Supply Chain Coordination with Buy-back and Wholesale-price Contracts under Random Demand

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1 Azama Rajapov e al. In. Jounal of Engineeing Reeah and Appliaion ISSN: , Vol. 6, Iue 2, Pa - 1) Febuay 216, pp RESEARCH ARTICLE OPEN ACCESS Supply Chain Coodinaion wih Buy-bak and Wholeale-pie Cona unde Random Demand Azama Rajapov, Jian Ming, Wang Nannan Shool of Tanpoaion and Logii, Souhwe Jiaoong Univeiy; Chengdu, China) Shool of Tanpoaion and Logii, Souhwe Jiaoong Univeiy; Chengdu, China) Shool of Tanpoaion and Logii, Souhwe Jiaoong Univeiy; Chengdu, China) ABSTRACT In ode o ineae upply hain oal pofi, we deign a upply hain laboaion mehanim beween one upplie and one eaile. Thoughou he pape we peen one upplie and one eaile e up wih andom demand in a deenalized upply hain. An inenive funion on he buy-bak o and wholeale pie o i peened and heme hough he buy-bak ona and wholeale-pie ona ha been developed o enih he eaile ake pa in he laboaion way. Fuhemoe, he pape how ha, how opoaion beween buy-bak ona and wholeale ona ould oodinae he upply hain in boh a deenalized and enalized upply hain. Laly, we e up numeial analye and he eul how he implied laboaion mehanim i no pemi he deenalized yem o ahieve he ame pefomane a he enalized deiion howeve i allow boh membe in he upply hain gain pofi haing by eing up he ona paamee. Keywod Supply hain, Collaboaion mehanim, Buy-bak ona, Wholeale ona. I. INTRODUCTION When we alk abou upply hain laboaion wih ona, we a o eview upply hain laboaion elaed eeah pape, wheein o deign effeive inenive heme an allow he deenalized yem of upply hain o ahieve opimal pefomane, ha been udied in eveal opeaion managemen lieaue Fawe e al. 212, Main e al. 21). Supply hain laboaion i diued la deade by aademi Baa e al. 24, Fawe e al. 212, Lambe e al. 24). In hee pape elaed o pay aenion o how o e up he effeive oodinaion mehanim by diffeen kind of ona. Haoya e al. 21) udied oodinaion ona fo upplie eaile poduing and elling fahionable podu exhibiing a ohai pie independen demand Giannoao e al. 24). A model of he upply hain ona wih evenue haing mehanim and hi model allow he yem effiieny alo i ould impove he upply hain membe pofi by ediing he ona paamee. Chahaooghi and Heydai develop an inenive heme aoding o edi opion ona o oodinae he eode poin and ode uaniy and impove he oveall upply hain pofiabiliy a well a eah membe pofiabiliy Chahaooghi e al. 21). Ozen e al. onenae on oodinaion of manufauing and he eaile wih buy-bak ona and pove buy-bak ona; in geneal, he diibuion yem anno ahieve he ame pefomane a he enalized yem Ozen e al. 26). Apa fom in aodane wih wha i done, hi pape onide laboaion of a wo-ehelon upply hain oniing of one upplie and one eaile baed on andom demand and find a laboaion mehanim wih he buy-bak ona and wholeale ona whih no only allow he deenalized yem o pefom ju well a a enalized one, bu alo povide a opion of o alloaion beween he membe of upply hain. Thi pape oni of following: In eion II, he bai aumpion of he model ae peened and he oal pofi funion of upplie and he eaile ae diued and alo wo exeme deenalized and enalized of upply hain yem o evaluae he laboaion mehanim above eion. Seion III he laboaion mehanim wih he buy-bak and wholeale ona unde uneain demand i developed. Numeial analyi i done in Seion IV. We will onlude he pape in Seion V. II. Model and Deiion Analyi In ou model we onide upply hain oodinaion wih boh membe a upplie and a eaile in a one-peiod eing. The eaile diove a ohai uome demand x and he ha o onide hi oking uaniy ha he ode fom he upplie, a he beginning of elling-peiod. When plaing hi ode, exa demand i unknown fo he eaile ha ealizaion bu know he diibuion of demand. Afe a eain peiod of ime, he good ae podued and hipped o by upplie o he eaile. Then he demand i ealized and aified a muh a poible. Thoughou he pape we ue following noaion: 29 P a g e

2 Azama Rajapov e al. In. Jounal of Engineeing Reeah and Appliaion ISSN: , Vol. 6, Iue 2, Pa - 1) Febuay 216, pp S p w u Supplie pofi Reaile pofi Salvage o fo eaile Ode uaniy fo eaile Selling pie fo eaile Poduion o fo he upplie Wholeale pie Shoage value fo eaile Fo he given value of he ode uaniy, upplie pofi i w ) 1) Fo he given value of he ode uaniy eaile pofi i p min{, x} S[ x] u[ x ] w 2) Whee [ x] max{, x}, [ x ] max{, x } When he eaile plae hi ode, only he diibuion of he demand Fx ) and he deniy funion f x ) ae known. By E. 2), he expeed pofi fo he eaile i A. Supply hain deenalized deiion In he upply hain deenalized deiion iuaion on he upplie and eaile opimize hei objeive funion individually and eaile define opimal ode uaniy ha ahieve highe expeed pofi depending on he wholeale pie of he upplie. E ) p xf x) dx p f x) dx S x) f x) dx u x ) f x) dx w 3) Fi deivaive of funion E ) on he ode uaniy i olved by de ) p f x) dx S f x) dx u f x) dx w 4) d Le he igh of he E. 4) eual o zeo hen opimal ode uaniy in deenalized iuaion an be alulaed olving he following euaion: F u p w u p S de de ) 5) Tha Fx ) epeen a oninuou pobabiliy diibuion of he andom vaiable x B. Supply hain enalized deiion In he upply hain enalized deiion iuaion he pofi deiion poblem apa fom deenalized deiion an be defined a a join opimal poblem. Thu he aual oal pofi of he upply hain i p min{, x} max{ h, S}[ x] u[ x ] 6) Wheein inend fo he aual oal pofi of he upply hain and h inend fo he alvage value fo he upplie. The expeed oal pofi funion i E ) p xf x) dx p f x) dx max h, S) x) f x) dx u x ) f x) dx 7) By olving he E. 7) he fi deivaive of funion E ) on he ode uaniy and le i eual o zeo he upplie and eaile define he opimal ode uaniy able o ge maximal en expeed pofi fo he enie upply hain. F u p u p max{ h, S} en) 8) Similaiy beween E. 5) and E. 8), nomally w, S max{ S, h}, hu i i wihou any doub an be veified ha he deenalized deiion iuaion moe ode uaniy will be plaed by eaile hen he deenalized deiion iuaion, ha lead o maximal oal pofi fo enie upply hain. Hene i i neeay ha laboaion mehanim would be e up o gain highe oal pofi of upply hain. 3 P a g e

3 Azama Rajapov e al. In. Jounal of Engineeing Reeah and Appliaion ISSN: , Vol. 6, Iue 2, Pa - 1) Febuay 216, pp III. Supply hain laboaion mehanim Aoding o he analye of upply hain enalized deiion and deenalized deiion epeen ha in he deenalized yem enaio wihou laboaion he expeed oal pofi of he enie upply hain i geneally lowe han in enalized enaio one. Salvaging a he upplie i moe pofiable han alvaging a he eaile beaue he upplie ould edie he unold o no-needed uni o he make of emanufaue and ge poiive evenue. Thi hane beome ignifian if he upplie offe buy-bak opion o he eaile Ozen e al. 21). Hene hee onide a vaiable buy-bak o ha he upplie pomie o offe o he eaile when he odeed amoun of he eaile moe han make demand a he end of he elling peiod. Le b denoe he buy-bak o he upplie ha o pay fo evey uni. Then he aual oal pofi of he upplie and eaile and alo oal pofi of he enie upply hain afe inoduing buy-bak o ae hen w ) h b)[ x] 9) p min{, x} u[ x ] w b[ x] 1) en p min{, x} h[ x] u[ x ] 11) By imilaiy beween E. 6) and E. 11), i uely ha oal pofi of upply hain in deenalized deiion and he oal pofi of he enie upply hain afe eneing inenive funion ae diffeen depending on he ode uaniy of he eaile alvaging a he upplie i moe pofiable han alvaging a he eaile, ha i h S By E. 1) he expeed pofi of he eaile i given by E ) p xf x) dx p f x) dx u x ) f x) dx w b x) f x) dx 12) Afe olving hi fi deivaive of funion E ) on he ode uaniy and le i eual o zeo, he eaile define he opimal ode uaniy ha an ge maximal expeed pofi. The opimal ode uaniy an be alulaed olving a following: F u p w u p b ) 13) Noe ha buy-bak ona anno make he diibuion yem an ge he ame pefomane a he enalized yem. Hene, i needed o inodue wholeale pie ona diffeen fom he buy-bak ona ealize he pefe upply hain laboaion. Le aep ha buy-bak pie b and wholeale pie w ha he deiion of he upplie pupoe o ge he highe oal pofi of he upply hain and he upplie die o he eaile o hooe oveall opimal deiion. To uely ha he eaile hooe hi value of ode uaniy ha inu oveall maximal expeed oal pofi, he upplie mu fix he buy-bak o b and wholeale pie w o ha following euaion hold: F ) F en ) u p w u p u p b u p max{ h, S} 31 P a g e 14) Afe olving hi E. 14) he opimal wholeale pie w a a funion of buy-bak pieb i ahieved: op u p w u p u p b) 15) u p max{ h, S} To enue ha in eah ombinaion he buy-bak o and wholeale pie ha aified he above euaion ha eaile hooe he oveall opimal ode poliy ha lead o maximal expeed oal pofi of he enie upply hain. Aoding o he upply hain membe agee wih laboaion model, afe inoduing inenive mehod he expeed pofi of upplie and eaile migh be moe han ha wih deenalized deiion iuaion, euie o fulfill a follow: E ) E ) and E de ) E de ) So fa, i ha been developed ha laboaion mehanim ha gain an oveall opimal pefomane of he enie upply hain. IV. Numeial analye In ode o diu he model and illuae he onluion moe lealy, hi eion hough numeial example analyi he upply hain deiion eul in boh deenalized and enalized yem ae alulaed and alo ge he bee undeanding laboaion model. In hi eion we ue following

4 Azama Rajapov e al. In. Jounal of Engineeing Reeah and Appliaion ISSN: , Vol. 6, Iue 2, Pa - 1) Febuay 216, pp paamee: wholeale pie of he elling podu i w 1$. The poduion o fo he upplie 6$. Fo he eaile elling pie pe uni fo he make p 13$ and hoage ou 85$. Fo he eaile alvaging pie pe uni S 35$ and alvaging pie fo he upplie h 5$. Random vaiable x i on he make ha we ge he ange of andom vaiable fom 1 o 2 o make eay numeial alulaion. Tha deniy funion i f x) 1/ 2 1) 1 x 2 17) Then pobabiliy diibuion beome: F x) x 1 / 2 1) 1 x 2 18) A. Deiion Analyi of Supply Chain Fom he E. 5) and E. 18) deniy funion opimal ode uaniy of he eaile in upply hain deenalized deiion an be olved a de 1 u p w 2 1 u p S 19) Afe hi imple alulaion he opimal ode uaniy beome 163 uni. Supplie expeed de pofi i E ) i 667 USD and expeed pofi of eaile E ) i eual o 288 USD afe ha he en oal enie upply hain E ) i eual o 955 USD, Then, opimal ode uaniy of he eaile in upply hain i eual 193 uni. Maximal en expeed pofi of enie upply hain E ) i eual o USD. Table 1 i illuaed ha expeed upply hain oal pofi in boh enalized and deenalized deiion. # Deenalized Cenalized deiion deiion E ) E ) en E ) Table 1. Supply hain deiion analye B. Running Collaboaion Mehanim Appliaion Opimal wholeale pie a funion of buy-bak o i op w 91.93b Whee he onain on he buy-bak o i aained E ) 288 E ) 667 Buy-bak o i alulaed fom E. 9), 12) and 17) 96 b 14 The eul of he oal ummaized numeial analye i hown in Table 2. I hown ha if buybak o i ineaing he wholeale pie o alo ineae a he expeed pofi of he upplie ineae and eaile expeed pofi i deeae. en b w E ) E ) E ) Table 2. Reul of numeial alulaion V. Conluion Thi pape illuaed he poblem of upply hain laboaion fo he deenalized wih one upplie and one eaile unde uneain and andom demand. The buy-bak and wholeale pie ona model ae ued. In ode o bee undeanding he laboaion model of upply hain he numeial analyi i done. Thi pape poved ha by adjuing he paamee of buy-bak and wholeale pie ona deign membe lead he upplie and he eaile o ineae pofi ompaing wih deenalized upply hain by eing he paamee. REFERENCES [1.] Fawe, S.E., Fawe, A.M., Waon, B.J. and Magnan, G. 212), "Peeking inide he Blak Box: Towad an Undeanding of Supply Chain Collaboaion Dynami", Jounal of Supply Chain Managemen, Vol.48, pp [2.] Main Albeh. Supply Chain Coodinaion Mehanim: New Appoahe fo Collaboaive Planning Leue Noe in Eonomi and Mahemaial Syem) 21h Ediion 32 P a g e

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