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1 Informaton Scences 0 (013) Contents lsts avalable at ScVerse ScenceDrect Informaton Scences journal homepage: A hybrd method of fuzzy smulaton and genetc algorthm to optmze constraned nventory control systems wth stochastc replenshments and fuzzy demand Ata Allah Talezadeh a,b, Seyed Tagh Akhavan Nak c,, Mr-Bahador Aryanezhad d, Nma Shaf e a Department of Industral Engneerng, Iran Unversty of Scence and Technology, Tehran, Iran b Department of Industral Management, Raja Unversty, Qazvn, Iran c Sharf Unversty of Technology, P.O. Box Azad Ave., Teharn, Iran d Iran Unversty of Scence and Technology, Tehran, Iran e Faculty of Engneerng of the Unversty of Porto, Rua Dr. Roberto Fras, Porto, Portugal artcle nfo abstract Artcle hstory: Receved 18 January 009 Receved n revsed form 1 March 01 Accepted 19 July 01 Avalable onlne 7 August 01 Keywords: Multproduct nventory control Partal back-orderng Stochastc replenshment Integer nonlnear programmng Fuzzy smulaton Genetc algorthm Mult-perodc nventory control problems are manly studed by employng one of two assumptons. Frst, the contnuous revew, where dependng on the nventory level, orders can happen at any tme, and next the perodc revew, where orders can only be placed at the begnnng of each perod. In ths paper, we relax these assumptons and assume the tmes between two replenshments are ndependent random varables. For the problem at hand, the decson varables (the maxmum nventory of several products) are of nteger-type and there s a sngle space-constrant. Whle demands are treated as fuzzy numbers, a combnaton of back-order and lost-sales s consdered for the shortages. We demonstrate the model of ths problem s of an nteger-nonlnear-programmng type. A hybrd method of fuzzy smulaton (FS) and genetc algorthm (GA) s proposed to solve ths problem. The performance of the proposed method s then compared wth the performance of an exstng hybrd FS and smulated annealng (SA) algorthm through three numercal examples contanng dfferent numbers of products. Furthermore, the applcablty of the proposed methodology along wth a senstvty analyss on ts parameters s shown by numercal examples. The comparson results show that, at least for the numercal examples under consderaton, the hybrd method of FS and GA shows better performance than the hybrd method of FS and SA. Ó 01 Elsever Inc. All rghts reserved. 1. Introducton and lterature revew The contnuous revew and the perodc revew are the man appled polces n mult-perodc nventory control models. However, the underlyng assumptons of these models restrct ther proper use n real-world envronments. In contnuous revew polcy, one has the freedom to act anytme and place orders based upon the avalable nventory level. Whle n the perodc revew polcy, the user s allowed to place orders only n specfc and predetermned tmes. Two of the wdely employed perodc revew polces are the so-called (R,T) and (R,nT) polces. In the frst one, at fxed predetermned ntervals, T, the nventory s revewed and an order s placed accordngly. The order quantty s determned by Correspondng author. Tel.: ; fax: E-mal addresses: Talezadeh@ust.ac.r, Talezadeh@raja.ac.r (A.A. Talezadeh), Nak@Sharf.edu (S.T.A. Nak), Mrarya@ust.ac.r (M.-B. Aryanezhad) /$ - see front matter Ó 01 Elsever Inc. All rghts reserved.

2 46 A.A. Talezadeh et al. / Informaton Scences 0 (013) subtractng the on hand nventory from a predetermned value R. If ths polcy s used n an n-echelon nventory system, t s called (R,nT) polcy. Further, the economc order quantty (EOQ) model along wth the (r,q) polcy are the two other major perodc revew nventory systems, where n the former the purchaser desres to determne the optmal quantty of the order whle n the latter the optmal values of the reorder pont and order quantty are sought. There s substantal research reported n the lterature n the area of mult-perodc nventory control. Some of these works are summarzed n Tables 1 and. Table 1 (except for Talezadeh et al. [39]) shows the man research efforts n the stochastc envronment dealng wth (R, T) and (r, Q) systems n whch demand and lead-tme are consdered stochastc varables. The man constrants shown n these works are servce level [30,39], order quantty [3], and jont order [13,33]. Although the decson varables such as order quantty, nventory level, reorder pont and perod length are smlar, varous assumptons have been made and dfferent models and procedures have been proposed. For example Chang [6] and Bylka [3] consdered emergency orders, Feng and Rao [14] assumed non-zero lead-tme, Eynan and Kropp [13] employed varable shortage cost, Mohebb [9] used a dscrete-pattern demand, and Qu et al. [33] consdered an ntegrated nventory transportaton system. Table shows the man research efforts n the fuzzy envronment performed on EOQ and economc producton quantty (EPQ) models. In ths category, the man constrants are budget [8,41], space [8,34,41], and servce level [41]. Whle the decson varables are smlar to the ones n the stochastc envronment, the demand and nventory costs are consdered fuzzy varables. In some of these research undertakngs such as [19,6,34] producton rate, prce, and deteroraton rate are consdered fuzzy varables as well. A careful observaton of the works lsted n Tables 1 and reveals that whle separate emphass has been devoted to the stochastc nature of demand and lead-tme, some real-world constrants of the systems have not been nvestgated smultaneously. For example, no work s reported where both demand and lead-tme are probablstc. Furthermore, some constrants have been partally studed, the decson varables have been consdered nteger, and constrants such as budget and space have not been nvestgated. In addton, many researchers have successfully used meta-heurstc methods to solve complcated optmzaton problems n dfferent felds of scence and engneerng. Some of these meta-heurstc algorthms are: fuzzy smulaton [41,44], genetc algorthms [], harmony search [3,17,40 43], smulatng annealng [1,39,44], ant colony optmzaton [10], neural networks [15], threshold acceptng [11], Tabu search [0], and evolutonary algorthm [,41]. Ths paper frst extends the perodc revew works n both stochastc and fuzzy envronments such that the replenshment ntervals become random and demands assume a fuzzy nature. Then, t presents a hybrd algorthm to solve the problem. To be more specfc, the extended model assumes a stochastc replenshment,.e., stochastc perod length, multple products that are stored n a sngle capacty-constraned warehouse, fuzzy customer demand, and nteger decson varables. Snce the tme between two replenshments ncludes the tme requred to order, the tme needed to provde or produce tems and the transportaton tme, the stochastc replenshment assumpton s closer to realty than the usual assumpton of a determnstc perod length. Furthermore, there are many stuatons n practce where the customer demand s fuzzy, especally n manufacturng where due to the machne breakdowns, shortage of raw materals, and fluctuatng rate of nonconformng producton, and so on, the demand for on hand nventores of parts and subassembles can be consdered fuzzy. The hybrd algorthm conssts of a GA for nventory control optmzaton and a method for fuzzy smulaton to evaluate dfferent solutons n the genetc optmzaton process. The models developed n ths research are useful for companes and manufacturers who are faced wth uncertan demands that do not follow a stochastc pattern. In other words, manufacturers who are unable to assume certan probablty dstrbutons for the uncertan demands of ther products can use fuzzy set theory to fnd sutable patterns. Addtonally, the proposed method s benefcal n stuatons where due to some lmtatons on the producton capacty, the supply of the raw materal, and the lke, the perod length may be uncertan and the goods may not be delvered on tme. As an example, when demand ncreases and producton capacty s lmted, n case of breakdowns or late recepts of mported raw materals (when delayed at customs) the lead-tme and hence the cycle length ncreases. Another example nvolves sale representatves that randomly vst retalers offerng them product replenshments. The stochastc nature of these factors causes the perod length to be stochastc. The rest of ths paper s organzed as follows. In Secton, the problem along wth ts assumptons s defned. The problem formulaton comes n Secton 3 after the parameters and the varables are defned. In ths secton, the sngle product problem s frst modeled and then t s extended to a multproduct formulaton. In the fourth secton of the paper, a hybrd algorthm s proposed to solve and analyze the problem at hand under specal condtons. By ncorporatng a numercal example, the soluton method s nvestgated n Secton 5. Secton 6 contans a senstvty analyss, and fnally the concluson and recommendatons for future research come n Secton 7.. Problem defnton and modelng Consder a perodc-revew nventory control model for one provder n whch the perod lengths are stochastc n nature,.e., the tmes between two replenshments are ndependent random varables followng ether Unform or Exponental probablty dstrbuton. Trangular fuzzy varables are used to model the demands of several products, and the partal backorderng polcy s employed for shortages,.e., a fracton of unsatsfed demands s lost and the rest s back-ordered.

3 Table 1 Lterature revew n stochastc envronment. Author Perodc revew Contnuous revew Mult products Constrant Dscount Fuzzy envronment Stochastc envronment Partal backorderng Lost sale or back-order Soluton method Decson varable Other consderatons Chang [6] (R,T) Demand B Dynamc Inventory level Emergency order programmng Mohebb and Posner [30] (r,q) Demand and lead tme L Heurstc Order quantty Multple replenshment Ouyang and Chang [3] Feng and Rao [14] (R,T) Servce level Demand Heurstc Revew perod (R,nT) (r,q) Demand B Heurstc Order quantty, reorder pont, nventory level Bylka [3] (R,T) Order and space Chang et al. [7] Eynan and Kropp [13] Mohebb [9] Qu et al. [33] Talezadeh et al. [39] Demand Heurstc and Markov process Regular and emergency order quantty Inventory level and order quantty Non-zero lead tme Emergency order and two supplers (R,T) Demand B and L Dynamc programmng Two models wth back orderng and lost sale (EOQ) Jont order Demand B Heurstc Perod length Varable shortage cost (r,q) Demand L Software (MATLAB) Reorder pont and order quantty (R,T) Jont order B Heurstc Rout, order quantty and perod length (R,T) Servce Perod Smulated Inventory level level and length annealng space Dscrete demand and unrelable suppler Integrated nventorytransportaton system A.A. Talezadeh et al. / Informaton Scences 0 (013)

4 Table Lterature revew n fuzzy envronment. Author Perodc revew Contnuous revew Multproduct Constrant Fuzzy envronment Lost sale or backorder Chang et al. [4] (r,q) Demand B and L Dfuzzfcaton and heurstc Das et al. (EOQ) Budget and Demand B Dfuzzfcaton and [8] space heurstc Hseh [19] (EPQ) Demand, Heurstc and producton rate, Lagrangan nventory costs Lu [5] (EOQ) Batch order Demand, nventory Possblty theory and costs geometrc programmng Mat and Mat [6] Mandal and Roy [8] Roy et al. [34] Talezadeh et al. [41] Yao et al. [46] Chen et al. [5] (r,q) Demand and prce Fuzzy smulaton and genetc algorthm (EOQ) Space Constrant goal, nventory costs Soluton method Decson varable Other consderatons Fuzzy geometrc programmng (r,q) Space Deteroraton rate Fuzzy smulaton and genetc algorthm (R,T) Budget, space and servce level Inventory costs Fuzzy smulaton and genetc algorthm Lead tme and order quantty Order and shortage quanttes Producton quantty Order quantty Reorder pont, order quantty, sellng prce, frequency of advertsements Order quantty Order quantty Inventory level (EOQ) Demand Heurstc Order quantty (EOQ) Demand, nventory costs B Functon prncple Order and shortage quanttes Varable lead-tme Tme varyng demand and producton rate All of the parameters are fuzzy Two storages, advertsement, sngle and mult objectve Mult objectve Stochastc perod length, two storage facltes, tme varyng demand Partal back orderng and ncremental dscount, stochastc perod length 48 A.A. Talezadeh et al. / Informaton Scences 0 (013)

5 A.A. Talezadeh et al. / Informaton Scences 0 (013) Moreover, when demands are hgher than replenshment levels, the back-ordered quanttes n the prevous cycle are carred out to the next cycle. In stuatons n whch demands are lower than replenshment levels, extra nventores are carred out to the followng cycle and are assumed to have an nsgnfcant mpact on the ndependence of the two perods. Assumng all the produced tems are sold, the costs assocated wth the nventory control system conssts of holdng, back-order, lost-sale, and purchase costs. Furthermore, the warehouse space s consdered a constrant and the decson varables are nteger. We need to dentfy the maxmum nventory levels n each cycle such that the expected proft s maxmzed. For the problem at hand, snce the tmes between two replenshments are ndependent random varables, n order to maxmze the expected proft over the plannng horzon one needs to consder only one perod. Moreover, snce the costs assocated wth the nventory control system are holdng and shortage (back-order and lost-sale), we need to calculate the expected nventory level and the expected requred storage space n each perod. Before dong ths, parameters and varables of the model are defned based on the works of Talezadeh et al. [39,41,4])..1. Defnng parameters and varables of the model For =1,,...,n, defne the parameters and the varables of the model as T h p W P D b I L B Q f F C h C b C l C p r Z Maxmum nventory level of the th product A random varable denotng the tme between two replenshments (cycle length) of the th product Holdng cost per unt nventory of the th product n each perod Back-order cost per unt demand of the th product Purchasng cost per unt of the th product Sellng prce per unt of the th product Constant demand rate of the th product Percentage of unsatsfed demands of the th product that s back-ordered Expected th product nventory multpled by the cycle tme Expected th product lost-sale n each cycle Expected th product back-order n each cycle Expected order sze of the th product n each cycle Requred warehouse space per unt of the th product Total avalable warehouse space Expected holdng cost per cycle Expected shortage cost n back-order state Expected shortage cost n lost-sale state Expected purchase cost Expected revenue obtaned from sales Expected proft obtaned n each cycle The pctoral representaton of the sngle-product problem s gven n Secton.. In Secton.3, we frst consder a sngleproduct problem, and then, the formulaton to the mult-product modelng s extended n Secton.4... Inventory dagram Accordng to Ertogal and Rahm [1] and consderng the tmes between replenshments stochastc varables, two cases may occur. In the frst case the tme between replenshments s less than the tme requred for the nventory level to reach zero (see Fg. 1), and n the second, t s greater (see Fg. ) [39,41,4]. Fg. 3 depcts the shortages n both cases. In the above fgures, t D denotes the tme at whch the nventory of the th product reaches zero. I D T t D t Fg. 1. Presentng the nventory cycle when T Mn 6 T 6 t D.

6 430 A.A. Talezadeh et al. / Informaton Scences 0 (013) I D T D T t D t DT R td T Fg.. Presentng the nventory cycle when t D < T 6 T Max. I D T DT R D t D T t D T β ( DT R ) ( β )( DT R ) 1 t Fg. 3. Presentng shortages n two cases of compact back order and lost sales..3. Sngle product model wth constant demand In ths secton, we frst model a sngle-product nventory problem wth constant demand where stochastc replenshments, back-orders, and lost-sales are allowed. Then, the model s extended n Secton.4 to contan several products wth fuzzy demands Calculatng the costs and the proft In order to calculate the expected proft n each cycle, we need to evaluate all the terms n the followng equaton [1]: Z ¼ r C p C h C b C l ¼ PQ WQ hi pb ðp WÞL Based on Fg. 3, L, B, I, and Q are evaluated by the followng equatons [39,41,4]: L ¼ð1 bþ Z TMax t D ðdt RÞf T ðtþdt; t D < T 6 T Max ðþ Z TMax B ¼ b ðdt RÞf T ðtþdt; t D < T 6 T Max ð3þ t D Z td I ¼ T Mn! RT DT f T ðtþdt þ Z TMax t D R D f TðtÞdt ð1þ ð4þ Z td Z TMax Q ¼ DTf T ðtþdt þ ðr þ bðdt RÞÞf T ðtþdt ð5þ T Mn t D.3.. Presentng the constrants Snce the total avalable warehouse space s F, the space requred for each unt of product s f, and the upper lmt for nventory s R, the space constrant wll be fr 6 F In short, the complete mathematcal model of the sngle-product nventory control problem wth stochastc replenshments, back-orders, lost-sales, and constant demand s ð6þ

7 Max " # Z R D Z TMax Z ¼ðP WÞ ðdtþf T ðtþdt þ ðr þ bðdt RÞÞf T ðtþdt T R Mn D " Z! R D h RT DT Z # TMax R f T ðtþdt þ T Mn R D f TðtÞdt D " Z # " TMax Z # TMax pb ðdt RÞf T ðtþdt ðp WÞð1 bþ ðdt RÞf T ðtþdt R D A.A. Talezadeh et al. / Informaton Scences 0 (013) s:t: : fr 6 F ð7þ R P 0; and Integer R D.4. Multproduct model wth fuzzy demand In the extendng phase of the sngle-product model of Secton.3 to the multple product formulaton of ths secton, the demands are assumed fuzzy and n addton to the two cases of back-order and lost-sales, ther combnaton s consdered as well. Let e D denote the fuzzy demand for the th product. Then, an extenson of (7) to nclude n products easly results n the multproduct model as Max s:t: : Zð ; D e Þ¼ Xn ½ðP W ÞQ h I p B ðp W ÞL Š X n f 6 F ¼ Xn 8 >< Z 6 ðp W Þ e 4 D ðd e T Þf T ðt Þdt þ >: T Mn Z h e D D T e! 6 T 4 f T ðt Þdt þ Xn Xn 6 p b 4 P 0; and Integer T Mn Z TMax e D 3 Z TMax e D Z TMax e D 39 >= ð þ b ðd e 7 T ÞÞf T ðt Þdt 5 >; R 3 7 D e f T ðt Þdt 5 ðd e 7 T Þf T ðt Þdt 5 Xn 6 ðp W Þð1 b Þ4 8 ¼ 1; ;...; n Z TMax e D 3 ðd e 7 T Þf T ðt Þdt 5 ð8þ In what follows, two probablty densty functons for T are assumed and hence two models are developed. In the frst model, T follows a unform dstrbuton, where the demands may occur n a fnte and specfc range (wthn an upper and a lower bound). In the second model, T follows an exponental dstrbuton, where the demands may ncrease sharply. Ths model s sutable for seasonal or new products T follows a unform dstrbuton 1 In ths case T follows a unform dstrbuton n the nterval ½T Mn ; T Max Š,.e., T U½T Mn ; T Max Š and f T ðt Þ¼T Max. T Mn Accordngly, (8) s changed to " # " # Max Zð ; D e Þ¼ Xn h 6D e ðt R 3 Xn ðp W Þð1 b Þþp b þ h T Max Max T Mn Þ D e R ðt Max T Mn Þ " # s:t: : X n f 6 F þ Xn þ Xn P 0; Integer; 4ðP W Þð1 b ÞT Max þ h T Mn þ p b T Max ðt Max T Mn Þ " h T 3 e Mn D þ 3ðP W Þðb T Max T Mn ÞD e # 3T e Max D ðp b þðp W Þð1 b ÞÞ 6ðT Max T Mn Þ 8 ¼ 1; ;...; n ð9þ

8 43 A.A. Talezadeh et al. / Informaton Scences 0 (013) T follows an exponental dstrbuton If T follows an exponental dstrbuton wth parameter k, then the probablty densty functon of T s f T ðt Þ¼k e k T.In ths case, the model s derved as Max >< Zð ; D e Þ¼ Xn 1 ½D k e ð1 b ÞðW P Þ p b e D Še >: e D k þ 1 ½D k e ðp W Þ h Šþ h D e 1 e k e D k >= C A >; s:t: : X n f 6 F P 0; and Integer 8 ¼ 1; ;...; n ð10þ In the next secton, a hybrd ntellgent algorthm s ntroduced to fnd near optmum solutons of the formulated problems n (9) and (10). 3. A hybrd ntellgent algorthm Snce analytcal solutons (f any) of the nteger-nonlnear models n (9) and (10) are hard to obtan [16], a hybrd ntellgent algorthm of fuzzy smulaton and genetc algorthm s developed n ths secton. Some related research that have employed the fuzzy smulaton approach along wth other meta-heurstc algorthms nclude [0,37,41,44]. In the next subsecton, a bref background n fuzzy smulaton s gven Some defntons n fuzzy envronment In ths paper, we adopt the concepts of the credblty theory ncludng possblty, necessty, credblty of fuzzy events, and the expected value of a fuzzy varable as defned n [1,4,47 51]. Defnton 1. Let n be a fuzzy varable wth the membershp functon l(x). Then the possblty, necessty, and credblty measures of the fuzzy event n P r can be represented, respectvely, by Posfn P rg ¼suplðuÞ upr Necfn P rg ¼1 suplðuþ u<r ð11þ ð1þ Crfn P rg ¼ 1 ½Posfn P rgþnecfn P rgš ð13þ Defnton. The expected value of a fuzzy varable s defned as E½nŠ ¼ Z 1 Crfn P rgdr Z Crfn 6 rgdr ð14þ Defnton 3. The optmstc functon of a s defned as n sup ðaþ ¼sup½rjCrfn P rg P aš; a ð0; 1Š ð15þ Defnton 4. If ~ n ¼ða; b; cþ s a trangular fuzzy number wth center b, left wdth a > 0, and rght wdth c > 0, then ts membershp functon has the followng form 8 r ðb aþ >< ; b a 6 r 6 b a lðrþ ¼ ðbþcþ r ; b 6 r 6 b þ c ð16þ c >: 0 : elsewhere Defnton 5. For the fuzzy varable descrbed n Defnton 4, the credblty of the event Cr{n 6 r} s defned based on the defnton n (13) as

9 8 0; r 6 b a >< r ðb aþ ; b a 6 r 6 b a lðrþ ¼ r ðb cþ ; b 6 r 6 b þ c c >: 1; elsewhere In ths research, the trangular fuzzy varable s used to model the fuzzy demand. 3.. Fuzzy smulaton A fuzzy smulaton technque s employed to estmate the fuzzy demands. Denotng ~ n by D e = ðd e 1 ; D e ;...; D e n Þ, l as the membershp functon of D, e and l as the membershp functons of D e, we randomly generate D k from the a-level sets of fuzzy varables D e, =1,,..., n and k =1,,..., K as D k ¼ D k 1 ; Dk ; ; Dk n and lðd k Þ¼l 1 ðd k Þ^l 1 ðdk Þ^;...; ^l n ðdk nþ, where a s a suffcently small postve number. Then, the expected value of the fuzzy varable s E½ZðR; e DÞŠ ¼ Z þ1 0 CrfZðR; e DÞ P rgdr Z 0 1 CrfZðR; e DÞ 6 rgdr Provded that O s suffcently large, for any number r P 0, Cr{Z(R,D k ) P r} can be estmated by CrfZðR; D k Þ P rg ¼ 1 Max fl k jzðr; D k Þ P rgþ1 Max fl k jzðr; D k Þ < rg k¼1;;...;o k¼1;;...;o And for any number r <0,Cr{Z(R, D k ) 6 r} can be estmated by CrfZðR; D k Þ 6 rg ¼ 1 Max fl k jzðr; D k Þ 6 rgþ1 Max fl k jzðr; D k Þ > rg k¼1;;...;o k¼1;;...;o The procedure of estmatng ZðR; e DÞ n (19) and (0) s shown n the followng algorthm: Algorthm 1. Estmatng ZðR; e DÞ A.A. Talezadeh et al. / Informaton Scences 0 (013) ð17þ ð18þ ð19þ ð0þ 1. Set E = 0 and ntalze K and O.. Randomly generate D k from a -level sets of fuzzy varables D e ; and set D k ¼ðD k 1 ; Dk ; ; Dk n Þ 3. Set a = Z(R, D 1 ) ^ Z(R, D ) ^^Z(R, D O ), b ¼ ZðR; D 1 Þ_ZðR; D Þ ZðR; D O Þ. 4. Randomly generate r from Unform [a, b]. 5. If r P 0, then E E þ CrfZðR; DÞrg. e Otherwse, E E CrfZðR; DÞrg. e 6. Repeat 4 and 5 for O tmes. 7. Calculate EðZðR; DÞÞ e ¼ a _ 0 þ b ^ 0 þ E b a O Genetc algorthm In the usual form of genetc algorthm (GA), descrbed by Goldberg [18], the best soluton s the wnner of the genetc game and any potental soluton s assumed a creature determned by dfferent parameters. Several authors have employed GA to solve complcated nventory control problems. A selecton of these works s demonstrated n Table 3. In what follows the man characterstcs of the genetc algorthm employed n ths research are descrbed Chromosomes A chromosome, an mportant part of GA, s a strng or tral of genes that s consdered the coded fgure of a possble soluton (proper or mproper). In ths paper, the chromosomes are strngs of the maxmum nventory levels of the products (R j ) that are ntegers [39,41,4]. Therefore, nteger numbers are randomly generated n the closed nterval [0, 1000] to represent the genes. Moreover, nfeasble chromosomes, the ones that do not satsfy the constrants of the models n (9) and (10), are not consdered. For an 8-product system, the chromosome structure s gven n Fg Populaton Each populaton or of chromosomes has the same sze that s known as the populaton sze denoted by N. Smlar to Talezadeh et al. [39,41,4], 50, 100, and 150 are chosen as dfferent populaton szes of the GA algorthm of the current research Crossover In a crossover operaton, matng pars of chromosomes creates offsprng. Crossover operates on the parents chromosomes wth the probablty of P c. If no crossover occurs, the offsprng s chromosomes wll be the same as ther parents

10 Table 3 Lterature revew of GA applcatons n nventory control. Author Area Varables Gene represents Intalzaton Mutaton Crossover Stoppng crtera Roy et al. [34] Suer et al. [38] Shahabudeen and Svakumar [36] Roy et al. [35] Nachapan and Jawahar [31] Talezadeh et al. [4] Talezadeh et al. [43] Integrated producton nventory system Capactated lot sze problem Kanban system Inventory model wth deteroraton tems Vendor managed nventory Inventory model wth random perod length Newsboy nventory system Cycle length, maxmum nventory level Producton quantty, human resource requrement, etc. Number of Kanban and extra cards Order quantty, cycle length Sales quantty of each buyer Maxmum nventory level Product type Product type Number of Kanban and extra cards Product type Sales quantty of buyers Product type Random Random Random Random Random Random Order quantty Product type Random Randomly by usng mutaton probablty Randomly by usng mutaton probablty Order based shft mutaton Randomly by usng mutaton probablty Randomly by usng mutaton probablty Randomly by usng mutaton probablty Randomly by usng mutaton probablty Sngle pont Multple chromosome Sngle pont Sngle pont Sngle pont Two pont crossover Two pont crossover Maxmum teraton number Maxmum number of Maxmum number of Maxmum number of Maxmum number of Maxmum number of Maxmum number of Hybrd by Fuzzy logc Other consderatons Fuzzy genetc algorthm s proposed Multple chromosome crossover s proposed GA compared by smulated annealng and performed better Fuzzy smulaton Necessty and possblty theores are consdered GA based heurstc s proposed Pareto and TOPSIS selectons Goal programmng TOPSIS s used to rank the Pareto GA solved a mult objectves problem 434 A.A. Talezadeh et al. / Informaton Scences 0 (013)

11 A.A. Talezadeh et al. / Informaton Scences 0 (013) R1 R R3 R 4 R5 R 6 R7 R8 Fg. 4. The structure of a chromosome. R j M=6 R j R j R j Fg. 5. The sngle-pont crossover operaton. [45]. Fg. 5 depcts a sngle-pont crossover operaton n whch R j shows the chromosome contanng the maxmum nventory levels of the products and the break pont s chosen at M = 6[39,41,4]. In ths research, a sngle pont crossover wth dfferent probabltes (P c ) of 0.6, 0.7, and 0.8 s utlzed. Note that nfeasble chromosomes do not move to the new populaton Mutaton Mutaton s the second operaton n GA to explore new solutons by operatng on each chromosome resulted from the crossover operaton, where genes are replaced wth randomly selected numbers wthn the boundares of the parameter [16,7]. To do ths, a random number RN between (0,1) s generated for each gene. If RN s less than a predetermned mutaton probablty P m, then the mutaton occurs n the gene. Otherwse, t does not. The usual value of P m s 0.1 over the numbers of genes n a chromosome. In ths research, 0.010, 0.015, and 0.00 are chosen as dfferent values of P m. Note that nfeasble chromosomes resulted by ths operaton do not move to the new populaton [39,41,4]. Fg. 6 depcts a mutaton operaton n whch P m s chosen to be 0.01 [39,41,4] Objectve functon evaluaton In a maxmzaton problem, the more adequate the soluton, the greater the objectve functon (ftness value) wll be. Therefore, the fttest chromosomes wll take part n offsprng wth a larger probablty. The fuzzy smulaton of Secton 3. s used to evaluate the objectve functon of ths research [39,41,4] Selecton Selecton plays a central role n GAs by determnng how ndvduals compete for survval. Selecton weeds out the bad solutons and keeps the good ones. Ths can be performed by proportonal ftness selecton that assgns a selecton probablty n proporton to the ftness of the gven ndvdual. The tournament selecton s the most commonly used method, n whch a number of randomly pcked ndvduals are compared to each other [9]. The fttest ndvdual s then selected to be a part of the next. The tournament sze determnes how many ndvduals are to be compared per selected populaton. Because of the randomness of the selecton method, most technques, ncludng tradtonal recombnaton and mutaton operators, cannot guarantee the survval of the current best soluton. In ths research, Eltsm s used to provde guarantee by explctly selectng the best ndvdual or group of ndvduals. The mplementaton of these two technques leads to duplcates of good ndvduals [9]. In ths paper, we move the fve best solutons to the next as eltes. After each, solutons are checked for feasblty n terms of satsfyng the constrant. If the constrant s satsfed, the correspondng chromosome wll mmgrate to the next populaton, otherwse the soluton wll be removed, and the wll contnue untl a suffcent number of chromosomes are produced. R j RN R j Fg. 6. A sample of mutaton operaton.

12 436 A.A. Talezadeh et al. / Informaton Scences 0 (013) Table 4 General data. Product h p P b f W D (7,10,13) (7,10,13) (7,10,13) (7,10,13) (18,0,) (18,0,) (18,0,) (18,0,) Stoppng crteron The last step n a GA s to check whether the algorthm has found a soluton that s good enough to meet the user s expectatons. Stoppng crteron s a set of condtons such that when satsfed, hnts at a good soluton. In ths research, snce the populaton szes of 50, 100, and 150 are used, t s better to stop the algorthm untl a maxmum number of 500 evaluatons (MN = 500) are performed [39,41,4]. In short, the steps nvolved n the GA algorthm used n ths research are 1. Set the parameters P c, P m, and N ntalze the populaton randomly (the ndvduals should satsfy the constrants). Evaluate the objectve functon for all chromosomes based on Flowchart (1) 3. Select an ndvdual for matng pool by tournament selecton method usng eltsms 4. Apply crossover to each par of chromosomes wth probablty P c 5. Apply mutaton to each chromosome wth probablty P m 6. Replace the current populaton by the resultng new feasble populaton (before replacng the old populaton, the feasblty of the newly generated chromosomes s checked and reproducton wll contnue untl a suffcent number of requred chromosomes s obtaned) 7. Evaluate the objectve functon 8. If the stoppng crteron s met, stop. Otherwse, go to step 4. In order to demonstrate and evaluate the performance of the proposed hybrd ntellgent algorthm, n the next secton we present three numercal examples that were orgnally used n Ertogal and Rahm [1]. In these examples, two cases of the unform and the exponental dstrbutons for the tme between two replenshments are nvestgated. To valdate the results obtaned, an exstng hybrd method of FS and SA [44] s employed as well. 4. Numercal examples Consder two multproduct nventory control problems wth dfferent numbers of products. The frst one has eght products, where ts general data s gven n Table 4. Tables 5 and 6 show the parameters of the unform and the exponental dstrbutons used for the tmes between two replenshments, respectvely. The total avalable warehouse space s 4800, and Table 7 shows dfferent values of the parameters of the GA method. In ths research all dfferent combnatons of the parameters of GA (P c, P m and N) gven n Table 7 are employed and usng the Max (Max) crteron the best combnaton of the parameters has been selected. Moreover, K = 15 and O = 100 are consdered n the fuzzy smulaton procedure. All runs are performed usng MATLAB on a Pentum4 computer wth. GHZ coreduo processor. In order to show the effectveness of the proposed hybrd method of FS and GA n solvng the complcated nventory problem of ths research, Talezadeh et al. s [44] hybrd method of FS + SA s also employed to solve the numercal examples. Tables 8 and 9 show the best results of the two approaches. The best combnatons of the GA algorthms are shown n Table 10. Furthermore, the convergence paths of the objectve-functon values of the FS + GA and FS + SA algorthms for unform and exponental dstrbutons are shown n Fgs The results n Tables 8 and 9 show that the hybrd FS + GA method provdes a better near-optmal soluton n terms of the objectve-functon value. Moreover, from Fgs. 7 to 10, one can observe that more s and teratons are requred to reach the best result n the case of unform compared to exponental dstrbuton. In the frst numercal example, to compare the performances of the two hybrd methods, whle the number of runs n each example s set at 5 for both methods; the sample means of the CPU tmes n reachng the best soluton n exponental Table 5 Data for unform dstrbuton. Product T Mn T Max

13 A.A. Talezadeh et al. / Informaton Scences 0 (013) Table 6 Data for exponental dstrbuton. Product k 1/30 1/30 1/60 1/60 1/30 1/30 1/60 1/60 Table 7 The parameters of the GA method. P c P m N Table 8 The best result for by FS + GA algorthm. Dstrbuton Product ZðR; DÞ e Unform ,400 Exponental ,80 Table 9 The best result for by FS + SA algorthm. Dstrbuton Product ZðR; DÞ e Unform ,366 Exponental ,550 Table 10 The best combnaton of the GA parameters. Numercal example wth P c P m N Unform dstrbuton Exponental dstrbuton x 104 Objectve Functon Value Generaton Number Fg. 7. The convergence paths of the best result by FS + GA n unform example.

14 438 A.A. Talezadeh et al. / Informaton Scences 0 (013) dstrbuton case are 9.35 and 9.59 s for FS + GA and FS + SA methods, respectvely. The correspondng sample varances are 0.5 and 0.7. In the unform case however, the sample means are and wth the sample varances of 0.6 and 0.65, respectvely. Ths shows that the proposed hybrd method has better performance n terms of the CPU tme to reach the best result n both dstrbutonal cases. Smlar results are obtaned for the next two numercal examples contanng 0 and 40 products. The summarzed CPU sample means n Table 11 show that, as expected, as the number of products ncreases, the requred CPU tme to reach the best soluton ncreases as well. Further, the proposed hybrd FS + GA provdes better results n terms of the objectve-functon value and CPU tme for both unform and exponental cases of the two dfferent problem szes. 5. A senstvty analyss To study the effects of parameter changes on the best result obtaned by the proposed method and the requred CPU tme, a senstvty analyss s performed to nvestgate the effect of ncrease or decrease of the parameters, one at a tme, by 0% and 40%. The parameters of the proposed method are the fuzzy demand (D ), the parameters of the dstrbuton of the perod length (T Max and k ), the crossover probablty (P c ), the mutaton probablty (P m ), the parameters of the fuzzy smulaton algorthm (K and O), the number of products (NP), and the maxmum number of the populaton szes (MN). Table 1 shows 1.7 x Objectve Functon Value Generaton Number Fg. 8. The convergence paths of the best result by FS + SA n exponental example. 4 x 104 objectve Functon Value Iteraton Number Fg. 9. The convergence path of the best result by FS + SA n the unform example.

15 A.A. Talezadeh et al. / Informaton Scences 0 (013) Objectve Functon Value 1.55 x Iteraton Number Fg. 10. The convergence path of the best result by FS + SA n the exponental example. Table 11 The summarzed results of the second and the thrd numercal examples. Example Unform Exponental FS + GA FS + SA FS + GA FS + SA Second example (0 products) Thrd example (40 products) Objectve functon value CPU tme (s) Objectve functon value CPU tme (s) Objectve functon value CPU tme (s) Objectve functon value CPU tme (s) 106, , , , , , , , Table 1 The effects of the parameter changes on the objectve-functon value. % Changes n parameters % Changes n Unform dstrbuton Objectve functon Exponental dstrbuton Objectve functon D T Max k the results of the senstvty analyss on the sample mean of the 5 best results obtaned for the unform and exponental dstrbuton cases. The results n Table 1 show that there s a drect relatonshp between the objectve-functon value and the changes n D, T Max and k, that s, ncrease or decrease of these parameters cause the objectve functon value to ncrease or decrease, respectvely. The numbers n Table 13 are the sample means of the 5 requred CPU tmes to solve the problem. The relatve percentages of ncrease or decrease n average CPU tme compared to the ones requred to acheve the results of Table 5 are also

16 440 A.A. Talezadeh et al. / Informaton Scences 0 (013) Table 13 The results of the senstvty analyss on CPU tme n FS + GA algorthm. Parameters Unform Exponental Value Dfference (%) Value Dfference (%) NP O K P c P m MN gven. The results n Table 13 show that n all stuatons the average CPU tme to solve the problem n a unform case s larger than that of the exponental dstrbuton. Furthermore, the fuzzy smulaton parameters, K and O, do not have much mpact on the CPU tmes. However, n both dstrbutons, the CPU tmes are very senstve to the changes n the number of decson varables. Fnally, the parameters of the GA have relatvely mld mpact on the requred CPU tme. 6. Concluson and recommendaton for future research In ths paper, a stochastc replenshment multproduct nventory model was developed. Two nteger-nonlnear programmng models for two cases of unform and exponental dstrbuton of the tme between two replenshments have been proposed. A hybrd method of FS and GA was developed to solve the problem and the results were valdated by both a senstvty analyss and a comparson wth an exstng hybrd method of FS and SA. The comparson results showed that at least for the selected numercal examples the proposed hybrd method of FS and GA had better performance n terms of objectve-functon values and requred CPU tme to obtan the best soluton. The models developed n ths research can help the practtoners who are faced wth uncertan demands that do not follow a probablty dstrbuton. Moreover, the models are helpful n stuatons n whch due to some lmtatons on the producton capacty, the supply of the raw materal, and the lke, the perod length may be uncertan and the supplers may not be able to delver the goods on tme. Some avenues for future works follow: 1. The demand or other parameters of the problem may take uncertan forms (stochastc or rough) as well.. Some other probablty densty functons rather than unform and exponental may be consdered for the tme between replenshments. 3. Some other meta-heurstc algorthms such as harmony search or partcle swarm may be employed to solve the problem. 4. Fuzzy dscount factor or fuzzy dscrete delvery orders may be consdered as well. 5. Dfferental evoluton can be consdered as an effectve technque to solve the problem. Acknowledgements The authors are thankful for constructve comments of revewers that sgnfcantly mproved the presentaton of the artcle. References [1] E.H.L. Aarts, J.H.M. Korst, Smulated Annealng and Boltzmann Machne, A Stochastc Approach to Computng, John Wley and Sons, Chchester, [] H. Al-Tabtaba, A.P. Alex, Usng genetc algorthms to solve optmzaton problems n constructon, Engneerng, Constructon and Archtectural Management 6 (1999) [3] S. Bylka, Turnpke polces for perodc revew nventory model wth emergency orders, Internatonal Journal of Producton Economcs (005) [4] H.C. Chang, J.S.H. Yao, L.Y. Ouyang, Fuzzy mxture nventory model wth varable lead tme based on probablstc fuzzy set and trangular fuzzy number, Mathematcal and Computer Modelng 39 (004)

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