As companies outsource more product design and manufacturing activities to other members of the supply

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1 MANAGEMEN SCIENCE Vol. 55, No. 7, July 2009, pp in ein inform doi /mnc INFORMS Quality Improvement Incentive and Product Recall Cot Sharing Contract Gary H. Chao College of Buine, Kutztown Univerity, Kutztown, Pennylvania 19530, Seyed M. R. Iravani Department of Indutrial Engineering and Management Science, Northwetern Univerity, Evanton, Illinoi 60208, R. Canan Savakan Managerial Economic and Deciion Science, Kellogg School of Management, Northwetern Univerity, Evanton, Illinoi 60208, A companie outource more product deign and manufacturing activitie to other member of the upply chain, improving end-product quality ha become an endeavor extending beyond the boundarie of the firm in-houe proce capabilitie. In thi paper, we dicu two contractual agreement by which product recall cot can be hared between a manufacturer and a upplier to induce quality improvement effort. More pecifically, we conider (i) cot haring baed on elective root caue analyi (Contract S), and (ii) partial cot haring baed on complete root caue analyi (Contract P). Uing inight from upermodular game theory, for each contractual agreement, we characterize the level of effort the manufacturer and the upplier would exert in equilibrium to improve their component failure rate when their effort choice are ubject to moral hazard. We how that both Contract S and Contract P can achieve the firt bet effort level; however, Contract S reult in higher profit for the manufacturer and the upply chain. For the cae in which the information about the quality of the upplier product i not revealed to the manufacturer (i.e., the cae of information aymmetry), we develop a menu of contract that can be ued to mitigate the impact of information aymmetry. We how that the menu of contract not only ignificantly decreae the manufacturer cot due to information aymmetry, but alo improve product quality. Key word: reliability; quality control; contract; product deign; upply chain coordination; information aymmetry Hitory: Received June 13, 2006; accepted December 15, 2008, by Paul H. Zipkin, operation and upply chain management. Publihed online in Article in Advance May 14, Introduction In 2004, a conumer reearch tudy in the auto indutry reported initial product quality a the econd mot important factor affecting conumer purchaing deciion after product price (J.D. Power and Aociate 2004). 1 he large number of product recall and lawuit in the auto indutry demontrate how undetected quality problem and related production delay can lead to a huge profit lo and degrade a company brand equity. For intance, in 2007, Ford concern about a deign related quality problem in cruie control witche reulted in a recall of 3.6 million vehicle manufactured between 1992 and 2004, increaing the total number of vehicle recalled for the ame quality problem to 9.6 million (Aociated Pre 2007). In the electronic indutry, in April 2007, Sanyo agreed to hare with the manufacturer Lenovo, the $17 million cot of recalling 205,000 Sanyo made 1 Initial product quality i meaured by the number of quality problem manifeting themelve during the firt 90 day after purchae. laptop battery pack that can overheat becaue of a flaw in the product deign (Nytedt 2007). In May 2007, the Conumer Product Safety Commiion, the National Highway raffic Safety Adminitration, and Evenflo Company Inc. announced a recall of Evenflo embrace infant car eat/carrier becaue of a malfunctioning handle. A total of 450,000 unit, manufactured in the United State and China, were old nationwide through department tore and baby item tore (CNNMoney 2007). hee are jut a few example that demontrate that recall are common in a variety of indutrie and often are aociated with ubtantial preent and future cot to a company. he cot and cale of recall neceitate a deeper undertanding of how to manage the quality improvement incentive of multiple upply chain partner to enure better product performance. Product recall reult from a lack of quality aurance in the manufacturing and/or deign procee of one or many upply chain partner and can affect a large number of product manufactured over extended period of time. For example, in a recent 1122

2 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract Management Science 55(7), pp , 2009 INFORMS 1123 tudy, Ford reported that 76% of the company quality problem tem from it firt tier upplier (Sherefkin 2002). oday, extended quality improvement effort take variou form. For intance, manufacturer in the auto indutry are more willing to involve upplier during the product development proce to enure early detection and elimination of quality problem (Kiiel 2007). In addition to preventive initiative, it i alo becoming a common practice among manufacturer to preent upplier with quality cot haring agreement to enure accountability of quality problem and to create incentive for proce improvement (Balachandran and Radhakrihnan 2005). In thi paper, we addre the optimal deign of a recall cot haring contract when both the upplier and the manufacturer quality improvement effort are ubject to moral hazard and when the manufacturer ha uncertainty regarding the quality of the upplier proce. In thi context, we dicu the optimal ue of product failure root caue analyi information in the deign of cot haring cheme. Previou reearch on contract deign in quality management ha tudied the fixed hare rate contract 2 (Contract F) a the external quality cot (uch a recall cot) haring cheme between a manufacturer and a upplier. In thi paper, we introduce two new contract format to hare product recall related external quality cot in the upply chain: (i) the elective root caue analyi contract (Contract S), which i characterized by a unit part price p, a fixed recall cot hare rate R, paid by the upplier, 3 and a threhold product failure time ( ); and (ii) the partial cot allocation contract (Contract P), which i characterized by a unit part price p, a fixed cot hare rate R m paid by the upplier if the manufacturer i reponible for the product failure, and a fixed cot hare rate R paid by the upplier if the upplier i reponible for the product failure. A critical component of both contract format i the root caue analyi information, which reveal the upply chain member who i reponible for the quality problem in the product. Contract S ue thi information only if the product failure occur before a threhold time ( ), which we will refer to a the root caue analyi threhold, and allocate the total recall cot to the party at fault. Otherwie, the cot i hared according to a fixed rate R. Under Contract P, root caue analyi information i alway ued in the cot allocation proce, the total recall cot i alway hared between the 2 Under a fixed hare rate contract, the upplier aume R percentage of the total external quality cot, while the manufacturer pay for the 1 R percentage, irrepective of who i at fault for the quality problem. 3 he upplier hare R percentage of the total recall cot, while the manufacturer pay the remaining 1 R percentage. partie, and the upply chain member who i reponible for the recall incur a larger hare of the recall cot. Conidering a ingle-manufacturer, ingle-upplier upply chain tructure, we addre the following reearch quetion regarding thee contractual agreement: How effective are elective root caue analyi and partial cot allocation contract in coordinating the manufacturer and the upplier quality improvement effort when the effort level are not obervable and therefore are ubject to moral hazard? Which contract format i optimal for the manufacturer? How do the fixed hare rate contract, elective root caue analyi contract, and the partial cot allocation contract compare with repect to the manufacturer profit a well a the quality of the final product? If the exact information about the upplier initial quality i not available, can the manufacturer deign a menu of elective root caue analyi contract to creen upplier type a well a to induce quality improvement effort? Under what circumtance doe knowing the upplier initial quality reult in ignificant aving for the manufacturer? How much doe the quality of the final product under the menu of contract differ from that under perfect information (i.e., when the manufacturer know the exact initial quality of upplier product)? In the following ection, we preent the contribution of our paper in the context of the exiting literature on quality and upply chain management. hen, we preent the baic modeling framework and the aumption of our model in 3. Section 4 develop a detailed analyi of Contract S and Contract P under the complete information aumption. We preent an extenive numerical tudy in 5, which examine the profitability of thee contract for the manufacturer a well a for the total upply chain. Section 6 invetigate the cae of information aymmetry and derive the optimal menu of elective root caue analyi contract. Section 7 preent a numerical tudy to compare the cot efficiency and product quality under different contract in cae of information aymmetry. A ummary of our finding and direction for future reearch are preented in Literature he main focu of thi reearch i on modeling the proce improvement incentive of upply chain member when their effort choice are not obervable and there i information aymmetry with regard to their exiting proce capability. herefore, by jointly modeling moral hazard and advere election iue, thi paper contribute to everal tream of reearch each of which we review below.

3 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract 1124 Management Science 55(7), pp , 2009 INFORMS In operation management, a group of paper dicue the deign of quality cot haring contract among manufacturer and upplier. In a game theoretic et-up, Reynier and apiero (1995a, b) and Lim (2001) model upplier choice of proce quality and manufacturer choice of inpection trategy. heir model characterize the Nah equilibrium of the upplier-manufacturer quality game in term of the cot haring parameter for internal (rework) and external (warranty) quality cot, auming a fixed rate for haring external quality cot between the partie. We, however, model a more general contract format for haring external quality cot reulting from a recall. A pecial cae of our contract of interet i the fixed hare rate contract tudied in the above paper. Baiman et al. (2000) analyze the relationhip between product quality, cot of quality, and the information that can be contracted on. In a rik neutral etting, the upplier invet in reducing the proce defect rate and the manufacturer invet in the inpection quality of the incoming part. Both deciion are ubject to moral hazard. Like Reynier and apiero (1995a, b) and Lim (2001), they alo aume that the external quality cot are hared at a fixed rate. In a ubequent paper (Baiman et al. 2001), the author invetigate the link between product deign, contractible information, and the upplier invetment in proce quality. In contrat to thi paper, which conider fixed hare rate contract, we focu on a broader et of contract format to hare external product failure cot and how that, even though the root caue analyi can perfectly determine the party reponible for product failure, it i not optimal for the upply chain to hare quality cot baed on thi information for all failure occurring during the contract period. We propoe a contract with elective root caue analyi which differentiate early failure from late failure to coordinate the quality improvement effort of upply chain member. In a ubequent paper, Baiman et al. (2003) examine a product tructure exhibiting the weaket link property and invetigate how the internal and external failure cot haring mechanim impact upplier election when there i an advere election problem. heir analyi conider moral hazard only on the upplier ide, wherea we model moral hazard both on the manufacturer and on the upplier ide. Furthermore, like previouly cited work, their analyi alo aume that the external quality cot are hared at a fixed rate, which i, in fact, a pecial cae of the contract we invetigate. Balachandran and Radhakrihnan (2005) conider a double moral hazard ituation in quality invetment effort, in which the final product conit of component made by a buyer and a upplier. Although their paper focue on the bet ue of incoming inpection information to achieve the firt bet effort level from the upply chain partner, in thi paper we invetigate the bet ue of root caue analyi information about external failure to achieve firt bet effort level from upply chain member. Furthermore, Balachandran and Radhakrihnan (2005) model the fixed hare rate contract for allocating the cot of internal failure, wherea we conider a more general contracting arrangement for external failure. In a recent paper, Zhu et al. (2007) look at a buyer who deign a product and own the brand, yet outource the production to a upplier. Both the buyer and the upplier incur quality related cot that are hared by a fixed hare rate contract. he Zhu et al. (2007) model capture the effect of the buyer involvement in enuring product quality. hey alo endogenouly model the effect of operational deciion uch a the buyer ordering quantity and the upplier production lot ize. Unlike Zhu et al. (2007), we look at a etting where the manufacturer i involved in the production proce and hi effort affect the final quality of the product and dicu two new contract format to hare external quality cot. A related upply chain management paper by Corbett and DeCroix (2001) dicue the ue of a hared aving contract (auming a fixed hare rate between a upplier and a buyer) to induce upplier and buyer effort that reduce indirect material conumption. Although the modeling of effort in our paper ha ome imilarity to their modeling contruct, we invetigate the ue of contractual format to hare external quality cot reulting from a recall rather than the cot of indirect material. Baed on data from the automotive and the pharmaceutical indutrie, a number of political economy reearch paper invetigate the real total cot of a recall for a manufacturer. For intance, Jarrell and Peltzman (1985), Barber and Darrough (1996), and Rupp (2004) tudy the cot attribute of recall in the U.S. automotive indutry and find that the indirect cot uch a brand equity lo, conumer goodwill lo, and lo in firm value are in fact much larger than the direct cot of a recall uch a product collection and repair cot. he finding of thi tream of empirical reearch erve a a bai for ome of our aumption regarding the manufacturer unit recall cot. In ummary, thi paper introduce two new contractual format for haring the external quality cot of product recall; in particular, we focu on the bet ue of root caue analyi information and it impact on the quality of the final product under both complete and aymmetric information aumption. In thi repect, our finding enrich the growing literature in thi area and help manager to better undertand the cot efficiency of thee contractual agreement and their impact on product quality.

4 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract Management Science 55(7), pp , 2009 INFORMS Modeling Framework We invetigate the implication of contract choice to hare product recall cot on manufacturer and upplier quality improvement effort and on upply chain profit. o thi end, we conider a manufacturer who produce a product that conit of two component, one of which he procure from a ingle upplier at a unit price p. he manufacturer procure a total of M component from the upplier and ue them to manufacture M unit of the good to be old in the market. he product generate a unit revenue of r for the manufacturer. We denote the manufacturer unit production cot by u m, and the upplier by u.at the time of contracting, both the manufacturer and the upplier know M. For example, when an auto manufacturer dicue a contract for, ay, the oil pump ued in the engine of a particular 2008 car model, he ha a good etimate of how many of thoe model car are planned for production in the year If M repreent thi number, then the quality cot haring contract cover M component. After the product i old to the cutomer, during it ueful life, it can fail to perform it function if any one of it component fail to do o. Baiman et al. (2003) define thi product failure behavior a the weaket link property. Component failure can reult from either a deign or a manufacturing related problem in the upplier or the manufacturer proce. In 2002, a tudy by A.. Kearney Inc., a global management conulting firm, of a large North American auto manufacturer external quality failure howed that although 25% of the external quality problem were deign iue, only 15% and 21% of the problem were related to the manufacturer aembly and the upplier manufacturing procee, repectively. he manufacturing and/or aembly related external quality problem are often eaily fixed by reintalling a nondefective component in the product. On the other hand, deign related external failure are more cotly to manufacturer becaue deign flaw often affect multiple product generation (model) and reveal themelve only after the cutomer ha ued the product for a period of time (Automotive New 2005). Relative to manufacturing related quality problem, deign flaw are more cotly to reolve a they may require redeigning multiple component and their interface in a product. Root caue analyi bear particular importance for deign related recall becaue the flawed deign deciion may not be readily obviou to upply chain partner. Given the high cot and challenge in reolving deign related quality problem, thi paper dicue contract to hare 4 Manufacturer uually have an etimate for the production quantity when they contact their upplier to order material; ee Production Planning and Control Hierarchy, Hopp and Spearman (2001), Nahmia (2000). external quality cot, more pecifically recall cot, reulting from product deign flaw. Current literature define product quality a the likelihood of producing a nondefective unit from either the manufacturer or the upplier proce (e.g., Reynier and apiero 1995a, b). hi way of modeling product quality i more relevant to manufacturing related defect that exit at the time of product purchae. We conider quality problem that unveil themelve during product uage and are due to unanticipated and undetected mode of deign failure. herefore, we model quality a the urvival likelihood of the product deign ubject to varying mode of uage during the ueful life of the product. Ex ante to procurement and production, the manufacturer and the upplier can chooe to exert cotly effort to reduce their component deign failure rate by carefully teting the deign characteritic. In practice, the proce of eliminating the many way in which a deign failure can occur i called failure mode and effect analyi (FMEA) and i performed by the manufacturer and/or the upplier during the product development tage prior to manufacturing (Stamati 2004). Below we preent the aumption underlying our model. he detailed decription and jutification of our aumption can be found in Online Appendix A (provided in the e-companion). 5 Aumption A1. We aume that M product are manufactured and old, and are either with the cutomer or in the ditribution channel when a recall i iued. Once a particular problem ha revealed itelf, and the recall i iued, the manufacturer fixe the particular quality problem in all M product. Aumption A2. We denote the recall cot per unit a and aume it to be independent of the root caue of the quality problem. At the time of contracting, both the manufacturer and the upplier agree on an etimate of. Aumption A3. We denote the unit cot of root caue analyi by c r = C r /M, where C r i the relevant fixed cot of the root caue analyi and M i the total number of product ubject to a recall. We aume that the root caue analyi perfectly identifie the component that caued the quality problem. Aumption A4. he manufacturer and the upplier have inherent proce capabilitie modeled by the initial failure rate of their component due to a deign related quality problem. he initial failure rate are common knowledge to both partie and are denoted by m 0 0 and for the manufacturer and the upplier, repectively. he failure rate are aumed to be time homogeneou (i.e., contant failure rate). 5 An electronic companion to thi paper (which contain all online appendice) i available a part of the online verion that can be found at

5 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract 1126 Management Science 55(7), pp , 2009 INFORMS In our model, we aume that the upplier and the manufacturer will exert cotly effort to improve the quality of their component and reduce the deign failure rate. We ue m and to denote the amount of quality improvement effort exerted by the manufacturer and the upplier, repectively, where 0 m 1 and 0 1. Under the effort level m (effort level ), the manufacturer (the upplier) can reduce the failure rate of hi (her) component from the initial value 0 m (value 0 )to 0 m 1 m (to 0 1 ). Given the contant failure rate aumption, the failure time ditribution of the product (after quality improvement) that reult in a recall will follow an exponential ditribution with a failure rate of 0 m 1 m he effort choice of the manufacturer and the upplier are not obervable. Conequently, neither the manufacturer nor the upplier can enforce a level of effort in the cot haring contract. We will refer to the game played between the manufacturer and the upplier a the quality improvement effort game. Aumption A5. In our model, the contract negotiated between the manufacturer and the upplier cover external quality cot (recall cot) for a duration of period, which will denote the duration that the product i in ue by the conumer. Without lo of generality, we normalize to 1. Given M and the exponential lifetime ditribution aumption, the probability of oberving the firt product failure that reult in a recall i given by 1 e M 0 m 1 m o further implify the expoition, we define the aggregate failure rate a 0 m = M 0 m and 0 = M 0. hen, the failure time probability ditribution implifie to 1 e 0 m 1 m Note that, if a product fail, the failure i going to be due to the upplier component with probability 0 1 / 0 m 1 m hi probability would be 0 m 1 m / 0 m 1 m for the manufacturer. We conider the manufacturer and the upplier procee to be tochatically independent, i.e., joint failure do not occur. Above, we aume that the manufacturer initiate a recall when the firt product failure i oberved. In practice, thi would be the cae when product failure lead to evere conequence for the conumer, and therefore delaying the recall deciion impoe a high product liability rik to the manufacturer. When uch rik are low, the manufacturer may wait to initiate a recall until N (where N>1) product failure are oberved (Rupp and aylor 2002). In 8, we elaborate on thi point and dicu the implication of thi aumption for our model inight. Aumption A6. Improvement in the product deign failure rate are cotly to both partie. More pecifically, the effort of and m reult in an effort cot of C able 1 Baic Notation r = elling price of the manufacturer product u m = unit production cot of the manufacturer u = unit production cot of the upplier M = total number of product ubject to recall = unit recall cot c r = unit root caue analyi cot p 0 = unit part procurement price under no cot haring p = unit part procurement price k = upplier effort level under contract k k m = manufacturer effort level under contract k = ueful life of the product in the market C = cot of effort for the upplier C m m = cot of effort for the manufacturer 0 0 m 0 0 m = upplier initial per unit failure rate = manufacturer initial per unit failure rate = upplier initial aggregate failure rate = manufacturer initial aggregate failure rate and C m m (per unit component) for the upplier and the manufacturer, repectively. We conider C and C m m to be twice continuouly differentiable on 0 1, and convex increaing in effort o that C >0 and C m m >0 for m and C >0, and C m m >0 for m , for the manufacturer and the upplier, repectively. In what follow, we would like to avoid boundary olution to the manufacturer and the upplier effort deciion. o thi end, we aume that there are low hanging quality improvement opportunitie for both partie. hi requirement i formalized by C 0 = 0 and C m 0 = 0. Alo, given the limited phyical, financial, and intellectual reource of firm, we will aume that it i prohibitively more expenive to how incremental effort at high effort level. More pecifically, we will aume lim 1 C = and lim m 1 C m m =. We ummarize our notation in able Complete Information About Component Quality In thi ection, we focu on the quality improvement effort game under complete information. Specifically, we aume that the manufacturer and upplier are fully informed about each other initial component failure rate (i.e., the initial tate of the component quality prior to quality improvement effort). hi uually occur when the upplier and the manufacturer have been working together for everal year, and therefore both partie have a good idea of each other proce capabilitie. A a benchmark, we tart by characterizing the total upply chain profit and the final product quality in a centrally coordinated ytem, in which both quality improvement effort deciion of the manufacturer and

6 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract Management Science 55(7), pp , 2009 INFORMS 1127 the upplier are made by a central planner. We refer to the total upply chain profit and the final quality of the product under thi cae a the firt bet. hen, in a decentralized upply chain, we dicu how the quality improvement effort game would be played out under Contract S and Contract P. he main focu of our paper i on deigning cot haring contract that maximize the manufacturer expected profit in a decentralized upply chain. he framework of our model i baed on a Stackelberg game in which the manufacturer i the leader. he manufacturer move firt and offer the cot haring contract to the upplier. hi i generally the cae in the automotive and electronic indutrie, where the manufacturer ha the power in the upply chain and often dictate the term of trade with hi upplier (Barkholz and Sherefkin 2007, Sherefkin and Armtrong 2003, BearingPoint Conulting 2008, Nytedt 2007). hat i the main reaon why many paper on quality management and upply chain contracting have alo made imilar aumption with regard to the manufacturer being the Stackelberg leader in the upply chain (ee Baiman et al. 2000, 2001, 2003; Lim 2001; Cachon 2003). In our decentralized model, the manufacturer take the initial tep and deign the contract. When deigning the contract, the manufacturer provide ufficient incentive for the upplier o that the upplier accept the cot haring contract. A we will explain later, the incentive provided to the upplier to have her accept a cot haring contract i an increae in the part price paid by the manufacturer for each component procured from the upplier. From the manufacturer profit maximization perpective, we invetigate whether it i ever optimal to induce firt bet effort level in a decentralized upply chain. Interetingly, we how in Lemma 1 that when cot of root caue analyi i negligible, i.e., c r = 0, the optimal contract for the manufacturer i alo the contract that coordinate the effort deciion of the manufacturer and the upplier and attain the firt bet quality and profit. We alo how that, for the decentralized etting, an effort coordinating Contract S or Contract P, or both, doe exit. Subequently, we preent an extenive numerical tudy that examine the manufacturer optimal contract when c r > 0. In the ret of the analyi, k i and k i will denote the profit function and the effort level of upply chain member i under model (contract) k. he ubcript i will take value of and m, denoting the upplier and the manufacturer, repectively. he upercript k will take the value of C, N, S, P, and F, denoting the centrally coordinated (firt bet), no haring, elective root caue analyi, partial cot allocation, and fixed hare rate model, repectively. For proof of all analytical reult ee the online appendix Centralized Supply Chain In thi ection, we conider the cae where a central planner maximize total upply chain profit by jointly electing the quality improvement effort and m. he central planner optimization problem i given by Max C = r u m u [ 1 e ] 0 m 1 m m C m m C (1) where r i the product elling price, and u m and u are the unit production cot of the manufacturer and the upplier, repectively. o enure concavity of the central planner maximization problem, we require 2 C / m 2 < 0 and 2 C / 2 < 0, which can be tranlated into lower bound on C m m and C. Specifically, let C m m > 0 m 2 and C > 0 2 for m , then: Propoition 1. If C m m > 0 m 2 and C > 0 2 for m hold, then there exit unique firt bet upplier and manufacturer effort ( C m C ) that atify the following firt order optimality condition: C m = 0 m e 0 m 1 C C = 0 e 0 m 1 C m C m C C m C m = 0 C C = 0 Note that 2 C / m = 0 m 0 e 0 m 1 C m C > 0, which implie complementarity between effort choice; i.e., m C ( C ) i increaing with C ( m C ), for all C, C m 0 1, and vice vera. he complementarity between effort choice increae with the unit recall cot and the initial failure rate of the upplier and the manufacturer. hi implie that the interaction effect between effort choice i more ignificant for newly deigned product, which are more likely to have a higher number of deign flaw (higher 0 m or 0 ) than product that have been on the market for a period of time and have already undergone quality improvement. Becaue a tronger interaction between effort deciion demand more coordinated deciion making, the effort coordinating contract dicued in thi paper may be of greater importance for product that have higher initial failure rate. In the above formulation, we aume that, upon a recall, all old and/or ditributed item (i.e., all M unit) are returned to the manufacturer for repair. hi type of conumer recall repone behavior i oberved for high value product (e.g., automotive and electronic) when product failure ha evere afety conequence for the cutomer. However, for a number of product categorie, it i not unuual to have le than

7 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract 1128 Management Science 55(7), pp , 2009 INFORMS a 100% repone rate. For example, low value product (low price product), old product, or product that conumer perceive to have low rik are aociated with low recall repone rate from conumer (Government Conumer Safety Reearch 2000). When only a fraction of old item i returned upon recall, the centrally coordinated model a well a the decentralized model under Contract S and Contract P can eaily be revied by multiplying by the anticipated repone rate. In the concluion ection, we further dicu how modeling le than a 100% repone rate from conumer would impact our inight. In the ret of the paper, we refer to the optimal effort level C m C and the final product quality in the centralized ytem, repectively, a the firt bet effort and firt bet quality. We alo call the optimal total expected upply chain profit a the firt bet profit. Note that, the firt bet effort (firt bet quality) repreent the optimal reolution of the trade-off between the quality improvement effort cot and the recall cot in a centralized ytem. We ue the firt bet quality a a benchmark to evaluate the quality of the final product in the decentralized upply chain ytem. Next, we examine decentralized upply chain model under different cot haring agreement (i.e., no cot haring, elective root caue analyi, and partial cot allocation), where the upplier and the manufacturer maximize their own profit. he fundamental quetion we addre are a follow: Under which contractual agreement() can a decentralized upply chain achieve the firt bet profit (or the firt bet quality)? If the contract cannot achieve the firt bet profit (or the firt bet quality), how do they perform relative to the firt bet benchmark? We tart firt with the no cot haring cae No Cot Sharing N In thi ection, we conider a decentralized etting in which the manufacturer internalize total recall cot, even though in ome cae the upplier may be at fault. In thi etting, although the manufacturer exert ome effort to improve hi component failure rate, the upplier ha no incentive to improve the quality of her component. he manufacturer and the upplier optimization problem are given repectively by Max N m = r u m p 0 [ 1 e ] 0 m 1 m m C m m (2) Max N = p 0 u C (3) where p 0 i the manufacturer part procurement cot. We conider p 0 u to enure that the upplier attain nonnegative profit under the no cot haring cenario. Note that p 0 doe not affect the effort choice of the manufacturer or the upplier but allocate the upply chain profit between the two partie. he next propoition ummarize optimal upplier and manufacturer effort level under no cot haring. Propoition 2. Under no cot haring, the upplier exert zero effort. he manufacturer underinvet in effort relative to the firt bet (i.e., m N < C m ) even though he fully internalize all cot aociated with hi effort choice. Under no cot haring, the optimal manufacturer effort increae with the effort exerted by the upplier. Specifically, the manufacturer bet repone function m N N i increaing in N. hi obervation, which follow from the poitive cro partial derivative of the manufacturer profit function in and m, how that there i complementarity between the effort choice of the manufacturer and the upplier. Under no cot haring, the upplier exert minimum effort becaue he doe not internalize any cot. herefore, N < C. From the complementarity of effort deciion, it follow that m N N = 0 < m N N = C = m C. Hence, the analyi of thi cae demontrate that even though the manufacturer internalize all cot aociated with hi effort choice, due to complementarity between effort choice, he underinvet in effort in equilibrium. Conequently, the no haring cheme not only directly affect the effort exerted by the upplier, but alo indirectly lead to le than the firt bet level of effort from the manufacturer Selective Root Caue Analyi Contract (Contract S) In thi ection, we tudy the optimal Contract S that maximize the manufacturer profit. Before we how thi, we would like to emphaize that a contract that lead to higher total upply chain profit alo lead to higher profit for the manufacturer, who i modeled a the Stackelberg leader in the upply chain. he conventional approach i that the manufacturer, being the Stackelberg leader, determine the allocation of the total upply chain profit to hi upply chain partner. Suppoe fraction of the total upply profit i allocated to the manufacturer and 1 to the upplier. Regardle of the exact value of, it i clear that for any given value of, a contract that reult in higher total expected upply chain profit alo reult in higher expected profit for the manufacturer and the upplier. herefore, the ranking of the contract with repect to the total upply chain profit alo mimic the ranking of the contract with repect to the manufacturer and the upplier profit. Interetingly, we find that when the root caue analyi cot i negligible, the optimal contract for the manufacturer (and the upply chain) i an effort coordinating contract, which we define a follow: Definition 1. An effort coordinating contract i a contract that, when implemented in a decentralized

8 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract Management Science 55(7), pp , 2009 INFORMS 1129 ytem, reult in the firt bet effort level from the manufacturer and the upplier and therefore attain the firt bet quality. Under an effort coordinating contract, when the manufacturer and the upplier maximize their own profit, the optimal effort level are found to be the firt bet effort. Note that, although in a centralized upply chain the firt bet effort C m C reult in the maximum total expected profit (firt bet profit): (i) there i no guarantee that thee effort level alo maximize total expected profit in a decentralized the upply chain, and (ii) even if in ome cae thee effort level maximize the total expected upply chain profit in a decentralized ytem, there i no guarantee that a contract exit that can induce thee effort level (i.e., there i no guarantee that an effort coordinating contract alway exit). In Lemma 1, we preent a condition under which the firt bet effort level alo maximize the total expected upply chain profit (and hence the manufacturer profit) in a decentralized ytem. In Propoition 3, we how that Contract S i flexible enough to be deigned a an effort coordinating contract. Lemma 1. In a decentralized upply chain, if the root caue analyi cot i negligible (i.e., c r = 0), then an effort coordinating contract maximize the manufacturer a well a the total upply chain profit and attain firt bet quality and profit. In practice, if the product architecture i eparable (a a reult of decoupling, no function haring, or modular deign, ee Baiman et al. 2001), then a product failure can eaily be traced to a particular component. In contrat, with noneparable product architecture, it i more difficult (if not impoible) to trace a product failure to a ingle component. herefore, for eparable product architecture, it i much cheaper to perform the root caue analyi when implementing a cot haring contract. Lemma 1 how, when thi i the cae, in a decentralized upply chain, an effort coordinating contract alo maximize the manufacturer and the total upply chain profit and attain the firt bet quality and profit. Next, we how that there exit an effort coordinating Contract S. Under Contract S, the cot allocation rule i defined a a function of the product time to failure that reult in product recall. If product failure occur before the root caue analyi threhold time, the party reponible for the quality problem i identified through root caue analyi and incur total recall cot. Otherwie (i.e., if product failure occur after ), the upplier only hare a percentage R of the total recall cot. In practice, we oberve a trend toward differentiating quality problem baed on their time of occurrence. For example, by centralizing part failure data collected from dealerhip, General Motor wa one of the firt to develop a monitoring ytem to differentiate early failure from late failure (White 1999). Early failure are claified a pecial caue quality problem. For thee type of product failure, the company purue a detailed root caue analyi. hi information i intantly fed into the deign proce to eliminate deign fault (White 1999). In thi paper, we identify way in which thi information can be ued for cot haring purpoe. Contract S i a more general and flexible contract format than the fixed hare rate contract, previouly modeled in the upply chain management literature. In fact, the fixed hare rate contract i a pecial cae of Contract S when = 0. A will be dicued below, unlike the fixed hare rate contract, the flexibility in the tructure of Contract S i critical to obtaining the firt bet level of effort from the upply chain member. Under Contract S, the manufacturer and the upplier olve the following optimization problem, repectively: Max S m = r u m p + c r G 1 e m 1 R e e C m m (4) Max S = p u + c r 1 G 1 e R e e C (5) where = 0 m 1 m and G = o m 1 m / o m 1 m + o 1. Note that p i the price under Contract S, where p p 0 and p p 0 i the incentive given to the upplier to accept the cot haring contract. Our interet i in undertanding whether one can deign a Contract S that achieve the firt bet effort level from upply chain partner. he next propoition decribe the equilibrium outcome of the effort game under Contract S, when the manufacturer and the upplier initial failure rate are not dratically different (i.e., when 0 = l 0 m, where l ). See Propoition 3(a) in the online appendix for detail. 6 Propoition 3. Poitive cro partial derivative of the manufacturer and the upplier objective function enure that the effort game i upermodular under the elective root caue analyi contract. Furthermore, (i) Supermodularity enure the exitence of at leat one Nah equilibrium; (ii) he bet repone function m S and S m are both increaing in their argument; 6 Note that l i a ufficient condition that guarantee the upermodularity of the game. In our numerical tudy we oberved cae that violated thi condition and the game wa till upermodular.

9 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract 1130 Management Science 55(7), pp , 2009 INFORMS (iii) he et of equilibria i a chain; i.e., if there are multiple equilibria, they can be ordered a follow: for any pair of equilibria m S, S and m S, S either m S S m and S S or m S S m and S S ; (iv) If there are multiple equilibria, then for any pair of equilibria m S S and m S, S, where m S S m and S S S S m, S S then S m S m S m, S S. S m S m S and Although upermodularity enure the exitence of at leat one Nah equilibrium, it doe not rule out multiple equilibria. However, the equilibria are Pareto rankable and there i a mot preferred and a leat preferred equilibrium by both partie. From part (iv) of Propoition 3, it follow that both partie prefer the equilibrium where they both how higher effort. herefore, in what follow, we will be focuing on the mot preferred equilibrium (Pareto optimal equilibrium) and avoid the iue aociated with multiple equilibria when analyzing the effort coordinating contract (Cachon and Neteine 2004). he next propoition preent cloed-form olution to the effort coordinating contract parameter that achieve the firt bet effort level in the aymmetric effort game with c r = 0 and the ymmetric effort game 7 with c r 0. he optimal contract parameter for the aymmetric effort game with c r 0 cannot be characterized in cloed-form olution; therefore, for clarity of expoition, we preent a detailed analyi of thi cae in the online appendix. Propoition 4. (i) In the aymmetric effort game with c r 0, there exit a unique effort coordinating Contract S defined by R, that achieve the firt bet effort level from the manufacturer and the upplier. (ii) In the aymmetric effort game with c r = 0, the effort coordinating Contract S i given by R = 0 1 C C and = Ln( 1 C e C C where C = 0 m 1 C m C and < 1. (iii) In the ymmetric effort game with c r 0, the effort coordinating Contract S i given by = Ln 1 C C e C where < 1 and R = 0 5. We gain the following inight from Propoition 4. Firt, notice that when both agent exert their firt bet effort level, the hare rate R i proportional to each party product failure rate at the firt bet effort level. Second, one can eaily how that the fixed hare rate 7 In the ymmetric game, the manufacturer and the upplier have identical effort cot function and initial product failure rate. ) contract ( = 0) and the cot haring contract that alway ue root caue analyi information ( = 1) cannot attain firt bet effort level. We ummarize thee inight in the following corollarie. Corollary 1. In an effort coordinating Contract S, optimal > 0; therefore, the fixed hare rate contract, which i a pecial cae of the elective root caue analyi contract when = 0, cannot achieve the firt bet effort level and coordinate the upply chain. Furthermore, etting = 0 (i.e., the fixed hare rate contract) lead to underinvetment in effort by the manufacturer and the upplier. Corollary 1 how that the fixed hare rate contract cannot coordinate the quality improvement effort level. More interetingly, in the next corollary, we point out that to achieve the firt bet quality level, one doe not need to alway ue the root caue analyi information, even if it were cotle and could perfectly identify the party at fault. Corollary 2. In an effort coordinating Contract S, optimal < 1; therefore, alway performing root caue analyi and allocating the recall cot to the party reponible for the quality problem, even if it were cotle to do o, would not attain the firt bet effort level. Furthermore, etting = 1 would lead to an overinvetment in effort by the manufacturer and the upplier. Latly, we oberve that d /d C < 0. hi implie that when the coordinated total aggregate failure rate at the firt bet i maller, (i.e., high effort i exerted at the firt bet), then the root caue analyi threhold,, i larger, reulting in a higher likelihood of haring recall cot baed on root caue analyi information. In other word, the more both partie improve the quality of their component, the more both ide are likely to determine the party at fault in cae of a failure that reult in a recall Partial Cot Allocation Contract (Contract P) In thi ection, we introduce an alternative cot allocation rule that can alo achieve the firt bet effort level from the manufacturer and the upplier. Under thi cot allocation cheme, the cot i alway hared between the manufacturer and the upplier. However, the haring rate are adjuted according to the root caue analyi information. More pecifically, we denote R m and 1 R m a the upplier and the manufacturer hare of the recall cot, repectively, when the manufacturer i at fault. Similarly R and 1 R are the upplier and the manufacturer hare of the recall cot, repectively, when the upplier i at fault. Under the partial cot allocation cheme, we have 0 R m 1 and 0 R 1 and R R m. he lat inequality enure that the upplier aume a larger fraction of the recall cot when he i at fault compared to the cae when the manufacturer i at fault.

10 Chao, Iravani, and Savakan: Quality Improvement Incentive and Product Recall Cot Sharing Contract Management Science 55(7), pp , 2009 INFORMS 1131 Analogou to the elective root caue analyi contract, we will conider an effort game under Contract P, in which the manufacturer and the upplier trategie are complement. Furthermore, we will focu on the Pareto optimal equilibrium outcome of thi game, where both partie how high effort. (See Online Appendix B for a dicuion of the upermodularity condition and elimination of multiple equilibria.) Propoition 5. (i) In the aymmetric effort game with c r = 0, if there exit R and R m, uch that R m = e 2 e 0 + e C {e 0 C m + e C 2 C m 1 C + e 0 C 0 C m 0 m } e Z R = R m + C 1 0 e C and 0 R, R m 1, then there exit an effort coordinating partial cot allocation contract that achieve the firt bet level of effort where 0 = 0 m + 0, C i = 0 i 1 C i for i = m, C = C + C m, = 0 C, and Z = C 2 m C C 0 m 0 + C 1 + C 0 C m 0 m 1 C. (ii) In the ymmetric effort game with c r 0, if there exit R and R m, uch that R m = A D B 0 m e C R = R m B C + A D A + A 0 e C A C 0 m e C A C and 0 R, R m 1, then there exit a coordinating partial cot allocation contract that achieve the firt bet level of effort. he expreion A = 1 + c r / G m H GH m, B = 1 + c r / G H GH, C = 1 + c r / H m, and D = 1 + c r / H m are evaluated at. Furthermore, m C C G = H = 1 e o m 1 m + o 1 H i = 0 i e 0 m 1 m o m 1 m o m 1 m + o 1 G m = K 1 G G = GK o o m and K = o m o m 1 C m + o 1 C In practice, it i not unuual to encounter ituation in which a upplier, particularly if it i a mall-ized company, face budget contraint that limit the maximum cot allocated to the firm. In uch ituation, even if the upplier i at fault, the manufacturer may chooe to refrain from allocating total cot to the upplier, a it may lead to her bankruptcy (Sherefkin and Armtrong 2003). One example i Bremi Auto- Elektrik, a upplier of ignition coil for Volkwagen AG. In February 2003, VW recalled about 500,000 VW and Audi vehicle in the United State, from the 2002 and 2004 model year, to replace their faulty ignition coil. he recall cot for VW wa $83 million, wherea Bremi annual revenue wa etimated at approximately $40 million (Sherefkin and Armtrong 2003). Under uch circumtance, our finding are particularly intereting in the ene that even if the upplier doe not internalize the full liability and cot aociated with her part failure, we how that the firt bet effort level and quality can till be attained if the hare rate are et optimally a demontrated in the above propoition. Although both effort coordinating Contract S and P reult in the firt bet profit for the total upply chain and maximize manufacturer profit when the root caue analyi i zero, thee effort coordinating contract lead to different total expected upply chain profit when the root caue analyi i greater than zero. Corollary 3. When the root caue analyi cot i ignificant (i.e., c r > 0), at the coordinated firt bet effort level, Contract S reult in higher total upply chain profit than Contract P. Notice that < 1, which implie that at the firt bet level of effort there i a maller likelihood that a root caue analyi will be performed under Contract S than under Contract P. hi reult in maller expected root caue analyi cot for Contract S than for Contract P. Becaue under the firt bet effort level, the effort and the expected recall cot are the ame for both contract, a maller expected root caue analyi cot lead to higher total upply chain profit for Contract S. A the root caue analyi threhold get larger, which occur when it i critical to induce higher effort and achieve lower failure rate at the firt bet effort, the cot difference between the two contract diminihe. 5. Efficiency of Contract In the previou ection, we howed that both Contract S and Contract P can be deigned to coordinate the manufacturer and the upplier quality improvement effort (attain the firt bet effort level) in the decentralized etting. Alo, we howed that when the root caue analyi cot i zero, the effort coordinating contract P and S reult in the maximum total upply chain profit (firt bet profit) a well a the maximum profit for the manufacturer. When the root caue analyi cot i not zero, Corollary 3 how that, at the firt bet effort level, Contract S reult in higher expected profit for the upply chain than Contract P. In thi ection, we examine whether,