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1 Contt in outouin Chnd, Him M pp, Sptmb 2006 Supvio: D. S. huli Vij Univitit Amtdm Fulty of Sin D olln HV Amtdm

2 Pf Th M-pp i on of th finl ompuloy ubjt of my tudy uin Mthmti nd nfomti (M t th vij Univitit in Amtdm. Th objtiv i to invtit th vilbl littu in fn to topi ltd to t lt two out of th th fild (Mthmti, Eonomi nd Comput Sin inttd in th tudy. Th ubjt fo thi pp w t in onulttion with d. S. huli of th Stohti oup of th Fulty of Sin. Hby, would lik to thnk my upvio, d. S. huli. Dpit hi buy hdul h md tim to uid m. m thnkful fo hi dvi nd ommnt. Him Chnd Sptmb

3 Summy Outouin i buin po tm fo hiin n xtnl ntity, n indpndnt ontto (ubontto, to do pifi tk o tk fo n oniztion in whih th oniztion ith do not hv th tim o th xpti to do on thi own. Outouin onit of two pti, th u ompny nd th ubontto, who hv onflitin intt. Fo xmpl, in ll nt outouin, th u ompny wnt to mk pofit nd iv optiml vi to hi utom nd th ubontto wnt to mk pofit with miniml ffot, ultin in poibly poo vi qulity. Tkin into ount th intt of both pti, oodintion i ny. Th on i tht by oodintion, th outouin upply hin n hiv th mximl pofit poibl. Thn, with pop ontt, th totl pofit n b plit btwn th u ompny nd th ubontto uh tht both pti btt off thn whn th outouin upply hin i not oodintd. n th fit pt of thi pp, ontt in ll nt outouin diud. Th onttin iu in n outouin upply hin onitin of u ompny nd ll nt, tht do outouin wok fo th u ompny, i dibd. H, th ll nt i modld n G/G/ quu with utom bndonmnt. Eh ll h vnu potntil, nd th ll nt vi qulity i modld by th pnt of ll vd nd olvd. Th ll nt mk two tti diion: how mny nt to hdul nd how muh ffot to xt to hiv th vi qulity. Of intt th ontt whih th u ompny n u to nfo th ll nt to both tff nd xt ffot t lvl tht optiml fo th outouin upply hin. n pi-ml (PM ontt, lo lld wholl ontt o lin ontt, th u ompny py th ll nt unit t b fo h ll vd. A pi-ml typ of vi ontt n oodint th tffin lvl of th ll nt. Howv, it i unbl to oodint th ffot lvl to hiv ytm-optimlity. n py-p-ll-olvd (PPCR ontt, th u ompny py th ll nt fo h ll vd nd olvd. Th PPCR ontt indu th ll nt to tff nouh popl o tht no ll lot, bu it vnu i ditly tid to th volum of ll vd. Futhmo, bu th ll nt t wdd only whn ll i vd nd olvd, it h n inntiv to xt ffot to in th vi qulity nd in th volum of ll olvd. Howv, quik ompion with th optiml ffot lvl und th inttd ytm vl tht th ffot lvl und th PPCR ontt i till not uffiintly hih to oodint th ll nt outouin upply hin. n th py-p-ll-olvd with ot hin (PPCR+CS ontt, th u ompny h th ll nt tffin nd ffot ot, whil th ll nt h th u ompny lo-of-oodwill ot. Thi oodintion hiv th mximl pofit in th outouin upply hin. u n bity plit i poibl, th i wy to h th totl pofit in uh wy tht both pti btt off thn whn th outouin upply hin i not oodintd. n th ptnhip, ontt th u ompny fit nd to h th tffin ot of th ll nt. Moov, in od to indu th ll nt to pnd mo ffot to impov vi qulity to mt ytm-optimlity, th ll nt pofit min nd to b djutd to mth tht of th whol outouin upply hin. Thi qui both pti to h ll of thi ot infomtion. n od to hiv oodintion, th ll nt nd th outouin ompny nd to ollbot loly. Th ontt ut tht lo ttntion h to b pid to vi qulity nd it onttibility in kin ll nt outouin. -2-

4 Th ond pt of thi pp diu ontt in th invntoy upply hin. Of th numou upply hin modl, th inl lotion b tok modl nd th two lotion b tok modl diud in th ond pt of thi pp. Th inl lotion b tok modl i tohti dmnd modl in whih th til iv plnihmnt fom uppli ft ontnt ld tim. Coodintion qui tht th til hoo hih tion, whih in thi modl i l b tok lvl. Th ot of thi hih tion i mo invntoy on v, but th uppli n vify th til invntoy nd thfo h th holdin ot of yin mo invntoy with th til. n th inl lotion modl th uppli oodint th upply hin with ontt tht h lin tnf pymnt bd on th til invntoy nd bkod. Coodintion with th infinit hoizon b tok modl i qulittivly th m oodintion in th inl piod nwvndo modl. n ptiul, oodintion vi holdin ot nd bkod ot tnf pymnt i imil to oodintion vi buy-bk ontt. Th two lotion b tok modl mk th uppli hold invntoy, lthouh t low holdin ot thn th til. Wh th fou in th inl lotion b tok modl i pimily on oodintin th downtm tion, in thi modl th uppli tion lo qui oodintion, nd tht oodintion i non-tivil. To b mo pifi, in th inl lotion modl th only itil iu i th mount of invntoy in th upply hin, but h th llotion of th upply hin invntoy btwn th uppli nd th til i impotnt wll. n th two lotion b tok modl dntlizd option lwy ld to ub-optiml pfomn, but th xtnt of th pfomn dtiotion (i.., th omptition pnlty dpnd on th upply hin pmt. f th fim bkod ot imil, thn th omptition pnlty i oftn onbly mll. Th omptition pnlty n b mll vn if th uppli do not bout utom vi bu h option ol in th upply hin my not b too impotnt. To oodint th til tion, th fim n to pi of lin tnf pymnt tht funtion lik th buy-bk ontt in th inl piod nwvndo modl. To oodint th uppli tion lin tnf pymnt bd on th uppli bkod i ud. With th ontt th optiml poliy i th uniqu Nh quilibium. Th ontt of th inl nd two lotion b tok modl mbl mot th py-p-ll-olvd with ot hin (PPCR+CS ontt of th ll nt. Out of thi it n b onludd tht ood ontt wd nd pnliz. -3-

5 Tbl of ontnt Pf... 1 Summy... 2 Tbl of ontnt ntodution Contt in ll nt ntodution Modl pmt An inttd outouin upply hin Dntlizd ytm of outouin Pi-ml (PM ontt Py-p-ll-olvd (PPCR ontt Py-p-ll-olvd with ot hin (PPCR+CS ontt Ptnhip ontt Contt in invntoy upply hin ntodution Coodintion in th inl lotion b tok modl Modl nd nlyi Coodintion in th two lotion b tok modl Modl Cot funtion hvio in th dntlizd m Coodintion with lin tnf pymnt Contt in nl Conluion Rfn Appndix

6 1 ntodution Thi pp diu oodintion of ontt in outouin, in ptiul, ontt in ll nt outouin nd ontt in invntoy upply hin outouin. Th tm outouin bm btt known lly bu of owth in th numb of hih-th ompni in th ly 1990 tht w oftn not l nouh to b bl to ily mintin l utom vi dptmnt of thi own [1]. Outouin o onttin out i buin po tm fo hiin n xtnl ntity, n indpndnt ontto (ubontto, to do pifi tk o tk fo n oniztion in whih th oniztion ith do not hv th tim o th xpti to do on thi own. Outouin involv tnfin inifint mount of mnmnt ontol nd diion-mkin to th outid uppli. An inin numb of ompni tnfin pt of thi wok to oth ounti; thi i lld offho outouin (offhoin. Philip hd ldy outoud pt of thi wok to oth ounti in th ihti. Mny ompni hv ind inifint ntiv publiity fo thi diion to u outoud lbo fo utom vi nd thnil uppot. n ft, fo om ompni, thi outouin tti bkfid nd thy hd to -vlut o vn bot thi outouin miion. On of th mot pominnt utom omplint i tht th outoud tff dliv low qulity of vi to utom. Outouin onit of two pti, th u ompny nd th ubontto, who hv onflitin intt. Fo xmpl, in ll nt outouin, th u ompny wnt to mk pofit nd iv optiml vi to hi utom nd th ubontto wnt to mk pofit with miniml ffot, ultin in potntilly poo vi qulity. Tkin into ount th intt of both pti, oodintion i ny. Th on i tht by oodintion th outouin upply hin n hiv th mximl pofit poibl. Thn, with pop ontt th totl pofit n b plit btwn th u ompny nd th ubontto uh tht both pti btt off thn whn th outouin upply hin i not oodintd. n oth wod, with oodintin ontt both pti n mk bi pi, nd h it in uh wy tht h t bi pi thn bfo. A ontt dtmin th ll pmt of th vi nd th ponibiliti of h pty nd mut py ttntion to th ombind vlu. Th dfinition of th poblm in thi pp n b wittn : How to oodint th diffnt ply in th hin (th u ompny nd th ubontto to hiv ytm optimlity? Fiu 1: Top on fo outouin [2] n th fit hpt of thi pp, ontt in ll nt outouin diud. Th onttin iu in n outouin upply hin onitin of u ompny nd ll nt, tht do outouin wok fo th u ompny, i dibd. H, th ll nt i modld n G/G/ quu with utom bndonmnt. Eh ll h vnu potntil, nd th ll nt vi qulity i modld by th pnt of ll vd nd olvd. Th ll nt mk two tti diion: how mny nt to hdul nd how muh ffot to xt to hiv th vi qulity. -5-

7 Of intt th ontt whih th u ompny n u to nfo th ll nt to both tff nd xt ffot t lvl tht optiml fo th outouin upply hin. Two ommonly ud ontt nlyzd fit: th pi-ml nd th py-p-ll-olvd ontt. Althouh thy n oodint th tffin lvl, thy ult in vi qulity tht i blow ytm optiml. Thn, two ontt popod tht n oodint both. Th followin hpt iv diption of ontt in invntoy upply hin outouin. An optiml upply hin pfomn qui th xution of pi t of tion. Unfotuntly, tho tion not lwy in th bt intt of th individul mmb in th upply hin, i.., th upply hin mmb pimily onnd with optimizin thi own objtiv, nd tht lfvin fou oftn ult in poo ovll pfomn. Howv, optiml pfomn i hivbl if th fim oodint by onttin on t of tnf pymnt uh tht h fim objtiv bom lind with th upply hin objtiv. Of th numou upply hin modl, th inl lotion b tok modl nd th two lotion b tok modl diud. Th inl lotion b tok modl i tohti dmnd modl in whih th til iv plnihmnt fom uppli ft ontnt ld tim. Coodintion qui tht th til hoo hih tion, whih in thi modl i l b tok lvl. Th ot of thi hih tion i mo invntoy on v, but th uppli n vify th til invntoy nd thfo h th holdin ot of yin mo invntoy with th til. Th two lotion b tok modl mk th uppli hold invntoy, lthouh t low holdin ot thn th til. Wh th fou in th inl lotion b tok modl i pimily on oodintin th downtm tion, in thi modl th uppli tion lo qui oodintion, nd tht oodintion i non-tivil. To b mo pifi, in th inl lotion modl th only itil iu i th mount of invntoy in th upply hin, but h th llotion of th upply hin invntoy btwn th uppli nd th til i impotnt wll. Thi hpt i followd by ontt in nl. Finlly, Chpt 5 iv ummy nd onluion on th findin. -6-

8 2 Contt in ll nt 2.1 ntodution A owin numb of ompni movin thi ll nt option offho. Th ovhd ot of utom vi typilly l whn outouin i ud, ldin to mny ompni to lo thi in-hou utom ltion dptmnt nd outouin thi utom vi to thid pty ll nt. Th loil xtnion of th diion w th outouin of lbo ov to ounti with low lbo ot. Du to thi dmnd, ll nt hv pun up in Cnd, Chin, Etn Euop, ndi, Si Lnk, l, lnd, Pkitn, th Philippin, nd vn in th Cibbn. Fiu 2: Offhoin dtintion woldwid [3] Outouin oftn bin immdit ot-vin. ut ompni hould kp in mind tht th n lo b om hiddn ot in outouin. Svi qulity ot i on of thm. Th ll nt i lly inviibl to th nd utom. Thfo, whn th ll nt und-pfom, it i th u ompny tht uff utom bklh. Fo thi on, th vi qulity of ll nt nd to b tkn into ount whn mkin outouin diion, nd nd to b fully mnd by th u ompny. Svi qulity i mud by th pnt of ll tht vd to th utom tiftion. u h vi nount i uniqu, it my not b poibl to ontt ditly on vi qulity. Thfo it i impotnt to find ontt tht n indu th ll nt to xt ffot to povid hih vi qulity. Thi hpt will nw th followin qution: 1. How fftiv th ll nt outouin ontt ommonly obvd in pti? A pi-ml ontt, py-p-ll-olvd ontt, nd py-p-ll-olvd plu ot hin ontt mon th ommonly obvd ontt in ll nt outouin. A thy pbl of induin ytm-optiml tffin lvl nd ffot (to impov vi qulity of th ll nt? 2. How to hiv oodintion, i.., ytm-optiml tffin lvl nd vi qulity, with onttin mhnim? n pti, th hv bn diffnt fom of obvd ll nt outouin mnt. Som ompni tk hnd-off ppoh, whil oth ompni fom ptnhip with thi outou, hin t-up nd optin ot. Wht fom of outouin hould on hoo? 2.2 Modl pmt Conid ll nt outouin upply hin tht onit of two ptly mnd ompni: ll nt (th outou nd u ompny. Th ll nt onidd h i typilly l -7-

9 (mployin hundd of nt, nd i modld multi-v quuin ytm (G/G/ with utom bndonmnt. Cotum iv to th ll nt with t λ, nd th ll nt tff v (i.., nt h with vi t µ. Cutom who not immditly vd upon ivl nt witin quu. Thy imptint, nd lv th ytm ft ndom mount of tim, whih h ontinuouly diffntibl Pobbility Dnity Funtion (PDF f nd Cumultiv Ditibution Funtion (CDF F. Th witin ot t i w, nd fo h utom tht bndon, th i ot of. Fo th utom who vntully vd, th i ftion p of thm who ll tiftoily olvd. Fo h ll tht i olvd, vnu i nd. Fo th t, ftion 1 p of th ll (vd but not olvd, th i lo of oodwill fo h ll. W um th ll olution pobbility p i non-ntiv ontinuou ndom vibl with uppot (0, 1 nd CDF G (p. W um tht p i inflund by th ll nt ffot, whih my b unobvbl o unvifibl by th u ompny. Th xptd vlu of th pnt ll olvd, dnotd p( fo ivn ffot lvl, i thn 1 [ 1 G( p ]. p ( = dp (1 0 Effot i otly to th ll nt with unit t of. Hiin tff lo ot th ll nt mony with unit t of. Lit of ymbol: λ Aivl t µ Svi t p( Pnt of ll olvd funtion of ffot Numb of v/nt T( Numb of otum vd in tdy tt W( Witin tim in tdy tt Q( Numb of otum witin (quu lnth L( Numb of bndonmnt in tdy tt Unit vnu Unit tffin ot Unit ffot ot Abndonmnt ot w Unit witin ot Lo of oodwill fom ll vd but not olvd 2.3 An inttd outouin upply hin W fit look t th inttd outouin upply hin wh th ll nt nd th u ompny ownd by th m ompny, nd mk th ntlizd diion on th optiml tffin lvl nd th ffot lvl. y tffin nt nd xtin n ffot, th ytm totl pofit i: (, = p(( T µ L( λw ( ( 1 p( T (. (2 vnu tffin & ffot o t bndonmnt o t w witin o t lo of oodwill o t W umin tht th ffot ot not dpndnt on th numb of v hid (th tffin lvl. Th inttd ytm olv th followin pofit mximiztion poblm: -8-

10 mx (,. (3, t i hd to find xt olution to G/G/+M quuin ytm, whih pohibit on to nlyz th optimiztion poblm (3. Thfo, on n u fluid ppoximtion fo nl G/G/ + G modl. A fluid ppoximtion llow fo nlytil ttbility, fom whih on n in impotnt mnil iniht. n th fluid ppoximtion, th numb of buy v in tdy tt i th minimum of th ivl t nd th numb of v: T ( = min( λ, µ. Th bndonmnt t i + L( = (λ µ = λ T (, (4 + ( x = mx 0, x. wh { } Futhmo, by uin Tylo i ppoximtion of th CDF F ound t = 0, nd with mild umption tht f (0 0, on n obtin ltivly impl ltionhip on tdy tt witin tim: L( W ( =. (5 f (0λ Thi ut tht in th fluid ppoximtion th witin tim i popotionl to th bndonmnt t. Of ou, th ltionhip do not ll pply to th oiinl tohti modl, but thy do ptu th fit-od fft of th quuin ytm. Pluin (1, (4, nd (5 into (2, nd tkin th xpttion w hv f (0 w (, = p( ( λ L( µ L( L( ( 1 p( ( λ L( = w [ p( ( 1 p( ] λ p( + + ( 1 p( L( µ f (0 To void th itution wh th xptd pofit i ntiv, w um tht p( 0 > + ( 1 p( 0. Tht i, th unit vnu i uffiintly l ompd to th ot. Thi wy, th inttd ytm h poitiv pofit min by optin it ll nt. Sin L( i pi-wi lin nd onvx in, th xptd pofit lo bhv nily with pt to. y olvin th fit-od ondition of (6 (th poof of it n b n in th Appndix th pofit-mximizin tffin lvl nd ffot fo th inttd ytm uh tht: = λ, (7 µ p ( =, if < p0; = 0 othwi. (8 + λ + λ ( ( Th olution in (7 indit tht th inttd ytm would tff nouh v to mt dmnd. Thfo, in tdy tt th i no utom bndonmnt o witin. On th oth hnd, th olution in (8 indit tht th ytm optimlly bln th ot nd bnfit of inin ffot. Fo xmpl, whn th vnu t in, it ll fo n in in ffot bu it xptd bnfit fom mo olvd ll bom l. Thi i tu fo th oodwill ot nd utom ivl t λ wll. Howv, whn th ot of ffot in, th ll nt will xt l ffot. f th tio i ov th thhold ( + λ xt no ffot. 0 p (ll tht p = p ( 0 0 = (6, th ll nt would -9-

11 At optimlity, th totl outouin upply hin pofit i (, = [ p + p ] λ. (9 With thi inttd modl bnhmk, th dntlizd of ll nt outouin will now b invtitd. 2.4 Dntlizd ytm of outouin n th dntlizd outouin upply hin, th u ompny nd th ll nt two indpndnt ntiti. Th u ompny off ontt to outou it ll nt funtion to th ll nt. f th ll nt pt th ontt, it hoo it tffin lvl wll it ffot lvl. Th ll nt v inomin ll fom th utom of th u ompny, nd t pid by th u ompny. Th u ompny, on th oth hnd, iv vnu fom tho ll vd nd olvd by th ll nt. u th ll nt i inviibl to th utom whn thy ll, it i th u tht b th ntiv onqun of utom bndonmnt, utom ditiftion fom witin, nd ll not bin olvd. Th mount th u ompny py to th ll nt i dnotd by Ψ. Ψ i pifid in th ontt nd ould b funtion of numb of vibl uh T, W, o L. Th tk of ontt din i to dtmin wht fto dtmin Ψ, nd how. Fom (2, th totl xptd outouin upply hin pofit i dompod into two pt: (, fo th ll nt u, fo th u ompny: nd ( u (, = E[ Ψ] µ, (, = p(( T L( λw ( ( 1 p( T ( E[ Ψ]. w Th ol i to idntify tho ontt tht n hiv two objtiv und th dntlizd ttin: (1 hiv ytm-optiml tffin nd ffot lvl, nd (2 hiv n bity plit of th ytm pofit btwn th u ompny nd th ll nt. Th dvnt of oodintion i tht th outouin upply hin n hiv th mximum pofit poibl. Thn, with pop ontt, th totl pofit n b plit btwn th u ompny nd th ll nt uh tht both pti btt off thn whn th outouin upply hin i not oodintd. Th ky h i to how th ll nt mk tffin nd ffot diion in th dntlizd ttin ompd to th inttd outouin upply hin. t obviouly dpnd on th pifi fom of th ontt Ψ. Fo xmpl, th implt fom of Ψ i fixd pymnt, i.., Ψ = σ, wh σ i ontnt, lthouh it i y to tht fixd pymnt will not indu th ll nt to tff o xt ffot t th ytm-optiml lvl. n th nxt tion, fou pifi fom of ontt will b dibd: pi-ml ontt, py-p-ll-olvd ontt, py-p-ll-olvd with ot hin ontt, nd ptnhip ontt Pi-ml (PM ontt A pi-ml ontt, lo lld wholl ontt o lin ontt, i th mot ommonly obvd ontt in th induty. Th u ompny py th ll nt unit t b fo h ll vd: Ψ = bt (. (10 n thi ontt, th u ompny only nd to did b nd th pymnt dpnd only on T(. n od fo th ontt to b ptbl to both pti, w hv < b <. Und thi ontt, th pofit fo both th outouin upply hin pti : PM u PM (, = bt ( µ, (, = p(( T L( λw ( ( 1 p( ( T bt ( w (11-10-

12 A pi-ml typ of vi ontt n oodint th tffin lvl of th ll nt. Howv, it i unbl to oodint th ffot lvl to hiv ytm-optimlity. Whn th ontt link th ll nt inom to th numb of ll vd, th ll nt h n inntiv to tff dqutly. Unfotuntly, th pi-ml ontt do not link th ll nt inom to th volum of ll olvd. A tionl pon, th ll nt h no inntiv to xt ny ffot to in it vi qulity. Conquntly, th outouin upply hin h optimlity only lon on of th two impotnt dimnion; it i not fully oodintd. Th om mnil iniht tht on n dw fom th ult bov. Fit, it indit tht if w look t ll nt on nd l, wh th dtminiti pt of fluid ppoximtion domint th tohti vition t th dtild quu lvl, thn lin pi-ml ontt n wok wll in tm of oodintin th tffin lvl. Plu, it h impl fom nd i y to implmnt. Thi miht xplin th populity of thi ontt in ll nt outouin. Howv, uh ontt do not dd th vi qulity in outouin. Thfo, whn vi qulity i untin nd dpnd on th ll nt pivt tion (i.., ffot, th u ompny nd mo thn jut pi-ml ontt to hiv th vi qulity it di Py-p-ll-olvd (PPCR ontt n od to tk th ll nt ffot into ount, ntul xtnion of th pi-ml ontt i to py th ll nt fo h ll vd nd olvd: ( T (. Ψ = bp (12 Unlik in th p-ml ontt, h b i th unit t th u py th ll nt fo h ll vd nd olvd. Th pymnt to th ll nt now dpnd on p(t(, whih i dtmind by both nd. Und thi ontt, th pofit fo both outouin upply hin pti : PPCR u PPCR (, = bp(( T µ, (, = p(( T L( λw ( ( 1 p( T ( bp((. T w (13 Th py-p-ll-olvd ontt n not only oodint tffin lvl, but lo motivt th ll nt to xt ffot to impov vi qulity. Simil to th p-ml ontt, th py-p-ll-olvd ontt indu th ll nt to tff nouh popl o tht no ll will b lot, bu it vnu i ditly tid to th volum of ll vd. Futhmo, bu th ll nt t wdd only whn ll i vd nd olvd, it h n inntiv to xt ffot to in th vi qulity nd in th volum of ll olvd. Howv, quik ompion with th optiml ffot lvl und th inttd ytm vl tht th ffot lvl und th py-p-ll-olvd ontt i till not hih nouh to oodint th ll nt outouin upply hin. Undlyin thi ult i th diffn in th pofit min of th inttd ytm nd tht of th ll nt. n th inttd ytm, by xtin n dditionl unit of ffot (t th minl ot of, th bnfit to th ytm i th in in vnu fom dditionlly olvd ll nd th dution in th lo-of-oodwill ot, ( + λ. Fo th ll nt und th PPCR ontt, howv, n dditionl unit of ffot would only in th vnu by b λ, whih i l thn ( + λ. Du to thi diffn in th pofit min, th ll nt will und-invt in th ffot. Thi undinvtmnt n b viwd doubl minliztion poblm nloou to tht obvd in n invntoy upply hin. t i wll known tht in n invntoy upply hin, doubl minliztion u th til to od l thn th upply hin optiml (with lin wholl ontt. -11-

13 Doubl minliztion ou whn th uptm (mnuftu nd downtm (til mkt not pftly omptitiv, nd th podut i tdd with unifom wholl pi. f monopoly mnuftu uppli ood to monopoly til nd h th til monopoliti pi, th ult i doubl minliztion piin [6]. Fiu 3 povid n illuttion. Fiu 3: Doubl minliztion oth th wholl nd th til mk up th pi bov thi full ot, uin ffiiny o ddwiht lo qul to th tinl A nd in Fiu 3. f th i on thin wo thn monopolit, it i two uiv monopolit. Monopolit h thi utom mkup bov ot. n of doubl minliztion, w hv mkup on top of mkup. Th inttin thin i, not only i doubl minliztion bd fo onum, but tully th fim thmlv my nd up with low pofit. Howv, uh poblm do not xit in th ll nt outouin upply hin, bu th ll nt, whih povid vi to nd utom, i pid by th u ompny, nd not by th utom. Whn th pofit min of th ll nt do not mth tht of th inttd outouin upply hin, th ll nt would tionlly xt l ffot, ultin in n infio vi qulity, ompd to tht in n inttd outouin upply hin. n oth wod, in tm of oodintin th ll nt ffot to impov vi qulity, typ of doubl minliztion xit in th outouin upply hin wll. th ontt tht n fix th doubl minliztion poblm in th outouin upply hin? Th nw i poitiv, but th fom of thi ontt will dpnd on whth ffot i onttibl. W tudy both blow Py-p-ll-olvd with ot hin (PPCR+CS ontt Mny ompni ldy u ot-plu typ of ontt, whih ll fo th hin of ot infomtion. n th, th u ompny n popo to h popotion of th ll nt ot in od to indu th ll nt to xt nouh ffot to oodint th outouin upply hin. Spifilly, onid th followin ontt. Th u ompny modifi th py-p-llolvd ontt by hin ftion ( 1 α of th ll nt ot, wh α = b /. n oth wod, th ll nt now py popotion α of th totl tffin nd ffot ot: α ( µ +. Moov, th ll nt py pnlty of α fo h ll vd but not olvd. n ummy, th u ompny h th ll nt tffin nd ffot ot, whil th ll nt h th u ompny lo-of-oodwill ot. Thfo, th onttul pymnt i -12-

14 Ψ = bp 1 α (0 (14 f w (( T + ( ( µ + α ( 1 p( ( T + L( + L(. W ll thi ontt py-p-ll-olvd plu ot hin ontt (PPCR+CS. A in th pyp-ll-olvd ontt, b i th unit t fo h ll vd nd olvd, nd it (o quivlntly α i th only diion vibl th u ompny h to onid in dinin th ontt. Und thi ontt, th outouin upply hin pti pofit : PPCR+ CS u PPCR+ CS w [ ] T ( µ L( L( (, = α p( ( 1 p( w [ ] T ( µ L( L(. (, = ( 1 α p( ( 1 p( f (0 f (0, (15 Clly, und th PPCR+CS ontt, th ll nt h of th totl ytm pofit i α, nd th u h i (1-α. Thfo th ll nt inntiv i ompltly lind with tht of th outouin upply hin. So it will tk th ytm-optiml tion in tffin nd ffot, nd th upply hin will b oodintd. y oodintion, th outouin upply hin n hiv th mximum pofit. u n bity plit i poibl, th i wy to h th totl pofit in uh wy tht both pti btt off thn whn th outouin upply hin i not oodintd. Th PPCR+CS ontt m quit diffiult bu th two pti nd to h infomtion bout mny ot itm. Not, howv, tht th bndonmnt nd witin ot in (14, L( nd w L(, v th m pupo: to punih th ll nt fo und-tffin. A lon on of f (0 thm i in th ontt, th ontt will ontinu to oodint th ytm. Thfo, by movin ith on of thm, w n implify th PPCR+CS ontt. Thi i not th only ot-hin ontt tht n oodint th upply hin. Fo xmpl, onid th followin ontt. Th u ompny h only th ffot ot, ( 1 α, but not th tffin ot. t n b hown tht if th u ompny lo t α = b / ( +, thn th ll nt would lo b indud to tff nd xt ffot t th ytm-optiml lvl. Howv, th dwbk of thi ontt i tht it nnot hiv n bity plit of pofit, bu th xptd pofit of th ll nt i no lon xtly α popotion of th totl ytm pofit Ptnhip ontt Oftntim th ll nt ffot nnot b obvd o vifid. Fo xmpl, it i oftn diffiult to mu ll nt mn ffot in upviin hi tff nd olvin dy-to-dy optionl poblm. f ontt li on th ll nt to tuthfully pot it mn upviion nd tinin ffot, thn th ll nt h n inntiv to ov-tt it ffot. n f of thi hlln, w popo th followin ontt, whih onit of two pt. Fit, th ll nt h o py f to v h ll, xludin th bndond ll, but t to kp ll th vnu fom vd nd olvd ll. Thi f i ( 1 α [ p( ( 1 p( ], wh [ 0,1] α i th only ontt pmt th u ompny nd to did. Th ll nt mut py pnlty of to th u ompny fo h ll vd but not olvd. -13-

15 Sond, th u ompny h th ll nt tffin nd ffot ot by pyin th ll nt 1 α µ + 1 α No ot will b hd whn th ll nt tff no v (i.., = 0. ( (. t i impotnt to not tht bu th ll nt ffot n not b obvd nd vifid, ll th pymnt in th ontt nnot b bd on th ll nt l ffot lvl. ntd, thy bd on th ll nt l tffin lvl (whih n b obvd nd th hin-optiml ffot lvl (whih n b lultd. Pt of th u pymnt to th ll nt, ( 1 α i fixd. Whn α i too mll, th fixd pymnt my b o hih tht it indu th ll nt to kp miniml tffin lvl nd xt no ffot. n thi, th ol pupo of th ll nt option i to ollt th fixd pymnt. Thi olution i lly imptil. To void thi tivil itution, w um in th followin diuion tht th u will hoo uffiintly l α. Th ontt h th flvo of tht of fnhi, in th n tht it lt th ll nt n vnu ditly fom th utom, nd py fnhi f bd on ll volum. ut it i mo thn ommon fnhi ontt, bu th u ompny lo h pt of th ot of th ll nt. Thfo it i mo lik ptnhip. Und thi ptnhip ontt (whih w dnot PART, th tnf pymnt i Ψ = p + ( T ( ( 1 α [ p( ( 1 p( ] T ( (( T 1 p( ( 1 α µ + ( 1 α α + f w ( 0 L (, (16 nd th pofit fo both pti in th outouin upply hin : PART u PART [ ] T ( ( 1 α [ p( ( 1 p( ] T ( (, = p( ( 1 p( α µ + ( 1 α ( 0 (, (, = ( 1 α [ p( ( 1 p( ] T ( ( 1 α ( 1 α µ ( 1 α. α + f w L + f w ( 0 L ( (17 Thi ptnhip ontt n lo u th pmt α to divid th pofit. n od to hiv th full ffiiny in ll nt outouin, lo ltionhip btwn th u ompny nd th ll nt h to b td. t i inttin to not tht th m to b dit oltion btwn th infomtion ontnt of ontt nd it fftivn in tm of hivin ytm ffiiny. n th pi-ml ontt, ll th u ompny nd to know i th ll nt thouhput T in od to lult it onttul pymnt. n th py-p-ll-olvd ontt, th u ompny nd to hv infomtion on pt, th ll nt olvd ll, o quivlntly, th vnu ntd by th ll nt. n th py-p-llolvd plu ot hin ontt, th u ompny lo nd to know th ot infomtion of th ll nt o tht h ould h pt of it. Moov, th u ompny h to h hi own ot with th ll nt o tht h ould hold th ll nt ponibl fo uboptiml tion. Collbotion nd infomtion hin i pilly impotnt whn th vi qulity i not ditly onttibl. A i dmonttd in th ptnhip ontt, th u ompny fit nd to h th tffin ot of th ll nt. Moov, in od to indu th ll nt to pnd mo ffot to impov vi qulity to mt ytm-optimlity, th ll nt pofit min nd to b djutd to mth tht of th whol outouin upply hin. Thi qui both pti to h ll of thi ot -14-

16 infomtion. Thi ppoh dmnd th lo ollbotion btwn th two pti. Alo in pti, fom n ountin point of viw, th tntion mo omplitd, nd h pty nd to kp lo tk of thi bookkpin. n hot, oodintion in outouin nnot b don in impl hnd-off fhion by om impl-tm ontt. Th hih ffiiny n outouin upply hin wnt to hiv, th lo ollbotion btwn outouin upply hin pti i ndd. -15-

17 3 Contt in invntoy upply hin 3.1 ntodution Optiml upply hin pfomn qui th xution of pi t of tion. Unfotuntly, tho tion not lwy in th bt intt of th mmb in th upply hin, i.., th upply hin mmb pimily onnd with optimizin thi own objtiv, nd tht lf-vin fou oftn ult in poo pfomn. Howv, optiml pfomn i hivbl if th fim oodint by onttin on t of tnf pymnt uh tht h fim objtiv bom lind with th upply hin objtiv. Th numou upply hin modl, but in thi hpt th inl lotion b tok modl nd th two lotion b tok modl diud. n h modl th upply hin optiml tion idntifid. n h, th fim ould implmnt tho tion, i.., h fim h to th infomtion ndd to dtmin th optiml tion nd th optiml tion fibl fo h fim. Howv, fim lk th inntiv to implmnt tho tion. To t tht inntiv, th fim n djut thi tm of td vi ontt tht tblih tnf pymnt hm. A numb of diffnt ontt typ idntifid nd thi bnfit nd dwbk illuttd. Thi hpt will nw th followin qution: 1. Whih ontt oodint th upply hin? A ontt i id to oodint th upply hin if th t of upply hin optiml tion i Nh quilibium, i.., no fim h pofitbl uniltl dvition fom th t of upply hin optiml tion. dlly, th optiml tion hould lo b uniqu Nh quilibium; othwi th fim my oodint on ub-optiml t of tion. 2. Whih ontt hv uffiint flxibility (by djutin pmt to llow fo ny diviion of th upply hin pofit mon th fim? f oodintin ontt n llot pofit bitily, thn th lwy xit ontt tht Pto domint non-oodintin ontt, i.., h fim pofit i no wo off nd t lt on fim i titly btt off with th oodintin ontt. 3. Whih ontt woth doptin? Althouh oodintion nd flxibl nt llotion dibl ftu, ontt with tho popti tnd to b otly to dminit. A ult, th ontt din my tully pf to off impl ontt vn if tht ontt do not optimiz th upply hin pfomn. A impl ontt i ptiully dibl if th ontt ffiiny i hih (th tio of upply hin pofit with th ontt to th upply hin optiml pofit nd if th ontt din ptu th lion h of upply hin pofit. 3.2 Coodintion in th inl lotion b tok modl Thi tion onid modl with pptul dmnd nd mny plnihmnt oppotuniti. Th b tok invntoy poliy i optiml: with b tok poliy fim mintin it invntoy poition (on-od plu in-tnit plu on-hnd invntoy minu bkod t ontnt b tok lvl. t i umd, fo ttbility, tht dmnd i bkodd, i.., th no lot l. A ult, xptd dmnd i ontnt (i.., it do not dpnd on th til b tok lvl. Optiml pfomn i now hivd by minimizin totl upply hin ot: th holdin ot of invntoy nd th bkod pnlty ot. n thi modl th uppli inu no holdin ot, but th uppli do bout th vilbility of h podut t th til lvl. To modl tht pfn, it i umd tht bkod t th til inu ot t th uppli. Sin th til do not onid tht ot whn hooin b tok lvl, on n how tht th til hoo b tok lvl tht i low thn optiml fo th upply hin, whih mn tht th til i too littl invntoy. Coodintion i hivd nd ot bitily llotd by povidin inntiv to th til to y mo invntoy. -16-

18 3.2.1 Modl nd nlyi Conid uppli tht ll inl podut to inl til. Lt L b th ld tim to plnih n od fom th til. Th uppli h infinit pity, o th uppli kp no invntoy nd th til plnihmnt ld tim i lwy L, no mtt wht th til od quntity i. (Th two fim, but only th til kp invntoy, whih i why thi i onidd inl lotion modl. Lt µ = E[ D ], wh D i th dmnd. Lt F nd f b th ditibution nd dnity funtion of D, ptivly: um tht F i titly inin, diffntibl, nd F ( 0 = 0 whih ul out th poibility tht it i optiml to y no invntoy. Th til inu invntoy holdin ot t t h > 0 p unit of invntoy. Fo nlytil ttbility, dmnd i bkodd if tok i not vilbl. Th til inu bkod pnlty ot t t β > 0 p unit bkodd. Th uppli h unlimitd pity, o th uppli do not nd to y invntoy. Howv, th uppli inu bkod pnlty ot t t β > 0 p unit bkodd t th til. n oth wod, th uppli inu ot whnv utom wnt to puh th uppli podut fom th til but th til do not hv invntoy. Thi ot flt th uppli pfn fo mintinin uffiint vilbility of h podut t th til lvl in th upply hin. Lt β = β + β, o β i th totl bkod ot t inud by th totl upply hin. Sl ou t ontnt t µ du to th bkod umption, no mtt how th fim mn thi invntoy. A ult, th fim only onnd with thi ot. oth fim ik nutl. Th til objtiv i to minimiz hi v invntoy holdin nd bkod ot p unit tim. Th uppli objtiv i to minimiz h v bkod ot p unit tim. Dfin th til invntoy lvl to b qul to th invntoy in-tnit to th til plu th til on-hnd invntoy minu th til bkod. (Thi h lo bn lld th fftiv invntoy poition. Th til invntoy poition qul hi invntoy lvl plu on-od invntoy (invntoy odd, but not yt hippd. Sin th uppli immditly hip ll od, th til invntoy lvl nd poition idntil in thi ttin. Lt ( y b th til xptd invntoy t tim t + L whn th til invntoy lvl i y t tim t : y y y y y = 0 0 ( y ( y x f ( x dx F ( x dx. = (18 ( y x f ( x dx = 1 f ( x dk dx = f ( k dk dx = F ( xdx 0 0 x Lt ( y y x b th nloou funtion tht povid th til xptd bkod: = ( y ( x y f ( x dx = y ( y. + y y µ (19 nvntoy i monitod ontinuouly, o th til n mintin ontnt invntoy poition. n thi nvionmnt it n b hown tht b tok poliy i optiml. With tht poliy th til ontinuouly od invntoy o tht hi invntoy poition lwy qul hi hon b tok lvl,. Lt ( b th til v ot p unit tim whn th til implmnt th b tok poliy : = h + β ( ( ( = β ( µ + ( h + β (. Givn th til b tok poliy, th uppli xptd ot funtion i -17-

19 Lt ( ( ( = β ( = β ( µ + (. b th upply hin xptd ot p unit tim, = + ( ( ( = β ( µ + ( h + β (. o i titly onvx, o th i uniqu upply hin optiml b tok lvl,. t tifi th followin itil tio qution o o β ( = F ( =. (21 h + β Lt b th til optiml b tok lvl. Th til ot funtion i lo titly onvx, o * * tifi * β ( =. F h + β * o Givn β < β, it follow fom th bov two xpion tht <, i.., th til hoo b tok lvl tht i l thn optiml. Hn, hnnl oodintion qui th uppli to povid th til with n inntiv to i hi b tok lvl. Suppo th fim to ontt tht tnf fom th uppli to th til t vy tim t ( y t ( y, t + wh y i th til invntoy lvl t tim t nd t nd t ontnt. Futhmo, onid th followin t of ontt pmtizd by λ ( 0,1], t = 1 λ h ( t = β λβ. (H w hoo to ul out λ = 0, in thn ny b tok lvl i optiml. Givn on of tho ontt, th til xptd ot funtion i now = β t µ + h + β t t (22 ( ( ( ( (. Th ontt pmt hv bn hon o tht β t = λβ > 0, nd h + β t t = λ h + β ( > 0. t follow fom (20 nd (22 tht with th ontt = λ (23 ( (. o Hn, minimiz th til ot, i.., tho ontt oodint th upply hin. n ddition, tho ontt bitily llot ot btwn th fim, with th til h of th ot inin in th pmt λ. Not, th λ pmt i not xpliitly inopotd into th ontt, i.., it i mly ud fo xpoitionl lity. Now, onid th in of th t nd th t pmt. Sin th ontt mut indu th til to hoo hih b tok lvl, it i ntul to onjtu t > 0, i.., th uppli ubidiz th til invntoy holdin ot. n ft, tht onjtu i vlid whn λ ( 0,1]. t i lo ntul to uppo < 0, λ 0,1 impli t, β (20 t i.., th uppli pnliz th til fo bkod. ut ( ] β i.., with om ontt th uppli ubidiz th til bkod ( t > 0 [, : in -18-

20 tho itution th uppli nou bkod by ttin t > 0, bu without tht noumnt th l invntoy ubidy ld th til to * o >. Th bov nlyi i minin of th nlyi with th nwvndo modl nd buy-bk ontt. Thi i not oinidn, bu thi modl i qulittivly idntil to th nwvndo modl. To xplin, bin with th til pofit funtion in th nwvndo modl (umin = = = υ = 0 : ( q = ps( q wq = ( p w q p( q. Th til pofit h two tm, on tht in linly in q, nd th oth tht dpnd on th dmnd ditibution. Now lt p = h + β nd w = h. n tht, ( q = β q ( h + β ( q = ( q + β µ. Hn, th i no diffn btwn th mximiztion of ( q nd th minimiztion of (. Now ll tht th tnf pymnt with buy-bk ontt i wq b( q, i.., th i pmt (i.., w tht fft th pymnt linly in th til tion (i.., q, nd pmt tht influn th tnf pymnt thouh funtion (i.., ( q tht dpnd on th til tion nd th dmnd ditibution. n thi modl t ( y + t ( y = ( t + t ( y + t ( µ y, o t i th lin pmt nd t + t i th oth pmt. n th buy-bk ontt th pmt wok indpndntly. To t th m fft in th b tok modl th uppli ould dopt tnf pymnt tht dpnd on th til invntoy poition, nd th til invntoy. Tht ontt would yild th m ult. 3.3 Coodintion in th two lotion b tok modl Th two lotion b tok modl build upon th inl lotion b tok modl diud in th pviou tion. Now th uppli no lon h infinit pity. ntd, h mut od plnihmnt fom h ou nd tho plnihmnt lwy filld within L tim (i.., h ou h infinit pity. So in thi modl th uppli njoy libl plnihmnt but th ld tim of th til plnihmnt dpnd on how th uppli mn h invntoy. Only if th uppli h nouh invntoy to fill n od, thn th til iv tht od in L tim. Othwi, th til mut wit lon thn L to iv th unfilld potion. Tht dly ould ld to dditionl bkod t th til, whih otly to both th til nd th uppli, o it ould ld to low invntoy t th til, whih hlp th til. n th inl lotion modl th only itil iu i th mount of invntoy in th upply hin. n thi modl th llotion of th upply hin invntoy btwn th uppli nd th til i impotnt wll. Fo fixd mount of upply hin invntoy th uppli lwy pf tht mo i llotd to th til, bu tht low both h invntoy nd bkod ot. (Rll tht th uppli i hd fo til bkod. On th oth hnd, th til pfn i not o l: l til invntoy mn low holdin ot, but lo hih bkod ot. Th lo ubtl inttion with pt to th totl mount of invntoy in th upply hin. Th til i bid to y too littl invntoy: th til b th full ot of hi invntoy but only iv potion of th bnfit (i.., h do not bnfit fom th dution in th uppli bkod ot. On th oth hnd, th i no l bi fo th uppli bu of two fft. Fit, th uppli b th ot of hi invntoy nd do not bnfit fom th dution in th til bkod ot, whih bi th uppli to y too littl invntoy. Sond, th uppli do not b th ot of th til invntoy (whih in lon with th uppli invntoy, whih bi th uppli to y too muh invntoy. Eith bi n domint, dpndin on th pmt of th modl. -19-

21 Evn thouh it i not l whth th dntlizd upply hin will y too muh o too littl invntoy (howv, it nlly i too littl invntoy, it i hown tht th optiml poliy i nv Nh quilibium of th dntlizd m, i.., dntlizd option i nv optiml. Howv, th omptition pnlty (th pnt of lo in upply hin pfomn du to dntlizd diion mkin vi onidbly: in om th omptition pnlty i ltivly mll,.., l thn 5%, wh in oth it i onidbl,.., mo thn 40%. Thfo, th nd fo oodintin ontt i not univl Modl Lt h, 0 < h < h, b th uppli p unit holdin ot t inud with on-hnd invntoy. (Whn h h th optiml poliy do not y invntoy t th uppli nd whn h 0 th optiml poliy h unlimitd uppli invntoy. Nith i inttin.. Th fim optin diion hv no impt on th mount of in-tnit invntoy, o no holdin ot i hd fo ith th uppli o th til piplin invntoy. Lt D > 0 b dmnd duin n intvl of tim with lnth L. (A in th inl lotion modl, D > 0 nu tht th uppli i om invntoy in th optiml poliy. Lt F nd f b th ditibution nd dnity funtion of tht dmnd. A with th til, um F i inin nd diffntibl. Lt [ ] µ = E D. Rtil od bkodd t th uppli but th i no xpliit h fo tho bkod. Th uppli till inu p unit bkod ot t t β fo bkod t th til. Th ompbl ot fo th til i till β. Evn thouh th no dit onqun to uppli bkod, th indit onqun: low til invntoy nd hih til bkod. oth fim u b tok polii to mn invntoy. With b tok poliy, fim i {, } od invntoy o tht it invntoy poition min qul to it b tok lvl, i. (Rll tht fim invntoy lvl qul on-hnd invntoy minu bkod plu in-tnit invntoy nd fim invntoy poition qul th invntoy lvl plu on-od invntoy. Th b tok polii opt only with lol infomtion, o nith fim nd to know th oth fim invntoy poition. Th fim hoo thi b tok lvl on nd imultnouly. Th fim ttmpt to minimiz thi v ot p unit tim. (Givn tht on fim u b tok poliy, it i optiml fo th oth fim to u b tok poliy wll. Thy both ik nutl. Th xit pi of b tok o o lvl, {, }, tht minimiz th upply hin ot. Hn, it i fibl fo th fim to optimiz th upply hin, but inntiv onflit my pvnt thm fom doin o. Th fit tp in th nlyi of thi modl i to vlut h fim v ot. Th nxt tp vlut th Nh quilibium b tok lvl. Th thid tp idntifi th optiml b tok lvl nd omp thm to th Nh quilibium on. Th finl tp xplo inntiv tutu to oodint th upply hin Cot funtion th fim nd th upply hin xptd ot inud t tim t + L t th til lvl whn th til invntoy lvl i y t tim t. Howv, in th two lotion modl th til invntoy lvl do not lwy qul th til invntoy poition,, bu th uppli my b out of tok. Lt i (, b th v t t whih fim i inu ot t th til lvl nd (, = (, + (,. To vlut i, not tht t ny ivn tim t th uppli invntoy poition i (bu th uppli u b tok poliy. At tim D o th uppli A in th inl lotion modl, ( y, ( y, nd ( y t + ith th uppli on-hnd invntoy i ( + L -20-

22 bkod qul ( + + ( D. So D. Thfo, th til invntoy lvl t tim t + L i i = i (, F ( ( + ( + x f ( x dx : t tim t + L th uppli n i th til invntoy lvl to F, othwi th til invntoy lvl qul + D. d on nloou onin, lt (, nd ( lvl: Lt (, i with pobbility (, b th til v invntoy nd bkod ivn th b tok (, = F ( ( + ( + x f ( x dx, (, = F ( ( + ( + x f ( x dx. i b fim i totl v ot t. Sin th til only inu ot t th til lvl, =, Lt ( ( (., b th uppli v invntoy. Anloou to th til funtion (dfind in th pviou tion Th uppli v ot i y ( y = F ( x dx. 0 (, = h ( + (,. Lt (, b th upply hin totl ot, ( = (, + (, hvio in th dntlizd m, Lt i ( j b n optiml b tok lvl fo fim i ivn th b tok lvl hon by fim j, i.., i ( j i fim i bt pon to fim j tty. Diffntition of h fim ot funtion dmontt tht h fim ot i titly onvx in it b tok lvl, o h fim h uniqu bt pon. With Nh quilibium pi of b tok, { *, * }, nith fim h pofitbl uniltl dvition, i.., * * * * = ( nd = (. Exitn of Nh quilibium i not ud, but in thi m xitn of Nh quilibium follow fom th onvxity of th fim ot funtion. (Thnilly it i lo quid tht th fim tty p hv n upp bound. mpoin tht bound h no impt on th nlyi. n ft, th xit uniqu Nh quilibium. To dmontt uniqun, tt by boundin h ply fibl tty p, i.., th t of tti ply my hoo. Fo th til it i not diffiult to how tht ( > ˆ > 0, wh ŝ minimiz ( y, i.., β F ( ˆ =. h + β n oth wod, if th til w to iv pftly libl plnihmnt, th til would hoo ŝ, o th til tinly do not hoo < ˆ if plnihmnt unlibl. (n oth -21-

23 wod, ŝ i optiml fo th til in th inl lotion modl diud in th pviou tion. Fo th uppli, ( > 0, bu (, / < 0 ivn F ( = 0 nd ( y < 0. Uniqun of th Nh quilibium hold if fo th fibl tti, > ˆ nd > 0, th bt ply funtion onttion mppin, i.., i ( j < 1. (24 Fom th impliit funtion thom nd ( ( = = F " ( + x f ( x " " ( ( + ( + x f ( x " dx ( + x f ( x dx " [ h ( ] f ( + ( + x f ( x " " Givn > ˆ nd > 0, it follow tht ( x > 0, ( y < 0, ( y > 0, nd F ( > 0. Hn, (24 hold fo both th uppli nd th til. A uniqu Nh quilibium i quit onvnint, in tht quilibium i thn onbl pdition fo th outom of th dntlizd m. (With multipl quilibi it i not l tht th outom of th m would vn b n quilibium, in th ply my hoo tti fom diffnt quilibi. Hn, th omptition pnlty i n ppopit mu of th p btwn optiml pfomn nd dntlizd pfomn, wh th omptition pnlty i dfind to b * * o o (, (,. o o (, n ft, th lwy xit poitiv omptition pnlty, i.., dntlizd option lwy ld to uboptiml pfomn in thi m. To xplin, not tht th til minl ot i lwy t thn th upply hin (, (, >, bu ( > (. Sin both (, nd (, titly onvx, it follow tht, fo ny th til optiml b tok i lwy low thn th upply hin optiml b tok. o o o o Hn, vn if th uppli hoo, th til do not hoo, i.., ( <. Althouh th Nh quilibium i not optiml, th mnitud of th omptition pnlty dpnd on th pmt of th modl. Whn th fim bkod pnlti th m (i.., β β = 1 th mdin omptition pnlty fo thi mpl i 5% nd th omptition pnlty i no t thn 8% in 95% of thi obvtion. Howv, vy l omptition pnlti obvd whn ith β β < 1 9 o β β > 9. Th til do not hv ton onn fo utom vi whn β β < 1 9, nd o th til tnd to y f l invntoy thn optiml. Sin th uppli do not hv dit to utom, th uppli n do littl to pvnt bkod in tht itution, nd o th upply hin ot i ubtntilly hih thn nd b. n th oth xtm, β > 9, th uppli littl bout utom vi, nd thu do not y nouh invntoy. β n tht itution th til n till pvnt bkod, but to do o qui ubtntil mount of invntoy t th til to ount fo th uppli lon ld tim. Th upply hin ot i ubtntilly hih thn th optiml on if th optiml poliy h th uppli y invntoy to dx, dx. -22-

24 povid libl plnihmnt to th til. Howv, th itution in whih th optiml poliy do not qui th uppli to y muh invntoy: ith th uppli holdin ot i nly hih th til (in whih kpin invntoy t th uppli iv littl holdin ot dvnt o if th uppli ld tim i hot (in whih th dly du to lk of invntoy t th uppli i nliibl. n tho th omptition pnlty i ltivly mino. n th inl lotion modl dntliztion lwy ld to l invntoy thn optiml fo th upply hin. n thi ttin th inttion btwn th fim mo omplx, nd o dntliztion nlly ld to too littl invntoy, but not lwy. Sin th til bkod ot t i low thn th uppli bkod ot t, fo fixd th til lwy i too littl invntoy, whih tinly ontibut to l thn optiml mount of invntoy in th ytm. Howv, th til i only pt of th upply hin. n ft, th uppli invntoy my b o l tht vn thouh th til i too littl invntoy, th totl mount of invntoy in th dntlizd upply hin my xd th upply hin optiml quntity. Suppo β i quit mll nd β i quit l. n tht, th til i vy littl invntoy. To ttmpt to mitit th build up of bkod t th til lvl th uppli povid th til with vy libl plnihmnt, whih qui l mount of invntoy, n mount tht my ld to mo invntoy in th upply hin thn optiml. Th min onluion fom th nlyi of th dntlizd m i tht th omptition pnlty i lwy poitiv, but only in om iumtn it i vy l. t i pily in tho iumtn tht th fim ould bnfit fom n inntiv hm to oodint thi tion Coodintion with lin tnf pymnt o Supply hin oodintion in thi ttin i hivd whn { }, i Nh quilibium. n th inl lotion modl th uppli oodint th upply hin with ontt tht h lin tnf pymnt bd on th til invntoy nd bkod. Suppo th uppli off th m nmnt in thi modl with th ddition of tnf pymnt bd on th uppli bkod: t + t, + t wh t, t, nd (, ( (, t ontnt nd ( ( y = µ y ( y. i th uppli v bkod: + Rll tht poitiv vlu fo th bov xpion pnt pymnt fom th uppli to th til nd ntiv vlu pnt pymnt fom th til to th uppli., n infomtion ytm i ndd fo th uppli to vify Whil both fim n ily obv ( th til invntoy nd bkod. Th fit tp in th nlyi povid om ult fo th optiml olution. Th ond tp dfin t of ontt nd onfim tht tho ontt oodint th upply hin. Thn, th llotion of ot i onidd. Finlly, it i hown tht th optiml olution i uniqu Nh quilibium. Th tditionl ppoh to obtin th optiml olution involv llotin ot o tht ll ot pvd. tok polii thn optiml nd ily vlutd. Howv, to filitt th ompion of th optiml poliy to th Nh quilibium of th dntlizd m, it i uful to vlut th optiml b tok poliy without tht tditionl ot llotion. Givn tht (, i ontinuou, ny optiml poliy with > 0 mut t th followin two minl to zo (, = F ( ( + ( + x f ( x dx, (25 nd o -23-

25 Sin ( > 0, nd ~1 (, = F ( h + ( + x f ( x dx. F th i only on poibl optiml poliy with { ~ 1 0,, ~ 1 > }, wh ~ 1 ~1 (, (26 ~1 tifi = (27 h 1 ~1 tifi ( = 0. (27 implifi to F ( =, h + β ~1 xit nd i uniqu. Sin (,, ~ ~1 ~1 o it i ppnt tht i titly onvx in, xit nd i uniqu. Th my lo xit n optiml poliy with 0. n tht, th ndidt polii ~ 2, ~ 2 wh ~ , ~ + ~, nd tifi { } = 0 ( x f ( x dx = 0. h + β (28 β Th bov implifi to P( D + D =, h + β o xit nd i uniqu. ~1 Givn tht (, i titly onvx in, ( ~ , ~ ( ~, ~ < ~1,0 < 0 ~1 Sin (, 0 i inin in fom (11, tht ondition hold whn <, othwi it do not. Thfo, { ~ 1, } ~1 ~1 i th uniqu optiml poliy whn <, othwi ny { ~ 2, ~ 2 } i optiml: { } { ~ } =, ~ ~ < ~ o o, ~ 2 2 ~ 1, ~ ~. 2 whnv (. { } Now onid th fim bhvio with th followin t of ontt pmtizd by ( 0,1] t ( λ h, λ, = 1 (29 t β λβ, (30 = o ( (. F t = 1 λ h (31 o F Th til ot funtion, djutd fo th bov ontt i ( y = ( h t ( y + ( β t ( y t ( = λ( y t ( nd o = λ, t (32 (, ( (. Rll (, = (, + (,, o (, = h ( + ( 1 λ (, + t ( (33 = ( h + t ( + ( 1 λ (, + t ( µ. o o o Th two to onid: ith > 0 o = 0. Tk th fit. f > 0, thn (27 impli o o o o o o (, λ (, (, = = λ = 0. 1 λ o o Futh, (, i titly onvx in nd (, i titly onvx in, o { o, } i indd Nh quilibium. n ft, it i th uniqu Nh quilibium. Fom th impliit funtion -24-

26 thom, ( i din, wh in th dntlizd m without th ontt, ( i bt pon to fim Hn, fo ( j tty, d ( = f " ( + x f ( x " " ( ( + ( + x f ( x = nd λ 1, th uppli minl ot i inin: ( ( dx dx i fim i j 0. ( + t t., = F ( h ( 1 λ ( ( tht tifi ( ( Thu, th i uniqu =, i.., th i uniqu Nh quilibium. Now uppo 0 0. t i tihtfowd to onfim tht ll of th { ~ 2, ~ 2 } pi tify th fim fit nd ond od ondition. Hn, thy ll Nh quilibi. Evn thouh th i not uniqu Nh quilibium, th fim ot idntil o th quilibi. Th ontt do llow th fim to bitily llot th til lvl ot in th ytm, but thy do not llow th fim to bitily llot ll of th upply hin ot. Thi limittion i du to th λ 1 tition, i.., it i not poibl with th ontt to llot to th til mo thn th optiml til lvl ot: whil th til ot funtion i wll bhvd vn if λ > 1, th uppli i not; with λ > 1 th uppli h ton inntiv to in th til lvl ot. Of ou, fixd pymnt ould b ud to hiv tho llotion if ny. ut in it i unlikly til would to uh budn, thi limittion i not too titiv. An inttin ftu of th ontt i tht th t nd t tnf pymnt idntil to th on ud in th inl lotion modl. Thi i mkbl bu th til itil tio diff o th modl: in th inl lotion modl th til pik uh tht β F = β (, + h wh in th two lotion modl th til pik uh tht β + h F ( =. β + h n th two lotion b tok modl dntlizd option lwy ld to ub-optiml pfomn, but th xtnt of th pfomn dtiotion (i.., th omptition pnlty dpnd on th upply hin pmt. f th fim bkod ot imil, thn th omptition pnlty i oftn onbly mll. Th omptition pnlty n b mll vn if th uppli do not bout utom vi bu h option ol in th upply hin my not b too impotnt (.., if h i l o if L i mll. To oodint th til tion, th fim n to pi of lin tnf pymnt tht funtion lik th buy-bk ontt in th inl piod nwvndo modl. To oodint th uppli tion lin tnf pymnt bd on th uppli bkod i ud. With th ontt th optiml poliy i th uniqu Nh quilibium. -25-

27 4 Contt in nl n thi hpt ontt in nl will b diud thouh ontt in th ll nt nd in th upply hin. Th ontt of th ll nt litd blow: Pi-ml (PM ontt: Ψ = bt (. Py-p-ll-olvd (PPCR ontt: Ψ = bp( T (. Py-p-ll-olvd with ot hin (PPCR+CS ontt: w Ψ = bp(( T + ( 1 α ( µ + α 1 p( T + L + L (0 f Ptnhip ontt: Ψ = p T 1 p T 1 α p 1 p T (( ( ( ( ( (. ( ( ( [ ( ( ( ] ( w + ( 1 α µ + ( 1 α α + L(. f ( 0 f th ontt of th upply hin wittn in th fom of th ontt of th ll nt thy will hv th followin fom: Th inl lotion b tok modl ontt: Ψ = t ( y + t ( y, wh y i th til invntoy lvl t tim t nd t nd t ontnt. ( y i th til xptd invntoy nd ( y i th til xptd bkod. Th two lotion b tok modl ontt: Ψ = t (, + t (, + t (, wh t ontnt nd ( i th uppli v bkod. t, t, nd Th ontt of th inl nd two lotion b tok modl mbl mot th py-p-ll-olvd with ot hin (PPCR+CS ontt of th ll nt. Out of thi w n onlud tht ood ontt wd nd pnliz. n th PPCR+CS ontt th u ompny ubidiz th ll nt tffin nd ffot ot nd th ll nt h to py bndonmnt nd witin ot to th u ompny, punihmnt fo und-tffin. n th inl lotion b tok modl ontt th uppli inu ot whnv utom wnt to puh th uppli podut fom th til but th til do not hv invntoy. Thi ot flt th uppli pfn fo mintinin uffiint vilbility of h podut t th til lvl in th upply hin. n th two lotion b tok modl pymnt fom th uppli to th til h to b i poitiv nd ntiv vlu pnt pymnt fom th til to th uppli. md if ( A nl ontt will hv th followin fom: Ψ, = h, + h, wh >, h 0 2 1( ( ( 1, 2 ( i 1, 2 i th ot funtion of th invntoy modl. h if ( < i 1, 2 in in 1 nd

28 5 Conluion Whn ompny outou it ll nt funtion to n xtnl ll nt, th ntlizd ytm hn into n outouin upply hin. n thi pp th oodintion of thi outouin upply hin with ontt h bn tudid. Spifilly, fom th u point of viw, it i inttin to tudy how to indu th ll nt to tff nouh nt nd xt nouh ffot to hiv hih vi qulity. Th ll nt i modld multi-v quu with utom bndonmnt. t i hown tht viou ontt btwn th ll nt nd th u ompny ult in diffnt tffin nd ffot lvl. n pi-ml (PM ontt, th u ompny py th ll nt unit t b fo h ll vd. A pi-ml typ of vi ontt n oodint th tffin lvl of th ll nt. Howv, it i unbl to oodint th ffot lvl to hiv ytm-optimlity. n py-p-ll-olvd (PPCR ontt, th u ompny py th ll nt fo h ll vd nd olvd. Th PPCR ontt indu th ll nt to tff nouh popl o tht no ll will b lot, bu it vnu i ditly tid to th volum of ll vd. Futhmo, bu th ll nt t wdd only whn ll i vd nd olvd, it h n inntiv to xt ffot to in th vi qulity nd in th volum of ll olvd. n th py-p-ll-olvd with ot hin (PPCR+CS ontt, th u ompny h th ll nt tffin nd ffot ot, whil th ll nt h th u ompny lo-of-oodwill ot. y oodintion, th outouin upply hin n hiv th mximum pofit. n th ptnhip ontt, th u ompny fit nd to h th tffin ot of th ll nt. Moov, in od to indu th ll nt to pnd mo ffot to impov vi qulity to mt ytm-optimlity, th ll nt pofit min nd to b djutd to mth tht of th whol outouin upply hin. Thi qui both pti to h ll of thi ot infomtion. n od to hiv oodintion in thi ttin, th ll nt nd th outouin ompny nd to ollbot loly. Th ontt ut tht lo ttntion h to b pid to vi qulity nd it onttibility in kin ll nt outouin. Of th numou upply hin modl, th inl lotion b tok modl nd th two lotion b tok modl diud in thi pp. Th inl lotion b tok modl i tohti dmnd modl in whih th til iv plnihmnt fom uppli ft ontnt ld tim. Coodintion qui tht th til hoo hih tion, whih in thi modl i l b tok lvl. Th ot of thi hih tion i mo invntoy on v, but th uppli n vify th til invntoy nd thfo h th holdin ot of yin mo invntoy with th til. Th two lotion b tok modl mk th uppli hold invntoy, lthouh t low holdin ot thn th til. Wh th fou in th inl lotion b tok modl i pimily on oodintin th downtm tion, in thi modl th uppli tion lo qui oodintion, nd tht oodintion i non-tivil. To b mo pifi, in th inl lotion modl th only itil iu i th mount of invntoy in th upply hin, but h th llotion of th upply hin invntoy btwn th uppli nd th til i impotnt wll. Th ontt of th inl nd two lotion b tok modl mbl mot th py-p-ll-olvd with ot hin (PPCR+CS ontt of th ll nt. n th PPCR+CS ontt th u ompny ubidiz th ll nt tffin nd ffot ot nd th ll nt h to py bndonmnt nd witin ot to th u ompny, punihmnt fo undtffin. n th inl lotion b tok modl ontt th uppli inu ot whnv utom wnt to puh th uppli podut fom th til but th til do not hv invntoy. Thi ot -27-