Can Contracts Replace Qualification in a Sourcing Process With Competitive Suppliers and Imperfect Information?

Transcription

1 Unversty of Nebraska - Lncoln DgtalCommons@Unversty of Nebraska - Lncoln Supply Chan Management and Analytcs Publcatons Busness, College of 2016 Can Contracts Replace Qualfcaton n a Sourcng Process Wth Compettve Supplers and Imperfect Informaton? Yue Jn Alcatel-Lucent, yue.1.jn@noka.com Jennfer K. Ryan Unversty of Nebraska - Lncoln, jennfer.ryan@unl.edu Follow ths and addtonal works at: Part of the Busness Admnstraton, Management, and Operatons Commons, Management Informaton Systems Commons, Management Scences and Quanttatve Methods Commons, Operatons and Supply Chan Management Commons, and the Technology and Innovaton Commons Jn, Yue and Ryan, Jennfer K., "Can Contracts Replace Qualfcaton n a Sourcng Process Wth Compettve Supplers and Imperfect Informaton?" (2016). Supply Chan Management and Analytcs Publcatons Ths Artcle s brought to you for free and open access by the Busness, College of at DgtalCommons@Unversty of Nebraska - Lncoln. It has been accepted for ncluson n Supply Chan Management and Analytcs Publcatons by an authorzed admnstrator of DgtalCommons@Unversty of Nebraska - Lncoln.

2 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST Can Contracts Replace Qualfcaton n a Sourcng Process Wth Compettve Supplers and Imperfect Informaton? Yue Jn and Jennfer K. Ryan Abstract Ths paper consders a manufacturer who outsources the producton of a product to multple competng supplers, who dffer n ther cost structures and n ther capabltes for producng hgh-qualty products. The manufacturer must desgn the sourcng process to ensure that the selected suppler has suffcent qualty capablty, whle encouragng competton among the supplers. We develop and analyze a mathematcal model of performancebased contractng, a sourcng method that s approprate when the manufacturer has mperfect nformaton regardng the supplers costs and capabltes. We compare the performance of performance-based contractng wth that of a two-stage sourcng process, an alternatve sourcng method that s more commonly used n practce. The theoretcal results and manageral nsghts derved from ths research can enable manufacturng frms to mprove the management of ther sourcng processes. In partcular, we demonstrate that performance-based contractng wth a symmetrc lnear penalty/reward functon wll always outperform the two-stage sourcng process from the perspectve of the buyer and that the optmal penalty/reward rate s less than or equal to the unt warranty cost. In addton, performance-based contractng generally leads to a hgher qualty level provded by the wnnng suppler. However, the wnnng suppler s generally better off under the two-stage sourcng process. Index Terms Aucton mechansms, procurement, sourcng. I. INTRODUCTION AND MOTIVATION RECENT years have seen an ncrease n the use of outsourcng n a varety of ndustres, wth the goal of reducng costs and obtanng operatonal effcences [1]. As a result, manufacturers must make strategc decsons regardng the desgn of ther sourcng processes, ncludng how to qualfy potental supplers and how to select among the set of potental supplers. Recent years have also seen an ncrease n the use of procurement auctons as part of the sourcng process [2]. Auctons can nduce compettve bddng, resultng n ncreased competton between supplers and reduced procurement costs [3]. Sourcng process desgn, ncludng the desgn of auctonbased mechansms, can be complex due to a number of Manuscrpt receved September 25, 2015; revsed March 14, 2016; accepted Aprl 20, Date of publcaton June 08, 2016; date of current verson July 15, Y. Jn s wth Bell Labs Ireland, Alcatel-Lucent, Dubln 15, Ireland (e-mal: yue.1.jn@noka.com). J. K. Ryan s wth the College of Busness Admnstraton, Unversty of Nebraska, Lncoln, NE USA (e-mal: jennfer.ryan@unl.edu). Ths paper has supplementary downloadable materal avalable at eeexplore.eee.org. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TEM factors. Frst, manufacturers often care about attrbutes besdes just prce, ncludng qualty, relablty, and payment terms. A second source of complexty arses when the manufacturer has mperfect nformaton regardng the supplers characterstcs. Fnally, sourcng process desgn s complex due to the fact that the nature of that process can nfluence the product characterstcs or attrbutes offered by the supplers. Supplers choose ther bddng strateges, ncludng how they set the nonprce attrbutes, such as product qualty, n order to mze ther chance of wnnng the buyer s busness, whle smultaneously mnmzng ther costs. Thus, how the buyer evaluates bds wll nfluence the supplers bddng strateges and product offerngs. A. Multattrbute Procurement Mechansms There are a number of alternatves for selectng among supplers when the buyer cares about multple attrbutes. Engelbrecht- Wggans et al. [4] provde the followng categorzaton. Under request for quotaton (RFQ) and request for proposal (RFP), the buyer provdes detaled specfcatons to the supplers. Under RFQ, the supplers are requred to meet the specfcatons and the buyer selects among those supplers based on cost. Under RFP, the supplers submt proposals whch are evaluated by the buyer, wth the contract awarded to the best overall suppler, as determned by the buyer, perhaps through the use of a score functon. Reverse auctons are structured versons of the RFQ and RFP mechansms. Prce-based (PB) reverse auctons are analogous to an RFQ. They are bndng,.e., the buyer commts n selectng the suppler wth the lowest bd prce. Buyer-determned (BD) reverse auctons are analogous to an RFP. They are not bndng,.e., the buyer can select the wnnng suppler as she sees ft. Multattrbute auctons allow supplers to bd on multple dmensons, rather than just prce. The buyer evaluates bds usng a score functon whch converts the bd nto a sngle number. These auctons are not wdely used n practce [5], [6]. Addtonal mechansms for selectng among supplers wth multple attrbutes nclude a two-stage sourcng process (TSP) and performance-based contractng (PBC). In a TSP, the frst stage s the qualfcaton stage, n whch the potental supplers are screened for varous nonprce capabltes. The second stage s the suppler selecton stage, n whch the qualfed supplers are nvted to compete n a prce-only procurement aucton [7], [8]. Under PBC, the supplers partcpate n a prceonly reverse aucton. After the contract s awarded, the wnnng suppler s assessed a penalty (or reward) based on hs IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See standards/publcatons/rghts/ndex.html for more nformaton.

3 284 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST 2016 realzed performance. PBC s useful when the supplers nonprce attrbutes are unknown to the buyer at the tme of bddng [9]. PBC s also referred to as a fxed prce aucton wth ncentves [10]. Gven the growth n the use of procurement auctons, there has been sgnfcant nterest n comparng the performance of these alternatve approaches, both theoretcally and emprcally. Engelbrecht-Wggans et al. [4] compare RFQ (or PB auctons) wth RFP (or BD auctons) and fnd that the buyer s choce wll depend on the number of potental supplers, as well as the correlaton between the cost and the nonprce attrbute. Chang [9] uses behavoral experments to compare PBC, PB auctons, and TSP and fnds that PBC provdes a hgher surplus to the buyer. B. Problem Statement and Related Lterature We study a manufacturer who outsources the producton of a product to competng supplers, who dffer n ther cost structures, as well as n ther capabltes for producng hgh-qualty products. The manufacturer has mperfect nformaton regardng the supplers costs and qualty capabltes. In contrast to much of the lterature, we model the supplers qualty endogenously,.e., the supplers select ther qualty based on the ncentves nduced by the sourcng process, as desgned by the buyer, subject to ther exogenously specfed qualty capabltes. We study how the manufacturer can desgn the sourcng process to 1) ensure that the selected suppler has suffcent qualty capablty to produce a hgh-qualty product, and 2) encourage competton among the potental supplers n order to reduce procurement costs. In addton, we consder how mperfect nformaton regardng the supplers qualty capabltes can be ncorporated nto procurement process desgn n practce. We do not focus on the desgn of a new procurement mechansm. Instead, we compare the performance of two mechansms that are commonly used n practce: 1) TSP; and 2) PBC. Both mechansms consder the supplers qualty capabltes, but stll allow suppler selecton to be conducted usng a prce-only procurement aucton. In addton, both TSP and PBC are approprate when, ntally, there s uncertanty regardng the supplers qualty capabltes. PBC assesses the penalty/reward after the bddng process, when the wnnng suppler s qualty level s realzed, whle TSP nvolves a prebddng qualfcaton stage n whch the buyer learns about the supplers qualty capabltes. Despte the smlartes between TSP and PBC, there are some dfferences. 1) TSP has been more wdely used, ncludng n defense procurement [11], as well as n the telecommuncatons [7], [8], pharmaceutcal [12], and automotve [13] ndustres. On the other hand, PBC s commonly used n federal and state government procurement [9], [10]. 2) Prevous lterature [7], [8] has demonstrated that TSP has a sgnfcant lmtaton: because suppler selecton s based only on prce, TSP provdes no ncentve for the supplers to offer qualty levels beyond the manufacturer s specfcaton. In contrast, PBC offers the supplers a penalty/reward based on ther realzed performance relatve to a qualty target. Thus, f properly desgned, PBC can be used to nduce the supplers to offer qualty levels that exceed the target. 3) TSP has a qualfcaton stage n whch qualty s assessed pror to suppler selecton. Under PBC, qualty s evaluated after the contract s awarded, based on realzed performance. Ths mples that any qualty problems may be dscovered when t s too late to make adjustments. Thus, poor qualty product may reach the consumer, damagng the frm s reputaton. Although the qualfcaton process (under TSP) and the potental for damaged reputaton (under PBC) are both costly, because these costs are dffcult to quantfy, we do not nclude them n our analyss. 4) TSP corresponds to the tradtonal organzatonal structure of many manufacturng frms. The qualfcaton process (nvolvng the producton or engneerng departments) s decoupled from suppler selecton (nvolvng the procurement department). Thus, decsons regardng qualty capabltes are separated from decsons regardng prcng. In contrast, PBC requres jont decson makng,.e., the departments must collaborate n settng an approprate penalty/reward rate. Our man contrbuton relatve to the prevous work on TSP [7], [8] s the ntroducton of a new model of PBC, for whch we provde analytcal results, as well as a detaled senstvty analyss, to understand whch nput parameters have the most mpact on the buyer s performance. In addton, we perform analytcal and numercal comparsons of PBC and TSP from the perspectve of the buyer and the supplers, wth the goal of understandng why TSP s more commonly used n practce. Chang [9] also compares TSP and PBC. However, there are a number of dfferences between that work and ths paper. Frst, the analyss n [9] s largely emprcal. In contrast, our analyss apples analytcal models, coupled wth smulaton. Second, [9] takes the supplers qualty levels as exogenous. Our analyss consders endogenous qualty. Ths dstncton s crtcal gven that PBC s a mechansm used to encourage the supplers to offer hgher levels of qualty. Fnally, [9] does not lnk the supplers unt costs to ther qualty levels. In contrast, n our analyss, we assume that the supplers producton costs are ncreasng and convex n ther qualty levels. Gupta and Chen [10] consder the desgn of ncentve functons, whch are analogous to desgnng a PBC mechansm. However, most of ther analyss consders exogenously specfed suppler qualty, they do not consder lmted suppler capabltes, and they do not compare the performance of PBC and TSP. There has been prevous work comparng PB and BD mechansms [4], [14]. Our research has some smlartes to ths prevous work. The TSP s smlar to a PB mechansm, snce whle PBC s smlar to a BD mechansm. However, our work dffers from ths prevous work by assumng endogenous suppler qualty. There has been some prevous work on procurement auctons wth endogenous qualty. For example, Branco [15] and Che [16] consder multattrbute procurement auctons wth endogenous qualty and derve the optmal score functon for the buyer. However, our model dffers from ths prevous work n that the supplers choose ther qualty levels subject to

4 JIN AND RYAN: CAN CONTRACTS REPLACE QUALIFICATION IN A SOURCING PROCESS WITH COMPETITIVE SUPPLIERS 285 exogenously specfed and heterogeneous qualty capabltes, whch sgnfcantly complcates the analyss. Our work also dffers from much of the prevous lterature by consderng a more complex, but more realstc, cost structure for the supplers. Much of the lterature on auctons wth qualty consderatons assumes the supplers unt costs are ndependent of ther qualty level [2], [9], [14]. Engelbrecht-Wggans et al. [4] allow for a correlaton between the supplers costs and qualtes. However, the relatonshp between cost and qualty s assumed to be lnear. In contrast, our unt producton cost s ncreasng and convex n qualty. C. Contrbutons and Manageral Insghts Ths research extends the lterature on sourcng process desgn to compare the performance of two mechansms that are approprate when the buyer has uncertanty regardng the supplers qualty capabltes: TSP and PBC. We do so usng a model that captures several complextes not consdered n the exstng lterature. In partcular, we assume the supplers endogenously determne ther qualty levels, n response to the sourcng process desgn, subject to exogenously specfed heterogeneous qualty capabltes, whch are (ntally) unknown to the buyer. In addton, we explctly model how the supplers unt producton costs vary as a functon of ther endogenous qualty levels usng a heterogeneous producton cost functon that s ncreasng and convex n qualty. Our models allow us to compare the performance of TSP and PBC to determne condtons under whch each s preferred by the buyer. Whle, as wll be seen, PBC outperforms TSP from the perspectve of the buyer, TSP s much more wdely used n practce, for the reasons outlned n Secton I-B. Thus, our analyss focuses on understandng the magntude of the performance gap between TSP and PBC and dentfyng condtons under whch usng PBC provdes the most value to the buyer, relatve to TSP. We fnd that PBC s most benefcal for buyers who faces sgnfcant uncertanty regardng the supplers costs, and for whom mantanng a hgh level of qualty s crtcal. We also fnd that the fnal delvered qualty s generally hgher under PBC than under TSP and that the gap between the qualty levels s largest when there s more uncertanty regardng the potental supplers costs and when the number of potental supplers and the unt warranty cost are small. Gven that the buyer prefers PBC to TSP, we also provde gudance regardng how the buyer should set the penalty/reward rate under PBC. We fnd that the optmal rate s always less than the unt warranty cost and that the optmal rate s largest when the number of potental supplers and the unt warranty cost are large. Fnally, whle the buyer always prefers PBC, the wnnng suppler s generally better-off under TSP. The suppler s preference for TSP s strongest when the number of supplers s large, the uncertanty n the supplers costs s small, and the unt warranty cost s large. II. TSP AND PBC: MODELS AND COMPARISON We consder a buyer (she) who sells a commodty-lke tem to consumers at a fxed prce. The buyer can purchase the tem from any of n potental supplers (he, denoted by =1,...,n), who TABLE I LIST OF NOTATION n Number of potental supplers, ndexed by =1, 2,...,n. Q Qualty threshold under TSP and PBC, a decson varable for the buyer. m (Q) Number of qualfed supplers under TSP gven threshold Q. C w (q) Buyer s qualty cost functon (cost of a sngle unt of product wth qualty level q). w Unt warranty cost n a lnear qualty cost functon,.e., C w (q) =w (1 q). C p (Q) Buyer s unt procurement cost under TSP gven qualfcaton threshold Q. C B Buyer s total cost functon. H (q ; Q) Penalty/reward functon used by buyer under PBC, H (q ; Q) =h(q Q). h Penalty/reward rate under PBC. p Suppler s bddng prce, a decson varable for suppler. q Suppler s qualty level, a decson varable for suppler, where 0 q 1. q Maxmum achevable qualty level of suppler, q Unform(q L,q H ), q H 1. f q ( ) Probablty densty functon for q. ˆq (h) Suppler s optmal unconstraned qualty level under PBC gven h. q (h) Suppler s optmal constraned qualty level under PBC gven h, q (h) =mn{ˆq,q }. E [q ] Expected delvered qualty by the wnnng suppler under PBC. c Unt producton cost for suppler to produce a unt wth q =1, c Unform(c L,c H ). c ( ) th smallest value n a random sample of sze n from the dstrbuton of c. z Constant parameter n the supplers producton cost functon, c q z. C S (q ) Suppler s qualty-related costs under PBC, C S (q )=c q z H (q ; Q). π, π S Suppler s proft, the wnnng suppler s optmal proft. C T Total system cost, C T = C B π S. dffer n ther capabltes for producng hgh-qualty products, as well as n ther producton costs. The buyer has chosen to snglesource and wll select a sngle suppler from the set of potental supplers. We normalze the buyer s volume (whch s fxed and known) to 1. The buyer cares about both procurement cost (prce charged by the suppler) and the qualty of the product. We let p and q denote suppler s bd prce and offered qualty. Both p and q are decson varables for suppler. We model qualty, q, as the probablty that a randomly selected unt of suppler s product s not defectve. Thus, q [0, 1], wth q =1 representng perfect qualty. The use of the defectve rate (or, equvalently, the conformance rate) to represent product qualty s farly common n the lterature [17], [18]. We assume that suppler s unt producton cost s c q z, where z>1 s a constant, and c can be thought of as suppler s cost to produce a unt of perfect qualty. Ths cost functon s ncreasng and convex, whch mples that the margnal cost of a unt of qualty s ncreasng n the qualty level,.e., hgher levels of qualty are ncreasngly costly to obtan. A larger value of z mples more curvature n the cost functon, wth z =1mplyng a lnear cost n qualty. A convex cost functon s commonly used n the lterature, e.g., the unt producton cost functon n [19] takes ths form. The c are assumed to vary across the supplers. In order to obtan closed-form results, we assume the c are ndependent draws from a unform[c L,c H ] dstrbuton, where c L > 0. We wll use c () to denote the order statstcs for the c, so that c (1) c (2) c (n). See Table I for a lst of notaton.

5 286 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST 2016 The supplers have dfferng capabltes for producng hghqualty products. Specfcally, let q denote suppler s qualty capablty,.e., suppler s mum achevable qualty level. Thus, suppler s decson problem ncludes the constrant q q. In practce, ths capablty wll depend on varous factors, ncludng the suppler s experence, access to sklled labor and access to technology. We assume the q are ndependent draws from a unform[q L,q H ] dstrbuton, where q H 1. The buyer desgns the sourcng process to mnmze the unt cost, ncludng the procurement cost (.e., the wnnng bd prce), the costs assocated wth poor qualty and, n the case of PBC, the penalty/reward payments made to the wnnng suppler. Whle more complex models are possble [7], [8], for smplcty, we model the buyer s cost of poor qualty as the expected unt warranty cost, whch can be wrtten as C w (q) =w(1 q), where 1 q s the probablty a randomly selected unt s defectve, and w s the warranty cost assocated wth a sngle defectve unt. Fnally, we assume that each suppler wll choose the qualty level and bd prce n order to mze hs expected unt proft, whch ncludes the unt producton cost, revenue from sales to the buyer (wnnng bd prce), and, n the case of PBC, the penalty/reward payments made to the buyer. A. TSP We frst consder a TSP, commonly used n ndustral and government procurement. In the qualfcaton stage, potental supplers are screened for a varety of capabltes, ncludng qualty and relablty. In the suppler selecton stage, the qualfed supplers are nvted to compete for the buyer s busness by partcpatng n a sealed-bd (closed) prce-only procurement aucton. Thus, each stage has a fundamentally dfferent objectve. The qualfcaton stage ensures that the buyer sources from only the most capable supplers, whle the suppler selecton stage nduces competton between the potental supplers. Jn et al. [8] present a model of a TSP n whch the buyer exerts effort n the qualfcaton stage n order to learn about the supplers qualty capabltes. We present only the smplest model n ther paper, n whch the qualfcaton process s costless and provdes the buyer wth perfect nformaton on the supplers capabltes. 1 Ths smplest model s the most favorable verson of TSP, and thus offers the most favorable comparson wth PBC. In the qualfcaton stage, the buyer specfes the qualfcaton threshold, Q, whch supplers must meet n order to partcpate n the suppler selecton stage,.e., only supplers wth q Q are nvted to partcpate n prce-only bddng. After the qualfcaton stage, some number, m(q) n, of the potental supplers wll be qualfed. Note that m(q) s a random varable whch depends on Q and on the dstrbuton of the q. These m(q) qualfed supplers wll submt sealed prce-only bds to the buyer, who wll select among them on the bass of prce. Usng standard results from aucton theory [20], the expected wnnng bd prce, whch s the buyer s expected unt procurement cost, can 1 Here, we do not nclude a detaled analyss of the TSP model wth non-fully capable supplers because that analyss has been publshed n prevous papers [7], [8]. be wrtten as C p (Q) =Q z 2 E[c H m (Q)+1 + c L m (Q) 1 m (Q)+1 ]. Ths result uses the fact that all of the qualfed supplers wll set ther qualty just equal to the qualfcaton threshold, Q, snce they have no ncentve to offer a hgher level of qualty. Thus, the buyer s problem s to choose Q to mnmze her total cost, C B (Q) =C p (Q)+C w (Q).Jnet al. [7], [8] show that the optmal threshold satsfes d C p d Q = d C w d Q = w,.e., at the optmal Q, the margnal ncrease n procurement costs due to an ncrease n Q just equals the margnal savngs n warranty costs due to an ncrease n Q. Although Jn et al. [7], [8] assume a closed aucton,.e., sealed bds, ther results also hold under an open aucton, due to revenue equvalence (see, for example, [20]). We can thus rewrte the results for the specal case n whch the supplers have unlmted qualty capabltes, and z =2, as follows: the qualfcaton threshold s Q =, the buyer s cost w 2E [c (2) ] s CB = w w 2 Q = w w E [c (2) ], and the wnnng suppler earns πs =(Q ) 2 (E[c (2) ] E[c (1) ]). B. PBC Under the TSP, the qualfed supplers compete only on the bass of prce. Thus, they have no ncentve to offer a qualty level greater than the qualfcaton threshold, Q. In constrast, under PBC, the buyer offers the supplers a penalty/reward based on the wnnng suppler s realzed qualty, relatve to the target qualty, Q. Thus, a properly desgned penalty/reward functon may encourage the supplers to offer qualty levels greater than Q. Another beneft of PBC s the potental for ncreased competton, relatve to TSP. Under TSP only the m(q) n qualfed supplers may partcpate n the suppler selecton stage. In contrast, under PBC, all n potental supplers are allowed to bd. 2 Under PBC, the buyer declares that the wnnng suppler s payment wll nclude a penalty/reward, denoted by H(q ; Q), where Q s the buyer s target qualty level. If q >Q, then H(q ; Q) > 0 represents a reward pad to the suppler; f q <Q, then H(q ; Q) < 0 represents a penalty charged to the suppler. Thus, suppler s proft, gven that he wns the buyer s busness, s π = p c q z + H(q ; Q). The buyer wll select among the n potental supplers by runnng a prce-only procurement aucton. We assume no cost for bddng and thus all n potental supplers partcpate n the aucton. Suppler wll bd n order to mze hs expected unt proft, subject to q q and p c q z H(q ; Q). Snce we assume prce-only bddng, the probablty that suppler wns s decreasng as p ncreases. Thus, the best q wll be the value that mnmzes suppler s total unt qualty-related costs, denoted by C S, where C S (q )=c q z H(q ; Q). Weletˆq denote that value of q that mnmzes C S (q ). However, suppler s constraned by hs mum achevable qualty level, q. Thus, suppler wll set hs qualty level equal to q = mn{ˆq,q }. 2 Of course, there are some settngs n whch t would not be feasble to allow all n supplers to bd, regardless of ther qualty capabltes, e.g., settngs n whch there s a hard mnmum on the qualty level that must be acheved. In such a settng, a pure PBC approach would not be approprate.

6 JIN AND RYAN: CAN CONTRACTS REPLACE QUALIFICATION IN A SOURCING PROCESS WITH COMPETITIVE SUPPLIERS 287 Whle alternatve forms for the penalty/reward functon are possble, we wll focus on a symmetrc and lnear functon, H(q ; Q) =h(q Q), where h s the penalty/reward rate, whch s a decson varable for the buyer. In ths case, we can solve C S q =0to fnd ˆq =( h zc ) 1 z 1. When z =2, ˆq = h 2c. As noted n [10], lnear ncentve functons are commonly used n practce. In Secton V, we wll demonstrate that the buyer does not see a sgnfcant ncrease n expected cost f she uses a symmetrc functon. Before analyzng PBC n detal, we argue that revenue equvalence apples, and thus the buyer s results for open and closed bddng wll be dentcal. The costs c for producng one unt of perfect qualty and the qualty capabltes q are ndependent across all supplers. The optmal qualty level for suppler, q, s equal to mn{( h zc ) 1 z 1,q }. The supplers optmal costs,.e., C S (q )=c (q )z h(q Q) for =1,...,n, are thus functons of c and q. For a gven h, these optmal costs are ndependent across the supplers. In addton, the costs follow dentcal dstrbutons snce c and q are drawn from dentcal dstrbutons. Thus, the aucton has a set of bdders wth ndependent and dentcally dstrbuted (..d.) costs. Followng the revenue equvalence result for forward auctons [20], for reverse auctons, all standard aucton forms, ncludng open auctons and frst-prce closed auctons, wth..d. costs, wll yeld the same expected cost for the buyer. C. Comparson of PBC and TSP The goal of ths work s to compare the performance of the TSP and PBC mechansms. We would lke to do so from three perspectves: those of the buyer, the supplers, and the system as a whole. It s straghtforward to argue that PBC wth a general penalty/reward functon always outperforms TSP. To do so, one can argue that the penalty/reward functon under PBC can always be desgned to acheve the results of TSP wth a specfed threshold, Q. Specfcally, the buyer can desgn the penalty/reward functon such that f a suppler s qualty capablty s at least equal to Q, then there s no penalty; otherwse, an nfnte penalty s assgned. Gven ths penalty/reward functon, only those supplers wth qualty capablty above the threshold Q would partcpate n the aucton, and the wnnng suppler would always set hs qualty equal to Q, thus replcatng the TSP outcome. Whle PBC wth a general penalty/reward functon, such as an asymmetrc lnear functon, wll always perform at least as well as the TSP, from the buyer s perspectve, t s not clear how the more practcal and easy-to-mplement symmetrc lnear penalty/reward functon wll perform relatve to the TSP snce the above argument does not apply for the symmetrc case. Thus, we next prove that PBC, wth a symmetrc penalty/reward rate, h, selected to mnmze the buyer s expected cost, always outperforms the TSP, wth qualfcaton threshold, Q, selected to mnmze the buyer s expected cost, from the perspectve of the buyer. The proofs of all theorems can be found n the techncal supplement. Theorem 1: Consder a settng n whch the potental supplers have lmted qualty capabltes,.e., suppler must satsfy q q, where q 1 for all. The buyer s expected cost under PBC, wth a symmetrc penalty/reward functon, H(q ; Q) =h(q Q), n whch the penalty/reward rate, h, s set optmally, s no greater than the buyer s expected cost under the TSP, n whch the qualfcaton threshold, Q, sset optmally. From the buyer s perspectve, PBC wll always outperform TSP,.e., PBC wll provde a lower expected cost. From the proof of Theorem 1, there are two factors contrbutng to ths result. Frst, the supplers qualty levels are endogenous and PBC provdes a greater degree of flexblty to the supplers n choosng ther qualty levels. Under TSP, the supplers must set ther qualty levels at least equal to the qualfcaton threshold Q. Snce the supplers expected costs are ncreasng n Q under TSP, all qualfed supplers wll choose to set ther qualty levels equal to exactly Q. Under PBC, although the penalty/reward functon specfes a target qualty level Q the supplers are able to choose ther optmal qualty level,.e., the qualty level that mnmzes ther costs, subject to ther qualty capabltes. Ths flexblty mples that each suppler may offer a dfferent qualty level,.e., the supplers wll offer dfferentated qualty levels to the buyer. Second, PBC allows all n potental supplers to bd n the aucton. On the other hand, under TSP, only the m(q) n qualfed supplers,.e., the supplers wth q Q,areallowed to bd n the aucton stage. Thus, there s less competton n the bddng stage under TSP than under PBC, leadng to a hgher wnnng bd prce for the buyer. III. PBC: DETAILED ANALYSIS We next perform a more detaled analyss of the buyer s decsons under PBC. Suppler wll set hs optmal qualty level, q, to mnmze C S(q ), subject to q q. One of the man goals of ths research s to study a settng n whch the supplers set ther qualty levels endogenously, based on the ncentves provded by the buyer, but subject to heterogeneous qualty capabltes. Thus, n Secton III-A, we consder a settng n whch the supplers do not have unlmted qualty capabltes, whch mples q = mn{ˆq,q } for all. Then, to derve addtonal results, n Secton III-B, we consder the case n whch all supplers have unlmted qualty capablty and thus q =ˆq,for all. A. Supplers are Not Fully Capable To model the supplers heterogeneous qualty capabltes, we assume the q are ndependent draws from a unform[q L,q H ] dstrbuton for =1,...,n, where q H 1 (and thus q 1 for all ). In ths case, assumng a symmetrc and lnear penalty/reward functon, H(q ; Q) =h(q Q), wth penalty/reward rate h, and z>1, suppler s optmal qualty level s q (h) = mn{ˆq (h),q }, where ˆq (h) =( h zc ) 1 z 1. When bddng n a prce-only aucton, suppler wll bd based on hs optmal cost CS(h) = c (q (h)) z H(q (h); Q) = c (q (h)) z h(q (h) Q). (1)

7 288 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST 2016 For ths model, determnng the wnnng suppler and the expected wnnng bd prce, and characterzng the buyer s optmal penalty/reward rate, are challengng. As wll be seen, n the case of fully capable supplers (wth q (h) =ˆq (h) for all ), suppler s total cost, C S (ˆq (h)), s decreasng n c, and thus the suppler wth the lowest unt cost wns the bddng. However, ths result does not hold when the supplers are not fully capable. Instead, the wnnng suppler wll be the one wth the lowest CS (h), whch depends n a complex manner on both c and,aswellash. To characterze the supplers bd prces, we need to fnd the probablty dstrbuton of CS (h), notng that snce the c and are..d., the CS (h) wll also be..d. Let F C ( ) denote the cdf CS (h). Furthermore, to characterze the buyer s expected cost, we must characterze the order statstcs of CS (h). Under open bddng, the buyer wll choose the suppler wth the lowest unt cost,.e., the lowest CS (h), and wll pay a bd prce equal to the second lowest value of CS (h). Consder a settng wth just two supplers, labeled as and j. Suppose that suppler wns the bddng. Ths mples that CS (h) C Sj (h). The buyer s total cost s q q C B (Q, h) = C Sj(h)+w(1 q (h)) + h(q (h) Q) = c j q j (h) z h(q j (h) q (h)) + w(1 q (h)). (2) Note that the Q cancels out of the buyer s cost functon. Hence, for the remander of ths secton, we wll wrte the buyer s cost as C B (h). We can now state the followng result. Theorem 2: Consder PBC wth a symmetrc penalty/reward functon, H(q ; Q) =h(q Q), when the supplers are not all fully capable,.e., when q 1 for all, wth z>1. 1) Gven CS (h) C Sj (h), then E[q (h)] E[q j (h)], E[q ] E[qj ] and E[c ] E[c j ]. Ths theorem holds for any par of supplers and j not just those wth the lowest and second lowest total unt cost. The theorem states that f suppler provdes the buyer wth a lower total unt cost than suppler j (CS (h) C Sj (h)), then the expected qualty from suppler must be no less than that from suppler j (E[q (h)] E[q j (h)]). Thus, the theorem mples that, for any h, the suppler who wns the bddng based on the costs CS (h) for =1,...,n, wll also provde the hghest expected level of delvered qualty. The fnal two results of the theorem mply that, n expectaton, the aucton process favors supplers wth hgher qualty capablty and lower unt costs. 1) Optmal Penalty/Reward Rate: We would lke to characterze the optmal penalty/reward rate, h. In order to understand the mpact of h on whch suppler wns the bddng, t s useful to consder two extreme cases. 1) When h, q (h) = mn{ˆq,q } = q and thus CS = c (q ) z h(q Q). In ths case, CS s domnated by the term h(q Q). The mpact of c s neglgble. Therefore, the suppler wth the hghest q wll wn the aucton. Thus, the expected delvered qualty s equal to the expected hghest q 1 draws, whch s n+1 q L + n among n random n+1 q H. Snce the mpact of c s neglgble, and snce c s ndependent of q, a suppler wth any c has an equal chance of wnnng the aucton. Thus, the expected unt cost of the wnnng suppler s c L +c H 2. 2) When h 0, q (h) = mn{ˆq,q } =ˆq =( h zc ) 1 z 1 and CS = hq (1 1 z )(h) z z 1 ( 1 zc ) 1 z 1. In ths case, CS s an ncreasng functon of c. Therefore, the suppler wth the lowest c wll wn the aucton and the expected unt cost of the wnnng suppler wll be n n+1 c L + 1 n+1 c H. Because c s ndependent of q, a suppler wth any q has an equal chance of wnnng the aucton. Thus, the expected delvered qualty level s q L +q H 2. Intutvely, when h =0, the lowest unt cost suppler wns. As h ncreases away from 0, supplers wth hgher unt cost starts to have a chance of wnnng the aucton, f they happen to have a hgh-qualty capablty. Fnally, when h becomes suffcently large, the most capable suppler wns. We are now ready to characterze the optmal penalty/reward rate for the buyer. Theorem 3: Under PBC, wth a symmetrc penalty/reward functon, H(q ; Q) =h(q Q), when the supplers are not all fully capable,.e., when q 1 for all, wth z>1, the optmal penalty/reward rate h satsfes h w, where w s the unt warranty cost. The theorem mples that the buyer does not share all of her warranty costs wth the wnnng suppler. Instead, t s optmal for the buyer to absorb some of the warranty costs. To understand ths result, note that Theorem 2 ndcates that the aucton process wll favor supplers who have lower unt costs and those who can provde a hgher level of delvered qualty. Thus, although a hgh penalty/reward rate h can be used to encourage the suppler s to offer a hgh level of qualty, t s not the only mechansm for dong so. Fnally, we note that ths result s consstent wth the fndngs n [10], although the assumptons and model settngs have a number of dfferences. 2) Evaluatng the Buyer s Cost: Whle Theorem 3 presents general results regardng the magntude of the optmal penalty/reward rate h provdng an exact expresson s more challengng. In fact, even dervng a useful closed-form expresson for the buyer s expected cost s dffcult. From (2), we can wrte the buyer s expected cost as C B (Q, h) = E[CS [2] (h)] + w +(h w)e[q [1](h)] hq, where we use the subscrpt [] to denote the order statstcs for the CS (h), so that CS [1] (h) C S [2] (h) C S [n] (h), and C S [2] (h) = c [2] (q[2] )z hq[2] + hq. The expected proft for the wnnng suppler,.e., suppler [1], wll then be π S (h) =E[CS [2] (h)] E[C S [1](h)]. (3) To evaluate the buyer s expected cost, we need to evaluate E[q[1] (h)] and E[C S [2](h)]. To do so, for smplcty, we wll focus on the case n whch q L ˆq (h) q H for all, where ˆq (h) =( h zc ) 1 z 1. For a gven c, usng the fact that q (h) = mn{ˆq (h),q }, wehave E q [q (h) c ]= ˆq (h) q L qh + ˆq (h) q f q (q )dq ˆq (h)f q (q )dq (4)

8 JIN AND RYAN: CAN CONTRACTS REPLACE QUALIFICATION IN A SOURCING PROCESS WITH COMPETITIVE SUPPLIERS 289 where f q (q ) denotes the densty functon for q, defned on [q L,q H ]. From (1), usng q (h) = mn{ˆq (h),q }, wehave E q [C S(h) c ]= + q H ˆq (h ) ( hq ˆq (h ) q L ( 1 1 z (c (q ) 2 hq ) ( ) 1 ) (h) z 1 z 1 z 1 zc + hq)f q (q )dq f q (q )dq. (5) We can now wrte the buyer s expected cost as C B (Q, h) = E c[2] {E q [C S(h) c [2] ]} + w +(h w)e c[1] {E q [q c [1] ]} hq (6) where E q [C S (h) c [2]] and E q [q c [1]] are computed usng (5) and (4), respectvely. We next present the results of an extensve numercal study to demonstrate the behavor of the optmal penalty/reward rate, as well as the buyer s expected cost. In Secton III-B, we consder the case n whch all supplers are fully capable, whch allows for closed-form results and addtonal nsghts. 3) Numercal Results: We fx z =2, so that the unt producton cost s quadratc n qualty. We have sx problem parameters to consder (c L, c H, q L, q H, w, and n). We set q H =1and the average unt cost to c L +c H 2 =2. We wrte c L =2 δand c H = 2+δ. Thus, we have four parameters to vary (q L, δ, w, and n). For each, we consder four values: q L {0.1, 0.3, 0.5, 0.7}, δ {0.5, 1, 1.5, 2}, w {0.25, 1, 2, 3}, and n {5, 10, 15, 20},resultng n a total of 256 experments. To set the range for the unt warranty cost w we consdered that n our experments the average unt cost s equal to 2. Thus, we nclude cases n whch the unt warranty cost s substantally less than the average unt cost, as well as cases n whch the unt warranty cost exceeds the average unt cost. In the latter case, the unt warranty cost ncludes a goodwll cost,.e., a cost of customer dssatsfacton due to the low qualty of the product. We next descrbe how we evaluate the buyer s cost for a gven penalty/reward rate h and search for the optmal h. We frst generate two sets of one mllon random numbers from a unform[0, 1] dstrbuton. For each combnaton of the parameters (c L,c H,q L,q H,w,n), we use the frst set to generate a set of one mllon c [c L,c H ] and the second set to generate a [q L,q H ]. For a gven h, we fnd the optmal qualty q = mn{ˆq (h),q } for each and compute the overall cost CS (h) =c (q )2 h(q Q) for each. We then repeat the followng process 5000 tmes. 1) Generate a set of ndexes (ntegers) from a unform dstrbuton between 0 and 1 mllon. The sze of the set s the number of supplers, n. set of one mllon q 2) Use the ndexes to extract the correspondng CS (h) and rank these costs n ascendng order. 3) Fnd the wnnng suppler (the suppler wth the lowest CS (h)) and the wnnng bd prce (the second lowest CS (h)). 4) Record the buyer s total cost, the delvered qualty and the wnnng bd prce. After 5000 repettons, we compute the average of the wnnng suppler s total cost, the delvered qualty and the wnnng bd TABLE II PBC SOLUTION FOR VARIOUS VALUES OF n C B E [q ] h C T π S (w h)/h n = % n = % n = % n = % TABLE III PBC SOLUTION FOR VARIOUS VALUES OF w C B E [q ] h C T π S (w h)/h w = % w = % w = % w = % TABLE IV PBC SOLUTION FOR VARIOUS VALUES OF q L C B E [q ] h C T π S (w h)/h q L = % q L = % q L = % q L = % TABLE V PBC SOLUTION FOR VARIOUS VALUES OF δ = c H c L C B E [q ] h C T π S (w h)/h δ = % δ = % δ = % δ = % prce. From these values, we compute the buyer s cost and the total system cost. Fnally, we need to fnd the h to mnmze the buyer s cost. Accordng to Theorem 3, h w. Thus, we search for the optmal h over the range [0.1, 3] wth an ncrement of Tables II V show the mpact of the parameters on the performance measures, ncludng the buyer s optmal cost CB,the wnnng suppler s expected proft πs, the expected total system cost, denoted by C T = CB π S, the expected qualty level provded by the wnnng suppler (referred to as the expected delvered qualty level ) E[q ], and the optmal penalty/reward rate h. Although the results n the tables are obtaned usng smulaton, we expect them to be qute accurate. Specfcally, for each of the 256 experments, for each performance measure, we computed the rato of the standard devaton to the mean (across the 5000 replcatons). For the buyer s expected cost (system cost), the average value of ths rato across the 256 experments was (0.0024), whle the mum value across the 256 experments was (0.0081). Tables II V show the mpact of one parameter. Each row shows the average

9 290 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST 2016 value of the performance measures across the 64 experments n whch the parameter took the specfed value. Table II shows the mpact of the number of potental supplers on the results. These results reflect the beneft of ncreased competton (larger n) for the buyer. Wth more potental supplers, the competton s more ntense, resultng the lower cost for the buyer and lower proft for the wnnng suppler. The buyer s also able to set a hgher penalty/reward rate, effectvely passng on more of the warranty cost to the wnnng suppler, and resultng n a hgher expected delvered qualty. Table III shows the mpact of the unt warranty cost. As w ncreases, CB ncreases, E[q ] ncreases, C T ncreases, πs ncreases, h ncreases, and (w h)/h ncreases. The mpact of the unt warranty cost on the buyer s cost, the penalty/reward rate and the delvered qualty are ntutve. Wth a hgher unt warranty cost, the buyer must set a hgher penalty/reward rate n order to nduce a hgher delvered qualty, leadng to a hgher expected cost for the buyer. A hgher unt warranty cost s benefcal for the wnnng suppler,.e., results n a hgher expected proft. The wnnng suppler s proft, as shown n (3), s equal to the dfference between the costs of the second lowest and lowest cost supplers. Ths dfference s ncreasng n h,.e., a larger h magnfes any dfferences between the supplers c and q. Snce the buyer sets a hgher penalty/reward rate as the warranty cost ncreases, the dfference between the second lowest and lowest supplers costs ncreases as the warranty cost ncreases, mplyng that the wnnng suppler s proft ncreases. Thus, the nformaton rent earned by the wnnng suppler ncreases wth the warranty cost, reflectng the ncreased power of the supplers. Table IV shows the mpact of the lower bound on the supplers qualty capabltes on the results. As q L ncreases, CB decreases, E[q ] ncreases, the expected system cost decreases, πs ncreases, h ncreases, and (w h)/h decreases. In our numercal examples, we fx the upper bound on the supplers qualty capabltes to q H =1. Thus, an ncrease n q L mples an ncrease n the average qualty level across the supplers. Ths ncrease leads to a hgher expected delvered qualty for the buyer and thus reduces the buyer s expected cost, as well as the expected system cost. However, the mpact of a larger q L on the wnnng suppler s expected proft s mxed. On one hand, an ncrease n the average qualty level across the supplers leads to an ncrease n the expected qualty capablty of the wnnng suppler, whch should beneft the wnnng suppler. On the other hand, a larger q L mples that the range (q H q L ) of qualty levels across the set of supplers s smaller,.e., there s less dfferentaton between the potental supplers, whle the number of supplers competng for the buyer s busness s fxed at n. Thus, as q L ncreases, the dfference between CS [2] (h) and C S [1] (h) decreases, mplyng that the wnnng suppler s expected proft, as gven n (3), decreases. In other words, less dfferentaton (tghter competton) between the supplers results n a lower expected proft for the wnnng suppler. Overall, consderng both factors, q L has lttle mpact on the wnnng suppler s expected proft. Table V shows the mpact of the spread n the supplers unt costs (δ = c H c L ) on the results. As c H c L ncreases, CB decreases, E[q ] ncreases, the expected system cost decreases, πs ncreases, h decreases, and (w h)/h ncreases. As we vary the spread (c H c L ), the average cost ( c H +c L 2 )skept constant. Thus, when the spread s ncreased, the supplers costs can take lower (and hgher) values. Due to the competton nduced by the aucton, as well as the fact that the aucton process favors lower cost supplers, the buyer s able to take advantage of the lower costs, wthout feelng the mpact of the hgher costs, resultng n a lower overall cost for the buyer and allowng the buyer to acheve a hgher expected delvered qualty level. The larger spread (c H c L ) also benefts the wnnng suppler. As noted above, the wnnng suppler s expected proft s equal to CS [2] (h) C S [1] (h), whch should be ncreasng n c H c L. In other words, greater dfferentaton between the supplers (larger c H c L ) mples less competton and hgher profts for the wnnng suppler. Whle Tables II V show the mpact of one parameter, Fg. 1 shows the jont mpact of the unt warranty cost w and the spread n the supplers unt costs, δ. Fg. 1(a) confrms the results shown n the tables,.e., the buyer s optmal cost s decreasng n δ and ncreasng n w. However, Fg. 1(b) demonstrates an nteracton not seen n the tables. Specfcally, when the unt warranty cost s small, the expected delvered qualty s much more senstve to the spread n the supplers costs than when the unt warranty cost s large. As explaned n the prevous paragraph, larger δ wll generally lead to larger E[q ]. However, when the unt warranty cost s large, ths effect s mnmal, snce the expected delvered qualty wll already be qute large due to the large penalty/reward rate nduced by the hgh warranty cost. Fnally, whle not shown here, we also created addtonal two-way graphs, for dfferent combnatons of nput parameters. All of these graphs are smlar to Fg. 1(a),.e., they confrm the results n Tables II V wthout demonstratng any nteractons between the parameters. B. All Supplers are Fully Capable To obtan further analytcal results, we consder the case n whch all supplers are fully capable,.e., q =1for all, wth z =2. When the supplers are all fully capable, gven the penalty/reward rate equal to h, suppler wll set hs qualty level equal to q (h) =ˆq (h) = h 2c. Dependng on the value of h,we may have ˆq (h) > 1. Gven our nterpretaton of qualty (.e., the probablty that a randomly selected unt s not defectve), we should defne q (h) = mn{ˆq (h), 1}. However, the goal of ths secton s to enable analytcal results for PBC. Thus, whle we wll dscuss the case n whch q (h) = mn{ˆq (h), 1} at the end of ths secton, for smplcty, we wll wrte q (h) = ˆq (h) = h 2c.Gvenq (h), suppler s optmal cost s C S (h) = C S (q (h)) = C s(ˆq (h)) = hq ( h 2 4c ). When bddng n the prce-only aucton, suppler wll bd based on CS (h). Snce the c are..d., the CS (h) wll also be..d. The supplers partcpate n an open aucton. Recall that the buyer wll select the suppler wth the lowest CS (h) and wll pay a bd prce equal to the second lowest CS (h). Fromthe expresson for CS (h), t s clear that the suppler wth the lowest c wll also be the suppler wth the lowest CS (h). Thus, the

10 JIN AND RYAN: CAN CONTRACTS REPLACE QUALIFICATION IN A SOURCING PROCESS WITH COMPETITIVE SUPPLIERS 291 Fg. 1. Impact of δ and w on buyer s expected cost and expected delvered qualty. (a) The buyer s expected cost. (b) The expected delvered qualty level. lowest cost suppler,.e., the suppler whose unt cost s c (1), wll wn wth a bd prce equal to p (1) (h) =C S (2)(h) =hq h 2 4c (2) and the qualty level equal to q(1) (h) = h 2c (1). The buyer s total expected unt cost C B (Q, h) conssts of the expected unt procurement cost (or expected wnnng bd prce), the expected unt warranty cost and the expected penalty/reward payment C B (Q, h) = E[p (1) (h)] + w(1 E[q (1) ]) + h(e[q (1)(h)] Q) [ ] ( [ ]) h 2 h = hq E + w 1 E ( + h E 4c (2) [ h 2c (1) [ = w h2 1 4 E c (2) ] ) Q 2c (1) ] + h [ ] 1 2 E (h w). c (1) Notce that the qualty threshold Q does not affect the buyer s cost. Thus, the buyer s problem s to select h to mnmze C B (h). We are now ready to characterze the buyer s optmal soluton. Theorem 4: Under PBC, wth a symmetrc penalty/reward functon, H(q ; Q) =h(q Q), when all supplers are fully capable, wth z =2, the optmal penalty/reward rate s h = w( 1 2 γ ), where γ = E [ 1 c ] (2) 1 1 and thus 0 h w. E [ c ] (1) The optmal delvered qualty level,.e., the qualty level provded by the wnnng suppler, s E[q(1) ]= h 2 E[ 1 c (1) ]= w 2 ( 1 2 γ )E[ 1 c (1) ]. The buyer s optmal expected cost s CB = w w 2 E[q (1) ]. The wnnng suppler earns expected proft equal to πs = (h ) E[ c (1) 1 c (2) ]. The total supply chan expected cost s C T = CB π S = E[c (1)(q(1) )2 + w(1 q(1) )]. The buyer s optmal expected cost satsfes C B w =1 E[q(1) ]. Thus, when E[q (1)] < 1, the buyer s optmal expected cost s ncreasng n the unt warranty cost w, whle the wnnng suppler s expected proft s ncreasng n the unt warranty cost, w. The optmal expected delvered qualty level E[q (1) ] s ncreasng n the unt warranty cost w. Ths theorem provdes the followng nsghts. 1) When desgnng PBC, the buyer must set the qualty target Q and the penalty/reward rate h. Q can take any value. However, the optmal h s always less than the unt warranty cost w. 2) For c L > 0, as the number of potental supplers n ncreases, t wll generally hold that E[ 1 c (2) ] E[ 1 c (1) ]. Thus, γ = E [ 1 c ] (2) 1 1, whch mples h w, asn E [ c ] (1). In other words, the optmal penalty/reward rate h approaches the warranty cost w as the number of supplers ncreases. 3) The wnnng suppler wll be the suppler wth the lowest unt cost c (1). That suppler wll provde qualty level q(1) = h 2c (1). Applyng the same argument as n the prevous bullet, E[q(1) ] wll generally be ncreasng n n. Thus, more potental supplers (more competton) s benefcal to the buyer, leadng to a hgher expected delvered qualty level and lower expected cost. 4) The expresson for the wnnng suppler s expected proft πs ndcates that ths proft should generally be decreasng as the number of supplers ncreases. To see ths, note that h s approxmately constant (equal to w) asn, 1 whle E[ c (1) 1 c (2) ] 0. 5) When E[q(1) ] < 1, a larger warranty cost w leads to hgher expected cost and a hgher expected delvered qualty level for the buyer. However, a larger warranty cost leads to hgher expected proft for the wnnng suppler. Ths mples that the supplers gan power over the buyer when the warranty cost s large, allowng the wnnng suppler to extract more proft from the buyer. 6) The expected delvered qualty level under PBC, as specfed n Theorem 4, s lower than n the system optmal soluton, whch can be obtaned by settng h = w.

11 292 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST 2016 TABLE VI PBC VERSUS TSP: IMPACT OF NUMBER OF POTENTIAL SUPPLIERS % C B % E [q ] % π S % C T q TSP q PBC C TSP T > C PBC T π TSP S > π PBC S n =5 7.55% 24.75% 1.41% 8.34% 92.19% % 57.81% n = % 17.60% 5.75% 10.39% 93.75% % 65.63% n = % 13.93% 9.26% 11.60% 87.50% 93.75% 84.38% n = % 11.49% 12.41% 12.21% 87.50% 93.75% 89.06% TABLE IX PBC VERSUS TSP: IMPACT OF SPREAD IN SUPPLIERS UNIT COSTS (δ = c H c L ) % C B % E [q ] % π S % C T q TSP < q PBC C TSP T > C PBC T π TSP S > π PBC S δ =1 1.10% 3.73% 10.87% 0.92% 68.75% 87.50% 89.06% δ =2 3.86% 9.67% 11.02% 3.66% 95.31% % 84.38% δ =3 9.57% 17.95% 8.02% 9.99% % % 68.75% δ = % 36.42% 1.07% 27.96% 96.88% % 54.69% TABLE VII PBC VERSUS TSP: IMPACT OF UNIT WARRANTY COST % C B % E [q ] % π S % C T q TSP q PBC C TSP T > C PBC T π TSP S > π PBC S w = % 28.41% -0.56% 4.96% 76.56% 87.50% 57.81% w =1 7.37% 14.23% 6.75% 8.19% 89.06% % 75.00% w = % 13.19% 8.07% 12.93% 95.31% % 75.00% w = % 11.93% 14.58% 16.45% % % 89.06% TABLE VIII PBC VERSUS TSP: IMPACT OF LOWER BOUND ON SUPPLIERS QUALITY CAPABILITIES % C B % E [q ] % π S % C T q TSP q PBC C TSP T > C PBC T π TSP S > π PBC S q L = % 18.89% 7.78% 9.91% 93.75% 96.88% 82.81% q L = % 17.33% 8.48% 10.67% 92.19% 96.88% 78.13% q L = % 16.74% 6.88% 11.31% 90.63% 96.88% 73.44% q L = % 14.80% 5.70% 10.65% 84.38% 96.88% 62.50% In summary, the optmal penalty/reward rate h s proportonal to the unt warranty cost w. Therefore, PBC wth a lnear penalty/reward functon s a form of shared warranty costs, where the porton of the warranty cost pad by the selected suppler ( 1 2 γ ) s ncreasng n the number of potental supplers. Thus, when there s more competton, the wnnng suppler s forced to bear a greater porton of the buyer s warranty costs. In addton, the optmal penalty/reward rate h wll generally approach w as the number of potental supplers ncreases. Thus, as the number of supplers ncreases, the optmal wnnng qualty level ncreases towards the system optmal qualty level. =1for all, the optmal qualty level should be wrtten as q (h) = mn{ˆq (h), 1}, rather than q (h) =ˆq (h). In ths case, whle the analyss s more complex than for Theorem 4, we can also demonstrate that a unque optmal h exsts and satsfes h w. As noted above, when q IV. COMPARISON OF PBC AND TSP In Secton II-C, we demonstrated analytcally that PBC always outperforms the TSP. However, TSP s more wdely used n practce. When qualty s of mportance to ther compettve advantage, e.g., when reputaton s crtcal, buyers prefer to contract wth supplers who are known to be capable of meetng the buyer s standards (e.g., through the qualfcaton stage of TSP), rather than contractng wth supplers of unknown qualty and then penalzng for poor performance (e.g., through PBC). Therefore, t s useful to understand the magntude of the performance gap between TSP and PBC and to dentfy condtons under whch buyers do not lose much by usng TSP. We consder the case n whch the supplers are not all fully capable,.e., the case n whch q 1 for all.thus,weassume the q follow a unform[q L,q H ] dstrbuton, wth q H 1, across the set of potental supplers. As dscussed n Secton III-A, analytcal results are not possble for PBC for ths settng. Therefore, we wll use numercal experments to compare PBC and TSP. Our numercal experments follow the expermental desgn outlned n Secton III-A3. For PBC, we use the smulaton approach outlned n Secton III-A3. For TSP, we follow the approach outlned n [7]. Tables VI IX follow the same format as Tables II V. In the tables, % C B represents the amount by whch the buyer s expected cost under TSP exceeds the buyer s expected cost under PBC. Smlarly, % E[q ] represents the amount by whch the expected delvered qualty under PBC exceeds the expected delvered qualty under TSP, whle % πs represents the amount by whch the wnnng suppler s expected proft under TSP exceeds the wnnng suppler s expected proft under PBC. Fnally, % C T represents the amount by whch the expected total system cost under TSP exceeds the expected system cost under PBC. In these tables, the columns labeled q TSP <q PBC, CT TSP >CT PBC and πs TSP >πs PBC show the percent of cases n whch the expected delvered qualty, system cost, and wnnng suppler s expected proft, have the relatonshp shown. Recall that the buyer s expected cost s always lower under PBC than under TSP. The results n Tables VI IX ndcate that PBC generally outperforms TSP from the perspectve of the buyer s expected cost, the system expected cost and the expected delvered qualty level. However, TSP generally outperforms PBC from the perspectve of the wnnng suppler s expected proft. We next consder some more detaled comparsons. Table VI shows the mpact of the number of potental supplers on the performance of PBC relatve to TSP. As n ncreases, the performance of PBC relatve to TSP mproves, n terms of both the buyer s expected cost and the system expected cost. The opposte s true for the wnnng suppler s expected proft,.e., as n ncreases, the suppler s proft under TSP mproves relatve to under PBC. To understand these results, note that under PBC all n supplers compete n the aucton process. Under TSP, only the subset of qualfed supplers s allowed to compete n the aucton. Thus, under TSP, the value of havng more potental supplers, n terms of ncreased competton n the aucton stage,

12 JIN AND RYAN: CAN CONTRACTS REPLACE QUALIFICATION IN A SOURCING PROCESS WITH COMPETITIVE SUPPLIERS 293 s tempered by the qualfcaton stage. Thus, the hgher degree of competton under PBC benefts the buyer and the system, whle the lower degree of competton under TSP benefts the wnnng suppler. Table VI also ndcates that, as n ncreases, the dfference between the expected delvered qualty under PBC and TSP decreases. As dscussed n [7], under the TSP there s a tradeoff between the optmal qualty level (threshold) and the number of supplers. Specfcally, when the number of potental supplers s small, the buyer wll set a low qualty level (threshold) n order to ensure that enough supplers are qualfed to mantan competton n the aucton stage. However, when n gets larger, the buyer s able to be more strngent n the qualfcaton stage, settng a hgher qualty level (threshold), whle stll mantanng competton n the aucton stage. Thus, havng a larger number of supplers wll reduce the qualty gap between PBC and TSP. Table VII shows the mpact of the unt warranty cost on the performance of PBC relatve to TSP. As w ncreases, the performance of PBC relatve to TSP mproves n terms of both the buyer s expected cost and the expected system cost. However, the opposte s true for the wnnng suppler s expected proft,.e., as w ncreases, the wnnng suppler s expected proft under TSP mproves relatve to under PBC. The explanaton here s smlar to that provded for Table VI. Under PBC, the number of supplers competng n the aucton,.e., the level of competton, s fxed. Under TSP, the buyer controls the level of competton n the aucton stage through the selecton of the qualty level (threshold). When the warranty cost s low, the buyer s wllng to sacrfce qualty for ncreased competton by settng a low qualty level (threshold). However, when the warranty cost ncreases, the buyer must focus more on the costs assocated wth poor qualty and thus wll set a hgher qualty level (threshold), whch leads to fewer qualfed supplers. Wth fewer supplers competng under TSP compared to PBC, the wnnng suppler s able to extract a hgher proft. Table VII also ndcates that as w ncreases, the dfference between expected delvered qualty levels under PBC and TSP decreases. Ths s most lkely due to the fact that when w ncreases the expected delvered qualty levels are ncreased under both PBC and TSP, along wth the fact that both expected delvered qualty levels are constraned by 1 (the mum qualty capablty). However, as w ncreases, the percentage of cases n whch PBC has a larger expected delvered qualty level than TSP ncreases. Table VIII shows the mpact of the lower bound on the supplers qualty capabltes on the performance of PBC relatve to TSP. Whle q L does not have a consstent mpact on the buyer s expected cost, the expected system cost or the wnnng suppler s expected proft, a larger q L mples that the suppler s less lkely to have hgher proft under TSP than under PBC. Also, a larger q L causes a decrease n the dfference n the expected delvered qualty levels under PBC and TSP. Ths result s lkely due to the fact that a larger q L mples less spread n the qualty levels across supplers. Table IX shows the mpact of the spread n the supplers unt costs (δ = c H c L ) on the performance of PBC relatve to TSP. As δ ncreases, the performance of PBC relatve to TSP mproves n terms of both the buyer s expected cost and the expected system cost. In addton, as δ ncreases, TSP becomes less preferred by the suppler. Overall, as δ ncreases, all partes become more lkely to prefer PBC. Fnally, as δ ncreases, the expected delvered qualty also mproves for PBC relatve to TSP. As noted n the dscusson of Table V, under PBC, a larger δ = c H c L leads to lower expected cost for the buyer and hgher expected proft for the wnnng suppler. The same result holds under TSP. However, the benefts to the buyer of a larger δ are more substantal under PBC than under TSP. The larger number of supplers competng n the aucton stage under PBC mples that the buyer can take more advantage of the potental for low unt costs for the supplers when δ s larger than she can under TSP, when fewer supplers compete n the aucton stage. Thus, as δ ncreases, the performance of PBC relatve to TSP, from the perspectve of the buyer s cost, mproves. Smlarly, as δ = c H c L ncreases, the performance of PBC relatve to TSP, from the perspectve of the wnnng suppler s proft, mproves. To understand ths, note that, from the perspectve of the supplers, TSP s less flexble than PBC. In other words, under TSP, every qualfed suppler sets ther qualty level equal to the threshold Q. As a result, as δ ncreases, the expected delvered qualty ncreases at a slower rate under TSP than under PBC. However, under PBC, the supplers can adjust ther delvered qualty wth greater flexblty,.e., the q wll vary across the supplers, whch leads to a greater dfferentaton between the supplers and a larger value of CS [2] (h) C S [1] (h).thus,as δ ncreases, the wnnng suppler s proft ncreases more quckly under PBC than under TSP. In summary, a larger δ = c H c L mples a greater degree of dfferentaton between potental supplers. Our results thus ndcate that PBC ncreasngly outperforms TSP, on all performance measures, as ths level of dfferentaton ncreases. Ths s an mportant nsght for buyer s when desgnng ther sourcng processes. Whle n some cases, for practcal reasons, the buyer may prefer TSP over PBC despte the lower cost assocated wth PBC, the buyer must be careful when selectng TSP f there s a hgh level of varaton n the potental supplers unt costs. V. PBC: ASYMMETRIC PENALTY/REWARD FUNCTION The above analyss assumes a symmetrc and lnear penalty/reward functon,.e., H(q ; Q) =h(q Q), where h s the penalty/reward rate. We next consder anasymmetrc penalty/reward functon of the form: H(q ; Q) =h 1 (q Q) f q Q and H(q ; Q) =h 2 (q Q) f q <Q, where h 2 >h 1. Thus, f q >Q, then H(q ; Q) > 0 represents a reward pad to the suppler. If q <Q, then H(q ; Q) < 0 represents a penalty charged to the suppler. The buyer must set h 1 and h 2,aswell as Q. For ths penalty/reward functon, the optmal qualty level and the cost for suppler are presented n the techncal appendx. Characterzng the wnnng suppler, and determnng the optmal penalty and reward rates for the buyer, are challengng. In addton, unlke for the symmetrc penalty/reward functon case, the qualty threshold Q does not cancel out of the buyer s

13 294 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 63, NO. 3, AUGUST 2016 TABLE X COMPARISON OF ASYMMETRIC AND SYMMETRIC PENALTY/REWARD FUNCTIONS % C B % C T % E [q ] % h Average 0.24% 0.37% 2.11% 1.39% Mnmum 0.00% 0.00% 0.00% 20.00% Maxmum 2.06% 3.79% 11.82% 10.00% cost functon, and thus the buyer has a three-dmensonal (3-D) strategy space (h 1,h 2,Q). Thus, analytcal results are not possble. We therefore conducted a set of numercal experments, followng the methodology and expermental desgn descrbed n Secton III-A3, n order to understand how the results for the asymmetrc penalty/reward functon wll dffer from those wth the symmetrc penalty/reward functon, as presented n Secton III. To determne the optmal h 1, h 2, and Q to mnmze the buyer s expected cost C B (h 1,h 2,Q), we performed an exhaustve search. Specfcally, we consdered h 1 [0,w], n ncrements of 0.05, h 2 [h 1,w], n ncrements of 0.05, and 20 values of Q, equally spaced between q L and q H. The results are shown n Table X. In the table, % C B represents the percentage dfference between the buyer s expected cost under the symmetrc and asymmetrc cases,.e., % C B =. Snce the symmetrc penalty/reward functon s a specal case of the asymmetrc penalty/reward functon, the buyer s optmal cost wll always be hgher for the symmetrc case. However, due to the use of smulaton to estmate the cost functons, as well as the lmtatons of the 3-D search process descrbed above, n some cases we have that C B (h ) s slghtly (no more than 2.06%) less than C B (h 1,h 2,Q ). Thus, we report the absolute value of the percent dfference. The C B (h ) C B (h 1,h 2,Q ) C B (h 1,h 2,Q ) columns labelled % C T and % E[q ] consder the dfference n the expected total system cost and the expected delvered qualty, and are calculated analogously to % C B. The fnal column compares the penalty/reward rates. Specfcally, we frst compute % h 1 = h h 1 h 1 and % h 2 = h h 2 h 2. We then let % h = % h 1 f E[q ] >Q and % h = % h 2 f E[q ] Q. Thus, we compare the symmetrc penalty/reward rate to the relevant rate for the asymmetrc case,.e., to the rate that s appled n the optmal soluton for the asymmetrc case. When E[q ] >Q (E[q ] Q ), the reward (penalty) wll be appled. Table X ndcates that, although the buyer s expected cost wll be lower under the asymmetrc penalty/reward functon, the buyer wll not see a sgnfcant loss f he chooses to mplement the smpler symmetrc functon. Our numercal results ndcate that the buyer s expected cost and realzed qualty, as well as the expected system cost, do not sgnfcantly dffer between the two types of functon. Fnally, n the techncal appendx, we provde some ntuton for the observaton that the performance of the symmetrc and asymmetrc penalty/reward functons are qute smlar. VI. CONCLUSION AND MANAGERIAL INSIGHTS In ths paper, we compare the performance of two mechansms that are approprate, and commonly used n practce, when the buyer has uncertanty regardng the potental supplers costs and qualty capabltes: the TSP and PBC. We do so usng a model settng that captures several complextes not generally consdered n the exstng lterature. Specfcally, our model captures the fact that supplers wll generally set ther qualty levels endogenously, n response to the ncentves provded by the sourcng process desgn. In addton, our model captures the realty that potental supplers wll often have dfferng qualty capabltes, whch constran the qualty levels they can offer to the buyer. Fnally, our model uses a heterogeneous producton cost functon to capture how the supplers producton costs vary wth qualty. We fnd that PBC always outperforms TSP from the perspectve of the buyer. Ths result s due, n part, to the fact that the supplers qualty levels are endogenous, set optmally n response to the buyer s sourcng process desgn. PBC provdes the supplers wth the flexblty to choose the qualty level to mnmze ther own costs, gven ther qualty capablty and the penalty/reward rate assessed by the supplers. TSP, on the other hand, mposes a mnmum qualty level across all of the potental supplers. Thus, TSP does not provde the same level of flexblty for the supplers to adjust ther qualty levels to ther own cost structure. In addton, PBC allows all potental supplers to compete n the bddng, whle TSP allows only the subset of qualfed supplers to bd. Ths ncreased competton n the bddng stage also favors PBC from the perspectve of the buyer. Whle PBC can be shown to outperform TSP, n practce TSP s more wdely used, for a number of practcal, but hard to quantfy, reasons. Thus, t s useful to understand the magntude of the performance gap between TSP and PBC n order to dentfy condtons under whch buyers do not lose much by choosng TSP. We fnd that PBC s most benefcal to the buyer when the spread n the potental supplers costs s large and when the unt warranty cost s large. Thus, a buyer who faces sgnfcant uncertanty regardng the supplers costs, and for whom mantanng a hgh level of qualty s crtcal, should gve extra consderaton n mplementng PBC, despte the practcal benefts of TSP. We also studed how the expected delvered qualty dffers under TSP and PBC. We fnd that the delvered qualty s generally hgher under PBC than under TSP and that the gap between the qualty levels s largest when the spread n the potental supplers costs s large and when the number of potental supplers and unt warranty cost are small. We fnd that the expected delvered qualty wll ncrease towards the system optmal qualty level as the number of potental supplers ncreases. Fnally, whle the buyer always prefers to source through PBC, we fnd that the wnnng suppler s generally better off under TSP. The suppler s preference for TSP s strongest when the number of supplers s large, the spread n the supplers costs s small, and the unt warranty cost s large. Gven that the buyer prefers PBC to TSP, we also consder how the optmal penalty/reward rate vares wth the problem parameters. We fnd that the optmal rate s always less than the unt warranty cost. In addton, when the supplers are fully capable, the optmal penalty/reward rate s proportonal to the unt warranty cost. Thus, PBC wth a symmetrc and lnear

14 JIN AND RYAN: CAN CONTRACTS REPLACE QUALIFICATION IN A SOURCING PROCESS WITH COMPETITIVE SUPPLIERS 295 penalty/reward functon s a form of warranty cost sharng. The optmal penalty/reward rate s largest when the number of potental supplers s large and the unt warranty cost s large. The former pont ndcates that more competton between the supplers enables the buyer to pass more of the warranty costs onto the wnnng suppler. The latter pont mples that when the warranty cost s large, the buyer must provde more ncentve to the supplers to provde hgh-qualty products. These results provde gudance regardng how to set the penalty/reward rate for frms that choose to source usng the PBC approach. Fnally, we note some future research drectons. It would be useful to consder a settng n whch the supplers have lmted capacty and thus the buyer may need to contract wth multple supplers. In addton, there may be some settngs n whch t would not be feasble to allow all supplers to bd, regardless of ther qualty capabltes, e.g., there may be a hard mnmum on the qualty level that must be acheved by the supplers. In such settngs, a pure PBC approach would not be approprate. However, a hybrd approach, combnng frst a qualfcaton process, perhaps wth the qualty threshold set to the mnmum acceptable level, followed by a PBC mechansm, may be a better approach. The desgn and performance of such hybrd mechansms s an mportant topc for future research. REFERENCES [1] R. McIvor, Outsourcng: Insghts from the telecommuncatons ndustry, Supply Chan Manage., Int. J., vol. 8, no. 4, pp , [2] T. I. Tunca, D. J. Wu, and F. V. Zhong, An emprcal analyss of prce, qualty and ncumbency n servce procurement auctons, Georga Inst. Technol., Atlanta, Georga, Workng paper, [3] W. J. Elmaghraby, Auctons wthn e-sourcng events, Prod. Oper. Manage., vol. 16, no. 4, pp , [4] R. Engelbrecht-Wggans, E. 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Mnnesota, Mnneapols, MN, USA, Workng paper, [11] W. E. Brown and L. D. Ray, Electronc reverse auctons n the Federal Government, MBA Prof. Report, Naval Postgrad. School, Monterey, CA, USA, Dec [12] J. Lampng, Aganst Better Judgement: Prequalfcaton n Procurement Auctons, [Onlne]. Avalable: Economcs/news/fall07-newsletter/fall07-lampng.html [13] M. Rordan, Contractng wth qualfed supplers, Int. Econ. Rev., vol. 37, pp , [14] J. Shachat and J. T. Swarthout, Procurement auctons for dfferentated goods, Decson Anal., vol. 7, no. 1, pp. 6 22, [15] F. Branco, The desgn of multdmensonal auctons, RANDJ.Econ., vol. 28, pp , [16] Y. Che, Desgn competton through multdmensonal auctons, RAND J. Econ., vol. 24, no. 4, pp , [17] D. J. Reyners and C. S. Tapero, The delvery and control of qualty n suppler-producer contracts, Manage. Sc.,vol.41,no.10,pp , [18] G. Tagaras and H. L. Lee, Economc models for vendor evaluaton wth qualty cost analyss, Manage. Sc.,vol.42,no.11,pp ,1996. [19] V. Choudhary, A. Ghose, T. Mukhopadhyay, and U. Rajan, Personalzed prcng and qualty dfferentaton, Manage. Sc.,vol.51,no.7,pp , [20] P. Klemperer, A survey of aucton theory, n Auctons: Theory and Practce. Prnceton, NJ, USA: Prnceton Unv. Press, 2004, pp Yue Jn receved the doctorate degree n ndustral engneerng and operatons research from the Unversty of Massachusetts Amherst, Amherst, MA, USA. Snce 2008, she has been workng as a Researcher wth the Advanced Analytcs Group at Bell Labs Ireland, Dubln, Ireland, where she was a Post-Doctoral Researcher. Her man research nterests nclude the optmzaton and coordnaton n supply chan and servce operatons management, suppler management, manpower management n servce systems, smart data prcng, and resource management n Cloud systems. She s constantly nvolved n nternal and external research collaboratons. She has also made major contrbutons n a number of projects that address practcal problems for Noka busness unts, ncludng the reconfguraton of regonal supply chans, the automaton of spot carrer selecton, and engneer schedulng n customer delvery unts. Jennfer K. Ryan receved the B.A. degree n mathematcs and the socal scences from Dartmouth College, Hanover, NH, USA, n 1990, and the M.S. and Ph.D. degrees from the Department of Industral Engneerng and Management Scences, Northwestern Unversty, Evanston, IL, USA, n She s a Professor of Supply Chan Management wth the College of Busness Admnstraton, Unversty of Nebraska Lncoln, Lncoln, NE, USA. Pror to jonng the Unversty of Nebraska, she served as a Faculty Member n the Department of Industral and Systems Engneerng, Rensselaer Polytechnc Insttute, Troy, NY, USA; the School of Busness, Unversty College Dubln, Dubln, Ireland; the Mendoza College of Busness, Unversty of Notre Dame, Notre Dame, IN, USA; and the School of Industral Engneerng, Purdue Unversty, West Lafayette, IN. Her research nterests nclude supply chan management. She teaches courses n operatons and supply chan management. Dr. Ryan has receved several Natonal Scence Foundaton (NSF) grants to support her research, ncludng an NSF CAREER grant. She currently serves as adepartment Edtor foriie Transactons, and as an Assocate Edtor for Naval Research Logstcs and OMEGA.