Research Article Supply Chain Contracts with Multiple Retailers in a Fuzzy Demand Environment

Transcription

1 Mathematcal Problems Egeerg Volume 3, Artcle ID 48353, ages htt://dx.do.org/.55/3/48353 Research Artcle Suly Cha Cotracts wth Multle Retalers a Fuzzy Demad Evromet Shegju Sag Deartmet of Ecoomcs, Heze Uversty, Heze 745, Cha Corresodece should be addressed to Shegju Sag; sagshegju@63.com Receved 7 March 3; Acceted July 3 Academc Edtor: Daoy Dog Coyrght 3 Shegju Sag. Ths s a oe access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch ermts urestrcted use, dstrbuto, ad reroducto ay medum, rovded the orgal work s roerly cted. Ths study vestgates suly cha cotracts wth a suler ad multle cometg retalers a fuzzy demad evromet. The market demad s cosdered as a ostve tragular fuzzy umber. The models of cetralzed decso, retur cotract, ad reveue-sharg cotract are bult by the method of fuzzy cut sets theory, ad ther otmal olces are also roosed. Fally, a examle s gve to llustrate ad valdate the models ad coclusos. It s show that the otmal total order quatty of the retalers fluctuates at the ceter of the fuzzy demad. Wth the rse of the umber of retalers, the otmal order quatty ad the fuzzy exected roft for each retaler wll decrease, ad the fuzzy exected roft for suler wll crease.. Itroducto Over the last decade or so, suly cha maagemet has emerged as a key area of research amog the racttoers of oeratos research. I recet years, coordato mechasm of suly cha cotracts has become oe of the most challegg ssues facg both racttoers ad scholars. Suly cha cotracts such as retur cotract ad reveuesharg cotract are strumets for suly cha coordato, whch shft the ucerta demad from the retaler to the suler, thus ecouragg the retaler to crease order quattes. A large body of lterature has exlored to coordate the suly cha wth retur cotract ad reveue-sharg cotract durg the last two decades. Pasterack []frst clamed that a arorate retur olcy ca fully coordate a sgle-suler sgle-retaler suly cha, whch was the exteded by Matrala ad Rama [] to the stuato where the retaler had several stores. Taylor [3] ad Lee et al.[4] studed the retur cotract wth effort-deedat demad. Theyshowedthatthsroblem,tattaedsulycha coordato combed wth feedback olcy. Yao et al. [5] aalyzedtheroftsofbothactorswhethemaufacturer adretalersharedorddotsharetheforecastformato returs olcy. Yue ad Raghuatha [6] dscussed the mact of a full retur olcy as well as formato sharg o the maufacturer ad the retaler uder formato asymmetry. Bose ad Aad [7] cosdered the wholesale rce as a exogeous rce to study returs olces for coordatg the suly cha. They showed that, geeral, a equlbrum returs olcy was ot Pareto effcet wth resect to a rce-oly cotract, but whe the wholesale rce was suffcetly hgh, the equlbrum returs olcy was Pareto effcet. These coclusos were cosstet wth those of Yao et al. [8]. Dg ad Che [9] studed the retur cotract ssues of a three-level suly a sgle-erod model. Yao et al. [] aalyzed the mact of rce-sestvty factors o characterstcs of retur cotract a sgle-erod roduct suly cha. Mollekof et al. []used a emrcalstudy to exlore how teret roduct returs maagemet systems affect loyalty tetos. Che ad Bell [, 3] showed that the customer returs affect the frm s rcg ad vetory decso ad roosed a agreemet betwee the maufacturer ad the retaler that cludes two buyback rces. Che [4] roosed a returs olcy wth a wholesale-rcedscout scheme that ca acheve suly cha coordato. A et al. [5] aalyzed the mlemetato of full returs olces the cha-to-cha cometto.

2 Mathematcal Problems Egeerg Reveue-sharg cotract has bee aled the vdeo cassette retal ad move dustry wth much success. Gaoccaro ad Potradolfo [6] showed that reveue-sharg couldcoordatememberstheewsboychaelwththree stages: suler, maufacturer, ad retaler. Cacho ad Larvere [7] tesvely dscussed a reveue-sharg cotract betwee a sgle suler ad a sgle retaler a sgleerod ewsboy roblem. Guta ad Weerawat [8] desged a reveue-sharg cotract to maxmze the cetralzed reveue by choosg a arorate vetory level. Yao et al. [9] vestgated a reveue-sharg cotract for coordatg a suly cha comrsg oe maufacturer ad two cometg retalers. Lh ad Hog [] studedareveue-sharg cotract a two-erod ewsboy roblem. Va Der Rhee et al. [] roosed a reveue-sharg mechasm multechelo suly chas. Ouardgh ad Km [] cosdered a sgle suler collaboratg wth two maufacturers o desgg qualty mrovemets for ther resectve roducts uder a reveue-sharg cotract. Krsha ad Wter [3] studed the role of reveue-sharg cotracts suly chas ad establshed a foudato algg cetves. Sheu [4] exlored reveue-sharg cotracts uder rce romoto to ed customers wth three tyes of romotoal demad atters. Zhag et al. [5] vestgated a reveuesharg cotract wth demad dsrutos a suly cha comrsg oe maufacturer ad two cometg retalers. Palsule-Desa [6] studed reveue-deedet cotracts ad reveue-deedet cotracts a two-erod model, ad they showed that both tyes of reveue-sharg cotracts could coordate the suly cha; however, there exsted stuatos whch reveue-deedet cotracts outerformed reveue-deedet cotracts. The covetoal studes have focused o the cases whch the demads are robablstc. I other words, the demads follow certa dstrbuto fucto. However, ractce, esecally for ew roducts, the robabltes are ot kow due to lack of hstory data. I ths case the demads are sutably descrbed subjectvely by lgustc terms, such as hgh, low, or aroxmately equal d, but deftely ot less tha a ad ot greater tha b. Thus, the ucerta theory, rather tha the tradtoal robablty theory, s well suted to the suly cha models roblem. Therefore, we assume that the exteral demad ca be aroxmately forecasted ad exressed as a tragular membersh fucto. I ths aer, the demads are aroxmately estmated by exerts ad regarded as fuzzy umbers. Retur cotract ad reveue-sharg cotract wth multle cometg retalers a fuzzy demad evromet wll be dscussed, ad the mact of the suler s roducto cost ad the umber of retalers o the models wll be aalyzed. The rest of the aer s orgazed as follows. Secto troduces some deftos ad roostos about fuzzy set theoryadotatosrelatedtothsaer.secto 3 develos three fuzzy suly models wth multle cometg retalers. Secto 4 rovdes a umercal examle to llustrate the result of the roosed cotracts.secto 5 summarzes the work doe ths aer.. Prelmares.. Fuzzy Set Theory Defto. The fuzzy set A =(a,a,a 3 ),wherea <a <a 3 ad defed o R,scalledthetragularfuzzyumber,fthe membersh fucto of A s gve by x a, f a μ A (x) = a a x a, a 3 x, f a a 3 a <x a 3, (), x (a,a 3 ), where a ad a 3 are the lower lmt ad uer lmt, resectvely, of the tragular fuzzy umber A. For x [a,a ],the left membersh fucto A L (x) = (x a )/(a a ) s a crease fucto of x.forx (a,a 3 ],therghtmembersh fucto A R (x) = (a 3 x)/(a 3 a ) s a decrease fucto of x. Defto. The tragular fuzzy umber A s called the ostve tragular fuzzy umber f a >. Defto 3. For ay α [,],theset A(α) = x μ A (x) α} s called the α cut set of A. A(α) s a oemty bouded closed terval cotaed the set of real umbers, ad t ca be deoted by A (α) =[ A L (α), A R (α)], () where A L (α) ad A R (α) are, resectvely, the left ad rght boudares of A(α),wth A L (α) = f x R : μ A(x) α}, (3) A R (α) = su x R : μ A(x) α}. Examle 4. For ay α [,],theα cut set of a tragular fuzzy umber A =(a,a,a 3 ) s A L (α) =a +(a a )α, (4) A R (α) =a 3 (a 3 a )α. Basedotheextesorclefuzzysets,wehavethe followg Proostos 5 ad 6. Proosto 5. For ay α [,],let A be a ostve tragular fuzzy umber ad let k be a ozero real umber; the k A (α) = [k A L [k A R (α),k A R (α)], k R+, (α),k A L (α)], k R. Proosto 6. For ay α [,],let B(α) = [ B L (α), B R (α)] ad C(α) = [ C L (α), C R (α)], resectvely, be the α cut set of theostvetragularfuzzyumbers B ad C;the B (α) + C (α) =[ B L B (α) C (α) =[ B L (α) + C L (α) C R (α), B R (α) + C R (α)], (5) (α), B R (α) C L (α)]. (6)

3 Mathematcal Problems Egeerg 3 Proosto 7 (see B. Lu ad Y.-K. Lu [7]). Let A be a ostve tragular umber; the exected value of A s E [ A] = [ A L (α) + A R (α)] dα. (7) Proosto 8 (seey.-k.luadb.lu[8]). Let A ad B be two deedet ostve tragular fuzzy umbers wth fte exected values. The for ay real umbers a ad b,oehas E[a A+b B] = ae [ A] + be [ B]. (8).. Problem Descrtos. Cosder a sgle-erod settg for a two-echelo suly, cosstg of a suler ad multle cometg retalers wth fuzzy demad. We assume that at the begg of the sellg seaso, the retaler ( =,,...,) has o vetory o had ad must decde the order quatty q from the suler. The, the retaler sells hs order of short-lfe roducts, such as ersoal comuters, cosumer electrocs, or fasho tems, wth hgh ucerta demad. The roducts are sold oly oe erod. As the lead tmes of such goods are much loger tha ther sellg seaso, the actors have o chace to lace a secod order. We cosder the total ucerta demad faced by the retalers as a ostve tragular fuzzy varable D =(d,d,d 3 ) wth the most ossble value d,where<d <d <d 3.The fuzzy demad D meas that the total demad s about d. d ad d 3 are the lower lmt ad uer lmt, resectvely, of the fuzzy demad D ad descrbed by a geeral membersh fucto μ D (x): L (x), x [d,d ], μ D (x) = R (x), x (d,d 3 ],, x (d,d 3 ). Let the total retal demad be dvded betwee the retalers roortoal to ther stockg quatty; that s, retaler s demad, D,s D =( q q ) D, () where q= = q. The followg otatos are used for a roduct the models: :theretalrce; w:thewholesalerce; c: the er ut roduct cost curred to the suler; b: the retur rce offered by suler retur cotract; Φ: the fracto reveue of the retaler reveuesharg cotract ad <Φ<; Π S : the fuzzy roft of the suler; Π R : the fuzzy roft of the retaler ; Π SC :thefuzzyroftofthesulycha. The suler ad the retaler areassumedtobersk eutral ad ursued maxmzato of ther fuzzy exected rofts. (9) 3. Model Aalyss 3.. Cetralzed Decso Makg wth Fuzzy Demad. Cosder a suly cha occued by a tegrated actor, whch ca also be regarded as the retalers ad the suler-makg cooerato. The fuzzy roft of two-stage suly cha ca be exressed as Π SC = m q, D} cq s.t. d q d 3. () Sce the fuzzy demad D = (d,d,d 3 ) () sa ostve tragular fuzzy umber, we kow that the order quatty q has two cases; that s, q [d,d ] or q (d,d 3 ]. Theorem 9. Whe c< c,theotmaltotalorderquatty q of the retalers s q =L ( ( c) ). () Proof. If q [d,d ], the the α cut set of mq, D} s [m q, D}] (α) = [L (α), q], α [, L (q)], [q, q], α (L (q), ]. (3) (a) For α [, L(q)], theα cut set of the suly cha s fuzzy roft s [ Π SC ] (α) =[L (α) cq,q cq]. (4) (b) For α (L(q), ],theresulttursto [ Π SC ] (α) = [q cq, q cq]. (5) By (7), we ca get the fuzzy exected roft E[ Π SC ] as E[ Π SC ]= L(q) (L (α) cq+q cq)dα + (q cq + q cq) dα L(q) = L(q) ( L (α) dα ql (q) + q) cq. (6) The frst ad secod dervatves of E[ Π SC ] (6) cabe obtaed as follows: de[ Π SC ] = dq ( L(q)) c, d E[ Π SC ] dq = L (q). (7)

4 4 Mathematcal Problems Egeerg Sce L(q) s a creasg fucto wth L (q) >, therefore d E[ Π SC ]/dq s egatve ad E[ Π SC ] s cocave q. Hece, the otmal total order quatty of the retalers ca be obtaed by solvg de[ Π SC ]/dq =,whchgves q =L ( ( c) ). (8) Sce <L(q ),thuswecagetc< c. The roof of Theorem 9 s comleted. Theorem. Whe >c,theotmaltotalorderquatty q of the retalers s q =R ( c ). (9) Proof. If q (d,d 3 ], the the α cut set of mq, D} s [m q, D}] (α) = [L (α),q], [L (α),r (α)], α [,R(q)], α (R(q),]. () (a) For α [, R(q)], theα cut set of the suly cha s fuzzy roft s [ Π SC ] (α) =[L (α) cq,q cq]. () (b) For α (R(q), ],theresulttursto [ Π SC ] (α) =[L (α) cq,r (α) cq]. () By (7), we ca get the fuzzy exected roft E[ Π SC ] as E[ Π SC ]= R(q) (L (α) cq+q cq)dα + R(q) (L (α) cq+r (α) cq)dα = ( R(q) R (α) dα +qr (q) + L (α) dα) cq. (3) The frst ad secod dervatves of E[ Π SC ] (3) cabe obtaed as follows: de[ Π SC ] dq = R (q) c, d E[ Π SC ] dq = R (q). (4) Sce R(q) s a decreasg fucto wth R (q) <,therefore d E[ Π SC ]/dq s egatve ad E[ Π SC ] s cocave q. Hece, the otmal total order quatty of the retalers ca be obtaed by solvg de[ Π SC ]/dq =,whchgves q =R ( c ). (5) Sce < R(q ),thuswecaget>c. The oof of Theorem s comleted. From (6) ad(3), Theorems 9 ad, wecaeasly obta the otmal fuzzy exected value of the roft for the tegrated suly cha, whch s gve by E[ Π SC ] = ( ( c)/ c/ L (λ) dλ, R (α) dλ+ L (α) dα), c < c, > c. (6) 3.. Retur Cotract wth Fuzzy Demad. I a retur cotract, the suler sets a wholesale rce w ad gves the retaler ( =,,...,)areturrceb for usold roducts at the ed of the seaso. The fuzzy roft of the retaler ca be exressed as follows: Π R (q,q )=m q, D }+bmax q D,} wq =m q, q q D} +bmax q q q D, } wq, (7) where q= = q ad q =q q. The retaler ( =,,...,)tres to maxmze ts fuzzy exected roft retur cotract by choosg the otmal order quatty q,whchsolvesthefollowg model: =E[m q, q q D} + b max q q q D, } wq ] q s.t. q d q q q d 3. (8) Theorem. Whe c< c, the otmal wholesale rce w retur cotract s w = ( b) (( c) ( ) ( ( c)/ L (α) dα L ( ( c) /) )). (9)

5 Mathematcal Problems Egeerg 5 Proof. If q [(q /q)d,(q /q)d ],thats,q [d,d ],the the α cut sets of mq,(q /q) D} ad maxq (q /q) D, } are d d q [m q, q q D}] (α) = [ q q L (α),q ], α (, L (q)], [q,q ], α (L(q),], [max q q q D, }] (α) = [, q q q L (α)], α (, L (q)], [, ], α (L(q),]. (3) = ( b) (q q q 3 L(q) L (α) dα+ q q L (q)). (34) Sce L(q) s a creasg fucto wth L (q) > ad >b, therefore d /dq s egatve ad s cocave q. Hece, there exsts a otmal order quatty for retaler for each q,whereq = j=,j = q j.asetof order quatty q = (q,...,q ) s a Nash equlbrum of the decetralzed system f each retaler s order quatty s a best resose. Thus, ay Nash equlbrum must satsfy each retaler s frst-order codto. Let d /dq =; we ca get (a) For α [, L(q)], the α cut set of the retaler s fuzzy roft s (α) =[ q q L (α) wq,q +bq b q q L (α) wq ]. (b) For α (L(q), ],theresulttursto (3) q =q ( ( w) b + L(q ) q L (α) dα L(q )) ( q L(q ) L (α) dα ). (35) (α) =[q wq,q wq ]. (3) By (7), we ca get the fuzzy exected roft as = L(q) ( q q L (α) wq +q +bq b q q L (α) wq ) dα + L(q) (q wq +q wq ) dα = L(q) ( b) (q q L (α) dα q L(q)) +( w)q. (33) The frst ad secod dervatves of (33)ca be obtaed as follows: d dq = ( b) (q q q L(q) L (α) dα L(q))+ w, Equato (35) gves each retaler s equlbrum order codtoal o q beg the equlbrum total order quatty. Hece, (35) descrbes a equlbrum oly f q =q. Substtute (35)toq =q ad smlfy L(q ) ( )( L(q ) q L (α) dα) = ( w). b (36) I order to fully coordate the suly cha, let L(q )=( c)/;wecaobta w = ( b) (L(q ) ( = ( b) (( c) )( L(q ) q ( ) L (α) dα)) ( ( c)/ L (α) dα L ( ( c) /) )). The oof of Theorem s comleted. (37)

6 6 Mathematcal Problems Egeerg Theorem. Whe >c, the otmal wholesale rce w retur cotract s By (7), we ca get the fuzzy exected roft as w =b+ ( b) ( c +( ) (38) = R(q) ( q q L (α) wq +q +bq c/ R (α) dα+ L (α) dα ( )). R (c/) Proof. If q ((q /q)d,(q /q)d 3 ],thats,q (d,d 3 ],the the α cut sets of mq,(q /q) D} ad maxq (q /q) D, } are [m q, q q D}] (α) [ q q L (α),q ], = [ q q L (α), q q R (α)], [max q q q D, }] (α) α [,R(q)], α (R(q),], [, q q q L (α)], α [, R (q)], = [q q q R (α),q q q L (α)], α (R(q),]. (39) (a) For α [, R(q)],theα cut set of the retaler s fuzzy roft s (α) =[ q q L (α) wq,q +bq b q q L (α) wq ]. (4) (b) For α (R(q), ],theresulttursto + R(q) = ( b) (q q b q q L (α) wq ) dα ( q q L (α) +bq b q q R (α) wq + q q R (α) +bq b q q L (α) wq ) dα R(q) R (α) dα + q q L (α) dα+q R(q)) (w b) q. (4) The frst ad secod dervatves of (4)ca be obtaed as follows: d dq = ( b) (q q q ( R(q) R (α) dα (w b), d dq L (α) dα) + R (q)) = ( b) ( q q 3 ( R(q) R (α) dα (α) = [ q q L (α) +bq b q q R (α) wq, q q R (α) +bq b q q L (α) wq ]. (4) L (α) dα) q q R (q)). (43) Sce R(q) s a decreasg fucto wth R (q) < ad >b, therefore d /dq s egatve ad s cocave q.

7 Mathematcal Problems Egeerg 7 Ay Nash equlbrum must satsfy each retaler s frstorder codto. Let d /dq =;wecaget q =q ( q R(q ) R (α) dα+ q L (α) dα +R (q ) (w b) b ) ( q R(q ) R (α) dα+ q Substtute (44)toq =q ad smlfy L (α) dα ). (44) Theorem 3. I fuzzy retur cotract, the retaler ( =,,...,)ad the suler atta ther otmal fuzzy exected value of the rofts at w fuzzy retur cotract, where =( b )E[ Π SC ], E[ Π S ] =( Proof. Cosder the followg. ( ) +b )E[ Π SC ]. =,,...,, (47) Case (d q d ).Whed q d,substtutgw ad L(q )=( c)/to (33), the fuzzy exected roft of the retaler (=,,...,)s gve as R(q )+( ) ( q R(q ) R (α) dα+ q L (α) dα) = (w b) b. (45) I order to fully coordate the suly cha, let R(q ) = c/;wecaobta = b ( c)/ L (α) dα = ( b )E[ Π SC ]. The fuzzy exected roft of the suler s E[ Π S ] =E[ Π SC ] = ( ( ) +b )E[ Π SC ]. (48) (49) w =b+ ( b) (R(q )+( ) Case (d q d 3 ).Whed <q d 3,substtutgw ad R(q )=c/to (4), the fuzzy exected roft of the retaler (=,,...,)s gve as ( q R(q ) R (α) dα+ q L (α) dα)) =b+ ( b) ( c +( ) = b ( c/ =( b )E[ Π SC ]. R (α) dα+ L (α) dα) The fuzzy exected roft of the suler s E[ Π S ] =E[ Π SC ] = ( ( ) +b )E[ Π SC ]. (5) (5) c/ R (α) dα+ L (α) dα ( )). R (c/) (46) The oof of Theorem s comleted. The oof of Theorem 3 s comleted Reveue-Sharg Cotract wth Fuzzy Demad. I a reveue-sharg cotract, the retaler (=,,...,)shares wth the suler a ercetage of hs reveue. Let ( Φ) be the fracto the suler ears, ad the Φ s the fracto the retaler kees.

8 8 Mathematcal Problems Egeerg Thus, we ca exress the fuzzy roft of the retaler as follows: Π R (q,q )=Φm q, D } wq =Φ m q, q q D} wq, (5) where q= = q ad q =q q. The retaler ( =,,...,)tres to maxmze ts fuzzy exected roft reveue-sharg cotract by choosg the otmal order quatty q,whchsolvesthe followg model: = E [Φ m q, q q D} wq ] s.t. q q d q q q d 3. (53) Theorem 4. Whe c< c, the otmal wholesale rce w reveue-sharg cotract s w =Φ Φ ( ( c) ( ) ( ( c)/ L (α) dα L ( ( c) /) )). (54) Proof. If q [(q /q)d,(q /q)d ],thats,q [d,d ],the the α cut set of mq,(q /q) D} s [m q, q q D}] (α) = [ q q L (α),q ], α (, L (q)], [q,q ], α (L(q),]. (55) (a) For α [, L(q)], the α cut set of the retaler s fuzzy roft s (α) =[Φ q q L (α) wq,φq wq ]. (56) (b) For α (L(q), ],theresulttursto (α) =[Φq wq,φq wq ]. (57) By (7), we ca get the fuzzy exected roft as = L(q) (Φ q q L (α) wq +Φq wq ) dα + L(q) (Φq wq +Φq wq ) dα = L(q) Φ (q q L (α) dα q L(q))+(Φ w)q. (58) The frst ad secod dervatves of (58)ca be obtaed as follows: d dq = Φ (q q q d dq = Φ (q q q 3 L(q) L (α) dα L(q))+Φ w, L(q) L (α) dα+ q q L (q)). (59) Sce L(q) s a creasg fucto wth L (q) > ad Φ >, therefore d /dq s egatve ad s cocave q. Hece, there exsts a otmal order quatty for retaler for each q.asetoforder quatty, q = (q,...,q ) s a Nash equlbrum of the decetralzed system f each retaler s order quatty s a best resose. AyNashequlbrummustsatsfyeachretaler sfrstorder codto. Let d /dq =;wecaget q =q (Φ w) /Φ + (/q ) L(q ) L (α) dα L(q ). (/q ) L(q ) L (α) dα (6) Equato (6) gves each retaler s equlbrum order codtoal o q beg the equlbrum total order quatty. Hece, (6) descrbes a equlbrum oly f q =q. Substtute (6)toq =q ad smlfy L (q ) ( )( L(q ) q L (α) dα) = (Φ w). Φ (6)

9 Mathematcal Problems Egeerg 9 I order to fully coordate the suly cha, let L(q )=( c)/;wecaobta w =Φ Φ (L(q ) ( )( L(q ) q L (α) dα)) (b) For α (R(q), ],theresulttursto (α) =[Φ q q L (α) wq,φ q q R (α) wq ]. (66) =Φ Φ By (7), we ca get the fuzzy exected roft as ( ( c) ( ) ( ( c)/ L (α) dα L ( ( c) /) )). (6) = R(q) + R(q) (Φ q q L (α) wq +Φq wq ) dα (Φ q q L (α) wq +Φ q q R (α) wq ) dα = Φ (q q R(q) R (α) dα The oof of Theorem 4 s comleted. Theorem 5. Whe >c, the otmal wholesale rce w reveue-sharg cotract s + q q L (α) dα+q R(q)) wq. (67) w = Φ ( c +( ) c/ R (α) dα+ L (α) dα ( )). R (c/) (63) Proof. If q ((q /q)d,(q /q)d 3 ],thats,q (d,d 3 ],the the α cut set of m q,(q /q) D} s [m q, q q D}] (α) The frst ad secod dervatves of (67)ca be obtaed as follows: d dq = Φ (q q q ( R(q) R (α) dα d dq L (α) dα) + R (q) ) w, [ q q L (α),q ], α [, R (q)], = [ q q L (α), q q R (α)], α (R(q),]. (64) (a) For α [, R(q)],theα cut set of the retaler s fuzzy roft s = Φ ( q q 3 ( R(q) R (α) dα L (α) dα) q q R (q)). (68) (α) =[Φ q q L (α) wq,q wq ]. (65) Sce R(q) s a decreasg fucto wth R (q) < ad Φ>, therefore d /dq s egatve ad s cocave q.

10 Mathematcal Problems Egeerg Ay Nash equlbrum must satsfy each retaler s frstorder codto. Let d /dq =;wecaget q =q ( q ( q R(q ) + q R(q ) R (α) dα Substtute (69)toq =q R(q )+( ) ( q R(q ) L (α) dα+r(q ) w Φ ) R (α) dα+ q ad smlfy R (α) dα+ q L (α) dα ). (69) L (α) dα ) = w Φ. (7) I order to fully coordate the suly cha, let R(q ) = c/;wecaobta w = Φ (R (q )+( ) ( q = Φ (c +( ) R(q ) R (α) dα + q L (α) dα)) c/ R (α) dα+ L (α) dα ( )). R (c/) (7) The oof of Theorem 5 s comleted. Theorem 6. I fuzzy reveue-sharg cotract, the retaler ( =,,...,) ad the suler atta ther otmal fuzzy exected value of the rofts at w fuzzy retur cotract, where Proof. Cosder the followg. Case (d q d ).Whed q d,substtutgw ad L(q )=( c)/to (35), the fuzzy exected roft of the retaler (=,,...,)s gve as = Φ ( c)/ L (α) dα = ( Φ )E[ Π SC ]. The fuzzy exected roft of the suler s E[ Π S ] =E[ Π SC ] = ( Φ )E[ Π SC ]. (73) (74) Case (d <q d 3 ).Whed <q d 3,substtutgw ad R(q )=c/to (44), the fuzzy exected roft of the retaler (=,,...,)s gve as = Φ ( c/ =( Φ )E[ Π SC ]. R (α) dα+ L (α) dα) The fuzzy exected roft of the suler s E[ Π S ] =E[ Π SC ] = ( Φ )E[ Π SC ]. The oof of Theorem 6 s comleted. (75) (76) Theorem 7. If b=( Φ), the the fuzzy exected rofts for retaler (=,,...,)ad suler retur cotract are equal to those reveue-sharg cotract: =, E[ Π S ] =E[ Π S ]. (77) Proof. Substtutg b=( Φ)to = (( b)/ )E[ Π SC ] ad E[ Π S ] = ((( ) + b)/())e[ Π SC ], we ca obta =( Φ )E[ Π SC ] =, =( Φ )E[ Π SC ], E[ Π S ] =( Φ )E[ Π SC ]. =,,...,, (7) E[ Π S ] =( Φ )E[ Π SC ] =E[ Π S ]. The oof of Theorem 7 s comleted. (78)

11 Mathematcal Problems Egeerg Table : Equlbrum value of the arameters for dfferet c ad retur cotract (b = 3.). c q w E[ Π S ] Table : Equlbrum value of the arameters for dfferet c ad reveue sharg cotract (Φ =.6). c q w E[ Π S ] Numercal Examle I ths secto, we ted to further elucdate the revously roosed two cotracts wth a umercal examle. We wll aalyze the effect of the umber of retaler o the other arameters. Suosethemostossblevalueofthedemadmarket s d = uts ad the maxmum ad mmum ossble values of the demad are, resectvely, d = uts ad d 3 = 3 uts; that s to say, the fuzzy demad s D = (,, 3).Let =.$erut. The otmal order quatty, wholesale rce, ad fuzzy exected roft of the actors retur cotract ad reveuesharg cotract ca be lsted Tables ad,resectvely. From Tables ad, we aalyze the fluece of arameters c ad o the otmal equlbrum values as show below. (a) It s obvous that the otmal order quatty for retaler wll decrease alog wth the rse of the suler s roducto cost c ad the umber of retalers whe the other arameters are fxed. Partculary, ths umercal examle, the otmal total order quatty q s equal to the most ossble value of fuzzy demad whe c = 5.Whec > 5 ad c < 5, the otmal total order quatty of retaler slocatedattheleftadrghtofthemostossble value of fuzzy demad D, resectvely.theotmal wholesale rce wll crease alog wth the suler s roducto cost c ad theumber of retalers whe the other arameters are fxed. (b) From Tables ad, tcabeotedthatwhethe other arameters are fxed both cotracts, the fuzzy exected roft of the retaler wll decrease alog wth the rse of arameters c ad. O the other had, the fuzzy exected roft of the suler wll decrease alog wth the rse of c ad crease alog wth the rse of. 5. Coclusos Ths aer formulates suly cha cotracts based o fuzzy set theory, where a suler ad multle cometg retalers adot retur cotract ad reveue-sharg cotract. I order to exame models erformace fuzzy demad, we use fuzzy cut sets method to solve ths roblem. The advatage of the roosed method s that t removes the eed for eumerato over alteratve values ad rovdes a better uderstadg of the relatos amog the decso arameters. The techque roosed ths aer s easer to mlemet ad requres less data. It s arorate whe the evromet s comlex, ambguous, or there s a lack of statstcal data. Ackowledgmets The author wshes to exress hs scerest thaks to the edtors ad aoymous referees for ther costructve commets ad suggestos o ths aer. Ths work was suorted by the Natoal Natural Scece Foudato of Cha (7975, 778) ad the Doctoral Foudato of Heze Uversty (XYBS3). Refereces [] B. A. Pasterack, Otmal rcg ad retur olces for ershable commodtes, Marketg Scece, vol. 7, o.,. 33 4, 8. [] M. K. Matrala ad K. Rama, Demad ucertaty ad suler s returs olces for a mult-store style-good retaler, Euroea Joural of Oeratoal Research, vol. 5, o.,. 7 84, 999. [3] T. A. Taylor, Chael coordato uder rce rotecto, mdlfe returs, ad ed-of-lfe returs dyamc markets, Maagemet Scece, vol. 47, o. 9,. 34,. [4] H.L.Lee,V.Padmaabha,T.A.Taylor,adS.Whag, Prce rotecto the ersoal comuter dustry, Maagemet Scece,vol.46,o.4, ,.

12 Mathematcal Problems Egeerg [5] D.-Q. Yao, X. Yue, X. Wag, ad J. J. Lu, The mact of formato sharg o a returs olcy wth the addto of a drect chael, Iteratoal Joural of Producto Ecoomcs, vol.97,o.,.96 9,5. [6] X. Yue ad S. Raghuatha, The macts of the full returs olcy o a suly cha wth formato asymmetry, EuroeaJouralofOeratoalResearch,vol.8,o., , 7. [7] I. Bose ad P. Aad, O returs olces wth exogeous rce, EuroeaJouralofOeratoalResearch, vol. 78, o. 3, , 7. [8] Z.Yao,Y.Wu,adK.K.La, Demaducertatyadmaufacturer returs olces for style-good retalg cometto, Producto Plag ad Cotrol, vol.6,o.7,.69 7, 5. [9] D. Dg ad J. Che, Coordatg a three level suly cha wth flexble retur olces, Omega, vol. 36, o. 5, , 8. [] Z. Yao, S. C. H. Leug, ad K. K. La, Aalyss of the mact of rce-sestvty factors o the returs olcy coordatg suly cha, Euroea Joural of Oeratoal Research, vol. 87, o.,. 75 8, 8. [] D. A. Mollekof, E. Rabovch, T. M. Laseter, ad K. K. Boyer, Maagg teret roduct returs: a focus o effectve servce oeratos, Decso Sceces,vol. 38,o.,. 5 5, 7. [] J. Che ad P. C. Bell, The mact of customer returs o rcg ad order decsos, Euroea Joural of Oeratoal Research,vol.95,o.,.8 95,9. [3] J. Che ad P. C. Bell, Coordatg a decetralzed suly cha wth customer returs ad rce-deedet stochastc demad usg a buyback olcy, Euroea Joural of Oeratoal Research,vol.,o.,.93 3,. [4] J. Che, Returs wth wholesale-rce-dscout cotract a ewsvedor roblem, Iteratoal Joural of Producto Ecoomcs, vol. 3, o.,. 4,. [5]X.Z.A,J.Che,H.X.Zhao,adX.W.Tag, Cometto amog suly chas: mlcato of full returs olcy, Iteratoal Joural of Producto Ecoomcs,vol.39,o.,.57 65,. [6] I. Gaoccaro ad P. Potradolfo, Suly cha coordato by reveue sharg cotracts, Iteratoal Joural of Producto Ecoomcs,vol.89,o.,.3 39,4. [7] G. P. Cacho ad M. A. Larvere, Suly cha coordato wth reveue-sharg cotracts: stregths ad lmtatos, Maagemet Scece,vol.5,o.,.3 44,5. [8] D. Guta ad W. Weerawat, Suler-maufacturer coordato caactated two-stage suly chas, Euroea Joural of Oeratoal Research,vol.75,o.,.67 89,6. [9] Z. Yao, S. C. H. Leug, ad K. K. La, Maufacturer s reveuesharg cotract ad retal cometto, Euroea Joural of Oeratoal Research,vol.86,o., ,8. [] C. T. Lh ad Y. Hog, Chael coordato through a reveue sharg cotract a two-erod ewsboy roblem, Euroea Joural of Oeratoal Research, vol.98,o.3,.8 89, 9. [] B. Va Der Rhee, J. A. A. Va Der Vee, V. Veugoal, ad V. R. Nalla, A ew reveue sharg mechasm for coordatg mult-echelo suly chas, Oeratos Research Letters, vol. 38,o.4,.96 3,. [] F. El Ouardgh ad B. Km, Suly qualty maagemet wth wholesale rce ad reveue-sharg cotracts uder horzotal cometto, Euroea Joural of Oeratoal Research, vol. 6, o., ,. [3] H. Krsha ad R. A. Wter, O the role of reveue-sharg cotracts suly chas, Oeratos Research Letters,vol.39, o.,. 8 3,. [4] J.-B. Sheu, Marketg-drve chael coordato wth reveue-sharg cotracts uder rce romoto to edcustomers, Euroea Joural of Oeratoal Research,vol.4, o., ,. [5]W.-G.Zhag,J.Fu,H.L,adW.Xu, Coordatoofsuly cha wth a reveue-sharg cotract uder demad dsrutos whe retalers comete, Iteratoal Joural of Producto Ecoomcs,vol.38,o.,.68 75,. [6] O. D. Palsule-Desa, Suly cha coordato usg reveuedeedet reveue sharg cotracts, Omega, vol. 4, o. 4, , 3. [7] B. Lu ad Y.-K. Lu, Exected value of fuzzy varable ad fuzzy exected value models, IEEE Trasactos o Fuzzy Systems, vol.,o.4, ,. [8] Y.-K. Lu ad B. Lu, Exected value oerator of radom fuzzy varable ad radom fuzzy exected value models, Iteratoal Joural of Ucertaty, Fuzzess ad Kowlege- Based Systems,vol.,o.,.95 5,3.

13 Advaces Oeratos Research Volume 4 Advaces Decso Sceces Volume 4 Joural of Aled Mathematcs Algebra Volume 4 Joural of Probablty ad Statstcs Volume 4 The Scetfc World Joural Volume 4 Iteratoal Joural of Dfferetal Equatos Volume 4 Volume 4 Submt your mauscrts at Iteratoal Joural of Advaces Combatorcs Mathematcal Physcs Volume 4 Joural of Comlex Aalyss Volume 4 Iteratoal Joural of Mathematcs ad Mathematcal Sceces Mathematcal Problems Egeerg Joural of Mathematcs Volume 4 Volume 4 Volume 4 Volume 4 Dscrete Mathematcs Joural of Volume 4 Dscrete Dyamcs Nature ad Socety Joural of Fucto Saces Abstract ad Aled Aalyss Volume 4 Volume 4 Volume 4 Iteratoal Joural of Joural of Stochastc Aalyss Otmzato Volume 4 Volume 4