Satisfying Warranty Claims on an Obsolete Product

Transcription

1 Dwayne Cole College of Business Adminisraion Florida A&M Universiy Tallahassee, FL 33 Burak Kazaz Marin J. Whiman School of Managemen Syracuse Universiy Syracuse, NY 344 Sco Webser W.P. Carey School of Business Arizona Sae Universiy Tempe, AZ 8587 ABSTRACT We invesigae a problem faced by a durable-goods manufacurer of a produc ha is no longer manufacured bu sill under warrany. A supplier announces ha a componen of he produc will be phased ou and specifies a deadline for he final order. A common response in radiional pracice is o place a final order sufficien o cover fuure warrany claims. We analyze and compare his policy wih a policy ha uses a rade-in program o supplemen he final order quaniy. KEYWORDS: Warrany, rade-in policies, susainabiliy INTRODUCTION We invesigae a problem faced by a durable-goods manufacurer of a produc ha is no longer manufacured bu sill under warrany. A supplier announces ha a componen of he produc will be phased ou and specifies a deadline for he final order. The manufacurer projecs he componen needs for he produc under warrany and considers a wo-sage decision problem: () he size of he final order and, in he even ha he final order is less han acual requiremens, () he design of a rade-in program for componen harvesing. The imporance and prevalence of his problem have increased over ime due o shrinking produc life-cycles and growh in ousourcing. These rends are especially pronounced in he compuer indusry where he high pace of change and echnical challenges favor supply chains of independen firms wih specialized experise (e.g., AMD and Inel for processors, Seagae and Wesern Digial and for hard drives, Cisco and D-Link for rouers, Flexronics and Selecron for assembly).

2 We consider he seing where he componen phase-ou announcemen (CPOA) occurs afer he manufacurer has disconinued manufacuring and sales of he paren produc. The paricular componen conribues significanly o he value of he produc and is no easily or inexpensively obained from alernaive suppliers (e.g., highly engineered and expensive componen). These feaures elevae he imporance of managerial aenion on an effecive response o he CPOA. We invesigae how a firm s opimal final order quaniy and rade-in program decisions are influenced by indusry and marke characerisics. Our main conribuion lies in wo observaions ha come from our analysis. Firs, a rade-in program has poenial o significanly reduce a firm s warrany liabiliy. Second, here are wo key indicaors ha savings from a rade-in program will be significan. One indicaor is he difference beween he componen cos and he marginal cos of he firs uni acquired via radein. A second indicaor is he expeced fracion of producs under warrany ha will fail. Boh of hese values should no be difficul for a firm o esimae. ELEMENTS OF THE COMPONENT PHASE-OUT ANNOUNCEMENT PROBLEM As a new generaion of a componen is inroduced and he volume of he previous generaion declines, a supplier evenually ceases o supply he older generaion componen and announces a ime-line for phase ou. While i is possible ha a CPOA may occur when he manufacurer is sill producing a produc wih he componen, we limi consideraion o he case where he produc is no longer being manufacured (as is consisen wih CPOA iming examples described o us by hose in indusry). Thus, he final componen purchase decision is driven by warrany obligaion consideraions. Durable-goods manufacurers commonly offer a limied-ime warrany o consumers. The CPOA problem can be viewed as a wo-sage decision problem. The firs-sage decision is he number of componens in he final order. Afer he final order is placed, componen demand is realized over ime. The second-sage decision, if necessary, is he price discoun o be offered on a rade-in. We assume ha he firm has access o cusomer-specific warrany daa. In hese seings, he rade-in offer can be argeed o specific cusomers based on produc age and ime remaining under warrany. We also assume ha warrany claim demand and rade-in reurn volume are known wih cerainy. Through his simplifying assumpion we are able o gain some insigh ino wha drives he value of a rade-in program. We leave consideraion of uncerainy for fuure research. LITERATURE REVIEW Mos consumer durables come wih eiher a pro-raa refund or a free repair/replacemen warrany policy (Blischke and Murhy 99). Murhy e al. (4) provide a comprehensive review of various issues associaed wih warrany managemen. Warrany claims are driven by he warrany populaion, usage characerisics, produc reliabiliy, and warrany erms. Seiz (7) repors ha he use of recovered componens o saisfy warrany claims is a common pracice in he auomobile and home appliance indusries. Cisco began using reurns o suppor warrany claims in 8. The iniiaive increased he recovered value from reurns by nine-fold, from 5% o 45% (Nidumolu e al. 9). There are hree sreams of research relaed o our wo-sage problem of how a manufacurer

3 deermines he size of he final order and designs he rade-in program for componen harvesing. One of he sreams relaes o he final order quaniy problem, which is he firssage decision in our model. Foruin (98) inroduces a model wherein he machines remaining operaing life is divided ino discree inervals and he number of componens ha fail in each inerval is random. He proposes a mehod for esimaing he componen sock-ou probabiliy as a funcion of he final order quaniy. Teuner and Hansveld (998) exend he single componen seing of he previous papers o a muli-componen ordering problem. They show he muli-componen problem can be decomposed ino independen single-componen ordering problems. Bradley and Guerrero (9) also consider he muli-componen final buy problem, hough in conras o Teuner and Hansveld (998) who assume all componens are phased ou a he same momen in ime, componens are phased ou gradually over ime. Teuner and Foruin (999) develop a single-sage dynamic program o deermine he opimal final order quaniy for a single componen. Their model allows for he possibiliy of harvesing componens from reurns as a source for spare-pars. However, he firm passively acceps used-produc reurns. This differs from our model where heir firm acively manages reurn volumes hrough he seing of rade-in discouns over ime. In sum, our work exends he lieraure on he final order quaniy problem by supplemening he final order quaniy wih produc acquired via a rade-in program he second sage in our model. A second relaed sream of research addresses he use of reurns as a source for spare pars. The lieraure on his opic is vas (e.g., see Kennedy e al. for a review). Wihin his lieraure, a number of researchers have sudied he problem of managing componen pars afer he paren produc has reached he end of is sales life-cycle. Minner and Kleber () examine a seing where he firm produces a new componen and remanufacures used componens o mee a deerminisic demand for spare pars. The auhors develop a dynamic invenory framework and use opimal conrol echniques o deermine an opimal producion and recovery sraegy for a firm ha passively acceps used producs. Spengler and Schröer (3) also develop a dynamic model ha inegraes componen harvesing. They use he model o sudy flows of new produc sales, spare pars demands, produc reurns, and recovery raes. Their model focuses on he behavior of he spare pars managemen sysem. Acquisiion cos and reurns flows are exogenous in heir model, and are endogenous in our model; hey are influenced by he firm s choices. Inderfurh and Mukherjee (8) develop a decision model where he firm mus mee a dynamic demand for spare-pars during he phase-ou period. Our model resembles ha of Inderfurh and Mukherjee (8) in he sense ha he firm chooses he final order quaniy and remanufacures componens from reurned producs o mee dynamic demand for spare pars. However, our model improves heir model in wo ways: () As opposed o an exogenous flow of reurns, we assume ha he firm proacively acquires used producs from is insall base; he firm ses rade-in discouns o influence he iming and quaniy of he reurn flow. () We accoun for he impac of reurns of produc under warrany on fuure warrany claims. A hird sream of relaed lieraure examines he relaionships beween new produc prices, rade-in rebaes, produc reurn volumes, and new produc purchases. Ray e al. (5) examine how a rade-in program for a produc ha is remanufacured can be used as a pricediscriminaion mechanism o increase profis. Bruce e al. (6) sudy rade-in programs for expensive durables purchased wih he aid of a loan (e.g., auomobiles). They examine he relaionship beween he magniude of he rade-in discoun and he durabiliy of he produc. Rao e al. (9) sudy he value of rade-in programs for producs in which used produc prices are negaively affeced due o informaion asymmery (e.g., a posiive probabiliy of buying a lemon ). A key difference beween hese papers and our work is ha here is no demand for

4 used componens ha mus be me, as is he case in our problem. The papers ha are mos closely relaed o our second-sage problem address he procuremen of end-of-use producs from he insall base. Guide and Van Wassenhove () consider a firm ha ses a buyback price o mach he supply of cores wih he demand for remanufacured componens. Bakal and Akcali (6) also consider a buyback price o mach supply wih demand. Galbreh and Blackburn (6) sudy he ineracion beween procuremen lo size and he firm s soring policies. Zikopoulos and Tagaras (7) consider he problem of ordering used producs from muliple supply sources wih correlaed recovery yield. Each of hese papers considers a singlesage decision environmen. Our work differs from his lieraure by including a firs-sage decision on he final order quaniy. In addiion, his lieraure focuses on reurns o suppor he remanufacured produc demand, no warrany claims. As a consequence, he models do no capure he negaive correlaion beween reurn volume and fuure demand. In summary, our sage-one problem is similar o final purchase quaniy problems in he lieraure. A key difference is he consideraion of a rade-in program ha leads o a wo-sage decision problem. The final purchase quaniy lieraure has no considered he design of radein programs as a mechanism for acquiring used componens. Our sage-wo problem is similar o he problem of designing a rade-in program ha is considered in he markeing and remanufacuring lieraure. As in his lieraure, we need o model how feaures of he rade-in program and oher facors influence reurn volume. However, as noed above, a key difference is ha we need o capure how reurn volumes influence fuure warrany claims. MODELS AND ANALYSES A firm has received a CPOA for a componen from a sole-source supplier and mus deermine he final order quaniy q ha will be received a ime =. The purchase cos per uni is c, he invenory holding cos rae is h, and he warrany claim service cos per uni is c w (e.g., disassembly, componen replacemen, reassembly, es, and shipping). The difference beween he firm s discoun rae and he rae of inflaion in operaing coss and margin is r. The las warrany expires a ime = (i.e., he uni of ime is seleced so as o normalize he warrany liabiliy horizon o one period). The componen demand rae a ime (due o warrany claims) is d(), he cumulaive demand hrough period is D(), i.e., D() = d x dx, () and he remaining warrany demand is D, i.e., D D D. () The componen demand rae is ne of any passive reurns of produc conaining a working componen. We assume deerminisic demand and focus on idenifying he drivers of performance in his seing. We le T (q ) denoe he ime ha componen invenory from he final order reaches zero, or he end of he warrany horizon, whichever is smaller, i.e.,

5 T (q ) = min min D q, The oal cos o service warrany claims is T q r w C q cq e h q D c d dc T q (3) where C () is he cos of saisfying warrany claims over ime inerval [, ] given ha he final order quaniy runs ou a ime. The firs erm in (3) is he componen purchase cos of he final order of q unis. The second erm in (3) is he invenory holding and warrany claim servicing cos, of which he parenheical erm in he inegrand is he invenory a ime ha is assessed a holding cos rae h. Second-Sage Trade-in Policy A firm offering a rade-in program specifies he discoun off he purchase price of a new model if he cusomer reurns he old model. The rade-in credi is offered only o cusomers wih produc under warrany. Conceivably a firm could offer he rade-in discoun o a cusomer wih a produc ha is no longer under warrany. While such a cusomer migh be willing o rade-in for a lower discoun, he acic of offering a rade-in discoun for produc no under warrany has wo drawbacks. Firs, here is a risk ha he componen in he reurned produc will be fauly. This risk is low for produc under warrany because, if i was fauly, he firm would have likely already received a claim. Second, he reurn of a produc under warrany reduces he firm s warrany liabiliy associaed wih he obsolee componen (i.e., he produc conaining he obsolee componen is raded in for a new model of he produc). The firm offers a ime-sensiive rade-in o some fracion of warrany holders in each period so as o mach he rae of supply wih he rae of demand. We refer o his policy as a maching rade-in policy. A maching rade-in policy is viable in seings where he firm has access o cusomer-specific warrany daa (i.e., cusomer conac informaion for producs under warrany). Firms ha sell direcly o cusomers are likely o have his level of deail in warrany daa. Before analyzing he rade-in policy, we describe how we model relaionships beween he rade-in discoun, rade-in volume, and rade-in cos. We begin wih wo assumpions ha allow us o define he fracion of cusomers who accep a rade-in offer as a funcion of he rade-in discoun: Assumpion (A). A cusomer receiving a rade-in offer receives a single ake-i-or-leave-i offer and acceps he offer if consumer surplus is posiive. Assumpion (A). The valuaion of he new model in exchange for he old model under warrany, denoed V, is independen of ime and is uniformly disribued and ordered by age of ownership wih range normalized o [, ]. An alernaive o A is o allow muliple rade-in offers o he same cusomer over ime. However, his promoes sraegic behavior ha grealy complicaes he analysis and may work agains he ineres of he firm (e.g., cusomer holds ou for a beer offer). Uniformly disribued

6 valuaion (A) is common in he lieraure (e.g., Mussa and Rosen 978, Purohi and Saelin 994) and resuls in reurn volume ha is linear in price. A also specifies ha a consumer who recenly purchased he produc will have a higher valuaion han a consumer who has owned he produc for a longer period of ime, i.e., cusomers wih older produc will accep a lower rade-in offer han cusomers wih newer produc. A firm offering a rade-in program mus selec he rade-in discoun and he rae a which cusomers are exposed o he rade-in offer (i.e., he rade-in offer rae), boh of which may vary wih ime. The rade-in discoun is c () and he rade-in offer rae is () (e.g., () is he number of cusomers receiving a rade-in offer in period ). The conribuion margin of a new model of he produc is m and he variable cos is c n, i.e., he new model selling price is p n = c n + m. Thus, he rade-in price is c n + m c () and, by A and A, he fracion of cusomers who accep he rade-in offer from among hose who receive i is P V c m c c m c c p n n n (4) Rewriing (4) in erms of he rade-in credi, c p. (5) n We see ha he rade-in price is he complemen of he accepance rae (), i.e., p c. n Noe ha he new model selling price should be more han he maximum valuaion of a rade-in exchange, i.e., p c m. (6) n n Condiion (6) reflecs he pracical realiy ha cusomers are unlikely o rade in a produc under warrany unless here is a rade-in discoun. For example, p n < would imply ha fracion p n of cusomers would be willing o reurn heir produc (ha is under warrany and funcional) and pay full price for he new model. The value of p n is a measure of rade-in resisance. This value is he minimum rade-in discoun ha is required before any cusomers will be willing o reurn heir uni. The larger he value of p n, he greaer he marke resisance o a rade-in offer, and herefore, he firm is pressured o increase is rade-in offer wih a higher value of c (). In (4), we see ha he difference beween he rade-in credi, c (), and he rade-in resisance, p n, gives he fracion of hose receiving he rade-in offer who accep he offer. Thus, he produc reurn rae s() is s c p n. In general, he specificaion of rade-in acquisiion cos can be challenging due o he effec of cannibalizaion. We model his effec hrough parameer. The inerpreaion of is relaively

7 sraighforward when he difference beween he firm s discoun rae and he rae of inflaion (in coss and margin) is zero (i.e., r = ): is he fracion of rade-in cusomers who would have purchased he new model a full price in he fuure if he rade-in program was no offered, or repea purchase rae. If r >, hen he value of he full margin in he fuure is lower due o he ime-value-of-money. All ime-value-of-money effecs and, more generally, all cannibalizaion effecs are incorporaed ino he value of parameer. Indeed, i is possible for o be negaive in some seings, e.g., by reducing secondary marke supply and hus cannibalizaion of new produc sales. Accordingly, he cos of a componen obained hrough a rade-in is he reducion in margin hrough a rade-in sale, which is where c c m (7) = ( )m (p n ). We refer o he value of as he rade-in poenial, which can be inerpreed as he difference beween he gain from locking-in disloyal cusomers via he rade-in offer, ( )m, and he marke resisance o a rade-in offer, p n. More generally, is he marginal profi on rade-in volume a he origin. For example, if >, hen rade-in poenial is posiive and rade-ins are profiable up o accepance rae (). On he oher hand, if <, hen rade-in poenial is negaive and rade-ins are cosly from he ge-go. Wihou loss of generaliy, we define he produc uni such ha he warrany populaion a ime zero is. In he absence of a rade-in program, he rae a which warranies expire a ime is given by n(), which is known wih cerainy (e.g., obained from company records). While n() can conceivably ake any funcional form, in he ineres of parsimony, we limi consideraion o he following form ha depends on a single parameer, n [, ]: n, n() =, n, i.e., warranies expire a rae n over ime inerval [, ) and n warranies expire a ime =. Figure illusraes hree alernaive warrany populaion funcions over ime in he absence of a rade-in program. Figure. Warrany populaion over ime a differen warrany expiraion raes n =,.5,. The case of n = reflecs a seing where monhly sales of he produc is relaively fla near he end of is life-cycle (e.g., warrany expires x monhs afer purchase). The cases of n < reflec seings where monhly sales of he produc is relaively fla near he end of is life-cycle excep n = n =.5 n =

8 for a jump in sales a he end hrough clearance pricing. The smaller he value of n, he larger he clearance sale volume relaive o volume prior o clearance discouning. Wih he warrany populaion characerized, he remaining conribuor o demand for componens o service warrany claims is he componen failure rae. We assume ha he failure rae funcion is consan a value. This assumpion is common in he lieraure (e.g., Murhy and Rodin 99, Zhou e al. 9). Assumpion 3 (A3). The componen failure rae is consan. Due o A3, in he absence of a rade-in program, he demand rae is d() = ( n) and demand funcions () and () become (8) D() = nxdx.5n.5 D D D n. In boh expressions, he erm in brackes reflecs he degree o which cumulaive demand and remaining demand are reduced when he warrany expiraion rae n is greaer han. We noe ha a reurned uni may conain some value beyond he componen ha has been phased ou. This value can be incorporaed ino our model as an addiional parameer ha does no change he srucure of he model or he resuls. In he ineres of parsimony, we do no inroduce a separae parameer; is value, if significan, is included in parameer m (e.g., if margin is m and savings generaed from oher componens in a reurned produc is s, hen m = m + s/( )). Maching Trade-In Policy The firm ses he rade-in credi c () and he rade-in offer rae () so ha componen supply maches componen demand over he remainder of he warrany horizon, i.e., s d (9) where () = c () is he rade-in accepance rae among hose cusomers exposed o he rade-in offer a ime (see (4)), or he rade-in fracion. Noe ha () mus be a valid fracion, i.e., () [, ], () and ha a cusomer receives a rade-in offer no more han once (see A), i.e., d. () The firm s choice of cusomers who will receive he rade-in offer over ime is influenced by A. Recall ha A implies ha cusomers wih a soon-o-expire warrany are more likely o accep a

9 rade-in offer han cusomers wih a more disan warrany expiraion dae. In recogniion of A, he firm sends he rade-in offer o cusomers in order of warrany expiraion dae. From (7), he componen acquisiion cos rae for he maching rade-in policy is c = c m. () n Thus, he cos of he maching rade-in policy is r, w C e c d d where funcions () and () saisfy (9) (). We wish o find he funcion () ha minimizes he second-sage cos subjec o he relevan consrains. The problem is r C min e cw d d: d, d,. The following proposiion characerizes he opimal soluion o he preceding problem. Proposiion. If.5r r e e, (3) hen he opimal rade-in fracion is.5r.5r e e, (4).5r he opimal rade-in offer rae is.5r.5re e.5r, e he demand (and supply rae) is d() = e, he oal number of unis raded in is q e, and he opimal second-sage cos is.5r.5r e e C = c w..5r r Table shows he maximum value of he failure rae ha saisfies condiion (3) for various values of he ne discoun rae r. Recall ha is he failure rae over he duraion of he second sage. For example, if he second-sage duraion is five years and he annual ne discoun rae is 5%, hen condiion (3) holds for a failure rae up o 5% per year (i.e., divide he figures in he row wih r = 5% by 5). In he compuer indusry ha moivaes his work, componen failure

10 raes end o be low (e.g., less han %) and he warrany duraion is on he order of hree o five years. In hese seings, he condiion given in (3) is highly likely o hold. r maximum value of saisfying (3) % % 549% % 33% 5% 49% 5% 88% % 3% % 79% Table. Upper limi on componen failure rae for differen ne discoun raes r. The following corollary gives he opimal soluion for he special case of r =. Corollary. If r =, hen he opimal rade-in fracion is e, he opimal rade-in offer rae is e, (5) e he demand (and supply rae) is d() = e, he oal number of unis raded in is q e, and he opimal cos is e e cw. (6) C = We see ha he opimal soluion has a simpler srucure when r =. In paricular, he opimal rade-in fracion () is independen of ime. This means ha he opimal rade-in discoun says consan over he warrany horizon, i.e., c p e, (7) n (obained by subsiuing () ino (5)). The erm in he parenheses in he righ-hand side of (5) is he fracion of he populaion exposed o he rade-in offer (i.e., e is he warrany populaion over ime), which also says consan over he warrany horizon. In conras, a r >, we see ha he opimal rade-in fracion () is increasing in ime (see (4)), and consequenly, he rade-in discoun is increasing in ime (e.g., he firm offers higher discouns laer in he horizon, which are less cosly for he firm due o he posiive discoun rae). Similarly, he fracion of he populaion exposed o he rade-in offer is decreasing over ime. Firs-Sage Problem Under a policy where he final order quaniy is se o mach oal demand, he oal cos is

11 .5 w r C cq e h q n c n d (8) and he final order quaniy is q D n.5 (see (3) and (8)). We refer o his policy, which is idenified by superscrip, as he benchmark policy. The opimal oal cos under he maching rade-in program is where Tq r C cq e hq D cwd dc T q Tq r arg min w q q cq e h q D c d d C T q. Clearly, C C wih equaliy if and only if he opimal second-sage rade-in quaniy is zero (i.e., C C q ). The following proposiion idenifies a simple indicaor of when i is profiable o supplemen he final order quaniy wih a rade-in program. r Proposiion. If c e, hen C C. A rade-in program clearly adds value when rade-in poenial () is nonnegaive (because c > ). In he even ha rade-in poenial is negaive, we can be assured ha rade-in programs save money if he presen value of he magniude of he rade-in poenial (discouned from he end of he warrany horizon) is less han he purchase cos per uni. Figure illusraes how he final order quaniy under he maching rade-in policy differs from he benchmark final order quaniy. Figure 3 illusraes he percen savings due o he maching rade-in policy relaive o he benchmark. Noe ha in figures and 3, a maching rade-in program is no used when n = and = -. (i.e., q q ).

12 Figure. Opimal final order quaniies under he maching rade-in policy as a percen of he benchmark final order quaniy. The rade-in poenial is =. in he lef plo and = in he righ plo. The oher parameer values, which are common o boh plos are c =., c w =., h =.7, r =. Figure 3. Percen savings in oal cos when he benchmark policy is replaced wih he maching rade-in policy. The rade-in poenial is =. in he lef plo and = in he righ plo. The oher parameer values, which are common o boh plos are c =., c w =., h =.7, r =. The lef plos of figures and 3 reflec he seing where c = -. In his seing, he rade-in acquisiion cos wih he rade-in credi is so low ha he cos of a single reurned uni is he same as he cos from he vendor. This seing is exremely unfavorable o a rade-in program and may rarely arise in pracice. Neverheless, even in his unfavorable seing, he rade-in policy is less expensive han he benchmark when n =. Figure shows ha he rade-in policy explois he flexibiliy of dividing he source of componens beween he final order quaniy and hose ha come from rade-ins. The fracion acquired from he vendor via he final order quaniy

13 increases as he failure rae increases. This is because he rade-in cos is sensiive o oal volume acquired, so a higher fracion of oal warrany demand is shifed o he final order quaniy. Figure 3 also illusraes ha he percenage savings due o he rade-in policy is significan when he aggregae failure rae is small, and diminishes as he failure rae increases (i.e., as he opimal final order quaniy covers an increasing fracion of oal demand). However, while percenage savings is decreasing in he failure rae, he cos of he benchmark policy is increasing in he failure rae. In paricular, he oal cos of he benchmark policy is proporional o he failure rae (see (8)). Depending on parameer values, he absolue savings may be eiher increasing or decreasing in he failure rae. For he parameer values in Figure 3, for example, absolue savings is nondecreasing in failure rae excep for he maching rade-in policy a n = =, which is iniially increasing, hen decreasing. SUMMARY OF LESSONS FOR MANAGERS We have considered wo policies for a CPOA response. The radiional policy, which we refer o as he benchmark policy, is o place a final order ha is large enough o cover componen warrany demand over he remainder of he warrany horizon. An alernaive is o be less aggressive on he final order quaniy and, as componen invenory approaches zero, acquire addiional componens hrough a rade-in program offered o consumers wih produc under warrany. A rade-in offer is made o only a fracion of he warrany populaion in each period, wih he fracions adding o % by he end of he warrany horizon. The rade-in credi and he number of cusomers who receive he offer in a period are designed o achieve a reurn response ha is sufficien o cover he warrany demand in he upcoming period. A maching rade-in program is viable for companies wih relaively accurae and complee informaion on he warrany populaion (e.g., cusomer conac daa). Our analysis leads o wo main lessons for managers. Firs, rade-in programs dominae he benchmark policy and have poenial o significanly lower cos. Trade-in programs can be perceived as a new sourcing opion, and herefore, incorporaing hem ino he se of alernaives will no increase cos. Wha is more perinen for managers are indicaors ha a rade-in program will generae significan savings. The single mos imporan indicaor is rade-in poenial. The value of rade-in poenial is he difference beween wo values: () he margin from a rade-in ransacion, or new-produc margin imes he probabiliy of a cusomer no purchasing from he firm if no for he rade-in offer, and () he reducion in new produc price required o ge a leas one cusomer o paricipae in he rade-in program. If rade-in poenial is posiive, hen he cos of acquiring componens via rade-in is negaive (a leas a low volumes); he firm earns money from acquiring a componen via rade-in raher han spending money buying a componen from he vendor. Managemen should examine he difference beween rade-in poenial and he cos of buying he componen from he vendor. These values should no be difficul o esimae, and a large difference is a srong indicaor of high savings from a rade-in program. A secondary indicaor of high savings from a rade-in program is he fracion of he warrany populaion expeced o fail prior o warrany expiraion. This indicaor draws on a more suble value proposiion han he difference in cos beween a rade-in-sourced uni and a vendorsourced uni: a rade-in program reduces warrany claims. Each uni raded in reduces he warrany populaion of he obsolee produc, which ranslaes ino fewer warrany claims. And he higher he failure rae, he greaer he reducion in warrany claims.

14 The overarching lesson from our analysis is ha he use of a rade-in program o suppor warrany claims should be considered by managemen. I is surprising o see ha he use of rade-ins o suppor warrany claims is no discussed in he indusry or academic lieraure. One possible reason is ha, relaive o he benchmark policy, a rade-in program is more difficul o implemen, i.e., he firm has o design and communicae he program. However, i is also he case ha rade-in programs are no unusual in pracice. Indeed, here is a wide lieraure in economics and markeing on his opic, and his may hin a he possibiliy of an organizaional barrier. Trade-in programs are ypically designed and adminisered by he markeing group for markeing reasons (e.g., price discriminaion, spur sales when a new generaion of a produc is inroduced, ec.). Reducing he cos of warrany claim processing is ofen ouside he mission of a markeing deparmen. Our work can moivae firms o lower such organizaional barriers. REFERENCES Bakal, I.S., & Akcali, E. (6). Effecs of random yield in remanufacuring wih price-sensiive supply and demand. Producion and Operaions Managemen, 5, Blischke, W., & Murhy, D. (99). Produc warrany managemen--i: A axonomy for warrany policies. European Journal of Operaional Research, 6(), Bradley, J. R., & Guerrero, H.H. (9). Lifeime buy decisions wih muliple obsolee pars. Producion and Operaions Managemen, 8(), 4-6. Bruce, N., Desai, P., & Saelin, R. (6). Enabling he willing: Consumer rebaes for durable goods. Markeing Science, 5(4), Fleischmann, M., van Nunen, J., & Gräve, B. (3). Inegraing closed-loop supply chains and spare-pars managemen a IBM. Inerfaces, 33(6), Foruin, L. (98). The all-ime requiremen of spare pars for service afer sales heoreical analysis and pracical resuls. Inernaional Journal of Operaions & Producion Managemen, (), Galbreh, M.R., & Blackburn, J.D. (6). Opimal acquisiion and soring policies for remanufacuring. Producion and Operaions Managemen, 5, Guide, V.D.R. Jr., & Van Wassenhove, L.N. (). Business aspecs of closed-loop supply chain managemen. V.D.R. Guide Jr., L.N. Van Wassenhove, eds. Business Aspecs of Closed-loop Supply Chain Managemen. Pisburgh, PA: Carnegie Mellon Universiy Press, 7-4. Inderfurh, K., & Mukherjee, K. (8). Decision suppor for spare pars acquisiion in pos produc life cycle. Cenral European Journal of Operaions Research, 6(), 7-4. Kennedy, W., Paerson, J.W., & Fredendall, L. (). An overview of recen lieraure on spare pars invenories. Inernaional Journal of Producion Economics, 76(), -5. Linon, J.D. (8). Assessing he economic raionaliy of remanufacuring producs. Journal of Produc Innovaion Managemen, 5(3), Minner, S., & Kleber, R. (). Opimal conrol of producion and remanufacuring in a simple recovery model wih linear cos funcions. OR Spekrum, 3, 3-4. Murhy, D. N. P., & Rodin, E.Y. (99). A new warrany cosing model. Mahemaical and Compuer Modelling, 3(9), Murhy, D., Solem, O., & Roren, T. (4). Produc warrany logisics: Issues and challenges. European Journal of Operaional Research, 56(), -6. Mussa, M., & Rosen, S. (978). Monopoly and produc qualiy. Journal of Economic Theory, 8(), Nidumolu, R., Prahalad, C.K., & Rangaswami, M.R. (9). Why susainabiliy is now he key driver of innovaion. Harvard Business Review, 87(9), 56-64

15 Purohi, D., & Saelin, R. (994). Renals, sales, and buybacks: Managing secondary disribuion channels. Journal of Markeing Research, 3, Rao, R., Narasimhan, O., & John, G. (9). Undersanding he role of rade-ins in durable goods markes: Theory and evidence. Markeing Science, 8(5), Ray, S., Boyaci, T., & Aras, N. (5). Opimal prices and rade-in rebaes for durable, remanufacurable producs. Manufacuring Service Operaions Managemen, 7(3), 8-8. Seiz, M.A. (7). A criical assessmen of moives for produc recovery: The case of engine remanufacuring. Journal of Cleaner Producion, 5(-), Spengler, T., & Schröer, M. (3). Sraegic Managemen of Spare Pars in Closed-Loop Supply Chains - A Sysem Dynamics Approach. Inerfaces, 33(6) 7-7. Teuner, R.H., & Foruin, L. (998). End-of-life service: A case sudy. European Journal of Operaional Research, 7(), Teuner, R.H., & Foruin, L. (999). End-of-life service. Inernaional Journal of Producion Economics, 59(-3), Teuner, R.L., & Haneveld, W.K. (998). The final order problem. European Journal of Operaional Research, 7(), Zhou, Z., Li, Y., & Tang, K. (9). Dynamic pricing and warrany policies for producs wih fixed lifeime. European Journal of Operaional Research, 96(3) Zikopoulos, C., & Tagaras, G. (7). Impac of uncerainy in he qualiy of reurns on he profiabiliy of a single-period refurbishing operaion. European Journal of Operaional Research, 8(), 5-5. APPENDIX Maching Trade-in Policy Cos for q r r r e he h e Cqq ch cw r r r r h e e h e n cw e r n r r r r r r r.5r.5r e e.5r When r =, he expression reduces o r w.5.5 w.33. C q q c h c h n c h n e e c w Proof of Proposiion. For his proof, we will no use he normalizaion of = in order o clarify he expressions under a general second-sage saring ime, expressions ha will appear in our analysis of he firs-sage problem. We iniially develop he resuls for he special case of n =. We will hen show ha he opimal soluion for his special case remains valid when n >. c w

16 Assume n =. Le N() denoe he warrany populaion a ime. Due o n =, we have N() = for all <. The demand and supply rae over ime inerval [, ] is d() = s() =N() and he warrany populaion funcion is N() = sxdxdxdx Nxdx,,. We obain an explici expression for N() by aking he limi of a discree-ime model as he ime inerval goes o zero. Given ime inerval >, he failure rae per ime inerval is, and we have d s N s s N s N s N N N s and, in general, for ineger i N i. i Le = = i. Taking he limi as approaches zero, N() = lim / lim / ln lim / e e e for [, ]. Thus, d() = N() = e for [, ]. By subsiuing () = d()/() = e / (9) ino (5), we see ha c () is decreasing in (). Therefore, we replace he inequaliy consrain () wih equaliy, and he consrain can be wrien as e d d, () and he second-sage problem can be wrien as

17 r e min w :, r C e e c d d We solve he following equivalen problem min :,. r e e d d, () bu we iniially relax he bound consrain () [, ] (i.e., unresriced problem), i.e., we solve min : r e e d d. () Afer solving he unresriced problem (), we idenify condiions on parameer values ha ensure he soluion is also opimal for he resriced problem (). To simplify noaion, we emporarily le = (we accoun for he impac of > laer). We define which implies x e y() = dx, x y() = e e y' Thus, problem () is. (3) min ' :, y r e y d y y, (4) which can be solved using calculus of variaions mehods. Le y * () denoe he opimal funcion. We express y() in erms of parameer a, y * (), and difference funcion h(), i.e., y() = y * () + ah() and hus y() = y * () + ah(). For any feasible y(), we mus have h() = h() = (i.e., in order o saisfy he boundary condiions, which are clearly saisfied by he funcion y * ()). Le r r * g(a) = e y' d e y ' ah' d.

18 Noe ha r * g(a) = e y ' ah' h' d. (5) Applying inegraion by pars and recognizing uv = due o h() = h() =, we have * ry ' ah' r g(a) = e h d. * 3 * y ' ah' y " ah" Since y * () is opimal, we can conclude ha for any h() (wih h() = h() = ), we mus have g() =. This implies ha he inegrand of he above (wih a = ) mus be equal o zero a all values of, i.e., we mus have ry * y * r * * 3 * e r y ' y ' y " h ' " * r y " y * ' Solving he differenial equaion, we ge *.5r y ' Ae, where A is obained from he boundary condiion, i.e., Thus * *.5r y y d A e y * '.5r '.5r e e.5r.5r. (6) The funcion y * ' is a unique exremal (i.e., no oher funcion yields g() = ). Taking he derivaive of (5) and evaluaing a a =, we ge r * 3 g() = e y ' h' d. Since y * () > for all [, ], i follows ha g() >, and hus Subsiuing (6) ino (3) yields opimal rade-in fracion y * ' solves (4).

19 .5 r.5 r.5 r.5 r e e e e e e y'.5r.5r, and accouning for > yields opimal rade-in fracion for he unresriced problem.5 r.5 r e e.5r. (7) Recall ha he difference beween he unresriced problem and he resriced problem is ha he resriced problem includes he consrain () [, ]. From (7) we see ha () is increasing in and ( ) [, ]. Thus, if (), hen he opimal soluion o he unresriced problem is also opimal for he resriced problem. Noe ha.5r.5r e e.5r r.5r e e. (8) Thus, () [, ] if and only if (8) holds. The opimal rade-in offer rae is obained by subsiuing (7) ino (9) and solving for (), he rade-in quaniy is obained from q d, and he opimal cos is obained by subsiuing he opimal accepance rae and rade-in offer rae funcions ino he cos funcion: q.5r e e e.5r.5r e.5r.5r e e.5r r r C e cw. (9) In he preceding, we derived he opimal soluion under he assumpion ha n =. We nex show ha he soluion is also opimal when n >. Noe ha warrany populaion a he beginning of he second sage when he rade-in program goes ino effec is n. Adaping he soluion in (9) o accoun for he fac ha a oal n are made during he second sage, we ge n e.5r e.5r.5r e, (3) i.e., due o our normalizaion of he populaion size o, (9) gives he opimal fracion of he warrany populaion ha receives he rade-in offer over ime. If n >, hen i is conceivable ha some warranies will expire during he second-sage prior o a

20 cusomer receiving a rade-in offer. If such a scenario canno occur under he opimal rade-in offer rae given in (3), hen he preceding analysis coninues o apply. Indeed, as we show below, his is he case. According o soluion (3), he oal number of rade-in offers during inerval [, ] is f() =.5rx.5r e x xdx e dx.5r e. Observe ha f() is a concave increasing funcion over he inerval [, ] wih f( ) = and f() = n. If here was no rade-in policy, he oal number of warranies ha would expire during inerval [, ] is n,, g() =. n, Thus, f() g() for all [, ], i.e., no warranies expire during he second sage prior o receip of a rade-in offer. Therefore, he srucure of he opimal soluion for he case of n = holds for he case of n >, hrough he expressions for (), q, and C are generalized o accoun for he lower warrany populaion a he sar of he second sage:.5 r.5 r e e. (3).5r.5r.5r e n e.5r, (3) e q n e (33).5r.5r e e r C e n c.5r r w. (34) Proof of Proposiion. The uni acquisiion cos under a rade-in program wih accepance rae is c = (see (7)), and he acquisiion cos a he origin ( = ) is c =. Compared o he benchmark, rade-in programs resul in lower invenory and fewer oal warrany claims. Thus, a necessary condiion for q, is c e -r (i.e., he firm canno reduce acquisiion cos by acquiring produc a he end of he warrany horizon via a rade-in), and he conraposiive of is r q c e c e q. r