demand is bounded, then S2t(p2t) will bea finite number. Consider when y is large enough so that the utility gener-ates revenue by selling excessive energy to the spot market. In that case,Gt(y) is negative, and the derivative of Gt(y) is more than −p2t. Thereforep2ty + Gt(y) is strictly increasing when y is large enough, thus S2t(p2t) isfinite.Theorem 2.4.6 In each period t, t = 1, . . . , T , given state (xt, ut, Rt, Pt),if4t ≤ st, then the optimal purchasing quantities of the utility in the forwardmarket, (y∗1, y∗2), can be characterized asy∗1t = y∗2t =SL1t(p1t, bt) if xt ≤ Lt(ut, Rt, Pt);s1t(xt, ut, Rt, Pt) if Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt);SH1t (p1t, st) if xt ≥ Ht(ut, Rt, Pt).(2.5)whereSL1 (p1t, bt) = arg miny≥0{(p1t − bt)y +Gt(y)}, (2.6)s1t(xt, ut, Rt, Pt) = arg miny≥0{p1ty +Gt(y)+ γE[Vt+1(xt + y, ut +Dt, Rt+1, Pt+1)]}, (2.7)232.4. Optimal PoliciesSH1t (p1t, st) = arg miny≥0{(p1t − st)y +Gt(y)}. (2.8)Moreover, the two thresholdsLt(ut, Rt, Pt) = wLt (ut, Rt, Pt)− SL1t(p1t, bt),Ht(ut, Rt, Pt) = wHt (ut, Rt, Pt)− SH1t (p1t, st), (2.9)where wLt (ut, Rt, Pt), wHt (ut, Rt, Pt) are the optimal REC levels at theend of period t when the utility buys or sells RECs in the REC marketrespectively.Theorem 2.4.6 specifies the optimal purchasing quantities for the casein Proposition 2.4.4 (b), and is presented in the second column of Figure2.3. If the intrinsic REC price 4t is less than the selling price of REC st,it is optimal for the utility to use the following strategy to comply with theRPS: supplying all of the customer demand with renewable energy. The op-timal purchasing quantity is given by (2.5). If the utility has bought (sold)RECs in the REC market, the optimal purchasing quantity is SL1 (p1t, bt)(SH1 (p1t, st)). These two quantities are independent of utility’s REC level orcumulative demand. If the utility has not done any REC trading, the op-timal purchasing quantity is s1t(xt, ut, Rt, Pt), which depends on its REClevel and cumulative demand.Note that the third and fourth assumptions we made about Gt(y) ex-clude some unreasonable boundary values for SL1 (p1t, bt) and SH1 (p1t, st).From the third assumption of Gt(y), we know that dGt(y)/dy is less than−p2t at y = 0. Therefore we have p1t−bt+dGt(y)/dy < p1t−bt−p2t = 4t−bt.Because the condition in Theorem 2.4.6 is 4t < st, we have 4t < bt. There-fore, (p1t − bt)y + Gt(y) is strictly decreasing when y = 0. Therefore fromthe definition we know that SL1 (p1t, bt) > 0. Similarly we can show thatSH1 (p1t, st) > 0. On the other hand, when y approaches infinity, from thefourth assumption of Gt(y), we know that dGt(y)/dy approaches 0, so thatthe first derivative of (p1t − bt)y + Gt(y) approaches p1t − bt. We did notmake any assumption regarding the relationship between p1t and bt becausethe electricity prices can be unpredictable. However, in most cases, it wouldbe reasonable to assume that in the same period, one unit of REC-bundledenergy should be more expensive than one unit of unbundled REC, i.e.,p1t > bt, then we have SL1 (p1t, bt) being finite. Similar argument can bemade about SH1 (p1t, st) if we assume p1t > st.242.4. Optimal PoliciesIn rare cases, if REC-bundled energy costs less than the buying priceof unbundled RECs in period t, i.e., p1t < bt, intuitively the utility shouldnever purchase any unbundled RECs. This is in line with the results inthe Theorem. Given p1t < bt, according to the descriptions (2.6) and (2.9),the optimal purchasing quantity SL1 (p1t, bt) −→ +∞, and the lower thresh-old Lt(ut, Rt, Pt) −→ −∞. Since the lower threshold approaches minusinfinity, the utility’s REC level will never be lower than that, and thus willnever purchase RECs. In other words, the case where z ≤ Lt(ut, Rt, Pt)will not happen. Similarly, if p1t < st, then according to the descriptions(2.8) and (2.9), the optimal purchasing quantity SH1 (p1t, st) −→ +∞, andthe higher threshold Ht(ut, Rt, Pt) −→ −∞. Since the higher threshold ap-proaches minus infinity, the utility’s REC level will be above that, and theutility should sell an infinite amount of unbundled RECs, since it can gainprofit by selling unbundled RECs and then purchasing REC-bundled energy.We also give another characterization of the two thresholds in the RECtrading policy. We define wt = x¯t + y1t as the REC level at the end ofperiod t. Define wLt (ut, Rt, Pt) as the optimal REC level at the end of pe-riod t given that the utility had purchased RECs to increase its REC levelto L(ut, Rt, Pt). Similarly, define wHt (ut, Rt, Pt) as the optimal REC levelat the end of period t given that the utility had sold RECs to decreaseits REC level to H(ut, Rt, Pt). If the utility has an REC level lower thanLt(ut, Rt, Pt), it would purchase some RECs to increase its REC level upto Lt(ut, Rt, Pt). Then since the utility will purchase SL1 (p1t, bt) amount ofREC-bundled energy, the end-of period REC level wLt (ut, Rt, Pt) should bea sum of Lt(ut, Rt, Pt) and SL1 (p1t, bt). Therefore, we have Lt(ut, Rt, Pt) =wLt (ut, Rt, Pt)−SL1t(p1t, bt). Similar analysis can be done to Ht(ut, Rt, Pt).Next we develop some monotonic properties of the optimal purchasingquantities given in Theorem 2.4.6.Proposition 2.4.7 (a) SL1t(p1t, bt) decreases in p1t and increases in bt.(b) SH1t (p1t, st) decreases in p1t and increases in st.(c) s1t(xt, ut, Rt, Pt) decreases in xt and increases in ut.(d) SL1t(p1t, bt) ≥ s1t(xt, ut, Rt, Pt) ≥ SH1t (p1t, st).If 4t ≤ st, the utility should purchase only REC-bundled energy. Asresults, when the price of REC-bundled energy goes up, the purchasing252.4. Optimal Policiesquantities go down. Thus both SL1t(p1t, bt) and SH1t (p1t, bt) decrease in p1t.When xt ≤ Lt(ut, Rt, Pt), the utility buys RECs. It also purchases theamount SL1t(p1t, bt) of REC-bundled energy. If the buying price of REC in-creases, the utility would have an incentive to gain RECs through buyingREC-bundled energy instead of buying unbundled RECs. This explains whySL1t(p1t, bt) increases in bt.When xt ≥ Ht(ut, Rt, Pt), the utility sells RECs. It also purchases theamount SH1t (p1t, st) of REC-bundled energy. If the selling price of REC in-creases, the utility would have an incentive to purchase more REC-bundledenergy so that it can sell into the REC market. This explains why SH1t (p1t, st)increases in st.When Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt), the utility neither buys norsells RECs. The only REC source is REC-bundled energy. If there is higherenergy demand, i.e., higher ut, the utility needs to buy more RECs asthe compulsory level of REC is proportional to the energy demand. Thuss1t(xt, ut, Rt, Pt) increases in ut. On the other hand, if the utility has moreRECs to start with, i.e., higher xt, the utility needs less RECs. Thuss1t(xt, ut, Rt, Pt) decreases in xt.Proposition 2.4.7 (d) implies that as the REC level increases, the amountof REC bundled energy the utility purchases decreases. This is reasonablebecause one purpose that the utility purchases REC-bundled energy is togain RECs, if it has more RECs to begin with, then it will need less.The following theorem demonstrates the optimal purchasing quantitiesunder the scenario corresponding to Proposition 2.4.4 (c). If st < 4t < bt,when the utility buys or sells unbundled RECs, it should do single sourcingas well, and the optimal purchasing quantities are independent of its REClevel and cumulative demand.Theorem 2.4.8 In each period t = 1, . . . , T , given state (xt, ut, Rt, Pt), ifst < 4t < bt,(a) When xt ≤ Lt(ut, Rt, Pt), it is optimal to purchase only REC-bundledenergy, and y∗1t = y∗2t = SL1t(p1t, bt);(b) When xt ≥ Ht(ut, Rt, Pt), it is optimal to purchase only regular energy,and y∗1t = 0, y∗2t = S2t(p2t).262.4. Optimal PoliciesMoreover, the two thresholdsLt(ut, Rt, Pt) = wLt (ut, Rt, Pt)− SL1t(p1t, bt), (2.10)Ht(ut, Rt, Pt) = wHt (ut, Rt, Pt). (2.11)Theorem 2.4.8 is presented in the third column of Figure 2.3. The re-sult is different from Theorem 2.4.5 and 2.4.6 in that the energy choicedepends not only on the relationship between the intrinsic REC price andthe REC prices, but also on the utility’s REC level. If the utility is in needof RECs, REC-bundled energy is more attractive, the utility should pur-chase SL1t(p1t, bt) amount of REC-bundled energy. In this case, the REClevel at the end of period t should be a sum of the lower threshold forREC trading and the amount of REC-bundled energy the utility purchased,thus wLt (ut, Rt, Pt) = Lt(ut, Rt, Pt) + SL1t(p1t, bt), thus Lt(ut, Rt, Pt) =wLt (ut, Rt, Pt)−SL1t(p1t, bt). If the utility sells RECs, regular energy is moreattractive, the utility should purchase S2t(p2t) amount of regular energy. Inthis case, the REC level at the end of period t should be the same as thehigher threshold, i.e., Ht(ut, Rt, Pt) = wHt (ut, Rt, Pt).Finally, we show how the utility should choose the product when st <4t < bt and Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt). This will complete ourcharacterization of the utility’s optimal policy.Theorem 2.4.9 In each period t = 1, . . . , T , given state (xt, ut, Rt, Pt), ifSt < 4t < bt, there exists a pair of thresholds (lt(ut, Rt, Pt), ht(ut, Rt, Pt))satisfyingLt(ut, Rt, Pt) ≤ lt(ut, Rt, Pt) ≤ ht(ut, Rt, Pt) ≤ Ht(ut, Rt, Pt), (2.12)such that the optimal purchasing quantities in the forward market, (y∗1t, y∗2t),can be characterized as(a) When Lt(ut, Rt, Pt) < xt ≤ lt(ut, Rt, Pt), it is optimal to purchase onlyREC-bundled energy, and y∗1t = y∗2t = s1t(xt, ut, Rt, Pt);(b) When lt(ut, Rt, Pt) < xt < ht(ut, Rt, Pt), it is optimal to purchase bothREC-bundled energy and regular energy, and y∗1t = w4t (ut, Rt, Pt) −xt, y∗2t = S2t(p2t);(c) When ht(ut, Rt, Pt) ≤ xt < Ht(ut, Rt, Pt), it is optimal to purchaseonly regular energy, and y∗1t = 0, y∗2t = S2t(p2t);272.4. Optimal PoliciesFigure 2.3: Optimal strategies in the forward market when single sourcing.Moreover, the two new thresholdslt(ut, Rt, Pt) =w4t (ut, Rt, Pt)− S2t(p2t),ht(ut, Rt, Pt) =w4t (ut, Rt, Pt),where w4t (ut, Rt, Pt) is the optimal REC level at the end of period twhen the utility does not trade RECs in the REC market.Note that Theorem 2.4.8 and Theorem 2.4.9 together characterize util-ity’s optimal policy in the forward market if St < 4t < bt, and these re-sults are presented in the third column of Figure 2.4. These two Theoremstogether states that if xt ≤ lt(ut, Rt, Pt), it is optimal to purchase onlyREC-bundled energy. If xt ≥ ht(ut, Rt, Pt), it is optimal to purchase onlyregular energy. It is only when lt(ut, Rt, Pt) < xt < ht(ut, Rt, Pt), bothregular energy and REC-bundled energy are purchased. The properties ofthe optimal purchasing quantities stated in Proposition 2.4.7 also applieshere.282.5. ConclusionFrom the third column of Figure 2.4, we can see that as the REC levelincreases, the utility is in less need of RECs, thus it is gradually switchingfrom REC-bundled energy to regular energy.Figure 2.4: Optimal strategies in the forward market .2.5 ConclusionA utility under the RPS faces the challenge to comply with the regulationwith minimum cost. The electricity prices and REC prices are stochasticand revealed at the beginning of each period. The utility observes theseprices, as well as its REC level and cumulative demand level, and make pe-riodic decisions in trading RECs and purchasing electricity in the forwardmarket. In specific, the optimal REC trading policy is a target intervalpolicy. If the utility’s REC level is lower than the lower threshold, then itshould purchase RECs to raise its REC level to the lower threshold. If theutility’s REC level is higher than the higher threshold, then it should sellRECs to reduce its REC level to the higher threshold. If the utility’s REClevel is between these two thresholds, it should not do REC trading.292.5. ConclusionAfter the REC trading, the utility purchases electricity from the for-ward market. The optimal purchasing quantities are summarized in Figure2.4. We identify the ”intrinsic” REC price, the price difference betweenREC-bundled energy and regular energy, as a critical value for utility’s en-ergy choice. When the intrinsic REC price is less than the selling priceof unbundled RECs, the utility should exclusively purchase REC-bundledenergy (the second column of Figure 2.4). When the utility trades RECs,the optimal purchasing quantities are irrelevant with the utility’s REC levelor cumulative demand. While when the utility is not trading RECs, theoptimal purchasing quantity would be based on the utility’s REC level andcumulative demand. What’s more, from the top to the bottom, the utility’sREC level increases, and it is in less urgent need for RECs, thus the optimalpurchasing quantity decreases. When the intrinsic REC price is more thanthe buying price of unbundled RECs, the utility should exclusively purchaseregular energy (the forth column of Figure 2.4). In this case, the purchasingquantity is irrelevant with the utility’s REC level, cumulative demand, orthe REC prices, it only depends on the price of regular energy. When theintrinsic REC price is between the selling price and the buying price of un-bundled RECs, the utility’s energy choice should depend on its REC level(the third column of Figure 2.4). We show that only when the utility’s REClevel is between the two additional thresholds, it is optimal for the utility topurchase both kinds of energy. In this case, from the top to the bottom, theutility’s REC level increases, the utility becomes less keen in RECs, thusthe utility gradually switches from REC-bundled energy to regular energy.We believe this paper can be a starting point for studying more sophisti-cated systems involve multiple utilities such as the market equilibrium withinelastic demand and the influence of REC price on the market equilibrium.One can also study a variation of our model with elastic demand and includepricing as a decision variable of the utility, since that is the vision of a futureelectricity market.30Chapter 3Leveraging Suppliers toCalibrate ProductSpecification3.1 Introduction“In many cases, the supplier simply executes the designspecifications from the manufacturer. If there is a design issue,a quality audit may not pick this up. It may be perfectlyproduced to a faulty design.”—Corporate Executive Board, (Gilligan, 2010).Setting right product specifications is a vital function for any firm, and in-correctly or inappropriately set production specifications can significantlyimpact a firm’s sales and reputation. As such, firms often closely exam-ine their internal design and engineering processes to ensure specificationsmatch what the market want. Specification flaws or mismatches, however,may persist even with best intentions from within the firm, and in this casethe firm may benefit from tapping its suppliers’ expertise to calibrate prod-uct specification.Globalization has afforded suppliers ample opportunities to learn, de-velop, and accumulate unique and often tacit product and process knowl-edge. Such supplier-held knowledge can help firms calibrate and refine theirproduct specifications to create successful products in the market. (Petrick,2012) It is especially valuable when product specifications interact subtlywith production process and technology choice: the supplier may perceivepotential specification problems based on material/manufacturing alterna-tives, local/regional market tastes, and/or different regulatory mandates,which the OEM firm may not be able to anticipate a priori.313.1. IntroductionPetersen et al. (2005) noted that “suppliers, because of their product andprocess knowledge or expertise, may have more realistic information on thetradeoffs involved in achieving particular goals. Such goals are not limited tocost but often include product performance characteristics (such as weight,size, speed, etc.). The buying company will have the ultimate authority ingoal setting, but the suppliers involvement can help in setting goals that areachievable.” Similarly, Ragatz et al. (2002) reported that ”using the knowl-edge and expertise of suppliers to complement internal capabilities may helpreduce concept to customer cycle time, costs, quality problems,” and that “interest in such efforts is growing.”From an industry’s perspective, for example, suppliers (contract manu-facturers) in chemical process industries can help to improve product specifi-cations by suggesting “alternative chemical pathways” to a product. (Graff,2014) Louis Assante, president of the Contemporary Cosmetic Group, notedthat as a contract manufacturer it often “make suggestions for improvementsor new technologies in personal care to our clients”. (Jeffries, 2004) Similarobservations can be found in many other industries as well. In particular,Petrick (2012) noted that there is ample empirical evidence that supplierheld knowledge can be important in creating successful products in the mar-ket place.One cannot take for granted, however, that suppliers will always be will-ing to point out potential specification problems by sharing their productprocess knowledge with the OEM firm. Often the OEM firm’s objective isnot perfectly aligned with that of the supplier’s, and hence the supplier maynot be willing to suggest improvements or point out specification flaws. Thesupplier, for example, may notice that a particular ingredient may nega-tively impact the product’s fit to the local market’s taste, but it may knowfor sure whether the OEM could rapidly engineer an alternative specification.In addition, specification flaws are often observed only after the volume pro-duction has begun, and hence the supplier could lose significant business if itpoints out the specification flaw but no immediate resolutions are available.In 2007, for example, Dell had to discontinue its “pearl white” color specswith XPS notebooks when dust contamination problem was found with vol-ume production runs not small test runs. (Cheng and Lawton, 2007) Herewe focus on the incentive of the suppliers. We do not discuss ethic issuesand reputation damage for suppliers.Given that suppliers may not always be willing to suggest improvements323.1. Introductionor point out specification flaws, we seek to understand what factors maymotivate the supplier to voluntarily help the OEM firm improve productspecifications. The extant literature in supplier integration has examinedthe pros and cons of including suppliers in project teams (Ragatz et al.,2002; Hoegl and Wagner, 2005; Koufteros et al., 2005; Das et al., 2006;Parker et al., 2008). This stream of literature focuses primarily on how theOEM firm should involve the suppliers, e.g., the timing and depth of sup-plier involvement, in product development effort. An implicit assumptionis that the suppliers will share tacit product and process knowledge withthe OEM firm as long as they are “included”. Relatively sparse attentionhas been paid, however, to whether the suppliers are willing to share theirinsights with the OEM firm even if they are included in the team.We first examine when it is in the supplier’s interest to voluntarilyhelp the OEM to improve product specifications or pinpoint specificationflaws. This question is particularly relevant when the OEM firm and itssupplier form a “white box” relationship, where “buyer consults with sup-plier on buyer’s design, discussion are held with suppliers about specifica-tions/requirements but the buying company makes all design and specifi-cations decisions. (Handfield and Lawson, 2007) Our research framework isin general not appropriate for the “black box” setting where “design is pri-marily supplier driven, based on buyer’s performance specifications. As wefocus on the “white box” type of relationships, we do not explore long termbusiness contracts with commodity type of products. In other words, we areinterested in industries with “fast clock speed”.Intuitively, the supplier would suggest to the OEM about potential speci-fication issues only if doing so helps the supplier’s current and/or near-futurebusinesses. Given that the OEM controls product specification, the suppliercannot be faulted for any quality problems associated with specificationproblems. Nevertheless, the supplier may still be motivated to help theOEM improve product specifications if the supplier’s current or near-futurebusiness is otherwise in jeopardy. Note that in this paper we ignore thesupplier’s outside options, which could either reinforce or diminish the sup-plier’s incentive to improve product specifications. As the supplier becomesless dependent on the OEM firm’s business, it becomes less concerned aboutlosing the particular business with the OEM firm and therefore may havea stronger incentive to suggest specification problems. On the other hand,however, the supplier also becomes less interested in securing the OEM firm’sbusiness, which could dampen the supplier’s incentives. The combined effect333.1. Introductioncould be ambiguous and oftentimes are influenced by specific organizationalculture and inter-firm relationships.We then study the OEM’s optimal strategy. The OEM’s profit dependson the supplier’s decision to inform or not inform. The OEM, however, doeshave the ability to design the contract to direct the supplier to choose toinform or not inform, whichever brings more profit to the OEM himself. Ifthe supplier informs, the benefit for the OEM is that he might be able torectify the flaw in time and thus will be able to satisfy the current perioddemand. In addition, the reputation of the products will bring a positivespillover effect to the demand in the future. The downside of the supplierinforming is that the OEM might not be able to rectify the flaw in time. Inthat case, the OEM may need to cancel the order and pays a cancellationpayment to the supplier. This cancellation payment might be expensive,especially given the fact that it might be just the incentive for the supplierto inform the OEM. What’s more, the delay of releasing the products mayhurt the OEM’s market demand in the future.If the supplier does not inform the OEM, the OEM will benefit by avoid-ing the cancellation payment. The OEM will be able to realize the flaw bycollecting feedback and response from the customer. If the demand for thecurrent period is fairly small and the cancellation payment is large, this maybe better for the OEM. The downside for the OEM if the supplier does notinform is obvious. The OEM may lose the demand in the current period ifthe customers return the products. The defected products in the current pe-riod may cause reputation damage, which will negatively affect the demandin the future. Again, if the demand for the current period is fairly small,this might not be a sufficient incentive for the OEM to direct the supplierto inform. With all these factors entangled, the OEM’s optimal strategy isambiguous.Therefore we solve two optimization problems from the perspective ofthe OEM. We first examine the optimal strategy for the OEM to maximizehis profit given that he does not want the supplier to inform. We thenexplore the optimal strategy for the OEM to maximize his profit given thathe wants the supplier to inform. After that, we compare the optimal profitsof the OEM in these two cases. We prove that it is strictly better for theOEM to design the contract so that the supplier will inform if she detectsany flaw. We give full description of the optimal solutions of the OEM andtherefore provide guidelines for the OEM to design the contract.343.2. Literature Review3.2 Literature ReviewOur paper is related to the supplier integration literature. Interestingly, thisstream of literature has found conflicting evidences on whether supplier in-tegration helps the OEM firm. Hoegl and Wagner (2005) empirically showthat involving supplier in product development project can positively in-fluence cost and schedule, but too much communication intensity may notbenefit the project. Das et al. (2006) also discuss pros and cons of supplierintegration. They find that too much investment in supplier integration isnot productive. Koufteros et al. (2005) explores the relationship betweeninternal integration and external integration, and find that internal inte-gration positively influences external integration and product developmentoutcomes. Parker et al. (2008) noted that cost of integrating suppliers acrossorganizational boundaries imply that the benefit of integration must be sig-nificantly higher to justify the supplier’s inclusion. From a general qualityimprovement perspective, Zhu et al. (2007) find that buyer’s involvementplays a critical role in improving product quality and supply chain profits.This study is also related to (but differs from) the extant literature onwarranty services. In that stream of literature, the output quality is in-fluenced by the supplier’s effort (which may not be observable) and/or theOEMs effort. The OEM could motivate the supplier to produce higher qual-ity product by sharing warranty cost with the supplier. Such shared war-ranty service is especially useful when the OEM cannot clearly disentanglethe supplier’s responsibility in the final product’s quality problems. Reyniersand Tapiero (1995) study how price and warranty influence the supplier’squality effort in a game theoretical setting, where the quality decision ismade by the supplier. Lim (2001) considers a similar problem, but incorpo-rates information asymmetry where the buyer does not know the supplier’squality type and therefore must offer a menu of contract with appropriatewarranty terms. Chao et al. (2009) explore warranty cost sharing contractbased on selective or complete root cause analysis. Interestingly, they provethat both approaches could achieve optimal efforts (as compared with anintegrated system), but cost sharing based on selective root cause analysiscould achieve a higher profit for the supply chain. From a somewhat differ-ent angle, Dai et al. (2012) explore how the length of warranty influencesthe product quality and system profit. In their model, it is the supplier thatdetermines the product quality level, but either party may determine thelength of warranty period. They found that the party that bears a higherfraction of warranty cost should be delegated to set the warranty period.353.3. The ModelHuang et al. (2008) study warranty service from an inventory managementpoint of view, and they do not consider interactions between the supplierand the OEM. Note that the use of warranty service has also received ex-tensive treatment in the new product development context, and we refer theinterested reader to Murthy and Djamaludin (2002) for an excellent reviewof the literature on new product warranty.Our research complements the above literature by considering how anOEM could motivate the supplier to voluntarily suggest potential specifica-tion issues, when the OEM is responsible for setting product specifications.In such a scenario, a shared warranty service would not be as effective be-cause the supplier cannot be held responsible for mismatches between cus-tomer demand and product specification problems.In closing, we note that Iyer et al. (2005) also explore the product spec-ification problem, but with a very different focus. In contrast to our paper,they consider a “black-box” relationship where the supplier owns the prod-uct specification, but the OEM may allocate resources to help the supplierimprove product specification. One can thus view our paper as complemen-tary to theirs.3.3 The Model3.3.1 A descriptive overviewWe consider an OEM firm sourcing a critical product (or component/subassembly)from an external supplier. The OEM firm determines the product featuresto be offered to the market and develops the corresponding specifications,whereas the supplier executes the OEM firm’s order based on the OEM’sspecifications. Product specifications influence the market demand, and in-correct or misaligned specifications leads to lower demand.The OEM firm may not be able to detect potential specification prob-lems. That is, the OEM firm may perceive the specifications to fit the markettaste well while in reality they may not. This could happen for several rea-sons. First, the distributed nature of production configuration often leads todispersed product and process knowledge beyond the OEM’s organizationalboundary. Second, the supplier’s process technology may interact subtlywith product specifications (and performance), and it can be difficult forthe OEM to access tacit supplier knowledge a priori. Third, the OEM may363.3. The Modelnot have prior expertise in certain (new) markets whereas the supplier hasaccumulated unique and tacit knowledge in market tastes/trends throughtheir relationship with other OEMs.The supplier may recognize the OEM’s specification problems, althoughthe supplier may not may not be sure about whether the problems can be re-solved timely. Such uncertainty exists if the supplier recognizes specificationproblems through its tacit production knowledge but does not have designand engineering capabilities to come up with a new set of specifications, oralternatively, if the supplier is unsure whether the OEM has the capabilityto re-engineer alternative specifications timely. Note that the OEM couldreduce such supplier’s uncertainty by sharing information about its engi-neering capabilities and/or collaborate more closely with the supplier.Once the supplier recognizes the OEM’s potential specification problems,it may either voluntarily point out the problem (and/or suggest improve-ments) to the OEM, or remain silent and simply execute the OEM’s orderto the print. In the former case, the OEM firm may or may not be able tofix the specification problems (or implement the suggested improvements)in time to satisfy current period market demand. If not, the OEM firm willcancel the order for the current period but will place another order for thenext period. We assume that if the specification problems cannot be fixedin the current period, it will be fixed in the next period. Demand that can-not be satisfied in the current period is lost. We will consider a two-periodmodel: such as model will best suit for fast-changing industries, and willserve as a starting point for other stable industries where the OEM firm hasplenty of time to iron out winkles in its product specifications.If the supplier remains silent and completes the OEM’s order to the print,the OEM will subsequently recognize the specification problems throughlower than expected market demand in the current period. We assume thatthe supplier’s production production process does not introduce other de-fects and therefore the OEM cannot fault the supplier for the lower thanexpected product performance. Implicitly, this means that even though thesupplier’s production process may interact subtly with the OEM’s specifi-cations, the performance problem can be solely attributed to the OEM’sspecification flaws as opposed to production errors. This is quite differentfrom the case where the root cause of the problem cannot be clearly disen-tangled (Kim and Tomlin, 2013).373.4. The Model SetupShould the OEM discover its specification flaws through lower marketdemand, the OEM may also suffer from reputation damage from its secondperiod demand. Such reputation damage is often referred to as productharm crises if the specification flaw is serious (Heerde et al., 2007). In sucha case, even if the OEM is able to correct its specification flaws by thebeginning of the second period, demand will still be lower due to customers’bad experience with its first period product offerings.3.4 The Model SetupHaving sketched the general aspects of the model, we are now in a positionto set up the model formally. In the following we refer the OEM as he andthe supplier as she.Let t = 1, 2 denote the time period. At the beginning of the first period,the OEM determines his product specification. We assume that with prob-ability θ there are flaws in the specification. Note that the OEM is aware ofthis probability but is not sure about the existence of flaws. The OEM thenoffers a contract (q1, T ) to the supplier on a “take it or leave it” basis, withq1 as the ordering quantity for the first period, and T as the cancellationpayment if the OEM cancels the order in the first period.The supplier receives the order and produces the products. If the OEM’sspecification is correct, then the supplier delivers to the OEM. After theOEM receives delivery from the supplier, he satisfies a market demand d1as much as possible. Unsatisfied demand is lost (with no penalty cost), butexcess inventory can be carried over to the next period (with a holding costh per unit). Note that the sale amount should be the minimum of d1 andq1. The OEM carries over (q1 − d1)+ to the second period. The successfuldelivery of the products in the first period will bring a positive spill overeffect. The market demand in the second period will be d2 + β(d1 ∧ q1),where d2 is a base market demand in the second period, β is a positivespillover effect that is only effective on sales. At the beginning of the secondperiod, the OEM will update his forecast on the demand in the secondperiod accordingly, and will hence order [d2 +β(d1∧q1)− (q1−d1)+]+. Thisscenario is presented as Case 0 in Figure 3.1.383.4. The Model SetupOEM offers (q1, T ) ✓1 ✓1G(q1)Supplier detects the flaw with probability G(q1)Supplier chooses  whether to inform or notOEM attempts to rectify flawNI↵1 ↵Supplier delivers      , OEM satisfies      and carries over any left inventory     q1d1Supplier delivers      ,but demand is lost, and OEM salvages   ↵1 ↵q1 q1OEM attempts to rectify flawOEM cancels the order,  and pays        to the supplierTSupplier delivers      , OEM satisfies       and carries over any left inventory     q1 d1OEM faces demand                    , orders       d2 + (q1 ^ d1)[d2 + (q1 ^ d1) (q1  d1)+]+Case 1OEM faces demand                       ,orders       Case 0d2 + (q1 ^ d1)[d2 + (q1 ^ d1) (q1  d1)+]+Case 2OEM faces demand          , orders       d2 d2Case 3Case 4OEM faces demand      0 , orders  0 OEM faces demand                            , orders       d2  (q1 ^ d1)d2  (q1 ^ d1)Period 1 Period 2Figure 3.1: Sequence of events and casesIf the OEM’s specification has flaws, the supplier may detect the flaws.We assume that the more the OEM orders, the better chance the suppliermay detect the flaws. One explanation is that if the OEM orders more,the supplier may conduct a more refined and thorough preparation, andtherefore has more chance to detect the flaw even before any production.Alternatively, imagine the supplier runs inspection on each individual unitas it produces. Assuming the passing rate for each individual unit is γ, thenif the supplier has produced q1 units, she can detect the flaw as long as oneof these q1 units failed the inspection. Therefore, the probability for thesupplier to detect the flaw after producing q1 unit is 1− γq1 . This functionis concave and increasing in q1. As q1 increases, the probability for thesupplier to detect the flaws increases. In addition, the marginal increase inthe detecting probability decreases as q1 increases. After certain value of q1,the marginal increase is fairly small, and the detecting probability is veryclose to 1. We give an example of 1 − γq1 with γ = 0.95 in Figure 3.2(a),where the detecting probability is larger or equal to 0.98 when q1 is atleast 80. In fact, for any value of γ and an arbitrarily small value , thereexists a threshold of q1, such that for any q1 larger than the threshold, the393.4. The Model Setupdetecting probability is larger than 1−. In other words, after producing theamount of the threshold, the marginal increase in detecting probability if thesupplier produces even more is negligible. In practice, the value of γ can varydepending on the property of the products and the OEM’s specifications, andthe threshold will vary accordingly. For the purpose of analysis, we normalizethis threshold as 1, and use a piece wise function G(q1) (Figure 3.2(b)) toapproximate the probability of the supplier to detect the flaws when sheproduces q1, whereG(q1) ={q1, q1 ∈ [0, 1]1, q1 ∈ (1,+∞),so that when q1 is less than 1, the detecting probability increases in q1 lin-early. After that, the detecting probability is 1. Producing more will notresult in a higher detecting probability.Figure 3.2: Probability of detecting flawsWe assume that d1 < 1, so that producing the market demand in thefirst period will not be enough for the supplier to detect the flaws with cer-tainty. We make this assumption to better examine under what conditionthe OEM will order more than the demand in the first period just for thesake of increasing the supplier’s detect probability.If the supplier detects any flaws, she may choose either to inform theOEM or remain silent. The supplier faces a dilemma because she is unsureabout whether an immediate resolution is available to correct the specifica-tion flaws. Even the OEM himself is not sure about this, because he was notaware of the specification flaws and thus can not guarantee the problem canbe resolved in time. We assume the OEM has communicated thoroughlywith the supplier. As results, they have a common knowledge that with403.4. The Model Setupprobability α the specification flaws can be corrected without significant de-lay (so that production can be completed in time for the current period’sdemand), and with probability 1 − α the specification flaw cannot be cor-rected until by the beginning of the next period. For the rest of the paper,we refer to α as the OEM’s capability to correct any potential specificationflaws. One could also alternatively interpret α as the supplier’s engineeringand process capability, but for exposition ease we use the former interpreta-tion throughout the paper, with the understanding that all results developedin the paper could be adapted to the latter interpretation.If the supplier detects the specification flaws and points out the specifi-cation issues to the OEM, there are two possible scenarios. One scenario isthat the OEM immediately corrects the specification flaws, then the supplierproduces the products and delivers to the OEM. The OEM then satisfiesthe demand d1 and carries over any left over inventory. In the second pe-riod, the demand will be positively affected, the OEM will therefore order[d2 + β(d1 ∧ q1) − (q1 − d1)+]+. This scenario is presented as Case 1 inFigure 3.1. Another scenario is that the OEM fails to find an immediateresolution. The OEM then cancels the first period’s order and pays a fixedcompensation fee T to the supplier. In this case, we assume that the OEMwill find a resolution with certainty until the beginning of the second period.Because the first period’s order is canceled, the demand in the second periodwill just be the base market demand d2. The OEM will therefore order d2in the second period. This scenario is presented as Case 2 in Figure 3.1.In contrast, if the supplier detects the specification flaws but remainssilent, she finishes production with the OEM’s original specifications. Inthis case, we assume the demand in the first period is lost. Customersreturn the defected products. The OEM discovers the specification flawsthrough the lost demand and salvages the products at s per unit. Untilthe beginning of the second period, the OEM has a chance of α again tofind a resolution. If he does correct the flaws, there will be a market de-mand in the second period. The demand in the second period, however, willbe negatively affected due to a reputation damage caused by the defectedproducts from the first period. We assume the negative spillover effect isthe same magnitude as the positive spillover effect. Therefore we can writethe demand in the second period as d2 − β(d1 ∧ q1). We assume d2 ≥ βd1,i.e., the base market demand in the second period is big enough so that thereputation damage from the first period will at most result in zero demandin the second period. Because there is no inventory carried over from the413.4. The Model Setupfirst period, the OEM will hence order d2− β(d1 ∧ q1). This scenario is pre-sented as Case 3 in Figure 3.1. On the other hand, if the OEM is not ableto correct the flaws until the beginning of the second period, the demand inthe second period will be lost. The OEM will update his demand forecastaccordingly and hence order nothing for the second period. This scenario ispresented as Case 4 in Figure 3.1.On the other hand, it is possible that there are specification flaws but thesupplier does not detect any. In this case, the supplier delivers the defectedproducts. The first period demand is lost. The OEM salvages those defectedproducts. In the second period, the OEM has a chance of α to correct thespecification flaws. If he succeed, he will satisfy a demand d2 − β(q1 ∧ d1).Otherwise the second period demand will also be lost. Note that the OEMcannot differentiate this scenario from the scenario where the supplier de-tects the flaws but choose to remain silent, nor can the OEM blame thesupplier for the specification flaws. This is precisely why we need to exploreand disentangle the incentives for the supplier to inform the OEM shouldthere be any flaws. Only having that clearly stated, we can then take thenext step to discuss from the perspective of the OEM under what condi-tions it is beneficial for himself to design the contract so that the supplierwill voluntarily inform should there be any flaws.In the best scenario for the OEM, i.e., there is no flaw in the speci-fication, the maximum total demand the OEM satisfies in two periods isd1 + d2 + βd1. We assume that if the OEM orders this amount at the be-ginning of the first period, then the supplier can detect the flaws for sure.We write this assumption as d1 + d2 + βd1 ≥ 1 given the structure of thedetecting probability G(q1).The OEM pays the supplier a unit wholesale price of w for all unitsdelivered, sells the product at a unit price of p, and the supplier incurs aunit production cost of c. We assume these prices are the same for twoperiods. In practice, these prices may be different for different periods. Inaddition, the supplier’s unit production cost may change as the OEM cor-rects the specification flaws. To focus on the supplier’s incentive to suggestpotential specification flaws and the OEM’s optimal strategy in designingthe contract, we ignore the possible changes of wholesale price, selling price,and production cost. In addition, we assume p > w > c > h ≥ s, andw − c − h > 0. Another thing to point out is that all of the parameters inour model is assumed to be common knowledge.423.5. Problem Formulation3.5 Problem FormulationWe first describe the supplier’s decision problem, and then the OEM’s deci-sion problem.3.5.1 The supplier’s decision problemThe supplier needs to make a decision only when there are flaws in thespecification and when she has detected the flaws. The supplier needs todecide whether to inform the OEM about the flaws. Let a ∈ {I,N} denotethe supplier’s action of either informing or not informing the OEM. If thesupplier informs the OEM, her expected profit isSI =α{(w − c)q1 + (w − c)[d2 + β(q1 ∧ d1)− (q1 − d1)+]+}+ (1− α)[T + (w − c)d2].We assume the supplier can fully salvage the production cost of thoseunits that have already been produced using the faulted specification. Thesupplier has many ways to salvage those units which are still on the produc-tion line. She can use them for production with the corrected specification.There will be extra cost, but the extra cost can be negligible if the modifi-cation is minor. She can also sell those units to a secondary market or toother OEMs who may have different standards. She can at least disassemblethose units and salvage them as raw materials.Under this assumption, with probability α, the OEM resolves the spec-ification flaws immediately. The supplier fully salvages the units that havealready been produced. The supplier then produces with the correct spec-ification. She incurs a production cost cq1 for the first period order, andreceives wq1 from the OEM. The OEM satisfies a demand d1, and carriesover (q1− d1)+ amount of inventory to the second period. At the beginningof the second period, the OEM updates his demand forecast for the secondperiod as d2 +β(q1∧d1) and orders [d2 +β(q1∧d1)−(q1−d1)+]+. Thereforethe supplier receives (w − c)[d2 + β(q1 ∧ d1) − (q1 − d1)+]+ in the secondperiod. On the other hand, with probability 1−α, the OEM cannot revolvethe specification flaws in time. He cancels the first period order, and pays acancellation fee T to the supplier. The supplier fully salvages the units thathave already been produced. In the second period, the OEM corrects the433.5. Problem Formulationspecification flaws, and update his demand forecast for the second period asd2. Since there is no inventory carried over from the first period, the OEMthen orders d2 from the supplier. Therefore, the supplier receives T in thefirst period, and (w − c)d2 in the second period.In contrast, if the supplier does not inform the OEM, her expected profitisSN = (w − c)q1 + α(w − c)[d2 − β(q1 ∧ d1)],where the supplier carries out production as usual in the first period (theOEM cannot blame the supplier for his specification flaws), but in the sec-ond period the supplier carries out production only if the specification flawscan be rectified at the beginning of the second period. Note that the OEM’ssecond period order d2−β(q1 ∧ d1) is based on an updated demand forecastdue to reputation damage caused by first period specification flaws.Given the OEM’s contract (q1, T ), the supplier’s problem is to decidea ∈ {I,N} to maximize {SI , SN}. Note that in the above formulation, weimplicitly assume that the supplier knows the OEM’s second period orderat the beginning of the first period. This is reasonable because all the pa-rameters in our model are assumed to be common knowledge. If, on theother hand, the supplier has to estimate the OEM’s second period order inadvance, then we need to replace the order quantity with supplier’s expec-tation on the OEM’s order quantity. If the supplier is risk neutral and herestimation is unbiased, then such as change will not affect our analysis. If,however, the supplier is risk averse or has biased estimation, then a differentmodel that incorporates risk aversion or asymmetric information would bemore appropriate.3.5.2 The OEM’s decision problemThe OEM needs to design the contract in the first period (q1, T ) to max-imize his total profit over two periods, where q1 is the order quantity forthe first period, and T is the cancellation payment if he cancels the order inthe first period. Note that the OEM does need to decide the order quantityin the second period as well. This decision, however, is straightforward.Once the OEM has the updated demand information, he just orders up tothe market demand in the second period, as we stated in the previous sec-tions. Therefore we focus on the OEM’s decision problem in the first period.443.5. Problem FormulationWe first write down the OEM’s profit in each case (Figure 3.1). We useV i to denote the OEM’s profit in Case i, i = 0, 1, 2, 3, 4.V 0 =V 1 = −wq1 + p(q1 ∧ d1)− h(q1 − d1)++ {−w[d2 + β(q1 ∧ d1)− (q1− d1)+]+ + p[d2 + β(q1 ∧ d1)]},V 2 = −T + (p− w)d2,V 3 = −wq1 + sq1 + (p− w)[d2 − β(q1 ∧ d1)],V 4 = −wq1 + sq1.Note that from the OEM’s perspective, Case 0 where there is no spec-ification flaw, is equivalent as Case 1 where there are specification flaws,the supplier detects and informs the OEM, and the OEM rectifies the flawsinstantly. In Case 1 and 2, the supplier detects the flaws after a certainamount of production, she informs the OEM about the flaws and fully sal-vages the production cost of those units. In Case 3 and Case 4, the defectedproducts have been delivered to the OEM and sold to the customers. It isthe OEM who salvages the defected products. We assume it is much harderfor the OEM to retrieve the cost. The salvage value is at s, lower than thesupplier’s production cost c.The OEM’s total expected profit in two periods will depend on the sup-plier’s decision.If the supplier decides to NOT inform the OEM even if she detects theflaw, the OEM’s expected profit isVN (q1, T ) = (1− θ)V 0 + θ{αV 3 + (1− α)V 4}If there is no flaw in the OEM’s specification, then it is Case 0. If there isflaw in the OEM’s specification, because the supplier has decided NOT toinform the OEM even if she detects the flaw, it does not matter whether ornot the supplier detects the flaw. The supplier carries out the productionaccording to the OEM’s original specification and delivers to the OEM. Thedemand in the first period is lost. The OEM realizes the flaws through thelost demand and salvages the defected products. At the beginning of thesecond period, the OEM has a probability of α to rectify the flaw.• If the OEM rectifies the flaws, the demand in the second period willbe d2 − β(q1 ∧ d1), and the OEM will order d2 − β(q1 ∧ d1). (Case 3).453.5. Problem Formulation• If the OEM is not able to rectify the flaws, the demand in the secondperiod will be lost, and the OEM will order nothing (Case 4).In contrast, if the supplier decides to inform the OEM if she detects theflaw, the OEM’s expected profit isVI(q1, T ) = (1− θ)V 0 + θ{G(q1)[αV 1 + (1− α)V 2] + [1−G(q1)][αV 3 + (1− α)V 4]}.If there is no flaw in the OEM’s specification, then it is Case 0. If there isflaw in the OEM’s specification, then the supplier has a probability of G(q1)to detect the flaw.• If the supplier detects the flaw, she then informs the OEM. The OEMhas a probability of α to rectify the flaws in a timely manner.– If the OEM rectifies the flaw in time, the demand in the firstperiod will not be affected, the OEM carries over inventory (ifany) to the second period. The demand in the second period willbe d2 + β(q1 ∧ d1), and the OEM will order [d2 + β(q1 ∧ d1) −(q1 − d1)+]+. (Case 1).– Otherwise, the OEM fails to resolve the flaws in time, he cancelsthe order. At the beginning of the second period, the OEM rec-tifies the flaw. The demand in the second period will be d2, andthe OEM will order d2. (Case 2).• If the supplier does not detect the flaw, then the demand in the firstperiod is lost. The OEM realizes the flaw through the lost demandand salvages the defected products. At the beginning of the secondperiod, the OEM again has a probability of α to rectify the flaw.– If the OEM rectifies the flaw, the demand in the second periodwill be d2 − β(q1 ∧ d1), and the OEM will order d2 − β(q1 ∧ d1).(Case 3).– Otherwise, the OEM is not able to rectify the flaw, the demandin the second period will be lost again, and the OEM will ordernothing (Case 4).The OEM can design the contract (q1, T ) to direct the supplier to informor not inform. Therefore we solve the OEM’s decision problems as follows.We will characterize the optimal solutions for the OEM to maximize hisexpected total profit given that he directs the supplier to not inform or in-form, respectively (section 2.4.1 and section 2.4.2). Then we will compare463.6. Analysisthe OEM’s optimal expected profit in these two cases (section 2.4.3). Wewill show that it is strictly better for the OEM to design (q1, T ) so thatthe supplier will inform should she detect any flaw. Finally, we will con-duct sensitivity analysis on the optimal solutions to gain some managerialinsights.3.6 Analysis3.6.1 Optimization problem NIn this section, we examine the optimal solutions of the OEM on the con-dition that the OEM directs the supplier NOT to inform when she detectsflaws. The OEM’s decision problem can be written asmaxq1≥0,TVN (q1) = VN (q1, T )s.t.SI ≤ SNSN ≥ 0T ≥ 0,where VN (q1) is the OEM’s expected total profit if he orders q1 in the firstperiod given that the supplier does NOT inform even if she detects anyflaw, SI ≤ SN is the incentive constraint for the supplier to NOT inform,SN ≥ 0 is the individual rationality constraint for the supplier, and T ≥ 0 isthe nonnegative constraint for the cancellation payment. Because we haveassumed that d2 ≥ βd1, the individual rationality constraint always holds.Note that the objective function in optimization problem N does notdepend on T . The OEM does not want the supplier to inform, thus hewill never try to rectify the flaws, and thus will never cancel the order andpay the cancellation payment T . Cancellation payment serves as an emptythreat in the contract. The only constraint on the cancellation payment isthat it needs to be small enough so that the supplier will not have incentiveto inform.We can summarize the optimal solutions for the OEM in optimizationproblem N as follows.Theorem 3.6.1 The optimal solutions for the OEM in optimization prob-lem N are as follows.473.6. Analysis• When d2 >1− α(1 + 2β)1− α d1, q∗1 = α(1+2β)d1+(1−α)d2, and T ∗ = 0.• When d2 ≤1− α(1 + 2β)1− α d1, the optimal solutions will depend on thevalue of θ.– When θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , q∗1 = d1, and T ∗ can beany value satisfying the incentive constraint. We set T ∗ = 0.– When θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , q∗1 =1− α1− α− 2αβd2,and T ∗ = 0.We can demonstrate Theorem 3.6.1 on the (d1, d2) map, as shown in Fig-ure 3.3 (a). Note that because we have assumed d2 ≥ βd1, if1− α(1 + 2β)1− α ≤β, theorem 3.6.1 degenerates to only the first case where q∗1 = α(1+2β)d1 +(1− α)d2 and T ∗ = 0, as shown in Figure 3.3 (b).Figure 3.3: The optimal solutions for optimization problem N3.6.2 Optimization problem IIn this section, we explore the optimal solutions of the OEM on the conditionthat the OEM directs the supplier to inform when she detects flaws. The483.6. AnalysisOEM’s decision problem can be written asmaxq1≥0,TVI(q1, T )s.t.SI ≥ SNSI ≥ 0T ≥ 0,where VI(q1, T ) is the OEM’s expected total profit if he orders q1 in the firstperiod and sets T as the cancellation payment on the condition that thesupplier chooses to inform should there be flaws, SI ≥ SN is the incentiveconstraint for the supplier to inform, SI ≥ 0 is the individual rationalityconstraint, and T ≥ 0 is the nonnegative constraint for the cancellationpayment.One observation we make here is that if the incentive constraint holds,then the individual rationality constraint also holds. Specifically, becausewe have assumed d2 ≥ βd1, then SN ≥ 0, therefore if SI ≥ SN , then SI ≥0. Thus the individual rationality constraint is redundant. Optimizationproblem I is equivalent asmaxq1,TVI(q1, T )s.t.SI ≥ SNT ≥ 0.There are two piece-wise functions in the objective function, one is[d2+β(q1∧d1)−(q1−d1)+]+ =d2 + βq1, q1 ≤ d1,(β + 1)d1 + d2 − q1, d1 < q1 ≤ (β + 1)d1 + d2,0, q1 > (β + 1)d1 + d2,the other isG(q1) ={q1, q1 ∈ [0, 1],1, q1 ∈ (1,+∞).We therefore divide the region of q1 into four subregions: [0, d1], (d1, 1],(1, (β+ 1)d1 + d2], and ((β+ 1)d1 + d2,+∞), and examine the local optimalsolutions in these four subregions receptively. After that, we will evaluatethe local optimal solutions in these subregions to get the global optimal so-lutions for the OEM in optimization problem I.493.6. AnalysisOptimal Solutions in the subregion [0, d1]We first study the local optimal solutions for the OEM in optimizationproblem I when q1 ∈ [0, d1).Proposition 3.6.2 In the subregion q1 ∈ [0, d1], the optimal solutions forthe OEM in optimization problem I are as follows.• When d2 >1− α(1 + 2β)1− α d1, T∗ = 0.– When (d1, d2) is above Line 1, q∗1 = d1.– Otherwise q∗1 = 0.• When d2 ≤1− α(1 + 2β)1− α d1:– When (d1, d2) is above Line 2, q∗1 = d1, and T ∗ =w − c1− α [(1−α−2αβ)d1 − (1− α)d2].– Otherwise q∗1 = 0, and T ∗ = 0.The equations for the Lines areLine 1: d2 =− [α(1 + 2β)1− α +w − s(1− α)(p− w) ]d1+ θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− w) ,Line 2: d2 =− [α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1+ θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− c) .We can demonstrate Proposition 3.6.2 on (d1, d2) map in Figure 3.4(a).Note that if 1− α(1 + 2β)1− α ≤ β, Proposition 3.6.2 degenerates to only thefirst part, as shown in Figure 3.4(b). Another thing to note is that whenθ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , both Line 1 and Line 2 has nonpositiveintercepts. In that case, the optimal solutions for the OEM in optimizationproblem I in the subregion [0, d1] are q∗1 = d1 and T ∗ = 0.503.6. AnalysisFigure 3.4: The optimal solutions for optimization problem I when q1 ∈[0, d1]Optimal Solutions in the subregion (d1, 1]We then explore the optimal solutions for the OEM in optimization problemI when q1 ∈ (d1, 1].Proposition 3.6.3 In the subregion (d1, 1], the optimal solutions for theOEM in optimization problem I are as follows.• When d2 ≤1− α(1 + 2β)1− α d1:– When (d1, d2) is above Line 6, q∗1 = 1, and T ∗ =w − c1− α [1−α(1 +2β)d1 − (1− α)d2].– Otherwise, q∗1 = d1, and T ∗ =w − c1− α [(1−α−2αβ)d1−(1−α)d2].• When d2 > −α(1 + 2β)1− α d1 +11− α , then T∗ = 0.– When it is above Line 3, q∗1 = 1.– Otherwise, q∗1 = d1.• When d2 ≤ −α(1 + 2β)1− α d1 +11− α :– When (d1, d2) is above both Line 4 and Line 5, q∗1 = 1 andT ∗ = w − c1− α [1− α(1 + 2β)d1 − (1− α)d2].513.6. Analysis– When (d1, d2) is below both Line 4 and Line 5, q∗1 = d1 andT ∗ = 0 .– When (d1, d2) is above Line 4 and below Line 5, q∗1 = α(1 +2β)d1 + (1− α)d2 and T ∗ = 0.– When (d1, d2) is below Line 4 and above Line 5, q∗1 could be d1or 1. If q∗1 = d1, then T ∗ = 0. If q∗1 = 1, then T ∗ =w − c1− α [1 −α(1 + 2β)d1 − (1− α)d2].The equations of the lines areLine 6: d2 =− [α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1+ θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .Line 3: d2 =− [α(1 + 2β)1− α +w − s(1− α)(p− w) ]d1+ h(1− θ + αθ)θ(1− α)(p− w) .Line 4: d2 =− [α(1 + 2β)1− α +w − s(1− α)(p− αh− s) ]d1+ θ(w − s) + h(1− θ)θ(1− α)(p− αh− s) .Line 5: d2 =− [α(1 + 2β)1− α +αh(1− α)(p− αh− s) ]d1+ θ(w − c) + h(1− θ + θα)θ(1− α)(p− αh− s) .Depending on the positions of these Lines, the division of (d1, d2) map canbe of various forms. One representative case is shown in Figure 3.5(a). Notethat If 1− α(1 + 2β)1− α ≤ β, the first case in Proposition 3.6.3 does not exist,as presented in Figure 3.5(b).523.6. AnalysisFigure 3.5: The optimal solutions for optimization problem I when q1 ∈(1, d1]Optimal Solutions in the subregion (1, (β + 1)d1 + d2]Next, we examine the optimal solutions for optimization problem I whenq1 ∈ (1, (β + 1)d1 + d2].Proposition 3.6.4 In the subregion q1 ∈ (1, (β+ 1)d1 + d2], the optimal q1for the OEM in optimization problem I is q∗1 = 1.• When d2 ≤ −α(1 + 2β)1− α d1 +11− α , then T∗ = w − c1− α [1−α(1+2β)d1−(1− α)d2].• Otherwise, T ∗ = 0.Optimal Solutions in the subregion ((β + 1)d1 + d2,+∞)Finally we study the optimal solutions for optimization problem I whenq1 ∈ ((β + 1)d1 + d2,+∞). If there is no specification flaw in the OEM’sdesign, or there is flaw but the supplier detects, informs, and the OEMcorrects the flaw in time, then the total demand for two periods would be(β + 1)d1 + d2. Thus (β + 1)d1 + d2 is the maximum total demand for the533.6. AnalysisOEM in two periods. Intuitively, the OEM has no incentive to order morethan (β + 1)d1 + d2 in the first period.Proposition 3.6.5 In the subregion q1 ∈ ((β+ 1)d1 +d2,+∞), the optimalsolutions for optimization problem I are q∗1 = (β + 1)d1 + d2, andT ∗ = (w − c)(β + 1)d1 +α1− α(w − c)(d2 − βd1).Optimal Solutions for optimization problem INow we are ready to state the optimal solutions for the OEM in optimizationproblem I. The possible optimal values of q∗1 are 0, d1, α(1+2β)d1+(1−α)d2,and 1.Theorem 3.6.6 When d2 ≤1− α(1 + 2β)1− α d1, the optimal solutions for op-timization problem I are as follows.(a) If θ ≤ (p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h , then• when (d1, d2) is above Line 6, q∗1 = 1 andT ∗ = w − c1− α [1− α(1 + 2β)d1 − (1− α)d2].• When (d1, d2) is below Line 2, q∗1 = 0 and T ∗ = 0.• When (d1, d2) is above Line 2 and below Line 6, q∗1 = d1 andT ∗ = w − c1− α [(1− α− 2αβ)d1 − (1− α)d2].(b) If θ > (p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h , then• When (d1, d2) is above Line 2, q∗1 = 1 andT ∗ = w − c1− α [1− α(1 + 2β)d1 − (1− α)d2].• When (d1, d2) is below Line 6, q∗1 = 0 and T ∗ = 0.• When (d1, d2) is above Line 6 and below Line 2, in the area that isabove Line 7, q∗1 = 1 and T ∗ = 0. In the part that is below Line 7,q∗1 = 0 and T ∗ = 0.Theorem 3.6.6 is presented in Figure 3.6. Note that Line 2 can have negativeintercept. Both Line 2 and Line 6 can have intercepts that is big enough sothat they do not contribute to divisions.543.6. AnalysisFigure 3.6: The global optimal solutions for the OEM in optimization prob-lem I when d2 ≤1− α(1 + 2β)1− α d1.Theorem 3.6.7 When d2 >1− α(1 + 2β)1− α d1, the optimal solutions for theOEM in optimization problem I is as follows.(i) If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then the optimal q1 for optimiza-tion problem I is restricted on [d1, 1]. The possible optimal value forq1 is d1, 1, and α(1 + 2β)d1 + (1− α)d2.(ii) If θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then the optimal q1 for optimiza-tion problem I is restricted on [0, 1]. The possible optimal value for q1is 0, d1, 1, and α(1 + 2β)d1 + (1− α)d2.• When q∗1 = 0 or α(1 + 2β)d1 + (1− α)d2, T ∗ = 0.• When q∗1 = d1, T ∗ = max{w − c1− α [(1− α− 2αβ)d1 − (1− α)d2], 0}.• When q∗1 = 1, T ∗ = max{w − c1− α [1− α(1 + 2β)d1 − (1− α)d2], 0}.We shown a representative case of the optimal solutions for the OEM inoptimization problem I in Figure 3.7. It is worth pointing out that underLine 1, the local optimal q1 in the subregion [0, d1] is q∗1 = 0, and the local553.6. Analysisoptimal q1 in the subregion (d1, 1] is q∗1 = d1, then the global optimal q1 isq∗1 = 0. This is implied by the continuity of the optimization problem. Inthe next section, we will further discuss the division of (d1, d2) map.Figure 3.7: The global optimal solutions for optimization problem I.3.6.3 Overall optimal solutions for the OEMGeneral StructureTheorem 3.6.8 The optimal objective value of optimization problem I isstrictly bigger than the optimal objective value of optimization problem N.This is a strong conclusion. This theorem implies that it is always betterfor the OEM to design the contract so that the supplier voluntarily informs.The optimal solutions for the OEM are the same as the optimal solutionsfor the OEM in optimization problem I.Having this theorem, next we focus on providing guidelines for the OEMto design the contract. In the following, we conduct sensitivity analysis onthe parameters θ, α, and β, to gain some insights from previous results.Sensitivity Analysis on θFirst, we want to isolate the influence of θ on the optimal solutions (q∗1, T ∗)of the OEM. We fix other parameters at reasonable values and explore how563.6. Analysisthe optimal solutions change as θ changes.Figure 3.8 presents the optimal solutions of the OEM in a case whenthe demands in two periods are comparable. When θ is small, meaning thatthere is a small chance the specification has flaws, the OEM should orderonly d1. Because if the OEM orders more than d1 in the first period, heincurs holding cost on the inventory. At this point, the supplier informsvoluntarily without the OEM providing cancellation payment. However, asθ increases, once θ reaches a certain level, the benefits of ordering morewhich is the increased detecting probability exceeds the cost of orderingmore which is the holding cost, and the OEM should order 1. At the sametime, because the OEM orders a large amount in the first period, if the sup-plier informs, she might loose the first period order and only gets the secondperiod demand, which is 0.5. Therefore, the supplier prefers to remain silentabout the flaws unless the OEM provides a positive cancellation payment.Because if the supplier remains silent, she gets the order of 1 in total in anycase.Figure 3.9 presents the optimal solutions of the OEM in a case whenthe second period’s demand is substantially larger than the fist period’s de-mand, and the spill over effect is strong. In this case, the OEM does notwant to risk losing the second period’s demand. Therefore as long as thereis a tiny probability that the specification has flaws, the OEM will order 1in the first period so that the supplier can detect the flaw with certainty andinforms the OEM. On the other hand, because the second period’s demandis significantly larger than the first period’s demand, the supplier wants toinform out of her own interest. Therefore the OEM does not need to providea cancellation payment.Figure 3.10 presents the optimal solutions of the OEM in a case when thesecond period’s demand is small, and the spill over effect is weak. When θ issmall, the OEM orders only d1 to avoid the holding cost. At the same time,because the first period’s demand is larger, the supplier has an incentive toremain silent, and the OEM needs to offer a positive cancellation paymentso that the supplier would inform the OEM should she detects any flaws.As θ increases, once θ reaches a certain level, the OEM starts to order 1 sothat the supplier can detect the flaws with certainty. The OEM needs toprovide a larger cancellation payment to maintain the supplier’s preferenceto inform.573.6. AnalysisThe case in Figure 3.11 is to show that the OEM’s order quantity doesnot necessarily jump among 0, d1 and 1. Sometimes the OEM will order anintermediate amount between d1 and 1, which is q∗1 = α(1+2β)d1+(1−α)d2.When he orders this amount, T ∗ = 0, meaning that the supplier will volun-tarily inform without a cancellation payment.Figure 3.8: The optimal solutions versus θ when demands are comparable(with d1 = d2 = 0.5, β = 1, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, andα = 0.3).Figure 3.9: The optimal solutions versus θ when d2 is large (with d1 = 0.8,d2 = 30, β = 20, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, and α = 0.3).583.6. AnalysisFigure 3.10: The optimal solutions versus θ when d2 is small (d1 = 0.8,d2 = 0.2, β = 0.01, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, and α = 0.3).Figure 3.11: The optimal solutions versus θ with intermediate q∗1 presented(with d1 = 0.46, d2 = 0.38, β = 0.8, p = 3, w = 2, c = 1, h = 0.3, s = 0.3,and α = 0.3).Sensitivity Analysis on αSecondly, we want to isolate the influence of α on the optimal solutions(q∗1, T ∗) of the OEM. We expect T ∗ to be decreasing in α. If α increases,the OEM has a better capability to rectify the flaws immediately, thereforeshould need less cancellation payment to motivate the supplier to inform.Figure 3.12 presents the optimal solutions of the OEM in a case when thedemands in two periods are comparable. As α increases, supplier becomesmore confident in the OEM’s capability to resolve the flaws in time. There-fore the OEM does not need to provide as much incentive for the supplierto inform. This explains why T ∗ decreases as α decreases.593.6. AnalysisFigure 3.13 presents the optimal solutions of the OEM in a case when thedemand in the second period is substantially larger than the demand in thefirst period, and the spill over effect is strong. In this case, the OEM doesnot want to lose the demand in the second period. Therefore the OEM willalways order 1 so that the supplier can detect the flaw. At the same time,because the second period’s demand is much larger than the first period’sdemand, the supplier will inform without cancellation payment.Figure 3.14 presents the optimal solutions of the OEM in a case whenthe demand in the second period is small, and the spillover effect is weak.In this case, when α is small, there is a big chance that the OEM is not ableto rectify the flaw in time, and because the demand in the second periodis very small, thus the OEM should order only d1. When α increases, theOEM has more confidence to rectify the flaw in time, then he should order1 so that the supplier can detect the flaw and inform him.An interesting observation in this particular case is that when α ap-proaches 1, T ∗ approaches infinity. When α is almost 1, meaning that theOEM in most cases can correct the flaws immediately. Suppose cancella-tion payment is zero, then the supplier would prefer not to inform. If thesupplier informs, the OEM will correct the flaws, the supplier will get thefirst period’s order, which is 1. In the second period, the demand will bed2 + βd1, but the OEM will order only βd1, for he carries over some inven-tory from the first period. The supplier in total gets order 1 + βd1 = 1.08.In contrast, if the supplier does not order, she gets the first period demand1. The OEM recognizes the flaws through lost demand. Defected productsare salvaged, not carried over. In the second period, the demand will bed2 − βd1, and the OEM will need to order d2 − βd1. The supplier in totalgets order 1 +d2−βd1 = 1.12. Therefore the supplier gets more order if shedoes not inform. In order to make the supplier inform, the OEM will need tofill this gap using the cancellation payment, so that the expected paymentfor the supplier to inform is better than not to inform. The cancellationpayment is only effective when the OEM cannot rectify the flaws. Becausethis probability 1−α approaches zero, the cancellation payment approachesinfinity.The case in Figure 3.15 is to show that sometimes the OEM will order anintermediate amount between d1 and 1, which is q∗1 = α(1+2β)d1+(1−α)d2.603.6. AnalysisFigure 3.12: The optimal solutions versus α when demands are comparable(with d1 = d2 = 0.5, β = 1, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, andθ = 0.5).Figure 3.13: The optimal solutions versus α when d2 is large (with d1 = 0.8,d2 = 30, β = 20, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, and θ = 0.5).Figure 3.14: The optimal solutions versus α when d2 is small (with d1 = 0.8,d2 = 0.2, β = 0.01, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, and θ = 0.5).613.6. AnalysisFigure 3.15: The optimal solutions versus α with intermediate q∗1 presented(d1 = 0.46, d2 = 0.38, β = 0.8, p = 3, w = 2, c = 1, h = 0.3, s = 0.3, andθ = 0.5).Sensitivity Analysis on βFinally, we want to isolate the influence of β on the optimal solutions (q∗1, T ∗)of the OEM. Intuitively, as β increases, the spillover effect is stronger, theOEM has more incentive to direct the supplier to inform so that the poten-tial flaws do not jeopardize the second period’s demand. On the other hand,the supplier herself would have stronger incentive to voluntarily inform be-cause she wants to secure the second period’s demand. The interests of theOEM and the supplier are aligned. What’s more, the optimal value of T ∗should decrease in β. Because if β is larger, the supplier has stronger in-centive to voluntarily inform and therefore does not need as much incentiveprovided from the OEM. Note that we have the assumptions that d2 ≥ βd1,and (β + 1)d1 + d2 > 1. Therefore for different values of α, d1, and d2, thefeasible value for β varies.Figure 3.16 presents the optimal solutions of the OEM in a case whenthe demands in two periods are comparable. In this case, the OEM willorder 1, and will need to provide positive cancellation payment so that thesupplier will inform.Figure 3.17 presents the optimal solutions of the OEM in a case whenthe second period’s demand is substantially larger than the first period’sdemand, and the spill over effect is strong. In this case, the supplier willvoluntarily inform without a cancellation payment.Figure 3.18 presents the optimal solutions of the OEM in a case when623.6. Analysisthe demand in the second period is small, and the spillover effect is weak.In this case, the demand in second period is not sufficient enough for thesupplier to inform voluntarily, the OEM will need to provide a positive can-cellation payment.The case in Figure 3.19 is to show that sometimes q∗1 = α(1 + 2β)d1 +(1 − α)d2. Note that T ∗ decreases in β. Because if the spillover effect isstronger, the supplier has a stronger incentive in inform voluntarily, so thatthe OEM does not need to provide as much incentive through T ∗.Figure 3.16: The optimal solutions versus β when demands are comparable(with d1 = d2 = 0.5, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, θ = 0.5, andα = 0.5).Figure 3.17: The optimal solutions versus β when d2 is large (with d1 = 0.2,d2 = 20, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, θ = 0.5, and α = 0.5).633.7. ConclusionFigure 3.18: The optimal solutions versus β when d2 is small (with d1 = 0.8,d2 = 0.2, p = 4, w = 2, c = 1, h = 0.1, s = 0.2, θ = 0.5, and α = 0.5).Figure 3.19: The optimal solutions versus β with intermediate q∗1 presented(with d1 = 0.46, d2 = 0.38, p = 3, w = 2, c = 1, h = 0.3, s = 0.3, θ = 0.5,and α = 0.5).3.7 ConclusionIn this research, we first explore potential factors that may motivate thesupplier to help the OEM improve product specifications by pointing outpotential specification flaws and /or suggest improvements in product spec-ifications. Our research is especially relevant when the supplier cannot befaulted for product quality issues arising from the OEM’s specification flaws,where some common approaches such as shared warranty services may notbe effective.We then solve the optimization problem for the OEM. We prove that itis strictly better for the OEM to design the contract so that the supplier643.7. Conclusionwill inform if she detects flaws. This is a strong conclusion. With this prin-ciple in mind, we characterize the optimal strategy of the OEM, includingorder quantities in each period, and cancellation payment in the first pe-riod. We find that the optimal solutions are very sensitive with regard tosome parameters. We perform sensitivity analysis on those parameters. Onething we show is that whenever the OEM is paying a positive cancellationpayment to the supplier, then the cancellation payment should decrease asthe OEM’s capability increases, and decrease as the spillover effect for thedemand increases.Leveraging supplier’s capabilities to improve product specifications andquality performance is an important area of research, and we hope thatfuture research in this direction, either by us or others, will provide furtherinsights in tapping the supplier’s knowledge to create successful products.65BibliographyS. Borenstein. 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Long-run equilibrium modeling of emis-sions allowance allocation systems in electric power markets. OperationsResearch, 58(3):529–548, 2010.69K. Zhu, R. Q. Zhang, and F. Tsung. Pushing quality improvement alongsupply chains. Management Science, 53(3):421–436, 2007.70Appendix AAppendix for Chapter 2In each period t, t = 1, · · · , T , the utility trades in the REC market first,then purchases electricity from the forward market. We write two cost-to-gofunctions for these two stages for ease of analysis.In stage two, the utility’s had adjusted its REC level to x¯t, the decisionvariables are y1t and y2t. Given state (x¯t, ut, Rt, Pt), the cost-to-go functionafter REC trading can be written asWt(x¯t, ut, Rt, Pt) = miny1t≥0y2t≥y1t{p1ty1t + p2t(y2t − y1t) +Gt(y2t)+ γEt[Vt+1(x¯t + y1t, ut +Dt, Rt+1, Pt+1)]}(A.1)In stage one, the decision variable is x¯t, the cost-to-go function beforeREC trading can be written asVt(xt, ut, Rt, Pt) = minx¯t{Ct(x¯t − xt, Rt) +Wt(x¯t, ut, Rt, Pt)} (A.2)In the appendix, for simplicity, we use “increase (decrease)” to indi-cate“nondecrease (nonincrease)”. Also, we omit the subscription t un-less there is confusion. For instance, in the following Lemma, we writeWt(x, u,R, P ) instead of Wt(xt, ut, Rt, Pt).Lemma A.0.1 Wt(x, u, R, P ) and Vt(x, u, R, P ) are jointly convex on(x, u).Proof of lemma A.0.1:We prove this lemma by induction in three steps:(i) VT+1(x, u) = pi(αu− z)+ is jointly convex on (x, u);(ii) If Vt+1(x, u, R, P ) is jointly convex on (x, u), then Wt(x¯, u, R, P ) isjointly convex on (x¯, u);71Appendix A. Appendix for Chapter 2(iii) If Wt(x¯, u, R, P ) is jointly convex on (x¯, u), then Vt(x, u, R, P ) isjointly convex on (x, u).We show the proofs of these three steps in the following.(i) Since pix+ is convex in x, VT+1(x, u) = pi(αu−z)+ = pi((α,−1)(u, z)T)+is jointly convex on (x, u) by the preservation of convexity under com-position with an affine mapping (Boyd and Vandenberghe, 2004).(ii) According to equation (A.1), by the preservation of convexity underminimization, it is sufficient to show:(a) {(y1, y2) : y1 ≥ 0, y2 ≥ y1} is a convex set;(b) p1y1 + p2(y2− y1) +Gt(y2) + γEt[Vt+1(x¯+ y1, u+D,Rt+1, Pt+1)]is convex on (x¯, u, y1, y2).(a) is easy to verify. Now we prove (b). Since Vt+1(x, u, R, P ) isjointly convex on (x, u), Vt+1(x¯+ y1, u+D,Rt+1, Pt+1) is jointly con-vex in (x¯, u) and (y1, u), also jointly convex in (x¯, y1) by the preser-vation of convexity under composition with an affine mapping. Thus,Vt+1(x¯+y1, u+D,Rt+1, Pt+1) is convex in (x¯, u, y1). Then Et[Vt+1(x¯+y1, u+D,Rt+1, Pt+1)] is convex in (x¯, u, y1) by preservation of convex-ity under nonnegative weighted sums (Boyd and Vandenberghe, 2004).Moreover, Gt(y2) is convex in y2. Thus, we’ve proved (b), and we’veproved (ii).(iii) According to equation (A.2), by the preservation of convexity underminimization, it is sufficient to show Ct(x¯− x, R) +Wt(x¯, u, R, P ) isconvex in (x, u, x¯). Since bx+ − s(−x)+ is convex in x, by the preser-vation of convexity under composition with an affine mapping, Ct(x¯−x,R) = b(x¯−z)+−s(z−x¯)+ = b((1,−1)(x¯, z)T)+−s(−(1,−1)(x¯, z)T)+is jointly convex on (x¯, z). Meanwhile, W (x¯, u, R, P ) is convex in(x¯, u) by assumption. Thus, Ct(x¯ − x, R) + W (x¯, u, R, P ) is convexin (x, u, x¯).Lemma A.0.2 Wt(x, u, R, P ) and Vt(x, u, R, P ) are submodular on (x, u).Proof of lemma 2.4.2:We prove this lemma by induction in three steps:(i) VT+1(x, u) is submodular on (x, u);72Appendix A. Appendix for Chapter 2(ii) If Vt+1(x, u,R, P ) is submodular on (x, u), then Wt(x¯, u,R, P ) is sub-modular on (x¯, u);(iii) If Wt(x¯, u,R, P ) is submodular on (x¯, u), then Vt(x, u,R, P ) is sub-modular on (x, u).We show the proofs of these three steps in the following.(i) Since pix+ is convex in x, VT+1(x, u) = pi(αu − z)+ is submodular on(x, u) (Topkis, 1998), Lemma 2.6.2.(ii) According to equation (A.1), by the preservation of submodularity(Topkis, 1998), Theorem 2.7.6, it is sufficient to show:(a) (a) {((x¯, u), (y1, y2)) : u ≥ 0, y1 ≥ 0, y2 ≥ y1} forms a lattice;(b) (b) p1y1+p2(y2−y1)+Gt(y2)+γEt[Vt+1(x¯+y1, u+D,Rt+1, Pt+1)]is submodular on ((x¯, u), (y1, y2)).(a) is easy to verify. Now we prove (b). Note p1y1 + p2(y2 − y1) islinear in (y1, y2) and Gt(y2) is a function of a single variable, thus,p1y1 + p2(y2 − y1) + Gt(y2) is submodular on (y1, y2). If we canshow Vt+1(x¯+y1, u+D,Rt+1, Pt+1) is submodular on ((x¯, u), y1), thenp1y1 +p2(y2−y1)+Gt(y2)+γEt[Vt+1(x¯+y1, u+D,Rt+1, Pt+1)] is sub-modular on ((x¯, u), (y1, y2)), because the sum of two submodular func-tions is also submodular. We know that Vt+1(x¯+y1, u+D,Rt+1, Pt+1)is submodular on (x¯, u) and (y1, u) by the assumption. Also, becauseV (x, u,R, P ) decreases in z, Vt+1(x¯+y1, u+D,Rt+1, Pt+1) is submod-ular on (x¯, y1). Thus we’ve proved (b), and we’ve proved (ii) .(iii) Since bx+ − s(−x)+ is convex in x, Ct(x¯ − x,R) = b(x¯ − z)+ − s(z −x¯)+ is submodular on (x¯, z) (Topkis, 1998), Lemma 2.6.2. BecauseWt(x¯, u,R, P ) is submodular on (x¯, u) by assumption, Ct(x¯ − x,R) +Wt(x¯, u, R, P ) is submodular on (x, u, x¯). By the preservation of sub-modularity (Topkis, 1998), Theorem 2.7.6, Vt(x, u,R, P ) is submodularon (x, u).Proof of Theorem 2.4.1:73Appendix A. Appendix for Chapter 2We can write equation (A.2) asVt(x, u,R, P ) = minx¯{Ct(x¯− x,R) +Wt(x¯, u,R, P )}= minx¯{minx¯≥z{bz¯ − bz +Wt(x¯, u,R, P )},minx¯≤z{−sz + sz¯ +Wt(x¯, u,R, P )}}= minx¯{−bz + minx¯≥z{bz¯ +Wt(x¯, u,R, P )}, (A.3)− sz + minx¯≤z{sz¯ +Wt(x¯, u,R, P )}}DefineLt(u, R, P ) = arg minx¯{bz¯ +Wt(x¯, u, R, P )}Ht(u, R, P ) = arg minx¯{sz¯ +Wt(x¯, u, R, P )}Since Wt(x¯, u, R, P ) is convex in x¯ (Lemma A.0.1) and b ≥ s, we haveLt(u, R, P ) ≤ Ht(u, R, P ). Consider the two sub-optimization problemsin (A.4). We refer to −bz + minx¯≥z{bz¯ + Wt(x¯, u,R, P )} as optimizationproblem I and −sz + minx¯≤z{sz¯ +Wt(x¯, u,R, P )} as optimization problemII.(i) If z ≥ Ht(u, R, P ), then z ≥ Lt(u, R, P ). Let’s consider optimiza-tion problem I first. From the definition of Lt(u, R, P ) and convexityof Wt(x¯, u, R, P ) on x¯, bz¯ + Wt(x¯, u,R, P ) is increasing in x¯ whenx¯ ≥ z. Thus, for optimization problem I, the optimal solution isx¯ = z, which gives an optimal value −bz + bz + Wt(x, u,R, P ) =Wt(x, u,R, P ). Now let’s consider optimization problem II. Note theoptimal solution for optimization problem I, x¯ = z, is a feasible solu-tion for optimization problem II. However, according to the definitionof Ht(u, R, P ), x¯ = Ht(u, R, P ) is the optimal solution to optimiza-tion problem II, which gives an optimal value −sz + sHt(u, R, P ) +Wt(Ht(u, R, P ), u,R, P ). From the definition of Ht(u, R, P ), we havesHt(u, R, P ) +Wt(Ht(u, R, P ), u,R, P ) ≤ sz +Wt(x, u,R, P ). There-fore −sz + sHt(u, R, P ) +Wt(Ht(u, R, P ), u,R, P )≤Wt(x, u,R, P ), meaning that the optimal value of optimization prob-lem II is less than the optimal value of optimization problem I. Thus,for optimization problem (A.4), x¯∗ = Ht(u, R, P ).(ii) If z ≤ Lt(u, R, P ), then z ≤ Ht(u, R, P ). Let’s consider optimizationproblem II first. From the definition of Ht(u, R, P ) and convexity of74Appendix A. Appendix for Chapter 2Wt(x¯, u, R, P ) on x¯, sz¯+Wt(x¯, u,R, P ) is decreasing in x¯ when x¯ ≤ z.Thus, for optimization problem II, the optimal solution is x¯ = z, whichgives an optimal value −sz+sz+Wt(x, u,R, P ) = Wt(x, u,R, P ). Nowlet’s consider optimization problem I.Note the optimal solution for optimization problem II, x¯ = z, is afeasible solution for optimization problem I. However, according to thedefinition of Lt(u, R, P ), x¯ = Lt(u, R, P ) is the optimal solution to op-timization problem I, which gives an optimal value−bz+bLt(u, R, P )+Wt(Lt(u, R, P ), u,R, P ). From the definition ofLt(u, R, P ), we have bLt(u, R, P ) +Wt(Lt(u, R, P ), u,R, P )≤ bz +Wt(x, u,R, P ). Therefore −bz + bLt(u, R, P ) +Wt(Lt(u, R, P ), u,R, P ) ≤ Wt(x, u,R, P ), meaning that the optimalvalue of optimization problem I is less than the optimal value of op-timization problem II . Thus, for optimization problem (A.4), x¯∗ =Lt(u, R, P ).(iii) If Lt(u, R, P ) ≤ z ≤ Ht(u, R, P ), bz¯+Wt(x¯, u,R, P ) is increasing in x¯when x¯ ≥ z, sz¯ +Wt(x¯, u,R, P ) is decreasing in x¯ when x¯ ≤ z. Thus,for both optimization problem I and II, the optimal solution is x¯∗ = z.Therefore, for optimization problem (A.4), x¯∗ = z.Lemma A.0.3 In period t, t = 1, ·, T , given state (x, u,R, P ), any feasibleaction (x¯, y1, y2) with y2 ≥ y1 ≥ 0 can be categorized into two types accord-ing to either x¯ ≥ z or x¯ ≤ z. We can represent these two types as follows.Type one: (−x,A,B)(x ≥ 0, A ≥ 0, B ≥ 0), which means selling xunits of unbundled RECs, buying A units of REC-bundled energy andB units of regular energy.Type two: (+y,A,B)(y ≥ 0, A ≥ 0, B ≥ 0), which means buying yunits of unbundled RECs, buying A units of REC-bundled energy andB units of regular energy.We make two observations as follows:(i) When 4t ≥ bt, if the optimal action is type one (−x,A,B) , thenx ≥ A.(ii) When 4t ≤ st, if the optimal action is type two (y,A,B), then y ≥ B.75Appendix A. Appendix for Chapter 2Proof of Lemma A.0.3:We prove this Lemma by contradiction.(i) When 4t ≥ bt, suppose a = (−x,A,B) is the optimal action andx < A. Consider another action a′ = (0, A − x,B + x). By takingeither action a or a′, the utility obtains A − x units of RECs andA + B units of electricity in period t. The costs of these two actions,however, are different. Notecost(a)− cost(a′) =− stx+ p1tA+ p2tB − [p1t(A− x) + p2t(B + x)]=− stx+ p1tx− p2tx=x(4t − st) ≥ 0.Thus a′ is a better action than a, contradicts with the optimality of a.(ii) When 4t ≤ st, suppose b = (y,A,B) is the optimal action and y < B.Consider another action b′ = (0, A+y,B−y). By taking either actionb or b′, the utility obtains (A + y) units of RECs and (A + B) unitsof electricity in period t. The costscost(b)− cost(b′) =bty + p1tA+ p2tB − [p1t(A+ y) + p2t(B − y)]=bty − p1ty + p2ty=y(bt −4t) ≥ 0.Thus b′ is a better action than b, contradicts with the optimality ofb.Proof of proposition 2.4.4:We prove this proposition by sample path and contradiction.(i) In period t, if 4t ≥ bt, we want to show that it is optimal to purchaseonly regular energy. Suppose the optimal action involves purchasingsome REC-bundled renewable energy. Note 4t ≥ bt implies p1t ≥p2t + bt. This means for every unit of REC-bundled energy, the utilitycan get the equivalent product by combining one unit of regular energyand one unit of REC, but at a cheaper price. Therefore if 4t ≥ bt, itis always better to purchase only regular energy.(ii) In period t, if 4t ≤ st, we want to show that it is optimal to purchaseonly REC-bundled energy. Suppose the optimal action involves pur-chasing some regular energy. Note that 4t ≤ st implies p2t ≥ p1t − st.76Appendix A. Appendix for Chapter 2This means for every unit of regular energy, the utility can get theequivalent product by purchasing one unit of REC-bundled energy andthen selling the REC comes with it. The resulting price is cheaper thanpurchasing regular energy directly. Therefore if 4t ≤ st, it is alwaysbetter to purchase only REC-bundled energy.(iii) In period t, if st < 4t < bt,when z ≤ Lt(u,R, P ), according to Theorem 2.4.1, it is optimalto purchase RECs. Therefore we can write the optimal action astype two b1 = (y,A,B). By Lemma A.0.3 we know that y ≥ B.Consider another action b2 = (y−B,A+B, 0) . By taking eitherof action b1 or b2, , the utility gains y + A units of RECs andA+B units of electricity in period t. Compare the costs of thesetwo actions, we havecost(b1)− cost(b2)=bty + p1tA+ p2tB − [bt(y −B) + p1t(A+B)]=btB + p2tB + p1t(−B)=B(bt −4t) ≥ 0.Thus action b2 is better than action b1, contradicts with the as-sumption that action b1 is optimal. Therefore it is optimal topurchase only REC-bundled energy.When z ≥ Ht(u,R, P ), according to Theorem 2.4.1, it is optimalto sell RECs. Therefore we can write the optimal action as typeone a1 = (−x,A,B). By Lemma A.0.3 we know that x ≥ A.Consider action a2 = (−(x − A), 0, A + B), By taking either ofaction a1 or a2, the utility gains −x+A units of RECs and A+Bunits of electricity in period t. Compare the costs of these twoactions, we havecost(a1)− cost(a2)=− stx+ p1tA+ p2tB − [−st(x−A) + p2t(A+B)]=− stA− p2tA+ p1tA=A(4t − st) ≥ 0.Thus action a2 is better than action a1, contradicts with theassumption that actiona1 is optimal. Therefore it is optimal topurchase only regular energy.77Appendix A. Appendix for Chapter 2Proof of Theorem 2.4.5:If4t ≥ bt, by Proposition 2.4.4, it is optimal to purchase only regular energy,i.e., y∗1 = 0. We can write the cost-to-go function after the REC trading(A.1) asWt(x¯t, ut, Rt, Pt) = miny2≥0{p2y2 +Gt(y2)+ γEt[Vt+1(x¯, u+Dt, Rt+1, Pt+1)]}Thusy∗2 = arg miny2≥0{p2y2 +Gt(y2)} 4= S2t(p2)Further, the convexity of Gt(y) implies that S2t(p2) decreases in p2.Proof of Theorem 2.4.6:If4t ≤ st, by Proposition 2.4.4, it is optimal to purchase only REC-bundledenergy, i.e. , y∗1 = y∗2. We consider three cases based on the utility’s REClevel at the beginning of period t. For ease of analysis, we write w = x¯+ y1as the REC level at the end of period t.(i) When z ≤ Lt(u,R, P ), by Theorem 2.4.1, it is optimal for the utilityto purchase RECs to increase its REC level up to x¯∗ = Lt(u,R, P ).Therefore, we can write the cost-to-go function asVt(x, u,R, P )= miny1≥0w{bt[(w − y1)− z)] + p1y1 +Gt(y1)+ γEt[Vt+1(w, u+D,Rt+1, Pt+1)]}=− btz + miny1≥0w{(p1 − bt)y1 +Gt(y1)+ btw + γEt[Vt+1(w, u+D,Rt+1, Pt+1)]}The objective function in the bracket is a separate convex functions on(y1, w), thusw∗ = arg minw{btw + γEt[Vt+1(w, u+D,Rt+1, Pt+1)]} 4= wLt (ut, Rt, Pt),(A.4)y∗1 = arg miny1≥0{(p1 − bt)y1 +Gt(y1)} 4= SL1t(p1, bt), (A.5)78Appendix A. Appendix for Chapter 2andLt(u,R, P ) = x¯∗ = w∗ − y∗1 = wLt (ut, Rt, Pt)− SL1t(p1, bt).(ii) When z ≥ Ht(u,R, P ), by Theorem 2.4.1, it is optimal for the utilityto sell RECs to decrease its REC level to x¯∗ = Ht(u,R, P ). Therefore,we can write the cost-to-go function asVt(x, u,R, P )= miny1≥0w{−st[z − (w − y1)] + p1y1 +Gt(y1)+ γEt[Vt+1(w, u+D,Rt+1, Pt+1)]}=− stz + miny1≥0w{(p1 − st)y1 +Gt(y1)+ stw + γEt[Vt+1(w, u+D,Rt+1, Pt+1)]}Thusw∗ = arg minw{stw + γEt[Vt+1(w, u+D,Rt+1, Pt+1)]} 4= wHt (ut, Rt, Pt),y∗1 = arg miny1≥0{(p1 − st)y1 +Gt(y1)} 4= SH1t (p1, st),Ht(u,Rt, P ) = x¯∗ = w∗ − y∗1 = wHt (ut, Rt, Pt)− SH1t (p1, st). (A.6)(iii) When Lt(u,R, P ) < z < Ht(u,R, P ), by Theorem 2.4.1, it is optimalfor the utility not to trade RECs, i.e., x¯∗ = z. Therefore, we can writethe cost-to-go function asVt(x, u,R, P )= miny1≥0{p1y1 +Gt(y1) + γEt[Vt+1(z + y1, u+D,Rt+1, Pt+1]}.Thusy∗1 = arg miny≥0{p1y +Gt(y) + γEt[Vt+1(z + y, u+D,Rt+1, Pt+1]}=s1t(x, u,R, P ).From the definitions above, s1t(Lt(u,R, P ), u,R, P ) = SL1t(p1, bt) ands1t(Ht(u,R, P ), u,R, P ) = SH1t (p1, st).79Appendix A. Appendix for Chapter 2Proof of proposition 2.4.7:(a) Since f(x, y) = xy is supermodular on (x, y), and (p1 − bt)y + Gt(y)is supermodular on (p1, y) and submodular on (bt, y), SL1t(p1, bt) (A.5)decreases in p1 and increases in bt by Topkis (1998) (Theorem 2.8.2).(b) Similar with (a) we can prove (b).(c) Define a function g(x, u,R, P, y) ass1t(x, u,R, P )= arg miny≥0{p1y +Gt(y) + γEt[Vt+1(z + y, u+D,Rt+1, Pt+1]}= arg miny≥0g(x, u,R, P, y).In order to show s1t(x, u, R, P ) decreases in z, it is sufficient to showthat g(x, y, u,R, P ) is supermodular on (x, y) (Topkis, 1998), Theorem2.8.2. Since {(x, y) : y ≥ 0} is a sublattice of R2, and Vt+1(x, u,R, P )is convex on z ∈ R, thus, Vt+1(z + y, u,R, P ) is supermodular on (x, y)(Topkis, 1998), Lemma 2.6.2. Thus, g(x, y, u,R, P ) is supermodular on(x, y). In order to show s1t(x, u, R, P ) increases in u, it is sufficient toshow that g(x, u,R, P, yt) is submodular on (u, y), which is true sinceVt+1(x, u,R, P ) is submodular on (x, u). Thus we’ve proved (c).(d) From the convexity of Gt(y) and the assumption st ≤ bt, we haveSL1t(p1, bt) ≤ s1t(x, u,R, P ) ≤ SH1t (p1, st).Proof of theorem 2.4.8:If st < 4t < bt,(a) when xt ≤ Lt(ut, Rt, Pt), we know x¯∗t = Lt(ut, Rt, Pt) (Theorem 2.4.1), and it is optimal to purchase only REC-bundled energy (Proposition2.4.4), i.e., y∗1t = y∗2t. Denote wt = x¯t + y1t, we can write the cost-to-gofunction asVt(xt, ut, Rt, Pt)= miny1t≥0wt{bt[(wt − y1t)− xt] + p1ty1t +Gt(y1t)+ γEt[Vt+1(wt, ut +Dt, Rt+1, Pt+1]}=− btxt + miny1t≥0wt{(p1t − bt)y1t +Gt(y1t)+ btwt + γEt[Vt+1(wt, ut +Dt, Rt+1, Pt+1]}80Appendix A. Appendix for Chapter 2Thus y∗1t = SL1t(p1t, bt), w∗t = wLt (ut, Rt, Pt). Moreover, we haveLt(ut, Rt, Pt) = x¯∗t = w∗t − y∗1t = wLt (ut, Rt, Pt)− SL1t(p1t, bt).(b) when xt ≥ Ht(ut, Rt, Pt), we know x¯∗t = Ht(ut, Rt, Pt) (Theorem 2.4.1),and it is optimal to purchase only regular energy (Proposition 2.4.4),i.e., y∗1t = 0. We can write the cost-to-go function asVt(xt, ut, Rt, Pt)= miny2t≥0x¯t{−st(xt − x¯t) + p2ty2t +Gt(y2t)+ γEt[Vt+1(x¯t, ut +Dt, Rt+1, Pt+1]}=− stxt + miny2t≥0x¯t{p2ty2t +Gt(y2t)+ stx¯t + γEt[Vt+1(x¯t, ut +Dt, Rt+1, Pt+1]}Thus y∗2t = S2t(p2t), Ht(ut, Rt, Pt) = x¯∗t = wHt (ut, Rt, Pt).Proof of theorem 2.4.9:If st < 4t < bt, when Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt), x¯∗t = xt (Theorem2.4.1). We can write the cost-to-go function asVt(xt, ut, Rt, Pt) = Wt(xt, ut, Rt, Pt)= minwt≥xty2t≥wt−xt{4t(wt − xt) + p2ty2t +Gt(y2t)+ γEt[Vt+1(wt, ut +Dt, Rt+1, Pt+1]}= minwt≥xty2t≥wt−xt{−4txt + p2ty2t +Gt(y2t)+4twt + γEt[Vt+1(wt, ut +Dt, Rt+1, Pt+1]}= minwt≥xty2t≥wt−xtf(xt, ut, Rt, Pt, wt, y2t). (A.7)Denote (w∗t , y∗2t) = arg min wt≥xty2t≥wt−xtf(xt, ut, Rt, Pt, wt, y2t) as the optimalsolutions to this optimization problem.Note f(xt, ut, Rt, Pt, wt, y2t) is a separate convex function on (wt, y2t). Definew4t (ut, Rt, Pt) = arg minw {4tw + γEt[Vt+1(w, u+D,Rt+1, Pt+1)]},81Appendix A. Appendix for Chapter 2then (w4t (ut, Rt, Pt), S2t(p2t)) is the global minimum of f on R2 plane.W.l.o.g, we assume this global optimum is unique.As xt increases from −∞ to +∞, the feasible area {(wt, y2t) ∈ R2 : wt ≥xt, y2t ≥ wt − xt} is moving towards right. In the following we divide theregion of xt (Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt)) into three sub-regions, sothat we can discuss whether or not the sub-region has the global minimumas an interior point.To this end, definelt(ut, Rt, Pt) =w4t (ut, Rt, Pt)− S2t(p2t),ht(ut, Rt, Pt) =w4t (ut, Rt, Pt).First we show when st < 4t < bt,Lt(ut, Rt, Pt) ≤ lt(ut, Rt, Pt) ≤ ht(ut, Rt, Pt) ≤ Ht(ut, Rt, Pt),so that we can divide Lt(ut, Rt, Pt) < xt < Ht(ut, Rt, Pt) into three sub-regions, Lt(ut, Rt, Pt) < xt ≤ lt(ut, Rt, Pt), lt(ut, Rt, Pt) < xt < ht(ut, Rt, Pt),and ht(ut, Rt, Pt) ≤ xt < Ht(ut, Rt, Pt).From the definition of lt(ut, Rt, Pt) and ht(ut, Rt, Pt), we know that it isequivalent to showLt(ut, Rt, Pt) ≤ w4t (ut, Rt, Pt)−S2t(p2t) ≤ w4t (ut, Rt, Pt) ≤ Ht(ut, Rt, Pt).Let us start with the first inequality. When st < 4t < bt, we have Lt(ut, Rt, Pt)= wLt (ut, Rt, Pt)−SL1t(p1t, bt). Thus in order to prove the first inequality, it issufficient to show wLt (ut, Rt, Pt) ≤ w4t (ut, Rt, Pt) and SL1t(p1t, bt) ≥ S2t(p2t).From the definition of wLt (ut, Rt, Pt), w4t (ut, Rt, Pt), and the submodularityof V (x, u,R, P ) on (x, u), we know that wLt (ut, Rt, Pt) ≤ w4t (ut, Rt, Pt). Onthe other hand, from the definition of SL1t(p1t, bt), S2t(p2t) and the convexityof Gt(y), we know that SL1t(bt, p1t) ≥ S2t(p2t). Thus we’ve proved the firstinequality.The second inequality is obvious since S2t(p2t) ≥ 0.Let us look at the third inequality. When st < 4t < bt, Ht(ut, Rt, Pt) =wHt (ut, Rt, Pt). Thus the third inequality is equivalent as w4t (ut, Rt, Pt) ≤wHt (ut, Rt, Pt). From the definition of w4t (ut, Rt, Pt), wHt (ut, Rt, Pt), and82Appendix A. Appendix for Chapter 2the submodularity of V (x, u,R, P ) on (x, u), we know that w4t (ut, Rt, Pt) ≤wHt (ut, Rt, Pt). Thus the third inequality holds.Now we’ve proved the legitimacy of dividing Lt(ut, Rt, Pt) ≤ xt≤ Ht(ut, Rt, Pt) into three sub-regions with lt(ut, Rt, Pt) and Ht(ut, Rt, Pt).In the following ,we discuss whether or not each of the three sub-regions hasthe global minimum as an interior point.(a) When Lt(ut, Rt, Pt) < xt ≤ lt(ut, Rt, Pt), the feasible area is on the left-hand-side of the global minimum (w4t (ut, Rt, Pt), S2t(p2t)) and does notinclude it as an interior point. Since f(xt, ut, Rt, Pt, wt, y2t) is jointlyconvex on (wt, y2t), the optimal solution to optimization problem (A.7),(w∗t , y∗2t), is on the right boundary of the feasible set. Thus y∗2t = w∗t −xt. Therefore y∗2t = y∗1t, the utility should purchase only REC-bundledenergy. We haveVt(xt, ut, Rt, Pt) = Wt(xt, ut, Rt, Pt)= miny1t≥0{p1ty1t +Gt(y1t) + γEt[Vt+1(xt + y1t, ut +Dt, Rt+1, Pt+1)]}.Thus y∗1t = y∗2t = s1t(xt, ut, Rt, Pt).(b) When lt(ut, Rt, Pt) < xt < ht(ut, Rt, Pt), the global minimum(w4t (ut, Rt, Pt), S2t(p2t)) is in the interior of the feasible set. Thus theglobal minimum is the optimal solution to optimization problem (A.7).In this case, the utility should purchase both REC-bundled energy andregular energy. We have (w∗t , y∗2t) = (w4t (ut, Rt, Pt), S2t(p2t)).Thus y∗1t = w4t (ut, Rt, Pt)− xt, y∗2t = S2t(p2t).(c) When ht(ut, Rt, Pt) ≤ xt < Ht(ut, Rt, Pt), the feasible area is on theright-hand-side of the global minimum (w4t (ut, Rt, Pt), S2t(p2t)) anddoes not include it as an interior point. Since f(xt, ut, Rt, Pt, wt, y2t)is jointly convex on (wt, y2t), the optimal solution to optimization prob-lem (A.7), (w∗t , y∗2t), is on the left boundary of the feasible set. Thusw∗t = xt. Therefore y1t = 0, the utility should purchase only regularenergy. We haveVt(xt, ut, Rt, Pt)= miny2t≥0,x¯t{p2ty2t +Gt(y2t) + γEt[Vt+1(x¯t, ut +Dt, Rt+1, Pt+1)]}.Thus y∗1t = 0, y∗2t = S2t(p2t).83Appendix BAppendix for Chapter 3B.1 Analysis for optimization problem NThere is a piece-wise function in the objective function[d2+β(q1∧d1)−(q1−d1)+]+ =d2 + βq1, q1 ≤ d1(β + 1)d1 + d2 − q1, d1 < q1 ≤ (β + 1)d1 + d20, q1 > (β + 1)d1 + d2.This piece-wise function divides the region of q1 into three subregions: [0, d1],(d1, (β + 1)d1 + d2], and ((β + 1)d1 + d2,+∞). Therefore in our analysis,we consider these three subregions respectively. Although we will solve thisoptimization problem based on subregions of q1, it is important to point outthat because the objective function and the constraints are all continuous inq1, the objective value at the division points are consistent regardless whichsubregion we include them in.Let us start with the first subregion q1 ∈ [0, d1].Proposition B.1.1 In the subregion q1 ∈ [0, d1]:• when d2d1> 1− α− 2αβ1− α , there is no feasible solution in this subregion.• when d2d1≤ 1− α− 2αβ1− α ,– when θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then q∗1 = d1, and T ∗can be any value satisfying the incentive constraint. We set T ∗ =0.– when θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then q∗1 =1− α1− α− 2αβd2,and T ∗ = 0.84B.1. Analysis for optimization problem NProof of proposition B.1.1With the restriction q1 ∈ [0, d1], we can write the the first order derivativeof the objective function w.r.t. q1 asV ′N (q1) = (1− θ)(p− w)(1 + β) + θ[−w + s− αβ(p− w)]= (p− w)(1 + β)− θ[(p− w)(1 + β + αβ) + w − s]Therefore VN (q1) is linear, but the sign of V ′N (q1) is indeterminate.• If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then V′N (q1) ≥ 0.• If θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then V′N (q1) < 0.On the other hand, SI ≤ SN can be written as(1− α)(w − c)d2 + (1− α)T ≤ q1(w − c)(1− α− 2αβ),Therefore the constraints SI ≤ SN and T ≥ 0 can be combined as(1− α)d2 ≤ (1− α− 2αβ)q1.• If (1 − α)d2 > (1 − α − 2αβ)d1, there is no feasible solution in thissubregion.• If (1 − α)d2 ≤ (1 − α − 2αβ)d1, the optimal solutions will depend onthe sign of V ′N (q1).– If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then VN (q1) increases inthis subregion, q∗1 = d1, and T ∗ can be any value satisfying theincentive constraint, i.e., 0 ≤ T ∗ ≤ 1− α− 2αβ1− α (w− c)d1− (w−c)d2.– If θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then VN (q1, T ) decreases inthis subregion, q∗1 =1− α1− α− 2αβd2, and T∗ = 0.Next we look at the second subregion q1 ∈ (d1, (β + 1)d1 + d2].Proposition B.1.2 In the subregion q1 ∈ (d1, (β + 1)d1 + d2],85B.1. Analysis for optimization problem N• when d2d1> 1− α− 2αβ1− α , then q∗1 = α(2β + 1)d1 + (1 − α)d2, andT ∗ = 0.• when d2d1≤ 1− α− 2αβ1− α , then q∗1 = d1, and T ∗ can be any valuesatisfying the incentive constraint. We set T ∗ = 0.Proof of proposition B.1.2With the restriction q1 ∈ (d1, (β + 1)d1 + d2], we can write the objectivefunction asVN (q1)=− q1[h(1− θ) + θ(w − s)] + d1{[(p− w)(β + 1) + h](1− θ)− αβθ(p− w)}+ d2(p− w)(1− θ + αθ)Thus the first order derivative of the objective function isV ′N (q1) = −(1− θ)h− θ(w − s) < 0.Therefore VN (q1) decreases in q1 in this subregion.On the other hand, SI ≤ SN can be written asq1(w − c) ≥ (1− α)T + (1− α)(w − c)d2 + α(w − c)(2β + 1)d1.Therefore the constraints SI ≤ SN and T ≥ 0 can be combined asq1(w − c) ≥ (1− α)(w − c)d2 + α(w − c)(2β + 1)d1q1 ≥ (1− α)d2 + α(2β + 1)d1Thus• If (1−α)d2 +α(2β+ 1)d1 ≤ d1, the optimal solutions are q∗1 = d1, andT ∗ can be any value satisfying the incentive constraint;• If d1 < (1 − α)d2 + α(2β + 1)d1 ≤ (β + 1)d1 + d2, then the optimalsolutions are q∗1 = (1− α)d2 + α(2β + 1)d1 and T ∗ = 0;• If (1−α)d2 +α(2β+1)d1 > (β+1)d1 +d2, there is no feasible solutionin this region.First let us compare compare (1 − α)d2 + α(2β + 1)d1 and (β + 1)d1 + d2.We can show that (1− α)d2 + α(2β + 1)d1 ≤ (β + 1)d1 + d2.(1− α)d2 + α(2β + 1)d1 ≤ (β + 1)d1 + d2⇔[α(1 + 2β)− (β + 1)]d1 ≤ αd286B.1. Analysis for optimization problem NHence if α(1 + 2β)− (β + 1) ≤ 0, this condition always holds.If α(1+2β)−(β+1) > 0, this condition is equivalent as d2d1≥ 2β+1− β + 1α .Because2β + 1− β + 1α − β = β + 1−β + 1α ≤ 0.Thus from d2d1≥ β we know that d2d1≥ 2β+1− β + 1α . Therefore (1−α)d2 +α(2β + 1)d1 ≤ (β + 1)d1 + d2 always holds.Secondly, let us compare (1− α)d2 + α(2β + 1)d1 and d1.(1− α)d2 + α(2β + 1)d1 > d1⇔(1− α)d2 > [1− α(2β + 1)]d1• when d2d1> 1− α− 2αβ1− α , then q∗1 = α(2β + 1)d1 + (1 − α)d2, andT ∗ = 0.• when d2d1≤ 1− α− 2αβ1− α , then q∗1 = d1, and T ∗ can be any valuesatisfying the incentive constraint. We set T ∗ = 0.Finally we examine the third subregion q1 ∈ ((β + 1)d1 + d2,+∞).Proposition B.1.3 In the subregion q1 ∈ ((β + 1)d1 + d2,+∞), q∗1 = (β +1)d1 + d2, and T ∗ can be any value satisfying the incentive constraint. Weset T ∗ = 0.Proof of proposition B.1.3With the restriction q1 ∈ ((β + 1)d1 + d2,+∞), we can write the objectivefunction asVN (q1) =(1− θ)q1(−w − h) + (1− θ)d1(p+ pβ + h) + (1− θ)d2p+ θ(−w + s)q1 + θα(p− w)(d2 − βd1).The first order derivative of the objective function isV ′N (q1) = −(1− θ)(w + h)− θ(w − s) < 0,thus VN (q1, T ) decreases in q1 in this subregion.On the other hand, SI ≤ SN can be written as(1− α)(w − c)q1 ≥ (1− α)T + (1− 2α)(w − c)d2 + α(w − c)βd1.87B.1. Analysis for optimization problem NTherefore the constraints SI ≤ SN and T ≥ 0 can be combined asq1 ≥(1− 2α)d2 + αβd11− α .Thus• If (1− 2α)d2 + αβd11− α > (β + 1)d1 + d2, then q∗1 =(1− 2α)d2 + αβd11− αand T ∗ = 0.• If (1− 2α)d2 + αβd11− α ≤ (β+1)d1+d2, then q∗1 = (β+1)d1+d2, and T ∗can be any value satisfying the incentive constraint. We set T ∗ = 0.Next we show that (1− 2α)d2 + αβd11− α ≤ (β + 1)d1 + d2 always holds.(1− 2α)d2 + αβd11− α ≤ (β + 1)d1 + d2 ⇔ [α(1 + 2β)− (β + 1)]d1 ≤ αd2.If α(1 + 2β)− (β + 1) ≤ 0, then this condition always holds.If α(1 + 2β) − (β + 1) > 0, then this condition is equivalent as d2d1≥ (1 +2β)− β + 1α .Because(1 + 2β)− β + 1α − β = β + 1−β + 1α ≤ 0.Therefore from d2d1≥ β we know that d2d1≥ (1 + 2β)− β + 1α .In summary, in the subregion q1 ∈ ((β+1)d1 +d2,+∞), q∗1 = (β+1)d1 +d2,and T ∗ can be any value satisfying the incentive constraint. We set T ∗ = 0.Proof of theorem 3.6.1We have the following results from previous analysis.• If d2 >1− α(1 + 2β)1− α d1,– In the subregion q1 ∈ [0, d1], there is no feasible solution (propo-sition B.1.1).– In the subregion q1 ∈ (d1, (β+ 1)d1 +d2], q∗1 = α(1 + 2β)d1 + (1−α)d2, and T ∗ = 0 (proposition B.1.2).88B.2. Analysis for optimization problem I– In the subregion q1 ∈ ((β + 1)d1 + d2,+∞], q∗1 = (β + 1)d1 + d2,and T ∗ can be any value satisfying the incentive constraint. Weset T ∗ = 0 (proposition B.1.3).Therefore if d2 >1− α(1 + 2β)1− α d1, the global optimal solutions foroptimization problem N are q∗1 = α(1 + 2β)d1 + (1−α)d2, and T ∗ = 0.• If d2 ≤1− α(1 + 2β)1− α d1,– in the subregion q1 ∈ [0, d1], according to proposition B.1.1,∗ If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then q∗1 = d1, and T ∗can be any value satisfying the incentive constraint. We setT ∗ = 0.∗ If θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then q∗1 =1− α1− α− 2αβd2,and T ∗ = 0.– In the subregion q1 ∈ (d1, (β + 1)d1 + d2], q∗1 = d1, and T ∗ canbe any value satisfying the incentive constraint. We set T ∗ = 0(proposition B.1.2) .– In the subregion q1 ∈ ((β + 1)d1 + d2,+∞), q∗1 = (β + 1)d1 + d2,and T ∗ can be any value satisfying the incentive constraint. Weset T ∗ = 0 (proposition B.1.3).Therefore if d2 ≤1− α(1 + 2β)1− α d1, the global optimal solutions foroptimization problem N will depend on the value of θ.If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then q∗1 = d1, and T ∗ can beany value satisfying the incentive constraint. We set T ∗ = 0. If θ >(p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , then q∗1 =1− α1− α− 2αβd2, and T∗ = 0.B.2 Analysis for optimization problem IFor convenience of analysis, we definef(q1, T ) = αV 1 + (1− α)V 2 − [αV 3 + (1− α)V 4],89B.2. Analysis for optimization problem Iand write optimization problem I asmaxq1,TVI(q1, T ) =(1− θ)V 0 + θ[αV 3 + (1− α)V 4] + θG(q1)f(q1, T )s.t.SI ≥ SN ,T ≥ 0.Proof of proposition 3.6.2In the subregion q1 ∈ [0, d1], G(q1) = q1, the optimization problem can bewritten asmaxq1,TVI(q1, T ) =q1{−θ(w − s) + (p− w)[(1 + β)(1− θ)− αβθ]}+ d2(p− w)(1− θ + θα) + θq21[w − s+ α(p− w)(1 + 2β)]+ θq1d2(1− α)(p− w)− (1− α)Tq1θs.t.(1− α)T ≥(w − c)[q1(1− α− 2αβ)− (1− α)d2],T ≥0,q1 <d1.• If d2 >1− α− 2αβ1− α d1, then in the region [0, d1], the incentive con-straint is always met, thus is redundant, and T ≥ 0 is binding at theoptimum. We plug T = 0 into the objective function and getVI(q1, 0) =q1{−θ(w − s) + (p− w)[(1 + β)(1− θ)− αβθ]}+ d2(p− w)(1− θ + θα) + θq21[w − s+ α(p− w)(1 + 2β)]+ θq1d2(1− α)(p− w).We derive the first order derivative and the second order derivative:dVI(q1, 0)dq1=− θ(w − s) + (p− w)[(1 + β)(1− θ)− αβθ]+ 2θq1[w − s+ α(p− w)(1 + 2β)] + θd2(1− α)(p− w),d2VI(q1, 0)dq21=w − s+ α(p− w)(1 + 2β) > 0.Thus VI(q1, 0) is convex in q1.Define q11,1 to be the root of the first order condition, thenq11,1 =θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ]− θd2(1− α)(p− w)2θ[w − s+ α(p− w)(1 + 2β)]90B.2. Analysis for optimization problem IWe compare q11,1 with the midpoint of [0, d1] to get the optimal solu-tions in this subregion.– If q11,1 ≤d12 , then q∗1 = d1;– If q11,1 >d12 , then q∗1 = 0.We can writeq11,1 ≤d12 ⇔ d2 ≥− [α(1 + 2β)1− α +w − s(1− α)(p− w) ]d1+ θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− w) .• If d2 ≤1− α− 2αβ1− α d1 (this implies that 1 − α − 2αβ > 0), then weneed to divide [0, d1] into two parts:– In the first part q1 ∈ [0,(1− α)d21− α− 2αβ ], the incentive constraint isalways met, thus is not binding at the optimum, and T ≥ 0 isbinding. We plug T = 0 into the objective function. We find thatVI(q1, 0) is convex in q1, and the root of the first order conditionis q11,1.We compare q11,1 with the midpoint of [0,(1− α)d21− α− 2αβ ] toget the optimal solutions in this part.∗ If q11,1 ≤12 ·(1− α)d21− α− 2αβ , then the local optimal point in[0, (1− α)d21− α− 2αβ ] is(1− α)d21− α− 2αβ .∗ If q11,1 >12 ·(1− α)d21− α− 2αβ , then the local optimal point in[0, (1− α)d21− α− 2αβ ] is 0.We can writeq11,1 ≤12 ·(1− α)d21− α− 2αβ⇔d2 ≥(1− α− 2αβ){θ(w − s)− (p− w)[(1 + β)(1− θ)− θαβ}θ(1− α)(p− s)(B.1)91B.2. Analysis for optimization problem I– In the latter part q1 ∈ ((1− α)d21− α− 2αβ , d1], the incentive constraintis binding, i.e., (1−α)T ∗ = (w− c)[q1(1−α− 2αβ)− (1−α)d2].We plug T ∗ into the objective function and getf(q1, T ∗) = q1[(p− c)α(1 + 2β) + c− s] + d2(1− α)(p− c),VI(q1, T ∗) =q1{(p− w)[(1− θ)(1 + β)− θαβ] + θ(−w + s)}+ d2(1− θ + θα)(p− w) + θG(q1)f(q1, T ∗).We derive the first order derivative and the second order deriva-tive of VI(q1, T ∗) w.r.t. q1,dVI(q1, T ∗)dq1=q12θ[(p− c)α(1 + 2β) + c− s]+ θd2(1− α)(p− c) + (p− w)[(1− θ)(1 + β)− θαβ]+ θ(−w + s),d2VI(q1, T ∗)dq21=2θ[(p− c)(α+ 2αβ) + c− s] > 0This implies that VI(q1, T ∗) is convex.Define q21,1 to be the root of first order condition,q21,1 =θ(w − s)− (p− w)[(1− θ)(1 + β)− θαβ]− θd2(1− α)(p− c)2θ[c− s+ (p− c)α(1 + 2β)] .We compare q21,1 with the midpoint of ((1− α)d21− α− 2αβ , d1] to getthe optimal solutions is this part.∗ If q21,1 ≤12[(1− α)d2(1− α− 2αβ) + d1], the local optimal q1in ( (1− α)d21− α− 2αβ , d1] is d1.∗ If q21,1 >12[(1− α)d2(1− α− 2αβ) + d1], the local optimal q1in ( (1− α)d21− α− 2αβ , d1] is(1− α)d2(1− α− 2αβ) .92B.2. Analysis for optimization problem IWe can writeq21,1 ≤12[(1− α)d2(1− α− 2αβ) + d1]⇔ d2 ≥−(1− α− 2αβ)[c− s+ (p− c)α(1 + 2β)](1− α)(p− s) d1+ (1− α− 2αβ){θ(w − s)− (p− w)[(1 + β)(1− θ)− θαβ}θ(1− α)(p− s)(B.2)Now we can summarize the case when d2 ≤1− α− 2αβ1− α d1. Noticethat inequality B.1 can imply inequality B.2.– Whend2 ≥(1− α− 2αβ){θ(w − s)− (p− w)[(1 + β)(1− θ)− θαβ]}θ(1− α)(p− s) ,in the first part [0, (1− α)d21− α− 2αβ ], the local optimal q1 is(1− α)d21− α− 2αβ ; in the latter part ((1− α)d21− α− 2αβ , d1], the local op-timal q1 is d1. Therefore in the region [0, d1], the optimal q1 isd1.– Whend2 ≤−(1− α− 2αβ)[c− s+ (p− c)α(1 + 2β)](1− α)(p− s) d1+ (1− α− 2αβ){θ(w − s)− (p− w)[(1 + β)(1− θ)− θαβ}θ(1− α)(p− s) ,in the first part [0, (1− α)d21− α− 2αβ ], the local optimal q1 is 0; inthe latter part q1 ∈ ((1− α)d21− α− 2αβ , d1], the local optimal q1 is(1− α)d21− α− 2αβ . Therefore in the region [0, d1], the optimal q1 is 0.93B.2. Analysis for optimization problem I– When− (1− α− 2αβ)[c− s+ (p− c)α(1 + 2β)](1− α)(p− s) d1+ (1− α− 2αβ){θ(w − s)− (p− w)[(1 + β)(1− θ)− θαβ]}θ(1− α)(p− s)< d2 <(1− α− 2αβ){θ(w − s)− (p− w)[(1 + β)(1− θ)− θαβ]}θ(1− α)(p− s) ,in the part q1 ∈ [0,(1− α)d21− α− 2αβ ), the optimal q1 is 0; in the partq1 ∈ ((1− α)d21− α− 2αβ , d1], the optimal q1 is d1. we need to comparethe objective value at 0 and d1.Overall, when d2 ≤1− α− 2αβ1− α d1, the optimal q1 is either q∗1 =0 (with T ≥ 0 binding) or q∗1 = d1 (with the incentive constraintbinding). We only need to compare the objective value at these twopoints to get the optimal solution.The objective value at q∗1 = 0 (with T ≥ 0 binding) isVI(0, 0) = d2(p− w)(1− θ + θα).The objective value at q∗1 = d1 (with the incentive constraint binding)isVI(d1, T ∗) =d1{(p− w)[(1− θ)(1 + β)− θαβ] + θ(−w + s)}+ d2(p− w)(1− θ + θα) + θd21[(p− c)α(1 + 2β) + c− s]+ θd1d2(1− α)(p− c).The difference of these two valuesVI(d1, T ∗)− VI(0, 0) =d1{(p− w)[(1− θ)(1 + β)− θαβ]+ θ(−w + s)}+ θd21[(p− c)α(1 + 2β) + c− s]+ θd1d2(1− α)(p− c).94B.2. Analysis for optimization problem IThusVI(d1, T ∗) ≥VI(0, 0)⇔ d2 ≥−(p− c)α(1 + 2β) + c− s(1− α)(p− c) d1+ θ(w − s)− (p− w)[(1− θ)(1 + β)− θαβ]θ(1− α)(p− c) .Thus we proved proposition 3.6.2.Proof of proposition 3.6.3 :In the subregion q1 ∈ (d1, 1], G(q1) = q1, the optimization problem I can bewritten asmaxq1,TVI(q1, T ) =− q1[(1− θ)h+ θ(w − s)] + d2(p− w)(1− θ + θα)+ d1{(1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]}+ θq1f(q1, T )s.t.(1− α)T ≥(w − c)[q1 − α(1 + 2β)d1 − (1− α)d2]T ≥0,d1 <q1 ≤ 1,wheref(q1, T ) =q1(−αh+ w − s) + d1α[(p− w)(1 + 2β) + h]+ d2(1− α)(p− w)− (1− α)T.Note the right hand side of the incentive constraint is nonnegative if andonly if q1 ≥ α(1 + 2β)d1 + (1 − α)d2. Depending on the position of α(1 +2β)d1 + (1− α)d2, we will have the following three scenarios:• When α(1 + 2β)d1 + (1 − α)d2 ≤ d1, then in the subregion (d1, 1],the right hand side of the incentive constraint is nonnegative, thus isbinding at the optimum.• When α(1 + 2β)d1 + (1− α)d2 > 1, then in the subregion (d1, 1], theright hand side of the incentive constraint is negative, thus T ≥ 0 isbinding at the optimum.• When d1 < α(1 + 2β)d1 + (1 − α)d2 ≤ 1, then we need to dividethe subregion (d1, 1] into two parts. In the first part q1 ∈ (d1, α(1 +2β)d1 +(1−α)d2], T ≥ 0 is binding at the optimum. In the latter partq1 ∈ (α(1 + 2β)d1 + (1− α)d2, 1], the incentive constraint is binding.95B.2. Analysis for optimization problem INext we discuss these three scenarios respectively.• When α(1 + 2β)d1 + (1 − α)d2 ≤ d1, i.e., d2 ≤1− α− 2αβ1− α d1, thenin the subregion (d1, 1], the incentive constraint is binding at the op-timum. We find that VI(q1, T ) is convex, and q11,2 is the root to thefirst order condition. We compare q11,2 with the midpoint of (d1, 1] toget the optimal solution.– If q11,2 ≤12(d1 + 1), q∗1 = 1.– If q11,2 >12(d1 + 1), q∗1 = d1.We can writeq11,2 ≤12(d1 + 1)⇔ d2 ≥ −[α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1 +θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .We define Line 6 asd2 = −[α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1 +θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .• When α(1 + 2β)d1 + (1 − α)d2 > 1, i.e., d2 > −α(1 + 2β)1− α d1 +11− α, in the subregion (d1, 1], T ≥ 0 is binding at the optimum. We plugT = 0 into the objective function, and derive the first order derivativeand second order derivative,dVI(q1, 0)dq1=− (1− θ)h− θ(w − s) + 2θq1(−αh+ w − s)+ θd1α[(p− w)(1 + 2β) + h] + θd2(1− α)(p− w),d2VI(q1, 0)dq21=− αh+ w − s > 0.Thus VI(q1, 0) is convex. Define q21,2 to be the root of the first ordercondition, thenq21,2 =θ(w − s) + h(1− θ)− d1θα[(p− w)(1 + 2β) + h]− θd2(1− α)(p− w)2θ[−αh+ w − s] .We compare q21,2 with the midpoint of (d1, 1] and get the optimal so-lution.96B.2. Analysis for optimization problem I– If q21,2 ≤12(d1 + 1), q∗1 = 1.– If q21,2 >12(d1 + 1), q∗1 = d1.We can writeq21,2 ≤12(d1 + 1)⇔ d2 ≥ −[α(1 + 2β)1− α +w − s(1− α)(p− w) ]d1 +h(1− θ + αθ)θ(1− α)(p− w) .Define Line 3 asd2 = −[α(1 + 2β)1− α +w − s(1− α)(p− w) ]d1 +h(1− θ + αθ)θ(1− α)(p− w) .• When d1 < α(1 + 2β)d1 + (1 − α)d2 ≤ 1, i.e., d2 ≤ −α(1 + 2β)1− α d1 +11− α .– In the region q1 ∈ (d1, α(1 + 2β)d1 + (1−α)d2], T ≥ 0 is binding.We plug T = 0 into the objective function. We find that VI(q1, 0)is convex, and q21,2 is the root of the first order condition. Wecompare q21,2 with the midpoint of (d1, α(1 + 2β)d1 + (1 − α)d2]to get the optimal solution.∗ When q21,2 ≤12[d1 + α(1 + 2β)d1 + (1 − α)d2], then q∗1 =α(1 + 2β)d1 + (1− α)d2.∗ When q21,2 >12[d1 + α(1 + 2β)d1 + (1− α)d2], then q∗1 = d1.We can writeq21,2 ≤12[d1 + α(1 + 2β)d1 + (1− α)d2]⇔ d2 ≥− [α(1 + 2β)1− α +w − s(1− α)(p− αh− s) ]d1+ θ(w − s) + h(1− θ)θ(1− α)(p− αh− s)Define Line 4 asd2 =− [α(1 + 2β)1− α +w − s(1− α)(p− αh− s) ]d1+ θ(w − s) + h(1− θ)θ(1− α)(p− αh− s)97B.2. Analysis for optimization problem I– In the region q1 ∈ [α(1 + 2β)d1 + (1 − α)d2, 1], the incentiveconstraint is binding. We plug the binding incentive constraintinto the objective function.f(q1, T ∗) =q1(−αh+ c− s)+ d1α[(p− c)(1 + 2β) + h] + d2(1− α)(p− c),VI(q1, T ∗) =− q1[(1− θ)h+ θ(w − s)]+ d1{(p− w)[(1 + β)(1− θ)− αβθ] + h(1− θ)}+ d2(p− w)(1− θ + αθ) + θq1f(q1, T ∗)We then derive the first order derivative and the second orderderivative.dVI(q1, T ∗)dq1=q12θ(−αh+ c− s) + d1θα[(p− c)(1 + 2β) + h]+ d2θ(1− α)(p− c)− θ(w − s)− h(1− θ),d2VI(q1, T ∗)dq21=2θ(−αh+ c− s) > 0.Thus VI(q1, T ) is convex. We define q11,2 as the root of the firstorder condition.q11,2 =θ(w − s) + h(1− θ)− d1θα[(p− c)(1 + 2β) + h]− θd2(1− α)(p− c)2θ(−αh+ c− s).– When q11,2 ≤ 12 [α(1 + 2β)d1 + (1− α)d2 + 1], then q∗1 = 1.– When q11,2 > 12 [α(1+2β)d1+(1−α)d2+1], then q∗1 = α(1+2β)d1+(1−α)d2.We can writeq11,2 ≤12[α(1 + 2β)d1 + (1− α)d2 + 1]⇔ d2 ≥− [α(1 + 2β)1− α +αh(1− α)(p− αh− s) ]d1+ θ(w − c) + h(1− θ + θα)θ(1− α)(p− αh− s) .Define Line 5 asd2 = −[α(1 + 2β)1− α +αh(1− α)(p− αh− s) ]d1 +θ(w − c) + h(1− θ + θα)θ(1− α)(p− αh− s) .98B.2. Analysis for optimization problem ITherefore when d2 ≤ −α(1 + 2β)1− α d1 +11− α :– In the area that is above both Line 4 and Line 5, in the first part(d1, α(1 + 2β)d1 + (1 − α)d2], q∗1 = α(1 + 2β)d1 + (1 − α)d2]. Inthe latter part (α(1 + 2β)d1 + (1− α)d2, 1], q∗1 = 1. Therefore inthe subregion (d1, 1], q∗1 = 1 and T ∗ =w − c1− α [1 − α(1 + 2β)d1 −(1− α)d2].– In the area that is below both Line 4 and Line 5, in the firstpart (d1, α(1 + 2β)d1 + (1 − α)d2], q∗1 = d1. In the latter part(α(1+2β)d1+(1−α)d2, 1], q∗1 = α(1+2β)d1+(1−α)d2. Thereforein the subregion (d1, 1], q∗1 = d1 and T ∗ = 0 .– In the area that is above Line 4 and below Line 5, in the first part(d1, α(1+2β)d1 +(1−α)d2], q∗1 = α(1+2β)d1 +(1−α)d2]. In thelatter part (α(1+2β)d1+(1−α)d2, 1], q∗1 = α(1+2β)d1+(1−α)d2.Therefore in the subregion (d1, 1], q∗1 = α(1 + 2β)d1 + (1 − α)d2and T ∗ = 0.– In the area that is below Line 4 and above Line 5, in the firstpart (d1, α(1 + 2β)d1 + (1 − α)d2], q∗1 = d1. In the latter part(α(1 + 2β)d1 + (1 − α)d2, 1], q∗1 = 1. Therefore in the subregion(d1, 1], q∗1 could be d1 or 1. If q∗1 = d1, then T ∗ = 0. If q∗1 = 1,then T ∗ = w − c1− α [1− α(1 + 2β)d1 − (1− α)d2].Thus we proved proposition 3.6.3.Proof of proposition 3.6.4:In the subregion q1 ∈ (1, (β+1)d1+d2], G(q1) = 1, the optimization problemcan be written asmaxq1,TVI(q1, T )s.t.(1− α)T ≥ (w − c)[q1 − α(1 + 2β)d1 − (1− α)d2],T ≥ 0,1 <q1 ≤ (β + 1)d1 + d2.Note that the right hand side of the incentive constraint is nonpositive ifand only if q1 ≤ α(1 + 2β)d1 + (1− α)d2.• If α(1 + 2β)d1 + (1 − α)d2 > (β + 1)d1 + d2, then in the subregion(1, (β + 1)d1 + d2], the incentive constraint is always met, thus is not99B.2. Analysis for optimization problem Ibinding at the optimum. Because the objective function decreases inT , T ≥ 0 is binding at the optimum. We plug T = 0 into the objectivefunction,VI(q1, 0) =− q1[(1− θ)h+ θ(w − s)] + d2(p− w)(1− θ + θα)+ d1{(1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]}+ θ{q1(−αh+ w − s) + d1α[(p− w)(1 + 2β) + h]+ d2(1− α)(p− w)}.We derive the first order derivativedVI(q1, 0)dq1= −(1− θ + αθ)h < 0.Therefore VI(q1, 0) decreases in q∗1, the optimal solutions are q∗1 = 1and T ∗ = 0.• If α(1+2β)d1 +(1−α)d2 ≤ 1, then in the subregion (1, (β+1)d1 +d2],the incentive constraint is binding at the optimum. We plug (1 −α)T ∗ = (w−c)[q1−α(1+2β)d1−(1−α)d2] into the objective function,VI(q1, T ∗) =q1[−θ(w − s)− h(1− θ)]+ d1{[(p− w)(β + 1) + h](1− θ)− αβθ(p− w)}+ d2(p− w)(1− θ + αθ) + θ{q1(−α+ c− s)+ d1α[(p− c)(1 + 2β) + h] + d2(1− α)(p− c)}.We derive the first order derivativedVI(q1, T ∗)dq1= −θ(w − c)− h(1− θ + θα) < 0.Thus VI(q1, T ∗) decreases in q∗1, the optimal solutions are q∗1 = 1 andT ∗ = w − c1− α [1− α(1 + 2β)d1 − (1− α)d2] .• If 1 < α(1 + 2β)d1 + (1 − α)d2 ≤ (β + 1)d1 + d2, then we need todivide the subregion (1, (β + 1)d1 + d2] into two parts, the first partis (1, α(1 + 2β)d1 + (1− α)d2], the second part is (α(1 + 2β)d1 + (1−α)d2, (β + 1)d1 + d2]. In the first part, at the optimum T ≥ 0 isbinding, we have shown that the objective function decreases in q1. Inthe second part, at the optimum the incentive constraint is binding, wehave shown that the objective function also decreases in q1. Thereforein the subregion(1, (β + 1)d1 + d2], the objective function decreases inq1, the optimal solutions are q∗1 = 1 and T ∗ = 0.100B.2. Analysis for optimization problem IThus we proved proposition 3.6.4.Proof of proposition 3.6.5:In the subregion q1 ∈ ((β + 1)d1 + d2,+∞), G(q1) = 1, the optimizationproblem can be written asmaxq1,TVI(q1, T )s.t.(1− α)T ≥ (1− α)(w − c)(q1 − d2) + α(w − c)(d2 − βd1)T ≥ 0q1 > (β + 1)d1 + d2.Note that (1 − α)(w − c)(q1 − d2) + α(w − c)(d2 − βd1) ≥ 0, thereforethe incentive constraint is binding at the optimum. We plug (1 − α)T ∗ =(1− α)(w − c)(q1 − d2) + α(w − c)(d2 − βd1) into the objective function,f(q1, T ∗) =q1[−αh+ (1− α)c− s] + d1[(α+ 2αβ)p+ αh− αβc]+ d2[(1− α)w − (1− 2α)c],VI(q1, T ∗) =q1[−w − h(1− θ) + θs] + d1[(p+ pβ + h)(1− θ)− αβθ(p− w)]+ d2[p(1− θ) + αθ(p− w)] + θf(q1, T ∗).We derive the first order derivative,dVI(q1, T ∗)dq1= −w − h(1− θ)− θαh+ θ(1− α)c < 0Therefore VI(q1, T ∗) decreases in this subregion, and the optimal q1 is q∗1 =(β + 1)d1 + d2. Thus we proved proposition 3.6.5.Proof of theorem 3.6.6:When d2 ≤1− α(1 + 2β)1− α d1, from proposition 3.6.4 and proposition 3.6.5,we know that the objective function of optimization problem I decreaseswhen q1 > 1. Thus the optimal q1 for optimization problem I will be in theregion q1 ∈ [0, 1]. From proposition 3.6.2, in the subregion q1 ∈ [0, d1],• When (d1, d2) is above Line 2, then q∗1 = d1 and T ∗ makes the incentiveconstraint binding.• When (d1, d2) is below Line 2, then q∗1 = 0 and T ∗ = 0.101B.2. Analysis for optimization problem IThe equation for Line 2 isd2 = −[α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1 +θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− c) .From proposition 3.6.3, in the subregion q1 ∈ (d1, 1],• When (d1, d2) is above Line 6, then q∗1 = 1 and T ∗ makes the incentiveconstraint binding.• When (d1, d2) is below Line 6, then q∗1 = d1 and T ∗ makes the incentiveconstraint binding.The equation for Line 6 isd2 = −[α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1 +θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .Notice that Line 2 and Line 6 has the same slope. Compare the interceptof Line 2 and Line 6,θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− c) ≤θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c)⇔θ(w − s)− (p− w)[(1 + β)(1− θ)− αβθ] ≤ θ(w − c) + h(1− θ + αθ)⇔θ ≤ (p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h.Note that we have(p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h >(p− w)(1 + β)(p− w)(1 + β + αβ) + w − s.Therefore we have the following results.• If θ ≤ (p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h , then Line 2’s interceptis less than Line 6’ intercept.– When (d1, d2) is above Line 6, in the subregion q1 ∈ [0, d1],q∗1 = d1 and T ∗ makes the incentive constraint binding. In thesubregion q1 ∈ (d1, 1], q∗1 = 1 and T ∗ makes the incentive con-straint binding. Therefore in the region q1 ∈ [0, 1], q∗1 = 1 and T ∗makes the incentive constraint binding.102B.2. Analysis for optimization problem I– When (d1, d2) is below Line 2, in the subregion q1 ∈ [0, d1], q∗1 = 0and T ∗ = 0. In the subregion q1 ∈ (d1, 1], q∗1 = d1 and T ∗makes the incentive constraint binding. Therefore in the regionq1 ∈ [0, 1], q∗1 = 0 and T ∗ = 0.– When (d1, d2) is above Line 2 and below Line 6, in the subre-gion q1 ∈ [0, d1], q∗1 = d1 and T ∗ makes the incentive constraintbinding. In the subregion q1 ∈ (d1, 1], q∗1 = d1 and T ∗ makes theincentive constraint binding. Therefore in the region q1 ∈ [0, 1],q∗1 = d1 and T ∗ makes the incentive constraint binding.• If θ > (p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h , then Line 2 has a big-ger intercept than Line 6. Then– When (d1, d2) is above Line 2, in the subregion q1 ∈ [0, d1],q∗1 = d1 and T ∗ makes the incentive constraint binding. In thesubregion q1 ∈ (d1, 1], q∗1 = 1 and T ∗ makes the incentive con-straint binding. Therefore in the region q1 ∈ [0, 1], q∗1 = 1 and T ∗makes the incentive constraint binding.– When (d1, d2) is below Line 6, in the subregion q1 ∈ [0, d1], q∗1 = 0and T ∗ = 0. In the subregion q1 ∈ (d1, 1], q∗1 = d1 and T ∗makes the incentive constraint binding. Therefore in the regionq1 ∈ [0, 1], q∗1 = 0 and T ∗ = 0.– When (d1, d2) is above Line 6 and below Line 2, in the subregionq1 ∈ [0, d1], q∗1 = 0 and T ∗ = 0. In the subregion q1 ∈ (d1, 1],q∗1 = 1 and T ∗ makes the incentive constraint binding. Thereforein the region q1 ∈ [0, 1], we need to compare the objective valueVI(0, 0) and VI(1, T ∗).VI(0, 0) =d2(p− w)(1− θ + θα),VI(1, T ∗) =− [(1− θ)h+ θ(w − s)] + d2(p− w)(1− θ + αθ)+ d1{(1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]}+ θ(c− s− αh) + θd1α[(p− c)(1 + 2β) + h]+ θd2(1− α)(p− c).ThereforeVI(1, T ∗)− VI(0, 0)=− [(1− θ)h+ θ(w − s)]+ d1{(1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]}+ θ(c− s− αh) + θd1α[(p− c)(1 + 2β) + h] + θd2(1− α)(p− c).103B.2. Analysis for optimization problem IWe haveVI(1, T ∗) > VI(0, 0)⇔ d2 > −[α(1 + 2β)1− α +αh(1− α)(p− c)+ (1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− c) ]d1 +θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .Define Line 7 asd2 =− [α(1 + 2β)1− α +αh(1− α)(p− c)+ (1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− c) ]d1+ θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .Recall that the equation for Line 6 isd2 = −[α(1 + 2β)1− α +c− s(1− α)(p− c) ]d1 +θ(w − c) + h(1− θ + αθ)θ(1− α)(p− c) .Note that Line 7 and Line 6 has the same intercept. Comparethe slope of Line 7 and Line 6, we haveαh(1− α)(p− c) +(1− θ)h+ (p− w)[(1 + β)(1− θ)− αβθ]θ(1− α)(p− c)< c− s(1− α)(p− c)⇔θ > (p− w)(1 + β) + h(p− w)(1 + β + αβ) + c− s+ (1− α)h.Therefore Line 6 is steeper than Line 7. Line 7 divides the regionbetween Line 6 and Line 2 into two parts. In the part that isabove 7, q∗1 = 1 and T ∗ = 0. In the part that is below Line 7,q∗1 = 0 and T ∗ = 0.We proved theorem 3.6.6.Proof of theorem 3.6.6:When d2 >1− α(1 + 2β)1− α d1, from proposition 3.6.4 and proposition 3.6.5,we know that the objective function of optimization problem I decreaseswhen q1 > 1. Thus the optimal q1 for optimization problem I will be in theregion q1 ∈ [0, 1].104B.3. Analysis for overall optimal solutions for the OEM• If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , both Line 1 and Line 2 has neg-ative intercepts, therefore according to proposition 3.6.2, in the sub-region q1 ∈ [0, d1], the optimal q∗1 = d1. Therefore the optimal q∗1 foroptimization problem I is in the region [d1, 1], and the possible valuesare d1, 1, and α(1 + 2β)d1 + (1− α)d2(proposition3.6.3).• If θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , according to proposition 3.6.2and proposition3.6.3, the possible values for optimal q∗1 are 0, d1, 1,and α(1 + 2β)d1 + (1− α)d2.We proved theorem 3.6.7.B.3 Analysis for overall optimal solutions for theOEMProof of theorem 3.6.8:First, let us compare the optimal expected profit for the OEM in optimiza-tion problem N and optimization problem I when d2 >1− α(1 + 2β)1− α d1.From theorem 3.6.1, we know that the optimal q1 for optimization problemN is q∗1 = α(1 + 2β)d1 + (1 − α)d2 and T ∗ = 0. At the optimum, the sup-plier is indifferent between inform or not inform. Next we show that givenq∗1 = α(1 + 2β)d1 + (1 − α)d2 and T ∗ = 0, the OEM can do strictly betterif the supplier informs. Therefore the OEM strictly prefers the supplier toinform. Note that d1 < α(1 + 2β)d1 + (1− α)d2 ≤ (β + 1)d1 + d2. BecauseVI(q1, 0) =− [h(1− θ) + θ(w − s)]q1 + d1{(p− w)[(1 + β)(1− θ)− αβθ]+ h(1− θ)}+ d2(p− w)(1− θ + αθ) + θG(q1)f(q1, 0).The difference between this twoVI(q1, 0)− VN (q1) = θG(q1)f(q1, 0),wheref(q1, 0) =q1(−αh+ w − s) + d1α[(p− w)(1 + 2β) + h]+ d2(1− α)(p− w).Because G(q1) > 0 and f(q1, 0) > 0, thus VI(q1, 0) > VN (q1). Thusit is optimal for the OEM to direct the supplier to inform when d2 >105B.3. Analysis for overall optimal solutions for the OEM1− α(1 + 2β)1− α d1.Secondly, let us compare the optimal expected profit for the OEM in opti-mization problem N and optimization problem I when d2 ≤1− α(1 + 2β)1− α d1.From theorem 3.6.1 and theorem 3.6.6, we have the following results.• If θ ≤ (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , the optimal solutions for theOEM in optimization problem N is q∗1 = d1 and T∗ can be any valuesatisfying 0 ≤ T ∗ ≤ w − c1− α [(1− α− 2αβ)d1 − (1− α)d2]. The optimalobjective value for optimization problem N isV ∗N =d1{(p− w)[(1 + β)(1− θ)− θαβ] + θ(−w + s)}+ d2(p− w)(1− θ + αθ).At q∗1 = d1 and T∗ =w − c1− α [(1−α− 2αβ)d1− (1−α)d2], the supplieris indifferent between inform or not inform. If we can show that whenq∗1 = d1 and T∗ =w − c1− α [(1 − α − 2αβ)d1 − (1 − α)d2], the objectivevalue of optimization problem N is less than the objective value ofoptimization problem I, then it is optimal for the OEM to direct thesupplier to inform.VI(d1, T ∗) =d1{(p− w)[(1 + β)(1− θ)− θαβ] + θ(−w + s)}+ d2(p− w)(1− θ + αθ) + θd21[(p− c)α(1 + 2β) + c− s]+ θd1d2(1− α)(p− c).Apparently VI(d1, T ∗) > V ∗N . Therefore it is optimal for the OEM todirect the supplier to inform.• If θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s , the optimal q1 for optimizationproblem N is q∗1 =1− α1− α(1 + 2β)d2 and T∗ = 0. The optimal objectivevalue for optimization problem N isV ∗N ={(p− w)[(1 + β)(1− θ)− θαβ] + θ(−w + s)}1− α1− α(1 + 2β)d2+ d2(p− w)(1− θ + αθ).106B.3. Analysis for overall optimal solutions for the OEMThe optimal solutions (q∗1, T ∗) for optimization problem I is (0, 0),(d1,max{w − c1− α [(1−α−2αβ)d1−(1−α)d2], 0}), or (1,max{w − c1− α [1−α(1+2β)d1− (1−α)d2], 0}). In order to prove optimization problem Idominates optimization problem N, it is sufficient to prove that at oneof these points, the objective value of optimization problem I is biggerthan the optimal objective value of optimization problem N. Next weshow VI(0, 0) is bigger than V ∗N .VI(0, 0) = d2(p− w)(1− θ + θα).V ∗N − VI(0, 0) ={(p− w)[(1 + β)(1− θ)− θαβ] + θ(−w + s)} 1− α1− α(1 + 2β)d2 < 0.The inequality comes from the condition that θ > (p− w)(1 + β)(p− w)(1 + β + αβ) + w − s .Therefore it is optimal for the OEM is to direct the supplier to informwhere d1 ≤1− α(1 + 2β)1− α d1. We’ve proved theorem 3.6.8.107

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