A Modelling Framework for the Acquisition and Remanufacturing of Used Products

Transcription

1 A Modellng Framework for the Acquston and Remanufacturng of Used Products Yanns Nkolads Department of Technology Management Unversty of Macedona, 5900 Naoussa, Greece tel: e-mal address: Abstract In recent tmes there has been a consstent need for companes to produce green products and offer green servces n order to contrbute to envronmental protecton. The utlzaton of used devces (extendng ther useful lfe cycle) s an excellent, ndrect way for companes to conform to ths requrement and, at the same tme, ncrease ther proft. Cell phones consttute one of the most nterestng cases of products, whch can be returned, remanufactured and reused: ther replacement rate s large, the avalable quantty for reuse s huge and, consequently, the proft potental s sgnfcant. Motvated by the real case of a company nvolved n the acquston and remanufacturng of used cell phones, a smple mathematcal programmng model s proposed n ths work that can help remanufacturng companes to make optmal decsons concernng the quanttes to be purchased and remanufactured. Its use, namely the smulaton of the model stochastc parameters and the optmzaton of the model, reveals not only that the explotaton of used products can be proftable, but also that as the product acquston system mproves, the economc benefts for any remanufacturng company can be even greater. Keywords: Reverse Logstcs, Remanufacturng, Mxed Integer Programmng, Cell Phones/Moble phones, Product Acquston 1

2 1. Introducton In the last decades, companes especally n the E.U. and the U.S.A. have been asked to produce envronmentally frendly products and offer green servces n order to contrbute to the nternatonal, large scale effort of envronmental protecton. One way of dong so s through the utlzaton of used devces, whch extends ther useful lfe cycle. Consequently, the number of products that are reused - through any of the dfferent optons of reuse (remanufacturng, reuse of components, recyclng etc.) - s gradually gettng bgger. Ths does not happen only because companes are encouraged (manly n the U.S.A.) or mandated (manly n the E.U., e.g. by the WEEE and RoHS drectves) to produce and sell green products, but also because companes become progressvely aware of the proftablty of reuse actvtes. Undoubtedly, the envronmental legslaton has already started to consttute a lever, whch makes companes serously examne the development of product recovery systems. Nevertheless, n recent years t has been realzed that these recovery systems should not be consdered as a cost centre, but as a proft one. Therefore, the enormous ncrease n research on remanufacturng (see comprehensve revews at Gude, 000, Fleschmann, 001 and Srvastava, 007) and, more recently, on closed loop supply chans (CLSC) s not at all surprsng. Van Wassenhove and Gude (00) focus on the busness aspects of creatng and managng cost effectve CLSC. They dentfy the most mportant processes requred by a CLSC ncludng product acquston, reverse logstcs, nspecton (consstng of testng, sortng and gradng), dsposton, remanufacturng, dstrbuton and sellng. CLSC have also been studed by the European network Reverse Logstcs and ts Effects on Industry, the major research results of whch are gathered and presented n the volumes edted by Dekker et al. (004) and Flapper et al. (005). A comprehensve revew and dscusson of several areas n CLSC s presented by Gude and Van Wassenhove (003), whle Gude et al. (003) crtcally dscuss mportant gaps n the research lterature; the areas of nspecton, dsposton, dstrbuton and sellng of remanufactured products have not been thoroughly examned from an academc pont of vew. On the contrary, the operatonal aspects of remanufacturng have receved the most attenton and there have been numerous contrbutons dealng

3 wth producton plannng and control (Gude and Srvastava, 1998, Gude, 000, Souza et al., 00), nventory control (Van der Laan, 1997, Inderfurth, 1997, Van der Laan et al., 1999, Toktay et al., 000) and materals plannng (Inderfurth and Jensen, 1999, Ferrer and Whybark, 001). In general, the decson to engage n reuse actvtes should be made based on an ntegrated economc analyss of the costs and benefts of such actvtes. Ths analyss should be done even when the law mandates recovery, because t can determne the best way to manage product returns. A company that has establshed a product recovery system reles on returned products, snce the latter consttute ts raw materals. Hence, product acquston management becomes of great mportance because t can affect several essental ssues: whether reuse actvtes are proftable; f they are, then, how proftable they can be; a number of operatonal ssues such as faclty desgn, producton plannng, control actvtes etc. However, the area of product acquston s another area of CLSC that has had a very lmted amount of research, as Gude and Van Wassenhove (001) ascertan. In addton, French and LaForge (006) arrve at the same concluson regardng product acquston n process ndustres. Nevertheless, some mportant contrbutons can stll be found n the area of product acquston. Galbreth and Blackburn (006) pont out that the condton of returned tems s often hghly varable and sortng s an mportant remanufacturng operaton. They examne the case of a remanufacturer who acqures unsorted used products from thrd party brokers and they propose some optmal acquston and sortng polces. A year later, Zkopoulos and Tagaras (007) examne procurement and producton decsons n reverse supply chans under yeld uncertanty. More specfcally, they nvestgate how the proftablty of reuse actvtes s affected by uncertanty n terms of the qualty of returned products. Note that our work s mostly focused on product acquston, tryng to fll n the aforementoned research gap. Gude and Van Wassenhove (001) menton that there are two systems for obtanng used products from end users: the waste stream system and the market drven system. In the context of the former, companes passvely accept any product return, not beng able ether to nfluence the qualty of returns or to acqure returned products only of a certan qualty. Inevtably, the companes nvolved consder the huge volumes of returned products as a costly nusance whose losses they try to mnmze. The adopton 3

4 of a rather chaotc system lke ths can be justfed only for companes facng mandatory product returns and just for a short adjustment perod, durng whch they should advance ther product acquston management to a hgher level, after the conducton of the economc analyss mentoned before. The market drven system s more popular among U.S. companes, because they have been persuaded about the proftablty of reuse actvtes, manly remanufacturng (Gude, 000). Besdes, by buldng a green profle voluntarly, U.S. companes ntend to keep the natonal legslaton loose - ndrectly, of course, because offcally they can not nfluence t - snce, for the tme beng, t smply encourages and does not force them to embody any reuse actvtes. Usually, companes whch have adopted ths system motvate end users to return used products, tryng to nfluence the products qualty level n varous ways: for example, by offerng fnancal ncentves especally for better qualty used products. Overall, t s a characterstc of market drven systems that the return rate, tmng and qualty of returned products are not regarded as varables of an extraneous process whch can not be controlled and nfluenced by the companes, but as a process wth wde margns of actvaton. One way or another, more and more companes fnd themselves ether wth used products n hand or wth offers n batches of returned products. Many of them get nvolved n reuse actvtes and ths s what makes the quanttatve mathematcal programmng model developed n ths work partcularly useful. Consequently, the major objectve of ths paper s to propose a smple, mxed nteger programmng (MIP) model, whch can help companes nvolved n remanufacturng actvtes to determne both the optmal quanttes to be purchased and remanufactured. Moreover, we apply the MIP model n practce to prove not only the proftablty of remanufacturng used products, but also the even greater proftablty of an evolved product acquston system (PAS). Ths double goal has been acheved frst, through the smulaton of the stochastc parameters of the model and, then, through the optmzaton of the approprate verson of the model. Fnally, we conduct senstvty analyss (of the approprate versons) of the MIP model, n order to fnd out whch of the model parameters affect the optmal soluton most and, thus, draw the attenton of any potental user of the presented optmzaton tool to ther careful determnaton. It should be mentoned that even f ths paper focuses on the acquston and remanufacturng of used 4

5 cell phones, the proposed mathematcal programmng approach s general enough to be used n other producton systems. The same strategy has been used by a few other researchers who focus on cell phones n order to study varous remanufacturng ssues. Frst, Selger et al. (004) and, then, Franke et al. (006) develop a generc remanufacturng plan for cell phones. Ther approach for the plannng of remanufacturng capactes and programs by means of combnatoral optmzaton and dscrete-event smulaton, supports any planner n the perodc adaptaton of an exstng remanufacturng system. Uncertantes regardng quantty and condton of moble phones, relablty of capactes, processng tmes and demand are consdered. Robots et al. (005) model the case of a reseller who acqures used products (e.g. cell phones) of old technology from an advanced market and then he ether (a) sells (a small fracton of) these used products n a developng market - where the old technology s acceptable - wthout any value-addng actvtes, or (b) adds value by remanufacturng these products to an as-new condton and then sells them n the developng market at a hgher prce. The paper s organzed as follows. A case of product recovery concernng a remanufacturer of cell phone unts s frst dscussed (Secton ). Then, n Secton 3 the mathematcal programmng model and ts versons are presented, whle Secton 4 ncludes some llustratve examples. The senstvty analyss of the proposed model s conducted n Secton 5. Fnally, some concludng remarks and drectons for future work are presented n Secton 6.. Problem defnton ReCellular, Inc. ( was establshed n 1991 as a global tradng operaton of remanufactured, graded and used cell phones, as well as other handheld electroncs. It offers remanufactured products as a hgh qualty, cost effectve alternatve to new cellular handsets. The company belongs to the cellular communcatons ndustry, whch s a hghly dynamc market where the demand for new telephones changes contnuously for many reasons: ntroducton of new technology, promotonal campagns, openng of new markets, etc. At the same tme, the factors that nfluence demand also affect the avalablty of used handsets: the supply of used cell phones s an unstable and volatle 5

6 market (Van Wassenhove and Gude, 00). The product acquston management of ReCellular also depends on the unknown future demand for remanufactured phones. ReCellular has to purchase and keep stocks of used phones to compete for sales, as the lead tmes for delvery, after the acquston agreement wth ts supplers, are often lengthy and uncertan. The company does not collect handsets drectly from the end users, but obtans them n bulk mostly from artme provders or thrd party collectors, such as chartable foundatons. All these may offer a varety of handsets n varyng condton (qualty level), for a wde range of prces and volumes. Thus, the company receves from ts supplers, almost on a daly bass, plenty of offers n batches of used cell phones of dfferent models, quanttes and qualty levels 1. Based on an estmated future demand for used handsets, Recellular has to decde whch offers to accept and then, from every batch that s selected to be purchased, the quanttes of used phones to remanufacture. All these daly decsons should be made n the context of maxmzng the total proft, whle satsfyng all operatng constrants. Table 1 about here In the proposed mathematcal model, t s assumed that ReCellular s not able, but manly wllng to nfluence the qualty of returned cell phones and, consequently, ther acquston cost. Frst, t s meanngless to offer fnancal ncentves to ts supplers n order to receve used phones of better qualty, because ReCellular does not acqure handsets drectly from the end users. Second, the volatlty of the market mentoned prevously makes the company cautous about offerng ths type of fnancal motvaton. Fnally, t s more mportant for the company to know the qualty of returned cell phones as accurately as possble, than to receve used devces of the hghest qualty; note that the latter s not absolutely certan that t s the most proftable potental for ReCellular. Ths actually depends on all the other prce and cost parameters, as t wll become clear through the llustratve examples n Secton 4. The PAS that s set up today between ReCellular - and generally between any remanufacturer - and ts supplers s rather mmature: t ncludes tradng of batches contanng cell phones n multple qualty 1 Table 1 ncludes a short verson of the descrpton of the varous qualty levels, presented ntally by Gude and Van Wassenhove (001). 6

7 levels, whle even the same suppler may offer batches of dfferent qualty mx from deal to deal. Besdes, no acceptance samplng s carred out before the acquston of any batch and generally there s sgnfcant uncertanty. It s nterestng to menton that n the past the system was even more prmtve, but enough progress has been made snce (Gude and Van Wassenhove, 001). For several reasons, further evoluton of the PAS should be temptng both for ReCellular and the supplers of used handsets. Fgure 1 represents the current state of the PAS (1) and the states towards whch t may and should move progressvely n the future (, 3 and 4). Evdently, the hgher the state of the PAS, the more mature t s. Fgure 1 about here If the aforementoned evoluton occurs, the expected beneft of ReCellular s evdent (and can be tremendous): any remanufacturng company would clearly prefer the PAS to become more determnstc, because ths would allow t to be more proftable, provdng ts customers the qualty they want at the lowest possble cost. Currently, Recellular may purchase cell phones of dfferent qualty level than t would lke to and be burdened much more than necessary. On the other hand, the most mportant ncentve for supplers to support the evoluton of the PAS s ther ablty to make ths busness more proftable for them as well. The ncreased proftablty would be the result of an ncrease n the prces of used cell phones; each suppler should be pad better for addng value to any batch (for example by sortng and/or gradng cell phones), before sellng t and for guaranteeng less or no uncertanty at all to ReCellular. Progressvely, the greater proftablty of ths busness for any partcpatng suppler wll attract others to jon the specfc market. The long term ncrease of supplers wll offer Recellular more alternatves, whch may even further contrbute to a proft ncrease. 3. Modellng framework As mentoned prevously the MIP model that s proposed n ths Secton ams at determnng the optmal quanttes to be purchased and remanufactured by any remanufacturng company lke ReCellular, every tme t receves several offers n batches of used products. Dependng on the state of the PAS, the 7

8 MIP model s modfed accordngly; n subsectons 3.1 to 3.4 we dscuss the varous alternatve versons of the model. Note that Pourmohammad et al. (004) desgn a materal network flow to mnmze the envronmental and operatonal costs of exchangng waste and by-product materals n a BB network. In order to represent ths problem, they formulate a mxed nteger lnear model. In addton, Lebreton and Tuma (006) use the same optmzaton approach n a smlar CLSC case: a lnear programmng model s developed to assess the proftablty of tre remanufacturng. Fnally, Wllems et al. (006) quantfy the dsassembly tme reductons requred to acheve economc feasblty of systematc product dsassembly. They use a modellng framework, based on lnear programmng, to nvestgate the effect of reducng the expected dsassembly tme and cost on the selecton of the optmal End-of-Lfe strategy. The MIP model presented here s developed for a sngle-perod settng and s related to a specfc type of used products (e.g. Noka 5100). In a partcular perod - deal, every suppler offers to the remanufacturng company just one batch of used products. The suppler quotes the prce of the latter accordng to () the qualty mx he usually offers based on all prevous deals and () the approxmate prces whch used products of certan qualty levels have n market (see Secton 4 for more detals). It s also assumed that the remanufacturng company does not try to nfluence ether the varous costs or the quanttes of the offered batches, whch s often a realstc assumpton as n the case of ReCellular that we mentoned earler. An estmated future demand for remanufactured products drves the remanufacturng company to order batches of used devces. As t usually happens n practce, the company has to purchase ether the whole quantty of an offered batch or nothng. When acceptance samplng s conducted (state of the PAS), the remanufacturng company s supposed to sample every offered batch before ts potental acquston. We also consder that there s no dsposal, because returned devces can be remanufactured regardless of ther ntal qualty level. On the other hand, every returned product needs remanufacturng n order to be sold n the secondary market: even the best qualty level of products, e.g. the one correspondng to devces whch have smply been out of package, s lower than the qualty level of a completely remanufactured product. Fnally, used products are remanufactured only f they are to be sold. 8

9 Let the ndex (= 1,, K) reflect the dentty number of the suppler offerng a batch of used products and the dscrete random varable J, takng the values j = 1,, L, reflect the qualty level of those products. Also assume that the best qualty level s L and the worst s 1. The mathematcal programmng formulaton reles on the followng parameters: D: estmated future demand for remanufactured products whch should be satsfed at the end of the examned perod; A : batch sze offered by suppler ; p s : p e : sellng prce of a remanufactured unt; salvage prce (correspondng to the salvage value) of an acqured, but non remanufactured unt; t s ndependent of the unt qualty level and the suppler who offered t. Evdently, t s p e < p s ; c a, : acquston cost of a used unt from suppler ; c r,j : remanufacturng cost of a used unt at qualty level j; c s : c c : q : shortage cost per unt of unsatsfed demand; samplng cost of a used unt; percentage of a batch offered by suppler that s gong to be sampled before the acquston of the batch. W: avalable budget; f (j): percentage of used unts n the batch offered by suppler, beng at qualty level j. These percentages are actually the values of the probablty mass functon f j Pr J j, whch are determned n practce through () extremely nsecure estmatons based, for example, on data collected from all prevous deals wth any suppler, f any (state 1 of the PAS), () estmatons based on the samplng outcome of batches (state of the PAS) and () relable estmatons based on data collected from prevous deals wth any suppler (state 3 of the PAS); The decson varables of the model are the followng: y : a bnary varable whch reflects the decson of the remanufacturng company to order or not the q could be potentally deemed as a decson varable. 9

10 batch offered by suppler ; t s 1 f the company should order the batch A y ; 0 otherwse x,j : the fracton of f j A whch should be remanufactured; t s 0 x,j 1 for every and j; The followng varables are also ntroduced to model the problem: C: total proft of the remanufacturng company, whch ncludes the opportunty cost as well; A: total quantty of used unts whch the company should order from ts supplers. It s A K 1 A y. R j : quantty of used unts beng at qualty level j whch should be remanufactured. It s calculated by: R j K 1 A f j x, j for j = 1,, L; (1) R: total quantty of used unts whch should be remanufactured (and sold). It s R L j 1 R j. () The objectve functon (OF) of the MIP formulaton expresses the total proft, whch should be maxmzed. It has the followng form: K L max C p R p A f s e c K c 1 q A 1 j 1 K 1 c a, j y x, j L K A y c A f r, j j1 1 jx c D R, j s (3) The frst two terms of (3) represent the company s revenue from sellng the remanufactured unts, as well as from salvagng the non remanufactured ones, respectvely. The negatve terms that follow, correspond to the total samplng cost, the total acquston cost, the total remanufacturng cost and the opportunty cost, respectvely. Note that the cost of samplng the batch offered by suppler s assumed to be proportonal to the sample sze q A, whch, n turn, s proportonal to the sze A of the specfc batch. The OF s subject to the followng constrants: x,j y for = 1,, K and j = 1,, L (4) 10

11 R D (5) c R L K j1 1 A f j K K cq A ca, 1 1 j1 1 x, j L K A y cr, j A f j x, j W (6) (7) x,j 0 for = 1,, K and j = 1,, L (8) y = 0 or 1 for = 1,, K (9) Constrants (4) ensure that f the remanufacturng company does not order the batch offered by suppler (.e., y = 0), then no products could be remanufactured from ths batch (.e., x,j = 0 for every j). Constrant (5) mposes that used products should be remanufactured only f they are to be sold and constrant (6) s auxlary and arses as a combnaton of () and (1). The MIP model can also nclude a budget constrant, lke (7). Omttng the constant term c s D and makng some modfcatons, then (3) becomes as follows: L K K K ps cs R c r, j pe A f jx, j A y pe ca, cc j1 1 1 ' max C q A (10) 1 where C C ' c D. Usng (1) and (), and lettng p = p s + c s, r j = c r,j + p e and a = c a, p e, then (10) s s further smplfed: max C ' L j1 K K p rj R j a A y cc 1 1 q A In addton, constrants (5) and (6) should be replaced by L R j D and (1) respectvely. 1 j In the above formulaton, t s nterestng to note that p expresses the proft from every remanufactured product, not only because t wll be sold (drect proft), but also because t wll not burden the company wth shortage cost (ndrect proft). Parameter r j reflects the cost of every remanufactured product at qualty level j: apart from the obvous remanufacturng cost, r j reckons that any remanufactured product 11

12 wll not be salvaged (ndrect loss), but sold. Fnally, a reflects the addtonal cost for acqurng every used product offered by a specfc suppler, snce the salvage prce guarantees a mnmum payback for every acqured used product. To enlghten the reader about the type of problems that can be solved wth (the approprate verson of) the MIP model presented above, let us consder the followng ntroductory nformaton of a fcttous case: n order to satsfy an estmated future demand for 1,000 remanufactured cell phones, a company nvolved n reuse actvtes examnes whch of the followng offers to accept: a batch offered by suppler 1 of 700 used phones and/or a batch offered by suppler of 1,100 phones. In the followng subsectons, addtonal nformaton regardng ths example s gven, whch s dfferentated dependng on the state of the PAS examned State 1 of the PAS In ths state, the percentages f (j) of used products beng at qualty level j, n the batch of any suppler, are ether completely unknown, e.g. n the case of a new suppler, or sgnfcantly dfferent from deal to deal. Consequently, although t s necessary to use certan values of those percentages for any suppler, n order to fnd the optmal MIP soluton of a specfc deal n practce, t should not be gnored that any estmatons used for ths purpose are not relable and the rsk of arrvng at a non-optmal soluton s hgh. Addtonally, n state 1 of the PAS t s q = 0 for any, snce no acceptance samplng s carred out before the acquston of any batch. In our llustratve example, usng the ntroduced notaton and the hypothetcal nformaton of all prevous deals wth supplers 1 and about the percentages f (j), we can consder that the remanufacturng company wants to satsfy a demand for D = 1,000 phones and based on the values of the prce and cost parameters (p s, p e, c a,, c r,j, c s ), t has to choose ether the batch of A 1 = 700 phones, contanng, let us say, 301 phones at qualty level 3 - assumng f 1 (3) = and 399 phones at the best qualty level - assumng f 1 (6) = or/and the batch of A = 1,100 phones, contanng, let us agan suppose, 97 phones at qualty level 5 - assumng f (5) = and 803 at the worst qualty level - assumng f (1) = As 1

13 mentoned prevously, the actual qualty mx of both offered batches could be slghtly or completely dfferent, a fact that renders acceptance samplng and transton to state of the PAS very temptng. 3.. State of the PAS In ths more advanced state, acceptance samplng would permt the remanufacturng company to have a better dea of the percentages f (j). Thus, both the MIP model and the decsons about the acquston and the remanufacturng of batches would become more accurate; they would be based on recent values of the percentages f (j) and not on old, and probably unrelable nformaton about them. Ths means that acceptance samplng of offered batches can be extremely useful for any company nvolved n remanufacturng actvtes. There are more than one ways of conductng and handlng the results of samplng. The only way to get a precse vew of the qualty mx of all offered batches, s by examnng the whole batches (100% nspecton) before acqurng any of them, whch means that n ths case t s q = 1 for any. A serous dsadvantage of ths procedure s the extremely large total nspecton cost. In our example, f the company had examned the whole batches before ther potental acquston, let us suppose that t would have found that the batch offered by suppler 1 contans 45 phones at qualty level 3 and 455 phones at the best qualty level, whle the batch offered by suppler contans 75 phones at qualty level 5 and 85 at the worst qualty level. Consequently, 100% nspecton would have gven the followng results: f 1 (3) = 0.35, f 1 (6) = 0.65, f (5) = 0.5 and f (1) = A smpler, faster and cheaper, but not so accurate alternatve, regardng the nformaton about the percentages f (j), would be to conduct random samplng n every offered batch and then use exclusvely the results of samplng n order to estmate those percentages. It s well known that the man dsadvantage of any samplng procedure s that n order to ncrease ts relablty, t s necessary, among other thngs, to take as large a sample as possble, whch means a hgh samplng cost. Besdes, by adoptng ths way of handlng the results of samplng, all prevous data reflectng the qualty hstory of supplers would not be taken nto consderaton. However, ths s not a dsadvantage at all, f the representatveness of the 13

14 samplng procedure s ensured and f the reader recalls that n ths partcular state of the PAS, the percentages f (j) may be dfferent from deal to deal. In our example, samplng e.g. 10% of both offered batches (q = 0.1 for any ),.e., 70 and 110 phones respectvely, could have gven the followng fcttous results: 5 phones offered by suppler 1 at qualty level 3 and 45 phones at qualty level 6, whle 8 phones offered by suppler at qualty level 5 and 8 phones at qualty level 1. These fgures would lead to the followng estmatons of the percentages f (j): f , f , f and f Consequently, the remanufacturng company would have concluded that the batch offered by suppler 1 contans 5 phones at qualty level 3 and 448 phones at the best qualty level, whle the batch offered by suppler contans 75 phones at qualty level 5 and 85 at the worst qualty level. The aforementoned dsadvantages of random samplng can be mtgated by calculatng a weghted estmaton, whch could combne the nformaton arsng both from the current sample and prevous deals. The offered quanttes of used products, the age of data or a combnaton of them could be used as weghts. However, the most effectve technque to explot the samplng procedure n order to update properly knowledge about the percentages f (j) s by usng Bayes theorem and the fact that regardless of the dstrbuton of the qualty level J, the sample mean J s dstrbuted normally for large sample szes. The dsadvantages of ths procedure are the followng: Makng use of past nformaton about any suppler s worthy only f the qualty mx he offers remans relatvely stable from deal to deal. Otherwse, the mportance of samplng becomes crucal and past nformaton has lmted value. In order to take advantage of Bayes theorem, t s necessary for the MIP model to nclude addtonal terms proportonal to the mean qualty level, such as form of the model. L j 1 jf j, whch do not exst n the partcular 14

15 3.3. State 3 of the PAS The verson of the MIP model that corresponds to state 3 of the PAS s dentcal to the one of state 1. However, there s an mportant dfference regardng ts accuracy, that arses from the fact that the percentages f (j) of all supplers do not change consderably between successve deals. Therefore, once a remanufacturng company obtans ther values after a number of deals, t can ncorporate them nto the MIP model, n any future deal. In ths state t s q = 0 for any, snce no acceptance samplng s carred out before the acquston of any batch State 4 of the PAS Let us now consder the followng varaton of the ntal example: n order for a remanufacturng company to satsfy an estmated future demand for D = 1,000 cell phones, t examnes whch of the subsequent offers to accept: a batch offered by suppler 1 contanng A 1 = 700 phones at qualty level 3 and/or a batch offered by suppler contanng A = 1,100 phones at qualty level 5. In ths state of the PAS, the mathematcal programmng formulaton s smplfed not only because t s q = 0 for every, but manly because for any suppler, f (j) = 1 for only one value of J and f (j) = 0 for any other value; n our example f 1 (3) = f (5) = Illustratve examples The presented MIP model has been appled to a number of practcal problems n order to show that remanufacturng may be sgnfcantly proftable. It has also been used to evaluate from an economc pont of vew the transton of the PAS from state 3 to 4. The proftablty of the transton from state 1 to or from to 3 largely depends on the uncertanty of the qualtes of returned products and, therefore, of the offered batches. Nevertheless, subsequently, apart from the transton of the PAS from state 3 to 4, whch s manly examned, we also dscuss the transton from state to 3. The methodology that has been used comprsed the followng steps: () determnaton of the range of 15

16 the parameters values, () smulaton of the parameters, () optmzaton of the approprate verson of the MIP model and (v) comparson and analyss of the emergng results. The tool that has been used n steps () and () was Mcrosoft Fortran PowerStaton 4 and ts lbrares; we have developed a number of programs specfcally for ths partcular paper. Addtonally, just for the sake of verfcaton, we have used another software combnaton: frst we have formed the approprate verson of the MIP model at Mcrosoft Offce Excel, then we have conducted smulaton of the parameters usng Crystal Ball, whch s a specal add n software of Mcrosoft Offce Excel and, fnally, the MIP model has been solved usng Solver of Mcrosoft Offce Excel. The data that have been used n the llustratve examples were collected manly from the web ste of ReCellular and are summarzed n Table. In the followng paragraphs we explan how the presented values have been chosen. Table about here The number of supplers and qualty levels of used phones s taken to be sx,.e., K = 6 and L = 6. In order to determne the percentages f (j) that are used for the evaluaton of deals made n state 3 of the PAS, each suppler s consdered to have access to a dfferent group of users; these users have dfferent habts and the qualty level of returned phones s hghly dependent on how the phones are treated durng ther use. For example: A collector that purchases phones from users n Scandnava and the artme provders offerng a 30 day return polcy, provde ReCellular wth unformly hgh qualty phones, e.g. f (3) f (4) f (5) f (6) 0.5 or f (4) f (5) f (6) A collector that deals wth users n the U.S.A. has a very mxed collecton: most phones show average wear, a few are n very poor condton (e.g. due to water damage) and a few are n lke new condton, e.g. f (1) 0.075, f () 0.175, f (3) 0.35, f (4) 0.75, f (5) 0.15, f (6) 0.05 (the specfc example s based on real data from a collector n the U.S.A.). A chartable foundaton provdes ReCellular wth very hgh qualty phones, e.g. f (5) f (6)

17 A suppler that deals wth commercal contracts (where the phones may have been heavly used and not mantaned, e.g. used by people n a bg constructon company) offers farly poor qualty phones, e.g. f (1) f () f (3) f (4) 0.5 or f (1) f () f (3) The varous types of supplers used to llustrate the applcablty of the MIP model are summarzed n Table 3. Table 3 about here The demand for remanufactured phones D vares unformly between 10 and 5,000 unts, accordng to real data from ReCellular. Usually, the sze of the offered batches s drectly connected to the demand D, because supplers take t nto account before determnng ther offers. Three cases have been studed regardng the values of A s of state 3: n the frst two ones, supplers are assumed to take the expected demand D nto account n order to determne ther offers, whle n the thrd one t s assumed that only the maxmum value of D,.e., 5,000 unts, s taken nto consderaton by the supplers. Consequently, accordng to the frst case A U (0, 0.8D), n order to take nto account that often the maxmum offered batch sze s somewhat smaller than the estmated demand. In the second case A U (0, 1.1D), consderng the possblty that the maxmum offered batch sze may be slghtly greater than the demand. Fnally, n the thrd case A U (0, 5,000). However, n order to evaluate the potental beneft from the transton of the PAS from state 3 to 4, t s necessary to determne smultaneously the sze of the offered batches for both examned states, namely () the sze of batches contanng used cell phones at a qualty mx, accordng to the percentages f (j) of any suppler (state 3), as well as () the sze of batches contanng phones at a specfc qualty level (state 4). The smultaneous determnaton of A s s based on the requrement that the total offered quantty of used phones per qualty level should be the same, regardless of the state of the PAS. Ths requrement can guarantee that the examned offers are equvalent and the results comparable. In Table 4 we present three ndcatve sets of batch szes for both examned states. In these sets, frst the values of all A s and all f (j)s of state 3 have been smulated - where A U (0, 1.1D) - and then every A of state 4 has been calculated 17

18 usng K 1 A f j ; n ths sum A refers to the batch sze of state 3. For example, the (hghlghted) quantty A 4 = (frst ndcatve set - state 4) arses as follows: A 4 = Table 4 about here Regardng the values of A s of state 4 n Table 4, t should be noted that the equvalency of offers between states 3 and 4 s acheved () n the frst two sets f the remanufacturng company receves offers for every qualty level and () n the thrd set f t receves offers for only four qualty levels. Consequently, the types of supplers partcpatng n a deal of state 3 affect the values of A s of state 4. Comng back to the percentages f (j) of state 3, two cases have been studed leadng to a total of sx scenaros presented n Table. As per the frst case, the types of supplers that partcpate n a deal arse randomly: for example, ReCellular may receve n a deal three offers from supplers of type I and three offers from supplers of type V (see the thrd ndcatve set n Table 4). As per the second case, one suppler of every type offers a batch to the remanufacturng company (see the frst ndcatve set n Table 4). In order to determne the values of the acquston cost c a, of a used cell phone accordng to ts qualty level j (state 4), frst the nformaton about the mnmum sum of money that ReCellular should have pad n order to acqure a returned Noka 5100 has been consdered for all sx qualty levels (Table 5). Then, t has been consdered that n practce c a, fluctuate unformly n the nterval (mn c a,, mn c a, ), for any qualty level j. The specfc range of values s consstent wth the fact that c a, should always be an ncreasng functon of j. On the other hand, the acquston cost c a, of a cell phone unt offered by suppler, when the deal s made n state 3, arses takng nto consderaton the need for equvalency between the compared states of the PAS and the assumpton mentoned at the begnnng of Secton 3; accordng to ths each suppler quotes the prce of the cell phones n hs batch, based on the qualty mx he usually offers,.e., accordng to hs percentages f (j). For example, a suppler of type I usng the mnmum c a, s of Table 5 should offer hs products n c a, = 91.5 $. 18

19 Table 5 about here The procedure of determnng the remanufacturng cost c r,j for every qualty level j s smlar. The mnmum remanufacturng costs c r,j s of a used Noka 5100 dependng on ts qualty, accordng to ReCellular, are gven n Table 5. In practce c r,j fluctuate unformly n the nterval (mn c r,j, mn c r,j ), for any qualty level j. ReCellular acqures used handsets and after ther remanufacturng process, sells them at a prce that goes beyond the varous costs (.e., acquston, remanufacturng etc.), assurng a reasonable proft for the company. Takng nto consderaton the fluctuaton of c r,j s and c a, s for all qualty levels, and a reasonable proft margn for ReCellular (.e., between 5% to 15%), t follows that the sellng prce p s of a remanufactured Noka 5100 vares unformly between 15 $ and 145 $. Concernng the salvage prce p e of a non remanufactured Noka 5100, t should be noted that there s always a common value for any cell phone, regardless of ts qualty level. Ths s typcally a very low value, whch vares unformly from 5% to 15% of the average acquston prce of cell phones n varous qualty levels. Shortage cost usually takes the form of lost sales cost or back order cost. It ncludes payment of overtme and/or specal admnstratve expenses due to stock outs, loss of sales and loss of goodwll. In our case, the shortage cost c s per cell phone unt of unsatsfed demand reflects a lost opportunty for proft. Therefore, usng p s and the average c r,j s and c a, s for any qualty level j, the shortage cost c s has been estmated to be dstrbuted unformly between 7.5 $ and 33.5 $. Fnally, n order to smplfy the MIP model, a very large value for the avalable budget W has been assumed. In ths way, constrant (7) becomes non-actve. Usng the data of Table, the parameters of the MIP model have been smulated 0,000 tmes for each one of the sx examned scenaros. Then, for each scenaro and smulated set of parameters values, the approprate verson of the MIP model (between those two that correspond to states 3 and 4 of the PAS) has been optmzed. In Table 6 we present a complete set of parameters values and the soluton of the MIP model for states 3 and 4 of the PAS, regardng the second (hghlghted) ndcatve set of A and f (j) values of Table 4. Note that C m represents the optmal - maxmum - total proft (OF) for states m = 3 and 4 of the 19

20 PAS. Ιn every other case, the parameters have been smulated and the optmal solutons have been determned accordngly. Table 6 about here The analyss of results determnes the percentage proft ncrease n state 4 compared to state 3,.e., ΔQ 4 C C 4 C 3 3 %, ranks combnatons accordng to ΔQ 4, determnes the maxmum, the mnmum and, manly, the average ΔQ 4 for every scenaro, detects and enumerates cases where ether C 3 < 0 or C 4 < 0 and fnally creates the hstogram of relatve frequences of ΔQ 4. In Table 7 and Fgure we summarze the most mportant results for all examned scenaros. Varous mportant conclusons can be drawn, such as: Table 7 about here Fgure about here C 3 or C 4 mght be negatve for many reasons. The most mportant one s related to the combnaton of szes of the offered batches, especally when every A U (0, 5,000), whch mght render the decson of the remanufacturng company to order the proper set of batches exceptonally dffcult: for nstance, f the company has to choose between sx very large (small) batches when D s very small (large), the, probably large, excess (shortage) n the optmal soluton and, consequently, the negatve values of C 3 or C 4 are unavodable. Consderng the small percentage (number) of sets where ether C 3 < 0 or C 4 < 0, the proftablty of remanufacturng actvtes - at least n the case of used cell phones - for every company takng acton n states 3 or 4 of the PAS becomes obvous. Evdently, ths happens manly when supplers take nto consderaton the estmated demand D before determnng ther batch sze (Scenaros 1-4); the ncreased dsperson of batch szes regardng the actual demand D (Scenaros 5-6) rase sgnfcantly the percentage (number) of unproftable deals. The percentage of sets where ΔQ 4 > 0 approaches (n the worst case) and exceeds (by far usually) 70%. Ths makes obvous that the proftablty of remanufacturng actvtes - at least at the cellular communcatons ndustry - wll become much more remarkable, f the PAS gradually evolves n the 0

21 future. More specfcally, the transton of the PAS from state 3 to 4 can ensure an average percentage proft ncrease ΔQ 4 that vares between 9.49% (Scenaro 6) and 46.1% (Scenaro 5). When supplers take the expected demand D nto account n order to determne ther offers (Scenaros 1-4) the average ΔQ 4 s not dfferentated serously; otherwse, the average ΔQ 4 takes more extreme values (see a more analytcal remark below). Undoubtedly, the maxmum and mnmum ΔQ 4 values are mpressve, ether postvely (manly) or negatvely. However, those values mght be msleadng, especally n Scenaros 5-6, where they often arse due to extremely small absolute values of C 3. A better dea about the dstrbuton of ΔQ 4 s gven n Fgure, where t s evdent that n the majorty of sets (for all scenaros) ΔQ 4 vares between 0% and 35%. Regardng the nfluence of the examned scenaros (bare n mnd that the varous scenaros are dfferentated accordng to the combnaton of the values of A s and the types of supplers partcpatng n deals made n state 3 - see Table for more nformaton) on the results, t should be noted that: o The bgger the potental maxmum sze of A s,.e., movng from the par of Scenaros 1 - to 3-4 and, then, to 5-6, the more mportant the way the percentages f (j) are determned; when every A U (0, 0.8D) (Scenaros 1 - ), then the types of supplers takng part n a deal do not affect sgnfcantly the average ΔQ 4. It remans almost stable at 17.5% at Scenaros 1 -. When every A U (0, 1.1D) (Scenaros 3-4) the dfferentaton of the average ΔQ 4 arses at.78% (= %) and fnally at Scenaros 5-6 the dfferentaton of the average ΔQ 4 s sgnfcant. o Comparng the scenaros n pars that are determned accordng to the maxmum value of A s, namely Scenaros 1 vs, 3 vs 4 and 5 vs 6, t seems that the sngle appearance of every type of suppler n a deal leads to an ncrease of the average ΔQ 4, when supplers take nto consderaton the estmated demand D n order to determne the sze of ther batches (Scenaros 1 - and 3-4). Ths can be manly attrbuted to the bgger unformty of the offered batches n Scenaros and 4 1

22 (n comparson wth Scenaros 1 and 3, respectvely), regardng the qualty of used phones they contan. On the contrary, when supplers partcpatng n a deal gnore the specfc value of D, the decrease of the average ΔQ 4 s remarkable: from 46.1 % t shrnks to 9.49 %. By havng a close look at the parameters values of Table t becomes clear that all returned handsets, even those of the poorest qualty, can be remanufactured proftably. Takng ths nto consderaton, f K 1 A D, then the optmzaton of the versons of the MIP model for states 3 and 4 of the PAS leads to the followng result: C 3 = C 4 and, thus, states 3 and 4 are equvalent regardless of the examned scenaro. Ths equvalency of states s a result of the fact that the optmal acquston polcy for the company n both states s to purchase and remanufacture every offered batch, due to the need to avod shortage, whch usually costs a lot. Consequently, there s no effect from the dfferent style of offers that appears at the two states of the PAS. Studyng the optmal solutons of the MIP model (for both states 3 and 4) whch recommend that the company should acqure an excess of products,.e., K 1 A y D, and, more specfcally, the optmal y s and x,j s, t can be seen that every acqured batch (.e., y = 1) wth sze A < D s completely remanufactured (.e., x,j = 1), except for one: ths batch can be consdered as the one that leads to excess and contans the products wth the lowest proftablty. In other words, n the optmal soluton when the choce of batches lead to an excess of products, one x,j < 1 can be found at most. Regardng the transton of the PAS from state to 3, the proftablty of two fcttous deals made between a remanufacturng company and ts supplers s compared; the frst one s made n state of the PAS, whle the second n state 3. We assume that n both deals each suppler offers a batch of sze A, wth the same qualty mx of used products, descrbed by some percentages f (j), whch are the same for both states of the PAS. Understandably, the two deals are almost dentcal and, therefore, comparable. They dffer only because n the case of the deal of state 3 of the PAS the company can rely on prevous knowledge of the percentages f (j) of any suppler, as they do not change sgnfcantly between

23 successve deals, whle n the case of the deal of state the qualty mx of any suppler s unstable from deal to deal and the company has to conduct acceptance samplng n order to estmate those percentages. In the deal occason where acceptance samplng gves absolutely accurate nformaton about the actual percentages f (j) of all supplers, then C K C c q A 3 c 1 and, therefore, t wll always be C < C 3. In the more realstc case where acceptance samplng leads to more or less naccurate estmatons of the percentages f (j), sub optmal solutons of the MIP model wll be obtaned, drectng the remanufacturng company to mproper choces of batches to acqure and remanufacture, thus n (much) lower total proft than the actual C. Hence, n any case t s C < C 3 and ths clearly reveals that the evoluton of the PAS can brng mportant economc benefts for any company nvolved n reuse actvtes. 5. Senstvty analyss of the proposed MIP model Wagner (1995) states that the values of varous parameters of any model are often guesstmated and that t s usually helpful to practtoners to understand the senstvty of a model to smultaneous varatons n several parameters. Therefore, the senstvty analyss of the versons of the MIP model correspondng to states 3 and 4 of the PAS s very nterestng, snce t reveals whch of the prce, cost and all other parameters affect the optmal solutons and, generally, the examned models most; thus, any potental user of the presented optmzaton tool wll be careful n ther determnaton. At frst, regardng the senstvty analyss of the verson correspondng to state 3, only Scenaros 1, 3 and 5 (where the types of supplers arse randomly) have been examned. Accordngly, n state 4 of the PAS t has been assumed that the qualty level of cell phones offered by each suppler arses randomly. The senstvty analyss has been carred out n the followng steps: () determnaton of the range of the parameters values (already done n Secton 4 - presented n Table ), () specfcaton of the senstvty analyss methodology, () smulaton of the parameters and optmzaton of the approprate verson of the MIP model and (v) analyss of results. As far as the senstvty analyss methodology s concerned, the one proposed by Wagner (1995) has 3

24 been mplemented. Among the three alternatve algorthmc methods that he presents, Monte Carlo estmaton of parameter senstvty has been selected n order to ascertan the relatve nfluence of the model parameters. Consderng the verson of the MIP model for state 4, t can be easly observed that parameters can vary,.e., A and c a, for = 1,, K = 6, c r,j for j = 1,, L = 6, as well as p e, p s, c s and D. All these mprecse parameters affect the OF and some of them affect constrant (6) as well. Snce all A s (as well as c a, s and c r,j s) are equvalent, the senstvty analyss of the specfc verson of the model n relaton to the varaton of just one A (c a, and c r,j ) s enough n order to fnd out the mpact of the varaton of any other A (c a, and c r,j ). A fnal remark about the mplemented senstvty analyss methodology s that due to the fact that the aforementoned mprecse parameters vary jontly, the one at a tme senstvty noton has been consdered as a more precse ndcator of global senstvty. Wagner (1995) mentons that ths noton s good at dscernng the parameters that nfluence strongly any examned model. The followng steps have been followed: One (at a tme) of the mprecse parameters has been consdered as known. Ths has been the condtonng parameter, p k, whch has been sampled S = 00 tmes. For each one of these 00 sampled values of p k, the remanng (1) parameters have been sampled T = 10 tmes smultaneously. For each one of the,000 combnatons of parameters values, s,t C for s = 1,,, S = 00 and t = 1, 4,, T = 10, namely the optmal total proft for state 4 of the PAS and for specfc values of s and t, has been determned. Consderng the T = 10 values of s,t C for each one of the 00 values of p 4 k at a tme (.e., for a specfc value of s at a tme), S = 00 averages of s,t C have been calculated from the followng formula: 4 Τ s 1 s,t avg C 4 C for s = 1,,, S = Τ t 1 4

25 Then, for s = 1,,, S = 00 the S. T =,000 resduals have been calculated from the followng formula: C s,t s avgc for a specfc value of s at a tme and for t = 1,,, T = 10. The value of τ has been estmated as the sample varance of the resduals, usng the followng k formula: τ k S 00 T 10 s1 t1 C s,t s - avgc 4 4 S T 1, and ndcates the varaton n the optmal total proft due to the jont varaton n all the other (1) parameters, gven only the value of p k. The varance of C 4, namely σ C 4, has been estmated usng σ C4 S 00 T 10 s1 t 1 S 00 T 10 s1 t 1 C s,t C 4 S T S T 1 s,t 4. The results have been transformed n terms of τ 1, σ k R through the followng formula: R τk C4 where R s the fracton of varaton n C τk 4 attrbutable to the condtonng parameter p k. The only dfference that can be detected at the senstvty analyss of the verson of the MIP model correspondng to state 3 of the PAS s that n ths case there are 58 mprecse parameters,.e., f (j) for = 1,, K = 6 and j = 1,, L = 6, A and c a, for = 1,, K = 6, c r,j for j = 1,, L = 6, p e, p s, c s and D, nstead of the mprecse parameters that can be found n state 4. The senstvty analyss results are summarzed n Table 8. Gven that () the larger the value of R, τk the more p k nfluences the varaton of the OF and () f R s greater than τk R, then the OF s more τj senstve to p k than p j, we conclude that among the mprecse parameters, t s by far the demand D and, secondarly, the sellng prce p s of a remanufactured cell phone unt that nfluence the MIP model the most. All the other parameters affect the model much less and almost equally. Ths concluson arses regardless of the examned state of the PAS and the dstrbuton of A s. The only nfluence that can be detected when ether of them dfferentates can be connected wth the varances σ and C 3 σ C 4,.e., n the 5