A Parameter-Tuned Genetic Algorithm to Solve Multi-Product Economic Production Quantity Model with Defective Items, Rework, and Constrained Space

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1 A Parameter-Tuned Genetc Algorthm to Solve Mult-Product Economc Producton Quantty Model wth Defectve Items, Rework, and Constraned Space Seyed Hamd Reza Pasanddeh, Ph.D., Assstant Professor Ralway Faculty, Iran Unversty of Scence and Technology, Tehran, Iran Phone: +98 (21) , Fax: +98 (21) , e-mal: Seyed Tagh Akhavan Nak, Professor Department of Industral Engneerng, Sharf Unversty of Technology, Tehran, Iran Phone: , Fax: , e-mal: Seyedeh Sameeh Mrhosseyn, M.Sc. Department of Industral Management, KAR Unversty, Qazvn Branch Phone: +98 (282) , Fax: +98 (282) , e-mal: Abstract The economc producton quantty (EPQ) model s often used n manufacturng envronments to assst frms n determnng the optmal producton lot-sze that mnmzes the overall producton-nventory costs. Whle there are some unrealstc assumptons n the EPQ model that lmt ts real-world applcatons, n ths research some of these assumptons such as 1) nfnte avalablty of warehouse space, 2) all of the produced tems beng perfect and 3) the exstence of one product type are relaxed. In other words, we develop a mult-product EPQ model n whch there are some mperfect tems of dfferent product types beng produced such that reworks are allowed and that there s a warehouse space-lmtaton. Under these condtons, we formulate the problem as a non-lnear nteger-programmng model and propose a genetc algorthm to solve t. At the end, a numercal example s presented to dentfy the optmal values of the genetc algorthm parameters and to llustrate the applcatons of the proposed methodology to more realstc real-world problems. Keywords: EPQ, mult-product; mperfect and scrap tems; constraned space; genetc algorthm 0

2 1. Introducton and lterature revew The economc producton quantty (EPQ) model can be consdered as an extenson to the well known economc order quantty (EOQ) model that was ntroduced by Harrs (1913). It s a technque to fnd the optmum producton quantty by consderng costs of procurement, nventory holdng, and shortages. In real-lfe manufacturng systems, there s usually more than one type of products, the demand for each product s a random varable, and that the generaton of defectve tems and random breakdowns of producton equpment are nevtable. As a result, the frst few assumptons of the EPQ model (sngle product, determnstc demand, producng only perfect tems, etc.) may not be realzed n many real-lfe problems. Many researchers have developed dfferent EOQ and EPQ models assumng the exstence of scrap tems. For nstance, Hayek and Salameh (2001) assumed that all of the produced defectve tems are reparable and derved an optmal operatng polcy of the EPQ model. By consderng rework tme n ther model, the basc assumptons of ther work were not only to allow backorders but also to permt all of the defectve tems to be reworkable to become perfect. In the research by Rosenblatt and Lee (1986), whch has been proposed an EPQ model for a producton system that contans defectve products, the man assumpton was that the producton system produces 100% non-defectve tems from the startng pont of the producton tme untl a tme pont that was consdered a random varable. At ths tme, the system becomes out of control and starts to produce defectve tems untl the end of the producton perod. Furthermore, they assumed that the dstrbuton of the tme lag untl the system becomes out of control s exponental. Km and Hong (1999) extended the Rosenblatt and Lee s model (1986) wth the assumpton of an arbtrary dstrbuton of the tme lag. Salameh and Jaber (2000) 1

3 developed an EPQ model n whch the shortages were not allowed and a unformly dstrbuted fracton of the ordered lot contaned mperfect qualty tems. Chu et al. (2007) presented a procedure to determne the optmal run tme for an EPQ model wth scrap, rework and stochastc machne breakdowns. Hou (2007) presented an EPQ model wth mperfect producton processes, n whch the set up cost and process qualty are functons of captal expendture. Ths model llustrates the relatonshp among producton run length, set up reducton, and process qualty mprovement n an mperfect producton system. He showed that nvestment n set up reducton would lead to the reducton n optmal producton run length and would reduce lot sze, whereas nvestment n process qualty mprovement would result n an ncrease n optmal producton run length and lot sze. At the end, he ponted out that t was very mportant to nvestgate the optmal allocaton of nvestment between both optons. Ln (1999) ntroduced an ntegrated EPQ model subject to an mperfect producton process and constrant on the raw materals. Hs basc model assumes that at the begnnng of each producton run, the producton faclty s n an n-control state. Then, after a perod of tme, the faclty shfts to an out-of-control state. The elapsed tme to the shft s a random varable havng an exponental dstrbuton wth a gven mean. Moreover, there s a constrant on the avalablty of the storage space for raw materals. Then he extended the model to ncorporate two cases that have a dynamc deteroraton n the producton process. In addton, he studed the model for the stuaton where elapsed tme to the shft and the percentage of defectve tems are functon of the producton setup cost. In several nstances of practce, producng new or recoverng defectve products take place on a common faclty. Consequently, t s necessary to coordnate the producton and 2

4 rework actvtes wth respect to the tmng of operatons and wth regard to approprate lot szes for both processes. Buscher and Lndner (2007) presented a lot sze model, whch addresses all of these aspects. In addton, they cted that t was very mportant to assgn completed unts at one stage to partal lots - called batches - for shpment to the next operaton. Lao et al. (2009) studed the mantenance and producton programs of an EPQ model for an mperfect process nvolvng a deteroratng producton system wth ncreasng hazard rate. The mperfect repar restores the system to an operatng state, but leaves t faled untl perfect preventve mantenance (PM) s performed. They ntroduced two types of PM, namely mperfect and perfect PM. The probablty that the perfect PM s performed depends on the number of mperfect mantenance operatons performed snce the last renewal cycle. In addton, they suggested that f the PM rate s estmated based on the actual data, analysts can use the learnng curves to project the PM costs n the ntegrated EPQ model. One of the most mportant aspects n extensons of the EOQ and EPQ models s to fuzzfy ther parameters. For nstance, Lee and Yao (1998) n ther research fuzzfed both the demand and the producton quantty to solve the problem of the economc producton quantty per cycle. Many researchers assumed that there s a large enough storage space to hold products. However, n realty, there s usually a lmtaton on avalable warehouse space for raw materals or fnshed goods. In addton, the cost of warehouse holdng sometmes outweghs the benefts of havng no lmtaton on space enormously and the manufacturers prefer to have lmted space. Hence, the storage space lmtaton wll surely affects the quantty of the lot-szes and needs to be consdered n the model. In ths paper, a mult-product EPQ model s consdered n whch there s lmted warehouse space. In addton, the rate of mperfect and scrap tems s known and 3

5 reworkng makes the mperfect products perfect. To solve ths problem, we frst defne the problem and the model n secton 2 and 3, respectvely. To solve the model, we present a genetc algorthm n secton 4. To demonstrate the applcaton of the proposed methodology, n secton 5, we present a numercal example n whch the parameters of the proposed GA are fne-tuned. Fnally, the concluson and some recommendatons for future research come n secton Problem defnton The proposed model of ths research s an appled one that s developed based on the real constrants and envronments of producton companes. Consder a manufacturng company that receves raw materals from a suppler to produce n products. All of the produced tems are nspected to be classfed as perfect, mperfect (defectve but reparable) and scrap (defectve and not reparable) products. Suppose the requred tme of the nspecton s ncluded wthn the producton tme such that t can separately be assumed zero. Ths assumpton s not far from realty, because n many nstances the nspecton task and producng an tem occur smultaneously. All of the mperfect products are reworked to be perfect and the scrap products are sold at reduced prce. The work n process nventory (WIP) conssts of three types of materals around the manufacturng machnes: 1) raw materal, 2) perfect products and 3) mperfect products. Furthermore, the warehouse space of the company for all perfect products s lmted, shortage and delay are not allowed, and that all parameters, such as the demand rate, the rate of mperfect and scrap tems producton, the set up cost, etc. are all known and determnstc. The objectve s to determne the optmal producton quanttes of the products that mnmze the total costs whle satsfyng the constrant. 4

6 Three man specfcatons of the proposed model of ths research that have led to ts novelty are 1) the allowance of several products, 2) rework and mperfect product are allowed, and 3) the warehouse space to store raw materals and fnshed goods s lmted. By allowng these condtons smultaneously, the created model s dfferent from the other models n the EPQ lterature. 3. Problem modelng In order to model the problem at hand, the classcal EPQ model wll be extended to contan the perfect, the mperfect and the scrap tems along wth the warehouse capacty. To do ths, we frst defne the parameters n secton 3.1. Then, the pctoral representaton of the nventory problem wll be gven by an nventory graph n secton 3.2. In secton 3.3, dfferent costs of the system wll be derved. Fnally, the model of the problem wll be presented n secton Parameters and notatons For products 1,2,..., n, we defne the parameters of the model as follow: n Q Number of products Order quantty of the th product P Producton rate of the th product D Demand rate of the th product A Set up cost per cycle of the th product h Holdng cost rate of the th product 5

7 M Raw materal cost per unt of the th product S Set up tme of the th product m Machnng tme per unt of the th product R Producton cost rate per unt tme of the th product c Average producton cost per unt of the th product v Average value added per unt of the th product w Average nvestment per unt of WIP of the th product I Average amount of warehouse nventory of the th product p 1 Imperfect producton percentage of the th product p 2 Scrap producton percentage of the th product s 1 Perfect producton cost of the th product s 2 Scrap producton cost of the th product T Cycle tme of the th product TP Total tme per cycle to produce the th product t Average producton tme per unt of the th product f Requred space per perfect unt of the th product F Total avalable warehouse space for all products TC P Total procurement cost of the th product TC O Total set up cost of the th product TC I Total nspecton cost of the th product 6

8 TC WIP Total holdng cost for WIP of the th product TC H Total holdng cost for perfect products of the th product TC Total annual cost of all products 3.2. The nventory graph In order to calculate all nventory costs, t s necessary to survey the work n process and warehouse nventory. For the problem at hand, the graph of raw materal quantty versus tme s demonstrated n Fg1.a. In addton, the graphs of the perfect and scrap WIP nventory versus tme are llustrated n Fgures 1.b and 1.c, respectvely. In ths problem, the rate of demand s constant and hence the graph of the fnal product quantty n the warehouse s smlar to the EOQ model and s gven n Fg 1.d. We note that n Fg 1.d, the amount of produced perfect products n the warehouse n each cycle s reduced based on the determnstc rate of demand. Insert Fgure (1) about here 3.3. Costs calculatons Snce the shortage and delay are not permtted, the total nventory costs of all products per year (TC ), s the sum of total procurement cost ( TC ), total set up cost ( TC ), total nspecton cost ( TC ), total holdng cost for WIP nventory ( TC for warehouse nventory ( TC I H P WIP ) for all products. In other words, we have: ) and the total holdng cost n P O I WIP H (1) 1 TC TC TC TC TC TC O 7

9 to In any cycle, snce the set up tme, the producton tme and the reworkng tme are equal S, equaton (2). mq and m p Q, respectvely, the total tme to produce product, ( TP ), s gven n TP S m Q m p Q S m Q p (2) Hence, the average producton tme for each unt of product s: TP S t m 1 p1 Q Q (3) Based on R whch s the rate of producton cost per unt tme, v and c are obtaned as: S v Rt R m 1 p Q S c M v M R m 1 p Q 1 1 (4) (5) Snce delays are not allowed, the supply and the demand quanttes are equal and we have: 1 p2 Q 1 p2 Q DT T (6) D As s1 and s 2 represent the prce of the perfect and the scrap tems, respectvely, the average revenue n unt tme s obtaned as: 1 p Q s p Q s p TR D s D s ; 1,2,..., n T 1 p2 (7) Note that for the problem at hand the revenue n unt tme does not depend on the lot sze. follows. Now, based on equatons (2) to (6), the nventory costs of equaton (1) are calculated as 8

10 Snce the annual rate of demand for each product s known, the total procurement cost for product per unt tme s obtaned as: mq md TCP = ; 1, 2,..., n T 1 p 2 (8) For each product, the setup process accrues only once and hence the set up cost per unt tme of the th product can be obtaned as: A AD TCO = ; 1,2,..., n T Q 1 p 2 (9) Assumng 100% nspecton and that all of the mperfect products transform to perfect ones after reworks, the nspecton of each product occurs once and ts assocated cost per unt tme s obtaned as IQ ID TCI ; 1,2,..., n T 1 p 2 (10) In order to calculate the holdng cost of WIP nventory of the th product, snce w denotes the average nvestment per unt of WIP nventory (ncludng raw materals, perfect and mperfect tems) and h represents the holdng cost rate of the th product, then TC h w (11) WIP The average raw materal nventory of each product s the total amount of raw materals (the surface under ts correspondng nventory graph) dvded by the cycle tme. Accordngly, the average nvestment value of the raw materal s obtaned by the product of the average raw materal nventory and the prce per unt of the raw materal. The average nvestment value of the perfect and mperfect products can be calculated smlarly. Hence, the average nvestment value per unt of the WIP nventory of product s gven n equaton (12). 9

11 1 1 1 QTP Q 1 p2 TP Qp2TP QTP w M c c M c T T T 2 T D RS S m 1 p1 Q M Rm p 21 p Q 2 (12) Hence, based on equatons (11) and (12), the average holdng cost of the WIP nventory of product becomes hd RS TC S m 1 p1 Q 2M R m 1 p1 ; 1,2,... n WIP 21 p2 Q (13) In order to calculate the holdng cost of the warehouse nventory, we frst need to estmate the average warehouse nventory. Regardng Fgure 4, we have 1 Q 1 p2 T 2 1 I Q 1 p2 (14) T 2 Hence, usng equaton (5) and (14) the holdng cost of the warehouse nventory for product becomes S 1 TCH hci h M R m p Q p Q (15) Fnally, the total annual nventory cost of all products descrbed n equaton (1) s gven n equaton (16). n TC TC TC TC TC TC 1 P O I WIP H 10

12 md AD ID 1p2 Q p p2 n hd RS S m 1 p1 Q M Rm p 1 21 p 2 Q (16) S 1 h M R m 1 p1 Q 1 2 p Q Problem formulaton As descrbed earler, the goal s to determne the economc producton quanttes of the products such that the total annual nventory cost obtaned n equaton (16) s mnmzed wthn the warehouse space lmtaton. Snce the space lmtaton can be modeled as n 1 p2 Qf F (17) 1 The mathematcal programmng model of the problem at hand becomes md AD ID 1p2 Q p p2 n hd RS Mn TC S m 1 p1 Q M Rm p 1 21 p 2 Q S 1 h M R m 1 p1 Q 1 2 p Q 2 s.t.: n 1p2 Q f F 1 (18) Q 0 and nteger ; 1, 2,... n In the next secton, an effcent algorthm s proposed to solve ths model. 11

13 4. A soluton algorthm In most EOQ or EPQ models that have been developed so far, researchers have tred to consder some constrants such as defectve tems, shortages, backorders, and so on such that the objectve functon of the model becomes concave and the model can easly be solved by some mathematcal approaches lke the Lagrangan or the dervatve methods. However, snce the objectve functon of the nonlnear nteger programmng model n (18) s a complex and sophstcated one, reachng an analytcal soluton (f any) s dffcult and tme-consumng (Gen & Cheng 1997). As a result, n ths secton a meta-heurstc stochastc search algorthm s used to solve the model. Many researchers have successfully used meta-heurstc methods to solve complcated optmzaton problems n dfferent felds of scentfc and engneerng dscplnes. Some of these meta-heurstc algorthms are smulatng annealng (Aarts and Korst (1989), Talezadeh et al. (2008)), threshold acceptng (Dueck and Scheuer (1990)), Tabu search (Joo and Bong (1996)), genetc algorthms (Pasanddeh & Nak (2006), Najaf & Nak (2006), Talezadeh et al. (2008b, 2009a, 2009b, 2009c), neural networks (Abbas & Nak (2007)) Gadock et al. (2002)), ant colony optmzaton (Dorgo and Stutzle (2004)), fuzzy smulaton (Talezadeh et al. (2009a), evolutonary algorthm (Laumanns et al. (2002), Talezadeh et al. (2009b), and harmony search (Lee and Geem (2004), Geem et al. (2001)). Among these meta-heurstc algorthms, the genetc algorthm has shown to be an effcent one to solve the nonlnear nteger programmng model of the problem at hand (Gen & Cheng 1997). The usual form of Genetc Algorthm (GA) was descrbed by Goldberg (1989). Snce then many researchers have appled and expanded ths concept n dfferent felds of study. Genetc algorthm was nspred by the concept of survval of the fttest. In genetc algorthms, the 12

14 optmal soluton s the wnner of the genetc game and any potental soluton s assumed to be a creature that s determned by dfferent parameters. These parameters are consdered as genes of chromosomes that could be assumed to be bnary strngs. In ths algorthm, the better chromosome s the one wth hgher ftness value. In practcal applcatons of genetc algorthms, populatons of chromosomes are created randomly. The sze of these populatons s dfferent n each problem. Some hnts about choosng the proper populaton sze exst n dfferent reports (Man et al. 1997). In the next subsectons, we descrbe the proposed GA to solve the model at hand Chromosomes In a GA, a chromosome s a strng or tral of genes, whch s consdered as the coded fgure of a possble soluton (approprate or none-approprate). Chromosome representaton s a very mportant part of the GA method descrpton. Whle n some researches a chromosome s compled n bnary code, a decmal (real)-mode code s used n others. The success of the codng format depends on the other routnes of GA, especally the crossover and the mutaton operatons (Gen & Cheng 1997). In ths paper, the chromosomes are strngs of the quanttes of the products ( Q j ) and are gven n real-mode code, and hence the crossover and the mutaton operators of ths research s based on the real-mode code of the chromosome that s descrbed later. Fgure (2) shows a typcal chromosome structure, n whch the genes are quantty of the products. Insert Fgure (2) about here 13

15 An nfeasble chromosome s defned as the one that does not satsfy the constrants of the model gven n equaton (18) Populaton A group of chromosomes s called populaton. One of the characterstcs of a GA s that nstead of focusng on a sngle pont of the search space (or one chromosome) t works on a populaton of chromosomes. Each populaton or generaton of chromosomes has the same sze whch s well-known as the populaton sze and s denoted by N. In ths research, the ntal populaton s randomly generated regardng the populaton szes that vary between 20 and Crossovers Crossover s the man genetc operator. In a crossover operaton, t s necessary to mate pars of chromosomes at a tme to create offsprng. Crossover operates on the parents chromosomes wth the probablty of P c. If no crossover occurs, the offsprng's chromosomes wll be the very same as ther parents. One smple way to acheve crossover s to randomly create a bnary chromosome correspondng to the chromosome at hand. Then the genes of the chromosome at hand that correspond to zeros n the bnary chromosome are not changed. However, those that correspond to ones are changed. Fgure (3) demonstrates the crossover operaton for four products. Insert Fgure (3) about here 14

16 In ths research, we use sngle pont crossover wth dfferent values of the P c parameter rangng between 0.45 and We note that an nfeasble chromosome that does not satsfy the constrants of the models (18) does not move to the new populaton Mutaton Mutaton s the second operaton n a GA method for explorng new solutons and t operates on each of the chromosomes resulted from the crossover operaton. Mutaton s a background operator, whch produces random change n chromosomes and may result n a chromosome wth hgher ftness value. In mutaton, we replace a gene wth a randomly selected number wthn the boundares of the parameter (Gen & Chen 1997). We create a random number RN between (0,1) for each gene. If RN s less than a predetermned mutaton probablty P m, then the mutaton occur n the gene. Otherwse, the mutaton operaton s not performed n that gene. Usual value of P m s 0.1 (sometmes 0.2) per chromosome or 0.1 (sometmes 0.05) over the numbers of genes n a chromosome. Based on the later approach, n ths research dfferent values between and 0.05 are chosen as dfferent values of P m. We note that an nfeasble chromosome that does not satsfy the constrants of the models n (18) does not move to the new populaton. Fgure (4) shows an example of the mutaton operator for four products n whch P m s chosen to be Insert Fgure (4) about here 15

17 4.5. Objectve Functon Evaluaton After producng the new chromosomes by crossover and mutaton operatons, we need to evaluate them. Whether a soluton (represented by a chromosome) s approprate or not depends on the objectve functon evaluaton. In a mnmzaton problem, the more approprate the soluton s the less the amount of the objectve functon (ftness value) wll be. The chromosomes that are the fttest wll take part n offsprng generaton wth more probablty. For the model at hand, the ftness value s the total nventory cost gven n equaton (18). For constraned optmzaton problems, the man ssue s to control the feasblty of the chromosomes. To do ths and to avod nfeasble chromosomes, the penalty polcy gven n Gen & Chen (1997) s employed. Snce we are faced wth a mnmzaton problem, a postve value s assgned to the penalty. Ths penalty s a squared functon of volaton of rght hand sde of the space constrant of the model. Thus, hgh penaltes are gven to more nfeasble chromosomes. For a feasble chromosome, the penalty s set to zero. In ths case, the ftness value of a chromosome s evaluated as the weghted sum of ts objectve functon Chromosome selecton After producng the offsprng wth crossover and mutaton operators and then evaluatng ther ftness value, the next populaton of sze N s made out of the chromosomes wth the hghest ftness. Ths selecton strategy s called the determnstc mechansm Stoppng crtera The last step n a GA method s to check f the algorthm has found a soluton that s good enough to meet the user s expectatons. Stoppng crtera s a set of condtons such that when 16

18 satsfed a good soluton s obtaned. Dfferent crtera used n lterature are as follows: 1) Stoppng of the algorthm after a specfc numbers of generatons, 2) reachng a maxmum number of evaluatons, 3) no mprovement n the objectve functon, and 4) reachng a specfc value of the objectve functon. algorthm. In ths research, a combnaton of the frst and the thrd crteron s used to stop the In short, the steps nvolved n the G.A algorthm used n ths research are: 1. Settng the parameters P c, P m and N 2. Intalzng the populaton randomly 3. Evaluatng the objectve functon for all chromosomes 4. Selectng ndvdual for matng pool 5. Applyng the crossover operaton for each par of chromosomes wth probablty P c 6. Applyng mutaton operaton for the genes of the chromosome wth probablty P m 7. Replacng the current populaton by the resultng new populaton 8. Evaluatng the objectve functon 9. If stoppng crtera s met, then stop. Otherwse, go to step 4 The proposed GA s coded usng the embedded algorthm n MATLAB. In order to demonstrate the applcaton of the proposed GA and to evaluate ts performance, n the next secton we brng a numercal example. 17

19 5. A numercal example Consder a four-product nventory control problem wth annual rate of producton P, annual demand rate of cycle of of D, annual nventory holdng cost per unt of h, fxed orderng cost per A, raw materal purchasng cost per unt of R, annual setup tme of S, machnng tme per unt of M, producton cost per unt per unt tme m, mperfect producton percentage of p 1, scrap producton percentage of p 2, and average amount of nventory I. The requred warehouse space per perfect unt s f and F s the total avalable warehouse space. The correspondng numercal data are gven n Table (1). The total avalable warehouse space s The goal s to fnd the nteger order quanttes of the products ( Q ; 1,2,3,4 as the decson varables) such that the total nventory cost s mnmzed whle the warehouse space constrant s satsfed. Insert Table (1) about here Snce the parameters of many meta-heurstc algorthms lke GA play mportant roles n the qualty of the soluton and that wthout tunng the parameters the algorthm may not work properly, they need to be adjusted such that a better soluton s reached. As a result, n ths research, a regresson analyss usng the SAS software s used to dentfy the sgnfcant parameters of the proposed GA and to fnd a proper relatonshp between the response (the qualty of the soluton defned as the total cost) and the parameters. At the end, the LINGO software wll be used to optmze the relatonshp functon and to fnd the optmal values of the sgnfcant GA parameters. 18

20 To fnd the optmal values of the GA parameters, usng MATLAB computer software, the algorthm s employed 110 tmes, each tme changng ts parameters n ther correspondng ranges and obtanng the response value. The crossover and mutaton operatons rates vary n the range of and , respectvely. Furthermore, dfferent nteger populaton szes between 20 and 60 are consdered n ths experment. Table (2) shows the results of the expermentaton. Insert Table (2) about here In the next step, all possble regresson procedure (Neter et al. 1999) was used to fnd the sgnfcant GA parameters. Table (3) shows the results of dfferent crtera for dfferent regresson functons. These results show that the model wth all GA parameters s the best. Insert Table (3) about here In order to study the nteracton effects between the GA parameters, consder the regresson equaton ncludng all man and nteracton effects n (19). E ( Y ) N Pc Pm NPc NPm PcPm NPcPm (19) Gven the data n Table (2), the Backward Elmnaton Procedure of the SAS software was then employed to study the regresson model n (19). The fnal analyss of varance table of ths procedure s gven n Table (4). Furthermore, the regresson parameter estmates are gven n Table (5). Moreover, the summary of the steps n backward elmnaton procedure s gven n Table (6). 19

21 Insert Table (4) about here Insert Table (5) about here Insert Table (6) about here Based on the results of Table (5), the estmated regresson functon s: Y N P NP P P NP P (20) m m c m c m The Lngo software solves the estmated regresson functon (the objectve functon) that needs to be mnmzed along wth the constrants (the space constrant and nteger chromosome) wthn the GA parameter ranges. The optmum values of the GA parameter are 0.85, 0.05 and 60 for crossover rate, mutaton rate and pop-sze respectvely. Table (7) shows the optmum results. Insert Table (7) about here Fnally, the proposed GA wth the optmal parameters gven n Table (7) employed to the problem at hand. The graph of the ftness value n terms of the generaton numbers s gven n Fgure (5). The results of Fgure (5) show that the mnmum cost s and the algorthm converges after 5 generatons. At ths cost, the optmal decson varables are gven n Fgure (6). Note that the meta-heurstc GA provdes a near optmal soluton and not necessarly the optmal one. 20

22 Insert Fgure (5) about here Insert Fgure (6) about here In order to demonstrate the proposed GA s an effectve procedure to solve the multproduct economc producton quantty model, example (1) s solved as a nonlnear nteger model by Lngo software as well. Fgure (7) shows the optmal values of decson varables along wth the optmal cost of These values are obtaned at the 978 th teraton of the software. However, we note that the optmum soluton was obtaned n the ffth generaton of the proposed GA. Ths means that whle the proposed GA s an effectve tool to provde a near optmum soluton, t s a faster approach. 6. Conclusons and recommendatons for future research One nevtable aspect of manufacturng systems s producng defectve products. In ths paper, we developed a mult-product EPQ model n whch defectve tems and reworkng are consdered. In addton, the warehouse space s lmted for all products. In ths condton, we formulated the problem as a nonlnear nteger programmng and solved t usng a genetc algorthm. At the end, a numercal example was presented to demonstrate not only the applcaton of proposed methodology, but also to evaluate the effectveness of the GA algorthm n solvng the mult-product economc order quantty model of ths research. In ths example, the optmum values of the GA parameters were obtaned usng regresson analyss along wth LINGO software. Some recommendatons for future research follow: 21

23 a) Some gradent-based algorthms such as sequental quadratc programmng or other heurstc search technques such as ant colony optmzaton, smulated annealng, etc. may be used to solve the problem at hand and compare the results wth the ones of the proposed genetc algorthm. b) Some other lmtatons such as delay, shortage, budget lmtaton, etc. can be augmented to the model. c) The model can be extended to nvolve stochastc or fuzzy natures of some parameters such as the demand rate. 7. Acknowledgement The authors are thankful for the constructve comments of the anonymous revewers that mproved the presentaton of the paper. 8. Refrences Aarts, E.H.L. and J.H.M. Korst, (1989), Smulated Annealng and Boltzmann Machne, A Stochastc Approach to Computng. John Wley and Sons, Chchester, USA. Abbas B. and S.T.A. Nak (2007), Montorng Hgh-Yelds Processes wth Defects Count n Nonconformng Items by Artfcal Neural Network, Appled Mathematcs and Computaton, 188: Buscher, U. and G. Lndner (2007), Optmzng a Producton System wth Rework and Equal Szed Batch Shpments, Computers and Operatons Research, 34:

24 Ln, CS (1999), Integrated producton-nventory models wth mperfect producton processes and a lmted capacty for raw materals, Mathematcal and computer modelng, 29: Chu, S.W., S.L.Wang and Y.S.P. Chu (2007), Determnng the Optmal Run Tme for EPQ Model wth Scrap, Rework and Stochastc Breakdowns, European Journal of Operatonal Research, 180: Dorgo, M. and T. Stutzle (2004), Ant colony optmzaton, MIT Press, Cambrdge, MA, USA. Dueck, G. and T. Scheuer, (1990), Threshold Acceptng: A General Purpose Algorthm Appearng Superor to Smulated Annealng, J Comput Phys, 90: Gaduk, A.R., Y.A. Vershnn, and M.J. West (2002), Neural networks and optmzaton problems. In Proceedngs of IEEE 2002 Internatonal Conference on Control Applcatons, 1: Geem, Z.W., J.H. Km, and G.V. Loganathan (2001), A New Heurstc Optmzaton Algorthm: Harmony Search, Smulaton, 76: Gen, M. and R. Cheng (1997), Genetc Algorthm and Engneerng Desgn, John Wley& Sons, New York, NY, USA. Goldberg, D. (1989), Genetc Algorthm n Search, Optmzaton and Machne Learnng, Addson-Wesley, Readng, MA, USA. Harrs, F.W. (1913), How many parts to make at once. Factory, The magazne of management, 10:

25 Hayek, P.A. and M.K. Salameh (2001), Producton Lot Szng wth the Reworkng of Imperfect Qualty Items Produced, Producton Plannng and Control, 12: Hou, K.L. (2007), An EPQ Model wth Set Up Cost and Process Qualty as Functons of Captal Expendture, Appled Mathematcal Modelng, 31: Joo, S.J. and J.Y. Bong (1996), Constructon of Exact D-Optmal Desgns by Tabu Search, Comput Stat Data Anal, 21: Km, C.H. and Y. Hong (1999), An Optmal Producton Run Length n Deteroratng Producton Processes, Internatonal Journal of Producton Economcs, 58: Laumanns, M., L. Thele, K. Deb and E. Ztzler (2002), Combnng Convergence and Dversty n Evolutonary Mult-Objectve Optmzaton, Evol Comput, 10: Lee, H.M. and J.S. Yao (1998), Economc Producton Quantty for Fuzzy Demand Quantty and Fuzzy Producton Quantty, European Journal of Operatonal Research, 109: Lee, K.S. and Z.W. Geem (2004), A New Structural Optmzaton Method Based on the Harmony Search Algorthm, Comput Struct 82: Lao, G.L., Y.H. Chen and S.H. Sheu (2009), Optmal Economc Producton Quantty Polcy for Imperfect Process wth Imperfect Repar and Mantenance, European Journal of Operatonal Research, n press. Man, K.F., K.S. Tang, S. Kwong and W.A. Halang (1997), Genetc Algorthms for Control and Sgnal Processng. Sprnger Verlag, London). 24

26 Najaf, A.A. and S.T.A. Nak (2006), A Genetc Algorthm for Resource Investment Problem wth Dscounted Cash Flows, Appled Mathematcs and Computaton, 183: Neter, J., M.H. Kutner, C.J. Nachtshem, and W. Wasserman (1996). Appled Lnear Regresson Models, The McGraw-Hll Companes, Inc., Chcago, USA. Pasanddeh, S.H.R. and S.T.A. Nak (2006), Mult-Response Smulaton Optmzaton Usng Genetc Algorthm wthn Desrablty Functon Framework, Appled Mathematcs and Computaton, 175: Rosenblatt, M.J. and H.L. Lee (1986), Economc Producton Cycles wth Imperfect Producton Processes, IIE Transactons, 18: Salameh, M.K. and M.Y. Jaber (2000), Economc Producton Order Quantty Model for Items wth Imperfect Qualty, Internatonal Journal of Producton Economcs, 64: Talezadeh, A.A., M.B. Aryanezhad, and S.T.A. Nak (2008a), Optmzng Mult-Product Mult- Constrant Inventory Control Systems wth Stochastc Replenshment, Journal of Appled Scences, 8: Talezadeh, A.A., S.T.A. Nak, and V. Hossen, (2008b), The Mult-Product Mult-Constrant Newsboy Problem wth Incremental Dscount and Batch Order, As J Appl Sc, 1: Talezadeh, A.A., S.T.A. Nak, and M.B. Aryanezhad (2009a), Mult-Product Mult-Constrant Inventory Control Systems wth Stochastc Replenshment and Dscount under Fuzzy Purchasng Prce and Holdng Costs, Amercan Journal of Appled Scences, 6:

27 Talezadeh, A.A., S.T.A. Nak, and M.B. Aryanezhad (2009b), A Hybrd Method of Pareto, TOPSIS and Genetc Algorthm to Optmze Mult-Product Mult-Constrant Inventory Systems wth Random Fuzzy Replenshment, Journal of Mathematcal and Computer Modellng, 49: Talezadeh, A.A., S.T.A. Nak, and V. Hossen (2009c), Optmzng Mult-Product Mult- Constrant B-Objectve Newsboy Problem wth Dscount by a Hybrd Method of Goal Programmng and Genetc Algorthm, Journal of Engneerng Optmzaton, 41:

28 Q Fg. a Fg. c pq 2 tme tme 1 p Q 1 p 2 Fg. b 2 Q Fg. d tme tme TP T TP T Fgure (1): Inventory graphs of raw materal (Fg. a), perfect products (Fg. b), scrap tems (Fg. c) and the product n the warehouse (Fg. d) 27

29 Q1 Q2 Q Qn Fgure (2): The structure of a chromosome 28

30 Bnary Chromosome [ ] [ ] [ ] [ ] [ ] Fgure (3): An example of a crossover operaton 29

31 [ ] [ ] Randomly Generated Chromosome [ ] Fgure (4): A graphcal representaton of the mutaton operator 30

32 Fgure (5): The graph of the convergence path 31

33 Q1 6 Q2 72 Q3 85 Q4 52 Fgure (6): The optmal values of the decson varables by the proposed GA 32

34 Q1 6 Q 2 70 Q3 88 Q 4 51 Fgure (7): The optmal values of the decson varables by LINGO 33

35 Table (1): Numercal data of the example p 1 p 2 A Product M D S m R h I f

36 Table (2): The expermental results No. N P c P m Ftness ( Y ) No. N P c P m Ftness ( Y )

37

38 Table (3): All possble regresson results No. of No. of Regresson Model Varables Parameters ( p ) Varables 2 R p SSR p SSE p MSE 2 p C R p p 1 2 N *10^ P c P m N, P c *10^ N, P m *10^ Pc, P m N, P, P *10^ c m 37

39 Table (4): The analyss of varance table Source DF SS MS F-Value Pr > F Model E < Error Corrected Total E11 38

40 Table (5): The parameter estmates Varable Parameter Estmate Standard Error Type II SS F-Value Pr > F Intercept < N < P m NP m PP c m NP P < c m 39

41 Table (6): The summary of backward elmnaton method Number Varable of Partal R- Model R- Step Removed Varables Square Square n 1 c C p F-Value Pr > F P NP c

42 Table (7): The Lngo optmum soluton * * * N P c P m Cost