Design of Extended Warranties in Supply Chains. Abstract

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1 Dsign of Extndd Waantis in Supply Chains Kunpng Li Univsity of Illinois at Ubana Champaign, Collg of Businss Dilip Chhajd Univsity of Illinois at Ubana Champaign, Collg of Businss Suman Mallik Univsity of Illinois at Ubana Champaign, Collg of Businss Abstact Consid a supply chain involving an indpndnt tail and an indpndnt manufactu. Th manufactu poducs a singl poduct and slls it xclusivly though th tail. Using this supply chain famwok, w dvlop a gam thotic modl to study two commonly obsvd pactics of slling xtndd waantis: th manufactu offs th xtndd waanty dictly to th nd consum, and th tail slling th poduct offs xtndd waanty. W show that, of th two dcntalizd systms, whn th tail offs an xtndd waanty, it is fo a long duation and gnats mo systm pofit. W compa and contast th two dcntalizd modls with a cntalizd systm wh a singl paty manufactus th poduct, slls to th consum and offs th xtndd waanty. W idntify th diffnt causs of infficincis in ach of th two dcntalizd modls and popos coodination mchanisms that liminat th infficincis. W also povid contacts to achiv both coodination and a Pato impovmnt ov a wholsal pic contact. Publishd: Sptmb 005 URL: 08.pdf

2 Dsign of Extndd Waantis in Supply Chains Kunpng Li Dilip Chhajd Suman Mallik Dpatmnt of Businss Administation Univsity of Illinois at Ubana-Champaign 350 Wohls Hall 06 South Sixth Stt Champaign, IL 680 Sptmb 005 i

3 Dsign of Extndd Waantis in Supply Chains Abstact Consid a supply chain involving an indpndnt tail and an indpndnt manufactu. Th manufactu poducs a singl poduct and slls it xclusivly though th tail. Using this supply chain famwok, w dvlop a gam thotic modl to study two commonly obsvd pactics of slling xtndd waantis: th manufactu offs th xtndd waanty dictly to th nd consum, and th tail slling th poduct offs xtndd waanty. W show that, of th two dcntalizd systms, whn th tail offs an xtndd waanty, it is fo a long duation and gnats mo systm pofit. W compa and contast th two dcntalizd modls with a cntalizd systm wh a singl paty manufactus th poduct, slls to th consum and offs th xtndd waanty. W idntify th diffnt causs of infficincis in ach of th two dcntalizd modls and popos coodination mchanisms that liminat th infficincis. W also povid contacts to achiv both coodination and a Pato impovmnt ov a wholsal pic contact. Ky wods: supply chain managmnt, xtndd waanty, gam thoy ii

4 . INTRODUCTION Slling xtndd waantis on poducts is a apidly xpanding businss. Duing th 980s, xtndd waantis w offd only on lag, xpnsiv itms. Now, xtndd waantis a offd on almost all consum lctonics and domstic appliancs anging fom laptop computs to simpl swing machins. An xtndd waanty is actually a svic plan und which th povid pomiss to pai, plac, o maintain th poduct fo f o at a low pic ov a ctain piod of tim aft th manufactu s oiginal waanty xpis. Th xtndd waanty may also off additional bnfits (such as tun and/o xchang pivilgs) that a not povidd by th manufactu s oiginal waanty. Extndd waantis a sold spaatly fom th poducts and usually cost xta mony. Gnally, an xtndd waanty is offd by a manufactu, a tail, o by a thid paty (Publication 53, Bst Businss Buau). Th tms of a typical xtndd waanty spcify th pic and th tim lngth duing which th poduct is covd. Th povid of th xtndd waanty incus costs latd to th waanty, fo xampl, th actual pai costs, costs associatd with th administation of th claim, xtndd waanty division stup and maintnanc costs. Th pimay focus of ou cunt sach is to analyz th dsign of xtndd waantis in a supply chain contxt. W study a simpl supply chain involving a singl manufactu and a singl tail. Th manufactu poducs a singl poduct and slls xclusivly though th tail. Th xtndd waanty could, howv, b offd ith by th manufactu o by th tail. Th paty offing th xtndd waanty dcids th tms of th policy in its bst intst and incus all costs associatd with administing th policy. Thus, th pai/administation costs dictly influnc th povid s xtndd waanty dcisions. Und such a stting, w us gam thotic modls to answ th following qustions. Which scnaio lads to a high total supply chain pofit, a tail offing th xtndd waanty o th manufactu? How do th optimum pic and waanty lngth vay und th two scnaios? How is th total supply chain pofit distibutd btwn th patis in th two scnaios? W also compa th two scnaios with a cntalizd systm in which a singl fim manufactus th poduct, slls dictly to th nd consum, and offs th xtndd waanty. Th two scnaios, a tail o a manufactu offing an xtndd waanty a quit common in pactic. Many manufactus off xtndd waantis dictly to th nd consums. GE Appliancs, a lading manufactu of majo appliancs, pats, and accssois,

5 with poducts anging fom consum lctonics (VCRs, camcods, CD plays, tc.) to appliancs (ovns, figatos, washs, tc.) offs GE xtndd waantis on almost all th poducts it poducs. Fims lik Fod, GM, JVC, and Appl hav dvotd whol divisions solly to managing and to sving xtndd waanty contacts (Padmanabhan 995). Manufactu offing xtndd waantis a also common fo offic machins such as copis, fax machins, and pints. Rtail stos such as Bst Buy, Cicuit City, and Hom Dpot off and pomot thi own xtndd waantis on most of th itms thy cay. Nomally, th xtndd waanty policy that is offd by th tail is calld svic plan. Oftn, ths plans xtnd th manufactu s oiginal waanty to a long piod and may off additional bnfits not povidd by th manufactu s oiginal waanty. Fo xampl, Bst Buy offs a th-ya xtndd pfomanc plan fo notbook computs in th pic ang $500 - $ (Cay- In) fo $ An xampl of a cntalizd systm slling a poduct as wll as managing th xtndd waanty is Dll Comput. A custom can choos fom a mnu of xtndd waantis whil customizing a comput at Dll s wbsit. In this pap, w us gam thotic modls to analyz th two scnaios: a tail offing th xtndd waanty (Modl R) and a manufactu offing th xtndd waanty (Modl M). W modl th xtndd waanty as a f pai svic ov th lngth of th contact. Th two modls a compad with spct to th total channl pofit and th tms of th xtndd waanty. Modl R givs is to high total channl pofit than Modl M and offs a long xtndd waanty. W discuss how th total channl pofit is distibutd btwn th manufactu and th tail in th two modls. In addition, w compa and contast th two modls with a cntalizd systm offing xtndd waanty. Ths sults futh xplain th diffnc btwn Modls R and M. W bnchmak th pfomanc of th dcntalizd modls with thos of a cntalizd systm and show that th cntalizd systm will off th longst xtndd waanty whil a systm in which th manufactu offs th xtndd waanty will hav th shotst xtndd waanty. W dvlop two paamts that influnc th pofitability of th xtndd waanty businss. W also idntify th souc of infficincy in Modls R and M and discuss ways to coodinat ou dcntalizd supply chains with xtndd waantis. A vnu shaing contact with sid paymnts not only achivs coodination but also offs a Pato impovmnt ov th wholsal pic contact fo th channl dscibd in

6 Modl R. A pofit-shaing-and-quantity-discount contact is shown to achiv coodination as wll as Pato impovmnt fo th channl dscibd in Modls M and R. Ou pap povids insights about th influnc of xtndd waantis on supply chain dcisions and pfomanc. It also hlps to undstand som of th uniqu fatus of th xtndd waanty, its pic and duation, as wll as its dmand dpndncy on th pimay poduct dmand. Not that xtndd waantis sold by thid patis, though common in pactic, is not a focus of ou cunt wok. Th contact btwn a thid paty waanty povid and a custom can asily b modld as an insuanc policy. Th dsign, picing, and analyss of insuanc policis hav bn wll studid in th conomics and insuanc and isk managmnt litatu (.g., Lutz and Padmanabhan 998, Manov 983, Schlsing 983, and Taylo 995). Unlik insuanc litatu, th xtndd waanty povid in ou pap not only slls and administs th policy, but also influncs th tail pic of th poduct dictly (in Modl R) o indictly (though th wholsal pic in Modl M). Ou modl thus allows us to study th intactions of th poduct and xtndd waanty dcisions simultanously in a supply chain. Th maind of th pap is oganizd as follows. Th nxt sction viws th latd litatu. W psnt ou modls and discuss th solution pocdus in Sction 3. In Sction 4, w analyz th sults and dvlop insights. Channl coodination is discussd in sction 5, whil Sction 6 summaizs and concluds th pap.. LITERATURE REVIEW Th sach on dsign and analyss of xtndd waanty policis is limitd. Howv, thois of poduct failu and waantis hav civd xtnsiv attntion in both conomics and opations sach litatu.. Economic litatu Th distinct thois hav bn poposd in th conomics litatu to xplain th xistnc of poduct waantis. Th insuanc thoy (fist addssd by Hal, 977) assums that consums a mo isk-avs than slls, so that waantis a povidd to consums as a fom of insuanc against poduct failu. That is, conomic litatu typically tats 3

7 waantis as compnsation paid to consums in cas of poduct failu. Th concpt may b th sult of consum htognity, as mntiond by Hollis (999) and as obsvd by Emons (989) and Padmanabhan (995). Th signaling thoy stats that a poduct waanty is a signal of poduct quality. A long and mo comphnsiv waanty usually indicats btt poduct quality. Spnc (977) was th fist to show th signaling lationship btwn th poduct and its waanty. Finally, th incntiv thoy is th indication of doubl moal hazads, i.., poduc s and consum s moal hazads, in which poduct liability could b affctd by actions not obsvabl by th oth paty. Th tms of th waanty contact dscib liabilitis fo both patis. Thfo, th manufactu has incntivs to poduc and maintain a ctain lvl of poduct quality, whil th consums hav incntivs fo pop us and ca of th poduct. Exampls of this sach a Coop and Ross (985), Emons (988), Mann and Wissink (988), and Dybvig and Lutz (993). Th spas litatu on xtndd waantis is dominatd by th insuanc thoy. Th xtndd waanty is usually modld as a cash paymnt, and th slf-slction mthod is adoptd to dal with th consum htognity. Padmanabhan (995) addsss consum moal hazad and htognity in poduct usag in th optimal choics of poduct pic and waanty paymnt. Lutz and Padmanabhan (998) study th influnc of xtndd waantis on manufactu s waanty policy und poduc moal hazad. In thi modl, waantis a modld as cash paymnts, and two poducts with diffnt qualitis a offd to htognous consums in th makt. Padmanabhan and Rao (993) mpiically dmonstat that th isk avsion influnc in choosing an xtndd waanty can b ducd by incasing th lngth of th bas waanty. Again, th waantis a considd as cash paymnts in th modl. Hollis (999) also uss th standad slf-slction mthod tying to distinguish th havyus and th light-us in th makt. Howv, his wok diffs fom th pvious sachs in th conomic litatu by modling waanty as duation, instad of cash paymnt. H agus that although xtndd waantis a a fom of insuanc, waanty contacts usually vay by duation ath than by amount of paymnt.. Opations Rsach litatu Th opations sach litatu on waantis focuss on mathmatical modls with considabl scop and dtails in th dsciption of waanty costs. Th poduct quality is usually 4

8 modld in conjunction with th poduct failu tim. Poduct lif-tim distibution and failu at function a vy impotant componnts of th fomulation of th waanty cost modl. Sahin and Polatogu (998) povid an xcllnt viw of vaious waanty policis and poduct failu modls. Wam (987) analyzs th tad-off btwn waanty and quality, and illustats th snsitivity of waanty costs to nvionmntal vaiabls. Andson (977) dvlops an optimization dcision modl fo th optimal choics of th waanty piod and th poduct pic. Balc and Sahin (986) div total poduct placmnt cost und both po ata and f placmnt waanty policis by assuming that th succssiv failu tims fom a nwal pocss. Opp t al. (003) consid a cost minimization poblm of outsoucing waanty svics to pai vndos und static allocation. Thy dvlop an fficint optimal algoithm and dmonstat that th optimal algoithm can handl industy-siz poblms and also pfoms much btt than common static allocation huistics. Anoth uniqu wok latd to waantis by Cohn and Whang (997) dvlop a poduct lif-cycl modl in which waanty cost is incopoatd in th pofit function of a fim sking to maximiz total lifcycl pofit fom a poduct. Thy assum that th waanty will un fo a fixd intval and that th waanty cost is lina with th manufactu s quality of aft-sals svic. Howv, th dsign o th managmnt of th waanty is not th focus of thi wok. As th ad will s in th nxt Sction, ou modl is an implmntation of th signaling thoy, which is flctd in th choic of ou xtndd waanty dmand function. Th following fatus distinguish ou modl fom th litatu citd. Consistnt with ou obsvation in pactic, w modl th xtndd waanty as duation of tim, instad of cash paymnt to th custom whn th poduct fails. W study th dsign of xtndd waantis in a supply chain stting, which allows us to study poduct picing dcisions and xtndd waanty dcisions within a unifid contxt. Ou modl, thus, allows a paty offing an xtndd waanty to incu loss fom th poduct sal if th pofit fom th waanty sal can compnsat th loss. To th bst of ou knowldg, th dsign of th xtndd waantis in a supply chain contxt has not bn studid bfo. 3. THE MODELS Consid a supply chain consisting of a singl manufactu and a singl tail. Th manufactu poducs a singl poduct (.g., TV, micowav, comput, o ovn) and slls it 5

9 xclusivly though th tail at a unit wholsal pic x. Th tail, in tun, slls th poduct to th nd consum at a unit tail pic p. Th manufactu offs th oiginal poduct waanty, whos lngth, without loss of gnality, is nomalizd to zo. This assumption nabls us to focus xclusivly on th analysis of th xtndd waanty. In lin with ou obsvations in pactic, two spaat modls fo xtndd waantis a considd. In Modl M, th manufactu offs th xtndd waanty, whil in modl R, th tail offs th xtndd waanty. In ith modl, th manufactu sts th wholsal pic whil th tail sts th tail pic. Th spcification of th xtndd waanty has two componnts, which a th dcision vaiabls fo th paty offing it: th lngth of th waanty in units of tim, dnotd by w ; and th pic, dnotd by p. Duing th liftim of th xtndd waanty, th povid commits to offing f pai svic fo a faild poduct and incus th associatd pai and administation costs. W will compa and contast th two dcntalizd modls (R and M) with a cntalizd systm, Modl C, in which th manufactu maks th poduct, slls dictly to th nd consum and offs th xtndd waanty. Figu schmatically dscibs ths th modls along with th lvant dcision vaiabls. Manufactu Manufactu (x) Rtail Manufactu (x) Rtail (p) (w, p ) (p) (w, p ) (p) (w, p ) Customs Customs Customs (a) Modl C (b) Modl R (c) Modl M Figu : Modls Whn Supply Chain Mmbs Off Extndd Waanty Dmand functions Two dmand functions a involvd in ou modl, th poduct dmand and th xtndd waanty dmand. Fist, consid th poduct dmand. W assum that th dmand is linaly dcasing in pic and is givn by: 6

10 q = bp, () wh, p is th tail pic of th poduct, q is its dmand, and b (0, ] is th pic snsitivity of th consums. Whn an xtndd waanty is offd, its dmand q dpnds on th factos: poduct dmand q, xtndd waanty pic p, and lngth of th xtndd waanty w. If no xtndd waanty policy is offd, q, by dfinition, quals to zo. W us th following dmand function to modl th dmand fo th xtndd waanty. q = ( bp ) b p + w, if > 0 () 0, if w = 0, w wh, b > b and b (0, ] is th xtndd waanty pic snsitivity, > 0 is th xtndd waanty lngth snsitivity. Not that only th consums who bought th poduct a potntial candidats fo buying th xtndd waanty, i.. q q. Simplifying this inquality using () and (), and w gt b p w. Thus, th maximum allowabl dmand fo an xtndd waanty fom () is simply ( bp), th dmand fo th poduct. Th assumption b > b is mad bcaus th xtndd waanty dmand is mo snsitiv to xtndd waanty pic changs, than that of th tail dmand fo th tail pic chang. Th xtndd waanty pic is gnally low than th poduct tail pic. Thus, if th xtndd waanty pic p has th sam amount of chang as th tail pic p, th xtndd waanty dmand q will b influncd mo than th tail dmand q. Th costs W assum that th poduction cost fo th poduct is constant and is nomalizd to zo, so that th wholsal pic is also th manufactu s poduct pofit magin in th dcntalizd modls. Two componnts of th xtndd waanty cost a considd. Fist, th avag unit pai and administation cost, c w, which is linaly popotional to th lngth of xtndd waanty. H c > 0 is th avag pai cost p unit of tim. Scond, a quadatic cost, cw /, that dpnds on th lngth of th waanty but is indpndnt of th dmand fo th 7

11 xtndd waanty. This componnt captus oth xognously dtmind cost componnts involvd in managing th xtndd waanty pocss, such as th cost of stup and maintnanc of a pai division. H is th cost paamt. A simila quadatic cost assumption can b found in Balasubamanian and Bhadwaj (004). Not that, whil it is intuitivly appaling to intpt as th lngth of th xtndd waanty, ou modling famwok is gnal nough to allow oth intptations of w. Fo xampl, w might also b intptd as th quality o th amount of covag includd in th waanty (.g. div tain vs. bump-to-bump covag in an auto). Claly, th high th includd covag in a waanty, th mo xtnsiv is th quid suppot facility. Th quadatic cost componnt is also consistnt with this intptation. Th quadatic cost of quality o svic lvl is wll documntd (. g., Moothy, 988 and Iy, 998). w c > 0 W assum that th is no infomation asymmty in th channl and that th manufactu acts as a Stacklbg lad in th gam. Th manufactu, thfo, can look ahad and anticipat th tail s picing and xtndd waanty dcisions. W will us subscipts and supscipts to facilitat th compaison of th optimal valus of th dcision vaiabls acoss th modls. Th supscipts C, R, o M will dnot Modls C, R, o M, spctivly; whil th subscipts, m, pod., wa., o sys. will dnot th tail, manufactu, poduct, xtndd waanty, and systm, spctivly. W will also us supscipt sys. * to dnot optimal quantitis. Thus, is th optimal total systm pofit in Modl R, whil C wa. is th pofit fom th xtndd waanty in modl C. W nxt dfin two paamts to simplify th xposition of ou pap. Dfin ( bc ) α α = as th xtndd waanty dsiability indx. It is asy to chck that < 0, b c c α α α < 0, < 0, and > 0. All ls bing qual, a paty offing an xtndd waanty will c b mak a low pofit fom th xtndd waanty fo a low valu of α than fo a high on. Thfo, fo a supply chain mmb, a small valu of α psnts lativly littl dsi to off an xtndd waanty; whil a lag valu of α psnts a lativly gat dsi to off on. 8

12 b Nxt, dfin = as th lativ pic snsitivity indx. Basd on ou assumption b b, b. A high valu of indicats that customs a much mo snsitiv to xtndd waanty pic than to poduct tail pic, and vis vsa. W will us th paamts α and xtnsivly thoughout th maind of th pap as wll as in th Appndix. 3. Modl C: Th cntalizd systm A cntalizd systm maximizs th total supply chain pofit by simultanously considing th poduct and xtndd waanty dmands. Max Π p, p, w s. t. C = p( bp) + ( p c w ) [( bp) b p cw + w ] w b p (4) p 0 (5) w > 0 (6) Th objctiv function in (3) incopoats th two typs of costs of xtndd waantis, whil constaint (4) nsus that th dmand fo an xtndd waanty cannot xcd th dmand fo th poduct. Th non-ngativity condition on pic in constaint (5) nsus th poduct dmand will not xcd. Constaint (6) guaants that th modl will indd off an xtndd waanty. If w = 0, w must hav, q = 0 indicating that w no long hav th xtndd waanty in ou modl. Modl C can b solvd asily by foming th Lagangian and using standad optimization tchniqus. Tabl psnts th optimal solution to Modl C, along with th optimal valus of all dcision vaiabls. (3) 3. Modl R: Rtail offs xtndd waanty In Modl R, bsids th tail pic, th tail also dcids th xtndd waanty policy by spcifying p and w. In th fist stag of th gam, th manufactu chooss th wholsal pic x that maximizs h pofit. Th optimization poblm of th manufactu is givn by: 9

13 R MaxΠ = x( bp). (7) x m In th scond stag, taking th wholsal pic as givn, th tail maximizs his own pofit. Th tail s poblm is as follows. Max Π p, p, w s. t. R = ( p x)( bp) + ( p c w ) [( bp) b p cw + w ] w b p (9) p 0 (0) w > 0. () Th intptations of th constaints (9) - () a simila to thos of (4) - (6), spctivly. Th is no quilibium solution to th poblm (7) - () fo p = 0. To s this, not, if w lt p = 0, thn th poduct dmand bcoms. Consquntly, th manufactu s optimization R poblm bcoms MaxΠ = x( bp) = x. Obviously, th wholsal pic x can go to infinity, x which will bing th tail s pofit down to ngativ infinity. So th tail will nv st th tail pic to zo, and th constaint on m (8) p bcoms stictly positiv. Howv, this dos not pclud th possibility that th tail will st th tail pic blow th wholsal pic. W solv th modl stating with th tail s poblm and woking backwads. Th Lagangian fo th tail s poblm is as follows: L cw = ( p x)( bp) + ( p c w ) [( bp) b p + w ] + λ ( b p w ). () W can find th tail s bst spons function using standad optimization pocdus. Two cass a possibl: λ = 0 and λ > 0. Consid th cas of λ = 0 fist. Solving fo th tail s bst sponss as a function of th wholsal pic ( α)( + bx) / p ( x) =, w b [( α ) / Substituting Max Π x R m x, w gt, ( bx) ( bc ) ( x) = b c [( α) /, ( bx) [( bc ) c + c] p ( x) =. b c [( α) / ( α)( bx) q ( x) = b( x) = into th manufactu s poblm, w gt, ( α) / ( α)( bx) = x( bp) = x. (4) ( α) / (3) 0

14 Th optimization poblm in (4) can b solvd to yild x R =. Optimal valus of all oth b dcision vaiabls can b obtaind by substituting th valu of x in (3). Th cas λ > 0 can b solvd similaly. Tabl dscibs th optimal solutions fo Modl R. 3.3 Modl M: Manufactu offs xtndd waanty Bing th Stacklbg lad in th gam, th manufactu anticipats th tail s picing dcision and maximizs h pofit accodingly. In th fist stag of th gam, th manufactu chooss th wholsal pic x, and th xtndd waanty policy ( p, w ). In th scond stag, th tail taks th manufactu s dcisions as givn and dcids th tail pic p. Th manufactu s optimization poblm is as follows. Max Π x, p, w s. t. M m = x( bp) + ( p c w ) [( bp) b p cw + w ] w b p (6) w > 0 (7) (5) In th scond stag, givn th manufactu s dcision, th tail solvs fo p by solving p ag max( p x)( bp) subjct to p 0. Again, w can show that p = 0 is not an quilibium solution. Th solution pocdu fo Modl M is simila to that of Modl R. W stat with th tail s poblm and wok backwads. Tabl 3 dscibs th optimal solution. 4. RESULTS AND ANALYSIS Ou analysis is basd on th sults summaizd in Tabls -3. All th xtndd waanty modls a valid only if th constaint w > 0 is satisfid, which nsus that xtndd waanty is, indd, offd. As w hav shown in th footnots in Tabls -3, th ncssay condition fo satisfying th constaint w > 0 is > b c. W stat this sult in th following poposition. Poposition : It is nv optimal fo any paty to off an xtndd waanty to th makt unlss th xtndd waanty snsitivity is at last b c.

15 Poposition implis that high waanty snsitivity is quid fo offing th xtndd waanty whn th unit pai cost c incass. Bcaus of th cosponding incasing pai costs and th shinking pofit magins on xtndd waanty slling, th povid is lss willing to off xtndd waantis unlss th customs hav a stong dsi fo it. Similaly, high waanty snsitivity is quid fo offing an xtndd waanty whn th xtndd waanty pic snsitivity b incass. A high valu of b implis a mo damatic chang in xtndd waanty dmand whn th xtndd waanty pic changs. Consquntly, a high xtndd waanty snsitivity,, is quid to compnsat fo this ffct. Poposition : Th following statmnts hold fo ach of th th modls (Modls C, R and M). (a) Th systm pofit, th xtndd waanty pic, th xtndd waanty lngth, th poduct dmand, and th xtndd waanty dmand a non-dcasing in α ; and a non-incasing in ; whil th poduct tail pic has a vsd lationship in α and. (b) Fo any paty offing th xtndd waanty, as α incass, th pofit fom slling th poduct dcass whil th pofit fom slling th xtndd waanty incass. Futhmo, th at of incas in xtndd waanty pofit is high than th at of dcas in poduct pofit, implying th povid s total pofit incass with α. Th poof of Poposition (a) follows fom obsving th signs of th cosponding fist p divativs with spct to α. Fo xampl, in Modl C, whn α <, = α b( α), s = α b( α) p α, = b( α ) s α and = b( α). Th poof of poposition (b), as wll as th poofs fo all oth sults, is includd in th Appndix. Accoding to Poposition (a), whn th valu of α bcoms small, th lngth of th xtndd waanty should b shot. A small valu of α indicats a lss favoabl condition fo offing an xtndd waanty, which, fo xampl, may b du to high pai costs. A cnt aticl in th Wall Stt Jounal pots that DaimlChysl is shotning th xtndd waanty on its vhicls bginning with th 006 modls bcaus of incasd pai costs

16 sulting fom high labo costs and mo complicatd tchnology (Saanow 005). Th findings of ou modl dictly suppot this action. As α incass, th makt bcoms mo favoabl fo a fim to off xtndd waantis. A mo favoabl makt mans that th povid will aliz high pofits fom th xtndd waanty makt. As α incass, th poduct tail pic dcass (Poposition a). This sults in a high poduct dmand but might duc th pofit fom th poduct makt. Howv, th xtndd waanty povid bnfits fom a low tail pic as a high poduct dmand implis a high potntial dmand fo th xtndd waanty. This, togth with th statgic choic of xtndd waanty pic and lngth, nabls th xtndd waanty makt to gnat mo pofit, which compnsats fo low pofit fom poduct sals and, hnc, th systm pofit incass with α. As α continus to incas, th pofits fom slling xtndd waantis will supass th pofits fom slling th poduct. To futh impov total pofit, th povid nds to low th poduct tail pic futh. In Modl R, th tail may st th tail pic so low that h will sustain a loss on th poduct in od to gain mo fom th xtndd waanty. As α incass, th manufactu in Modl M ducs th wholsal pic so that sh can bnfit fom a low tail pic st by th tail. Whn th lativ pic snsitivity indx incass, customs bcom mo snsitiv to xtndd waanty pic chang. Small changs in th xtndd waanty pic could sult in lativly lag fluctuations in th xtndd waanty makt. Thus, th makt bcoms lss stabl fo th xtndd waanty businss. Similaly, a dcasing implis a mo stabl dmand that allows asi dmand manipulation in th poduct makt though th tail pic. This povids an intuitiv xplanation fo th snsitivity analysis sults with spct to in Poposition (a). Compaing th tail pic p and th wholsal pic x in Modl R, w not that whn + R (, ) and α (, ), 0 < p < x. In this scnaio, th tail chooss to los mony on th poduct in od to gain high pofit fom xtndd waantis. Not that a small valu of indicats that th poduct dmand is mo sponsiv to tail pic changs, and a high α psnts a vy favoabl xtndd waanty makt. Whn is low, th tail is asily abl to incas poduct dmand by stting a low tail pic. This may duc 3

17 th pofit coming fom poduct sals. Howv, a high dmand fo th poduct also contibuts to a high dmand fo th xtndd waantis. Whn α is high, a high pofit could b divd fom th xtndd waanty makt. Th ang of valus fo α and povids a sufficint condition fo xtndd waanty pofits to dominat th poduct pofit. Not that α > in this ang. Howv, a simila scnaio of th manufactu incuing a loss on th poduct (i.. x < 0 ) dos not ais in Modl M. Th condition fo x < 0 in Modl M a (0.46, 0.853) and α (, ). Howv, th assumptions in ou modl dictat that. This 4 implis that th manufactu in Modl M is lss ffctiv in influncing poduct dmand and can only do so though th choic of th wholsal pic. 4. Pofit analysis In this sub-sction, w compa th total channl pofit fo th th modls, as wll as look at th division of total pofits btwn th manufactu and th tail within th modls. Thom stats ou fist sult. Thom : C R (a) Whn 0 α, all th modls a fasibl and sys. > sys. > sys.. (b) Whn < α, only Modl M has fasibl solutions. 4 (c) No fasibl solution xists fo any modl whnα >. 4 Thom (a) shows that, as xpctd, fo a low valu of th dsiability indx α, a cntalizd systm offs th highst systm pofit. It is intsting to not that Modl R always offs a high systm pofit than Modl M. Why? In Modl R, th tail dcids p, p and w, whil th manufactu dcids th wholsal pic x. Thus, th tail can manipulat all th vaiabls und his contol simultanously to maximiz pofits. Ths th vaiabls allow him to influnc th poduct as wll as th xtndd waanty dmand dictly. On th oth hand, in Modl M th manufactu can dictly choos p and w, and can only indictly influnc p 4

18 though th choic of th wholsal pic x. Thus th manufactu in Modl M can nith influnc th poduct dmand dictly, no manipulat th vaiabls p, p and w simultanously, sulting in a low systm pofit. Thom (b) and (c) a divd fom th scond-od conditions of th spctiv optimization poblms. Fo intmdiat valu of α, only Modl M is fasibl, which might xplain why it is mo common fo th manufactu to off xtndd waantis than th tail in ctain industis, such as th automobil industy. Gnally, th valu of α is lativly high in th auto industy du to consums stong dsi (high valu of ) fo th xtndd waanty. Compad with oth poducts, xtndd waantis fo autos a also mo xpnsiv. Thfo, consums a lss snsitiv to unit pic changs fo th xtndd waanty (small valu of b ), which contibuts to high valu of α. W nxt look at th division of pofit btwn th manufactu and th tail within a modl, as wll as btwn th two modls. Figu plots th pofits as a function of th dsiability indx α. Th solid lins in Figu psnt Modl R, whil th dashd lins psnt Modl M. Pofit m m -/() -/(4) α Figu : Pofits of th tail and th manufactu in Modl R and Modl M Admittdly, thid paty xtndd waantis a also common in th automobil industy. Th wbsit discusss vaious asons fo this. Howv, xtndd waantis povidd by thid patis a byond th scop of ou cunt wok. 5

19 Th following poposition is divd fom th pofit compaison of th tail and th manufactu within and btwn Modl R and Modl M. Poposition 3 (a) Compaison btwn Modl R and Modl M in thi common fasibl gion: Modl R has both high tail pofits and high manufactu pofits than Modl M, i.., > M and m. > M m (b) Compaison within Modl R and Modl M: (i) In Modl R, th optimal pofit of th manufactu is twic as much as that of th tail, i.., R m =. (ii) In Modl M, th optimal pofit of th manufactu is gat than that of th tail, but is lss than twic th optimal th pofit of th tail, i.., > m >. Moov, th diffnc btwn th two pofits ( ) is dcasing with α. M m In Thom w hav shown that Modl R gnats mo systm pofit compad with Modl M. Poposition 3(a) futh stats that both th tail and th manufactu a btt off with Modl R than Modl M. Modl R, thfo, maks a Pato impovmnt ov Modl M. This insight can sv as a usful qualitativ guidlin fo th pactitions sponsibl fo stting up o unning xtndd waanty businsss. Poposition 3(b)(i) stats that th manufactu gts twic th pofit of th tail in Modl R. Intstingly, th sam lationship holds fo a dcntalizd supply chain involving a singl manufactu and a singl tail that offs no xtndd waanty. As dscibd in th discussion following Thom, th tail has full contol ov th xtndd waanty makt in Modl R. Th tail s only intaction with th manufactu is though th wholsal pic, which is vitually th sam as that in th modl without an xtndd waanty. Howv, th manufactu in Modl M dos not hav full contol ov xtndd waanty dmand, giving is to th lationship dscibd in Poposition 3(b)(ii). What s mo, in Modl ( m ) M, < 0, i.., th pofit diffnc btwn th manufactu and th tail is α dcasing as α incass. As α incass, offing th xtndd waanty bcoms mo attactiv. Th manufactu taks this oppotunity to duc th wholsal pic in hops that 6

20 th tail will duc th tail pic, cating a high dmand fo th poduct. Th tail obligs th manufactu. Howv, th tail uss this as an oppotunity to impov his own pofit as wll, thus sulting in a gat pofit incas than that of th manufactu. As a sult, dcass with α. M m 4. Extndd waanty dsign Tabls -3 dscib th optimal valus fo th dsign vaiabls of xtndd waantis fo th th modls discussd. In this sub-sction, w us ths Tabls to dvlop insights about th dsign of th xtndd waanty as wll as poduct picing. Th following poposition dscibs ou sult. Poposition 4: Among th th modls discussd, Modl C offs th longst xtndd C waanty, whil Modl M offs th shotst, i.., > >. A simila lationship holds fo xtndd waanty pic, dmand fo th xtndd waanty, and pofit fom th C C C xtndd waanty, i..: > >, > >, and > >. p p p q q w q w w wa. wa. wa. Poposition 4 shows that th lngth of th xtndd waanty is diffnt fo ach of th th modls, vn though th unit xtndd waanty pic is sam fo all th th modls. Modl C, bnfiting fom a cntalizd stuctu, is abl to off th longst waanty and captus th most xtndd waanty dmand and pofit. Compad with Modl M, Modl R offs a long xtndd waanty sinc th tail can influnc th xtndd waanty dmand by dictly choosing th tail pic. Extndd waanty dmand and pofit show a simila lationship. Th intuitiv xplanations a simila to that of th waanty lngth. Th pics, on th oth hand, hav a vs lationship in th th modls as dscibd by Poposition 4. W nxt compa th optimal valus of th wholsal pic x, th tail pic p, and th poduct dmand q. W summaiz th sults in th following poposition: Poposition 5: (a) Th wholsal pic in Modl M is low than that in Modl R, i.., x > x. 7

21 (b) Th optimal valus of poduct pic and poduct dmand in th th modls hav th following lationship: C p < p < p, and C q > q > q. Th dcisions facing th manufactu in Modl R a idntical to thos in a singlmanufactu-singl-tail supply chain with no xtndd waanty. Not supisingly, w find x R = /(b), which is also th optimal wholsal pic in th no-xtndd-waanty modl. In Modl M, in od to cat high xtndd waanty dmand fom a low tail pic, th manufactu has to st th wholsal pic low than that in Modl R, i.., x < /(b). This xplains Poposition 5(a). Compad with Modl M, th tail pic in Modl R is low dspit th high wholsal pic. Th tail in Modl R has mo dcision vaiabls und his contol. As a sult, h can jointly dcid on all of th vaiabls, sulting in a high poduct dmand and low poduct pic. High poduct dmand, in tun, also givs is to a high dmand fo th xtndd waanty. 5. CHANNEL COORDINATION AND IMPROVEMENT Ou discussion so fa has focusd on a wholsal pic contact btwn th manufactu and th tail in modls R and M. In this sction, w xplo altnativ contacts to coodinat th dcntalizd channls in ths two modls. W say that channl coodination is achivd whn th systm pofit fom a dcntalizd channl quals that of th cntalizd channl. In addition to achiving coodination, w also xplo th issu of making a Pato impovmnt ov th wholsal pic contact. A nw contact achivs a Pato impovmnt ov th wholsal pic contact whn both th manufactu and th tail an pofits at last qual to that of th wholsal pic contact. Indd, as w show, it is possibl to dsign a contact that simultanously achivs coodination and Pato impovmnt. 5. Th Rvnu Shaing Contact and Sid Paymnts In Sction 4, w discussd how in Modl R th tail can manipulat th vaiabls p, p and w simultanously. Not that in ou modl ths a th only th vaiabls that affct th pofit fom th xtndd waanty. Th manufactu can, of cous, influnc th tail pic 8

22 though h choic of th wholsal pic. In addition, th dcision vaiabl fo th manufactu in Modl R is only th wholsal pic. Thfo, w can think of Modl R as dcntalizd in poduct sals and somwhat lik a cntalizd systm fo th xtndd waanty sals. Th main souc of infficincy in modl R, thus, is th doubl maginalization in th poduct makt. This, howv, no long holds tu in Modl M. In modl M, th manufactu slls th xtndd waanty whil th tail contols its dmand dictly though th tail pic (call that th dmand fo xtndd waantis dpnds on poduct dmand). Thfo, Modl M can b thought of as bing dcntalizd in th poduct as wll as in th xtndd waanty makt. Thus th souc of infficincy in Modl M coms fom th doubl maginalization of th poduct makt, as wll as fom th dcntalization in th xtndd waanty makt. This implis that any contact that mdis th doubl maginalization in a singl-manufactusingl-tail dcntalizd supply chain should coodinat th channl dscibd in Modl R, but not th channl dscibd in Modl M. Obsvation : A vnu shaing contact in which only th poduct vnu is shad btwn th manufactu and th tail can coodinat th channl in Modl R, but no such contact xists fo th channl in Modl M. W nxt ask th qustion of whth it is possibl to achiv both coodination and Pato impovmnt in Modl R using a vnu-shaing contact. Consid a vnu-shaing contact with sid paymnts. Such a contact has two paamts: φ, th faction of poduct sals vnu allocatd to th tail (th manufactu thus gts φ ); and F, th sid paymnt th manufactu offs to th tail. Poposition 6: A poduct vnu shaing contact with sid paymnt (φ, F) simultanously achivs coodination and Pato impovmnt fo th dcntalizd channl dscibd in Modl, 0 α ( α ) R, wh, F [, m ], and φ =. α, < α /( ) 9

23 Rcall that w hav dfind and as th optimal pofit of th tail and th m manufactu, spctivly, in Modl R und th wholsal pic contact. As dscibd in Obsvation, a sid paymnt is not quid fo coodination. A vnu-shaing contact (with φ as shown in Poposition 6 and F=0) alon is sufficint to coodinat th chain. Und this pu coodination famwok, th manufactu gts th whol systm pofit, and th tail gts zo pofit. That is possibl sinc th tail s svation pofit is assumd to b zo, and bing th Stacklbg lad, th manufactu offs a contact that xtacts th nti suplus fom th tail. C Und th vnu shaing schm with sid paymnts, th manufactu s pofit is F, whil that of th tail is simply th fixd sid paymnt F. Any abitay distibution of th pofit is possibl btwn th manufactu and th tail using th sid paymnts. In paticula, th sid paymnt dscibd in Poposition 6 allows us to achiv Pato impovmnt. Givn that both patis will b btt off und this schm, th contact sms quit pactical and asy to implmnt. Th actual choic of F pobably dpnds on th two patis lativ bagaining pow. W hav shown in Sction 4 that und a wholsal pic contact, th tail gts only half of th manufactu s pofit in Modl R. Und th schm dscibd in Poposition 6, a powful tail can ctainly ngotiat a btt dal, whil making a Pato impovmnt fo th manufactu as wll. sys. 5. Pofit Shaing and Quantity Discounts W hav shown that a vnu shaing contact can coodinat th channl dscibd in Modl R. Howv, such a contact is unabl to coodinat th channl dscibd in Modl M. W now dscib a contact involving pofit shaing fom th sals of xtndd waantis and a quantity discount schm that achivs coodination and Pato impovmnt simultanously fo both Modls M and R. In this schm, th paty offing an xtndd waanty shas th pofit fom xtndd waanty with th oth paty. Thus, th manufactu will sha th xtndd waanty pofit with th tail in Modl M. Lt φ dnot th faction of th xtndd waanty pofit allocatd to th tail, ispctiv of th paty offing th waanty. Th manufactu, thus, gts φ of th xtndd waanty pofit. In addition, dpnding upon th tail s od quantity q, and th valu of th paamt φ, th manufactu offs a quantity 0

24 discount schm fo th wholsal pic. As shown in Poposition 7, such a schm achivs both coodination and Pato impovmnt fo Modls M and R. Poposition 7: An xtndd waanty pofit shaing and quantity discount contact (φ, x(φ,q)) achivs both coodination and Pato impovmnt fo both Modls M and Modl R, wh, ( q) x( φ, q) = ( φ) and b in Modl R φ [, 4 ] in Modl M, and φ ( [( / (, ], whn 0 α < [4( / 4( / [ ( [, ], othwis. (4 4 α Th schm in Poposition 7 allows th manufactu to align th tail s pofit with that of th cntalizd channl ( = φ ). Th schm is abl to coodinat th supply chain fo both c modls bcaus it links th poduct and th xtndd waanty makts, allowing simultanous optimization of both makts. Th spcific choics of th contact paamt φ nsu Pato impovmnt. This schm can achiv coodination only with φ = 0. In Modl M, φ = 0 mans that no xtndd waanty pofit shaing is quid to achiv channl coodination. In Modl R, φ = 0 implis that th manufactu gts th nti pofit fom th xtndd waanty businss vn though th tail offs th waanty. Similaly to Poposition 6, th tail s zo svation valu and th manufactu s lad position nsus that th nti systm pofit gos to th manufactu und this pu coodinating contact. In Modl M s impovmnt-coodination contact, th manufactu distibuts th fficincy gains though xtndd waanty pofit shaing. Fo Modl R, th tail shas only pat of th xtndd waanty pofits in od to achiv both coodination and Pato impovmnt.

25 Thotically, ou contact impovs and coodinats both Modls R and M pfctly. Howv, th pofit shaing contacts a oftn had to implmnt in pactic, as it quis monitoing of sals as wll as associatd cost paamts. Nvthlss, th pu coodination famwok using a quantity discount schm in Modl M is pactical and asy to implmnt. 6. SUMMARY AND CONCLUSION In this pap, w hav dvlopd a gam-thotical modl to compa diffnt schms fo xtndd waantis in a supply chain contxt. Spcifically, w considd a supply chain involving a singl manufactu and a singl tail. W studid two modls with th tail and th manufactu, spctivly, bing th povid of th xtndd waanty. W compad th pfomancs of th two modls in tms of systm pofits, tail pofits, manufactu pofits, poduct picing dcisions, and xtndd waanty dcisions. W also bnchmakd th pfomancs of th two modls with that of a cntalizd systm. Ou sults show that offing xtndd waantis though a tail gnats mo systm pofit than offing it though th manufactu. This is bcaus th tail can simultanously optimiz all lvant dcision vaiabls and, thus, acts as a cntal plann in th xtndd waanty makt. Howv, a simila situation dos not occu whn th manufactu offs th xtndd waanty. W hav also analyzd and compad th optimal valus of th dcision vaiabls in th two modls. W show that a cntalizd systm will off th longst xtndd waanty whil a systm in which th manufactu offs th xtndd waanty will hav th shotst xtndd waanty. W dvlopd two paamts that influnc th pofitability of th xtndd waanty businss. W popos channl coodination mchanisms to liminat th causs of infficincy in ach of th two modls. Doubl maginalization in th poduct makt is shown to b th pimay souc of infficincy in th channl dscibd in Modl R. On th oth hand, th infficincy in Modl M aiss bcaus of th doubl maginalization in th poduct makt and bcaus of th dcntalization of th xtndd waanty makt. W poposd a poduct vnu shaing contact with sid paymnts that not only coodinats th supply chain but also achivs Pato impovmnt ov th wholsal pic contact in Modl R. Fo Modl M, an xtndd waanty pofit shaing contact along with a quantity discount schm is shown to achiv both coodination and Pato impovmnt.

26 Ou pap maks th following contibution to th opations managmnt litatu. Fist, it povids a uniqu dmand function fo th xtndd waantis that xplicitly taks into account poduct dmand. In lin with ou obsvation in pactic, w modl th xtndd waanty as a svic poduct with pic and tim duation. Scond, whil a majoity of th litatu in xtndd waantis focuss on consum moal hazad and htognity issus, ou wok addsss its ol in th channl comptition, pfomanc and coodination. By adopting a gam-thotic appoach, w study th statgic intaction btwn th manufactu and th tail as wll as th intaction btwn th poduct sals and xtndd waanty sals. Th channl coodination and impovmnt mchanisms w psnt addss th diffnt causs of channl infficincy in diffnt modls. To th bst of ou knowldg, ou pap is th fist to addss ths issus lating to xtndd waantis in a supply chain contxt. Thid, ou modl implis that xtndd waantis a not mly a souc of vnu. Bcaus of th uniqu chaact of its dmand, which is dpndnt on poduct dmand, xtndd waantis could b usd statgically in channl choics to impov systm pofits and poduct picing dcisions. Ths sults povid usful insights as wll as qualitativ guidanc to pacticing manags involvd in dsigning, stting up, and managing an xtndd waanty businss. Lik any oth modl in th opations managmnt litatu, ou modl is basd on a st of assumptions. Fo simplicity, w studid th xtndd waanty businss using a monopolistic stting. This allowd us to isolat th impact of xtndd waantis on total supply chain pofits. An impotant xtnsion of ou modl would b to includ comptition in th modl. Such comptition can b ith btwn manufactus o btwn tails. Incopoating dmand unctaintis might b anoth usful xtnsion and might qui a nw st of coodinating contacts. Ou modl assums no infomation asymmty. A tail might hav pivat infomation about consum dmand. In such a scnaio, th tail might us this infomation statgically to impov his pofit. Anoth way to xtnd ou modl could b to incopoat poduct dsign lmnts into ou modl. Fo xampl, poduct quality can b a dcision vaiabl in th modl, which affcts both poduct cost and th cost of th xtndd waanty. REFERENCES Andson, E. (977), Poduct Pic and Waanty Tms: An Optimization Modl, Opational Rsach Quatly, Vol. 8, No. 3,

27 Balasubamanian, S. and P. Bhadwaj. (004), Whn Not All Conflict is Bad: Manufactuing Makting Conflict and Statgic Incntiv Dsign, Managmnt Scinc, Vol. 50, No. 4, Balc, Y. and I. Sahin. (986), Rplacmnt Costs Und Waanty: Cost Momnts and Tim Vaiability, Opations Rsach, Vol. 34, No. 4, Cohn, M. and S. Whang. (997), Compting in Poduct and Svic: A Poduct Lif-Cycl Modl, Managmnt Scinc, Vol. 43, No. 4, Coop, R. and T.W. Ross. (985), Poduct Waantis and Doubl Moal Hazad, Rand Jounal of Economics, Vol. 6, No., Dybvig, P. and N. A. Lutz. (993), Waantis, Duability, and Maintnanc: Two-sidd Moal Hazad in a Continuous-Tim Modl, Rviw of Economic Studis, Vol. 60, Emons, W. (988), Waantis, Moal Hazad and th Lmons Poblm, Jounal of Economic Thoy, Vol. 46, Emons, W. (989), On th Limitation of Waanty Duation, Th Jounal of Industial Economics, Vol. 37, No. 3, Hal, G. (977), Guaants and Risk Shaing, Rviw of Economic Studis, Vol. 44, Hollis, A. (999), Extndd Waantis, Advs Slction, and Aftmakts, Th Jounal of Risk and Insuanc, Vol. 66, No. 3, Iy, G. (998), Coodinating Channls Und Pic and Nonpic Comptition, Makting Scinc, Vol. 7, No. 4, Lutz, N.A. and V. Padmanabhan. (995), Why Do W Obsv Minimal Waantis?, Makting Scinc, Vol. 4, No. 4, Lutz, N.A. and V. Padmanabhan. (998), Waantis, Extndd Waantis, and Poduct Quality, Intnational Jounal of Industial Oganization, Vol. 6, Mam, J. (987), Discountd and P Unit Costs of Poduct Waanty, Managmnt Scinc, Vol. 33, No. 7, Mann, D. and J.P. Wissink. (988), Mony-back Contacts with Doubl Moal Hazad, Rand Jounal of Economics, Vol. 9, No., Manov, M. (983), Povid Insuanc, Th Bll Jounal of Economics, Vol. 4, No.,

28 Moothy, K. S. (988), Poduct and Pic Comptition in a Duopoly, Makting Scinc, Vol. 7, No., Opp, M., I. Adan, V. Kulkani, and J. Swaminathan. (003), Outsoucing Waanty Rpais: Static Allocation, Woking Pap, Univsity of Noth Caolina, Chapl Hill. Padmanabhan, V. (995), Usag Htognity and Extndd Waantis, Jounal of Economics & Managmnt Statgy, Vol. 4, No., Padmanabhan, V. and R.C. Rao. (993), Waanty Policy and Extndd Svic Contacts: Thoy and An Application to Automobils, Makting Scinc, Vol., No. 3, Sahin, I. and H. Polatogu. (998), Quality, Waanty, and Pvntiv Maintnanc, Kluw Acadmic Publishs. Saanow, J. (005), DaimlChysl Gos in Rvs on Waantis, Th Wall Stt Jounal, May 6, 005, D4. Schlsing, H. (983), Nonlina Picing Statgis fo Comptitiv and Monopolistic Insuanc Makts, Th Jounal of Risk and Insuanc, Vol. 50, No., Spnc, M. (977), Consum Mispcption, Poduct Failu, and Poduc Liability, Rviw of Economic Studis, Vol. 44, Taylo, G. (995), An Equilibium Modl of Insuanc Picing and Capitalization, Th Jounal of Risk and Insuanc, Vol. 6, No. 3, Tiol, J. (988), Th Thoy of Industial Oganization, MIT Pss. 5

29 Tabl : Optimal solutions fo Modl C Optimal solution Modl C: Cntalizd Systm Offs Extndd Waanty + + Whn < < Whn < α α < α 0 < α < α α / α α / α 0 b [ ( / b ( α) b [ ( / b ( α) Condition 0 < C p C q C sys. C pod. α ( / α α b [ ( / b ( ( ( α / ) b [ ( / ( (/ ) C wa. b [ ( / c ( bc ) + c p b c [ ( α) / C ( b ( α b ( ( / b c b c ( α) ) α b α ( / α α b [ ( / b ( ( ( α / ) 0 α b ( b c ) b c b [ ( / ( (/ ) b [ ( / c ( b c ) + c b c [ ( α) / b ( ( α b ( ( / b c b c ( α) ) bc w b c [ ( α) / C / b c c ( α) b c b c b c b c [ ( α) / / b c c ( α) p w C C C q c ( b c ) + c b c ( / b α b c ( b c ) + c b c ( / b α Not: () ( bc ) α = is th Extndd Waanty Dsiability Indx; = is th Rlativ Pic Snsitivity bc b Indx. () Modl C s scond od condition is satisfid whn α. (3) w > 0 quis condition > b c. b 6

30 Tabl : Optimal solutions fo Modl R Optimal Solution Modl R: Rtail offs xtndd waanty Rtail s Pofit Systm Pofit Condition 0 < α < p q x m p w p w q / pod. / wa. sys. pod. wa. α 3 / ( / 3 / α b [ ( / b ( α) α [ ( / ( b b α 8b [ ( / 8b ( ( [( / 4b [ ( / ( (/ ) ( 4b ( α 8b [ ( / 8b ( α 4b [ ( / 4b ( 3 ( 3 8b [ ( / 8b ( ( [3 / ( / b [ ( / ( (/ ) 8b [ ( / c ( b c ) + c b c [ ( α) / b c [ ( α) / c b c ( b c ) + c b c (3 / b ( α 8b ( ( / b c b c ( α) / b c c ( α) [ ( / ( b Not: () ( bc ) α = and =. bc b () Modl R s scond od condition is satisfid whn α. (3) w > 0 quis condition > b c. b ) 7

31 Tabl 3: Optimal solutions fo Modl M Optimal Solution Modl M: Manufactu offs xtndd waanty Condition 0 < α < p q x Manufactu s Pofit Systm Pofit p w p w q m m / pod. m / wa. α 4 3 ( / 3 α b [4 ( / b (4 α) α 4 ( / 4 α ( / α b [4 ( / b (4 α) ( α) b [4 ( α) / b ( 4 α) α b [4 ( / b (4 ( [ ( / b [4 ( / ( (/ ) b [4 ( / ( [6 ( / sys. pod. wa. b b [4 ( / ( [3 ( / b [4 ( / ( (/ ) b [4 ( / c ( b c ) + c b c [4 ( α) / b c [ 4 ( α) / c b c ( b c ) + c b c 4 ( / Not: () ( bc ) α = and =. bc b () Modl M s scond od condition is satisfid whn α. 4 (3) w > 0 quis condition > b c. ( b (4 α b (4 (6 b (4 (3 b (4 α b (4 ( / b c b c (4 α) / b c c (4 α) b 4 α ) 8

32 Appndix: Poof of th Rsults Poof of Poposition W psnt th poofs fo Modls R and M. Th poofs fo Modl C a simila and can b divd fom Tabl. Rtail s Pofit Snsitivity R / pod. α Expssion < α α 0 < α < / > 0 8b [ ( / (/ ) / pod. < 0 3 α 4b [ ( α) / [ ( + / (/ ) / wa. > 0 3 R / wa. α 8b [ ( / Valu (Modl R) α 8b ( α 4b ( ( + α ) 8b ( / < ( α < > 3 < 3 > Expssion Manufactu s Pofit Snsitivity M m / pod. α < α α α 0 < α < / m > 0 b [4 ( / (/ ) m / pod. < 0 3 α M m / wa. b [4 ( α) / [4 ( + / (/ ) m / wa. > 0 3 b [4 ( / Valu (Modl M) α 4 b (4 α b (4 (4 + α ) b (4 / < 4( α < 4 > 3 < 3 > Poof of Thom C Us th xpssions fom Tabls -3. Fo 0 < α <, > is obvious. > 8 ( / > 0, which is tu und condition 0 < α <. So und condition 0 < α <, sys. sys. sys. sys. 9

33 C + w hav sys. > sys. > sys.. Nxt consid α. Consid two cas: (i) (, ), and (ii) + + C (i) (, ), h sys. has two sts of solutions und α. C (i.a) Consid α <. > is obvious. > α < 8, which is tu und sys. sys. C condition α <. Thfo, > >. sys. sys. sys. sys. sys. (i.b) Nxt, consid C α. sys. > sys. 0.5 α <. 5. Sinc /(). 5, it is tu und cunt condition; > α < 8, obviously it is sys. sys. tu und cunt condition. Thfo, fo C α, w hav sys. > sys. > sys.. (i.c) Finally, whn α, only sys. has fasibl solution. 4 + C (ii), h sys. has only on st of solution und α. C (ii.a) Consid, α. As shown in (i.a) abov, sys. > sys. > sys.. (ii.b) Again, whn < α, only sys. has fasibl solution. 4 Poof of Poposition 3 (a) Whn 0 < α <, > ( b / ) 0, which is tu. Th poof fo > is b > m m staightfowad. Whn α /( ), > α > 0, which is again tu. Th poof fo > is staightfowad. m m (b) (i) Fom Tabl, it is obvious that = fo Modl R. m (ii) In Modl M, whn 0 < α <, > m ( > /, lationship holds. Whn α /(4 ), > α <, lationship holds. Oth lationships can b divd ath asily. m 30

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