Optimizing the Fuzzy Closed-Loop Supply Chain for Electrical and Electronic Equipments

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1 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 9 Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens Phan guyen Ky Phuc, Vncen F. Yu, and Shu-Yan Chu Absrac The rapd develpmen f elecrcal and elecrnc equpmen (s) mare n recen years creaes bh psve and negave mpacs n he envrnmen and human. In rder acheve he susanable develpmen, he ms crcally negave mpac,.e., wase s (Ws) mus be prperly handled such as dspsed, remanufacured, recycled name a few. Amng suggesed sluns fr reslvng hs ssue, clsed-lp supply chan (CLSC) managemen has been prved as a hghly penal canddae due s effcency, effecveness and ecnmcs. Ths paper presens a mdel f mul s CLSC sysems wh fuzzy parameers. The prpsed mdel s ransfrmed n he equvalen auxlary crsp mdel by an apprprae apprach, and he fnal preferred cmprmse sluns are als fund. Fnally, a numercal example s cnduced llusrae he applcably and prvde beer nsghs abu he prpsed mdel. Keywrds: lecrcal and lecrnc equpmens, fuzzy, mul prduc, mul-echeln, supply chan managemen.. Inrducn Laes echnlgcal advances have sgnfcanly shrened elecrcal and elecrnc equpmens (s) lfecycles. Currenly, ld mdels are rapdly replaced by he new nes wh much mre advanced funcns and desgns. As a resul f hs, a numerus quany f Ws s dscarded he envrnmen every year. In Crrespndng Auhr: Phan guyen Ky Phuc s wh he Deparmen f Indusral Managemen, anal Tawan Unversy f Scence and Technlgy, 43 Keelung Rd., Tape 67, Tawan. -mal: pn_yphuc@yah.cm Vncen F. Yu s wh he Deparmen f Indusral Managemen, anal Tawan Unversy f Scence and Technlgy, 43 Keelung Rd., Tape 67, Tawan. -mal: vncen@mal.nus.edu.w Shu-Yan Chu s wh he Deparmen f Indusral Managemen, anal Tawan Unversy f Scence and Technlgy, 43 Keelung Rd., Tape 67, Tawan. -mal: sychu@mal.nus.edu.w Manuscrp receved 9 Oc. 22; acceped 3 Mar. 23. urpe Unn (U), 6.5 mlln ns f Ws s dscarded every year, and hs number s expeced ncrease 6-28% every 5 years []. Accrdng he recrds f Chna gvernmen, n 29 he quanes f dspsed TVs, refrgerars, washng machnes, ar cndners, cmpuers, prners and mble phnes are 25 mlln, 5.4 mlln, mlln, mlln, 2 mlln, 6 mlln and 4 mlln, respecvely [2]. Kea e al, [3] predced ha hese numbers ncrease 3 5% per annum. In 27, 2.25 mlln ns f Ws was generaed n he USA, cmprsng f nearly 27 mlln TVs, 99 mlln scpled, name a few. Hwever, he numbers f Ws clleced fr recyclng was nly ne ffh whle he res was dspsed f n landfll [2]. Under he pressure f he cnsderable surge f Ws, several effrs have been spen fnd an effecve mehd fr reang and recyclng hem. Generally, Ws cmprse f bh hghly xc and hgh value maerals and cmpnens where he hazardus wase mus be apprprae reaed and valuable cmpnens shuld be reused and recycled [4-6]. In rder mvae envrnmenal respnsbles f manufacurers wh her prducs, many cunres have mpsed nernanal sandards n Ws reamens. Fr example, Resrcn f Hazardus Subsances drecves have been mpsed n U regulae manufacurers and dsrburs ae-bac end-f-lfe prducs and mee sandards fr recyclng and recvery. Accrdng he drecves, manufacurers f s have guaranee a rae n he range f 5% and 8% fr reuse, recyclng and recvery, dependen n he ype f s [7]. Cnsumers are als her bjecs f e-wase regulans. Accrdng [8], a ae-bac sysem n whch cnsumers pay fees dspse f e-wase was esablshed n Japan. In he Sae f Calfrna cnsumers are charged advanced recyclng fees fr dsplay devces a he purchase pn (Calfrna Sae Bard f qualzan, 27). Due he mpsn f hese drecves and regulans, elecrnc scrap recyclng feld has radcally changed. Glbal cmpanes are currenly aware f he urgency f desgnng an effecve s clsed-lp supply chan newr acheve susanable develpmens. T asss cmpanes reslve hs ssue, varus CLSC mdels have been prpsed [9-7] Hwever, ms f hese mdels are deermnsc and nly address specf- 23 TFSA

2 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 cally Ws prblems. In realy, CLSCs are he dynamc and nn-deermnsc sysem wh many facrs havng he hgh degree f uncerany such as demands, prces, hrd pary lgsc prvders, r supplers [8-2]. T deal wh hs prblem, Lses & Deer [2] nrduced her schasc mx neger prgrammng mdel n recyclng sand ndusry maxmze he ne value under dfferen scenars. Salema e al. [22] suggesed a schasc mdel fr he mul-prduc lgsc newr under he mpacs f uncerany f prduc demands and reurns. l-sayed e al. [23] develped a mul-perd mul-echeln frward reverse lgscs newr under rs mdel maxmze he al expeced prf. Ths mdel adped schasc mxed neger lnear prgrammng apprach, where bh deermnsc and schasc cusmer demands were all pssble. Pshvaee e al. [24] prpsed a schasc mx neger lnear prgrammng mdel fr sngle perd, sngle prduc, mul-sage CLSC newr wh he ably cpe wh he uncerany n he quany and qualy f reurned prducs, demands and varable css. Hwever, he applcably f he schasc apprach n realy ssues, fr nsance, mdelng s CLSC newr s cnsderably resrced fr w reasns. The frs s he lac f hsrcal daa, whch creaes radcal dffcules n banng exac randm dsrbuns f unceran parameers. Heavy and cmplex cmpuans f usng hgh number f scenars n mdelng he uncerany s a secnd reasn ha lm he applcably f he schasc apprach n mdelng CLSC newr. Wh he ably f handlng varus ypes f uncerany ncludng fuzzy parameers due lac f nwledge r epsemc uncerany, fuzzy apprach prves self as a hghly penal alernave n descrbng he vagueness f he supply chan sysem. Selm & Ozarahan [25] prpsed a frward supply chan newr wh he neracve fuzzy gal prgrammng based slun apprach deermne he preferred cmprmse slun. Wang & Shu [26] prpsed a pssblsc supply chan newr emplyng genec algrhm fr mnmzng he al supply chan cs and maxmzng he pssbly f fulfllng he arge servce level. Selm e al. [27] presened her wrs n applyng fuzzy apprach n explrng he cllabrave prducn dsrbun plannng prblem n supply chan sysem. Trab & Hassn [28] ffered a supply chan mdel cmprsng f mulple supplers, ne manufacurer and mulple dsrbun ceners. By cnverng hs pssblsc mdel n an auxlary crsp mul-bjecve lnear mdel, preferred cmprmse sluns was prperly baned. Pshvaee & Trab [29] suggesed a sngle prduc pssblsc CLSC mdel. Then he mdel was slved by he cmbnans f dfferen slun appraches The res f he paper s rganzed as fllws. The descrpn f s CLSC sysem s delvered n Secn 2. Secn 3 prpses he pssblsc mxed neger mdel, whle secn 4 presens he algrhm fr slvng he sysem. umercal example and cmpuan expermens are shwed n Secn 5. Fnally, secn 6 s deved fr cnclusn and fuure research. 2. Prblem Descrpns Due he mpran rle f he reverse lgsc newr framewr n decdng hw effecve s clsed-lp newr s, many effrs have been pu cnsruc a frame wr fr he reverse flw n he supply chan [9-, 3, 3]. Fleschmann e al., [9] generalzed he reverse flws by fur prcesses ncludng cllecn, nspecn/ separan, re-prcessng, dspsal and re-dsrbun. Lu e al. [32] suggesed a sraegy fr handlng bslee ems as reuse, servce, remanufacure, recycle, and dspsal. Oher frame wrs f hese prcesses were presened n wrs f [7, 33-35]. Fllwng he wr f [3], n hs sudy, Ws are reaed n hree dfferen ways, namely, reuse, recyclng, and dspsal. Besde he framewr, he perfrmance f he nverse lgsc newr s heavly affeced by several facrs le reurn rae, nerdependences beween usage phase and qualy f dscarded prduc, qualy f prduc and recvery pns and s n. Tae he case f reurn rae as an example. Ws are usually classfed as end-f-lfe reurns, ndcang ha hse prducs are reurned a he end f her ecnmc r physcal lves. The reurn raes f Ws are srngly nfluenced by he mean prduc lfeme, rae f echncal nnvan, and falure rae f cmpnens [36]. Accrdng [36], based n hsrcal sales daa, prduc dspsal dsrbun S () can be cnsruced, and s a funcn f hsrcal sales daa D () ver a lmed me frame D( ), dsrbued as a nrmal dsrbun funcn (Ds) wh a sandard dervan ( ) afer an average usage me ( ). Mrever, her sudes f nerdependency beween he qualy f prduc and recvery pn were shwed n wrs f [, 6, 37-39]. These facrs are capured n hs mdel ban mre praccal relevance. In ur CLSC newr mdel, due large varans n demands f cusmer znes, recyclng mare and secndary mare, sme fracns f prducs and reurned prducs are hld a plans and cllecn ceners as scs. Ths ncurs sme hldng css hese facles. Ws frm cllecn ses are frs ranspred dsassembly cener. Afer dsassembly acves, cmpnens are dvded n 3 grups ncludng: recvery maerals, dspsal, and reusable. A he frs grup, ems

3 Phan guyen Ky Phuc e al.: Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens are ransfrmed n raw maerals and sld crrespndng cusmers a apprprae mares. Hazardus wases and dspsals are dspsed a landflls whle ems belnged he reusable grup are sen nspecn ceners. Iems faled n he qualy es a nspecn ceners wll be ransferred and prcessed a sub dsassembly ceners. Sme passed ems sll can be ranspred hese ceners, dependng n he demands f maeral and secndary mares. The srucure f he prpsed clsed-lp supply chan newr mdel s shwn n Fgure. As a resul f he reamens a dsassembly cener, he al shppng vlume f he reverse supply chan vares sgnfcanly. A he frs and secnd grups, ems are usually cmpressed befre beng shpped he crrespndng desnans. As a resul f hs cmpressn, al shpped vlume usually decreases when we cmpare wh he sum f he vlume f each cmpnen. Hwever, a he hrd grup, because each em n hs grup need mre space be well mananed, he shpped vlume n hese grups fen ncrease. These vlume varans affec heavly n ranspran cs, whch s a crucal facr n a supply chan newr. Whu lss generales, he lcans f plans are usually assumed be predeermned, especally n he feld f manufacurng s. The cnsderable ncrease n a number f rgnal brand manufacurers (OBMs) n recen years manly accuns fr hs assumpn. Reled heavly n rgnal desgn manufacurers (ODMs), OBMs mus accep ha plan lcans are predeermned by ODMs, when desgnng her wn supply chan newrs. The mpressn and he ncmpleeness f daa are als her prmnen feaures n any CLSC newr desgn. The fac f lac f nwledge s refleced hrugh he uncerany f newr parameers. ven f he nfrman s cmplee, elmnang he uncerany n he desgn s sll mpssble, parcularly n case f lng-erm and dynamc sysem. Tae he maxmum capacy fr a specfc prduc f a facly as an example, n realy, several facrs can affec hs value dependng n hw he facly s rganzed. In he ms cmmn layu, where dfferen ypes f prducs are sred n her crrespndng znes and an exra zne s served as reserve space fr all ypes f prducs, he uncerany f he maxmum capacy f a prduc s a funcn f he avalable space n he exra zne. Fxed cs fr penng facles and ranspran css are als he her uncerany surces ha have be cped wh n he desgn prcess. In hs sudy, he ambgues f he newr are mdeled hrugh apprprae pssbly dsrbuns. Va examnng he quany f flws beween facles n dfferen echelns, he prpsed mdel prperly depcs he neracn beween sraegc newr desgn decsns and accal plannng decsns. m: ndex f prducs n: ndex f reusable em : ndex f recyclng maeral p: ndex f dspsal em Cusmer zne (C) Q3 mcd Cllecn cener (D) Q4 mde Dsassembly cener () Q7 peh Landfll (H) I 2 md Q2 mbc Q5 nef Dsrbun Cener (B) Re-prcessng cener (K) Q9 nf Inspecn cener (F) Q6 e Maeral mare (J) Q mab Q nl Q8 nfg Q3 j Q pgh Frward supply chan Reverse supply chan Plan (A) Secndary mare (L) Sub dsassembly cener (G) Recyclng cener (I) I ma Q2 g Fgure. The srucure f he prpsed clsed-lp supply chan newr. 3. Mdel Develpmen Assumpns: The fllwng assumpns are cnsdered n he prpsed mdel.. The mdel s a mul-echeln, mul-prduc CLSC newr. 2. Lcans f plans, cusmers, maeral mares, secndary mares and landflls are predeermned. 3. The capaces f varus facles are lmed. 4. The ranspran cs s prprnal delvery quany and he ravellng dsance. 5. Predeermned values are emplyed as average reurn raes fr dfferen cmpnens baned frm a reurned prduc. ans: In he nan sysem, fuzzy symbls are ndcaed by a lde. Indces: a ndex f fxed lcans f plans, a,..., A. b ndex f canddae lcans f dsrbun ceners, b,..., B. c ndex f fxed lcans f cusmer znes, c,..., C. d ndex f canddae lcans f cllecn ceners, d,..., D.

4 2 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 e ndex f canddae lcans f dsassembly ceners, e,...,. f ndex f canddae lcans f nspecn ceners, f,..., F. g ndex f fxed lcans f sub dsassembly ceners, g,..., G. h ndex f fxed lcans f landflls, h,..., H. ndex f canddae lcans f recyclng ceners,,..., I. j ndex f fxed lcans f maeral mare, j,..., J. ndex f fxed lcans f re-prcessng ceners,,..., K. l ndex f fxed lcans f secndary mares, l,..., L. m ndex f prducs, m,..., M. n ndex f reusable ems, n,...,. ndex f recyclng maerals,,..., O. p ndex f dspsal ems, p,..., P. ndex f me perds,,..., T. Parameers: e mc demand fr prduc m f cusmer zne c n perd. e 2nl maxmum demand fr reusable em n f secndary mare l n perd. e 3j maxmum demand fr recyclng maeral f maeral mare j n perd. he lfecycle lengh f prduc m m. n average fracn f reusable prduc n passng he qualy es a nspecn ceners. r mc rae f reurn percenage f prduc m frm cusmer zne c n perd. r 2mc amun f reurned prduc m frm cusmer zne c n perd r 2 mc r mc emc,. m mn average number f reuse em n baned frm reurned prduc m a dsassembly ceners. 2m average uns f recyclng maeral baned frm reurned prduc m a dsassembly ceners. 3mp average uns f dspsal p exraced frm reurned prduc m a dsassembly ceners. n average uns f recyclng maeral baned frm reuse em n a sub-dsassembly ceners. 2np average uns f dspsal p exraced frm reusable em n a sub-dsassembly ceners. b fxed cs fr penng dsrbun cener b 2d fxed cs fr penng cllecn cener d. 3e fxed cs fr penng dsassembly cener e. 4 f fxed cs fr penng nspecn cener f. 5g fxed cs fr penng sub-dsassembly cener g. 6 fxed cs fr penng recyclng cener. 7 fxed cs fr penng re-prcessng cener. ma un hldng cs f prduc m a plan a. 2 md un hldng cs f reurned prduc m a cllecn cener a. m m un ranspran cs f prduc m. n n un ranspran cs f reusable em n. un ranspran cs f recyclng maeral. p p un ranspran cs f dspsal p. d ab dsance beween plan a and dsrbun cener b. d 2 bc dsance beween dsrbun cener zne b and cusmer zne c. d 3 cd dsance beween cusmer zne c and cllecn cener d. d 4 de dsance beween cllecn cener d and dsassembly cener e. d 5 ef dsance beween dsassembly cener e and nspecn cener f. d 6 e dsance beween dsassembly cener e and recyclng cener. d 7 eh dsance beween dsassembly cener e and landfll h. d 8 fg dsance beween nspecn cener f and nspecn cener g. d9 f dsance beween nspecn cener f and re-prcessng cener. d l dsance beween re-prcessng cener and secndary mare l. d gh dsance beween sub-dsassembly cener g and landfll h. d 2 g dsance beween sub-dsassembly cener g and recyclng cener. d 3 j dsance beween recyclng cener and maeral mare j. ma ma maxmum manufacurng capacy fr prduc m f plan a n each perd. mb mb maxmum prcessng capacy fr prduc m f dsrbun cener b n each perd. md md maxmum capacy fr prduc m f cllecn cener d n each perd.

5 Phan guyen Ky Phuc e al.: Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens 3 me me maxmum prcessng capacy fr prduc m f dsassembly cener e n each perd. mf nf maxmum prcessng capacy fr reusable em n f nspecn cener f n each perd. m n maxmum prcessng capacy fr reusable em n f re-prcessng cener n each perd. mg ng maxmum prcessng capacy fr reusable em n f sub-dsassembly cener g n each perd. mh ph maxmum prcessng capacy fr dspsal p f landfll h n each perd. m maxmum prcessng capacy fr recyclng maeral f recyclng cener n each perd. pa ma un manufacurng cs f prduc m a plan a. pb mb un prcessng cs f prduc m a dsrbun cener b. pd md un prcessng cs f prduc m a cllecn cener d. pe me un prcessng cs f prduc m a dsassembly cener e. pf nf un prcessng cs f reusable em n a nspecn cener f. pg ng un prcessng cs f reusable em n a sub-dsassembly cener g. p n un prcessng cs reusable em n a re-prcessng cener. p un prcessng cs f recyclng maeral a recyclng cener. ph ph un prcessng cs f dspsal p a landfll h. sj j un sellng prce f recyclng maeral a maeral mare j. sl nl un sellng prce f reusable em n a secndary mare l. Decsn varable: I ma Invenry level f prduc m a plan a a he begnnng f perd. I 2 md Invenry level f prduc m a cllecn cener d a he begnnng f perd. X quany f prduc ma m manufacured a plan a n perd. Q mab quany f prduc m shpped frm plan a dsrbun cener b n perd. Q 2 mbc quany f prduc m shpped frm dsrbun cener b cusmer zne c n perd. Q 3 mcd quany f reurned prduc m shpped frm cusmer zne c cllecn cener d n perd. Q 4 mde quany f reurned prduc m shpped frm cllecn cener d dsassembly cener e n perd. Q 5 nef quany f reusable em n shpped frm dsassembly cener e nspecn cener f n perd. Q 6 e quany f recyclng maeral shpped frm dsassembly cener e recyclng cener n perd. Q 7 peh quany f dspsal p shpped frm dsassembly cener e landfll h n perd. Q 8 nfg quany f reusable em n shpped frm nspecn cener f sub dsassembly cener g n perd. Q 9 nf quany f reusable em n shpped frm nspecn cener f re-prcessng cener n perd. Q nl quany f reusable em n shpped frm re-prcessng cener secndary mare l n perd. Q pgh quany f dspsal p shpped frm sub-dsassembly cener g landfll h n perd. Q 2 g quany f recyclng maeral shpped frm sub-dsassembly cener g recyclng cener n perd. Q 3 j quany f recyclng maeral shpped frm recyclng cener maeral mare j n perd. O b bnary varable, Ob f a dsrbun cener s pened a lcan b ; Ob herwse. O 2 d bnary varable, O2d f a cllecn cener s pened a lcan d ; O2d herwse. O 3 e bnary varable, O3e f a dsassembly cener s pened a lcan e ; O3e herwse. O 4 f bnary varable, O4f f an nspecn cener s pened a lcan f ; O4f herwse. O 5 g bnary varable, O5g f a sub-dsassembly cener s pened a lcan g ; O5g herwse. O 6 bnary varable, O6 f a recyclng cener s pened a lcan ; O6 herwse. O bnary varable, O7 f a re-prcessng cener 7 s pened a lcan ; O7 herwse. The bjecve f he prpsed mdel s mnmze he al cs f he enre CLSC newr, ncludng

6 4 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 fxed penng css, varable ranspran css and prcessng css. The cmplee mahemacal mdel s as fllws. B D S bob 2dO2d 3eO3e Mn b d e F G I K fo go O 7O7 f g f g M A T M A T pa X I ma ma m a m a M D T 2 I 2 md m d M A B T m a b M B C T m b c M C D T m c d M D T m d e e F T n e f O I T e P H T p e H F G T n f g F K T n f K L T n l P G H T p g h O G I T g O I J T md ma ( pb m d ) Q mb m d 2 Q2 m bc mbc m ab mab ( pd m d3 ) Q3 md m cd mcd ( pe m d 4 ) Q4 me m de mde ( pf n d5 ) Q5 nf n ef nef ( p d6 ) Q6 e e ( ph p d7 ) Q7 ph eh ( pg n d8 ) Q8 ng p peh n fg nfg ( p n d9 ) Q9 n n f nf ( n nd sl nl) Q l ( ph p d ) Q ph gh ( p d2 ) Q2 p g g ( d3 j sjj ) Q3 () j j Subjec ma ma ; ma,, (2) A Xma nl pgh Q mab mbmbo ; mb,, (3) b a I2 md mdo2 ; md,, (4) D md d 4 Q mde memeo3 ; me,, (5) e d ma 5 Q nef mf nf O4 ; n, f, (6) f e F 8 Q nfg mgngo5 ; ng,, (7) g f G 6 2 Q e Q g mo6 ;,, (8) e g F 9 Q nf mno7 ; n,, (9) f G 7 Q peh Q pgh mhph ; ph,, () e g D 3 Q mcd r2mc ; mc,, () d B 2 Q mbc emc ; mc,, (2) b K Q nl e2nl ; nl,, (3) I 3 Q j e3j ;, j, (4) B I I X Q ;,, ma, ma, ma mab b C md, md mcd mde c e A C ma (5) (6) I 2 I2 Q3 Q4 ; m, d, Q Q2 ; mb,, (7) mbc mab a c C Q3 Q4 ; md,, (8) mde mcd c e F L Q9 Q ; n,, (9) nl nf f l G K Q5 Q8 Q9 ; n, f, (2) nef nfg nf e g G J Q6 Q2 Q3 ;,, (2) e g j e g j M D F mn Q4mde Q5nef m d f ;,, ne (22) M D I 2 m Q4mde Q6 e,, e m d (23) M D H 3 mp Q4mde Q7 peh; p, e, m d h (24) I n Q8nfg Q2 g;, g, n f (25) H 2 np Q8nfg Q pgh; p, g, n f h (26) K 9 Q nf n Q5 nef; n, f, (27) e

7 Phan guyen Ky Phuc e al.: Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens 5 I2 ; md, (28) md O, b O2 d, O3, e O4 f, O5 g, O6, O 7 are bnary, bde,,, f, g,, (29) Q mab, Q2 mbc, Q3 mcd, Q4mde Q5 nef, Q6 e, Q7 peh, Q8nfg, (3) abcde,,,,, f, gh,,, jlmn,,,,,, p, Q9 nf, Q nl, Q pgh, Q2g Q3 j, I ma, I2md Xma, (3) abcde,,,,, f, gh,,, jlmn,,,,,, p, Cnsran (2) maes sure ha he ugng flw f each ype f prduc frm a plan des n exceed he prcessng capacy f he plan. Cnsrans (3)-() assure ha he al amun f ncmng flws f any cmpnen n a facly s smaller han he facly s capacy fr he cmpnen. Cnsran () spulaes ha all reurned prducs f cusmers mus be clleced. Cnsrans (2)-(4) guaranee ha he man cusmer demands mus be sasfed and he quanes f reusable prducs and recyclng maerals sld cann exceed her mare demands. Cnsrans (5)-(6) express he relanshps beween w cnsecuve nvenry levels a plans and cllecn ceners, respecvely. Cnsrans (7)-(2) requre he flw balances a dfferen ypes f facles. The ucmes f dsassembly and sub-dsassembly ceners are descrbed by cnsrans (22)-(26). Cnsran (27) blges ha he ncmng flws f reurned prducs n a re-prcessng cener cann be greaer han he number f reurned prducs passed he qualy es. Cnsran (28) gves nal cndns fr nvenry level a cllecn ceners. Cnsran (29) ndcaes ha decsn varables are bnary. Cnsrans (3-3) mae sure ha he delvery quanes beween w lcans n he newr, he nvenry levels, and he number f prducs manufacured are nn-negave. 4. The Prpsed Slun Apprach The prpsed CLSC newr mdel n hs sudy s a pssblsc mxed neger lnear prgrammng prblem and can be ransfrmed effecvely n an equvalen auxlary crsp mdel by applyng he mehd f [29]. Generally, hs mehd s a cmbnan f he appraches f [4, 4]. The mehd s superr n ha he lneary f he rgnal pssblsc mdel s preserved and can be easly appled varus ypes f fuzzy numbers, eher lnear r nnlnear nes. The mehd s summarzed as fllws. Gven a rapezdal fuzzy number a ( a, a2, a3, a4), he membershp funcn f a s gven by x a f ( x), a xa2 a2 a, a2 xa3 a (32) a4 x g( x), a3 xa4 a4 a3, herwse Accrdng [42], he expeced nerval (I) and expeced value (V) f a fuzzy number a are cmpued as: a a Ia ( ) [, 2] f ( xdx ), g ( xdx ) (33) ( aa2), ( a3 a4) 2 2 a a 2 a a2 a3 a4 V ( a) (34) 2 4 Fr any par f fuzzy numbers a and b, he degree n whch a s bgger han b s gven by [38]: a b, 2 a b 2 a b a b M ( ab, ), [ 2, 2 ] (35) a b a b 2 ( 2) a b, 2 When ( ab M, ), s sad ha a s bgger han, r equal b a leas n degree and dened by a b. If ( M, ) 2 ab, a s sad be ndfferen (equal) b n degree f [4]. Hence, gven 2 a fuzzy mahemacal prgrammng wh he frm T Mn z c x Subjec ax b,,..., l ax b, l,..., M x, (36) cnsrans ax b and ax b can be ransfrmed n he equvalen frms, respecvely, as ax b 2,,..., l, (37) ax ax b b 2 2 ax b 2, l,..., M (38) ax ax b b Accrdng [38], he feasble slun x amng all feasble decsn vecr x s an -accepable pmal slun f he mdel f and nly f he fllwng equan s sasfed: cx /2 cx (39) Applng q. (37), hs equan can be rewren as cx cx cx cx 2 2. (4) 2 2 Fnally, he mdel f q. (36) can be cnvered n

8 6 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 an equvalen crsp - paramerc mdel as fllws: Mn V() c T x Subjec a a b b 2 x2 ( ),,..., l a a b b 2 x 2 ( ), l,..., M (4) a a b b 2 x( ) 2, l,..., M In rder balance he cnflc beween decreasng he bjecve funcn value and enhancng he degree f sasfacn cnsrans, Jmenze e al. [4] suggesed an neracve apprach ban he pmal sluns. Le x be he -accepable pmal slun baned by slvng (4), where. Accrdng (4), he crrespndng fuzzy number represenng he bjecve value s cmpued as T z ( ) c x ( ). The se Q cnanng dscree values f fr evaluang z ( ) s deermned as fllw: Q.,,..., (42). where s an arbrary value decded by he decsn maer (DM),. In he nex sep, afer bservng all values f z ( ), he DM decdes a value gal G and s lerance hreshld G whch are emplyed cnsruc a fuzzy se G evaluae he DM s degree f sasfacn fr he bjecve value. The membershp funcn f G and he degree f he sasfacn f fuzzy gal G by each z ( ) are gven as fllw: z G, z G ( z) G z G, (43) G G G z G, K G ( ( z). ( z) dz z ( ) G z ( )) (44) z ( ) ( zdz ) The degree f balance f each slun crrespndng s evaluaed hrugh: ( x ( )) * ( ( )) R K z (45) G where * represens a -nrm such as he mnmum, he algebrac prduc, name a few. The pmal slun x* s he slun havng hghes degree f balance ( x* ) max * K ( z ( )) (46) R Q Fllwng (33-46), he prpsed clsed-lp supply G chan newr mdel can be cmpleely ransfrmed n an equvalen crsp - paramerc mdel and hen effecvely slved as a mxed-neger prgrammng prblem. 5. umercal xample Ths secn prvdes a numercal example llusrae he valdy f he prpsed mdel. In general, he sze f he real-wrld prblem s fen remendus whch maes nearly mpssble verfy he calculan prcess. Therefre, mae hs example mre raceable fr he readers, he scale f he prblem s carefully seleced. The enre mahemacal mdel s cded n MATLAB R29, a pwerful sfware fr slvng pmzan prblems. The supply chan newr s desgned serve fr w perds. The sze f he es prblem and her relevan daa are gven n Tables -. Table. The sze f he es prblem. A B C D F G H I J K L M O P T Table 2. Cmpnens f each prduc. Prducs Reusable ems Recyclng maerals Dspsals m m 2 2m 2m 2 a 3m (.5,.6,.7) (.6,.7,.8) (.9,,.) (.6,.7,.8) (.4,.5,.6) 2 (.6,.7,.8) (,.,. 2) Reusable ems (.8,.9,) (,.,.2 ) Prducs Dspsals n n 2 2 n (.2,.3,.4) (.6,.7,.8) (,.,.5) 2 (.8,.9,) (,.,.2 ) (,.,.5) (.9,2,2.) Table 3. The un ranspran cs f each em. Prduc Reusable ems 2 n n 2 (.4,.6,.8) (.3,.5,.6) (.2,.4,.5) (.2,.3,.4) Recyclng maerals Dspsal 2 p (.3,.4,.6) (.2,.4,.6) (.2,.4,.6) The pssbly dsrbuns f bjecve values are evaluaed fr each dscree value f n se Q.4,.5,.6,.7,.8,.9,. I s suppsed ha afer bservng he pssbly dsrbuns f he bjecve values, he decsn maer esablshes G and G as

9 Phan guyen Ky Phuc e al.: Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens and 2, respecvely. Usng he -nrm mnmum and fllwng (43-45), we can easly cmpue he cmpably ndex and he degree f balance f each slun. The resuls are presened n Table. Accrdng he Table, he slun f he equvalen crsp - paramerc mdel wh.7 has he hghes degree f balance. The values f decsn varables f hs slun are shwn n Table 2. Table 4. The un hldng css, sellng prces, lfe cycles and he pass raes f reusable ems. Prducs Un hldng css Lfe cycle Pass rae m m 2 n n (.8,.9,2.) (,.2,.5) (.69,.7,.7) 2 (.8,.9,2.) (,.4,.6) (.74,.75,.76) Prducs Sellng prces sj sj 2 sl n sl n 2 (.2,.3,.4) (.5,.6,.8) (2,2.5,3) (2,3,3.5) 2 (.3,.4,.5) (.4,.5,.6) (2,2.5,3.5) (2.5,3,3.5) Table 5. The un prcessng cs a each facly. m Dsrbun Dsassembly pb pb m2 pe m pe m2 (.2,.4,.6) (,.3,.5) (,.,.3) (,.2,.2) 2 (.3,.5,.7) (.3,.6,.7) (.9,.3,.5) (,.2,.4) Plan Cllecn Sub Ds. Landfll m pa pd m pg n ph h (2.5,3,3.5) (.8,.2,.3) (2.5,3.,3.5) (.9,2.3,2.5) 2 (2.7,3.2,3.5) (,.,.2) (2.2,3.,3.5) _ n Inspecn cener Re-prcessn Recyclng pf pf n2 p ph h (.,.2,.3) (.9,.3,.5) (.8,2.,2.6) (.5,.9,2.3) 2 (,.,.2) (.8,.4,.5) (.9,2.,2.5) (.6,.9,2.) Table 6. The fxed css fr penng facles (x 2 ). Dsrbun cener Cllecn cener Dsassembly cener Inspecn cener b 2d 3e 4 f (7,8,9) (3,5,6) (8,,2) (6,9,) 2 (5,6,8) _ (7,,2) (8,9,) Subdsassembly cener 5g Recyclng cener 6 Re-prcess ng cener 7 (7,9,) (5,7,9) (9,23,24) 2 _ Table 7. The maxmum capaces f facles (x 2 ). Dsrbun cener m Dsassembly cener mb mb m 2 me m me m 2 (8,9,) (8,9,) (8,9,) (8,9,) 2 (7,8,9) (7,8,9) (7,8,9) (7,8,9) Inspecn cener Plan Cllecn cener mf n mf n 2 ma m md m (8,9,2) (8,9,2) (4,5,6) (4,5,6) 2 (,2,3) (2,3,4) (3,4,5) (4,5,6) Sub dsassembly cener n Re-prcessn g cener Landfll Recyclng cener mg m n mh p m (4,5,6) (2,22,23) (6,7,8) (34,35,37) 2 (9,,) (2,22,23) _ (34,35,37) Table 8. The dsances beween ses. Ds. A C C 2 Ds. D Ds. K B C 64 L 57 B C 2 6 L 2 52 Ds. I Ds. I Ds. K G Ds. H J 44 G 55 F G 55 J 2 4 F Ds. D F F 2 I H Table 9. The demands a cusmers, maeral mares and secndary mares. Mare (j=) Maeral mares Mare 2(j=2) e 3 e 3 2 e 3 2 e 3 22 (,,) (,,) (,,) (,,) 2 (2,23,25) (22,23,25) (2,23,25) (22,23,25) 3 (2,23,25) (22,23,25) (2,23,25) (22,23,25) Cusmer znes Zne(z=) Zne(z=2) e e 2 e 2 e 22 (6,7,8) (4,5,6) (8,9,) (8,9,) 2 (7,8,9) (5,6,7) (7,8,9) (8,9,) 3 (5,6,7) (5,6,7) (5,6,7) (8,9,) Secndary mares Mare (l=) Mare 2(l=2) e 3 e 3 2 e 3 2 e 3 22 (,,) (,,) (,,) (,,) 2 (,2,5) (8,9,2) (,2,5) (8,9,2) 3 (,2,5) (8,9,2) (,2,5) (8,9,2)

10 8 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 If he decsn maer s n sasfed wh hs slun, s/he can adjus he values f G and G. Ths adjusmen can sll ulze he resuls baned frm he equvalen crsp -paramerc mdel. Therefre, generally, des n enhance he cmplexy f he enre prcess. Table. Raes f reurn percenage f prducs. Raes f reurn percenage f prducs Zne (c=) Zne 2(c=2) r r 2 2 r 2 r 22 (,,) (,,) (,,) (,,) 2 (.4,.6,.7) (.3,.4,.6) (.4,.5,.7) (.3,.5,.7) 3 (.3,.5,.6) (.5,.6,.8) (.5,.6,.8) (.4,.6,.7) Feasbly degree Table. α-accepable pmal sluns. Pssbly dsrbun f bjecve value.4 (882427,49362,873857).5 (94,44824,92669).6 (969,4763,9526).7 (933537,53866,987564).8 (949936,53458,22462).9 (96635,55924,26832) (9824,587246,298986) The cmpably ndex f each slun The degree f balance f each slun Table 2. Values f decsn varables a α=.7.. Q mab Q 2 mbc Q 3 mcd Q 4 mde Q5 nef Q 6 e Q 7 peh Q 8 nfg Q 9 nf Q pl O b. Q pgh Q 2 g Q3 j O2 d O 3 e O 4 f O 5 g O6 O7 2 _. X ma I ma I 2 md Cnclusns Ths paper prpses a relavely cmplee mul-echeln, mul-prduc CLSC newr mdel cmprsng f several recyclng prcesses fr varus ypes

11 Phan guyen Ky Phuc e al.: Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens 9 f W based n her characerscs. In rder mnmze he al cs f he enre sysem under he effecs f uncerany and ncmpleeness f daa, he prblem s mdeled as a pssblsc mxed neger prgrammng prblem and hen effecvely slved by he ransfrman apprach f [29]. Snce bh balanced and unbalanced effcen sluns can be fund by hs apprach, decsn maers are gven mre flexbly cme u wh he ms apprprae newr desgn based n bjecve cndns. Alhugh pssblsc mxed neger prgrammng prblems are very prmsng appraches, perfrmng sensvy analyses fr hese ypes f prblems s raher dffcul. Furher sudes n he sably f pssblsc prgrammng are necessary n rder prvde decsn maers beer nsghs. In addn, several pssble fuure research drecns may be cnsdered exend ur mdel. Fr example, ncludng he ranspran flee capacy n he prpsed mdel r cnsderng css f shrage r baclg may mae he mdel mre cmprehensve and clser realy. References [] I. Dalrymple e al., An negraed apprach elecrnc wase (W) recyclng, Crcu Wrld, vl. 33, n.2, pp , 27. [2] F. O. Ongnd, I. D. Wllams, and T. J. Cherre, Hw are W dng? A glbal revew f he managemen f elecrcal and elecrnc wases, Wase Manage., vl. 3, n. 4, pp.74-73, 2. [3] H. Kea, L. L, and D. Wenyng, Research n recvery lgscs newr f Wase lecrnc and lecrcal qupmen n Chna, Indusral lecrncs and Applcans. ICIA 28. 3rd I Cnference, pp , 28. [4]. Dmraas e al., Small W: Deermnng recyclables and hazardus subsances n plascs, J. Hazard. Maer., vl. 6, n. 2-3, pp , 29. [5]. Trumann and H. Rechberger, Cnrbun resurce cnservan by reuse f elecrcal and elecrnc husehld applances, Resur. Cnserv. Recy., vl. 48, n. 3, pp , 26. [6] H. Y. Kang and J. M. Schenung, lecrnc wase recyclng: A revew f U.S. nfrasrucure and echnlgy pns, Resur. Cnserv. Recy., vl. 45, n. 4, pp , 25. [7] W. He e al., W recvery sraeges and he W reamen saus n Chna, J. Hazard. Maer., vl. 36, n.3, pp , 26. [8] K. aan e al., valuang he reducn n green huse gas emssns acheved by he mplemenan f he husehld applance recyclng n Japan, In. J. Lfe Cycle Ass., vl. 2, pp , 27. [9] M. Fleschmann e al., A characersan f lgscs newrs fr prduc recvery, Omega, vl. 28, n. 6, pp , 2. [] H. R. Kre, A. van Haren, and P. C. Schuur, Busness case Reb: recvery sraeges fr mnrs, Cmpu. Ind. ng., vl. 36, n. 4, pp , 999. [] R. Zudwj and H. Kre, Sraegc respnse reurns: Prduc ec-desgn r new recvery prcesses?, ur. J. Oper. Res., vl. 9, n. 3, pp , 28. [2] P. Gergads and M. Besu, Susanably n elecrcal and elecrnc equpmen clsed-lp supply chans: A Sysem Dynamcs apprach, J. Clean. Prd., vl. 6, n. 5, pp , 28. [3] D. Hammnd and P. Beullens, Clsed-lp supply chan newr equlbrum under legslan, ur. J. Oper. Res., vl. 83, n. 2, pp , 27. [4] I. H. Hng and J. S. Yeh, Mdelng clsed-lp supply chans n he elecrncs ndusry: A realer cllecn applcan, Transpr. Res.. Lg., vl. 48, n. 4, pp , 22. [5] S. Mra, Invenry managemen n a w-echeln clsed-lp supply chan wh crrelaed demands and reurns, Cmpu. Ind. ng., vl. 62, n. 4, pp , 22. [6] J. Ösln,. Sundn, and M. Björman, Imprance f clsed-lp supply chan relanshps fr prduc remanufacurng, In. J. Prd. cn., vl. 5, n. 2, pp , 28. [7] Q. Zhu, J. Sars, and K.-H. La, Green supply chan managemen mplcans fr clsng he lp, Transpr. Res.. Lg., vl. 44, n., pp. -8, 28. [8] W. Klb, A. Marel, and A. Guun, The desgn f rbus value-creang supply chan newrs: A crcal revew, ur. J. Oper. Res., vl. 23, n. 2, pp , 2. [9] Z. Chang and G. H. Tzeng A hrd pary lgsc prvder fr he bes slun n fuzzy dynamc decsn envrnmens, In. J. Fuzzy Sys., vl., n., pp. -9, 2. [2] Z. Chang and G. H. Tzeng Suppler evaluan and selecn usng axmac fuzzy se and DA mehdlgy n supply chan managemen, In. J. Fuzzy Sys., vl. 4, n. 2, pp , 22. [2] O. Lseş and R. Deer, A schasc apprach a case sudy fr prduc recvery newr desgn, ur. J. Oper. Res., vl. 6, n., pp , 25. [22] M. I. G. Salema, A. P. Barbsa-Pva, and A. Q. vas, An pmzan mdel fr he desgn f a

12 2 Inernanal Jurnal f Fuzzy Sysems, Vl. 5,., March 23 capacaed mul-prduc reverse lgscs newr wh uncerany, ur. J. Oper. Res., vl. 79, n. 3, pp , 27. [23] M. l-sayed,. Afa, and A. l-kharbly, A schasc mdel fr frward reverse lgscs newr desgn under rs, Cmpu. Ind. ng., vl. 58, n. 3, pp , 2. [24] M. S. Pshvaee, F. Jla, and J. Razm, A schasc pmzan mdel fr negraed frward/ reverse lgscs newr desgn, J. Manuf. Sys., vl. 28, n. 4, pp. 7-4, 29. [25] H. Selm and I. Ozarahan, A supply chan dsrbun newr desgn mdel: An neracve fuzzy gal prgrammng-based slun apprach, In. J. Adv. Manuf. Technl., vl. 36, n. 3, pp. 4-48, 28. [26] J. Wang and Y.-F. Shu, A pssblsc decsn mdel fr new prduc supply chan desgn, ur. J. Oper. Res., vl. 77, n. 2, pp. 44-6, 27. [27] H. Selm, C. Araz, and I. Ozarahan, Cllabrave prducn dsrbun plannng n supply chan: A fuzzy gal prgrammng apprach, Transpr. Res.. Lg., vl. 44, n. 3, pp , 28. [28] S. A. Trab and. Hassn, An neracve pssblsc prgrammng apprach fr mulple bjecve supply chan maser plannng, Fuzzy Ses Sys., vl. 59, n. 2, pp , 28. [29] M. S. Pshvaee and S. A. Trab, A pssblsc prgrammng apprach fr clsed-lp supply chan newr desgn under uncerany, Fuzzy Ses Sys., vl. 6, n. 2, pp , 2. [3] M. Fleschmann e al., Quanave mdels fr reverse lgscs: A revew, ur. J. Oper. Res., vl. 3, n., pp. -7, 997. [3] I. Bereel e al., W reamen sraeges evaluan usng fuzzy LIMAP mehd, xper Sys. Appl., vl. 38, n., pp. 7-79, 2. [32] Z. F. Lu e al., Recyclng sraegy and a recyclably assessmen mdel based n an arfcal neural newr, J. Maer. Prcess. Technl., vl. 29, n.-3, pp. 5-56, 22. [33] A. Gungr and S. M. Gupa, Dsassembly sequence plannng fr prducs wh defecve pars n prduc recvery, Cmpu. Ind. ng., vl. 35, n. -2, pp. 6-64, 998. [34] A. Muha and S. Pharel, Sraegc newr desgn fr reverse lgscs and remanufacurng usng new and ld prduc mdules, Cmpu. Ind. ng., vl. 56, n., pp , 29. [35] M. C. Therry e al., Sraegc prducn and perans managemen ssues n prduc recvery managemen, Calf. Manage. Rev., vl. 37, pp. 4-35, 995. [36] Y. Umeda, K. Shnsue, and S. Taash, Prpsal f Margnal Reuse Rae fr evaluang reusably f prducs, Inernanal Cnference n ngneerng Desgn, Melburne, pp. 5-8, Aug. 25. [37] H. B. Lee,. W. Ch, and Y. S. Hng, A herarchcal end-f-lfe decsn mdel fr deermnng he ecnmc levels f remanufacurng and dsassembly under envrnmenal regulans, J. Cleaner Prd., vl. 8, n. 3, pp , 2. [38] A. Xanhpuls and. Iavu, On he pmal desgn f he dsassembly and recvery prcesses. Wase Manage., vl. 29, n. 5, pp. 72-7, 29. [39] C. Zpuls and G. Tagaras, Impac f uncerany n he qualy f reurns n he prfably f a sngle-perd refurbshng peran, ur. J. Oper. Res., vl. 82, n., pp , 27. [4] M. Jménez e al., Lnear prgrammng wh fuzzy parameers: An neracve mehd reslun, ur. J. Oper. Res., vl. 77, n. 3, pp , 27. [4] M. A. Parra e al., Slvng a mulbjecve pssblsc prblem hrugh cmprmse prgrammng, ur. J. Oper. Res., vl. 64, n. 3, pp , 25. [42] S. Helpern, The expeced value f a fuzzy number, Fuzzy Ses Sys., vl. 47, n., pp. 8-86, 992. Phan guyen Ky Phuc receved hs B.. degree n Mechancal n 28 frm H- ChMnh Cy Unversy f Technlgy, and hs M.S. degree n Indusral Managemen n 2 frm anal Tawan Unversy f Scence and Technlgy. He s currenly Ph. D canddae n Indusral Managemen deparmen a anal Tawan Unversy f Scence and Technlgy. Hs recen research neress nclude nvenry managemen, ranng fuzzy numbers, dynamc prgrammng, mea heursc algrhm. Vncen F. Yu s an asscae prfessr f Indusral Managemen a he anal Tawan Unversy f Scence and Technlgy. He receved hs Ph.D. n Indusral and Operans ngneerng frm he Unversy f Mchgan, Ann Arbr. Hs curren research neress nclude nfrman managemen, perans research, lgscs/supply chan managemen, and sf cmpung. He had publshed arcles n Cmpuers & Indusral ngneerng, Cmpuers & Operans Research, urpean Jurnal f Operanal Research, Managemen Decsn, and Servce Indusres Jurnal.

13 Phan guyen Ky Phuc e al.: Opmzng he Fuzzy Clsed-Lp Supply Chan fr lecrcal and lecrnc qupmens 2 Shu-Yan Chu s a dsngushed prfessr f ndusral managemen and he drecr f he Cener fr Inerne f Thngs Innvan (CITI) a anal Tawan Unversy f Scence and Technlgy and als hlds appnmens a Graduae Insue f Auman and Cnrl and Graduae Insue f Technlgy Managemen n he same unversy. He s currenly a vsng schlar a agya Unversy n Japan. Hs research neress nclude Inerne f Thngs, echnlgy-enabled servces, nellgen sysem mdelng and applcan, RFID, supply chan managemen, and gemerc algrhms. Dr. Chu was he dean f nernanal affars, he nanal crdnar f he urpean Unn Framewr Prgrammer anal Cnac Pn Tawan Offce and he edr-n-chef f he Jurnal f Chnese Insue f Indusral ngneers publshed by Taylr and Francs. He was a vsng schlar a Unversy f Washngn and a Hng Kng Unversy f Scence and Technlgy. Dr. Chu has been very acve n nernanal cperan, havng served as he general char fr C29, 2 IFORMS Servce Scence Cnference and MCP AP2 as well as he rganzer f many nernanal evens. He receved hs BBA n ndusral managemen frm anal Cheng-Kung Unversy, Tawan n 983, hs MS and PhD n ndusral and perans engneerng frm he Unversy f Mchgan n 987 and 992 respecvely.