A Reverse Logistics Model for the Distribution of Waste/By-products. Hamid Pourmohammadi, Maged Dessouky*, and Mansour Rahimi

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1 A Reverse Logscs Model for he Dsrbuon of Wase/By-producs Hamd Pourmohammad, Maged Dessouy*, and Mansour Rahm Unversy of Souhern Calforna Epsen Deparmen of Indusral and Sysems Engneerng 375 McClnoc Avenue, GER 240 Los Angeles, Calforna Emal: Tel: (23) , (23) , (23) Fa: (23) * correspondng auhor

2 A Reverse Logscs Model for Dsrbung Wases/By-producs Absrac: In hs sudy, a reverse logscs model s developed o mnmze he envronmenal and operaonal coss of echangng wase and by-produc maerals n a busness o busness newor. One of he goals of he research s o nvesgae he opporunes of subsuon he raw maeral wh value added wase/by-produc maeral colleced from he regon. The newor conans frms, collecon ceners, value added process ceners (e.g. dsassembly, recyclng, or remanufacurng), dsposal ceners, and vrgn maeral mare. The developed reverse logscs model s a med neger lnear model. Snce he problem s NP-hard, we develop a Genec Algorhm (GA) o effcenly solve he model for large problem nsances. Key words: Reverse Logscs, Med Ineger Lnear Program, Genec Algorhm

3 . Inroducon Increasng world populaon and sandards of lvng have magnfed resource consumpon and he dsposal rae. In a ypcal day, humans add 5 mllon ons of carbon o he amosphere, desroy 5 square mles of ropcal ranfores, creae 72 square mles of deser, elmnae beween 40 and 00 speces, erode 7 mllon ons of opsol, add 2700 ons of CFCs (Chlorofluorocarbons) o he envronmen and ncrease populaon by 263,000 (Orr, 992). Growng concerns abou clmae changes, local and regonal mpacs of ar, ground and waer polluon from ndusral acves have sgnfcanly epanded he neracon beween envronmenal managemen and operaons, leadng o he area ermed as reverse logscs (Corbe, 200b). There are economcal and polcal usfcaons ha hghlgh he necessy of nvesmen n hs area of research. Publc pressure on reducng he envronmenal mpacs of ndusral operaons has resuled n seng non-fleble sandards and penales for envronmenally nensve ndusral operaons (Corbe, 200a). On he oher hand, processng wase maerals and end-oflfe goods o be subsued for raw resources wll save money boh n erms of purchasng fewer raw maerals and less dsposal. In Europe, EU regulaon ncreases producer responsbly or produc sewardshp for several branches of ndusry (Kre e al., 200). These rules force he Orgnal Equpmen Manufacurers (OEMs) o se-up a ae-bac and recovery sysem for dscarded producs. Producer responsbly s supplemened by measures such as ncreased dsposal arffs, dsposal bans, resrcons on wase ransporaon, wase prevenon, and emsson conrol. Consumers 2

4 demand for clean manufacurng and recyclng s also ncreasng. Consumers epec o be able o rade an old produc when hey buy a new one. From anoher perspecve, realers also epec OEMs o esablsh a proper envronmenally responsble reverse logscs and recovery sysem. A well-managed reverse logscs program should also be able o provde mporan cos savngs n procuremen, dsposal, nvenory carryng and ransporaon. Emssons durng ransporaon are ofen recognzed as havng he greaes envronmenal mpac on all acves n a produc s lfe cycle (Corbe, 200b). There has been a sgnfcan growng neres n he subec of reverse logscs (Kre, 998; Sars, 200; Fleschmann, 200). Mos of he models developed n hs feld are smlar o he radonal locaon problems, n parcular locaon-allocaon models (Kroon and Vrens, 995; Ammons e al., 997; Spengler e al., 997; Marn and Pelegrn, 998; Jayaraman e al., 999; Kre e al., 999, 200; Fleschmann e al., 200). In mos of he models, ransporaon and processng coss were mnmzed whle he envronmenal coss assocaed wh he desgned newor were ofen negleced. Curren leraure on reverse logscs concenraes on he end-of-lfe produc s recovery sysems (Busness-o-Consumer, B2C, newor). Despe end-of-used maerals ha have been he subec of mos reverse logscs relaed sudes, few wors (e.g. Mondschen and Schlru, 997) have addressed he recovery process for wase/by-producs sreams n an echange newors among ndusres, whch share a consderable amoun of wase sreams. For eample, Los Angeles Couny, by self, produces 2.2 mllon ons of by-produc/wase maerals n he manufacurng secor. By-producs and wase maerals are poenally valuable npus for a varey of ndusral 3

5 processes. Thnng of wases and by-producs as poenally valuable feedsoc may allow for he desgn of a hgh degree of susanably no hem. Ths may also creae mares specfcally amed a capalzng on he use and reuse of hese maerals as npus. Ths drec use of hgh qualy wases as npus benefs he suppler, he cusomer, and he envronmen as well as sgnfcanly eends he lfespan of a gven by-produc, delayng s ulmae fae a he landfll and reducng he consumpon of he vrgn source maeral(s) for whch has been subsued. In hs sudy, managng he recovery of wase and by-produc sreams n an ndusral echange newor (Busness-o-Busness, B2B, newor) s nvesgaed. Due o he mporance of he logscs ssues (e.g. nvenory level) n a B2B maeral echange newor, hey are negraed o he proposed approach o buld up a novel comprehensve reverse logscs model. In he ne secon, he leraure on reverse logscs s brefly revewed. In secon 3, a reverse logsc newor s frs defned for he dsrbuon of wase and by-producs. Then, a mahemacal model s developed o deermne he locaon of he facles and he flows of maeral among hem. Due o he combnaoral naure of he problem, we develop a heursc approach based on a Genec Algorhm (GA) o effcenly solve he problem for large sze problem nsances. The developmen of he GA approach s presened n secon 4. In secon 5, we demonsrae he effecveness of he soluon on randomly generaed daa ses. 2. Leraure Revew Durng he las decade, reverse logscs has receved ncreasng aenon from boh academc researchers and ndusral praconers. Serous and perssen envronmenal concerns and governmen regulaons have creaed a movaon o pursue furher research n hs feld. Durng 4

6 he early nnees, he Councl of Logscs Managemen publshed wo sudes on reverse logscs. Frs, Soc (992) proposed he applcaon of reverse logscs n busness and socey n general. One year laer, Kopc e al. (993) elaboraed he opporunes on reusng and recyclng. In he lae nnees, several oher sudes on reverse logscs were compleed. Kosec (998) dscussed mareng aspecs of reuse and ssues nvolvng he eenson of produc lfe cycle. Soc (998) nvesgaed how o sar and carry ou reverse logscs programs. Rogers and Tbben-Lembe (999) demonsraed a collecon of reverse logscs busness pracces usng a comprehensve quesonnare among US ndusres. Reverse logscs sudes can be dvded no several caegores. Dowlashah (2000) denfed fve caegores as follows: global conceps of reverse logscs, quanave models, logscs (dsrbuon, warehousng, and ransporaon), company profles, and applcaons. Recenly, many researchers have concenraed on he opmzaon and quanave models n reverse logscs. Mos of he proposed models are smlar o radonal facly locaon models, and are n he shape of a med neger lnear program for a sngle perod of me (Kroon and Vrens, 995; Ammons e al., 997; Spengler e al., 997; Barros e al., 998; Marn and Pelegrn, 998; Jayaraman e al., 999; Kre e al., 999; Fleschmann e al., 200). Oher researchers suded problems wh a sngle nbound commody ecep for Spengler e al. (997) and Jayaraman e al. (999). Louwers e al. (999) proposed he desgn of a recyclng newor for carpe wase. The goal of her sudy was o deermne he locaons and capaces of he regonal recovery ceners o mnmze nvesmen, processng, and ransporaon coss. They developed a nonlnear model and solved opmally wh sandard sofware. A comprehensve revew on varous cases can be found n Bro e al. (2002). 5

7 We found lle wor addressng he envronmenal coss of maeral echange newors. Loclear (200) elaboraed several echnques ha can be appled o deermne he value of envronmenal coss. One approach s Conngen Valuaon, where eernal coss are based on how much he publc s wllng o pay for proecon of he envronmen. Shadow Prcng s anoher echnque, whch uses esng regulaons o esmae he coss ha he socey s wllng o accep for he reducon of polluon. In 990, Tellus Insue conduced an analyss o esmae he eernal coss for seven dfferen componens of ar emssons ncludng CO 2 and NO (Loclear, 200). Ther esmaons are based on he Conngen Valuaon mehod and have been frequenly ced n he leraure. Accordng o her resuls, n US dollars per pound, values for CO 2 and NO are 0.02 and 3.4 respecvely. Saleem (200) repored cos esmaes for hree manufacurng scenaros n he Heang Venlaon and Ar Condonng (HVAC) ndusry. In he frs scenaro, coss were calculaed for HVAC producon usng vrgn maerals. In he second one, coss for producon usng maerals from secondary mnng (recyclng) processes were esmaed, whch refleced an 82% cos reducon compared o he frs alernave. In he hrd approach, maerals acqured from dsassembly of value eraced HVAC uns were used, whch resuled n 88% cos savngs as compared o he use of maeral from prmary eracon. Mahews (999) used a Leonef npu-oupu (IO) model o evaluae he envronmenal mpac on he enre economy resulng from he producon processes. In addon, hs model consders envronmenal mpacs. He generaed a subsanal daa se lnng releases of crera polluans 6

8 and greenhouse gases wh manufacurng acves n each ndusral secor. The oal ar polluon releases found for each commody were combned wh a range of envronmenal damage valuaon sudes o esmae he eernal coss of hese acves. 3. Problem Formulaon In hs secon, a mahemacal model for he problem s presened. Frs, elemens of he model are defned. A dagram oulnng dfferen elemens of he newor and her connecvy s depced n Fg.. Ths dagram demonsraes a general regonal recovery newor whn a sysem boundary (e.g. LA Couny regonal recovery newor). Remanufacurng and recyclng ceners are eamples of value-added process ceners. In hs model he locaons of plans, collecon ceners, and value-added process ceners are nsde he sysem boundares, whle he locaons of vrgn maeral mares and dsposal ceners can be boh nsde and ousde of he boundares. Also, he locaons of vrgn maeral mares, plans, and dsposal ceners are fed. The oher facles locaons are deermned by he model. As Fg. llusraes, he wase and by-producs generaed by a frm are ransferred o he collecon ceners or f her qualy are accepable, hey wll be consumed drecly by anoher frm. In he collecon ceners, afer qualy nspecon, he maerals are passed o oher facles such as he VAP ceners or he dsposal ceners. Based on he qualy of he generaed maerals, hey may be sen o plans o be used as a subsue for raw maerals. Afer performng valueadded processes, he maerals are sen o he downsream frms. A poron of he maeral flow deemed unusable s sen o a dsposal cener. 7

9 Vrgn Maeral Mares Value-added Process Ceners Plans Collecon Ceners Dsposal Ceners REGIONAL BOUNDARY Fg. : A model dagram llusrang he dsrbuon of he wase/by-producs. The prmary goal of he newor s o provde enough raw maerals for he plans from he maeral echange newor, bu f more maerals are requred (due o low qualy maeral generaed n he newor) he vrgn maeral mare can be consdered as anoher source of supply. The mahemacal model mnmzes wo caegores of coss: producon coss (ncludng facly openng, ransporaon, processng, and nvenory coss) and envronmenal coss (ncludng energy, waer, and ar polluon coss, eernal envronmenal coss of producng from vrgn maerals, dsposal coss ncludng ppng fees and effecs on local communes). The mnmzaon of he obecve funcon s subec o a se of consrans, namely, maeral balance 8

10 a facles, demand consrans, shppng from open facles, capacy consrans, doman consrans, and non-negavy consrans. The followng ses and ndees are used o defne he parameers and varables of he model. Ses: V: Se of vrgn maeral mares P: Se of plans I: Se of collecon ceners J: Se of value added process ceners D: Se of dsposal ceners K: Se of maeral ypes TP: Se of me perods Indees: v: Inde for vrgn maeral mares p, r: Inde for plans : Inde for collecon ceners : Inde for value added process ceners d: Inde for dsposal ceners : Inde for maeral ypes : Inde for me perods Model Parameers: CT vp : Transporaon cos per mle per un of maeral from vrgn maeral mare v o plan p. 9

11 CT p : Transporaon cos per mle per un of maeral from plan p o collecon cener. CT p : Transporaon cos per mle per un of maeral from collecon cener o plan p. CT pr : Transporaon cos per mle per un of maeral from plan p o plan r. CT : Transporaon cos per mle per un of maeral from collecon cener o value-added process cener. CT d : Transporaon cos per mle per un of maeral from collecon cener o dsposal cener d. CT p : Transporaon cos per mle per un of maeral from value-added process cener o plan p. CT d : Transporaon cos per mle per un of maeral from value-added process cener o dsposal cener d. CN vp : Envronmenal cos of ransporaon per mle per un of maeral from vrgn maeral mare v o plan p. CN p : Envronmenal cos of ransporaon per mle per un of maeral from plan p o collecon cener. CN p : Envronmenal cos of ransporaon per mle per un of maeral from collecon cener o plan p. CN pr : Envronmenal cos of ransporaon per mle per un of maeral from plan p o plan r. CN : Envronmenal cos of ransporaon per mle per un of maeral from collecon cener o VAP cener. CN d : Envronmenal cos of ransporaon per mle per un of maeral from collecon cener o dsposal cener d. 0

12 CN p : Envronmenal cos of ransporaon per mle per un of maeral from VAP cener o plan p. CN d : Envronmenal cos of ransporaon per mle per un of maeral from VAP cener o dsposal cener d. CP : Un processng cos of maeral ype a collecon cener. CP : Un processng cos of maeral ype a value-added process cener. CP d : Un processng cos of maeral ype a dsposal cener d. h p : Invenory cos per un per perod for maeral ype a plan p. h : Invenory cos per un per perod for maeral ype a collecon cener. h : Invenory cos per un per perod for maeral ype a value-added process cener. π : Bacorder cos per un per perod for maeral ype a plan p. p π : Bacorder cos per un per perod for maeral ype a collecon cener. π : Bacorder cos per un per perod for maeral ype a value-added process cener. F : Cos of openng collecon cener. F : Cos of openng VAP cener. CD d : Un dsposal cos (ppng fee) a dsposal cener d for maeral ype. CV v : Eernal envronmenal cos of producng a un of maeral ype by vrgn maeral mare v. CE : Energy consumpon cos a value-added process cener for a un of maeral ype. CW : Envronmenal cos of dsposng a un of maeral no waer a value-added process cener. CW d : Envronmenal cos of dsposng a un of maeral no waer a dsposal cener d. CA : Envronmenal cos of dsposng a un of maeral n ar a value-added process cener.

13 CA d : Envronmenal cos of dsposng a un of maeral no ar a dsposal cener d. T vp : The dsance beween vrgn maeral mare v and plan p. T p : The dsance beween plan p and collecon cener. T pr : The dsance beween plan p and plan r. T : The dsance beween collecon cener and value-added process cener. T d : The dsance beween collecon cener and dsposal cener d. T p : The dsance beween value-added process cener and plan p. T d : The dsance beween value-added process cener and dsposal cener d. CAP p : The capacy of plan p for maeral ype. CAP : The capacy of collecon cener for maeral ype. CAP : The capacy of value-added process cener for maeral ype. CAP d : The capacy of dsposal cener d for maeral ype. S p : Toal supply of maeral ype a plan p n me perod. R p : Demand of plan p for maeral ype n me perod. T: Number of plannng perods. B : A large number. w : Fracon of maeral dsposed o waer a value-added process cener. w : Fracon of maeral dsposed o waer a dsposal cener d. d a : Fracon of maeral dsposed o ar a value-added process cener. a : Fracon of maeral dsposed o ar a he dsposal cener d. d α : A mulpler o adus maeral ype balance n he consrans. β : A mulpler o adus maeral ype balance n he consrans. δ : Mnmum fracon of npu maeral o he collecon ceners ha can be dsposed. 2

14 γ : Mamum fracon of maeral ha eners he collecon cener ha can be used by he plans. η τ : Mnmum fracon of maeral n VAP ceners ha can be dsposed. : Mamum fracon of maerals n he plans ha can drecly be used by he oher plans. Decson Varables: vp : The flow of maeral ype from vrgnal maeral mare v o plan p n perod. pr : The flow of maeral ype from plan p o plan r n perod. p : The flow of maeral ype from plan p o collecon cener n perod. : The flow of maeral ype from collecon cener o VAP cener n perod. d : The flow of maeral ype from collecon cener o dsposal cener n perod. p : The flow of maeral ype from collecon cener o plan p n perod. p : The flow of maeral ype from VAP cener o plan p n perod. d : The flow of maeral ype from VAP cener o dsposal cener d n perod. Y : The ndcaor of openng collecon cener. Y : The ndcaor of openng value-added processng cener. INV p : Invenory level of maeral ype a plan p a he end of perod. INV : Invenory level of maeral ype a collecon cener a he end of perod. INV : Invenory level of maeral ype a value added process cener a he end of perod. BOR p : Bacorder of maeral ype a plan p a he end of perod. BOR : Bacorder of maeral ype a collecon cener a he end of perod. BOR : Bacorder of maeral ype a value-added process cener a he end of perod. 3

15 Obecve Funcon The obecve funcon mnmzes wo caegores of coss: producon coss (Z ) and envronmenal coss (Z 2 ). A wegh, λ and - λ, s assgned o each par of he obecve funcon, o dfferenae he degree of sensvy. The obecve funcon s as follows: Mn Z λz λ) ( Z 2 In he obecve funcon, he producon coss (Z ) nclude facly openng (), ransporaon (2), processng (3), and nvenory/bacorder coss (4): Z I p p p pp I K pp rr K I pp K I dd K J pp K ( h ( h J [ T F Y I K K I K J CP CT CT ( F CT CT p p d p pp INV INV Y d p p ) J K π BOR p d p I J K vv pp K J ddk π BOR ) CP )] ( CT I K pp CT CT CT vp d ( h pr ) p pr vp d dd K INV p vp d pr CP p d ( I π BOR p d ) J d ) Envronmenal coss (Z 2 ) nclude envronmenal ransporaon (5), energy (6), waer (7), ar polluon cos (8), eernal envronmenal coss of producng from vrgn maerals (9), and dsposal coss ncludng ppng fees (0). Eernal envronmenal cos s he era money ha a frm s charged when refuses o subsue he vrgn maeral mare by an accepable recycled maeral. The mahemacal formulaon s as follows: 4

16 Z 2 [ T p p p pp I K pp rr K I pp K I dd K J pp K J K J K J K vv pp K dd K CE CW CA CD CN CN CN CN d ( ( w ( a CV ( p d p I I ) I I v p d p vp d d p p ) ) J I J K vv pp K J ddk dd K dd K d )] CW CA d CN CN CN d d CN a w vp d ( pr d ( I vp d I pr d d vp pr d J J d d ) ) Consrans Consran () ndcaes ha he oal supply of maerals from each plan mus be equal o he oupu flows. Consrans (2) o (4) guaranee he balance of maeral (npu flows n he curren me perod plus he avalable nvenory up o hs perod n one sde and demand or oupu flows plus nvenory o be ep n he curren perod from he oher sde) n collecon ceners, VAP ceners, and plans accordngly. Mananng balance of maeral n VAP ceners s more comple han he oher facles due o he possble chemcal reacons. Therefore, by nroducng mulplers ( α, β ), equaon (3) can be modfed based on dfferen scenaros. Consrans (5) and (6) ensure ha maerals flow hrough he acve facles. Capacy consrans for plans, collecon, VAP, and dsposal ceners are lsed from (7) o (0). The ne four ses of consrans are added o he model n order o provde flebly n dfferen real word scenaros. Consrans () and (3) assgn he leas dsposal rae for each collecon cener and VAP cener 5

17 6 accordngly, based on hsorcal daa. Consrans (2) and (4) lms he amoun of reused maeral provded by a collecon cener and oher plans accordngly. Consrans (5) and (6) denfy he doman of he decson varables. (6),...,T,,, 0 (5) J I, {0,}, (4) ) *( (3), * (2) * () *,...,T (0) D, d K, (9),...,T J, K, (8),...,T I, K, (7),...,T P, p K, (6),...,T J, K, (5),...,T I, K, (4),...,T P, p K, (3),...,T J, K, (2),...,T I, K, (),...,T P, p K, I P p I P p P p I d ' ' ' ' ' ' I d P I K V v D, d J I, P, p,,,,,, Y Y T K, P, p T K J, T K, I, T K, I, CAP CAP BOR INV CAP BOR INV CAP BOR INV B Y Y B BOR INV R BOR INV BOR INV BOR INV BOR INV BOR INV S vp d pr, p d p p P r J p I p pr P r pr D d I d P p p P p p p p D d d d J d I d p p p p V v vp R r rp J p p D d d p D d d J p p p p p p V v vp R r rp J p p D d K P p p K D d J P p p p p p R r pr p τ η γ δ β α

18 4. Soluon Approach The developed mahemacal model s a med neger lnear program (MILP), whch belongs o he Facly Layou and Locaon caegory of problems. Smlar problems n he leraure are addressed as Dscree Facly Locaon or Fed Charged Locaon problems and due o her combnaoral naure, hey are denfed as NP-Complee problems (Hller and Leberman, 990; Ignzo and Cavaler, 994; Klose and Drel, 200). Goldberg (989) proposed Genec Algorhm (GA) as one of he effcen mea-heursc approaches o solve combnaoral problems. We appled GA o solve he MILP model presened n he prevous secon. In he developed algorhm, GA s frs used o generae a se of zero-one values. Then, CPLEX 8. s employed o solve he correspondng Lnear Programmng (LP). Fg. 2 demonsraes he seps of hs approach. F Facles(GA) Solve Transporaon and Invenory Problem (LP) Compue he Toal Cos Sop Y Gen<Ma-Gen N Fg. 2: Seps of he algorhm, developed o solve he MILP model wh GA. 7

19 Fg. 3 shows he seps of he GA procedure n deal. The nal se of zero-one values (soluon o bnary varables) s randomly generaed. The sze of hs se depends on he number of bnary varables (collecon ceners and VAP ceners) and ermed as he populaon sze. To oban he maeral flow varables for each se of bnary varables a LP s solved usng an opmzaon commercal sofware CPLEX 8.. Afer solvng he LP sub-problems for he whole populaon, he solved sub-problems are sored based on he obecve funcon n order o deermne whch sream of bnary values has he beer obecve value. In he ne consecuve generaons, he curren se of sored soluons (we refer hem as a paren sream) s used o generae a new populaon se (offsprng se). There are several mehods o form an offsprng se, such as crossover, reproducon, and muaon (Goldberg, 989). Based on an assgned probably (0.8 for crossover, 0. for reproducon, and 0. for muaon) one of hese echnques s seleced. In he crossover procedure, wo members of he parens se are chosen and med o generae wo new zero-one modules n he offsprng se. Three dfferen echnques wh equal chance of occurrng are employed o m he orgnal bnary soluons. They are random, sem-random, and ournamen. In random selecon, wo members of he parens se are randomly chosen and crossover or par-wse echange s made on a randomly seleced dg. The probably of choosng from beer soluons s hgher n sem-random selecon. In ournamen, wo members of he parens se are compared. The one wh he beer obecve funcon value s ep and he oher one s reurned o he se. Usng he same process he oher paren srng s deermned. In reproducon, he bes n answers from he parens se are moved o he offsprng se. For eample, n can be deermned randomly from a poron of he populaon sze. Our procedure 8

20 uses a poron of.0. Ths echnque gves more wegh for good soluons o be nvolved n he ne generaons. To perform he muaon, one of he dgs of he bnary se s randomly changed. Ths echnque prevens rappng n a local opmum soluon (Goldberg, 989). The ermnaon creron n GA s defned as he number of repeons or generaons. The mamum number of generaons s subecve and s based on he sze and srucure of he problem. Modfed Genec Algorhm In order o mprove he qualy of he soluons gven by he GA, we modfy he nal seed soluon. Insead of randomly generang he nal bnary se, he zero-one consrans are relaed and he model s solved. Then, based on predefned rules, zero-one values are assgned o he bnary varables. Dfferen roundng boundares are appled o defne he rules. For eample, 0.90 can be consdered as one of he roundng boundares, whch means ha f he oucome of he model for a bnary varable s equal or greaer han 0.90 he correspondng varable s se o one. Oherwse, s assgned o a value of zero. 9

21 Fg. 3: The Genec Algorhm. Parameers used n he sample problem are as follows: M20, Ma-Gen0, Pm.0, Pc0.20, and PourPrandPsem_ram

22 5. Epermenal resul We esed he proposed soluon approach on randomly generaed problem ses. We compare he performance of he GA agans opmal soluons, whch are found by solvng he med neger lnear model usng a commercally avalable opmzaon sofware pacage, CPLEX 8.. The epermens were run on a Penum IV, 2.8 G sysem wh 52 M RAM. The problem parameers used n he epermens are lsed n Table 3. In he daa ses, one vrgn maeral mare, one dsposal cener, weny plans, en me perods and wo maeral ypes (K and K2) are consdered. We esed dfferen combnaons of collecon and VAP ceners. The smalles run problem sze had 3 collecon ceners and 2 VAP ceners whle he larges had 25 collecon ceners and 25 VAP ceners. λ was se o Therefore, boh pars of he obecve funcon were equally weghed and also no chemcal converson was assumed n he value added process ceners ( α β 0 ). The dsposal and ransporaon (envronmenal and operaonal) cos parameers gven n he Table 3 were obaned from he leraure (Lman, 999; Spengler e al., 997). The res of he parameers lsed n able 3 were developed based on assumpons. For eample, n order o have a dsncenve for bacorders, a hgh cos was assgned for he relaed parameers. Also we assumed ha annual fed cos of havng a VAP cener s $50000, whle for he collecon cener s $ The dsances beween he facles were unform random values beween 5 and 90. I s also assumed ha a leas 8% of all npu maerals o collecon ceners can be dsposed (δ), no more han 5% of he maeral ha eners he collecon ceners are qualfed o be used by he plans (γ), a leas 5% of he maeral n VAP ceners can be dsposed (η), and a mos 5% of he maerals n he plans can be drecly used by he oher plans (τ). 2

23 Table 3: Parameers esmaon Parameer defnon Value Transporaon of wase maeral from Plans o Collecon ceners per mle $.5 Transporaon among all oher facles per mle $.25 Transporaon (Envronmenal) among facles per mle $. Fed cos o buld he collecon cener $00000 Fed cos o buld he VAP cener $50000 Invenory holdng cos per perod for each on (Collecon and VAP ceners) $5 Invenory holdng cos per perod for each on n plans (K,K2) ($5,$8) Bacorder cos for each on n each perod $000 Cos of waer polluon for each on a VAP and Dsposal ceners (K,K2) (0,$5) Cos of ar polluon for each on a VAP and Dsposal ceners (K,K2) ($4, 0) Cos of Energy consumpon for each on a VAP cener $0.5 Processng cos a collecon ceners (K,K2) per on ($,$2) Processng cos a VAP ceners for each on $5 Processng cos a dsposal cener for each on $35 Dsposng cos (ppng fee) for each on $0 Eernal cos of purchasng from Vrgn Maeral Mare per on $200 Plans and VAP ceners capacy n all perods for boh maeral ypes 5000 Collecon cener capacy n all perods (K,K2) (0000, 20000) Fracon of maeral (K,K2) dsposed o ar/waer a value-added process cener (0, 0.05) Fracon of maeral (K,K2) dsposed o waer a dsposal cener d (0, 0.05) Fracon of maeral (K,K2) dsposed o ar a dsposal cener d. (0.0, 0) Demand/Supply of maeral ype K for plan p (p,..20) n all perods 000 Demand/Supply of maeral ype K2 for plan p (p,..20) n all perods 2000 To mplemen he GA, he populaon sze and he oal number of generaons were se o 20 and 0 accordngly. The crossover, reproducon, and muaon probables were se o.80,.0, and.0 accordngly. In crossover hree echnques of random generaon, sem-random generaon, and ournamen were employed wh an equal chance of occurrence. In he sem-random echnque, one of he bnary values s seleced randomly and he bes soluon n he prevous generaon s chosen as he second bnary srng o parcpae n he crossover. The number of bnary soluons ha ransfer o he new generaon n reproducon s randomly chosen from he bes 0% of he populaon sze. For he modfed GA, he roundng boundares ha were used 22

24 o generae he nal soluon se n hs eample were 0.90, 0.80, 0.70, 60, 0.55, 0.50, 0.40, 0.30, 0.20, and 0.0. The resuls are shown n Table 4. The frs column shows he number of collecon and VAP ceners ha were n he problem se. The ne hree columns ls he number of consrans, he number of zero-one varables, and he number of connuous varables. For he CPLEX resuls, we ls he opmal soluon OP and he CPU me n seconds o oban he opmal soluon (T). For he wo GA resuls, we ls he obecve value a he me of ermnaon (OBJ), he CPU me n seconds o oban he soluon (T), and he percenage gap from he opmal soluon (GAP). Table 4: Comparson of Dfferen Soluon Approaches # Coll/VAP # zero-one varables # of Consrans # of Connues Varables CPLEX GA Modfed GA OP T(sec) Ob T Gap% Ob T Gap% 3/ e e e / e e e / e e e / e e e / e e e / e e e / e e e / e e e / e e e / e e e / e e e / e e e

25 The resuls show ha he GA algorhm and especally he modfed GA algorhm are effecve procedures n denfyng near-opmal soluons. The modfed GA was able o fnd he opmal soluon n four of he problem ses and n all problem ses, he soluon gven by he modfed GA were whn 2% of he opmal soluon. Fg. 4 shows a plo of he gap as a funcon of he number of bnary varables. As Fg. 4 shows, he modfed GA performs much beer han he GA. By ncreasng he number of bnary varables he gap beween he GA soluon and he opmum oucome also ncreases, whle hs gap says a he same level for he modfed GA. 6 Gap from Opmum value GA Modfed GA Number of Bnary Varables Fg. 4: Comparson of performance of GA and Modfed GA approaches. Fg. 5 llusraes he eecuon me as a funcon of he number of bnary varables. For he smaller problem szes, he CPLEX eecuon me s smaller han he GA approaches. However, as Fg. 5 demonsraes, an ncrease n he number of bnary varables wll eponenally ncrease 24

26 he eecuon me when CPLEX s used o solve he model opmally. As for he oher wo mehods, he eecuon me changes slowly and whn a small range Tme (sec) CPLEX GA Modfed GA Number of Bnary Varables Fg 5. : Comparson of eecuon me for dfferen mehods. 4. Summary and Concluson In hs sudy, a reverse logscs model wh a concenraon on maeral echange n a B2B newor was developed. Envronmenal ssues as well as operaonal coss were also negraed o he model. Due o he combnaoral naure of he model, he eecuon me o denfy an opmal soluon wll ncrease eponenally. Therefore, Genec Algorhm was employed o solve he model more effcenly for large sze problem nsances. In order o mprove he qualy of he soluons, he radonal GA was modfed and promsng resuls were obaned by applyng a modfed genec algorhm. The resuls showed on a number of es problems he modfed GA found soluons whn 2% of he opmal soluon. 25

27 Acnowledgemens: The wor was suppored by NSF s Maeral Use, Scence, Engneerng and Socey (MUSES) program and he Cener for Susanable Ces (Unversy of Souhern Calforna). We would le o han Rober Vos, Davd Rgby, and Berna Yenc-Ay for her conrbuons on hs MUSES proec. References: Ammons J.C., Realff M.J., and Newon D Reverse Producon Sysem Desgn and Operaon for Carpe Recyclng. Submed for publcaon. Barros A.I., Deer R., and Scholen V A Two-Level Newor for Recyclng Sand: A Case Sudy. European Journal of Operaonal Research, 0: Bro M.P., Flapper S.D.P., and Deer, R Reverse Logscs: A Revew of Case Sudes. Economerc Insue Repor EI Corbe C. and Klendrofer P.R. 200a. Inroducon o he Specal Issue o he Envronmenal Managemen and Operaon (Par : Manufacurng and Eco-logscs). Producon and Operaons Managemen, 0(2). Corbe C. and Klendrofer P.R. 200b. Inroducon o he Specal Issue o he Envronmenal Managemen and Operaon (Par 2: Inegrang Managemen and Envronmenal Managemen Sysems). Producon and Operaons Managemen, 0(3). Dowlashah S Developng a Theory of Reverse Logscs. Inerfaces, 30(3): Fleschmann M., Bloemhof-Ruwaard J. M., Deer R., Van der Laan E., Van Nunen J.A.E.E., and Wassenhove L.V Quanave Models for Reverse Logscs: A Revew. European Journal of Operaonal Research, 03:

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