IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 A anufacturer Stackelberg Game n Prce Cometton Suly Chan under a Fuzzy Decson Envronment SHNA WANG Abstract In a to-echelon suly chan models comosed of a manufacturer actng as the leader and to retalers actng as folloers under a fuzzy decson envronment. The arameters of demand functon and manufacturng cost are all characterzed as fuzzy varables. To retalers are assumed to act n colluson and the otmum olcy of the exected value and chance-constraned rogrammng models are derved. Fnally numercal examles are resented to llustrate the theoretcal undernnng of roosed models. It s shon that n fuzzy models the confdence level of the rofts for suly chan members affects the fnal otmal solutons. Index Terms Suly chan rce cometton game theory fuzzy theory I I. INTRODCTION N today s hghly comettve market more and more frms realze that rce s mortant behavor and cometng frms often carry a rce ar to attract customers. In suly chan cometton retalers comete th each other on determnng ther retal rces and order quanttes to maxmze ther rofts. There s a large body of lteratures that deals th rce cometton n suly chan. Cho [] used the lnear and constant elastcty demand functons to study the rce cometton n a to-manufacture and one-retaler suly chan th to Stagckelberg and one Nash games. Ingene and Parry [] consdered the coordnaton of the suly chan th to retalers cometng n rce. Yang and Zhou [] nvestgated to duoolstc retalers three knds of comettve behavors: Cournot Colluson and Stackelberg. Sang [] researched the rcng and retal servce decsons n a to-stage suly chan comosed of one manufacturer and one retaler under an uncertan envronment. Xao and Q [5] studed the coordnaton models of cost and demand dsrutons for a suly chan th to cometng retalers. Yao et al. [6] nvestgated a revenue sharng contract for coordnatng a suly chan comrsng one manufacturer and to cometng retalers. They shoed that the ntensty of anuscrt receved Arl 8 6; revsed August 5 6. Ths ork as suorted n art by Shandong Provncal Natural Scence Foundaton Chna (No. ZR5GQ) and the Proect of Shandong Provncal Hgher Educatonal Humanty and Socal Scence Research Program (No. J5WB). Shuna Wang s th the Deartment of Economc anagement Heze nversty Heze 75 Chna (hone: +86 875987; e-mal: shuna_ang@6.com). cometton beteen the retalers leaded to a hgher effcency but t ould hurt the retalers themselves. Anderson and Bao [7] consdered n suly chans rce cometng th a lnear demand functon. Farahat and Peraks[8] studed the effcency of rce cometton among mult-roduct frms n dfferentated olgooles. Zhao and Chen [9] nvestgated a coordnaton mechansm of a suly chan that conssts of one suler and duooly retalers from the ersectve of oeratng uncertanty. Cho and Fred [] studed rcng strateges n a market channel comosed of one natonal brand manufacturer and to retalers. Wang et al. [] studed a marku contract for coordnatng a suly chan comrsng to comettve manufacturers and a common domnant retaler. Kaakatsu et al. [] dscussed a quantty dscount roblem beteen a sngle holesaler and to retalers. All studes mentoned above dscussed the rce cometton models under a crs envronment such as a lnear or robablstc market demand and knon roducton cost. Hoever n real orld esecally for some ne roducts the relevant recse date or robabltes are not ossble to get due to lack of hstory data. oreover n today s hghly comettve market shorter and shorter roduct lfe cycles make the useful statstcal data less and less avalable. Thus the fuzzy set theory rather than the tradtonal robablty theory s ell suted to the suly chan roblem. In recent years more and more researchers have aled the fuzzy sets theory and technque to develo and solve the suly chan models roblem. Huang and Huang [] studed rce coordnaton roblem n a three-echelon suly chan comosed of a sngle suler a sngle manufacturer and a sngle retaler. Xu and Zha [-5] assumed the demand to be a trangular fuzzy number and dealt th the nesboy roblem n a to stage suly chan. Zhou et al. [6] consdered to-echelon suly chan oeratons n a fuzzy envronment hch comosed of one manufacturer and one retaler. We and Zhao [7] consdered a fuzzy closed-loo suly chan th retaler s cometton. Ye and [8] develoed a Stackelberg model th fuzzy demand. Recently Zhao et al. [9] consdered a to-stage suly chan here to dfferent manufacturers cometed to sell substtutable roducts through a common retaler. We and Zhao [] nvestgated the decsons of reverse channel choce n a fuzzy closed-loo suly chan. Zhao et al. [] studed a dstrbuton system n hch to manufacturers cometton under servce and rce suled to substtutable roducts to one common retaler n fuzzy envronments. Yu et al. [] develoed the ont otmal (Advance onlne ublcaton: February 7)
IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 rce-nventory decsons n a fuzzy rce-settng nesvendor model. Sang [] nvestgated suly chan contracts th a suler and multle cometng retalers n a fuzzy demand envronment. In ths aer e ll concentrate on rce cometton here a manufacturer ho sell hs roduct to to retalers under a fuzzy decson envronment. We also erform senstvty analyss of the confdence level of the rofts for suly chan members of the models. The rest of aer s organzed as follos. In secton the fuzzy set theory n our models s descrbed. Secton s the roblem descrtons. Secton develos the fuzzy to-echelon suly chan models th a manufacturer and to comettve retalers. Secton5 rovdes numercal examles to llustrate the result of the roosed models. The last secton summarzes the ork done n ths aer and further research areas. II. PREIINARIES Ths secton begns th some concets and roertes of fuzzy varables hch ll be used n the rest of the aer. et ξ be a fuzzy varable on a ossblty sace P Pos (for the concet of the ossblty sace see Nahmas []) here Θ s a unverse P s the oer set of Θ and Pos s a ossblty measure defned on P. Defnton (u [5]) A fuzzy varable ξ s sad to be nonnegatve f Pos. Defnton (u [5]) et ξ be a fuzzy varable and (]. Then nf{ r Pos{ r} } and R su{ r Pos{ r} } are called the α-essmstc value and the α-otmstc value of ξ. Defnton The fuzzy set ( abc ) here ab c and defned on R s called the trangular fuzzy number f the membersh functon of A s gven by x a f a x a a a ( ) a x x f a x a a a otherse. here a and a are the loer lmt and uer lmt resectvely of the trangular fuzzy number.the trangular fuzzy number s called the ostve trangular fuzzy number f a. Examle et ξ= (a b c) be a trangular fuzzy varable then ts α-essmstc value and α-otmstc value are resectvely b a( ) And b c( ). Prooston (u and u [6] and Zhao et al. [7]) et ξ and η be to nonnegatve ndeendent fuzzy varables. Then for any (] (a) and ; (b) f λ > and ; (c) ( ) and ( ) ; (d) and. Prooston (u and u [6]) et ξ be a fuzzy varable th the fnte exected value E[ξ] Then e have d. E Prooston (u and u [6]) et ξ and η be to ndeendent fuzzy varables th fnte exected values. Then for any real numbers a and b e have E a b ae be. Examle et ξ= (a b c) be a trangular fuzzy varable then ts exected value E[ξ] s a b c E[ ] (b a( ) ba c( ))d Defnton et ξ and η be to nonnegatve ndeendent fuzzy varables f and only f for any (] and. Defnton 5 et ξ and η be to nonnegatve ndeendent fuzzy varables f then E[ξ]>E[η]. III. PROBE DESCRIPTIONS Ths aer consders a to-echelon suly chan consstng of a manufacturer sellng hs roduct to to comettve retalers ho n turn retal t to the customers. The nteracton beteen to echelons s assumed that the manufacturer acts as a leader and sets a unted holesale rce to the to retalers then the to comettve retalers share a common sharng resond ndeendently by settng the sale rce and the corresondng order quantty. The notatons used n ths aer are gven as follos: the sale rce charged to customers by retaler ; the holesale rce er unt charged to the retalers by the manufacturer; c unt manufacturng cost; Q determnstc demand faced by retaler or quantty ordered by retaler ; the fuzzy roft for retaler ; R the fuzzy roft for manufacturer. We assume that retaler faces the smlar market demand functon hch s gven by Q D. here D and are ostve and ndeendent fuzzy varables. The arameter D reresents the market base the arameter reresents the measure of senstvty of retaler- s sales to changes of the retaler- s rce and the arameter reresents the degree of substtutablty beteen retalers and reflects the macts of the marketng mx decson (Advance onlne ublcaton: February 7)
IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 of retaler on customer demand. The arameters and are assumed to satsfy and. Snce there s no negatve demand n the real orld e Pos D. The quantty ordered by assume retaler can be exressed as Q E Q. Q s called a fuzzy lner demand functon n ths aer. et the cost c be a ostve fuzzy varable and be ndeendent of arameters D and. The fuzzy rofts of the manufacturer and the retaler ( ) can be exressed as c D D. R IV. FZZY SPPY CHAIN ODES IN PRICE COPETITION In ths secton e develo the fuzzy to-echelon suly chan models th a manufacturer and to comettve retalers hch can tell both the manufacture and the retalers ho to make ther decsons hen the duoolstc retalers actng n colluson n a fuzzy decson envronment. In ths condton to cometton retalers agree to act n unon n order to maxmze ther total fuzzy exected rofts hence the fuzzy otmal model n ths condton can be formulate as belo: max E ( ) E c D s. t. Pos c arg max E R max E R E D s. t. PosD. Theorem et E R be the total fuzzy exected values of the rofts for to retalers. A holesale rce chosen by the manufacturer s fxed. If E D PosD and E E then the otmal resonse functons and retaler and are () of the ED. () E E Proof. Note that fuzzy varables D and are ostve and ndeendent th each other. By Prooston e have E R E E E E D E E E D. () From () e can get the frst-order dervatves of E R th resect to and as follos: E R E E E D E E () E R E E E D E E. (5) Therefore the Hessan matrx of E R s E E H. (6) E E Note that the Hessan matrx of E R s negatve defnte snce are ostve fuzzy varables and. Consequently E R concave n and functons s ontly. Hence the otmal resonse and ( ) of the retaler and are can be obtaned by solvng E R and E R hch gve (). The oof of Theorem s comleted. Havng the nformaton about the decsons of the retalers the manufacturer ould then use those to maxmze hs fuzzy exected roft. So e get the follong Theorem. Theorem et E be the fuzzy exected Pos value of the roft for manufacturer. If c E D E c c d and Pos D the otmal solutons of model () are E D d E c c c (7) E E 6E D E c c c d. (8) 8E E Proof. Note that fuzzy varables c D and are ostve and ndeendent th each other. By Proostons and e have E c D c D d (Advance onlne ublcaton: February 7)
IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 c D c D d. (9) Substtutng ( ) and ( ) n () nto (9) e can get E E E E D E c c c d E D E c E D c c d E E c D c D d. () Thus from () e can get the frst-order and second-order dervatves of E follos: th resect to are as E E E E D E c c c d () E E E. () Note that the second-order dervatve of E s negatve defnte snce are ostve fuzzy varables and. Consequently E s concave n. Hence the otmal holesale rce of manufacturer can be obtaned by solvng E hch gve (7). Substtutng n (7) nto () e can get (8). The oof of Theorem s comleted. Combnng (7) and (8) th () and (9) ll easly yeld the otmal fuzzy exected rofts for retaler and manufacturer. The chance-constraned rogrammng hch as ntroduced by u and Iamura [8-9] lays an mortant role n modelng fuzzy decson systems. Its basc deal s to otmze some crtcal value th a gven confdence level subect to some chance constrants. otvated by ths deal the follong maxmax chance-constraned rogrammng model for the to-echelon suly chan can be formulated n the colluson soluton: max s. t. Pos cd Pos c arg max R max R s. t. Pos D R PosD. () here α s a redetermned confdence level of the rofts for the manufacture and the retalers. For each fxed feasble R should be the total maxmum value of the roft functon for retalers hch least ossblty α and R acheves th at should be maxmum value of the roft functon for manufacture hch acheves th at least ossblty α. Clearly the model () can be transformed nto the follong model () n hch the manufacture and the retalers try to maxmze ther otmal α-otmstc rofts and R resectvely by selectng the best rcng strateges c D max s. t. Pos c arg max R max s. t. PosD. R D Theorem et R () be the total α-otmstc value of the roft for to comettve retalers. A holesale rce chosen by the manufacturer s fxed. If PosD D and then the otmal resonse functons and are and ( ) of the retaler D. (5) Proof. Note that fuzzy varables D and are ostve and ndeendent th each other. By Prooston e have R ( ) D D. (6) From (6) e can get the frst-order dervatves of R th resect to and as follos: R D (7) R D. (8) Therefore the Hessan matrx of R s (Advance onlne ublcaton: February 7)
IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 H. (9) Note that the Hessan matrx of R negatve defnte snce are ostve fuzzy varables. Consequently R and concave n and functons and s s ontly. Hence the otmal resonse of the retaler and are can be obtaned by solvng R and R hch gve (5). The oof of Theorem s comleted. Havng the nformaton about the decsons of the retalers the manufacturer ould then use those to maxmze hs α-otmstc value of the roft. So e get the follong Theorem. of the roft for manufacturer. If c D c and PosD. The otmal solutons Theorem et are D c be the α-otmstc value Pos of model () () D c. () Proof. Note that fuzzy varables c D and are ostve and ndeendent th each other. By Prooston e have ( ) c D ( ) c D. () Substtutng ( ) and ( ) n (5) nto () e can get D c c D. () Thus e can get the frst-order and second-order dervatves of follos: th resect to are as D c (). (5) Note that the second-order dervatve of s negatve defnte snce are ostve fuzzy varables and. Consequently s concave n. Hence the otmal holesale rce of manufacturer can be obtaned by solvng hch gve (). Substtutng n () nto (5) e can get (). The oof of Theorem s comleted. Combnng () and () th (6) and () ll easly yeld the otmal α-otmstc value of rofts for the to cometton retalers and manufacturer as follos R R 6 c c D D (6). (7) The mnmax chance-constraned rogrammng model for the to-echelon suly chan hen to retalers act the colluson soluton can also be formulated as bello: max mn s. t. Pos cd Pos c arg max mn R R max mn R R s. t. Pos D R PosD. (8) Where α s a redetermned confdence level of the rofts for the manufacture and the retalers. For each fxed feasble R should be the total mnmum value of the roft functon for retalers hch least ossblty α and R acheves th at should be mnmum value of the roft functon for manufacture hch acheves th at least ossblty α. The model (8) can be transformed nto the follong model (9) n hch the manufacture and to cometton retalers try to maxmze ther otmal α-essmstc rofts and R by selectng the best rcng strateges resectvely (Advance onlne ublcaton: February 7)
IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 c D max s. t. Pos c arg max R max s. t. PosD. R D Theorem 5 et and R (9) be the otmal α-essmstc value of the rofts for retaler and Pos c D c manufacturer. If and PosD. The otmal solutons of model (8) are D c () D c. () Proof. Smlar to the roof of Theorem. The otmal α-essmstc value of rofts for the to cometton retalers and manufacturer are as follos R R 6 D c c D (). () Remark hen α= t s clear the manufacturng cost c the market base D the demand change rate and the degree of substtutablty beteen retalers degenerate nto crs real numbers the man result n Theorems and 5 can degenerate nto: Dc Dc (). (5) There are ust the conventonal results n crs soluton. V. NERICA EXAPE In ths secton e resent a numercal examle hch s amed at llustratng the comutatonal rocess of the fuzzy suly chan models establshed n revous secton. We ll also erform senstvty analyss of the arameter α of these models. Here e consder that D s about D 5866 9 s 6 s about about 5 56 and c s about c 9 resectvely. oreover the α-otmstc values and α-essmstc values of D and c are as follos D 58 D 6 9 6 c 9 c. The exected values of arameters are 58 6 6 ED 6 9 E 5 6 E 5 9 Ec. We can calculate that 6 E c c c d 98 c c d 596 c D c D d. Based on the above analyss the otmal exected values α-otmstc values and α-essmstc values for the fuzzy suly chan models above can be lsted n Table I. TABE I OPTIA EQIIBRI VAE OF THE PARAETERS FOR DIFFERENT α IN FZZY SPPY CHAIN E R E Exected value.5 5. 8.6.67 α-otmstc value α-essmstc value..5 5. 8.75 75..95.7 5. 859.5 9..9.98 5.9 876. 5..85. 5. 89.65 57.6.8.8 5.58 99.5 67.8.75.7 5.7 96.5 76...5 5. 8.75 75..95.6.86 87.9.7.9..7 8.5 9.8.85.8.58 796.99 87.96.8.57.5 78.86 7.5.75.5. 766.96 67.8 Based on the results shoed n Table I e fnd: (a) The th and 9th ros n TableⅠ sho the solutons for fuzzy models at α= hch are ust the results n crs case. (b) Because of domnatng the exected values α-otmstc values and α-essmstc values of the rofts for manufacturer are more than that of the total rofts for to retalers. It ndcates that the actor ho s the leader n the suly chan holds advantage n obtanng the hgher exected (Advance onlne ublcaton: February 7)
IAENG Internatonal Journal of Aled athematcs 7: IJA_7 8 rofts. oreover the rofts of retaler are equal to those of retaler n the Colluson soluton. (c) The α-otmstc values of the otmal rcng strateges and otmal rofts for the retaler and the manufacturer decrease th the ncreasng of the confdence level α. Wth the ncreasng of the confdence level α the α-essmstc values of the otmal rcng strateges and the rofts for the retaler and the manufacture ll ncrease. VI. CONCSIONS Ths aer rooses a fuzzy model for to-echelon suly chan management here to comettve retalers ursue the Colluson soluton. The rcng solutons for manufacturer and to retalers n exected value and chance-constraned rogrammng models are rovded. We fnd that the roosed fuzzy models can be reduced to the crs models and the confdence level of the rofts for the manufactures and the retaler affects the fnal otmal solutons. Our study manly concentrates on one manufacture and to cometng retalers hen the fuzzy demand functon s lnear. Therefore other forms of fuzzy demand functon and th multle comettve retalers or manufacturers are the mortant drectons for the future research. REFERENCES [] S.C. Cho Prce cometton n a channel structure th a common retaler arketng Scence vol. no. 7 96 99. [] C.A. Ingene and.e. Parry Channel coordnaton hen retalers comete arketng Scence vol. no. 6 77 995. [] S. Yang and Y. Zhou To-echelon suly chan models: Consderng duoolstc retalers dfferent comettve behavors. Internatonal Journal of Producton Economcs vol. no. 6 6. [] Shengu Sang "Otmal Prcng and Retal Servce Decsons n an ncertan Suly Chan" IAENG Internatonal Journal of Aled athematcs vol. 6 no. 68-7 6. [5] T. Xao and X. Q Prce cometton cost and demand dsrutons and coordnaton of a suly chan th one manufacturer and to cometng retalers Omega vol. 6no 5. 7 75 8. [6] Z. Yao S.C.H. eung and K.K. a anufacturer s revenue-sharng contract and retal cometton Euroean Journal of Oeratonal Research vol. 86 no. 67 65 8. [7] E. Anderson and Y. Bao Prce cometton th ntegrated and decentralzed suly chans Euroean Journal of Oeratonal Research vol. no. 7. [8] A. Farahat and G. Peraks On the effcency of rce cometton Oeratons Research etters vol. 9 no 6. 8. [9] Q. Zhao and H. Chen Coordnaton of a suly chan ncludng duooly retalers under suly-chan rle effect by an oeratng uncertanty: rce cometton Advances n Informaton Scences and Servce Scences vol. no 6.. [] S. Cho and K. Fred Prce cometton and store cometton: Store brands vs. natonal brand Euroean Journal of Oeratonal Research vol. 5 no. 5 58. [] C.J. Wang A.. Wang and Y.Y. Wang arku rcng strateges beteen a domnant retaler and comettve manufacturers Comuters & Industral Engneerng vol. 6 no. 5 6. [] H. Kaakatsu T. Homma and K. Saada An otmal quantty dscount olcy for deteroratng tems th a sngle holesaler and to retalers IAENG Internatonal Journal of Aled athematcs vol. no. 8-86. [] Y. Huang and G. Q. Huang Prce cometton and coordnaton n a mult-echelon suly chan Engneerng etters vol.8 no.99-5. [] R. Xu and X. Zha Otmal models for sngle-erod suly chan roblems th fuzzy demand. Informaton Scences vol.78 no 7. 7 8 8. [5] R. Xu and X. Zha Analyss of suly chan coordnaton under fuzzy demand n a to-stage suly chan. Aled athematcal odelng vol. no. 9 9. [6] C. Zhou R. Zhao and W. Tang To-echelon suly chan games n a fuzzy envronment. Comuters & Industral Engneerng vol.55 no. 9 5 8. [7] J. We and J. Zhao Prcng decsons th retal cometton n a fuzzy closed-loo suly chan. Exert Systems th Alcatons vol. 8 no 9. 9 6. [8] F. Ye and Y. A Stackelberg sngle-erod suly chan nventory model th eghted ossblstc mean values under fuzzy envronment. Aled Soft Comutng vol. no 8. 559 557. [9] J. Zhao W. Tang and J. We Prcng decson for substtutable roducts th retal cometton n a fuzzy envronment. Internatonal Journal of Producton Economcs vol. 5 no.-5. [] J. We and J. Zhao Reverse channel decsons for a fuzzy closed-loo suly chan. Aled athematcal odellng vol. 7 no. 5-5. [] J. Zhao W. u and J. We Cometton under manufacturer servce and rce n fuzzy envronments. Knoledge-Based Systems vol. 5 no. -. [] Y. Yu J. Zhu and C. Wang A nesvendor model th fuzzy rce-deendent demand. Aled athematcal odellng vol. 7 no 5. 6-66. [] S. Sang Suly Chan Contracts th ultle Retalers n a Fuzzy Demand Envronment. athematcal Problems n Engneerng vol. no. -. [] S. Nahmas Fuzzy varables. Fuzzy Sets and Systems vol. no. 97 978. [5] B. u Theory and ractce of uncertan rogrammng Physca-Verlag Hedelberg. [6] B. u and Y. u Exected value of fuzzy varable and fuzzy exected value model IEEE Transactons on Fuzzy Systems vol. no. 5 5 6. [7] R. Zhao W. Tang and H. Yun Random fuzzy Reneal rocess. Euroean Journal of Oeraton Research vol. 69 no. 89 6. [8] B. u and K. Iamura Chance constraned rogrammng th fuzzy arameters Fuzzy Sets and Systems vol. 9 no. 7 7 998. [9] B. u and K. Iamura A note on chance constraned rogrammng th fuzzy coeffcents Fuzzy Sets and Systems vol. no. 9 998. (Advance onlne ublcaton: February 7)