Mahemaical Problems in Engineering Volume 216, Aricle ID 36565, 9 pages hp://dx.doi.org/1.1155/216/36565 Research Aricle A Composie Conrac for Coordinaing a Supply Chain wih Price and Effor Dependen Sochasic Demand Yu-Shuang Liu, Yun Shang, and Hong-yan Su School of Mahemaics and Physics, Qingdao Universiy of Science and Technology, Qingdao 26661, China Correspondence should be addressed o Yu-Shuang Liu; uslys@126.com Received 25 January 216; Acceped 9 June 216 Academic Edior: Emilio Insfran Copyrigh 216 Yu-Shuang Liu e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. As he demand is more sensiive o price and sales effor, his paper invesigaes he issue of channel coordinaion for a supply chain wih one manufacurer and one reailer facing price and effor dependen sochasic demand. A composie conrac based on he uaniy-resriced reurns and arge sales rebae can achieve coordinaion in his seing. Two main problems are addressed: (1) how o coordinae he decenralized supply chain; (2) how o deermine he opimal sales effor level, pricing, and invenory decisions under he addiive demand case. Numerical examples are presened o verify he effeciveness of combined conrac in supply chain coordinaion and highligh model sensiiviies o parameric changes. 1. Inroducion Channel coordinaion via designing perfec conrac is an imporan opic in supply chain managemen. I is well known ha if boh he manufacurer and he reailer maximize heir own profis, double marginalizaion, a phenomenon firs idenified by Spengler [1], will prevail. Tha is, he reailer will order less han he inegraed supply chain s opimal invenory level, which negaively affecs he supply chain performance. A properly designed conrac can compleely eliminae he problem of double marginalizaion. Tha is, he supplychainconracsareusedodriveheoalexpecedprofi of he decenralized supply chain oward ha achieved under an inegraed sysem. In his paper, we refer o his pracice as supply chain coordinaion. In he majoriy of supply chain conrac lieraure, he reailer impacs sales only hrough his socking decision, bu in realiy a reailer can spur sales hrough many differen aciviies. Probably he firs influenial one is he reail price. When he reail price is given exogenously, conracs such as reurns (buy-back), revenue sharing, uaniy flexibiliy, and salesrebaehavebeenshownobesuccessfulincoordinaing a wo-echelon supply chain [2]. When he demand is sensiive o he reail price, he reailer is needed o decide he opimal reail price and order uaniy joinly. Unforunaely, coordinaion is hard o achieve by he fac ha he incenives o align he reailer s uaniy decision may disor he reailer s price decision. Apar from he reail price, he reailer s sales effor is also imporan in influencing demand. A reailer can hire more sales people, improve heir raining, increase adverising, provide aracive shelf space, and guide consumer purchases wih sales personnel. All of hose aciviies are cosly. A conflic exiss beween he manufacurer and reailer, because hose aciviies benefi boh firms bu are cosly o only one. If he demand is considered only o be dependen on he reailer s sales effor, Cachon [2] demonsraes ha boh reurns policy and sales rebae conrac fail o coordinae a supplychaininhisseing,becausehereailer sopimal effor level is lower han ha of he inegraed supply chain under he buy-back conrac, while he reailer exers oo much effor under he sales rebae conrac. As he demand is more sensiive o price and sales effor, reail price and sales effor are key decisions in reailing. Some lieraure such as [3 5] sudy he supply chain wih he price and effor dependen demand; however, he demand is assumed o be deerminisic. In his paper, we consider a sochasic demand for a produc a he reailer during a single selling season ha is price and effor dependen. If he reail price, sales effor, and order uaniy are decision variables
2 Mahemaical Problems in Engineering ogeher, here is a lack of effecive coordinaing conracs in he lieraure. He e al. [6] have shown ha coordinaion canno be achieved by a single conrac mechanism, for reurns or arge sales rebae conrac alone. They find ha only he properly designed reurns policy wih sales rebae and penaly conrac are able o achieve channel coordinaion and lead o a Pareo improving win-win siuaion for supply chain members. Differen from [6], we will ransform full reurns ino uaniy-resriced reurns and sales rebae and penaly ino argesalesrebae.thiscomposieconracalsocanachieve perfec coordinaion. The main idea behind his composie conrac is o use he wo disincive conracs o align he reailer s hree decisions wih he opimal decisions of he cenralized sysem. Conseuenly, in his paper, our analyical resuls lend insigh ino how a manufacurer should use uaniy-resriced reurns and arge sales rebae provisions in a conrac in order o coordinae he reailer who makes hree decisions simulaneously. Afer ha, we use an addiive demand model o show ha here exis he opimal pricing, sales effor, and ordering decisions uniuely in he cenralized supply chain. The paper is organized as follows. Secion 2 provides a survey of relaed research on composie conracs. In Secions 3 and 4, we inroduce he composie conrac in deail, and hen we presen he basic models for a cenralized and a decenralized supply chain under he price and effor dependen demand cases, respecively. We invesigae he role of he composie conrac for supply chain coordinaion and profi allocaion. For he addiive demand model, Secion 5 proves ha here exiss he opimal effor-pricing-ordering join decision uniuely and uses numerical example o illusrae our resuls. Secion 6 provides concluding remarks. 2. Relaed Lieraure on Composie Conracs Supply chain conracs have received much aenion from researchers; comprehensive reviews of he supply chain conracing lieraure are provided by Cachon [2]. One of he commonly used supply chain conracs for shor-life-cycle producs is he reurns policy, wherein he manufacurer agrees o buy back unsold goods from he reailer a a prespecified price ha is less han he wholesale price. The main objecive of he reurns policy is o miigae he risk of oversocking faced by he reailer, which is caused by he uncerain demand. Pasernack [7] is he firs o sudy a reurns conrac, he demonsraes ha reurns policy (e.g., parial reurns wih full buy-back credi or full reurns wih parial buy-back credi) can achieve channel coordinaion wih price-independen demand. Sales rebae is a converse of reurns policy in ha he manufacurer pays he reailer based on he reailer s sales performance. There are wo common forms of channel rebaes. One is linear rebae, in which he manufacurer pays he reailer a fixed rebae foreachunisold.theoherformisargesalesrebae,in which he reailer will be graned a rebae for each uni sold beyond a specified arge level. As Taylor demonsraed in [8], a properly designed arge sales rebae achieves channel coordinaion and a win-win oucome, while he linear rebae fails o coordinae when demand is no influenced by sales effor. In conras, here is a limied research on hybrid conracs. To fill his research gap, we invesigae he role of some combined conracs for miigaing he double marginalizaion effec and improving supply chain performance. Wih regard o he reailer s decision-making behavior, Gan e al. [9] consider a supply chain wih a risk-neural supplier and a downside-risk-averse reailer and design a risk-sharing conrac ha offers he desired downside proecion o he reailer. Ineresingly, he conrac hey propose is a composie conrac, which involves adding a reurn policy o he iniial conrac. Noe ha he reurn policy is differen from ha inpasernack[7]inhesensehaheamounofrefundable invenory in heir conrac does no exceed a cerain level. Wang and Webser [1] consider a supply chain wih a riskneural manufacurer and a loss-averse reailer. They idenify a class of disribuion-free coordinaing conracs by combining gain/loss-sharing and buy-back conracs, which arbirarily allocae he expeced supply chain profi beween he manufacurer and reailer. Under he general sochasic demand, Xiong e al. [11] propose a composie conrac based on he buy-back conrac and uaniy flexibiliy conrac. They discuss is advanages over he buy-back conrac and uaniy flexibiliy conrac in erms of coordinaion, profi allocaion, and risk allocaion. Liu e al. [12] also sudy a selling sysem consising of one supplier and one reailer, which has a shor life cycle, bu is lead ime of producion is longer han is selling cycle; a he same ime, is selling price is decreased gradually o he end cusomer under uncerain demand. They develop a wo-period dynamic model o depic he above case and design a combined conrac wih markdown money and reurn policy o coordinae he channel. When demand is influenced by reailer s sales effor, Taylor [8] designs a conrac by combining a arge rebae conrac and a reurns conrac o achieve supply chain coordinaion and a win-win oucome. As reail pricing is an imporan vehicle o enhance supply chain revenue, Chiu e al. [13] show ha a policy ha combines he use of wholesale price, channel rebae, and reurns (PRR) can coordinae a channel wih boh addiive and muliplicaive price-dependen demands. In fac,chiueal. sconracisahybridoffullreurnsandarge sales rebae. Tha is, he manufacurer allows he reailer o reurn all he unsold producs a he end of season. He e al. [6] invesigae he issue of channel coordinaion wih a hybrid conrac for a supply chain facing sochasic demand ha is sensiive o boh sales effor and reail price. Based on [6], we explore he fac ha anoher composie conrac also can achieve channel coordinaion in his seing. 3. Model Assumpions and he Inegraed Supply Chain Consider a supply chain model wih one manufacurer M and one reailer R.Lep be he reail price, le c be he producion cos of manufacurer, le V be he salvage value, and le be he reailer s order uaniy. The reailer faces a random
Mahemaical Problems in Engineering 3 demand ha is sensiive o boh reail price and sales effor. To model reail effor, we suppose a single effor level e which summarizes he reailer s aciviies and le g(e) be he reailer s cos of exering effor level e,whereg() =, g (e) >,and g (e) >. X denoes he random demand during a shor selling season, which is decreasing in price and increasing in effor. Given he price p and he effor level e, f(x p, e) and F(x p, e) are he probabiliy densiy funcion and disribuion funcion. S(p, e, ) is expeced sales given he reail price, he effor level, and he socking uaniy; ha is, S(p,e,)= dx. (1) Then he expeced lef over invenory is I(p, e, ) = S(p, e, ). A he beginning of he selling season, M offers a conrac o R, and hen R, knowing he manufacurer s conrac, deermines he order uaniy, he reail price, and effor level. The purpose of he manufacurer offering a composie conrac o he reailer is o coordinae he supply chain and maximize he enire supply chain s profi. To do so, we sar our analysis wih he cenralized supply chain. The inegraed channel s profi is π (p, e, ) =ps(p, e, ) + VI (p, e, ) c g(e). (2) In order o faciliae he analysis, (2) can be rearranged as follows: π (p, e, ) = ps (p, e, ) dx dx c g(e). Furhermore, here is no need o show ha he inegraed channel profi funcion is concave or unimodal. Le (e,p, ) be he opimal decision for he cenralized supply chain. The parial derivaives of π(p, e, ) wih respec o each ofhedecisionvariablesareasfollows: π(p,e,) π(p,e,) π(p,e,) S (p, e, ) =p dx dx g (e), S (p, e, ) =S(p,e,)+p dx dx, S (p, e, ) =p + VF( p,e) c. (3) (4) Seing hese derivaives o zero, (e,p, ) saisfy he following firs-order opimaliy condiions: π (p,e, ) = π (p,e, ) =. = π (p,e, ) 4. The Decenralized Supply Chain under Composie Conrac 4.1. The Composie Conrac. In his secion, M offers a composie conrac, denoed by C(w,, r, b), byorganicallycombining wo popular conracs: reurns policy and arge sales rebae conrac. A he beginning of he selling season, M makes four decisions: he uni wholesale price w, he arge sales level, herebaevaluer perunisoldbeyondhehreshold,andhe buy-back price b for each unsold uni reurned by R.Forany uni of produc sold by he reailer which exceeds arge sales level,hereailerwillbegranedarebae.buheuaniyha canbereurnedohemanufacurerisresriced.morespecific,ifheacualsalesarenogreaerhan,onlyhevolume ha exceeds he arge level can be reurned o he manufacurer wih he buy-back price b. To he remaining unsold producs, R hasnochoicebuoreceivehesalvagevaluev per uni a he end of he selling season. Clearly, above composie conrac wih four parameers is a hybrid of uaniy-resriced reurns conrac and arge sales rebae conrac, which is similar in some sense o he one sudied by Chiu e al. [13]. The only difference is o resric uaniies of reurns, while in Chiu e al. s conrac he unsold producs are full reurns. As Chiu e al. said in [13], in he reailing indusries such as Marahon Spor (MarahonSporisoneofhelargessporswearreailers in Hong Kong who sells muliple brands of spors apparels; differen manufacurers such as Nike or Adidas offer differen arge sales rebae programs wih differen duraions ha vary beween 1 and 2 monhs; Marahon Spor can earn a rebae value for each iem only when Marahon Spor sells more han a prespecified sales arge; o reduce he risk of oversocking because of he rebae program, Marahon Spor can reurn unsold iems (up o a cerain percenage of he order uaniy) a he end of he season), i is a common pracice o be provided wih he reurns and arge sales rebae. On he oherhandialsodiffersfromheeal. sconrac.thekey difference beween He e al. s conrac is ha for uaniy below arge here is neiher a penaly nor a buy-back and i is he reailer s responsibiliy. In addiion, no only does he composie conrac wih uaniy-resriced reurns and arge sales rebae help alleviae he double marginalizaion phenomenon, bu also he conrac parameers are free of demand funcion. For any given composie conrac C(w,, r, b), R has he objecive of deermining joinly he socking uaniy, he uni selling price p, and he level of effor e o maximize her expeced profi. The arge sales rebae conrac is ineresing only if i achieves supply chain coordinaion for [2]; (5)
4 Mahemaical Problems in Engineering we herefore assume ha in he following model. The reailer s profi funcion under he composie conrac is Π c R (p,e,,x)=p(min (, x)) + r (min (, x) )+ The expeced revenue is +b( max (, x)) + + V ( x) + w g(e). π c R (p, e, ) = ps (p, e, ) dx Then, + (b r) dx (w r) r g(e). π c R (p, e, ) S (p, e, ) =p + (b r) g (e), dx dx π c R (p, e, ) S (p, e, ) =S(p,e,)+p + (b r) dx dx, π c R (p, e, ) S (p, e, ) =p + (b r) F( p,e) (w r). We now verify ha he composie conrac is capable of coordinaing he supply chain. The composie conrac designed by M is said o coordinae he supply chain if i is able o enice he reailer o se heir opimal decision as (e,p, ).Thus,ifheopimalsoluion(e,p, ) for he inegraed channel is o be adoped by he reailer, i mus opimize heir expeced profi funcion π c R (p, e, ), and hence i mus saisfy he following firs-order condiions: π c R (p,e, ) = πc R (p,e, ) = πc R (p,e, ) =. By comparing (4) wih (8), we find ha (e,p, ) saisfies (9) only if b and r saisfy he cerain condiions as follows: b =w c+v, (1) r =w c. (6) (7) (8) (9) In oher words, if (1) holds, he composie conrac can coordinae he enire supply chain. I is significan ha he parameers for coordinaion are no dependen on he sochasic demand funcion. Now consider he allocaion of channel profi. Subsiuing (1) ino (7), he reailer s profi funcion given a coordinaing composie conrac is π c R (p,e, )=π(p,e, ) r. (11) Hence, he manufacurers profi funcion is π c M (p,e, )=r=(w c). (12) The manufacurer earns a profi margin for each uni up o he sales arge and has a zero profi margin for each reurned uni because w b c+v =.Insummary,wegivehe following proposiion. Proposiion 1. The composie conrac C(w,, r, b) based on uaniy-resriced reurns and arge sales rebae can achieve channel coordinae if and only if he manufacurer ses conrac parameers as b =w c+v and r =w c,andheoalsupply chain profi can be spli as [(w c), π(p,e, ) (w c)] beween he manufacurer and he reailer. Noice ha he opimal parameers for coordinaion in Proposiion 1 are similar o He e al. s model. This connecion beween he wo conracs can be inerpreed as follows. In he decenralized supply chain, he reailer s profi depends on he locaion of <wih respec o he sochasic demand X. Specifically, he sochasic demand X has he following hree cases: (1) <X<,(2) <<X,and (3) X<<. In case (1), since he final sales uaniy min{, X} is always greaer han he arge, M gives R he rebae r for each uni sold beyond. A he same ime, R can fully reurn heir unsold producs o he manufacurer. In case (2), he reailer s sales uaniy always exceeds he arge and all he producs are sold ou. Hence, R can only ge he rebae from he manufacurer. Obviously, in hese wo cases hey are euivalen. In case (3), as X < <,hereailercan never exceed he arge. Thus, he reailer could no ge rebae and can receive parly reurns. In paricular, only he volume beyond he arge can be reurned o M wih he buy-back price b. To he remaining unsold producs, R receives he salvage value V peruniaheendofhesellingseason.as a conseuence, he reailer s revenue of unsold producs x is b( ) + V( X).WihHeeal. sconrac,hereailercan fully reurn heir unsold producs o he manufacurer and will need o pay he manufacurer a penaly: a paymen of r foreachuniofunsoldproducbelow.inhisseing,he reailer s revenue of unsold producs Xis b( X) r( X), which can be rearranged as b( )+(b r)( X). Therefore, he reailer s revenue of unsold producs Xis exacly he same if we se b r=v. Sincewehaveesablishedhesamecondiionsforachieving coordinaion in he supply chain under a decenralized seing, however, i is realisic when his composie conrac resrics he reurns only for he uaniy in excess of he sales
Mahemaical Problems in Engineering 5 arge. Meanwhile, limiing he number of buy-backs is much easier o accep han a penaly. In fac, he uaniy flexibiliy conrac is also a kind of uaniy-resriced reurns in which he uaniy ha can be reurned is resriced as δ wih he wholesale price. Webser and Weng [14] also consider uaniy-resriced reurns. From his poin of view, he composie idea proposed in his paper seems more naural as an alernaive conrac under he price and effor dependen sochasic demand. 4.2. Pareo Improvemen. In his secion, we proceed o give he analyical condiion for seing he sales arge ha can achieve Pareo improvemen under he composie conrac.thewholesalepriceconracusuallycanbeusedasa benchmark for comparing he channel performance. In order o induce he reailer o accep he composie conrac, he manufacurer can choose an aracive sales arge o achieve Pareo improvemen so ha boh he manufacurer and he reailer can obain a higher expeced profi han ha of under he wholesale price conrac. Under a wholesale price conrac, he manufacurer charges he reailer w for each uni, he reailer hen decides his opimal effor level e d,reailingpricep d, and order uaniy d. The manufacurer s and he reailer s profi funcions are π d R (p,e,,w)=ps(p,e,)+vi(p,e,) w g(e), π d M (p,e,,w)=(w c). (13) The very basic aims of he reailer and he manufacurer wouldbeokeepheirprofihigherhanhainhewholesale price conrac; ha is, π c R (p,e, ) π d R (pd,e d, d,w), π c M (p,e, ) π d M (pd,e d, d,w). (14) 5. Opimal Soluions and Numerical Analysis 5.1. The Opimal Decisions under he Addiive Demand Model. FromSecion4,weknowhaifhemanufacureroffersabove composie conrac, he reailer will choose he inegraed supply chain s opimal decisions. Therefore, we will use he inegraedsupplychainoderiveheopimalefforlevele, pricing p,andsocking. Because he sochasic demand is sensiive o boh reail price and sales effor, only he addiive-error/linear-demand funcion will be discussed [6, 15]. Specifically, he random single-period oal demand X is of he form X = y(p, e) + ε, y(p, e) = α βp + ke, (16) where α,β,k>.thais,heacualdemandishesummaion of he deerminisic funcion and he sochasic facor. y(p, e) is modeled using he downward-sloping linear funcion wih respec o boh price and sales effor, where α is he primary demand when boh price and effor are se a zero, β is he price sensiiviy of demand, and k is he sales effor sensiiviy of demand. ε is a random variable defined on he range [A, B] wih PDF φ( ) and CDF Φ( ).Thereailer ssales effor cos is assumed as an increasing and convex funcion of sales effor e, defined as g(e) = μe 2 /2,whereμ(μ>)is a consan, called sales effor cos parameer. Such a uadraic cos funcion is commonly used in previous lieraure. The following mild assumpions are necessary for obaining euilibrium soluions. Assumpion 2. One has β>k 2 /μ. Assumpion 3. The disribuion of he random variable ε has an increasing failure rae (IFR) propery: h( ) = φ( )/(1 Φ( )) is increasing in x. Defining he socking facor z as z = y(p, e), iwas inroduced by [15] and widely used by [6, 16] as a decision variable insead of. Then he cenralized expeced profi is wrien as follows: From (14), he arge sales level canbechoseninhe following range: π (p, e, z) = (p c) (y(p, e) + z) (p V)Λ(z) g(e), (17) d π(p,e, ) π d R (pd,e d, d,w). (15) w c If (15) holds, he coordinaing composie conrac can yield a higher expeced profi for boh he manufacurer and he reailer han he euilibrium wholesale price conrac. In oher words, he manufacurer can se he arge level by (15). If he arge reaches he lef boundary, he manufacurer s revenueunderhecomposieconracishisincomeinhe wholesale price. Wih he increase of, he manufacurer s profi is increasing, whereas he reailer s profi is decreasing. If he arge reaches is upper bound, he reailer s revenue reduces o he profi of he wholesale price conrac. where Λ(z) = z (z u)φ(u)du represens he expeced number of lefovers. The objecive is o find he opimal e, p, A and z ha maximize π(p, e, z), saedinhefollowingproposiion. Proposiion 4. For he cenralized supply chain and any given socking facor z(a z B),hen (1) heuniuebesresponseefforlevele is given by e =e(p) = k(p c) ; (18) μ
6 Mahemaical Problems in Engineering (2) if β>k 2 /μ, he uniue bes response price p is given by p =p(z) = α+c(β k2 /μ) + z Λ (z) 2β k 2 ; (19) /μ (3) if β > k 2 /μ and ε has IFR, hen he reailer s bes response socking facor z is uniuely deermined as he soluion of α cβ+z Λ(z) Φ (z) = α cβ+(c V) (2β k 2 /μ) + z Λ (z). (2) Proof. Firs, for any given z and p, we ake he parial derivaive of (17) wih respec o e and ge π (p, e, z) 2 π (p, e, z) 2 =(p c) y(p, e) g (e), =(p c) 2 y(p, e) 2 g (e). (21) Since 2 π(p, e, z)/ 2 <, π(p, e, z)/ = implies ha e =e(p)=k(p c)/μ. Subsiuing (18) ino (17): π(p,e, z) = (p c) (y(p, e )+z) (p V)Λ(z) (22) g(e ). Second, for any given z, wewanoderivep which maximizes (22). By he chain rule, we have dπ (p, e (p), z) dp = π (p, e,z) de dp + π (p, e,z). (23) Since e isheuniuemaximumvalueofπ(p, e, z), hefirs erm π(p, e,z)/ is. Thus, dπ (p, e (p), z) dp =α+c(b k2 μ )+z Λ(z) (2β k2 μ )p. (24) From dπ(p, e (p), z)/dp =, (19)isgiven.Meanwhile,if β > k 2 /μ, dπ(p, e (p), z)/dp > for all p < p(z) and dπ(p, e (p), z)/dp < for all p > p(z).sop is he uniue maximum value of π(p, e (p), z). Finally, we will prove ha z exiss uniuely. Subsiuing p ino (22): π(p,e,z)=(p c)(y(p,e )+z) (25) (p V)Λ(z) g(e ). By he chain rule, we have dπ (p,e,z) dz = π (p,e,z) de dz + π (p,e,z) dp dz + π (p,e,z). z (26) Likewise, he firs and second erms are : dπ (p,e,z) dz =p c (p c)φ(z) = g (z) 2β k 2 /μ, (27) where g(z) = α βc+z Λ(z) [α βc+(c V)(2β k 2 /μ) + z Λ(z)]Φ(z). Under Assumpion 2, we have 2β k 2 /μ >, which implies ha he opimal z saisfies g(z )=. We herefore give (2). Such a z always exiss in he suppor inerval [A, B] of ε,sinceg(z) is coninuous in [A, B],andg(A) = α βc+a >, g(b) = (c V)(2β k 2 /μ) <. To verify he uniueness of z, we need g (z) = [1 Φ(z)] {1 Φ (z) h(z) [α βc+(c V) (2β k2 μ )+z Λ(z)]}, g (z) = h(z) g (z) + [1 Φ(z)] { φ (z) h (z) [α βc+(c V) (2β k2 μ )+z Λ(z)] h(z) [1 Φ(z)]}. (28) According o Assumpions 2 and 3, g (z) < a g (z) =, which implies ha g(z) is a unimodal funcion in [A, B], which guaranees he uniueness of z. 5.2. Numerical Analysis. Given he cenral objecive of he paper, his secion operaes wih he inegraed supply chain and he wholesale price conrac o derive he opimal soluions. For noaional convenience, le z, p, e,and denoe he opimal decisions under he cenralized supply chain, and π denoes he inegraed channel profi. Le z d, p d, e d,and d be he reailer s decisions a euilibrium under he wholesale price conrac. π d M and πd R are he respecive profis of he manufacurer and reailer given ha he manufacurer charges w and he reailer ses p d, e d,and d.as aresul,π d denoes he aggregae profi of he decenralized sysem; ha is, π d =π d R +πd M. We se he parameer values as α=5, μ=1, c=5,andv =1. The random perurbaion on he demand, ε, is assumed o follow he uniform disribuion on [, 2]. In Table 1, he values of w are fixed a w=1;hentable1 summaries, a euilibrium, he reailer s opimal decisions under cenralized and decenralized decision models by varying he values of β and k, respecively. The reailer s opimal socking facor and sales effor level under he wholesale price conrac are less han ha in he cenralized channel model, bu he opimal selling price changes in an opposie way.
Mahemaical Problems in Engineering 7 Table 1: The effec of β and k on he opimal decisions and profi. z p e π z d p d e d d π d R π d M π d β=2, k=2 1.48 16.37 2.27 23.29 23.51 1.36 18.58 1.72 17.63 128.42 88.16 216.57 17.63 2.42 β=3, k=2 1.23 11.4 1.28 19.59 113.25.96 13.7.74 11.34 36.29 56.68 92.97 11.34 15.39 β=3, k=3 1.27 12.3 2.11 21.51 124.51 1.1 14.7 1.22 12.46 39.94 62.31 12.25 12.46 16.91 β=3, k=1 1.2 11.7.61 18.59 17.42.93 13.51.35 1.75 34.4 53.76 88.16 1.75 14.6 Table 2: The effec of w on he opimal decisions and profi. z p e π z d p d e d d π d R π d M π d w=8 1.48 16.37 2.27 23.29 23.51 1.42 17.7 1.94 19.9 165.8 59.7 225.5 19.9 21.57 w=1 1.48 16.37 2.27 23.29 23.51 1.36 18.58 1.72 17.63 128.42 88.16 216.57 17.63 2.42 w=12 1.48 16.37 2.27 23.29 23.51 1.3 19.47 1.49 15.36 95.68 17.49 23.17 15.36 19.26 Hence,heaggregaeprofiofhedecenralizedsysemπ d is less han he inegraed channel profi π. Furher, Table 1 also gives he range of he arge sales level.ifhemanufacurer ses he same as Table 1, he composie conrac can yield a higher expeced profi for boh he manufacurer and he reailer han he euilibrium wholesale price conrac. For example, a =19,wehave π c R (p,e, ) = 135.51 > π d R (d,p d,e d,w) 35 3 25 2 15 = 128.42, π c M (p,e, )=95>π d M (d,p d,e d, w) = 88.16. (29) 1 5 InTable2,weincreasehemanufacurer swholesaleprice from 8 o 12 while keeping β = 2 and k = 2,respecively. I is shown ha he reailer s euilibrium order uaniy d and heir expeced profi π d R under he wholesale price conrac decreases as w increases. This is inuiive; hus, we will mainly give he effec of w on he sales arge. The lower limi of he sales arge is he reailer s euilibrium order uaniy under he wholesale price conrac. Since he arge reaches is upper bound, he reailer s revenue reduces oheprofiofhewholesalepriceconrac.thaisosay, helowerandupperboundof decrease as w increases. However, he range of would become larger wih he increase of w. From Proposiion 1, he oal supply chain profi can be spli arbirarily beween he manufacurer and he reailer by varying for a given w. Hence, he manufacurer, as he Sackelberg leader, can raise his expeced revenue by increasing he wholesale price w. On he oher hand, accordingoourdemandmodel,p d mus saisfy α βp, so he manufacurer could no se w arbirarily. Because we consider an addiive demand model which is a funcion of he reail price p and he sales effor e, he price sensiiviy of demand β and he sales effor sensiiviy of demand k are wo major facors which affec marke demand. In order o properly evaluae he effec of one parameer on he euilibrium resuls, i is necessary o isolae i from oher parameers by keeping oher parameers consan. As a resul, Figure 1 describes he effec of β on he opimal decisions wih fixed k = 2, and Figure 2 describes he effec of k on he opimal decisions wih fixed β = 2.Weobserveby 1 2 3 4 5 6 7 8 p e Figure 1: The effec of β on he opimal decisions. comparison ha he reailer s opimal decisions are decreasing in β, while he reailer s opimal decisions are increasing in k. Figures 3 and 4 explore he effec of β and k on he arge sales level. FromFigure3,heargesaleslevel becomes lower wih he increase of β, buherangeof becomes greaer.asshowninfigure4,heargesaleslevel is increasing in k;however,herangeof changes lile. Figures 3 and 4 indicae ha he parameer β has greaer flexibiliy han k in spliing he oal supply chain profi beween he manufacurer and he reailer. 6. Conclusion As demand becomes price and effor dependen, a key uesion is wheher he conracs ha coordinae he reailers order uaniy also coordinae he reailers pricing and effor level. As a way o improve he coordinaion beween he manufacurer and he reailer, here is an increasing ineres in examining wha ype of supply conracs can be used o achieve channel coordinaion in he sense ha each supply β
8 Mahemaical Problems in Engineering 35 3 25 2 15 1 5.5 1 1.5 2 2.5 3 3.5 4 3 25 2 15 1 p e Figure 2: The effec of k on he opimal decisions. k 5 1 1.5 2 2.5 3 3.5 4 32 3 28 26 24 22 2 18 Figure 3: The effec of β on he arge sales level. 16 1 1.5 2 2.5 3 3.5 4 k Figure 4: The effec of k on he arge sales level. β chain parner s objecive becomes aligned wih he supply chain s objecive. In his paper, we show ha here exiss a composie conrac ha can achieve coordinaion when he random demand is sensiive o boh reail price and sales effor. Then a properly designed arge sales rebae conrac and parial reurns conrac can achieve channel coordinaion (i.e., maximizes he profiabiliy of he enire supply chain of boh manufacurer and reailer) and win-win (i.e., boh paries are made beer off). One imporan feaure worh noing is ha he conrac parameers for coordinaion in he composie conrac are independen of he sochasic demand disribuion. As a fuure research direcion, i is ineresing o explore he composie conrac in coordinaing a supply chain wih price-dependen demand and risk sensiive agens ogeher. I will also be ineresing o invesigae he performance of he composie conrac in considering sraegic cusomer behavior. Compeing Ineress The auhors declare ha hey have no compeing ineress. Acknowledgmens This work is suppored by Science and Technology Program of Qingdao, China (Gran no. 14-2-4-56-jch). References [1] J. J. Spengler, Verical inegraion and anirus policy, Journal of Poliical Economy,vol.58,no.4,pp.347 352,195. [2] G. P. Cachon, Supply chain coordinaion wih conracs, Working Paper, Universiy of Pennsylvania, Philadelphia, Pa, USA, 22. [3] B. C. Giri, S. Bardhan, and T. Maii, Coordinaing a woechelon supply chain hrough differen conracs under price and promoional effor-dependen demand, Sysems Science and Sysems Engineering,vol.22,no.3,pp.295 318,213. [4] D. D. Wu, Bargaining in supply chain wih price and promoional effor dependen demand, Mahemaical and Compuer Modelling,vol.58,no.9-1,pp.1659 1669,213. [5] J.H.Gao,H.S.Han,L.T.Hou,andH.Y.Wang, Pricingand effor decisions in a closed-loop supply chain under differen channel power srucures, Cleaner Producion, vol. 112, par 3, pp. 243 257, 216. [6] Y.He,X.Zhao,L.D.Zhao,andJ.He, Coordinaingasupply chain wih effor and price dependen sochasic demand, Applied Mahemaical Modelling, vol.33,no.6,pp.2777 279, 29. [7] B. A. Pasernack, Opimal pricing and reurn policies for perishable commodiies, Markeing Science, vol. 4, no. 2, pp. 166 176, 1985. [8] T. A. Taylor, Supply chain coordinaion under channel rebaes wih sales effor effecs, Managemen Science,vol.48,no.8,pp. 992 17, 22. [9] X. Gan, S. P. Sehi, and H. Yan, Channel coordinaion wih a risk-neural supplier and a downside-risk-averse reailer, Producion and Operaions Managemen, vol.14,no.1,pp.8 89, 25.
Mahemaical Problems in Engineering 9 [1] C. X. Wang and S. Webser, Channel coordinaion for a supply chain wih a risk-neural manufacurer and a loss-averse reailer, Decision Sciences,vol.38, no.3,pp.361 389, 27. [11] H. C. Xiong, B. T. Chen, and J. X. Xie, A composie conrac based on buy back and uaniy flexibiliy conracs, European Operaional Research,vol.21,no.3,pp.559 567,211. [12] B.Liu,J.Chen,S.F.Liu,andR.Zhang, Supply-chaincoordinaion wih combined conrac for a shor-life-cycle produc, IEEE Transacions on Sysems, Man, and Cyberneics Par A: Sysems and Humans,vol.36,no.1,pp.53 61,26. [13] C.-H. Chiu, T.-M. Choi, and C. S. Tang, Price, rebae, and reurns supply conracs for coordinaing supply chains wih price-dependen demands, Producion and Operaions Managemen,vol.2,no.1,pp.81 91,211. [14] S. Webser and Z. K. Weng, A risk-free perishable iem reurns policy, Manufacuring & Service Operaions Managemen, vol. 2,no.1,pp.1 16,2. [15] N. C. Peruzzi and M. Dada, Pricing and he newsvendor problem: a review wih exensions, Operaions Research, vol. 47, no. 2, pp. 183 194, 1999. [16] F. J. Arcelus, R. Gor, and G. Srinivasan, Price, rebae and order uaniy decisions in a newsvendor framework wih rebaedependen recapure of los sales, Inernaional Producion Economics,vol.14,no.1,pp.473 482,212.
Advances in Operaions Research Advances in Decision Sciences Applied Mahemaics Algebra Probabiliy and Saisics The Scienific World Journal Inernaional Differenial Euaions Submi your manuscrips a Inernaional Advances in Combinaorics Mahemaical Physics Complex Analysis Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics Discree Mahemaics Discree Dynamics in Naure and Sociey Funcion Spaces Absrac and Applied Analysis Inernaional Sochasic Analysis Opimizaion