Research Article Competition and Integration in Closed-Loop Supply Chain Network with Variational Inequality

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1 Hndaw Publshng Cororaton Mathematcal Problems n Engneerng Volume 2012, Artcle ID , 21 ages do: /2012/ Research Artcle Cometton and Integraton n Closed-Loo Suly Chan Network wth Varatonal Inequalty Gengu Zhou, 1 Yuxang Yang, 2 and Jan Cao 1 1 School of Busness Admnstraton, Zheang Unversty of Technology, Hangzhou , Chna 2 College of Economcs and Management, Chna Jlang Unversty, Hangzhou , Chna Corresondence should be addressed to Gengu Zhou, ggzhou@zut.edu.cn Receved 11 Arl 2012; Acceted 25 May 2012 Academc Edtor: Yung Wang Coyrght q 2012 Gengu Zhou et al. Ths s an oen access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch ermts unrestrcted use, dstrbuton, and reroducton n any medum, rovded the orgnal work s roerly cted. A closed-loo suly chan network nvolves the manufactured and remanufactured homogeneous roducts. It comrses oeraton lnks to reresent busness actvtes ncludng manufacturng/remanufacturng actvtes, treatment actvtes for EOL roducts, transortaton actvtes, and storage actvtes, whch are erformed by the frms. Among all closed-loo suly chan roblems, the horzontal merger of olgoolstc frms s so mortant and attractng to both busnessman and researchers. In ths aer, the nteracton of the comettve frms ror to horzontal merger s analyzed. Three networks ncludng ror to horzontal merger, ostartal merger, and comlete merger are studed. Smultaneously, three economcal models for these networks on dfferent condtons of mergers are establshed and dscussed. The varatonal nequalty formulatons are used for these three models, whose solutons gve out the roducton quantty of new roducts, and remanufactured roducts, the roduct flows for new roducts, remanufactured roducts and end-of-lfe roducts at every ath, the demand quantty, the recovery quantty of end-of-lfe roducts and the equlbrum rces. Fnally, numercal examles are tested and llustrated for the roosed models. 1. Introducton In lght of ncreasng envronmental conscousness and strcter legslaton, dsosal of lentful EOL end-of-lfe roducts becomes a crtcal roblem. EOL roduct recovery ams to mnmze the amount of waste sent to landflls by recoverng materals and arts from old or outdated roducts by means of recyclng and remanufacturng 1. Nowadays, EOL roduct recovery has become an oblgaton to the envronment and to the socety tself. Hence, more and more countres extend manufacturers resonsblty and requre them to take back the EOL roducts. Ths concern for the envronment has motvated ncreased

2 2 Mathematcal Problems n Engneerng nterest n reverse logstcs arose from EOL roduct recovery, whch has become the subect of growng attenton over the last decade 2. In an effort to curb olluton and acheve the goal of sustanable develoment, some measures have been roosed n Chna to romote the traded-n of the automobles and home alances coverng fve tyes of commodtes: TV, refrgerator, washng machne, ar condtoner, and comuter. The tral onts of traded-n comrse Beng, Shangha, Tann, Jangsu, Zheang, Shandong, Guangdong, Fuzhou, and Changsha. The consumers who urchase new home alances and meet the requrements of subsdy can be rovded a subsdy worth 10 ercent of the rces of the new roducts. The hghest subsdes wll be determned by ther tyes. The transortaton cost of the roducts to the recovery centers for treatment can be subsdzed wthn a certan range. It shows the stronger background and mortant meanng to deal wth the EOL roducts n Chna. Manufacturers have faced ncreasng ressures from both governmental regulatons and from consumers and become more envronmentally resonsble 3. Under ths background, manufacturers not only focus on ther forward logstcs, but also have to consder reverse logstcs. Ths may lead to oen systems f the recovered content of the orgnal roducts leaves the orgnal suly chans and s used by other manufacturers to manufacture roducts servng a dfferent urose, but t may also encourage manufacturers to have reverse flows mlemented nto ther own suly chans, gvng rse to a closed-loo suly chan 4. The study of closed-loo suly chan s a relatvely new feld of research. More recently, a varety of closed-loo suly chan network otmzaton models have been develoed. Schultmann et al. 5 roosed a hybrd aroach to establsh a closed-loo suly chan for sent batteres. It combnes an otmzaton model for lannng a reversesuly network and a flow-sheetng rocess model that enables a smulaton talored to otental recyclng otons for sent batteres n the steelmakng ndustry. Easwaran and Üster 6 resented a mxed-nteger lnear rogrammng to determne the otmal locatons of the collecton centers and remanufacturng facltes along wth the ntegrated forward and reverse flows and devsed two Tabu Search heurstcs sequental and random neghborhood search rocedures to solve the roblem. Wang and Hsu 7 consdered the ntegraton of forward and reverse logstcs. A generalzed closed-loo model for the logstcs lannng was roosed by formulatng a cyclc logstcs network roblem nto an nteger lnear rogrammng model, and a revsed sannng-tree-based genetc algorthm was develoed. Moreover, concernng the uncertan factors, a stochastc rogrammng model was develoed by Chounard et al. amng at evaluatng macts of randomness related to recovery, rocessng, and demand volumes on the desgn decsons 8. Francas and Mnner resented a two-stage stochastc rogrammng model wth normally dstrbuted demands and returns 9. In addton, a multerod nteger rogrammng model 10 and a multerod and multroduct network model 11 were roosed n recent years. Among all closed-loo suly chan roblems, the horzontal merger of olgoolstc frms wth closed-loo suly chan networks s so mortant and attractng to both busnessman and researchers. Mergers and acqustons have become an nsearable art of nternatonal busness and a maor means for comanes to grow n sze, go nternatonal, or obtan knowhow 12. Comanes seek roft mrovement by the rad exanson of sales va mergers. In addton, a merger motve can be to buld a conglomerate 13. Every comany ursues contnuously roft growth and exects to take away more market share from ts comettors. Thus, the ursut of those merger advantages mght exlan recent horzontal mergers such as those of Kmart and Sears n the retal ndustry, the merger of

3 Mathematcal Problems n Engneerng 3 Interbrew and Ambev and that of SAB-Mller and Molson Coors as well as the merger of InBev and Anheuser-Busch n the beer ndustry, the merger of Gome and Yongle n retal ndustry n Chna. Many researchers have focused ther attenton on the study of horzontal merger. Chakravarty 14 used the Debreu-Farrell-Färe measure of effcency of merger to comare economc effcences of alternatve merged enttes n a homogeneous good ndustry. Nlssen and Sørgard 15 analyzed the nterdeendence of merger decsons over tme. The Fudenberg-Trole taxonomy of busness strateges was used to dscuss merger decsons made n sequence by dsont grous of frms and a smle olgooly model roosed to these strateges. Ivald and Verboven 16 used the arameter estmates to comute the ostmerger equlbrum and the mled rce and welfare changes under varous alternatve scenaros. Davdson and Mukheree 17 analyzed horzontal mergers n a smle model n whch free entry and ext and rovded a general analyss of the mact of mergers on the long-run ndustry structure. In contrast to the aforementoned models, Nagurney 18 resented a new theoretcal framework for the quantfcaton of strategc advantages assocated wth horzontal mergers through the ntegraton of suly chan networks. In ths aer, wthn the context of closed-loo suly chan network, the frms n the network nclude manufacturng/remanufacturng lants and dstrbuton/recovery centers and serve the demand markets. Manufacturers/remanufacturers are nvolved n the roducton of a knd of homogeneous roducts from the raw materals and recovered EOL roducts. We consder that each frm as a network of busness actvtes reresented by forward lnks, and reverse lnks. The forward lnks consst of manufacturng lnks, remanufacturng lnks, forward transortaton lnks and forward storage lnks, whle reverse lnks nclude treatment lnks for EOL roducts, reverse transortaton lnks, and reverse storage lnks. Three networks ncludng ror to horzontal merger, ost artal merger and comlete merger are studed. Smultaneously, we roose three economcal models for these networks on dfferent condtons of mergers. Theory of varatonal nequaltes s used to derve the varatonal nequalty formulatons of those models. Fnally, several examles are studed to descrbe the effects of arameters on equlbrum results and analyze what mact the costs of horzontal merger wll have on rofts of the frms ater artal merger and comlete merger. Ths aer s organzed as follows. In Secton 2, we analyze the nteracton of the comettve frms ror to horzontal merger and roose an equlbrum model for the network. In Secton 3, the artal frms are merged nto one. We develo an economcal model for the network after artal merger. In Secton 4, a nonlnear otmzaton model for the closed-loo suly chan network after all frms merged s gven. In Secton 5, we rovde numercal examles to resent the effects of arameters on the equlbrum results n the case of dfferent merger scenaros and analyze what mact the costs of horzontal merger wll have on rofts of the frms after artal merger and comlete merger. At last, the aer ends wth conclusons n Secton Prehorzontal Merger Economcal Model We resent a closed-loo suly chan network ror to horzontal merger assocated wth a number of frms n the same ndustry, denoted by Frm 1, 2,...,I. Each Frm has m manufacturng/remanufacturng lants, n dstrbuton/recovery centers, and serves J demand markets. The lants roduce the same homogeneous manufactured and remanufactured roducts. The dstrbuton centers are resonsble for transortng the manufactured and

4 4 Mathematcal Problems n Engneerng remanufactured roducts from manufacturng/remanufacturng lants, and storng these roducts, then transortng these roducts to the demand markets. The recovery centers are resonsble for recoverng and storng the EOL roducts and transortng these roducts to remanufacturng lants to be treated and to be ready for remanufacturng. Let G 1 N 1,L 1 denote the network, where N 1 denotes the set of nodes and L 1 denotes the set of oeraton lnks ncludng forward and reverse lnks. The forward lnks are the lnks whch are located on the forward suly chans, and the reverse lnks are the lnks whch are located on the reverse suly chans. We assume that each frm s reresented as a network of many oeraton lnks whch reresent busness actvtes ncludng manufacturng/remanufacturng actvtes, treatment actvtes for EOL roducts, transortaton actvtes, and storage actvtes erformed by the frms, as dected n Fgure 1. Product flows on the forward lnks are classfed nto two tyes: the roducts remanufactured from recovered EOL roducts and the roducts manufactured from new materals. The roduct flows on the reverse lnks are the recovered EOL roducts. The forward lnks between the frst layer nodes wth the second layer nodes are the manufacturng/remanufacturng lnks reresentng manufacturng and remanufacturng oeratons, and the reverse lnks between them are the treatment lnks reresentng treatment oeratons for the EOL roducts. The lnks between second layer nodes wth thrd layer nodes are transortaton lnks reresentng transortaton oeratons between the manufacturng lants wth dstrbuton centers. The next layer lnks are storage lnks reresentng storage oeratons for the manufactured/remanufactured roducts and EOL roducts. Moreover, there are transortaton lnks connectng dstrbuton center nodes wth all demand market nodes. The maorty of the requred notaton s gven n Table 1. Suose that q N denotes the demand for the roducts at demand market. The quantty of roducts transacted for Frm s equal to that of manufactured roducts and remanufactured roducts. Therefore, we have the followng conservaton of roduct flows: q N P 1N x 1 y Let q R denote the quantty of EOL roducts recovered from the demand market. Then, we have the followng conservaton of roduct flow: q R z 1. P 1R 2.2 Let μ denote the fracton of the roduct that can be remanufactured from one unt of recovered EOL roduct. Therefore, we have: x 1 μ z 1. P 1N P 1R 2.3 Let ρ N denote the sale rce of roducts at demand market, andthesalerce functon s assocated wth the quantty of roducts at each demand market, and we assume that ρ N ρ N q N. In the reverse suly chans, demand markets act as a source of EOL

5 Mathematcal Problems n Engneerng 5 FIRM 1 FIRM I 1 I M 1 1 M 1 m1 M I 1 M I mi D 1 1 D 1 n1 D I 1 D I ni D 1 1 D 1 n1 D I 1 D I ni C 1 C J Fgure 1: The closed-loo suly chan network ror to horzontal merger. roducts. Increasng the recovery rce, more consumers are ersuaded to return roducts. Let ρ R denote the recovery rce of EOL roducts at demand market, and the rce functon s assocated wth the suly quantty of EOL roducts, that s, ρ R ρ R qr.letσ denote the rato of EOL roducts recovered to total quantty sold at the demand market; we have q R σq N. 2.4 For any forward lnk a L 1, we denote by h 1 a, the remanufactured roduct flow on lnk a. Letδ a f the lnk a artcates n forward ath, andδ a 0, otherwse. Hence, the followng conservaton equaton should be satsfed for the remanufactured roduct flow on forward aths and on oeraton lnks: h 1 a P 1N δ a x For any forward lnk b L 1, we denote by h 1 the manufactured roduct flow on lnk b b. Letδ b 1 f the lnk b artcates n forward ath, andδ b 0, otherwse. Hence, the followng conservaton equaton should be satsfed for the manufactured roduct flow on forward aths and on oeraton lnks: h 1 b δ b y P 1N

6 6 Mathematcal Problems n Engneerng Notaton L 1N L 1R P 1N P 1R P 1N P 1R P 1N P 1R P 1N M 1N P 1R M 1R x 1 x 1 y 1 y 1 z 1 z 1 Table 1: Notaton for the closed-loo suly chan network ror to horzontal merger. Defnton The set of all forward lnks for the Frm, where 1,...,I The set of all reverse lnks for the Frm,where 1,...,I Aathofthenetwork The set of all forward aths connectng the frst layer node wth demand market, where 1,...,I, 1,...,J The set of all reverse aths connectng the node wth demand market, where 1,...,I, 1,...,J The set of all forward aths between all I nodes wth demand market,where 1,...,J The set of all reverse aths between all I nodes wth demand market, where 1,...,J The set of all forward aths between the node wth all demand markets, where 1,...,I The set of all reverse aths between the node wth all demand markets, where 1,...,I The set of all forward aths The number of all forward aths The set of all reverse aths The number of all reverse aths The nonnegatve remanufactured roduct flow on the ath P 1N M 1N -dmensonal vector of all remanufactured roduct flows on the aths n P 1N The nonnegatve manufactured roduct flow on the ath P 1N M 1N -dmensonal vector of all manufactured roduct flows on the aths n P 1N The nonnegatve recovered EOL roduct flow on the ath P 1R M 1R -dmensonal vector of all EOL roduct flows on the aths n P 1R X 1 The set of all roduct flows, X 1 x 1,y 1,z 1 For any reverse lnk c L 1, we denote by h 1 c the roduct flow on lnk c. Letδ c 1f the lnk c artcates n forward ath,andδ c 0, otherwse. Smlarly, we have h 1 c P 1R δ c z We defne the cost on each lnk n erformng the corresondng task, and the cost functon s a generc cost functon of ts roduct flow, that s, e a e a h 1 a, e b e b h 1 b,and e c e c h 1 c. Therefore, the roft of Frm s TP ρ N a L 1N q N P 1N x 1 y 1 b L 1N ρ R e a h 1 a e b h 1 b e c h 1 c. c L 1R q R P 1R z We assume that the above roft functon s contnuously dfferentable and convex, all frms comete n a noncooeratve manner, and each one s a roft maxmzer. The otmalty

7 Mathematcal Problems n Engneerng 7 condtons for all frms smultaneously, under the above assumtons, concde wth the soluton of the followng varatonal nequalty 19 : determne X 1 x 1,y 1,z 1 K 1 satsfyng P 1N TP x 1 x 1 x 1 X 1 K 1 P 1N TP y 1 y 1 TP y 1 { x 1,y 1,z 1 P 1R z 1 z 1 z 1 0, x 1,y 1,z 1 R N 1 and hold }, 2.9 where N 1 2M 1N M 1R. Uon usng , varatonal nequalty 2.9 may be reexressed as 20 : determne X 1 x 1,y 1,z 1,λ 1 1,λ1 2 RN 1 I J, such that e a P δ ax 1 1N δ a ρ N x 1 y 1 P 1N a L 1N ρ N P 1N P 1N b L 1N h 1 a q N ρ N P 1R μ P 1R σ P 1N z 1 c L 1R x 1 ρ R x 1 y 1 P 1N x 1 P 1N e b P δ by 1 1N δ b ρ N P 1N h 1 b q N h 1 c x 1 y 1 y 1 λ 1 P 1N x 1 1 σλ1 2 P 1N e c P δ cz 1 1N δ c ρ R P 1N P 1R q R x 1 y 1 P 1R z 1 P 1R λ 1 1 λ1 1 z 1 z 1 μλ 1 λ 1 2 λ1 2 P 1R 1 λ1 2 0, X 1 x 1 y 1 z 1 y 1 x 1 x 1 σλ 1 2 z 1 z 1 y 1 y 1 x 1,y 1,z 1,λ 1 1,λ1 2 R N 1 I J, 2.10

8 8 Mathematcal Problems n Engneerng FIRM K FIRM S S M 1 1 M 1 m1 M K 1 M K mk M S 1 M S ms D 1 1 D 1 n1 D K 1 D K nk D S 1 D S ns D 1 1 D 1 n1 D K 1 D K nk D S 1 D S ns C 1 C J Fgure 2: The closed-loo suly chan network after horzontal merger of artal frms. where λ 1 1 λ1 11 λ1 12 λ 1 1I T and λ 1 2 λ1 21 λ1 22 λ 1 2J T are the vectors of Lagrange multlers assocated wth constrants 2.3 and Postartal Horzontal Merger Model In ths secton, we consder the artal horzontal merger case, that s, K frms n I frms are horzontal merged nto a new Frm 1 and K S 1 I, as dected n Fgure 2. The new frm shares manufacturng/remanufacturng lants and dstrbuton/recovery centers of orgnal K frms, that s, the manufactured/remanufactured roducts at any of these lants can be dstrbuted by any of these dstrbuton centers, and the recovered EOL roducts at any of these recovery centers can be transorted to any of these lants. We denote the new network by G 2 N 2,L 2, where N 2 denotes the set of nodes and L 2 denotes the set of oeraton lnks n ths network. The maorty of the requred notaton for the network ost horzontal merger of artal frms s gven n Table 2. For any forward lnk a L 2, we denote by h 2 a the remanufactured roduct flow on lnk a. For any forward lnk b L 2, we denote by h 2 the manufactured roduct flow on lnk b. For b any reverse lnk c L 2, we denote by h 2 c the EOL roduct flow on lnk c. The cost functons are e a e a h 2 a, e b e b h 2 b,ande c e c h 2 c. Snce we have assumed that each frm s a roft maxmzer, smlarly, the otmzaton roblem of Frm can be exressed as follows. Maxmze TP ρ N a L 2N q N P 2N b L 2N x 2 y 2 ρ R e a h 2 a e b h 2 b e c h 2 c. c L 2R q R P 2R z 2 3.1

9 Mathematcal Problems n Engneerng 9 Table 2: Notaton for the closed-loo suly chan network after horzontal merger of artal frms. Notaton L 2N L 2R P 2N P 2R P 2N P 2R P 2N P 2R P 2N M 2N P 2R M 2R x 2 x 2 y 2 y 2 z 2 z 2 Defnton The set of all forward lnks for the Frm, where 1, 2,...,S The set of all reverse lnks for the Frm,where 1, 2,...,S The set of all forward aths connectng the frst layer node 1, 2,...,S wth demand market, where 1, 2,...,J The set of all reverse aths connectng the demand market wth the frst layer node 1, 2,...,S The set of all forward aths between all S nodes wth demand market The set of all reverse aths between all S nodes wth demand market The set of all forward aths between the node 1, 2,...,S wth all demand markets The set of all reverse aths between the node 1, 2,...,S wth all demand markets The set of all forward aths The number of all forward aths The set of all reverse aths The number of all reverse aths The nonnegatve remanufactured roduct flow on the ath P 2N M 2N -dmensonal vector of all remanufactured roduct flows on the aths n P 2N The nonnegatve manufactured roduct flow on the ath P 2N M 2N -dmensonal vector of all manufactured roduct flows on the aths n P 2N The nonnegatve recovered EOL roduct flow on the ath P 2R M 2R -dmensonal vector of all EOL roduct flows on the aths n P 2R X 2 The set of all roduct flows, X 2 x 2,y 2,z 2 Subect to: q N q R P 2N P 2N x 2 y 2,, P 2R z 2, x 2 μ P 2R, z 2, q R σq N,, 3.5 h 2 a δ a x, 2 a L 2N, P 2N 3.6 h 2 b δ b y, 2 b L 2N, P 2N 3.7 h 2 c δ c z 2, c L 2R, P 2R 3.8 x 2 0, y 2 0, P 2N ; z 2 0, P 2R. 3.9

10 10 Mathematcal Problems n Engneerng We assume that the above roft functon s contnuously dfferentable and convex, S frms comete n a noncooeratve manner, and each one s a roft maxmzer. The otmalty condtons for all frms smultaneously, under the above assumtons, concde wth the soluton of the followng varatonal nequalty: determne X 2 R N 2 S J, such that P 2N a L 2N ρ N e a P δ ax 2 2N δ a ρ N P 2N h 2 a q N x 2 y 2 P 2N x 2 P 2N x 2 y 2 y 2 λ 2 x 2,y 2,z 2,λ 2 1,λ2 2 1 σλ2 2 x 2 x 2 P 2N b L 2N e b P δ by 2 2N δ b ρ N h 2 b P 2N x 2 y 2 ρ N P 2N q N x 2 y 2 P 2N x 2 y 2 σλ 2 2 y 2 y 2 P 2R c L 2R e c P δ cz 2 2N δ c ρ R h 2 c P 2R z 2 μ P 2R σ P 2N z 2 x 2 ρ R P 2N P 2R q R x 2 y 2 P 2R z 2 P 2R λ 2 1 λ2 1 z 2 z 2 μλ 2 λ 2 2 λ2 2 X 2 1 λ2 2 0, z 2 z 2 x 2,y 2,z 2,λ 2 1,λ2 2 R N 2 S J, 3.10 where N 2 2M 2N M 2R, λ 2 1 λ2 11 λ2 12 λ 2 1S T and λ 1 2 λ1 21 λ1 22 λ1 2J T, are the vectors of Lagrange multlers assocated wth constrants 3.4 and 3.5.

11 Mathematcal Problems n Engneerng 11 FIRM I M 1 1 M 1 m1 M I 1 M I mi D 1 1 D 1 n1 D I 1 D I ni D 1 1 D 1 n1 D I 1 D I ni C 1 C J Fgure 3: The ntegrated closed-loo suly chan network after comlete horzontal merger of all the frms. 4. Postcomlete Horzontal Merger Integraton Model In ths secton, we develo a ostcomlete merger model n the closed-loo suly chan network. I frms merge nto a new frm, and the new one shares manufacturng/ remanufacturng lants and dstrbuton/recovery centers of orgnal I frms. The closedloo suly chan network ror to horzontal merger s ntegrated, as dected n Fgure 3. There s the frst layer Node 1 whch reresents the ntegraton of all frms n terms of ther closed-loo suly chan network wth addtonal oeraton lnks connectng Node 1 to the nodes of orgnal frms. Moreover, we add the addtonal oeraton lnks connectng the manufacturng/remanufacturng lants of each frm wth the dstrbuton/recovery centers of the other frms. We denote the new network by G 3 N 3,L 3, where N 3 N 1 Node 1 and L 3 L 1 the addtonal oeraton lnks. The maorty of the requred notaton for the network after comlete horzontal merger s gven n Table 3. For any forward lnk a L 3, we denote by h 3 a the remanufactured roduct flow on lnk a. For any forward lnk b L 3, we denote by h 3 the manufactured roduct flow on lnk b. For b any reverse lnk c L 3, we denote by h 3 c theroductflowonlnkc. We defne that the cost functons are e a e a h 3 a, e b e b h 3 b,ande c e c h 3 c. Snce all frms are merged nto a frm, there s no longer cometton. Hence, the roft of the new frm can be exressed as follows.

12 12 Mathematcal Problems n Engneerng Table 3: Notaton for the closed-loo suly chan network after comlete horzontal merger. Notaton Defnton L 3N The set of all forward lnks L 3R The set of all reverse lnks P 3N The set of forward aths between the frst layer node 1 wth the demand market, where 1,...,J P 3R The set of reverse aths between the demand market wth the frst layer node 1 P 3N The set of all forward aths M 3N The number of all forward aths P 3R The set of all reverse aths M 3R The number of all revere aths x 3 The nonnegatve remanufactured roduct flow on the ath P 3N x 3 M 3N -dmensonal vector of all remanufactured roduct flows on the aths n P 3N y 3 The manufactured roduct flow on the ath P 3N y 3 M 3N -dmensonal vector of all manufactured roduct flows on the aths n P 3N z 3 The EOL roduct flow on the ath P 3R z 3 M 3R -dmensonal vector of all EOL roduct flows on the aths n P 3R X 3 The set of all roduct flows, X 3 x 3,y 3,z 3 Maxmze TP ρ N q N P 3N a L 3N e a h 3 a x 3 y 3 b L 3N e b ρ R h 3 b e c h 3 c. c L 3R q R P 3R z Subect to: q N q R P 3N x 3 y 3,, P 3R z 3,, x 3 μ z 3, P 3N P 3R q R σq N,, 4.5 h 3 a δ a x, 3 P 3N a L 3N, 4.6 h 3 b δ b y, 3 P 3N b L 3N, 4.7 h 3 c δ c z 3, P 3R c L 3R, 4.8 x 3 0, y 3 0, P 3N ; z 3 0, P 3R. 4.9

13 Mathematcal Problems n Engneerng 13 The obectve of the model shown n 4.1 s to maxmze the total roft of the frm after comlete merger. Constrant 4.2 reresents that the quantty of manufactured and remanufactured roducts shed to demand market s equal to the demand of the market. Constrant 4.3 reresents that the quantty of recovered EOL roducts from demand market s equal to the suly of them. Constrant 4.4 restrcts that the quantty of remanufactured roducts come from the EOL roducts. Constrant 4.5 reresents that the quantty of EOL roducts recovered must not exceed ther suly at the demand market. Constrant 4.6 reresents that conservaton equaton for the remanufactured roduct flow on forward aths and on oeraton lnks. Constrant 4.7 reresents that conservaton equaton for the manufactured roduct flow on forward aths and on oeraton lnks. Constrant 4.8 reresents conservaton equaton for the EOL roduct flow on reverse aths and on oeraton lnks. Constrant 4.9 reresents the nonnegatve restrcton on roduct flows. Smlarly, we assume that the above roft functon s contnuously dfferentable and convex wth resect to varable X 3. The otmalty condtons for the frm, under the above assumtons, concde wth the soluton of the followng varatonal nequalty: determne X 3 x 3,y 3,z 3,λ 3 1,λ3 2 RN 3 J 1 satsfyng P 3N a L 3N ρ N e a P 3N δ ax 3 h 3 a P 3N P 3N b L 3N P 3R μ c L 3R z 3 P 3R σ P 3N q N ρ N ρ R x 3 y 3 δ a ρ N P 3N e b P 3N δ by 3 h 3 b P 3N q N x 3 e c P δ cz 3 3R h 3 c P 3R q R x 3 P 3N x 3 z 3 y 3 P 3R P 3N x 3 δ b ρ N y 3 P 3R λ 3 1 λ3 1 z 3 δ c ρ R z 3 x 3 y 3 y 3 λ 3 P 3N P 3N x 3 μλ 3 λ 3 2 λ3 2 P 3R 1 σλ3 2 x 3 y 3 z 3 1 λ3 2 X 3 0, y 3 x 3 x 3 σλ 3 2 z 3 z 3 y 3 y 3 x 3,y 3,z 3,λ 3 1,λ3 2 R N 3 J 1, 4.10

14 14 Mathematcal Problems n Engneerng where N 3 2M 3N M 3R, λ 2 1 and λ2 2 λ2 21 λ2 22 λ2 2J T are the vectors of Lagrange multlers assocated wth constrants 4.4 and Dscusson and Numercal Analyss In ths secton, four sets of numercal examles are tested and dscussed on the roosed models. The models are comuted by usng modfed roecton method 21 and mlemented n MATLAB. The convergence crteron used s that the absolute value of the roduct flows between two successve teratons dffered by no more than The constant ste length wthn the modfed roecton method s set to All solutons obtaned satsfy the otmalty condtons wth good accuracy. Examle Set 1. We consder that a closed-loo suly chan network ror to horzontal merger conssts of three frms, each wth ts manufacturng/remanufacturng lants, dstrbuton/recovery centers, and a demand market. Then, Frm 1 and Frm 2 merge nto a new frm. Fnally, the three frms merge comletely. The networks ror to merger, ostartal merger and comlete merger, are dected n Fgure 4. The cost functon on the lnks for the closed-loo suly chan network ror to merger s as follow. The cost functons on the frst layer lnks are e a h 1 a 1.5 h 1 a 2 h 1 a, e b h 1 b 2 h1 b 2 1.5h 1 b,ande c h 1 c 0.2 h 1 c 2. The cost functons on the second layer lnks are e a h 1 a 0.5 h 1 a 2 h 1 a, e b h 1 b 0.5 h1 b 2 h 1 b,ande c h 1 c 0.3 h 1 c 2 0.5h 1 c. The cost functons on the thrd layer lnks are e a h 1 a 0.5 h 1 a 2, e b h 1 b 0.5 h1 b 2,ande c h 1 c 0.3 h 1 c 2.The cost functons on the fourth layer lnks are e a h 1 a 0.1 h 1 a 2, e b h 1 b 0.1 h1 b,ande c h 1 c 0.1 h 1 c 2. The sale rce functon s ρ N 1 qn qn 1. The recovery rce of EOL roducts at demand market s ρ R 1 0.5qR 1. When the frst two frms merge, we set the cost on the new lnks connectng the frst layer Node 1 wth the second layer Nodes 1 and 2, equal to zero, and the cost functons on the new lnks connectng manufacturng/remanufacturng lants wth dstrbuton/recovery centers are dentcal to the cost functons on the lnks of the network ror to merger. When all frms merge, we assume that the cost on the new lnks connectng the frst layer Node 1 wth the second layer Nodes 1, 2, and 3, equal to zero, the cost functons of other new lnks are dentcal to the orgnal functons. Frstly, we set μ 0.6andσ 0. There are three forward aths and three reverse aths n the network ror to merger. The equlbrum results ror to merger can be obtaned by varatonal nequalty 2.10, andx 1 0 for all P 1N, y for all P 1N, z 1 0 for all P 1R. The revenue for each frm s The cost for each frm s , and the roft for each frm s There are fve forward aths and fve reverse aths n the network after artal merger. The equlbrum results after artal merger can be obtaned by varatonal nequalty 3.10, andx 2 0 for all P 2N, z 2 0 for all P 2R. However, y 2 for all P 2N has changed, the manufactured roduct flow on each ath connectng Frm 2 wth the demand market s 15.64, and the roduct flows on the other aths are The revenue of the new Frm 1 s , and the revenue of the new Frm 2 s The cost of the new Frm 1 s , and the cost of the new Frm 2 s The roft of the new Frm 1 s , and the roft of the new Frm 2 s There are nne forward aths and nne reverse aths n the network after comlete merger. The equlbrum results ost comlete merger can be obtaned by varatonal nequalty 4.10, andx for all

15 Mathematcal Problems n Engneerng 15 Frm 1 Frm 2 Frm Frm Frm 1 1 Frm Fgure 4: The networks ror to merger, ostartal merger, and comlete merger. P 3N, y for all P 3N, z for all P 3R. The total revenue s The total cost s The total roft s Secondly, we set μ 0.6, σ 0.2. The equlbrum results for the network ror to merger are x for all P 1N, y for all P 1N, z for all P 1R. The revenue for each frm s The cost for each frm s The roft for each frm s The equlbrum results for the network ost artal merger are obtaned. The remanufactured roduct flow on each forward ath connectng the new Frm 2 wth the demand market s 2.80, and the roduct flows on the other aths are 0.81, resectvely. The manufactured roduct flow on each forward ath connectng the new Frm 2 wth the demand market s 15.42, and the roduct flows on the other ath are The EOL roduct flow on each reverse ath connectng the new Frm 2 wth the demand market s 4.67, and the roduct flows on the other aths are The revenue of the new Frm 1 s , and the revenue of the new Frm 2 s The cost of the new Frm 1 s , and the cost of the new Frm 2 s The roft of the new Frm 1 s , and the roft of the new Frm 2 s The equlbrum results for the network after comlete merger are x for all P 3N, y for all P 3N and z for all P 3R. The total revenue s The total cost s The total roft s Fnally, we set μ 0.6, σ 0.7. The equlbrum results for the network ror to merger are x for all P 1N, y for all P 1N, z for all P 1R. The revenue for each frm s The cost for each frm s The roft for each frm s The equlbrum results for the network ost artal merger are obtaned. The remanufactured roduct flow on each forward ath connectng the new Frm 2 wth the demand market s 5.61, and the roduct flows on the other aths are The manufactured roduct flow on each forward ath connectng the new Frm 2 wth the demand market s 13.74, and the roduct flows on the other aths are The EOL roduct flow on each

16 16 Mathematcal Problems n Engneerng reverse ath connectng the new Frm 2 wth the demand market s 9.34, and the roduct flows on the other aths are The revenue of the new Frm 1 s , and the revenue of the new Frm 2 s The cost of the new Frm 1 s , and the cost of the new Frm 2 s The roft of the new Frm 1 s , and the roft of the new Frm 2 s The equlbrum results for the network after comlete merger are x for all P 3N, y for all P 3N,andz for all P 3R. The total revenue s The total cost s The total roft s The other equlbrum results ncludng the demand quantty, the quantty of recovered EOL roducts, the rces, the total revenues, the total costs, and the total rofts for the networks ror to merger, ostartal merger, and comlete merger from three dfferent ntal arameter settng are resented n Table 4. It s mortant to note that, n each of arameter settng, all frms ganed from the artal merger. The merged Frm 1 has hgher total roft than the sum of those two orgnal frms rofts, and the unmerged Frm 2 orgnal Frm 3 obtans also more roft. It s also worth notng that the total revenues and the total costs are lower n these artal merger examles. Moreover, the new frm after comlete merger obtans more roft than the sum of those three orgnal frms rofts, also more than the sum of the two frms rofts n artal merger case. In addton, n every examle for each arameter settng, the demand for remanufactured and manufactured roducts, the quantty of recovered EOL roducts, and the recovery rce decrease slghtly; however, the demand rce at demand market ncreases. Examle Set 2. The second set of examles uses the same network and cost functons as n Examle Set 1 but assumes that dfferent μ and σ are faced. We assume that μ ncreases from 0.1 to 0.5 and ncreases to 0.8. The rato σ s set to 0.6. When μ 0.1, the revenue for each frm ror to merger s , the cost for each frm s , and the roft for each frm s The revenue of the new Frm 1 after artal merger s , and the revenue of the new Frm 2 s The cost of the new Frm 1 after artal merger s , and the cost of the new Frm 2 s The roft of the new Frm 1 after artal merger s , and the roft of the new Frm 2 s The revenue for the frm after comlete merger s , ts cost s , and ts roft s When μ 0.5, the revenue for each frm ror to merger s , the cost for each frm s , and the roft for each frm s The revenue of the new Frm 1 after artal merger s , and the revenue of the new Frm 2 s The cost of the new Frm 1 after artal merger s , and the cost of the new Frm 2 s The roft of the new Frm 1 after artal merger s , and the roft of the new Frm 2 s The revenue for the frm after comlete merger s , ts cost s , and ts roft s When μ 0.8, the revenue for each frm ror to merger s , the cost for each frm s , and the roft for each frm s The revenue of the new Frm 1 after artal merger s , and the revenue of the new Frm 2 s The cost of the new Frm 1 ost artal merger s , and the cost of the new Frm 2 s The roft of the new Frm 1 ostartal merger s , and the roft of the new Frm 2 s The revenue for the frm after comlete merger s , ts cost s , and ts roft s The other equlbrum results ncludng the demand quantty, the quantty of recovered EOL roducts, the equlbrum rces, the total revenues, the total costs, and the total rofts for the networks ror to merger, ostartal merger, and comlete merger from three dfferent ntal arameter settngs are resented n Table 5.

17 Mathematcal Problems n Engneerng 17 Table 4: Results of Examle Set 1. μ 0.6 σ 0 μ 0.6 σ 0.2 μ 0.6 σ 0.7 Results Pror merger Partal merger Comlete merger Pror merger Partal merger Comlete merger Pror merger Partal merger Comlete merger q N q R ρ N ρ R Total revenue Total cost Total roft

18 18 Mathematcal Problems n Engneerng Table 5: Results of Examle Set 2. μ 0.1 σ 0.6 μ 0.5 σ 0.6 μ 0.8 σ 0.6 Results Pror merger Partal merger Comlete merger Pror merger Partal merger Comlete merger Pror merger Partal merger Comlete merger q N q R ρ N ρ R Total revenue Total cost Total roft

19 Mathematcal Problems n Engneerng 19 Due to the ncreased μ, all frms choose to recover more roducts. As a result, market share ncreases, and the total rofts for the networks ror to merger, ostartal merger, and ostcomlete merger are ncreased. Examle Set 3. The examle uses the same network and cost functons as n Examle Set 1, μ 0.5, σ 0.6, but assumes that the cost functons of the new lnks connectng the frst layer Node 1 wth the second layer Nodes 1 and 2 are e a h 2 a gh 2 a, e b h 2 b gh2 b, e c h 2 c gh 2 c, whch can be regarded as the cost of horzontal merger ncludng drect costs and ndrect costs assocated wth merger. In order to analyze what mact the costs of horzontal merger wll have on rofts of the new Frm 1 and Frm 2 after artal merger, t s feasble to vary the arameter g from 0 to 10, then to 20, and so on, whch means dfferent from the cost of horzontal merger. The results from dfferent arameter settngs are resented n Fgure 5. We fnd that the arameter s set to 140 then the roft of the new Frm 1 after artal merger s zero and t makes no economc sense for such a merger, but the roft of the new Frm 2 reach the maxmum Wth further exermentaton, note that the arameter s equal to 2, then the roft of the new Frm 1 after artal merger s that s greater than the sum of the rofts for the Frm 1 and Frm 2 ror to merger cf. Examle Set 2. Thus,the arameter s equal to or smaller than 2 then, the merged frms can obtan gan from such a merger. However, after the arameter s set to 3, the roft of the new Frm 1 after artal merger s , whch s smaller than the sum of the rofts for the Frm 1 and Frm 2 ror to merger cf. Examle Set 2. Therefore, the arameter s equal to or greater than 3, then the merged frms cannot obtan roft growth from such a merger. Examle Set 4. The examle uses the same network and cost functons as n Examle Set 1, μ 0.8, σ 0.6, but assumes that the cost functons of the new lnks connectng the frst layer Node 0 wth the second lager Nodes 1, 2, and 3 are e a h 2 a gh 2 a, e b h 2 b gh2 b, e c h 2 c gh 2 c. Further, we analyze what mact the costs of horzontal merger wll have on rofts of the new frm after comlete merger. Smlarly, we vary the arameter g from 0 to 10, then to 20, and so on. The results from dfferent arameter settngs are resented n Fgure 6. We fnd that the arameter s set to 160 then the roft of the new frm after comlete merger s zero and the merger has no economc sense. Further, after the arameter s set to 6, the roft of the new frm after comlete merger s , whch s stll greater than the sum of the rofts for all frms ror to merger cf. Examle Set 2. Thus, the arameter s equal to or smaller than 6, then, the merged frms can obtan gan from such a comlete merger. However, after the arameter s set to 7, the roft of the new frm s , whch s smaller than the sum of the rofts for all frms ror to merger cf. Examle Set 2. Therefore, the arameter s equal to or greater than 7, then the merged frms cannot obtan roft growth from ths comlete merger. 6. Concluson Ths aer has studed the closed-loo suly chan network ncludng a number of frms n the same ndustry. The network s reresented by forward and reverse oeraton lnks ncludng manufacturng/remanufacturng lnks, treatment lnks for EOL roducts, transortaton lnks, and storage lnks. The economcal models for the networks ror to merger, ostartal merger, and comlete merger are develoed. The research work has resented the varatonal nequalty formulatons of these models, whose solutons yeld the roducton quantty of new roducts and remanufactured roducts, the roduct flows

20 20 Mathematcal Problems n Engneerng Proft g TP1 TP2 Fgure 5: The total rofts of the Frm 1 and Frm 2 after artal merger from dfferent arameter settngs Proft g TP Fgure 6: The roft of the Frm 1 after comlete merger from dfferent arameter settngs. for new roducts, remanufactured roducts and EOL roducts at every ath, the demand quantty, the quantty of recovered EOL roducts, and the equlbrum rces. The modfed roecton method s aled for all numercal examles tested. In addton, ths aer can be extended n several drectons, such as more merger forms and the models for multroduct closed-loo suly chan network. Acknowledgments Ths aer s suorted by Natonal Nature Scence Foundaton of Chna no , no Ths aer s sonsored by Zheang Provncal Key Research Base for Humantes and Socal Scences Standardzaton and Intellectual Proerty Management and Zheang Provncal Key Research Base for Phlosohy and Socal Scences Industral Develoment Polcy Research Center.

21 Mathematcal Problems n Engneerng 21 References 1 A. Gungor and S. M. Guta, Issues n envronmentally conscous manufacturng and roduct recovery: a survey, Comuters & Industral Engneerng, vol. 36, no. 4, , M. Fleschmann, J. M. Bloemhof-Ruwaard, R. Dekker, E. van der Laan, J. A. E. E. van Nunen, and L. N. van Wassenhove, Quanttatve models for reverse logstcs: a revew, Euroean Journal of Oeratonal Research, vol. 103, no. 1,. 1 17, V. Jayaraman, Producton lannng for closed-loo suly chans wth roduct recovery and reuse: an analytcal aroach, Internatonal Journal of Producton Research, vol. 44, no. 5, , D. Hammond and P. Beullens, Closed-loo suly chan network equlbrum under legslaton, Euroean Journal of Oeratonal Research, vol. 183, no. 2, , F. Schultmann, B. Engels, and O. Rentz, Closed-loo suly chans for sent batteres, Interfaces, vol. 33, no. 6, , G. Easwaran and H. Üster, Tabu search and Benders decomoston aroaches for a caactated closed-loo suly chan network desgn roblem, Transortaton Scence, vol. 43, no. 3, , H. F. Wang and H. W. Hsu, A closed-loo logstc model wth a sannng-tree based genetc algorthm, Comuters & Oeratons Research, vol. 37, no. 2, , M. Chounard, S. D Amours, and D. Aït-Kad, A stochastc rogrammng aroach for desgnng suly loos, Internatonal Journal of Producton Economcs, vol. 113, no. 2, , D. Francas and S. Mnner, Manufacturng network confguraton n suly chans wth roduct recovery, Omega, vol. 37, no. 4, , B. M. Beamon and C. Fernandes, Suly-chan network confguraton for roduct recovery, Producton Plannng & Control, vol. 15, no. 3, , M. I. G. Salema, A. P. Barbosa-Povoa, and A. Q. Novas, Smultaneous desgn and lannng of suly chans wth reverse flows: a generc modellng framework, Euroean Journal of Oeratonal Research, vol. 203, no. 2, , P. L. Anslnger and T. E. Coeland, Growth through acqustons: a fresh look, Harvard Busness Revew, vol. 74, no. 1, , J. L. Bower, Not all M&As are alke and that matters, Harvard Busness Revew, vol. 79, no. 3, , S. R. Chakravarty, Effcent horzontal mergers, Journal of Economc Theory, vol. 82, no. 1, , T. Nlssen and L. Sørgard, Sequental horzontal mergers, Euroean Economc Revew, vol. 42, no. 9, , M. Ivald and F. Verboven, Quantfyng the effects from horzontal mergers n Euroean cometton olcy, Internatonal Journal of Industral Organzaton, vol. 23, no. 9-10, , C. Davdson and A. Mukheree, Horzontal mergers wth free entry, Internatonal Journal of Industral Organzaton, vol. 25, no. 1, , A. Nagurney, A system-otmzaton ersectve for suly chan network ntegraton: the horzontal merger case, Transortaton Research Part E, vol. 45, no. 1,. 1 15, M. S. Bazaraa, H. D. Sheral, and C. M. Shetty, Nonlnear Programmng: Theory and Algorthms, John Wley & Sons, New York, NY, USA, A. Auslender and M. Teboulle, Lagrangan dualty and related multler methods for varatonal nequalty roblems, SIAM Journal on Otmzaton, vol. 10, no. 4, , G. M. Korelevch, The extragradent method for fndng saddle onts and other roblems, Matekon, vol. 13, no. 4, , 1977.

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