Coordination Contracts for Competitive Two Echelon Supply Chain With Priceand-Promotional

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1 Mngeent Science nd Engineering Vol 0 No OI:03968/834 ISSN [Print] ISSN X [Online] wwwcscndnet wwwcscndorg Coordintion Contrcts for Coetitive Two Echelon Suly Chin With Pricend-Prootionl Effort eendent end TNG onging [] ; WU Zhihong []* [] School of Business nd dinistrtion South Chin University of Technology Gungzhou Chin * Corresonding uthor Received 0 Februry 06; cceted 9 My 06 Published online 6 June 06 bstrct With the deeening of globliztionirove the rket shre is becoing ore nd ore iortntto ny enterrises rootionl effort is good choicebsed on rice-nd-rootionl effort deendent dendwe develoed odel consists of doinnt nufcture nd two retilersnd cored the otil strtegies under the centrlized nd decentrlized scenrios To coordinte the suly chin nd irove the rofits of the ebers of the suly chin two coordintion contrcts re roosed we find tht these contrcts iroved rofits of both sides Finlly we vlidted the effectiveness of these contrcts through nuericl exles Key words: Prootion effort; Price; end; Suly chin coordintion Tng P & Wu Z H (06) Coordintion Contrcts for Coetitive Two Echelon Suly Chin With Pricend-Prootionl Effort eendent end Mngeent Science nd Engineering 0() 7-3 vilble fro: URL: htt://wwwcscndnet/indexh/se/rticle/view/834 OI: htt://dxdoiorg/03968/834 INTROUCTION It is iortnt for suliers nd retilers to deterine the dend In relity there re ny fctors influence dends ong the the effect of rice on dend is extensively cceted However s the rid develoent of econoic globliztion nd rket coetition it is urgent to increse the rket dend s well nd the ost vlued of those is the rootion of dend tht cn rise dend in the rket Both nufcturers nd retilers re hoing to get ore rofits by the rootion of dend For instnce nufcturers like Coc-Col nd Pesi-Col increse rket dend for their roducts by the rootion of dend nd retilers like Suning nd Goe increse the rket shre of roduct sles by ny kinds of rootion en Therefore it is iortnt for us to study on the effect of the rket ricing nd rootion strtegies on the decision de by suly chins nd their ebers t resent there hve been ny scholrs crried out ny studies bout it For instnce Hung nd Li (00) nd Hung et l (00) nd Li et l (00) ll discussed the decision roble bsed on Stckelberg ge nd Nsh equilibriu ge s the nufcturer dointe the rket nd tteted to redistribute the rofit for every suly chin ebers In order to coordinte the suly chin Xie et l (009) studied on the otil ricing nd dvertising strtegy in the single nufcturer nd single retiler suly chin He et l (009) considered suly chin contrct nd coordintion robles on the rndo dend of downstre retilers under the influence of rootion nd rice Tso et l (00) studies on ultilevel suly chin collbortion under the influence of credit level nd efforts in rketing de by retilers Wng et l (0) studied the cooertive dvertising odel tht contins one nufcturer nd two retilers in the suly chin under the condition of four ossible ge In existing reserch there re lot of studies bout cooertive dvertising of nufcturers nd retilers nd retilers rooting by their own In this er we consider the ricing nd rootion roble of two-tier suly chin coosed by doinnt nufcturer nd two retilers In tht dends re not only influenced by rice lso deend on the rootion effort of the roducts In our odel in order to eet their own dends the nufcturers roote to irove the overll dend 7 Coyright Cndin Reserch & eveloent Center of Sciences nd Cultures

2 Coordintion Contrcts for Coetitive Two Echelon Suly Chin With Price-nd-Prootionl Effort eendent end while the two retilers ke decisions for their own sles rices In ddition we lso consider coordintion of the nufcturers nd retilers in the suly chin so s to ttin reto iroveents PROBLEM ESCRIPTION Consider two-tier suly chin coosed by doinnt nufcturer nd two retilers in which the nufcturer is the leder while the retilers re the followers nd ccord to the Nsh equilibriu of ge between retilers Totl rket size of roducts de by the nufcturers is + in which resectively reresent rket size of retiler nd retiler Mnufcturers cost of roducing the roducts is c who sells roducts to retilers t wholesle rice w while retiler nd retiler resectively sell their roducts t rice nd Then the ctul dend of retiler nd retiler resectively is - +θ - +θ in which reresents the ctul dend influence coefficient of to retiler nd retiler θ reresents the influence coefficient of cross rice between retilers nd - θ >0 In rctice nufcturers often ke rootion by dvertising nd other ens which rise the overll size of the rket esecilly in the consuer industry For instnce P&G nd Unilever will invest lot of oney on the dvertising to exnd the size of the rket We ssue tht the rootion of nufcturer for their roducts is ρ(ρ ) so the overll size of the rket will be incresed fro to ρ Becuse the rootion is intened for the whole rket therefore we ssue tht the size of the rket of retiler nd retiler is iroved to ρ nd ρ nd the ctul dend of retiler & will turn into ρ - +θ ρ - +θ See Figure + Size of rket ρ Retiler Mnufcturer ρ ρ-+θ Custoer ρ Retiler ρ-+θ Figure Role of Mrketing nd the rootion cost of nufcturer is corresonding to its rootionl efforts we refer to forul given by Hrish (004) nd ssue the cost of king rootion is C(ρ )k(ρ-) In which k>0 is constnt nd reresents ttrction to custoers tht is the difficulty of exnd the rket on the bsis of the existing rket scle In ddition we ssue tht the infortion for the rootion cost is syetricl between ustre nd downstre BSIC ECISION MOEL N THE COORINTION ecentrlized ecision Model In decentrlized decision king ode nufcturers nd retilers ke decisions ccording to stckelberg ge while the retilers ly ge with ech other ccording to Nsh equilibriu Mnufcturer should deterine wholesle rices w of roducts sold to retiler nd retiler nd efforts ρ to roote its roducts; while retiler nd retiler should ke decisions for their selling rices ccording to it s result we hve to confir the rofit function for ech Prootion ρ other Profit function for nufcturer: ( w c)( ( w c)[ ρ( + ) C( ρ ) + ) + ( θ ) Profit function for retiler : + ( θ ) ] k( ρ ) () w) ( w)( ρ + θ ) () r ( Profit function for retiler : w) ( w)( ρ + θ ) (3) r ( ecisions of Retiler For the ossible rootionl efforts ρ nd wholesle rice w decided by nufcturer retiler nd retiler should decide their own otil sle rice nd Retiler nd retiler ly ge with ech other ccording to Nsh equilibriu Thus resectively tke the first rtil derivtives to nd of Forul () nd (3) then ke it to zero Below is the siultneous equtions: r ρ + θ + w 0 r ρ + θ + w 0 Coyright Cndin Reserch & eveloent Center of Sciences nd Cultures 8

3 TNG onging; WU Zhihong (06) Mngeent Science nd Engineering 0() 7-3 To solve the bove equtions ( + θ ) w + ( + θ (4) ( + θ ) w + ( + θ (5) Proerty For retiler nd retiler there exists otil sle rice i (i) which xiizes their own rofits nd the decision of i (i) is corresonding to the nufcturer s which eets Foruls (4) nd (5) ccording to roerty retil sles rice decided by nufcturer is directly roortionl to the wholesle rice nd the rootionl efforts re decided by retilers Tht is there exist i > 0 i > 0 In ddition the w ρ reltionshi between the sles rice nd the rootionl efforts is ffected by the rket size of these two retilers ecisions of Mnufcturer In decentrliztion decision-king odels nufcturer dointes so we nlyse nufcturers nd retilers by the stckelberg ge tht is nufcturer kes decisions for wnd ρ fter considering the ossible counteresures of two retilers Thus under this circustnces we substituted Foruls (4) nd (5) into forul () get w + R k( ) ( w ( w c)( R ρ (6) ( + ) θ + θ ( + ) In tht R R + ( + ) θ Tke the first rtil derivtives to w nd ρ of Forul (6) then ke it to zero siultneously solving the eqution nd get 4kR cr R ρ 4kR + R ckr cr kr w + 4kR + R < Proerty : If 4kR + R 0 is et w nd ρ tht is solved is the nufcturer s otil decision Pr:by -θ >0 ttin ( + ) θ + θ ( + ) < ( + ) 0 tht is R <0 < ( w Rw + Rρ cr w ( w ( w c) R k( ρ ) ρ ( w ( w R < 0 R w w ρ ( w k < 0 ρ To find otil solution of the functionit ust exist > < 4kR R 0 tht is 4kR + R 0 QE Then if roerty is et under the decentrlized circustnces the otil sle rice decided by retiler nd retiler is resectively: ( ( + θ ) w + θ ) w + ( + ( θ + θ + θ Thus we cn esily see when retiler nd retiler ke decision for their selling rices they not only ffected by the rice-interction-influence dend coefficient between retilers but lso ssocited with the rice-influence dend coefficient of other retilers; enwhile the selling rice is lso ssocited with the rket size of both sides Besides it is notble tht the decisions of king selling rice interction between the retilers Under this circustnces the ctul dend nd rootionl cost of retiler retiler nd nufcturer re: ( θ + θ ( θ + θ ) w ) w + ( + θ + ( + θ [θ + ( + ) θ ( + )] w + [ + ( + C(ρ )k(ρ -) Resectively the otil rofit corresonding to retiler retiler nd nufcturer re: r ( w) r ( w) ) θ ] ρ ( w c) Q C( ρ ) 9 Coyright Cndin Reserch & eveloent Center of Sciences nd Cultures

4 Coordintion Contrcts for Coetitive Two Echelon Suly Chin With Price-nd-Prootionl Effort eendent end Centrlized ecision Model In centrlized ode the nufcturer nd two retilers re regrded s whole to ke decisions Tht is to xiize the rofit of the suly chin by the overll decision Under this circustnces the whole suly chin need to ke decisions for the sles rices nd the rootion ρ Thus in the first we ust deterine the rofit function of the suly chin ( -c) +( -c) -k(ρ-) (7) We tke the first rtil derivtives to nd ρ of Forul (7) then ke it to zero foring forul includes three equtions nd obtin: ρ ( ) c θ ρ ( ) + kρ ( c k ) Solve the bove forul nd obtin the otil rofit nd rootion under this circustnces: ρ + ( c 4k)( + θ θ ) c 4k( ) c ( + θ ) + ρ (8) c ( + θ ) + ρ (9) Proerty 3: Under centrlized decision-king situtions the sles rice nd the rootion is linerly relted which eet Foruls (8) nd (9) Under this circustnces the dends nd rootion costs of retilers & nd nufcturers re: Then corresonding otil rofit of suly chin is: ( c) + ( c) C( ρ ) 3 Suly Chin Coordintion Under centrlized decision-king sitution the overll rofit is ( -c) +( -c) -C(ρ ); while under decentrlized decision-king situtiton the su of the rofit r of retiler the rofit r of retiler nd the rofit of nufcturer is r + r + ( -c) +( -c) -C(ρ ) Fro bove it is esy to see tht r + r + tht is overll rofit under centrlized decision-king sitution is greter thn the su of the rofits of retilers nd nufcturer under decentrlized decision-king sitution which ke roo for nufcturers to cooerte with retilers to increse their rofits In other words rofits of ech rty re not less thn the otil rofits r r nd de by retilers nd nufcturer under decentrlized decisionking sitution Becuse the incresed rofits re cquired by nufcturer with dends rootion it requires nufcturers y ore rootion cost Then in order to retilers & should give certin rootionl subsidies to encourge nufcturers to intensify the rootion 3 The Wholesle Price Incentives In order to encourge nufcturer ke decisions ccording to rootion under centrlized-decision situtions we ssue tht retiler nd retiler hve given certin wholesle rice incentives to nufcturer resectively tht is iroving the wholesle rice to γ nd γ ties while the selling rice of retilers re nd nd the wholesle rice of nufcturer is turned into γ w nd γ w t this tiethe rofits of ech rty re: r ( γ w ) r ( γ w ) γ w c) + ( γ w c) C( ρ ) ( nd it is required to eet: γ w ) r ( γ w ) r ( ( γ w c) + ( γ w c) k( ρ ) Solve the bove inequity nd ttin: r w r w + k( ρ ) + c γ γ w + k( ρ ) + c γ γ w Fro the bove reltions it is esy for us to see tht there re links between γ nd γ deterined by retiler nd retiler which is exctly confored to relity Soe retilers give wholesle rice suort to nufcturer nd it will influence the others For retiler nd retiler when γ nd γ tke vlues in the bove rnge under the condition which cn xiize the rofits of the whole suly chin the rofits of ech rty re not less t lest For retilers nd retilers when nd where within the rnge vlues cn ke the whole suly chin rofit xiiztion under the condition of the rofits t lest worse 3 Prootion Cost-Shring The cost of nufcturer will rise s the rootionl iroveent nd this rt of cost should be coensted We ssue tht the wholesle rices of nufcturer intins the levels s is under decentrlized decisionking situtions nd the roortion of retiler nd retiler shres the costs is λ nd λ resectively t this tie the rootion nd the wholesle rice levels decided by nufcturer is ρ nd w while the selling rice of retiler nd retiler re ρ nd ρ t this oint the rofits of ech rty re: Coyright Cndin Reserch & eveloent Center of Sciences nd Cultures 30

5 TNG onging; WU Zhihong (06) Mngeent Science nd Engineering 0() 7-3 nd it is required to eet: r ( w ) λc( ρ ) r ( w ) λc( ρ ) w c) ( λ λ ) C( ρ ) Solve the bove inequity nd ttin ( w ) λk( ρ ) r ( w ) λk( ρ ) r ( ( w c) ( λ λ ) k( ρ ) Fro the bove reltions it is esy for us to see tht there re links between λ nd λ deterined by retiler nd retiler tht is the rootion cost shred by soe retilers which will influence the level of the cost shred by other retilers For retiler nd retiler when λ nd λ tke vlues in the bove rnge under the condition which cn xiize the rofits of the whole suly chin the rofits of ech rty re not less t lest 3 CLCULTING-EXMPLES NLYSIS ccording to the bove nlysis we cn get the decisions nd results under three different situtions Thus we use the following instnce dt to siulte ssue tht the overll rket size of the roduct de by nufcturer is 30 in tht the rket shre of retiler nd retiler re 0 nd 0 resectively It is known tht dends of retilers re influenced by its own selling rice the selling rice of its coetitors nd the rootion of the nufcturer in tht the ctul dend of retiler i is i ρ i - i i +θ j θ05 For the nufcturer the function of rootion cost is C(ρ )50(ρ-) By the bove dt we choose the vlues of γ nd γ re 8 nd ; while the vlues of λ nd λ re 035 nd 0 Clculting the decisions nd results of ech rty we find tht the ctul dend of the rket is rised fro to 547 fter we ke the coordintion which increses the dends indeed The secific results re shown in Tble Is clculted in three cses of the rties to the decision-king nd s result we found tht the rel needs of the rket fter coordintion gretly incresed fro to 547 relly chieve the growth of dend the secific results re shown in Tble : Tble Clculting-Exles nlysis Under Three ifferent Situtions Situtions * * w * ρ * r r ecentrliztion Centrliztion Coordintion It is esy to find tht the rofit of the whole suly chin is not only rised fter coordintion but lso the rofits of ech rty re iroved gretly which indeed chieves the coordintion of the suly chin CONCLUSION s to the cooertion ge odel tht there re single nufcturer nd two retilers we consider the influence of the nufcturer rootion nd retiler ricing We solve this roble under the sitution tht ccord to stckelberg ge between nufcturer nd retilers nd ccord to Nsh equilibriu between retilers nd find tht the rofits of the suly chin in under this cse re less thn king decisions regrd the suly s whole rt Therefore we consider three-sides collbortion of the two kinds of situtions: One is nufcturer iroves its rootionl efforts while ech of retiler subsidizes nufcturer bsed on wholesle rice the other is nufcturer irove the rootion while retilers subsidize the rootionl cost We find tht in the certin rnge of wholesle ricing subsidy nd rootion cost-shring we cn chieve the gol of the coordintion of the retilers nd nufcturer in the suly chin to irove their rofits 3 Coyright Cndin Reserch & eveloent Center of Sciences nd Cultures

6 Coordintion Contrcts for Coetitive Two Echelon Suly Chin With Price-nd-Prootionl Effort eendent end REFERENCES He Y Zho X Zho L & He J (009) Coordinting suly chin with effort nd rice deendent stochstic dend lied Mtheticl Modelling Hung Z & Li S X (00) Co-o dvertising odels in nufcturer retiler suly chins: ge theory roch Euroen Journl of Oertionl Reserch 35(3) Hung Z Li S X & Mhjn V (00) n nlysis of nufcturerretiler suly chin coordintion in cooertive dvertising ecision Sciences 33(3) Krishnn H Kuscinski R & Butz (004) Coordinting contrcts for decentrlized suly chins with retiler rootionl effort Mngeent Science 50() Li S X Hung Z & Zhu J (00) Cooertive dvertising ge theory nd nufcturer-retiler suly chins Oeg 30(5) Tso Y C (00) Mnging ulti-echelon ulti-ite chnnels with trde llownces under credit eriod Interntionl Journl of Production Econoics Wng S Zhou Y W Min J & Zhong Y G (0) Coordintion of cooertive dvertising odels in onenufcturer two-retiler suly chin syste Couters & IndustrilEngineering Xie J & Wei J C (009) Coordinting dvertising nd ricing in nufcturer retiler chnnel Euroen Journl of Oertionl Reserch Yng S L & Zhou Y W (006) Two-echelon suly chin odels: Considering duoolistic retilers different coetitive behviors Int J Production Econoics Coyright Cndin Reserch & eveloent Center of Sciences nd Cultures 3

OXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved.

OXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved. Renshw: Mths for Econoics nswers to dditionl exercises Exercise.. Given: nd B 5 Find: () + B + B 7 8 (b) (c) (d) (e) B B B + B T B (where 8 B 6 B 6 8 B + B T denotes the trnspose of ) T 8 B 5 (f) (g) B