Research Article A New Extended MILP MRP Approach to Production Planning and Its Application in the Jewelry Industry

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1 Mathematcal roblems n Engneerng Volume 2016, Artcle ID , 18 pages Research Artcle A New Extended MIL MR Approach to roducton lannng and Its Applcaton n the Jewelry Industry Erhan YazJcJ, 1 Gülçn Büyüközkan, 2 and Murat Baskak 1 1 Industral Engneerng Department, Istanbul echncal Unversty, Istanbul, urkey 2 Industral Engneerng Department, Galatasaray Unversty, Istanbul, urkey Correspondence should be addressed to Erhan Yazıcı; erhan.yazc@technoroma.com Receved 8 October 2015; Accepted 18 January 2016 Academc Edtor: Anna M. Gl-Lafuente Copyrght 2016 Erhan Yazıcı et al. hs s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. It s mportant to manage reverse materal flows such as recyclng, reusng, and remanufacturng n a producton envronment. hs paper addresses a producton plannng problem whch nvolves reusng of scrap and recyclng of waste that occur n the varous stages of the producton process and remanufacturng/recyclng of returns n a closed-loop supply chan envronment. An extended materal requrement plannng (MR) s proposed as a mxed nteger lnear programmng (MIL) model whch ncludes besde forward these reverse materal flows. he proposed model s developed for the jewelry ndustry n urkey, whch uses gold as the prmary resource of producton. he am s to manage these reverse materal flows as a part of producton plannng to utlze resources. Consderng the mostly unpredctable nature of reverse materal flows, the proposed model s lkewse transformed nto a fuzzy model to provde a better revew of producton plan for the decson maker. he suggested model s examned through a case study to test the applcablty and effcency. 1. Introducton In terms of producton plannng, MR has a key role whch generates outputs such as producton orders, capacty requrements, and raw and semmateral requrements by usng customer orders, bll of materals (BOM), routngs for capacty requrements as man data, and nventory status records as ntal plannng status [1, 2]. Classcal MR only consders forward materal flows. Many papers that nclude detals about the MR process can be found n the lterature. In addton to classcal MR process, there s also another type that handles reverse materal flows whch s called reverse MR [3] that takes nto account the dsassembly of the products as a separate process alone. Addtonally, another approach takes both forward and reverse movements nto consderaton together at producton plannng whch extends MR approach that s proposed by Grubbström [4] usng Laplace transformaton for manufacturng and remanufacturng of product at the same tme. Grubbström s approach was extended wthn producton recyclng n addton to fnshed product s recyclng by Kovačć and Bogataj [5]. he jewelry ndustry has a dstnctve characterstc due to ts use of precous metals as raw materals. Usng metals as a raw materal s a specfc characterstc needed to be analyzed n the vew of producton plannng as precous metals are recyclable and reusable n producton. Usng precous metals such as gold also has ts fnancal value. Consderng the customer and suppler-based materal movements, jewelry producton can be consdered as a closed-loop supply chan. he extended MR approach s capable of gvng a basc soluton to the producton plannng problem of jewelry ndustry. But t needs to be expanded and extended to consder the specfc producton characterstcs of jewelry. he exstng research on extended MR approach whch s proposed Grubbström [4] and mproved by Kovačć and Bogataj [5] uses an nput-output model by usng matrces to solve the MR process by bascally usng only BOM and predefned ratos, perodcty beng excluded. he cost s not takenntotheconsderatonforreusngorrecyclngdecsons and there s no cost mnmzaton for recyclng, supplyng, or reusng decsons. here should be a tme nterval of the scraps collected for economc recyclng level. Scrap and loss

2 2 Mathematcal roblems n Engneerng ratosofproductonarenotsodetermnstcandhaveto be managed as an uncertanty. Furthermore, there should be a decson mechansm to remanufacture or recycle a returned product. A fnancal deducton s regstered as gold onthesdeofthesuppler,asgoldsusedasacurrencyn fnancal reports. Besdes, there s a perodcty n producton and capacty constrants. hese are the motvatons of ths paper to resolve for the jewelry sector of urkey. It s antcpated that ths paper may open new perspectves for other researchers, other ndustres, and countres seekng gudance for dealng wth ther smlar problems. When the lterature s revewed, there s no smlar research for ths type of problem. Moreover, t s not probable to obtan MR approach for jewelry. he man objectve of ths paper s to reach a soluton for jewelry producton plannng problem n urkey. he producton process nvolves back and forth movements smultaneously. Gold s used as a raw materal and fnancal nstrument at the same tme. Collectons and returns of gold from customers, ts purchasng, and ts recyclng from supplers and everythng n producton actvtes relate to materal movements of gold and have effects on producton plannng. Cost coeffcents are used as ratos of producton amount n the unt of gold. hese are requrements whch seem specfc to the urksh jewelry ndustry. A mathematcal model has been proposed to resolve these problems, whch has been observed to be nonlnear due to the descrpton of some producton processes. Nonlnear mathematcal problems are hard to solve and there s no global optmal soluton. hus, usng a lnearzaton technque, ths problem s defned as a mxed nteger lnear programmng (MIL) mathematcal model, whch s solved to fnd an optmal soluton. In summary, ths example s transformed wth fuzzy coeffcents for reverse materal flows such as recyclng, whch s dffcult to forecast and ncludes uncertanty. hs uncertanty s dscussed n a number of papers on fuzzy MR, especally by Mula et al. [6 11]. hey used fuzzy methods to manage uncertanty n MR for mathematcal modellng. artcularly, Serna et al. proposed usng parametrc lnear programmng for producton plannng [12] and for protectng the lnearty n order to obtan an optmal soluton. her approach s appled to the proposed extended MR model n ths paper to get a more relable decson on producton plannng by handlng uncertantes. hus, the proposed model s transformed nto a fuzzy extended MR model. he rest of the paper s arranged as follows. Relevant lterature about MR wth recyclng and remanufacturng and MR ncludng fuzzy solutons s revewed n Secton 2. In Secton 3, the jewelry producton and the proposed new extended MR MIL model for the producton plannng problem are presented. hen, the proposed model s transformed wth a fuzzy approach whch s frst descrbed n general and then n partcular for the recyclng process n Secton 4. In Secton 5, the proposed determnstc and fuzzy models are solved for a real example n the jewelry ndustry. Fnally, concludng remarks and drectons for further research are provded n Secton Lterature Revew Currently, there are a bunch of studes about producton plannng and MR n the lterature. he lterature was surveyed by consderng the characterstcs of the problem, producton plannng wth recyclng, reusng, and remanufacturng. he concept of fuzzy producton plannng s also revewed due to handlng uncertanty of recyclng n producton plannng roducton lannng wth Recyclng and Remanufacturng. here are varous researches on producton plannng and MR n the lterature. here are also studes proposng mathematcal models for MR. Regardng cost mnmzaton n a capacty and resource constraned envronment, some authors reported mathematcal programmng models for MR [6, 13]. Bllngton et al. [13] proposed general mathematcal modellng for MR and presented an extensve revew. Mula et al. [6] modfed ths model and presented a determnstc MR model for optmzaton of producton plannng. Besdes general MR concept, recyclng and remanufacturng n producton plannng ssues are also revewed n the lterature. here have been a number of lterature surveys on remanufacturng and producton plannng. Junor and Flho [14] publshed a lterature revew about producton plannng and control for remanufacturng. hey tred to defne the complexty of remanufacturng and gave a perspectve to researchers on that ssue. Morgan and Gagnon [15] also reported a well-classfed lterature survey about remanufacturng schedulng. hey suggested that reverse MR can be a soluton for the problem related to nfnte capacty and sngle product remanufacturng schedulng problem. Lately, MR models tend to seek solutons usng cost-based nteger programmng models. Omar and Yeo [16] proposed a model for nventory system that consdered both producton and repar of products wth tme varyng demand and multple setups. hey modelled new tems and used tems producton and repar runs at each tme nterval. L et al. [17] studed the dynamc lot szng problem wth product returns and remanufacturng. hey suggested an algorthm to fulfll customer demand for products and mnmze the cost at a plannng perod by consderng demand and returns for manufacturng and remanufacturng decsons. Km et al. [18] developed a hybrd model to coordnate manufacturng and remanufacturng and effectve dsposal of new products. hey made use of a Markov decson process to nvestgate the optmal polcy. Deuy et al. [19] ntroduced a producton plannng method ncludng remanufacturng. hey descrbed an MR approach by dscussng a case study, whch ncluded probabltes about returns and remanufacturng tmes. Coromnas et al. [20] generated a jont aggregate plannng of a system for manufacturng and remanufacturng. Frst, they proposed a nonlnear mathematcal model wth some breakdown functon defntons and then used a pecewse lnearzaton to transform t nto a lnear model. We et al. [21] consdered an nventory and producton plannng problem about remanufacturng of returns at uncertan demand and n a fnte plannng horzon. Robust optmzaton approach

3 Mathematcal roblems n Engneerng 3 wth a robust lnear programmng model was employed to handle the uncertanty of demand and yelds. AccordngtoGudeJr.etal.[22],recyclngsamore valuableoperatonthanremanufacturngandrequresamore complex approach than tradtonal manufacturng. hey amed to defne dfferent and more complex structures n a lterature survey and ponted out future perspectves for researchers nterested n that ssue. Addtonally, they suggested that there was no MR research about materal recyclng and recycle rato uncertanty. Waste management and Green MR are another way of recyclng at producton phase. Melnyk et al. [23] offered a Green MR approach consderng waste management wthn producton plannng and appled ths approach on the Amercan automotve ndustry. Mrzapour Al-e-hashem et al. [24] bult up a stochastc programmng model for multpont, multproduct, multplant producton plannng problem n case of uncertan demand. hey ncluded waste management n ther model and used numerc technques for the lnearzaton of nonlnear breakdown functons. Modellng of reverse materal flows such as recyclng, reusng, and remanufacturng n productonplannnggenerallyendsupwthnonlnearmathematcal models due to processes that nvolve f-then-else structures. Coromnas et al. [20] also used a pecewse lnearzaton technque to convert nonlnear models nto lnear. When MR lterature wth recyclng and remanufacturng processed s revewed, reverse MR and extended MR approaches are observed. Grubbström [4] and Kovačć and Bogataj [5] and Barba-Gutérrez et al. [3, 25] have publshed several studes on these ssues. Kovačć and Bogataj [5] proposed an nput-output model for materal requrement plannng, consderng both the fnshed product and work n process materals recyclng. hs publcaton s based on Grubbström s [4] work whch ncluded just the recyclng of the fnshed product. Barba-Gutérrez and Adenso-Díaz [3] revewed the MR concept n reverse logstcs. Jewelryproductonsmostlydoneasmaketoorder producton, due to ts use of valuable raw materals, such as gold, whch s a frequent bottleneck n jewelry producton. Durng the producton plannng process, gold always should be at the core, turnng the producton plannng of jewelry nto a gold-centrc producton plannng. Everythng about gold s mportant wth all ts detals. When the lterature s analyzed, no paper on ths knd of producton plannng s observed, nether for jewelry nor for any smlar ndustry. As an outler, Süer et al. [26] have a few works about the producton, ncludng cases of jewelry, however not as an MR ssue. As a smlar ndustry, alumnum manufacturng, whch uses recyclable materals s dscussed n a study by Davd et al. [27]. hey demonstrated the benefts of enterprse resource plannng (ER) and MR systems usefulness n alumnum converson ndustry. Includng customer and suppler-based materal movements n the producton plannng process, jewelry producton s consdered as a closed-loop supply chan producton, as t controls materal movements of gold n every stage. Inderfurth et al. [28] reported a research about product recovery systems n a closed-loop supply chan, presented them n producton plannng, and demonstrated on a case study. Another extensve study by He [29] offered a closedloop supply chan model wth recyclng, reusng, and remanufacturng decsons for cost mnmzaton. Despte all these researches n related areas, there s not anyresearchonmrwthrecyclngandremanufacturng n the lterature. artcularly for jewelry ndustry, the MR model s not researched by ncludng aspects lke recyclng, remanufacturng, supplyng, collectng, and returns. In ths paper, recyclng and remanufacturng are studed n a mathematcal model for MR and returned products and work n process materals s consdered as recycle resource. Addtonally, remanufacturng and recyclng decson s ncluded n the proposed model roducton lannng and Fuzzness. In spte of determnstc characterstc of most studes, there are many uncertantes n real cases as a part of producton plannng process. Dealng wth problems that contan uncertantes, fuzzy sets, and fuzzy logc, as proposed by Zadeh [30], s of great mportance for researchers. Fuzzy approaches are more suted for resolvng complex problems wth uncertanty by defnng human choces and thoughts on problems. he proposed model s therefore transformed nto a fuzzy model to provde a better vew of producton plannng for the decson maker. hs transformaton s carred out to deal wth the mostly unpredctable nature of reverse materal flows such as recyclng, reusng, and remanufacturng. Lterature revew papers about producton plannng and uncertanty are as follows: Guffrda and Nag [31] provded a survey of the applcaton of the fuzzy set theory n producton management research. hs revew conssted of 73 journal artcles and nne books and classfed fuzzy applcatons n producton management research. Mula et al. [7] revewed the lterature about producton plannng under uncertanty and proposed a determnstc MR for the optmzaton of producton plannng. Dolgu and rodhon [32] constructed a detaled lterature revew on supply plannng under uncertantes n an MR envronment. In that respect, there are varous studes about MR wth uncertantes. Fuzzy MR problems are mostly examned by Mula et al. [9 11] and Mula et al. [6 11]. Mula et al. [6] proposed a fuzzy mathematcal programmng model for producton plannng under uncertanty. hs model ncludes fuzzy constrants and fuzzy coeffcents. Grabot et al. [33] suggested a gudance to explctly model the uncertanty and mprecson of the demand, allowng to pass through all the MR steps. Mula et al. [9] provded a new lnear programmng model for medum term producton plannng n a capacty constraned MR. hs ncluded a multproduct, multlevel, and multperod manufacturng envronment wth three fuzzy models wth flexblty n the objectve functon, n the market demand, and n the avalable capacty of resources. Fgueroa-García et al. [34] presented a general modelofamxedproductonplannngproblemwthfuzzy demand. Fuzzy lnear programmng model (soft constrant model) has been used wth an nterval fuzzy set approach to defne uncertanty. edro et al. [10] tred to prove the effcency of a fuzzy mathematcal programmng approach

4 4 Mathematcal roblems n Engneerng to model a supply chan producton plannng problem wth uncertanty n demand. Addtonally, there are fuzzy producton plannng problems whch occur n a supply chan. For example, Blgen [35] presented a soluton approach usng fuzzy operators for producton allocaton and dstrbuton problems n a supply chan network. Lu et al. [36] proposed a novel fuzzy multobjectve mxed nteger lnear programmng model to multproduct multstage ntegrated producton plannng problem. Weghted average method and fuzzy rankng method are used for defuzzfcaton of fuzzy constrants at frst phase. hen, an nteractve resoluton method s used wth satsfacton degree of objectves consderng decson makers preference. In addton to tradtonal producton plannng, some authors used a fuzzy approach wth reverse movement. shvaeeandorab[37]proposedabobjectvepossblstc mxed nteger programmng whch ntegrates the network desgn decsons n both forward and reverse supply chan networks. An nteractve fuzzy soluton approach, whch was bascallyaresolutonmethod,wasdevelopedtosolveproblems nteractvely wth the decson maker. Barba-Gutérrez et al. [25] presented an MR algorthm for schedulng the dsassembly of dscrete parts characterzed by a well-defned product structure n an uncertan envronment usng reverse materal requrements. Olugu and Wong [38] studed a fuzzyrule based performance evaluaton system for closed-loop supply chan, whch conssted of reverse movements. Addng fuzzy constrans or coeffcents to mathematcal models generally makes models nonlnear. Mostly, α-cuts fuzzy parametrc lnear programmng approach s utlzed to keep mathematcal model lnear, consderng the dffculty of solvng nonlnear mathematcal problems. hose papers that use α-cuts are as follows: Serna et al. [12] amed at provdng a materals requrement plannng (MR) problem wth uncertanty n the automotve ndustry. hey solved the problem by usng fuzzy parametrc lnear programmng. arra et al. [39] proposed a method for solvng multobjectve possblstc problems through a fuzzy compromse programmng approach. her soluton concept was founded on soft preference and ndfference relatonshps and on the canoncal representaton of fuzzy numbers by means of α-cuts. Besde lnear mathematcal modellng solutons, some papers used algorthms to solve the fuzzy producton plannng problem. L et al. [17] showed a fuzzy programmng model wth recourse based on credblty theory whch ncludes fuzzy varable coeffcents related to the market demand andtheuntcostofthefabrc.heydesgnedahybrd algorthm whch combnes an approxmaton approach (AA) and partcle swarm optmzaton (SO). Lan et al. [40] studed a class of multperod producton plannng and sourcng problem wth credblty servce levels and desgned an algorthm whch s a combnaton of approxmaton approach, SO, and neural networks. Srvastava and Nema [41] proposed a fuzzy parametrc programmng model for a multobjectve recyclng problem under uncertanty. Zhang et al. [42] also formulated a generalzed producton plannng problem under uncertanty wth fuzzy ntervals usng α-cuts fuzzy parametrc lnear programmng and possblty degrees of decson makers. Chen and Huang [43] and Madad and Wong [44] used α-cuts fuzzy parametrc lnear programmng on aggregate producton plannng problem. On that pont there are also other papers handlng uncertantes wth lnear results. orab et al. [45] dealt wth a herarchcal producton plannng and schedulng problem whch contans uncertantes and used an effectve method to defne fuzzy objectve functon. hey made use of the weghted average method for defuzzfcaton; then the problem s solved as lnear model. edro et al. [10] proposed a fuzzy mathematcal programmng model for supply chan plannng whch consders supply, demand, and process uncertantes. he model was formulated as a fuzzy mxed nteger lnear programmng model where data are llknown and modelled by trangular fuzzy numbers. Kundu et al. [46] proposed a new parametrc lnear programmng method for type-2 fuzzy ntervals at dfferent α-cuts levels tosolvefxedchargetransportatonproblem.srnvasanand Geetharaman [47] used α-cuts levels wth degree of satsfacton of the constrants where resources and technology coeffcents are defned as type-2 fuzzy ntervals. Yager [48] used a rankng functon and α-cuts to solve problems keepng them n lnear form. Klr and Yuan [49] and Kacprzyk and Orlovsk [50] proposedα-cuts and an acceptance level-based soft constrants approach as a framework to solve problems whch contan fuzzy parameters. anaka et al. [51] also used parametrc solutons for fuzzy lnear models. Lang [52] and Lang and Cheng [53] used a weghted average method for defuzzfcaton to solve a fuzzy MOL problem. Wang and Lang[54]alsontroducedaweghtedaveragemethod. In ths paper, the presented extended MR approach s transformed wth fuzzy coeffcents nto a fuzzy extended MR approach. hs paper extends the lterature by proposng a new approach for extended MR on a jewelry case from urkey whch ncludes recyclng, reusng, and remanufacturng decsons as a part of producton plannng processes. Addtonally, due to the unpredctable nature of these reverse materal movements, the proposed approach s also transformed nto the fuzzy MR applcaton to handle uncertantes over ths knd of process n producton plannng, contrbutng to the lterature. 3. roblem Descrpton and Formulaton hsresearch smanmotvatonstheproductonplannng problemencounteredbyagoldjewelrycompanynurkey, whch has to manage recyclng, reusng, and remanufacturng n coordnaton wth the producton process. hs cost mnmzaton producton plannng problem contans multple products, multple plants, and capacty constrants of more than one perod producton envronment, regardng only one bottleneck raw materal, gold, and recyclng, reusng, and remanufacturng of raw materal gold and products contanng gold as raw materal. he problem nvolves materal flows both for n process and out of process such as supply, demand, return, and collecton of gold. Jewelry producton can be consdered as a closed-loop supply chan producton as materal movements are both customer- and suppler-based so that materal movements of

5 Mathematcal roblems n Engneerng 5 roducton Suppler 1 Raw materal suppler balance MR Manufacturng and remanufacturng 6 Customer 3 Collecton as raw materal Demand 2 Recycle at suppler Waste Raw materal Raw materal Reuse n process 4 Scrap 5 Returned product Fnshed product Raw materal as gold Raw materal as gold used from suppler as lablty Returned product made by gold from customer Collecton as raw materal gold from customer Fnshed product made by gold at customer Scrap gold n process reusable (physcal recycle process needed for reuse) Waste gold sent to suppler for recycle (chemcal recycle process needed for reuse) Fgure 1: Jewelry producton envronment and materal flows. gold can be controlled n every stage of producton plannng. he problem s detals and assumptons are gven below for ths jewelry closed-loop supply chan. In roducton Related () Each product can be manufactured n dfferent plants. Capacty s needed n dfferent plants to produce the defned products. If capacty s not avalable n a plant, a penalty cost s determned. () All cost estmates are computed as the producton of grams of gold. Gold tself s not regarded as a cost of producton despte ts raw materal stuaton. () Only gold alone s consdered as a raw materal n the plannng due to the bottleneck stuaton. Other raw materals lke alloys and zrcon stones are not consdered as a part of plannng as they are managed wth safety stocks. If requred, they can be easly appended to the model. (v) Scrap and waste come from the raw materal (gold) flow used durng the producton. (v) Scrap s collected from the producton as reusable peces. Waste occurs n the producton process as granulated from and gathered wth specal methods. (v) Scrap s reused by collectng peces physcally and then just meltng and reusng agan n producton. (v) Waste recyclng conssts of collecton of gold dust wth specal flters from the ar, water, and surfaces, applyng a chemcal process obtaned from a specalst suppler and recyclng as pure gold, whch ncurs a recyclng cost. (v) he economc recovery amount of the accumulated waste for recyclng s requred. (x) Scrap reusng s lmted and can be reprocessed n a certan cycle tme due to ts effects on the end product qualty. At the end of ths reusng cycle tme, scrap must be commtted to recyclng. It s rather dffcult to separate that porton of the gold used n output. Gold resource planners follow up wth the scraps collected from producton unts and send them for recyclng accordng to the qualty of gold. After a suffcent perod of tme, t s assumed that the used gold s sent to recyclng by consderng the parameter reusable perod. Suppler Concerned () Gold supply from supplers s restrcted wth the debt balance lmt, snce gold s a valuable materal whch atthesametmecanbeusedasafnancalnstrument. hspresentsabalancerskwhentsgventothe customer as debt. () he chemcal recyclng process s performed by the suppler as t requres specalzaton, economc sze, and a hgh ntal nvestment. Customer Related () roducton s made by demand of customers. ()heresacostforthebackloggeddemandnproducton plannng duraton. Backlogged demand s not desrable at the end of the plannng horzon. () Returns from customers are decded to be sent for recyclng or remanufacturng. (v) ayments are collected from customers as raw materal gold. hat s an nterestng feature of the urksh jewelry ndustry, whch needs to be consdered as part of the producton plannng. All the above provded detals of jewelry producton are llustrated n Fgure 1, showng producton, recyclng, reusng, and remanufacturng processes, together wth materal flows. Descrptons of the cons are presented n the legend of Fgure 1. In summary, the materal flows, especally those that arenumberedareexplanedasfollows: () Number 1 ndcates the supply of gold from the vendor as pure gold.

6 6 Mathematcal roblems n Engneerng () Number 2 represents recycle of waste whch ncludes gold. () Number 3 dsplays collecton from customers as gold. (v) Number 4 refers to the reuse of scrap gold, whch occurs n producton. (v) Numbers 5 and 6 are about returned product s reuse, recycle, or remanufacturng decson. Number 6 stands for the decson of remanufacturng and number 5 shows the reusng decson. In the next secton, the problem s formulated wth mathematcal modellng roblem Formulaton. For ths producton plannng problem, a MIL MR model s developed for the gold jewelry ndustry n urkey wth recyclng, reusng, and remanufacturng. he proposed model consders reverse materal flows both n processes, such as recyclng, reusng, and remanufacturng, and out of processes, such as supplerbased recyclng, returns, and collectons from customers as raw materal gold. Man outputs of the proposed model are the man producton schedule consstng of the products to produce and raw materal needed; product and raw materal stock quantty end of every perod; amount of demand met; capacty usages; recycle, scrap, and waste quanttes or raw materal; returned product recyclng and remanufacturng quanttes; and the raw materal supply amount for each perod. he man ams of the model are mnmzng the total producton cost; plannng of scrap and waste recyclng; plannng return remanufacturng and recyclng; plannng of raw materal supply (manly gold supply n ths model); mnmzng demand backlog; mnmzng fnshed product and raw materal stock level; effectve resources usage; consderng scrap, waste and return recyclng and collectons as raw materal gold to mnmze supply cost and consderng returned product remanufacturng to mnmze total producton cost. Constrants of the model are the stock, producton, and demand balance constrants for the fnshed product; raw materal stock and supply constrants; operatonal resource capacty constrants and nonnegatvty, bnary, and nteger subjectons for the decson varables. he proposed MIL model s sets, decson varables, cost coeffcents as parameters, and needed ntal data defntons used to formulate are descrbed as follows Notaton Sets : set of plannng perods (t=1,2,...,). I: set of raw materals ( =1)(I=1s gold as raw materal). :setofproducts(,2,...,). J: set of producton resources/plants (j=1,2,...,j). arameters Data cp p :varablecostofproductonofauntofthe product p. cr p :varablecostofreprocessofauntoftheproduct p. cc p :varablecostofrecycleofauntoftheproduct p. c p :nventorycostofauntoftheproduct p. crr :recyclecostofauntoftherawmateral. cru : reuse cost of a unt of the raw materal. crl : recycle-lost cost of a unt of the raw materal. cobl : over balance cost of supplers for the raw materal. cdb p :backloggeddemandcostofauntoftheproduct p. cuc j :unusedcapactyuntcostoftheresource j. coc j : overtme capacty unt cost of the resource j. d pt : market demand of the product p nperod t. g p : requred quantty of the raw materal to produce a unt of the product p, =1as gold so t can be notated as g p. y p : recyclable quantty waste of the raw materal comes from producton of a unt of the product p. s p :reusablequanttyscrapoftherawmateral comes from producton of a unt of the product p. t p : requred producton perod of the product p. Rcl : recyclable quantty lmt of the raw materal. Rup : reuse perod of the raw materal. Bls : suppler balance lmt of the raw materal. Rp pt : returned quantty of the product p nperod t. Col t :collectedquanttyoftherawmateral n perod t. c pj :requredcapactyuntoftheresource j forunt of producton of the product p. cu pj : requred capacty unt of the resource j for unt of reprocess of the product p. Rac jt :avalablecapactyuntoftheresource j n perod t. Inv p0 :nventoryoftheproduct p nperod0. Inv 0 : nventory of the raw materal nperod0. Rcyp 0 :recyclablenventoryoftherawmateral n perod 0. Reuc 0 : reusable nventory of the raw materal n perod 0. Bd p0 : backlogged demand of the product p n perod 0.

7 Mathematcal roblems n Engneerng 7 Bs 0 : suppler balance of the raw materal nperod t. M: suffcently hgh number, used for lnearzaton. Decson Varables rd pt :quanttytoproductonoftheproduct p n perod t. Inv pt :nventoryoftheproduct p attheendof perod t. Inv t : nventory of the raw materal attheendof perod t. Bd pt :demandbacklogoftheproduct p attheend of perod t. Ucr jt : unused capacty of the resource j nperod t. Ocr jt : overtme capacty of the resource j nperod t. Sm t : supply quantty of the raw materal nperod t. Ruc t :reusedandrecycledquanttyoftherawmateral nperod t. Rcy t : recycle quantty of the raw materal n perod t. Reu t :reusequanttyoftherawmateral nperod t. Reuc t :nreusngcycle,totalquanttyoftheraw materal nperod t. Rcyp t : n pendng recycle, total quantty of the raw materal nperod t. Bs t : balance of supplers for the raw materal n perod t. Bols t : over balance lmt of supplers for the raw materal nperod t. Rprt pt : quantty to remanufacture of the returned product p nperod t. Rcrt pt : quantty to send reusng process of the returned product p nperod t. Rt pt : quantty returned of the product p attheend of perod t. rd pjt : quantty to manufacture of the product p n perod t onresource j. Rprt pjt : quantty to remanufacture of the returned product p nperod t onresource j. r t : recycle process exstence (1 or 0) of the raw materal nperod t, used for lnearzaton. o t : over balance lmt of supply exstence (1 or 0) of the raw materal nperod t, used for lnearzaton. u jt : undercapacty use exstence (1 or 0) of the resource j nperod t, used for lnearzaton. Due to I beng defned only for gold as raw materal, t can be used for all notatons wthout ndces. he model s especally defned wth ndces to be extensble for future researches n smlar ndustres whch need to handle more than one recyclable materal Objectve Functon. Consder Mnmze (1) (cp u rd pt + c u Inv pt + cdb u Bd pt ) g p t=1 J + (cuc j Ucr jt + coc j Ocr jt ) j=1 t=1 I + ((crr + crl ) Rcy t + cru Reu,t Rup 1) + =1 t=1 I (cc p Rcrt pt + cr p Rprt pt + c p Rt pt ) g p t=1 + cobl Bols t. =1 t=1 (1a) (1b) (1c) (1d) (1e) Equaton (1) defnes the total producton cost to be mnmzed and contans 5 dfferent terms. erm (1a) and term (1b) represent the order, producton, nventory, and resource usage cost. In addton, when the supply of raw materals, recyclng, and remanufacturng costs s added to the model, reverse movements wll be ntegrated to t, as desred. erm

8 8 Mathematcal roblems n Engneerng (1c) shows the recyclng and reusng cost of the raw materal. erm (1d) shows the return, recyclng, and remanufacturng cost of products. erm (1e) represents the supply cost and enforces balance between supply and reusng, recyclng, and remanufacturng takng nto account supply cost. erm (1a) and term (1d) contan a product by g p because, as mentoned before, all cost coeffcents are used as ratos of producton amount n gold unts. In term (1a), term(1d), and term(1e), the decson varables are already determned as the amount of gold. he objectve s the mnmzaton of the total of these costs. In addton to producton, nventory, and capacty utlzaton, the costs of recyclng, reusng, lost and remanufacturng costs, and raw materal procurement costs are ncluded. Constrants assocated wth the proposed model are descrbed as follows, respectvely Model Constrants Inventory, roducton, and Demand Balance Constrants for roduct. Consder Inv pt = Inv p,t 1 + rd p,t tp d pt Bd p,t 1 + Bd pt + Rprt pt p, t, (2) Rt pt = Rt p,t 1 + Rp pt Rcrt pt Rprt pt p, t, (3) Reu t = U s p rd pt, t, (7) Reuc t = Reuc,t 1 + Reu t Reu,t Ruph 1, t, (8) Rcyp t = y p rd pt + Rcyp,t 1 Rcy t, t, (9) Rcy t = Rcyp,t 1, Rcyp,t 1 Rcl 0, Rcyp,t 1 < Rcl, t. (10) Inventory balance constrants for raw materals nclude the supply, reusng, and recyclng costs. Equaton (5) shows the nventory balance for raw materals. Equatons (6) (10) show the recyclng process. Recyclng constrant Equaton (6) Ruc t can drectly be substtuted nto (5). Equaton (7) collects the reusable quantty of producton. Wth (8), reusng process s managed by regardng the reuse perod lmt. Equaton (9) represents the accumulated amount of gold for the recyclng process. Rcy t n (10) enforces the economcally recyclable quantty to accumulate. If t reaches the amount defned as Rcl, t can be sent to recyclng. Here, by defnton, Rcy t ncludes a nonlnear expresson. In order to convert (10) from nonlnear to lnear, the lnearzaton of Rcy t s carred out as follows: Rcl Rcyp,t 1 M (1 r t ),t, ( (rd p,t tp +Rprt pt ) d pt )=0. (4) t=1 t=1 Rcyp,t 1 Rcl M r t, t, Rcyp,t 1 Rcy t M (1 r t ),t, (11) Equaton (2) corresponds to the nventory balance for products. he nventory of product at the end of the perod plannng horzon wll be equal to the sum of stock from the prevous perod, exstng perod producton, backlogged demand for the next perod, and remanufacturng of exstng perod, where the prevously backlogged demand and the demand of the exstng perod are to be subtracted. Equaton (3) manages product return s recyclng, reusng, and remanufacturng operatons of products. It bears on the decson of returnng product recyclng, keepng n stock, or remanufacturng decsons. Equaton (4) establshes the demand and producton balance. Inventory Balance Constrants for Raw Materals. Consder Inv t = Inv,t 1 + Sm t + Ruc t (g p +y p +s p ) rd pt Reuc,t 1, t, (5) Ruc t =(1 crl 1000 ) Rcy t + Reuc t, t, (6) Rcy t M r t Rcy t Rcyp,t 1, t,, t. When Rcy t, the recyclable waste amount, approaches Rcl, the economc recyclable quantty lmt, and recyclng can be performed. r t represents the exstence of a recyclng processthatshouldbemanaged.equaton(11)controlsthe recyclng process wth economc sze. he frst two terms represent the exstence of the recyclng procedure because of the accumulated amount that reached the economc recyclng lmt, Rcl. he thrd term ensures that the exstng collected waste amount s reset once the recyclng operatons are completed. he fourth term enforces the recyclable amount to zero when t does not reach the economc sze. Eventually, the last term guarantees that the recyclng amount s equal to the accumulated amount, wth coordnaton of the second condton. Supply Balance Constrants of Raw Materal. Consder Bs t = Bs,t 1 Reu,t Rup 1 Col t + Sm t, t, (Rcrt pt g p ) (12)

9 Mathematcal roblems n Engneerng 9 Bs t Bls, t, (13) Bols t = Bs t Bs,t 1, Bs t Bs,t 1 >0 0, Bs t Bs,t 1 0, t. (14) Equatons (12) (14) show supply balance constrants for raw materals. Equaton (11) defnes the supply balance and the Reu,t Rup 1 varable s added as representng raw materal should be sent to suppler after Rup perod reused for recyclng and/or as a payment to the suppler. hus, t decreases the balance of the suppler. he Rcrt pt varable also represents the recycled returned products to raw materal. he Col t varable represents the raw materal gold that s collected as payment from customers, ncreasng ts exstng nventory. Hence, there wll be no need of debt from suppler. Equaton (13) restrcts the suppler balance lmt to Bls. Equaton (14) shows the ncrease of debt balance of gold at the suppler. Increasng of debt means the use of suppler s gold. At that pont, there should be a prce for that. Besdes, ths s specfed n the objectve functon n (1) s term (1e). Here, by defnton, Bols t ncludes a nonlnear expresson. In order to convert (14) from nonlnear to lnear, lnearzaton equatons of Bols t are presented as follows: Bs t Bs I,t 1 M o t I, t, Bs,t 1 Bs t M (1 o t ) Bs,t 1 Bs t + Bols t M (1 o t ) Bols t M o t Bols t Bs,t 1 Bs t, t,, t.,t,,t, (15) IftheresnotanyncreasenBs t Bs,t 1 on the debt balance of gold at the suppler sde, there should be no dfference n the frst term. he second term guarantees the vce versa stuaton of the frst term. he thrd term calculates the ncrease amount, Bols t, at debt balance wth coordnaton of last term. he fourth term ensures that t s zero f there s no ncrease n the balance. Resource Capacty Constrants for lants. Consder (c pj rd pjt + cu pj Rprt pjt ) Ocr jt + Ucr jt J = Rac jt j, t, (16) rd pjt = rd pt p, t, (17) j=1 J Rprt pjt = Rprt pt p, t, (18) j=1 Ocr jt = Ocr jt, Ucr jt =0 0, Ucr jt >0 Ucr jt = Ucr jt, Ocr jt =0 0, Ocr jt >0 j, t, j, t. (19) (20) Equatons (16) (20) show the resource capacty balance for producton plants. In addton to normal manufacturng tmes, remanufacturng tmes n (16) manage the capacty usage of plants. Equatons (17) and (18) sum up all plants manufacturng and remanufacturng amounts of products to the total producton amount. Equatons (19) and (20) dentfy the capacty usage of producton plants, whether they are operatng at over- or undercapacty. Here, by defnton, Ocr jt and Ucr jt nclude a thought whch s expressed n nonlnear form. In order to convert (19) and (20) from nonlnear to lnear, the lnearzaton equatons of Ocr jt and Ucr jt used n (19) and (20) are formulated as follows: Ocr jt M (1 u jt ) j,t, (21) Ucr jt M u jt j, t. Both (19) and (20) are managed wth one bnary parameter, because whenever one of them exsts, the other one wll not. Nonnegatvty, Bnary, and Integer Constrants. Consder rd pt, Inv pt, Bd pt, Rprt pt, Rcrt pt, rd pjt, Rprt pjt, Rt pt, Ucr jt, Ocr jt 0, Inv t, Ruc t, Rcy t, Reu t, Reuc t, Rcyp t 0, Sm t, Bs t, Bols t 0, Bd p =0, rd pt, Inv pt, Bd pt, Rprt pt, Rcrt pt, rd pjt, Rprt pjt, Rt pt Ζ, r t, o t, u jt 0, 1},,j,p,t. (22) Fnally, (22) guarantees the nonnegatvty, nteger, and bnary constrants of decson varables. 4. Fuzzy Approach for Recycle rocess In addton to the determnstc model whch s presented n the precedng secton, the proposed approach also transformed the model nto a fuzzy MIL model for MR to

10 10 Mathematcal roblems n Engneerng handle uncertantes, so that the unpredctable nature of reverse materal movements lke recycle process effcency n producton plannng can be taken nto account. he proposed determnstc model certanly defnes all aspects of the problem. However, the effectveness of the adopted recyclng process n the proposed model s dffcult to predct determnstcally. Besde that uncertanty, also as our lterature survey suggests, there exst other uncertantes based on demand, supply, process, and envronment. he scope of ths paper s lmted to recyclng uncertanty only, because of the hgh fnancal value of gold as the raw materal n the jewelry ndustry. Although recyclng s not a hundred percent effcent process, the amount of loss remans uncertan. he proposed MIL approach s modeled nonlnearly by defnton at frst hand. herefore, there s no optmal soluton and the model s dffcult to solve. o manage ths challenge, bnary lnearzaton s used to transform t nto a MIL model, meanng that addng fuzzy defntons to the modelalsowllchangemodeltononlnear.odealwth ths stuaton, the proposed MIL s transformed accordng to lnear programmng fuzzy solutons. As the fuzzy set approach to handle uncertanty, parametrc lnear programmng s used wth α-cuts and an nteractve resoluton method wth decson maker whch s proposed by Jménez etal.[55],edroetal.[10],andmulaetal.[6].forthe fuzzy nterval approach, the parametrc lnear programmng method proposed by Kundu et al. [46] s used ype-1 Fuzzy (Fuzzy Set) Approach. In the proposed MIL model, the recyclng process effcency s defned as uncertan accordng to (6) and also (1), because t s cost coeffcent at the same tme. crl s the recycle-lost cost of a unt of the raw materal, only gold n ths case. In the jewelry ndustry, unt cost measure s determned as a porton of gold weght. It s set as a percentage of recyclable gold collected as waste n producton. crl, the recycle-lost cost as rato of gold weght, s used n the objectve functon as a cost coeffcent and n a constrant to calculate recycled quantty. hat s because of the nature of gold, whch s both a raw materal and a fnancal nstrument at the same tme. crl recycle-lost cost s presented as a trangular fuzzy number (FN): crl =(crl a, crl b, crl c ). he membershp functon to descrbe a fuzzy set s proposed n Gen et al. [56]: μ crl (x) =μ crl (x : crl a, crl b, crl c ) x crl a, crl b crl a f crl a x crl b = crl c x, crl c crl b f crl b x crl c 0, f x>crl c or x<crl a. (23) he nteractve resoluton method used here contans α-cuts whch represents the acceptable feasblty degree of the fuzzy coeffcent. α 0 stands for the mnmum coeffcent feasblty degree that the decson maker s wllng to accept. hen, the feasblty nterval of α s α 0 α 1.Adscrete scale proposed by Jménez et al. [55] s appled wth 0.1 nterval steps, whch ranges from unacceptable to completely acceptable soluton. Accordngly, the correspondng ordnary lnear program for each α at ntervals of 0.1 cuts are solved. he α-parametrc lnear programmng transformaton of fuzzy model s carred out by applyng the proposed method by Jménez et al. [10, 55] wth (24) as an equvalent of (1) and (6) whch conssts of recycle-lost cost. Consder the followng: (cp p rd pt + c p Inv pt + cdb p Bd pt ) g p + (cuc jt Ucr jt + coc j Ocr jt ) t=1 J j=1 t=1 I + + =1 t=1 ((crr +((1 α) (crl b + crl c ) 2 (cc p Rcrt pt + cr p Rprt pt + c p Rt pt ) g p + cobl Bols t, t=1 +α (crl a + crl b ) )) Rcy 2 t + cru Reu,t Rup 1 ) I =1 t=1 (24) Ruc t (1 ((1 α) ((crl b + crl c )/2)+α ((crl a + crl b )/2)) ) Rcy 1000 t + Reuc t, t. After solvng the α-parametrc lnear programs for acceptable feasblty degrees of the fuzzy coeffcent, the decson maker specfes satsfacton degrees of each soluton of the parametrc lnear program wth α-cuts. Usng these satsfacton degrees, the decson maker obtans an nsght about dfferent α acceptable feasblty degrees mpact on the total cost of producton and wll make hs/her plannng accordngly, regardng that uncertanty s effect on the producton plan ype-2 Fuzzy (Fuzzy Interval) Approach. Usng the fuzzy set approach, the uncertanty of the recyclng process s assumed to only have one dmenson to consder n the proposed model. hs type of uncertanty s descrbed as type-1 uncertanty. On the other hand, accordng to the sze of uncertanty dmenson more dmensons of uncertanty can be taken nto account when examnng t, called type-n uncertanty by Zadeh [57, 58]. For recyclng process losses,

11 Mathematcal roblems n Engneerng 11 uncertanty can vary by plants and dfferent product groups n producton plannng term. If the decson maker wants to consder these stuatons together, then recyclng losses can be defned as a type-2 [57, 58] uncertanty. In the precedng fuzzy set approach, recyclng cost s defned as a trangular fuzzy number. However, n fuzzy nterval approach, t s defned as a trangular fuzzy area. he type-2 fuzzy nterval of crl can be represented as crl = (crl 1, crl2, crl3 ;θ l,,θ r, ) type-2 FN [46]. Here, crl 1,crl2,andcrl3 are real numbers and represent type-1 uncertantes. θ l, and θ r, show type-2 trangular fuzzy area varables type-1 uncertanty dstrbuton ntervals. At type- 2 level, the uncertanty dstrbuton nterval membershp functon defned as (x) sgvenwth(25)[46].inallcases, μ crl membershpsaregvenastype-1fuzzysets: ( x crl1 x crl 1 x crl 1 x crl 1 crl 2 crl 1 θ l crl 2 crl 1, crl 2 crl 1, crl 2 crl 1 ( x crl1 crl 2 x x crl 1 x crl 1 crl 2 crl 1 θ l crl 2 crl 1, crl 2 crl 1, crl 2 crl 1 (x) = μ crl ( crl3 x x crl 2 crl 3 x crl 3 x crl 3 crl 2 θ l crl 3 crl 2, crl 3 crl 2, crl 3 crl 2 ( crl3 x crl 3 x crl 3 x crl 3 x crl 3 crl 2 θ l crl 3 crl 2, crl 3 crl 2, crl 3 crl 2 x crl 1 +θ r crl 2 crl 1 crl 2 x +θ r crl 2 crl 1 x crl 2 +θ r crl 3 crl 2 crl 3 x +θ r crl 3 crl 2 ), If x [crl 1, crl1 + crl 2 2 ), If x [ crl1 + crl 2 2 ], crl 2 ] ), If x [crl 2, crl2 + crl 3 2 ), If x [ crl2 + crl 3 2 ], crl 3 ]. (25) When applyng the defuzzfcaton method proposed by Kundu et al. [46] for dfferent α value nterval cases as presented n precedng membershp functon, the equvalent crsp parametrc equatons of (6) become the form presented below. Also for (1), the crsp equvalent parametrc objectve functon can be rewrtten usng the same defuzzfcaton method. Case 1 (0 < α 0.25). Consder Case 4 (0.75 < α 1). Consder Ruc t (1 ( (2α 1 + (4α 3) θ r,) crl 3 +2(1 α) crl 2 1+(4α 3) θ r, )) Rcy t + Reuc t, t. (29) Ruc t (1 ( (1 2α + (1 4α) θ r,) crl 1 +2 crl 2 1+(1 4α) θ r, )) Rcy t + Reuc t, t. Case 2 (0.25 < α 0.5). Consder Ruc t (1 ( (1 2α) crl1 +(2α+(4α 1) θ l, ) crl 2 1+(4α 1) θ l, )) Rcy t + Reuc t, t. Case 3 (0.5 < α 0.75). Consder Ruc t (1 ( (2α 1) crl3 +(2(1 α) + (3 4α) θ l, ) crl 2 1+(3 4α) θ l, )) Rcy t + Reuc t, t. (26) (27) (28) he other constrants and coeffcents stll reman the same as the determnstc model. hen, the nteractve resoluton method s used to solve the problem for every acceptable feasblty degree of α, usng the related case s equatons (1) and (6). 5. Applcaton to a urksh Jewelry Company wth a Case Study 5.1. Background of the Case Company. he proposed model developed for the producton plannng problem s appled on one of urkey s leadng jewelry manufacturers, mporters, and exporters whch s called ABC Jewelry. he name s not dsclosed to keep anonymty. ABC Jewelry Company s engagednmanufacturngfnegoldjewelryproductssnce 1992 and has 25 tons per year manufacturng capacty of fne gold n ts facltes wth 23,000 square meters closed area. ABC Jewelry Company exports nearly half of ts producton to 45 countres around the world, ncludng USA and the European Unon. he company also sells ts products n the domestc market, supplyng to about 2000 local jewelry retalers. he company uses a producton plannng system whch s based on standard MR. he company takes orders from both domestc and foregn customers. Weekly producton plans

12 12 Mathematcal roblems n Engneerng Descrpton Formulaton Implementaton Resoluton Evaluaton Descrpton of the producton plannng problem Formulaton of the problem as mathematcal model Determnstc MIL MR model approach Recyclng uncertanty extenson to the problem Fuzzy type-1 defnton extenson Fuzzy type-2 defnton extenson AIMMS modellng language Implementaton of the proposed models Applcaton wth real data Resoluton of the problem MIL MR model computatonal effcency Recyclng and remanufacturng management n MR run CLEX, GUROBI, CBC, MOSEK, XA solvers Determnstc and fuzzy approaches comparson Evaluaton of the results Fgure 2: Flow chart of the problem resoluton. aremadeonadalybass,bytakngthedeadlnesoforders receved nto account. Orders are planned by consderng the producton cost and producton capacty n the plants, snce products can be produced n dfferent plants. As the raw materal, gold s the man bottleneck n plannng and must be used n the most effectve manner. In jewelry the plannng of the raw materal wth all aspects s the bass of producton plannng. he purpose of ths secton s to demonstrate that the producton plannng can be carred out by consderng recyclng, reusng, and remanufacturng wth the proposed model Implementaton and Resoluton. In addton to the gven detals and assumptons provded n the prevous sectons, specfc ssues to be consdered n ths case are as follows: () here are 4 manufacturng plants. () lannng perod s 2 weeks and producton plannng s done on a daly bass. () Manufacturng plants are operated 5 days per week durng normal work hours. If there s a need for overcapacty, overtme s possble. (v) here exst 45 dfferent products of 7 dfferent product groups n the plannng horzon. All are taken nto account. (v) Supplers of gold are consdered as one suppler to manage balance n total. (v) It assumed that no work n producton stock exsts n that plannng. (v) he data used n ths study are obtaned from the producton plannng software whch s used by the case company, provded as an appendx (see Supplementary Materal avalable onlne at to ths paper. (v) he uncertanty of the recyclng process s determned as FN and FV by decson makers, who are producton managers. Fgure 2 llustrates the steps of the problem-solvng process untl the results of the problem are found. he MIL MR model proposed n ths paper was mplemented usng the modellng language AIMMS 4.1 x64 [59] and solved by the solvers CLEX 12.5, CBC 2.7.5, GUROBI 5.5, MOSEK 6.0,andXA15toevaluatemodelperformance.henput dataandmodelsaremanagedonaimms4.1x64[59]as dfferent cases of determnstc and fuzzy problems. he computer envronment used for evaluaton has an Intel M 1.70 GHz Dual Core rocessor and 4 GB Ram runnng on Wndows 7 x64.

13 Mathematcal roblems n Engneerng 13 able 1: Effcency of applcaton for dfferent plannng perods and solvers. erod (w) Solver Iteraton Constrant Varable Integer Nonzero CU tme (s) CLEX , CBC GUROBI 5.5 1,171 2,631 6, , MOSEK 6.0 5, XA 15 1, CLEX , CBC , GUROBI 5.5 4,488 3,901 10, , MOSEK , XA 15 5, CLEX , CBC , GUROBI ,701 5,171 13, , MOSEK , XA 15 2,970,445 2, Obtaned Results wth Determnstc Approach. As the ntal step, the proposed determnstc MIL MR model s taken nto account. As the plannng horzon, a two-week perod s used prmarly for plannng. Because of the fnancal value of the gold, customers prefer to work wth short-range order nstead of long-term stock. he model performance s therefore also assessed for longer plannng horzons wth threeandfour-weekperodplannng.heamstoensure the effcency of the computatonal performance for the proposed model, ndependent from the plannng horzon. able 1 shows ths computatonal effcency of the proposed determnstc MIL MR model. he data comprse the number of teratons, constrants, varables, ntegers and nonzero elements, and solvng tme of solvers. Fgure 3 presents the results for comparng the computatonal performance of the proposed model for ncreased plannng perod at dfferent solvers usng gven CU tmes at able 1. he results ndcate that all solvers are able to fnd a soluton to the proposed determnstc MR problem, as gven n able 1. CLEX 12.5 and GUROBI 5.5 solvers are the best ones n terms of solvng performance by solvng tme for the proposed model, as presented n Fgure 3. In addton to the computatonal effcences, the proposed MIL MR model also provdes nsghts on how recyclng, reusng, and remanufacturng can be managed n the producton plannng. In able 2, the raw materalbased decson varables that are run for a four-week based producton plannng problem are presented. All values n able2arecalculatedasagramofgoldwththendces I andndces p varables are gven as peces quantty. he producton plannng for a four-week perod s appled to especally emphasze the reverse materal flows. he ntent of presentng able 2 s to demonstrate how recyclng, reusng, and remanufacturng operatons are covered n the proposed extended MIL MR model. In the frst CU tme (s) CLEX 12.5 CBC GUROBI lannng horzon (w) MOSEK 6.0 XA 15 Fgure 3: Dfference of CU tmes of solvers wth respect to dfferent plannng horzons. two columns, the plannng perod s gven as weeks and days. Decson varable Rcyp t represents the accumulated quantty of recyclable raw materal gold. he economc sze of recyclng s determned as 2000 grams of gold (Rcl = 2000, see Supplementary able Coeffcents&Int (RawMateral)). So,attheendofday4,theaccumulatedgoldnwastereaches ths threshold and on day 5 the perod plannng shows a recyclng process wth r t and Rcy t decson varables. It also ncreases the varable Ruc t, representng total of reused and recycled quantty and as seen at varable Inv t the model adds that value to stock of gold. Varables Reuc t and Reu t ndcate the reusng process n coordnaton wth the recyclng process on varable Ruc t. erodcally, enough perod used gold s sent to the recyclng process by regardng Rup whch s reusable perod. Here, an example s also gven for the remanufacturng decson for returned products. At perod 18, one

14 14 Mathematcal roblems n Engneerng able 2: roducton plannng decson varables values for raw materal (gold). erod (w) (d) Rcyp t Rcy t Ruc t Reuc t Reu t Inv t Sm t Col t r t Bs t Rp pt Rprt pt Rt pt Rcrt pt ,000 3, , ,577 3, , ,362 4,572 4, , ,788 5,423 5, , ,087 3,021 3, ,748 3, ,087 4,627 2, , ,648 1, ,717 10,995 10, ,451 1, , ,790 1, , ,392 2,502 2, , ,675 3,005 3, , ,689 2,378 2, ,384 10,655 10, ,006 2,075 2, , ,006 4,029 2, , ,360 2, , ,287 3,071 3, , ,684 3,231 3, ,840 10,634 10, ,000 3,016 3, , ,000 2,155 2,155 28, ,000 1,417 1,417 28, , ,389 productwth35pecessreturnedbythecustomerandthe returned products are forwarded to remanufacturng to meet the demand of that product, as presented wth varable Rprt pt whch stands for the remanufactured product quantty (see Supplementary able Demand&Return (roduct)) Resoluton wth Fuzzy Approaches. In the second step, uncertanty of the recyclng process s consdered n the proposedmodel.recyclnglosscost crl s defned by the decson maker n the jewelry company as FN crl = (100, 20, 0) as a type-1 fuzzy set. It s defned as 20 n the determnstc model (crl = 20; see Supplementary able Coeffcents&Int (RawMateral)). In Fgure 4, the representaton of recyclelost cost crl s shown as FN. he feasblty degrees α of recycle process, whch represent the values that the decson maker s wllng to admt, are determned n set M 1,wth M 1 = 0.7, 0.8, 0.9, 1} for fuzzy type-1 soluton by decson maker usng solutons n able 2. As the consecutve step, the recycle-lost cost s revewed as second order fuzzy dmenson. It s determned by the decson maker n the jewelry company as type-2 trangular fuzzy varable (FV) wth crl = (100, 20, 0; 0.7, 0.3). he feasblty degrees α of the recyclng process, n whch the decson maker s wllng to admt, are the same as n the type-1 fuzzy problem. Accordng to the acceptable feasblty degrees, case 3 needs to consder (28) and case 4 needs to consder (29) as parametrc lnear equatons to solve. In Fgure 5, the recycle-lost cost crl s depcted as the footprnt μ crl (x) crl Fgure 4: FN presentaton of recycle-lost cost crl. of uncertanty (FOU) by usng the gven uncertanty dstrbuton nterval membershp functon μ crl (x). he feasblty degrees α of the recyclng process, n whch the decson maker s wllng to admt, are determned n set M 2,wth M 2 = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8} for fuzzy type-2 soluton by decson maker usng solutons n able 2. In order to compare the mpacts of the determnstc model and fuzzy approaches on the total cost of the recyclng process, solutons obtaned by all approaches are summarzed as follows. he determnstc model s solved usng determnstc data and fuzzy approaches also use the same data except

15 Mathematcal roblems n Engneerng 15 able 3: otal producton cost for determnstc, fuzzy type-1, and fuzzy type-2 model solutons. Feasblty degree otal cost (determnstc) otal cost (fuzzy type-1) otal cost (fuzzy type-2) 0 1,569, ,649, ,559, ,622, ,549, ,589, ,539, ,545, ,529, ,511, ,519, ,488, ,508, ,482, ,498, ,473, ,488, ,462, ,477, ,449, ,467, ,446, Determnstc 1,488, for the recyclng process. For the recyclng process, type-1 fuzzy approach s used wth a FN for defnng uncertanty related to recyclng lost cost. he type-2 approach also uses the same FN wth an nterval area whch was gven before n ths secton. he soluton obtaned by usng α parametrc equatons and the soluton method wth acceptable feasblty degrees and satsfacton of dfferent α cut solutons whch are determned by the decson maker are explaned n the precedng secton Evaluaton of the Obtaned Results. In order to evaluate the effcency of the model soluton, the proposed model s run for perods longer than 2 weeks as well, such as 3 and 4 weeks. Nevertheless, the actual plannng perod n thecasecompanys2weeks,aslongerperodsarenotvery realstc wthout many revsons. o dentfy the effect on the results and to compare wth the determnstc soluton, fuzzy models are run only for the 2-week plannng perod. For ths comparson, CLEX 12.5 solver s used for all evaluatons, because t s the most effectve solver for the proposed determnstc model. Solver tmes are not provded for ths comparson, snce the soluton for fuzzy models s already obtaned wth parametrc lnear equatons, meanng that t salsodetermnstcandhasalmostthesamecutmes. Fuzzy model solutons are found wth CLEX 12.5 solver. able 3 summarzes the planned total producton costs for the determnstc, fuzzy type-1, and fuzzy type-2 model solutons. Fgure 6 presents the total producton cost changes accordng to feasblty degrees α between determnstc and fuzzy model solutons. Wth ths presentaton, the decson maker gans valuable nsght nto the uncertanty of recyclng processes mpact on total producton costs, wth whch the most satsfactory soluton can be selected. As seen n Fgure 5, the determnstc cost s a crsp soluton, so that the decson maker s not able to analyze the mpact of recyclng loss on producton plannng. Fuzzy type-1 model s parametrc soluton gves a better nsght to decson makers. he resoluton range for feasblty degree α provdes a better pontofvewtothedecsonmaker.fgure5alsoshows that the fuzzy type-2 approach offers a better nsght about uncertanty for the decson maker regardng the recyclng u crl Fgure 5: ype-2 trangular fuzzy nterval presentaton; FOU of recycle-lost cost crl. otal cost ,700 1,650 1,600 1,550 1,500 1,450 1, Feasblty degree α otal cost (determnstc) otal cost (fuzzy type-1) otal cost (fuzzy type-2) Fgure 6: otal cost comparson of determnstc and fuzzy models by feasblty degree α. loss cost. It consders the spread of uncertanty around the determnstc soluton more accurately. Consequently, the decson maker can select the most satsfactory feasblty degree α soluton as a plannng decson.

16 16 Mathematcal roblems n Engneerng he outcomes ndcate that the proposed MIL MR model satsfes the plannng problem of jewelry producton, whch ncludes reverse materal flows such as recyclng, reusng and remanufacturng. he man am of ths producton plannng s to effectvely manage gold nventory n coordnaton of manufacturng, reuse, recycle, and remanufacturng processes smultaneously, at the same tme meetng other standard producton plannng requrements nvolvng demand and capacty. 6. Concluson hs paper proposes an extended materal requrement plannng (MR) model usng a mxed nteger lnear programmng (MIL) technque for the producton plannng problem of jewelryndustrynurkeybyregardngrecyclng,reusng, and remanufacturng n addton to standard producton plannng. he proposed model covers both forward and reverse materal flows n a closed-loop jewelry supply chan whch consders process, supply, and customer sde smultaneously. Furthermore, lnear fuzzy programmng methods are appled takng the unpredctable nature of the recyclng process nto account wth fuzzy type-1 and type-2 uncertanty. hese are the man contrbutons of ths paper whch extend lterature. he proposed model and fuzzy applcatons are examned usng data from ABC Jewelry Company. he computatonal performance of the proposed determnstc model s presented for dfferent solvers and plannng perods. he results fortheproductonplannngrunndcatehowtheproposed model handles reverse materal flows. Determnstc and fuzzy model solutons are compared n terms of total cost varatons for dfferent feasblty degrees. rojectons of fuzzy approaches for decson makers are presented. It s demonstrated that the fuzzy approach gves a better perspectvetodecsonmakers.hemanobjectvestheeffcent utlzaton of gold, a challenge for producton managers, where the model proposes a soluton to ths man bottleneck of producton. hs effectve producton plannng approach mostlysatsfescompanyshareholdersandproductonplanners. Applcaton of the proposed model ndcates that t s practcal and usable both for achevng solutons n terms of computatonal effort and for effectve producton plannng. Fnally, potental drectons for further researches are ponted as below: () Besde the man bottleneck, whch s the raw materal, consderaton of all blls of materals should be helpful, as n the standard producton plannng. () Other raw materals whch are also recyclable (.e., precous stones and alloys for jewelry) can be consdered to extend the model, wth recyclable and nonrecyclable materals. Dfferent recycle rules should also be set as needed. () All supplers, customers, and plants can be defned separately n the model to manage closed-loop supply chan n partcular. (v) Use of dfferent uncertanty handlng approaches and fuzzy models can be helpful to reflect a better explanaton of decson maker s experence about fuzzy varables. (v) Defnton of other uncertan varables can help produce a more realstc soluton. As a gudance, n process, demand and supply-based varables can be consdered. (v) he proposed model n ths paper can be adapted to jewelry ndustres n other countres, or to smlar ndustres that are usng metals n producton. Conflct of Interests he authors declare that there s no conflct of nterests regardng the publcaton of ths paper. References [1] D. N.. Murthy and L. Ma, MR wth uncertanty: a revew and some extensons, Internatonal Journal of roducton Economcs,vol.25,no.1 3,pp.51 64,1991. [2] G. lenert, Focusng materal requrements plannng (MR) towards performance, European Journal of Operatonal Research,vol.119,no.1,pp.91 99,1999. 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