Research on Closed-Loop Supply Chain Network Equilibrium. with Two-Type Suppliers, Risk-Averse Manufacturers. and Capacity Constraints

Transcription

1 Jounal of Industial Engineeing and anageent JIE, 205 8(2): Online ISS: Pint ISS: Reseach on Closed-Loop Supply Chain etwok Equilibiu with Two-Type Supplies, Risk-Avese anufactues and Capacity Constaints Sun Hao *, Li Jian 2, Zhang Gui-Tao Qingdao Univesity, School of anageent Science and Engineeing (China) 2 anjing Agicultual Univesity, College of Engineeing (China) * Coesponding autho: ivaldoking@gail.co, t e l o ance6@gail.co, zhangguitao@qdu.edu.cn Received: ovebe 204 Accepted: Januay 205 Abstact: Pupose: The ai of this pape is to investigate the closed-loop supply chain (CLSC) netwok equilibiu with the consideation of thee pactical factos: two copleentay types of supplies, isk-avese chaacte of the anufactue and capacity constaints of the supplies. Design/ethodology/appoach: The equilibiu of vaious decision akes including the supplies, the anufactues, the etailes, the collectos and the deand akets ae odeled via finite-diensional vaiational inequality, espectively. Then the govening CLSC netwok equilibiu odel is established. The logaithic-quadatic poxial pediction-coection algoith is designed to solve the vaiational inequality odel. ueical exaples ae given to analyze the ipact of etun ate, isk-avese degee and capacity constaints on the netwok equilibiu unde diffeent poduct BOs. Findings: With the incease of etun ate, the pofits of vaious channel ebes and the pefoance of the CLSC syste will ipove. Thee is a contadiction between pofit axiization and isk iniization fo the anufactues. oeove, the econoic behavio of the CLSC is likely to be liited by the capacity constaints of the supplies. Oiginality/value: Pio to this pape, few papes have addessed with the CLSC netwok equilibiu consideing soe pactical factos. They assue all the supplies ae identical and -509-

2 Jounal of Industial Engineeing and anageent all the decision-akes ae isk neutal. Futheoe, the poduction capacities of all supplies ae assued to be infinite o lage enough. To fill the gap, this pape exaines the influences of two-type supplies, isk avesion and capacity constaints upon the CLSC netwok equilibiu. eywods: closed-loop supply chain netwok, vaiational inequalities, ulti-type supplies, isk avese, capacity constaints. Intoduction Reanufactuing and closed-loop supply chain (CLSC) anageent can not only potect envionent, poote sustainable developent and but also ake the fis ean oe pofits (Guide & Wassenhove, 2006; Atasu, Guide & Wassenhove, 2008). Theefoe, they have eceived geat attention fo the entepeneus in ecent yeas. any faous copanies, such as BW, IB, odak and Fuji Xeox, just to nae a few, have established effective endof-life (EOL) poduct collection and eanufactuing systes besides thei taditional poduction and distibution facility. Coespondingly, in acadeia, any scholas also have done plenty of eseach on CLSC anageent to guide pactice. In geneal, ost of the liteatue deals with tactical and opeational issues, such as evese channel stuctue, picing stategies of new and eanufactued poducts, inventoy contol and logistics netwok design. Savaskan, Bhattachaya and Wassenhove (2004) analyzed the evese channel choice poble in CLSC and poved that etaile-collection ode is optial. Shi, Zhang and Sha (20) developed a atheatical odel to axiize the oveall pofit of the CLSC syste by siultaneously deteining the selling pice, the poduction quantities fo band-new poducts and eanufactued poducts. Golinska and awa (20) povided a faewok fo anageent of evese flow of ateials in autootive industy, and utilized agent-based technology to find the optial solutions fo the dynaic netwok configuation. Zhu and Xu (202) built an integated optiization odel fo a CLSC netwok unde uncetainty, and solved the odel by hybid genetic algoith. But they failed to ecognize the CLSC as a business pocess and ignoed the ipotant stategic issue of copetition. Unde a copetitive envionent, ajude and Goenevelt (200) pesented a two-peiod odel of eanufactuing in the face of copetition between an oiginal equipent anufactue (OE) and a local eanufactue. Savaskan and Wassenhove (2006) investigated the optial evese channel design in a CLSC with one anufactue and two copeting etailes. Reseach by (Webste & ita 2007; ita & Webste, 2008) exained the ipact of take-back laws and govenent subsidies within an OE/eanufactue copetitive faewok, espectively. Othe elated eseach includes Debo, Toktay and Wassenhove (2005), Feguson and Toktay (2006), Chen and Chang (202), Ösdei, eahlıoğlu-ziya and -50-

3 Jounal of Industial Engineeing and anageent Palaktük (204) and so on. Although the liteatues lay a solid foundation fo futue eseach, they only discuss the copetition between an OE and a eanufactuing o between two etailes who ae dealing with a single deand aket. In fact, as we know, the CLSC is likely to have a wide vaiety of channel ebes and coplex copetitive elations. Fo exaple, a coplete CLSC ay have seveal utual copetitive supplies, copetitive anufactues, copetitive etailes, copetitive collectos and seveal deand akets. As noted in aguney, Dong and Zhang (2002), such type of copetition could be the ain dive of the supply chain pofitability and theefoe, needs to be analyzed oe in depth. Thee ae a few papes studying the copetition aong ultiple kinds of channel ebes in evese supply chain netwok o CLSC netwoks. aguney and Toyasaki (2005) constucted a ultitieed netwok equilibiu faewok fo electonic waste ecycling using the vaiational inequality ethod. Haond and Beullens (2007) expanded the odel of aguney and Toyasaki (2005) to build an oligopolistic CLSC netwok odel including anufactues and deand akets unde WEEE legislation. Yang, Wang and Li (2009) used vaiational inequality ethod to odel the five-echelon CLSC netwok including supplies, anufactues, etailes, ecovey centes and consue akets. Qiang, e, Andeson and Dong (203) established a CLSC netwok odel consideing the copetition, distibution channel investent and uncetainties. Wakolbinge, Toyasaki, owak and aguney (204) foulated the e-waste netwok flow odel as a vaiational inequality poble to analyze how technical, aket, and legislative factos influence the total aount of e-waste that is collected, ecycled, expoted and (legally and illegally) disposed off. Howeve, the above entioned netwok odels didn t conside the case of ulti-type supplies. In eality, the anufactues need coopeate with any diffeent types of povides, such as ulti-type aw ateial o coponent supplies, so the peise that all supplies ae identical ay be inappopiate; In addition, ost liteatue assued the poduction capacity of all supplies ae infinite o lage enough, but in fact the quantity of aw ateials supplied to the anufactues ae usually liited by the capacity of the supplies. Futheoe, the chaacteistics of decision-akes isk attitudes ae seldo consideed in the pevious odels. To fill the gap, this pape studies the CLSC netwok equilibiu odel with the consideation of thee distinct factos: two copleentay types of supplies, capacity constaints of the supplies and the anufactues isk-avese chaacte, then analyzes the ipact of these factos on the equilibiu solutions cobined with nueical exaples and povides seveal eaningful anageent insights. The est of this pape is oganized as follows: in Section 2, we deive the optiality conditions of the vaious decision-akes by finite-diensional vaiational inequality theoy and foulate the whole CLSC netwok equilibiu odel. In Section 3, we pesent the coputational algoith fo solving the ash equilibiu. We conduct nueical exaples and -5-

4 Jounal of Industial Engineeing and anageent sensitivity analysis in Section 4 to geneate anageial insights fo ou odel. Finally, conclusions and suggestions fo futue eseach ae given in section Closed-Loop Supply Chain etwok odel The geneal CLSC netwok investigated is illustated in Figue, which consists of the fowad supply chain netwok and the evese supply chain netwok. In the fowad netwok, two goups of J and J 2 supplies povide two copleentay types of aw ateials espectively fo anufactues who ae involved in the poduction of hoogeneous poducts, which ae then shipped to etailes, who, in tun, sell the poducts in deand akets. In the evese netwok, I collectos pick up the EOL poducts fo the deand akets, gathe, clean and delive the back to the anufactues fo eanufactuing. In Figue, the eal lines expess the fowad tansactions between the supplies and the anufactues, the anufactues and the etailes, the etailes and the deand akets; the dashed lines illustate the evese tansactions between the deand akets and the collectos, the collectos and the anufactues. oeove, it is assued the eanufactued poducts and the new poducts ae indistinguishable. In othe wods, thee ae no peceived quality and eliability depeciation of the two kinds of poducts fo the consues. Figue. The stuctue of CLSC netwok with two-type supplies 2. The Equilibiu of the Supplies and thei Optiality Conditions Let q j R +,q 2 j2 R +, denote the aw ateial shipent between a typical supplie j, j 2 and a typical anufactue, and goup the shipents between all pais of supplies and anufactues into the colun vecto (Q,Q 2 ) R + J +J 2. Let f j (q J ) and f 2 j 2 (q J2 ) denote the poduction cost of supplie j and j 2 espectively, which eans the cost of supplie j, j 2 depend not only on his own aw ateial output but also on those of the othe supplies. The tansaction cost undetaken by the supplie between supplie j, j 2 and anufactue is -52-

5 Jounal of Industial Engineeing and anageent denoted by c (q j j ),c (q j2 2 j 2 ) espectively, and the tansaction pice associated with supplie j, j 2 and anufactue is denoted by ρ j, ρ j 2. Let C j and C 2 j2 epesent the uppe bound of the poduction capacity fo the st-type supplie j and 2nd-type supplie j 2, espectively. oeove, assue f j (q J diffeentiable and convex functions. ),f 2 j2 (q J2 ) and c (q j j ),c (q j2 2 j 2 ) ae continuous, Given the notations and assuptions entioned above, the pofit axiization poble of each supplie j can be expessed as axπ j (q J )= q j 0 ρ j q j f j (q J ) c (q j ) j () s.t. q j C j, q j q j (2) Equation () states that the pofit of the st-type supplie is equal to sales evenues of aw ateials inus the costs associated with poduction and tansactions. The fist inequality of constaint set (2) eans the poduction quantity of the st-type aw ateial by the supplie should be no oe than the uppe bound of his capacity. The second inequality of constaint set (2) eans the tansaction volues between the supplie and all the anufactues ust not exceed his poduction quantity. The pofit axiization poble of each supplie j 2 can be expessed as ax q j 2 0 π j2 (q J2 )= ρ j 2 q 2 j2 f 2 j (q J2 2 ) c (q j2 2 j 2 ) (3) s. t. q 2 j2 C 2 j 2, q 2 j 2 q 2 j2 (4) Siila as Equation () and constaint set (2), we can explain Equation (3) and constaint set (4) fo the 2nd-type supplie. Because the sae type supplies copete in a non-coopeative fashion, the optiality conditions fo both type of supplies can be expessed siultaneously using the following vaiational inequality: deteine the optial solution (q J,q J2,Q,Q 2 ) Ω S, satisfying J + j J 2 j 2 [ c (q j j ) q j [ c (q j 2 2 j 2 ) q 2 j2 (q J,q J2,Q,Q 2 ) Ω S ρ j ρ j 2 Whee Ω ={ s (q J, q J2, Q, Q 2 J ) R +J 2 +J +J 2 + J f x[q j q j j + (q J ) x[q q j q j j x[q 2 j2 q 2 j 2 q j j + j 2 C j,q 2 j 2 J 2 f 2 j2 (q 2 J 2 ) x[q q 2 j 2 q 2 j j 2 C 2 j 2, q j q J, q 2 j2 q 2 J } 2 (5) -53-

6 Jounal of Industial Engineeing and anageent The Equilibiu of the anufactues and thei Optiality Conditions Each unit of poduct is ade up of b unit of st-type aw ateial and b 2 unit of 2nd-type aw ateial, which is the so-called Bills of ateials (BO). Let q denote the non-negative poduction output by aw ateials of anufactue, and goup the poduction output of all anufactues into the colun vecto q R +. The poduct shipent between anufactue and etaile n is denoted by q n; the poduct shipents between all pais of anufactues and etailes ae gouped into a colun vecto Q 3 R +. Denote ρ n as the pice chaged fo the poduct by anufactue to etaile n. The EOL poduct tansaction volue between collecto i and anufactue is denoted by q i; the EOL poduct tansaction volues between all pais of anufactues and collectos ae gouped into a colun vecto Q 4 R + I. Let q u denote the quantity of the EOL poducts that anufactue decides to eanufactue, and goup all these q u into the colun vecto q u, then it is staightfowad to obtain q u i I q i. Denote ρ i as the pice chaged fo the EOL poduct by collecto i to anufactue. Each anufactue is face with the poduction cost of the aw ateials f (q) and the eanufactuing cost of the EOL poducts f u (q u ) espectively. Assue these two functions ae continuous, diffeentiable and convex. The tansaction cost undetaken by the anufactue between anufactue and etaile n is a function of c n=c n(q n). oeove, in ode to expess copetition aong anufactues, it is assued that f (q) depends not only on the poduction output of anufactue but also on those of the othe anufactues. Given the above notations and assuptions, we can expess the citeion of pofit axiization fo anufactue as: ax π (q, q u, Q, Q 2,Q 3,Q 4 )= J 2 j 2 ρ 2 j2 I q 2 j 2 i n ρ i q i f u (q u ) ρ n q n f (q) n J c n (q n ) j ρ j q j (6) whee Ω ={(q,q,q, q,q u +J j j 2 n,q i ) R + J I + n J q n q +q u, β q q j j J 2 β 2 q q,q 2 j 2 j 2 u I i q i } (7) Equation (6) indicates that the pofit of the anufactue is equal to sales evenues of new and eanufactued poducts inus the poduction and eanufactuing costs, the buy-back costs of EOL poducts and the tansaction costs with the supplies and the etailes. The fist inequality of constaint set (7) eans the shipents of poducts between the anufactue and all the etailes should be equal to o less than the su of the poduction output and the -54-

7 Jounal of Industial Engineeing and anageent eanufactuing output. The second inequality of constaint set (7) eans that the anufactue s poduction quantity of new poducts should be constained by the quantity of st-type aw ateial puchased fo the supplies dividing by the nube of the st-type aw ateial in each poduct accoding to the BO. The explanation of the thid inequality is siila as the second one. The fouth inequality of constaint set (7) eans the quantity of the EOL poducts that the anufactue intends to eanufactue is less than o equal to the tansaction volues between hi and all the collectos. oeove, we assue that with the consideation of the isk associated with puchasing aw ateials fo the two-type supplies, shipping the poducts to the vaious etailes and obtaining EOL poducts fo the collectos, all the anufactues ae chaacteistic of isk avesion. Theefoe, besides the citeion of pofit axiization each anufactue is concened with isk iniization. The isk level peceived by a typical anufactue ay be dependent not only on the flows he contols but also upon those contolled by othe anufactues. Hence, the second citeion of anufactue can be expessed as (Zsidisin, 2003; aguney, Cuz, Dong & Zhang, 2005): in = (Q, Q 2,Q 3,Q 4 ), (8) q j 0, q j 2 0,q n 0,q i 0 Each anufactue associates a non-negative isk-avese degee l with the isk iniization citeion (8). Hence, the ulti-citeion decision-aking poble fo anufactue can be tansfoed into a single objective optiization poble (eeney & Raiffa, 993): ax U (q,q u,q,q 2,Q 3, Q 4 )= n J 2 j 2 ρ 2 j2 I q j 2 i ρ n q n f (q) n ρ i q i f u (q u ) λ (Q, Q 2, Q 3, Q 4 ) J c n (q n ) j ρ j q j (9) We pesue that the tansaction cost functions and the isk functions fo each anufactue ae continuous and convex. Because all the anufactues copete non-coopeatively, the optial conditions fo all the anufactues can be descibed as the following vaiational inequality: deteine the optial (q *, q u*, Q *, Q 2*, Q 3*,Q 4* ), satisfying: f (q ) q J + j J 2 + j 2 [ ρ j [ ρ 2 j2 I + [ ρ i i x[q q + n [ c (q n n) q n +λ (Q,Q 2, Q 3, Q 4 ) q j +λ (Q,Q 2, Q 3, Q 4 ) q 2 j2 +λ (Q, Q 2,Q 3,Q 4 ) (q,q u,q,q 2,Q 3,Q 4 ) Ω q i ρ n x[q j x[q 2 j 2 x[q i q i +λ (Q, Q 2,Q 3,Q 4 ) q n q j q 2 j2 f (q u ) x[q u u q 0 + q u x[q n q n (0) -55-

8 Jounal of Industial Engineeing and anageent Whee Ω ={(q, q u, Q,Q 2,Q 3,Q 4 2+J ) R +J 2 + +I n + J q n q +q u,β q q j j J 2 β 2 q q,q 2 j2 j 2 I u i q i }. 2.3 The Equilibiu of the Retailes and thei Optiality Conditions The etailes tansact with the anufactues as well as the consues. Specifically, they need to decide how uch to ode fo the anufactues in ode to deal with the deand akets while seeking to axiize thei pofits. Let s n = q n, in tun, denote the total ode quantity at etaile n that he obtains fo all the anufactues, and goup these puchased aounts into the colun vecto s R +. The etaile n is faced with what we te a handling cost, which ay include the display and stoage cost associated with the poduct. Let c n(s n) denote this cost function and it is continuous, diffeentiable and convex. The poduct shipent between etaile n and deand aket k is denoted by q nk; the poduct shipents between all pais of etailes and deand akets ae gouped into a colun vecto Q 5 R +. The etaile n has associated tansaction costs in egad to tansacting with the deand aket k. Let c nk(q nk) epesent this cost. In addition, the tansaction pice between etaile n and deand aket k is denoted by ρ nk Fo etaile n, his axiu pofit odel is ax π n (Q 3, Q 5 )=ρ nk k q nk c n (s n ). ρ n q n c nk (q nk ) () k whee Ω R n ={ (q,q ) R + n nk + s = n q n, k q nk q n} (2) Equation () indicates that the pofit of the etaile is equal to sales evenues of new and eanufactued poducts inus the handling cost, the puchasing costs paid to the anufactues and the tansaction costs with the collectos. The equality in constaint set (2) eans that the handling quantity at the etaile s should be equal to the quantity of the poducts he puchases fo all the anufactues. The inequality in constaint set (2) eans that the total sale volues fo the etaile to all the deand akets should be no oe than the etaile obtains fo all the anufactues. -56-

9 Jounal of Industial Engineeing and anageent Because all the etailes copete in a non-coopeative fashion, the optiality conditions fo all etailes can be expessed as the vaiational inequality: deteine the optial (Q 3,Q 5 ) R, satisfying: k k n Whee Ω R ={ [ c (q ) nk nk q nk ρ nk x[q nk q nk (Q3,Q 5 ) R + + s = n + n q n, k [ρ n + c (s ) n n q n q nk q n}. x[q n q n 0, (Q 3,Q 5 ) Ω R (3) 2.4 The Equilibiu of the Deand akets The consues in the deand akets tansact with the etailes as well as the collectos. Specifically, in the fowad supply chain, the consues need to decide the puchasing quantity of the poducts and the pice they ae willing to pay. In the evese supply chain, the consues should decide the quantity of EOL poducts that will be etuned to the collectos. In the fowad supply chain, let k denote the pice of the poducts at deand aket k and goup all these k into a colun vecto ρ R k +. Futhe, denote the deand of the poducts at deand aket k as d k(), which is a onotone deceasing function with espect to the pices of all the akets. In addition, ^c nk (q nk ) is intoduced to epesent the tansaction costs undetaken by the consues in deand aket k while buying the poducts fo etaile n. ow we can give the following equations accoding to the spatial pice equilibiu theoy (aguney et al., 2002). Fo etaile n, and fo deand aket k, ρ nk + ^c nk (q nk d k (ρ ){= n ) { =ρ, ρ k nk >0 ρ k, ρ nk =0 n q nk, ρ k >0 q nk, ρ k =0 (4) (5) Equation (4) states that if the consue at deand aket k puchases the poducts fo etaile n, then the pice chaged by the etaile fo the poduct plus the unit tansaction cost undetaken by the consue does not exceed the pice that the consue is willing to pay. Equation (5) states that if the pice that the consue at deand aket k is willing to pay is positive, and then the quantities puchased of the poduct should exactly be equal to the deand (aguney et al., 2002). Let q ik denote the EOL poduct tansaction volue between collecto i and deand aket k, and goup all these q ik into a colun vecto Q 6 R I +. In the evese supply chain, the consues behavios can be chaacteized by Equation (6) subject to constaint (7). -57-

10 Jounal of Industial Engineeing and anageent Equation (6) eans that the custoes at deand aket k need to choose the quantities of EOL poducts etuned to collecto i coesponding to the value of the buy-back pice. Heein a k(q 6 ) epesents the consue disutility function at deand aket k fo handing ove the EOL poducts, which is a onotone inceasing function depending on the collection quantity. That is, the oe EOL poducts collecto i wants to obtain, the highe buy-back pice he has to offe. Constaint (7) eans that the aount of the EOL poducts deand aket k decides to etun should be less than o equal to the aount puchased in the fowad supply chain ultiply the etun atio (Haond & Beullens, 2007; Yang et al., 2009). a k (Q 6 ) { =ρ ik,if q ik >0 ρ ik,if q ik =0 (6) I s.t. i q ik μ n q nk, k (7) Cobining consue behavios in both fowad and evese supply chains, the equilibiu conditions fo all the deand akets can be expessed as the following vaiational inequality: deteine the optial (, Q 5,Q 6 ) C, satisfying: k + k [ n I i q nk d k (ρ ) x(ρ ρ k k)+ k n [a k (Q 6 ) ρ ik x[q ik q ik 0 [ρ nk + ^c nk (q nk ) ρ k x [q nk q nk (8) Whee Ω C ={ (ρ,q5,q 6 + ) R + +I I i q ik μ q n nk}. 2.5 The Equilibiu of the Collectos and thei Optiality Conditions The collectos tansact with the deand akets as well as the anufactues. They need to decide the collection quantity of the EOL poducts fo the deand akets and the tansaction volues with the anufactues. Let t i = k q ik, in tun, denote the total volue of the EOL poducts at collecto i that he obtains fo all the deand akets, and goup these aounts into the colun vecto t R I +. The collectos clean, inspect and classify the EOL poducts befoe tansacting with the anufactues. The coesponding pocessing cost of collecto i depends not only on his own but also on those of the othe collectos in ode to ebody copetition. Let c i(q i) denote the tansaction cost function undetaken by the collecto between collectos i and anufactue. L et b 3 epesent the popotion that the quantity of eanufactuable poducts accounts fo that of all the EOL poducts and satisfy b 3 (0,. oeove, the govenent povides the collectos with unit subsidy u fo each EOL poduct collected to -58-

11 Jounal of Industial Engineeing and anageent stiulate the developent of evese logistics. Theefoe, collecto i can obtain the aggegate subsidy υ k q ik. Fo collecto i, his axiu pofit odel is ax π i (Q 4, Q 6 )= ρ i q i +υ k q ik k ρ ik q ik c i (q i ) c i (t ) (9) s.t. q i β 3 k q ik, i (20) Equation (9) indicates that the pofit of the collecto is equal to sales evenues of EOL poducts plus the govenent subsidies inus the collection costs fo the deand akets, the tansaction costs with the anufactues and the pocessing costs. Constaint (20) eans that the tansaction volues between the collecto and all the anufactues should be no oe than the aount that he tansacts with all the deand akets ultiplies the atio of the eanufactuable EOL poducts. Because all the collectos copete in a non-coopeative fashion, the optiality conditions fo all collectos can be expessed as the vaiational inequality: deteine the optial (Q 4, Q 6 I ) R +I +, satisfying: I i [ c (q ) i i q i (Q 4,Q 6 ) Ω I ρ i x[q i q i + I i k [ c (q ) ik ik q ik + c i (t ) q i υ +ρ ik x[q ik q ik 0, (2) Whee Ω I ={ (Q4, Q 6 I ) R + +I q i β 3 q k ik}. 2.6 The Govening Equilibiu Condition of the CLSC etwok In equilibiu state, the optiality conditions fo all supplies, all anufactues, all etailes, all collectos and the deand aket equilibiu conditions ust be satisfied siultaneously. Hence, we can obtain the govening equilibiu condition of the CLSC netwok. Theoe The govening equilibiu condition of the CLSC netwok is given by the following vaiational inequalities: deteine (q J,q J2, q,q,q 2,Q 3, Q 4,Q 5,Q 6,q u, ρ ) Ω, satisfying: -59-

12 Jounal of Industial Engineeing and anageent J j J + f j (q J ) j J 2 + q j j 2 J 2 f x[q j q j + 2 j2 (q 2 J 2 x[q q 2 j2 q 2 j 2 2 j 2 [ c (q j j q j [ c (q j 2 2 j 2 q 2 j2 f + (q ) x[q q q + n I + i k n I + i k [ c (q n n) ) j 2 ) +λ (Q,Q 2,Q 3,Q 4 ) q j ) +λ (Q,Q 2,Q 3,Q 4 ) q 2 j2 x[q j x[q 2 j2 + c (s n ) +λ (Q,Q 2,Q 3,Q 4 ) q n q n [ c (q i i ) +λ (Q q i ) [ c nk (q nk q nk ) [ c (q ki ik + c (t i q ik f + (q u ) u q q n,q 2,Q 3,Q 4 ) q i +c nk (q nk ) ρ k x[q nk q nk ) q i x[q u q u + υ+α k (Q 6 )x[q ik q ik k (q J,q J2,Q,Q 2,q,Q 3, Q 4,Q 5,Q 6, q u,ρ ) Ω x[q i q i [ n q nk d n (ρ )x[ ρ k ρ k 0 q j q 2 j 2 x[q n q n (22) whee = s x x x C x I. Poof With soe algebaic anipulation, it can be seen that vaational inequality (22) and its constaint set is indeed the su of inequalities (5), (0), (3), (8) and (2) and thei constaint sets. Let ζ =(ζ,...,ζ j,...,ζ J ) T J R +, δ =(δ,...,δ j 2,...,δ J 2 ) T J R 2 +, ε=(ε,...,ε j,...,ε J ) T J R +, ψ =(ψ,...,ψ j2,...,ψ J2 ) T J R 2 +, ϕ =(ϕ,...,ϕ,...,ϕ ) T R +, χ =( χ,..., χ,..., χ ) T R +, ω =(ω,...,ω,...,ω ) T R +, γ =(γ,...,γ,...,γ ) T R +, η=(η,...,η n,...,η ) T R +, τ =(τ,...,τ k,...,τ ) T R +, θ =(θ,...,θ i,...,θ I ) T R I + be Laganian ultiplies colun vecto with espect to the constaints q j C j, q 2 j2 C 2 j 2, q q j j, q q j2 2 j 2 J, β q q, j j β 2 q q, 2 j 2 q J 2 j 2 I u i q i, n q n q +q u, k q nk q n, I i q ik μ q nk n a n d q i β 3 q ik espectively. k When the govening netwok odel is deteined, the endogenous pice vaiables can be etieved by eans of the ethod intoduced in aguney et al. (2002) and Yang et al. (2009), -520-

13 Jounal of Industial Engineeing and anageent that is: ρ j = c (q j j )/ q j +ε j ;ρ j 2 = c (q j2 2 j 2 )/ q 2 j 2 +ψ j 2 ; ρ n = c n (q n )/ q n + γ +λ (Q, Q 2, Q 3, Q 4 )/ q n ; ρ nk = c nk (q nk )/ q nk + η n ; ρ ik =α k (Q 6 )+τ k ;ρ i = c i (q i )/ q i +θ i. 3. etwok Equilibiu Algoith Fo easy foulation in the following, we goup the tes of the ultiplication signs in inequality (22) into a colun vecto F (y)={f j,f j2,f j, F j2,f 5,F 6 n,f 7 i,f 8 nk,f 9 ik, F 0,F k } j and intoduce, j 2,, n,k,i y= { q J, q J2,Q, Q 2,q,Q 3,Q 4, Q 5,Q 6,q u,ρ } Ω ; thus we can ewite the vaiational inequality poble (22) in the standad fo: deteine y = { q J,q J 2,Q,Q 2, q,q 3,Q 4,Q 2, Q 6,q u, ρ } Ω, satisfying: F (y) T (y y ) 0, y Ω (23) The feasible doain of vaiational inequality (23) is defined on a polyhedal set of nonnegative othant. As a consequence, the poduction capacity constaints induce ineffectiveness of the odified pojection ethod poposed by opelevich (976). Fotunately, the logaithic-quadatic poxial pediction-coection (LQP-PC) ethod developed by He, Xu and Yuan (2006) can be applied fo solving this vaiational inequality poble. LQP-PC ethod povides an iteative solution faewok that can not only get the optial solutions and Lagangian ultiplies siultaneously, but also has faste convegence and highe accuacy copaed with the odified pojection ethod. Theefoe, in this pape we eploy LQP-PC ethod to solve vaiational inequality (23). To ipleent LQP-PC ethod effectively, it is essential to tansfo the constaint set into the fo of patitioned atix fistly, and then cay out the pediction and coection pocedues while adjusting the step size adaptively. We efe the eades to the wok such as eng, Huang and Cheu (2009) to each a copehensive undestanding. 4. ueical Exaples ow conside a CLSC netwok which consists of two copetitive st-type supplies, two copetitive 2nd-type supplies, two copetitive anufactues, two copetitive etailes, two copetitive collectos and two deand akets. The aw ateial cost functions of fou supplies ae f (q J and f 22 (q J2 )=2q q 22 )=2.5 q 2 +q q 2 +q, f 2 (q J )=2.5 q 2 2 +q q 2 +q 2, f 2 (q J 2 )=2q 2 2 +q 2 q 22 +q 2, q 2 +q 22, espectively. The tansaction cost functions undetaken by the -52-

14 Jounal of Industial Engineeing and anageent supplies between supplie j, j 2 and anufactue ae c (q j j )=(q j ) 2 +0., j, and c (q j 2 2 j 2 )=(q 2 j2 ) 2 +0., j 2,, espectively. The poduction cost function of anufactue is f (q)=4q 2 +q q 3 +2q,,2. The eanufactuing cost function of anufactue is f (q u )=(q u ) 2 +q u u q 3 +q u,,2. The tansactions cost function undetaken by the anufactue between anufactue and etaile n is c n (q n )=4q 2 n +2.5q n,,2; n,2. The tansactions cost function undetaken by the collecto between collecto i and anufactue is c i (q i )=0.5q 2 i +0.5q i, i,2 ;,2. The handling cost of etaile n is c n (s n )=8s n 2, n,2. The pocessing cost of collecto i is c i (t )=0.2( k q ki ) 2 +0., i,2. The unit subsidy fo each EOL poduct collected is u=2. 2 The popotion the quantity of eanufactuable poducts accounts fo that of all the EOL poducts is b 3=0.96. The tansactions cost function undetaken by the deand aket between etaile n and deand aket k is ^c nk (q nk )=q nk +5, n,2, k,2. The consue disutility function at deand aket k is a k (Q 6 )=0.5( i 2 2 k q ik )+5. The tansactions cost function undetaken by the etaile between etaile n and deand aket k is c nk (q nk )=4q 2 nk +5, n,2, k,2. The deand functions at deand akets ae d ()= and d 2()= Ipleenting the LQP-PC ethod fo the exaple can yield the equilibiu solutions of the CLSC odel with and without capacity constaints. The convegence citeion of the LQP-PC algoith is set that the absolute value of diffeence of decision vaiables and Lagange ultiplies between two steps is lowe than o equal to 0-8. Hee we list and illustate the esults of the followed fou cases. Exaple : Set the isk-avese degee of the anufactues l =0.3,,2. Investigate the ipact of etun ate on the tansaction pices, tansaction volues and the pofits of two types of supplies, anufactues, etailes and collectos unde diffeent poduct BOs, as shown in Table

15 Jounal of Industial Engineeing and anageent q j q 2 j 2 b = 2, b 2 = b =, b 2 = 2 b = 2, b 2 = 2 = 0.4 = 0.5 = 0.6 = 0.4 = 0.5 = 0.6 = 0.4 = 0.5 = q u q q n q nk q ik q i k π j π j p p n p i Table. The Ipact of Retun Rate on The CLSC etwok Equilibiu unde Diffeent Poduct BOs Accoding to Table, it can be obseved that no atte which kind of poduct BO is, as the incease of the etun ate, the quantities of aw ateials poduced by the two type supplies decease; on the contay, the tansaction volues of EOL poducts between anufactues and collectos, the quantity of EOL poducts used to eanufactue and the total tansaction volues of poducts (including new poducts and eanufactued poducts) between anufactues and etailes incease. oe concetely, as the etun ate inceases, the inceased output of eanufactued poducts geneated fo oe EOL poducts available fo euse is lage than the deceased output of new poducts due to the eduction of aw ateials povided by the supplies. The aveage poduction cost coes down as a esult of the incease of the EOL poducts so that the anufactues pefe to poduce oe poducts. Futhe, the etailes obtain oe poducts fo the anufactues and sell the at a elatively lowe pice, which is benefit fo the consues at the deand akets. All the channel ebes and the whole CLSC can get oe pofits except the supplies. Base on the above esults, we can daw the conclusions that the incease of the etun ate can educe the input of aw ateials, stiulate the tansactions of the CLSC netwok and boost the econoy, so it is accodance with the philosophy of the sustainable developent

16 Jounal of Industial Engineeing and anageent Exaple 2: This exaple is conducted fo Exaple as follows. We keep the data in Exaple but set the axial poduction capacity of the two-type supplies. On this basis, we investigate the ipact of etun ate on the equilibiu of the CLSC netwok unde diffeent poduct BOs, as shown in Table 2. q j q 2 j2 b = 2, b 2 =, C = 0.5, C 2 = 0.25 b =, b 2 = 2, C = 0.25, C 2 = 0.5 b = 2, b 2 = 2, C = 0.5, C 2 = 0.5 = 0.4 = 0.5 = 0.6 = 0.4 = 0.5 = 0.6 = 0.4 = 0.5 = q u q q n q nk q ik q i k π j π j p p n p i Table 2. The Ipact of Retun Rate on the CLSC etwok Equilibiu unde Diffeent Poduct BOs with Poduction Capacity Constaints of Supplies In Table 2, copaing the equilibiu solutions between the fist two syetic poduct BOs (b =2, b 2 and b, b 2=2), we can find that with the syetic capacity constaints, the poduction quantity of new and eanufactued poducts, the tansaction volues between adjacent echelons and the aket pices unde the two BOs ae identical when the sae etun ate is set. The pofits of the etailes and the collectos ae equal too. This is because the capacity constaints liit the supply of the supplies, which, in tun, akes the quantity of the new poducts ade by aw ateials pecisely be equal to oeove, the sae etun ate unde the two cases of BOs coesponds to the sae quantities of EOL poducts collected and eanufactued poducts. So thee is no diffeence of tansaction volues between the etailes and the anufactues, the collectos and the anufactues, which futhe leads to the sae picing and pofits of the etailes and the collectos in the two BOs. The eason that the pofits of the anufactues and the supplies ae diffeent in two BOs lies in that the cost functions of the two-type supplies ae distinct. Fo the last poduct BO (b =2, b 2=2), when the etun ate is elatively low (=0.4), the poduction capacity constaints liit the supply of aw ateials; instead when the etun ate is elatively high (=0.5 and =0.6), the quantities of aw ateials povided by the supplies ae equal to that without capacity constaints. It is because the anufactues have two diffeent channels to eet the aket deand in CLSC. If the EOL poducts available fo eanufactuing ae sufficient, the quantity of the aw ateials puchased by the anufactues need not be up to the axiu capacities of the supplies

17 Jounal of Industial Engineeing and anageent Exaple 3: This exaple is conducted fo Exaple as follows. We fix the etun ate = 0.5, keep the othe data in Exaple but assue the two anufactues ae isk-avese. They ae not only concened with pofit axiization but also isk iniization. Thei isk functions ae given by =( n q n 0.6) 2. This isk easue can be explained as follows: the two anufactues wish to incease the tansaction volues with the etailes in ode to sell oe poducts. On this basis, we investigate the ipact of isk-avese degee of the anufactue on the equilibiu of the CLSC netwok unde diffeent poduct BOs, as shown in Table 3. q j q 2 j 2 b = 2, b 2 =, = 0.5 b =, b 2 = 2, = 0.5 b = 2, b 2 = 2, = 0.5 l = 0.2 l = 0.3 l = 0.4 l = 0.2 l = 0.3 l = 0.4 l = 0.2 l = 0.3 l = q u q q n q nk q ik q i k π j π j p p n p i Table 3. The Ipact of Risk-avese Degee of the anufactues on the CLSC etwok Equilibiu unde Diffeent Poduct BOs Accoding to Table 3, it can be obseved that as the isk-avese degee l inceases, the poduction quantity of aw ateials, new poducts and eanufactued poducts, the collection quantity of EOL poducts and the tansaction volues between adjacent echelons incease too. All the channel ebes and the whole CLSC can get oe pofits except the anufactues. We can explain the esults as follows: accoding to the equilibiu solutions in the absence of isk avesion in Table, it can be found that the optial tansaction volue of the poducts between the anufactue and the etaile is lowe than 0.3 unde each poduct BO when =0.5 is set, so the tansaction volue between one anufactue and the two etailes is less than 0.6. Theefoe, based on the given isk functions, the anufactues need to puchase oe aw ateials fo the supplies, buy-back oe EOL poducts fo the collectos and ake oe poducts to hedge the isk. The behavios of the anufactues ake the tansaction volues of the CLSC netwok incease and ipove the pofits of the othe ebes. Howeve, fo the anufactue hiself, thee is a conflict between the two objectives of pofit axiization and isk iniization. In othe wods, if the anufactues can give up a pat of inteests, then the pefoance of CLSC will be bette

18 Jounal of Industial Engineeing and anageent Exaple 4: This exaple takes the two factos of isk avesion of the anufactues and capacity constaints of supplies into account. On this basis, we investigate the ipact of iskavese degee on the equilibiu of the CLSC netwok unde diffeent poduct BOs, as shown in Table 4. q j q 2 j2 b = 2, b 2 =, = 0.5, C = 0.5, b =, b 2 = 2, = 0.5, C = 0.25, b = 2, b 2 = 2, = 0.4, C = 0.5, C 2 = 0.25 C 2 = 0.5 C 2 = 0.5 l = 0.2 l = 0.3 l = 0.4 l = 0.2 l = 0.3 l = 0.4 l = 0.2 l = 0.3 l = q u q q n q nk q ik q i k π j π j p p n p i Table 4. The Ipact of Risk-avese Degee of the anufactues on the CLSC etwok Equilibiu unde Diffeent Poduct BOs with Poduction Capacity Constaints of Supplies Accoding to Table 4, it can be obseved that unde the fist two kinds of poduct BOs (b =2, b 2= a n d b =, b 2=2), the equilibiu poduction and eanufactuing quantities, the equilibiu tansaction volues and the pofits of vaious channel ebes don t vay with the change of the etun ate due to the effect of the capacity constaints of the supplies. Unde the last poduct BO (b = 2, b 2=2), when l =0.2, the optial quantities of aw ateials ade by the supplies don t each the uppe bound of thei capacities; wheeas as l continues to incease to 0.3 and 0.4, the capacity constaints will take effect. 5. Conclusions We study a CLSC netwok equilibiu poble consideing two-type supplies, the isk-avese chaacte of the anufactues and the capacity constaints of the supplies copehensively. Based on finite-diensional vaiational inequality theoy, we analyze the optial behavios of vaious channel ebes and establish the govening CLSC netwok equilibiu foulation. The LQP-PC algoith is utilized to solve the vaiational inequality poble. ueical exaples ae given to analyze the ipact of etun ate and isk-avese degee upon the netwok equilibiu unde diffeent BOs. The esults show that with the incease of etun ate, the pofits of vaious channel ebes and the pefoance of the CLSC syste will ipove. With the incease of isk-avese degee, the anufactues will bea benefit losses. In othe wods, thee is a contadiction between pofit axiization and isk iniization fo the -526-

19 Jounal of Industial Engineeing and anageent anufactues. oeove, the econoic behavio of the CLSC is likely to be liited by the capacity constaints of the supplies. In the futue, we would like to extend ou odel to study the ulti-peiod CLSC netwok equilibiu poble involving ulti-type supplies and isk avesion. Acknowledgent The wok is suppoted by the ational atual Science Foundation, China (720242). Refeences Atasu, A., Guide, V.D.R., & Wassenhove, L.. (2008). Poduct Reuse Econoics in Closed-Loop Supply Chain Reseach. Poduction and Opeations anageent, 7(5), Chen, J.., & Chang, C.I. (202). The co-opetitive stategy of a closed-loop supply chain with eanufactuing. Tanspotation Reseach Pat E: Logistics and Tanspotation Review, 48(2), Debo, L.G., Toktay, L.B., & Van Wassenhove, L.. (2005). aket segentation and poduct technology selection fo eanufactuable poducts. anageent Science, 5(8), Feguson,.E., & Toktay, L.B. (2006). The effect of copetition on ecovey stategies. Poduction and opeations anageent, 5(3), Golinska, P., & awa, A. (20). Reanufactuing in autootive industy: Challenges and liitations. Jounal of Industial Engineeing and anageent, 4(3), Guide, V.D.R., & Wassenhove, L.. (2006). Closed Loop Supply Chains: An Intoduction to the Featue Issue (Pat ). Poduction and Opeations anageent, 5(3), Haond, D., & Beullens, P. (2007). Closed-loop supply chain netwok equilibiu unde legislation. Euopean Jounal of Opeational Reseach, 83(2), He, B.S., Xu, Y., & Yuan, X.. (2006). A logaithic-quadatic poxial pediction-coection ethod fo stuctued onotone vaiational inequalities. Coputational Optiization and Applications, 35(), eeney, R.L., & Raiffa, H. (993). Decisions with ultiple objectives: pefeences and value tade-offs. London: Cabidge univesity pess

20 Jounal of Industial Engineeing and anageent opelevich, G.. (976). The extagadient ethod fo finding saddle points and othe pobles. atecon, 2, ajude, P., & Goenevelt, H. (200). Copetition in eanufactuing. Poduction and Opeations anageent, 0(2), eng, Q., Huang, Y., & Cheu, R.L. (2009). Copetitive facility location on decentalized supply chains. Euopean Jounal of Opeational Reseach, 96(2), ita, S., & Webste, S. (2008). Copetition in eanufactuing and the effects of govenent subsidies. Intenational Jounal of Poduction Econoics, (2), aguney, A., Cuz, J., Dong, J., & Zhang, D. (2005). Supply chain netwoks, electonic coece, and supply side and deand side isk. Euopean Jounal of Opeational Reseach, 64(), aguney, A., Dong, J., & Zhang, D. (2002). A supply chain netwok equilibiu odel. Tanspotation Reseach Pat E: Logistics and Tanspotation Review, 38(5), aguney, A., & Toyasaki, F. (2005). Revese supply chain anageent and electonic waste ecycling: a ultitieed netwok equilibiu faewok fo e-cycling. Tanspotation Reseach Pat E: Logistics and Tanspotation Review, 4(), Ösdei, A., eahlıoğlu Ziya, E., & Palaktük, A.. (204). Copetitive quality choice and eanufactuing. Poduction and Opeations anageent, 23(), Qiang, Q., e,., Andeson, T., & Dong, J. (203). The closed-loop supply chain netwok with copetition, distibution channel investent, and uncetainties. Oega, 4(2), Savaskan, R.C., Bhattachaya, S., & Van Wassenhove, L.. (2004). Closed-loop supply chain odels with poduct eanufactuing. anageent science, 50(2), Savaskan, R.C., & Van Wassenhove, L.. (2006). Revese channel design: the case of copeting etailes. anageent Science, 52(), Shi, J., Zhang, G., & Sha, J. (20). Optial poduction planning fo a ulti-poduct closed loop syste with uncetain deand and etun. Coputes & Opeations Reseach, 38(3), Wakolbinge, T., Toyasaki, F., owak, T., & aguney, A. (204). When and fo who would e- waste be a teasue tove? Insights fo a netwok equilibiu odel of e-waste flows. Intenational Jounal of Poduction Econoics, 54,

21 Jounal of Industial Engineeing and anageent Webste, S., & ita, S. (2007). Copetitive stategy in eanufactuing and the ipact of take-back laws. Jounal of Opeations anageent, 25(6), Yang, G.F., Wang, Z.P., & Li, X.Q. (2009). The optiization of the closed-loop supply chain netwok. Tanspotation Reseach Pat E: Logistics and Tanspotation Review, 45(), Zhu, X., & Xu, X. (202). An integated optiization odel of a Closed-Loop Supply Chain unde uncetainty. Jounal of Syste and anageent Sciences, 2( 3), Zsidisin, G.A. (2003). anageial peceptions of supply isk. Jounal of supply chain anageent, 39(4), Jounal of Industial Engineeing and anageent, 205 ( Aticle's contents ae povided on a Attibution-on Coecial 3.0 Ceative coons license. Reades ae allowed to copy, distibute and counicate aticle's contents, povided the autho's and Jounal of Industial Engineeing and anageent's naes ae included. It ust not be used fo coecial puposes. To see the coplete license contents, please visit