Multi-Product Multi-Constraint Inventory Control Systems with Stochastic Replenishment and Discount under Fuzzy Purchasing Price and Holding Costs

Transcription

1 Amercan Journal of Appled Scences 6 (): -, 009 ISSN Scence Publcaons Mul-Produc Mul-Consran Invenory Conrol Sysems wh Sochasc eplenshmen and scoun under Fuzzy Purchasng Prce and Holdng Coss Aa Allah alezadeh, Seyed agh Akhavan Nak and Mr-Bahador Aryanezhad eparmen of Indusral Engneerng, Iran Unversy of Scence and echnology, Iran eparmen of Indusral Engneerng, Sharf Unversy of echnology, Iran Absrac: hle n mul-perodc nvenory conrol problems he usual assumpon are ha he orders are placed a he begnnng of each perod (perodc revew) or dependng on he nvenory level hey can happen a any me (connuous revew), n hs research, we relax hese assumpons and assume ha he perods beween wo replenshmens of he producs are ndependen and dencally dsrbued random varables. Furhermore, assumng he purchasng prce are rangular fuzzy varables, he quanes of he orders are of neger-ype and ha here are space, budge and servce level consrans, ncremenal dscoun s consdered o purchase producs and a combnaon of back-order and los-sales are aken no accoun for he shorages. e show ha he model of hs problem s a fuzzy mxed-neger nonlnear programmng ype and n order o solve, a hybrd mehod of fuzzy smulaon and genec algorhm approach s used. A he end, a numercal example s gven o demonsrae he applcably of he proposed mehodology n real world nvenory conrol problems. Key words: Invenory conrol, sochasc replenshmen, dscoun, fuzzy mxed-neger nonlnear programmng, fuzzy smulaon, genec algorhm INOUCION In mul-perodc nvenory conrol models, he connuous revew and he perodc revew are he maor vasly used polces. However, he underlyng assumpons of he proposed models resrc her correc usage and ulzaon n real-world envronmens. In connuous revew polcy, he user has he freedom o ac a anyme and replensh orders based upon he avalable nvenory level. hle n he perodc revew polcy, he user s allowed o replensh he orders only n specfc and predeermned mes. he mul-perodc nvenory conrol problems have been nvesgaed n deph n dfferen research. Chang [] consdered a perodc revew model n whch he perod s parly long. he mporan aspec of hs sudy was o nroduce emergency orders o preven shorages. He employed a dynamc programmng approach o model he problem. Mohebb and Posner [] nvesgaed an nvenory sysem wh perodc revew, mulple replenshmen and mul-level delvery. hey assumed ha he sochasc demand followed Posson dsrbuon, shorages were allowed and ha he los sale polcy could be employed. Feng and ao [3] consdered a (, n) model n whch n he frs level a sochasc demand enered he sysem and he oal unsasfed demand were back-ordered a he second level. Ouyang and Chuang [4] nvesgaed a (, ) model n whch he perod-lengh and lead-me were he decson varables, demand was a random varable and he servce level was a consran. Chang [5] analyzed a perodc revew problem n wo cases of back-order and los sales and employed he (, ) polcy. Qu e al. [6] nvesgaed a ransporaon model negraed wh an nvenory model wh a perodc revew polcy. Eynan and Kropp [7] have propounded he assumpon of sochasc demand and varan warehousng coss on a perodc revew sysem, whle assumng nonzero lead-me and safey sock. Bylka [8] nvesgaed a model wh consrans on he amouns of orders and back-order shorages n whch he lead-me was consan and demand was sochasc and Mohebb [9] consdered demand a compound Posson random varable. Furhermore, alezadeh e al. [0] nvesgaed a sochasc replenshmen mul-produc nvenory sysem and proposed wo models for wo cases of unform and exponenal dsrbuon of he me beween wo replenshmens. hey showed ha he models were of neger non-lnear ype and proposed a smulaon annealng algorhm o solve hem. Correspondng Auhor: Seyed agh Akhavan Nak, eparmen of Indusral Engneerng, Sharf Unversy of echnology, Iran

2 Am. J. Appled Sc., 6 (): -, 009 In he leraure revew of he fuzzy nvenory models, Hseh [] nroduced wo models wh fuzzy parameers for crsp and fuzzy producon quanes. In oy e al. [] research an nvenory model for a deerorang em wh sock dependen demand was developed under wo sorage facles over a random plannng horzon. hen, for he crsp deeroraon rae on one hand, he expeced prof was derved and maxmzed va Genec Algorhm (GA). On he oher hand, when deeroraon rae was mprecse, he opmsc/pessmsc equvalen of fuzzy obecve funcon was obaned usng possbly/necessy measure of fuzzy even. Yao e al. [3] consdered an nvenory problem whou backorder where boh he order and he oal demand quanes were rangular fuzzy numbers. as e al. [4] formulaed mul-em sochasc and fuzzy-sochasc nvenory problems under oal budgeary and space consrans. Chang e al. [5] nvesgaed he fuzzy problems of he mxure nvenory sysems nvolvng varable lead-me wh backorders and los sales. In order o maxmze he average prof, Mandal and oy [6] formulaed a mulem dsplayed nvenory problem under shelf-space consran n a fuzzy envronmen. In anoher research n hs area, Ma and Ma [7] developed a mul-em nvenory model wh wo-sorage facles and adversemen where he prce and dsplayed nvenory level-dependen demand, he purchase cos, he nvesmen amoun and sorehouse capacy were mprecse. he problem was formulaed as a sngle/mul-obecve programmng problem under fuzzy consran. Lu [8] developed a soluon mehod o derve he fuzzy prof of he nvenory model when he demand quany and he un cos were fuzzy numbers. Fve man specfcaons of he proposed model of hs research ha have led o s novely are he sochasc perod lengh, he allowance of mulproducs mul-consran, he purchasng prce beng fuzzy varable, ncorporang dscouns o purchase producs and he fac ha he decson varables are neger. By deployng hese condons smulaneously, he problem becomes more realsc and he creaed model s dfferen from he oher models n he perodc revew leraure. A BIEF BACKGOUN IN FUZZY ENVIONMEN In hs research, we adop he conceps of he credbly heory ncludng possbly, necessy and credbly of fuzzy even and he expeced value of a fuzzy varable defned as [9] : efnon : Le ξ be a fuzzy varable wh he membershp funcon µ(x). hen he possbly, necessy and credbly measure of he fuzzy even ξ r can be represened, respecvely, by: { } Pos ξ > r sup µ (u) () u r { } Nec ξ r sup µ (u) () u< r Cr r Pos r Nec r { ξ } { ξ } + { ξ } (3) efnon : he expeced value of a fuzzy varable s defned as: [ ξ ] { ξ } { ξ } E Cr r dr Cr r dr 0 0 (4) efnon 3: he opmsc funcon of α s defned as: sup { } (5) ξ ( α ) sup r Cr ξ r α, α (0,] POBLEM EFINIION Consder a perodc nvenory conrol model for one provder n whch he mes requred o order each of several avalable producs are sochasc n naure. Le he me-perods beween wo producreplenshmens be dencal and ndependen random varables; he purchasng prce of he producs o be rangular fuzzy varables, he demands are crsp and n case of shorage, a fracon are consdered back-order and a fracon as los-sale. he coss assocaed wh he nvenory conrol sysem are holdng (a percenage of he purchasng cos), back-order, los-sales and purchasng coss. Furhermore, he ncremenal dscoun polcy s used, he servce level of each produc, warehouse space and budge are consdered consrans of he problem and he decson varables are neger dgs. e need o denfy he nvenory levels n each cycle such ha he expeced prof s maxmzed. POBLEM MOELING For he problem a hand, snce he me-perods beween wo replenshmens are ndependen random varables, n order o maxmze he expeced prof of he plannng horzon we need o consder only one perod. Furhermore, snce we assumed ha he coss

3 Am. J. Appled Sc., 6 (): -, 009 assocaed wh he nvenory conrol sysem are holdng and shorage (back-order and los-sale), we need o calculae he expeced nvenory level and he expeced requred sorage space n each perod. Before dong hs, le us defne he parameers and he varables of he model. he parameers and he varables of he model: For,,,n, le us defne he parameers and he varables of he model as: : : F ( ): q : : : : he nvenory level of he h produc A random varable denong he me-perod beween wo replenshmens (cycle lengh) of he h produc he Probably densy funcon of he h dscoun pon for he h produc he crsp purchasng cos per un of he h produc whou dscoun he crsp purchasng cos per un of he h produc a he h dscoun pon he fuzzy purchasng cos per un of he h produc a he h dscoun pon : he weghed expeced purchasng cos of he h produc FI : A fracon of he purchasng cos of he h produc used o calculae s holdng cos h : he holdng cos per un nvenory of he h produc n each perod h : he crsp holdng cos per un nvenory of he h produc n each perod ( h FI * ) h : he fuzzy holdng cos per un nvenory of he h produc n each perod ( h FI * ) Q : he order quany of he h produc a he h dscoun prce π : he back-order cos per un demand of he h produc ˆπ : he shorage cos for each un of los sale of he h produc P : he sale prce per un of he h produc : he consan demand rae of he h produc SL : he lower lm of he servce level for he h produc : he me a whch he nvenory level of he h produc reaches zero β : he percenage of unsasfed demands of he h produc ha s back-ordered I : he expeced amoun of he h produc nvenory per cycle 3 L : he expeced amoun of he h produc lossale n each cycle. B : he expeced amoun of he h produc backorder n each cycle Q : he expeced amoun of he h produc order n each cycle f : he requred warehouse space per un of he h produc F: oal avalable warehouse space B: oal avalable budge C h : he expeced holdng cos per cycle of he h produc. C b : he expeced shorage cos n back-order sae of he h produc C : he expeced shorage cos n los-sale sae l of he h produc C p : he expeced purchase cos of he h produc r : he expeced revenue obaned from sales Z(,, h) : he expeced prof obaned n each cycle For sake of smplcy, we frs consder a sngleproduc problem n whch he purchasng prces and holdng coss are crsp and here s no dscoun. hen, we are devoed for a sngle-produc problem wh ncremenal and oal dscoun polces, respecvely. e dscuss he cases n whch he demands are fuzzy random varables. Fnally, we exend he sngle-produc o he mul-produc modelng. However, le us nroduce he pcoral represenaon of he sngleproduc problem. Invenory dagram: Accordng o Erogal and ahm [0] and consderng he fac ha he me-perods beween replenshmens are sochasc varables, wo cases may occur. In he frs case he me-perod beween replenshmens s less han he amoun of me requred for he nvenory level o reach zero (Fg. ) and n he second case, s greaer (Fg. ). Sngle produc model-back order and los sales cases: In hs secon, we frs model he coss, he prof and he consran of a sngle-produc nvenory problem wh crsp demand where here s no dscoun on purchasng producs. he replenshmens are sochasc and back-order and los-sales are allowed. Calculang he coss and he prof: In order o calculae he expeced prof n each cycle, we need o evaluae all of he erms n Eq. 6 [0] : Z r C C C C P Q p h b l Q h I π B πˆ L (6)

4 Am. J. Appled Sc., 6 (): -, 009 I Max L ( β ) ( )f ( )d ; < (7) Max MAx (8) B β ( )f ( )d ; < < Max Fg. : Presenng he nvenory cycle when I Mn Max (9) I f ( )d + f ( )d Mn ( ) Q f ( )d Mn Max ( ( )) + + β f ( )d (0) - - Presenng he consrans: As he oal avalable warehouse space s F, he space requred for each un of he h produc s f and he nvenory level of he h produc s, he space consran wll be: Fg. : Presenng he nvenory cycle when < I Max f F () Snce he oal avalable budge s B, he cos for each un of produc s and he order quany s Q, he budge consran s: Q B () - - β ( - ) (- β )( - ) Fg. 3: Presenng shorages n wo cases of back order and los sales Based on Fg. 3, L, B, I and Q are evaluaed by he followng equaons: Knowng ha he shorages only occur when he cycle me s more han and ha he lower lm for he servce level s SL, hen: ( ) Max (3) P > f ( )d SL In shor, he complee mahemacal model of he sngle produc nvenory problem wh crsp demand and no dscoun s: 4

5 Am. J. Appled Sc., 6 (): -, 009 Max Max Z (P ) ( ) f ( )d + ( ) + β ( ) f ( )d Mn + Max h f ( )d f ( )d Mn Max Max πβ ( ) f ( )d ( ( ) ) ˆ f ( )d β π s..: f F Max f ( )d + ( + β( ))f ( )d B Mn ( ) Max P > f ( )d SL 0 and Ineger (4) Sngle produc model-back ordered and los sales cases wh dscoun: In hs secon, we assume ha an ncremenal dscoun polcy s applcable o purchase he produc. In ncremenal dscoun polcy, he purchasng cos for each un of he h produc depends on s order quany and s assumed o be: 0 < Q q q < Q q q Q (5) he purchasng cos assocaed wh hs polcy s calculaed as follows: C p Q 0 < Q q q + (Q q ) q < Q q q + (q q ) + + (Q q ) q Q (6) where, for,,,, q and are he dscoun pons and he purchasng coss for each un of he h produc ha corresponds o he h dscoun break pon, respecvely. In order o nclude he dscoun polcy n he nvenory model, usng Eq. 6, he purchasng cos wll be modelled as: 5

6 Am. J. Appled Sc., 6 (): -, 009 C Q + Q Q p Q Q + Q Q qy Q qy ( ) ( ) q q Y3 Q q q Y 0 Q MY M s a large dg Y Y Y Y 0,;,,..., (7) By hs modelng, he nvenory model of he sngle produc problem wh ncremenal dscoun polcy becomes: Max Z PQ h I πb πˆ L Q π β ( ) f ( )d Mn P Maz + ( + β ( )) f ( )d f ( )d Mn h Max + f ( )d Max ( ) f ( )d πˆ ( β ) f ( )d Q s..: Max ( ) f F Q B Max f ( )d SL Q Q + Q Q qy Q qy ( q q ) Y3 Q ( q q ) Y 0 Q MY M s a large dg Y Y Y Y 0,;,,..., 0 and Ineger (8) 6 Sngle-produc model wh dscoun, fuzzy purchasng and holdng coss: he sngle-produc nvenory model wh crsp purchasng and holdng cos and ncremenal dscoun of (8) can be easly exended o sngle produc models wh fuzzy purchasng and holdng cos as follows: Max Z(,,h ) P Q h I πb πˆ L Q s.: P Mn Max f ( )d + ( ) f ( )d ( ) ( + β ) f ( )d Mn h Max + f ( )d π β Max ( ) f ( )d πˆ ( β ) f ( )d Q Max ( ) f F Q B Max f ( )d SL Q Q + Q Q qy Q qy (q q )Y3 Q (q q )Y 0 Q MY M s a large dg Y Y Y Y 0, 0, Ineger (9) In he nex secon, we exend he models n (9) o mul-produc models. Mul-produc models: he sngle-produc nvenory models of (9) can be easly exended o a mulple

7 Am. J. Appled Sc., 6 (): -, 009 produc. In hese models, we consder wo probably densy funcons for as follow: follows a unform dsrbuon: In hs case he probably densy funcon of s Accordngly, (9) wll change o (0) as: f ( ) max mn. n h 3 Max Z(,,h ) 6 (Max Mn ) n ( β )p + h Max + π β + πˆ ( β) (Max Mn ) n p ( β )Max + h Mn + ( π β + πˆ ( β)) Max + ( Max Mn ) 3 3p ( βmax Mn ) h Mn 3( πβ + πˆ ( β))max + 6(Max Mn ) n n Q s..: n n f Q F Max Mn ( β ) + ( B Max SL :,,...,n ( ) Q,,,...,n ( ) ( ) Max ( β )) + ( βmax Mn ) Max Mn,,, n n ( ) q Y Q q Y,,,..., q q Y Q q q Y,,...,n,,,..., 0 Q MY n,,,...,, M s a large dg Y Y Y,,,, Y 0,,,,,,,,, n 0, Ineger :,,...,n (0) I follows an exponenal dsrbuon: If follows an exponenal dsrbuon wh parameer λ, hen he probably densy funcon of wll be f ( ) λ e λ. In hs case, he model s shown n () as: 7

8 Am. J. Appled Sc., 6 (): -, 009 n λ (p ˆ )( ) Max : Z(, + π β + π β,h ) e λ n λ n n h p h + ( e ) + Q λ λ s..: n Q B n f F λ e SL :,,...,n λ Q ( )e β +,,,...,n λ qy Q qy,,,..., ( ) ( ) q q, Y Q q q, Y,,,...,n,,,..., 0 Q MY,,,..., and M s a large dg n n Y Y Y,,,, n Y 0,,,,,,,,, n 0, Ineger :,,...,n () In he nex secon, we wll nroduce a hybrd nellgen algorhm o solve he model. A HYBI INELLIGEN ALGOIHM Snce he models n (0) and () are fuzzy mxed neger-nonlnear n naure, reachng an analycal soluon (f any) o he problem s dffcul []. In order o solve he model under dfferen crera, we develop a hybrd nellgen algorhm of fuzzy smulaon and genec algorhm. Fuzzy smulaon: In order o esmae he unceran purchasng prce and holdng cos of he fuzzy model, snce he holdng cos s a funcon of s correspondng purchasng cos, an esmae of he former cos wll provde an esmae of he laer cos. As a resul, n he smulaon echnque used for he esmaon, denong,,...,, µ as he membershp by ( n) funcon of and µ are he membershp funcons of, we randomly generae from he α-level ses k 8 of fuzzy varables,,,,n,,,, and k,,,k as k ( k, k,, nk ) and µ ( ) µ ( ) µ ( ),, µ ( ), where α k k k n nk s a suffcenly small posve number. Based on he defnon n Eq., he expeced value of he fuzzy varable s: { } E Z(,, h) Cr Z(,, h) r dr { } Cr Z(,, h) r dr () hen, provded O s suffcenly large, for any r 0, Cr Z(,,h) r can be esmaed by: number { } { µ k } Max Z(,, h) r k,,...,o Cr{ Z(,, h) r} + Max { µ k Z(,, h) < r} k,,...,o (3)

9 Am. J. Appled Sc., 6 (): -, 009 and for any number r < 0, Cr{ Z(,,h) r} esmaed by: { µ k } can be Max Z(,, h) r k,,...,o Cr{ Z(,, h) r} + Max { µ k Z(,, h) > r} k,,...,o (4) However, he procedure of esmang Z(,, h) n (3) and (4) s shown n algorhm (). Se E 0 andomly generae k from -level ses of fuzzy varables and se k ( k, k,, nk ), Se a Z(,,h) Z(,,h) Z(, O,h) b Z(,,h) Z(,,h) Z(, O,h) andomly generae r from Unform [a,b] r 0, hen E E + Cr Z(,, h) r, oherwse, If { } E E Cr{ Z(,, h) r} epea 4 and 5 for O mes b a Calculae E(Z(,,h)) a 0 + b 0 + E * O Algorhm (): Esmang Z(,, h). Genec algorhm: he man nformaon un of any lvng organsm s he gene, whch s a par of a chromosome ha deermnes specfc characerscs such as eye-color, complexon, har-color, ec. he fundamenal prncpal of Genec Algorhms (GA) frs was nroduced by Holland []. Snce hen many researchers have appled and expanded hs concep n dfferen felds of sudy. Genec algorhm was nspred by he concep of survval of he fes. In genec algorhms, he opmal soluon s he wnner of he genec game and any poenal soluon s assumed o be a creaure ha s deermned by dfferen parameers. hese parameers are consdered as genes of chromosomes ha could be assumed o be bnary srngs. In hs algorhm, he beer chromosome s he one ha s nearer o he opmal soluon. In appled applcaons of genec algorhms, populaons of chromosomes are creaed randomly. he number of hese populaons s dfferen n each problem. Some hns abou choosng he proper number of populaon exs n dfferen repors by Man e al. [3]. Genec algorhms mae he evoluonary process of speces ha reproduce. hey herefore do no operae 9 on a sngle curren soluon, bu on a se of curren soluons called populaon. New canddaes for he soluon are generaed wh a mechansm called crossover ha combnes par of he genec parmony of each paren and hen apples a random muaon. If he new ndvdual, called chld or offsprng, nhers good characerscs from hs parens he probably of s survval ncreases. hs process wll connue unl a soppng creron s sasfed. hen, he bes offsprng s chosen as a near opmum soluon. In hs research, he chromosomes are srngs of he nvenory levels of he producs ( ). Each populaon or generaon of chromosomes has he same sze whch s well-known as he populaon sze and s denoed by N. If N s relavely small, hen a small search space wll be nvesgaed and he GA algorhm wll be very slow. In hs research, 0, 00 and 500 are chosen as dfferen populaon szes. In a crossover operaon, s necessary o mae pars of chromosomes o creae offsprng. here are hree ypes of crossover operaons: sngle-pon, mul-pon and unform []. In hs research, we employ he sngle-pon crossover ha s appled o paren chromosomes wh he possbly of P c 0.8, 0.85 and 0.9. Muaon s he second operaon n a GA mehod for explorng new soluons and operaes on each of he chromosomes resuled from he crossover operaon. In muaon, we replace a gene wh a randomly seleced number whn he boundares of he parameer []. e creae a random number N beween (0,) for each gene. If N s less han a predeermned muaon probably P m, hen he muaon occur n he gene. Oherwse, he muaon operaon s no performed n ha gene. More precsely, assume ha for a specfc gene such as a n a chromosome he generaed random number s less han P m and hence he gene s seleced for muaon. hen, we change he value of a o he new value * a accordng o Eq. 5 and 6, randomly and wh he same probably: * a a + (u a ) r ( ) (5) max gen * a a (a l ) r ( ) (6) max gen where, l and u are he lower and upper lms of he specfed gene, r s a unform random varable beween 0 and, s he number of curren generaon and max gen s he maxmum number of generaons. Noe ha he value of a s ransferred o s rgh or lef randomly by Eq. 5 and 6 respecvely and r s hs percenage.

10 Am. J. Appled Sc., 6 (): -, 009 Furhermore, s an ndex wh a value close max gen o one n he frs generaon and close o zero n he las generaon ha makes large muaons n he early generaons and almos no muaon n he las generaons. In hs research, 0.076, and 0. are employed as dfferen values of he P m parameer. Furhermore, Algorhm () of secon 5. s used o evaluae he obecve funcon of hs research. he las sep n a GA mehod s o check f he algorhm has found a soluon ha s good enough o mee he user s expecaons. Soppng crera s a se of condons such ha when sasfed a good soluon s obaned. fferen crera used n leraure are as follows: () Soppng of he algorhm afer a specfc number of generaons, () no mprovemen n he obecve funcon and (3) eachng a specfc value of he obecve funcon. In hs research, we sop when a predeermned number of consecuve generaons s reached. he number of sequenal generaons depends on he specfed problem and he expecaons of he user. In shor, he seps nvolved n he hybrd mehod of fuzzy smulaon and GA algorhm used n hs research are: Seng he parameers P c, P m and N Inalzng he populaon randomly Evaluang he obecve funcon for all chromosomes based on Algorhm () Selecng ndvdual for mang pool Applyng he crossover operaon for each par of chromosomes wh probably P c Applyng muaon operaon for each chromosome wh probably P m eplacng he curren populaon by he resulng mang pool Evaluang he obecve funcon If soppng crera s me, hen sop. Oherwse, go o sep 5 In order o demonsrae he proposed Hybrd nellgen algorhm and evaluae s performance, n he nex secon we brng a numercal example used n Erogal and ahm [0]. In hs example, wo cases of he unform and he exponenal dsrbuons for he meperod beween wo replenshmens are nvesgaed. NUMEICAL EXAMPLES Consder a mul-produc nvenory conrol problem wh egh producs and general daa gven n able. able shows he parameers of boh he exponenal and unform dsrbuons used for he meperod beween wo replenshmens. he oal avalable warehouse space and oal budge are F 000 and B , respecvely. able 3 shows he bes combnaon and dfferen values of he GA parameers used o oban he soluon. In hs research, all he possble combnaons of he GA parameers (P c, P m and N) are employed and usng he max(max) creron he bes combnaon of he parameers has been seleced. able 4 shows he bes resul for he unform and exponenal dsrbuons. Furhermore, he convergence pahs of he bes resul of he obecve funcon values n dfferen generaons of he unform and he exponenal dsrbuons are shown n Fg. 4 and 5, respecvely. able : General daa Produc FI π ˆπ β f (65,70,75) (65,70,75) (65,70,75) (65,70,75) (65,70,75) (65,70,75) (65,70,75) (65,70,75) q q q SL P

11 Am. J. Appled Sc., 6 (): -, 009 able : Parameers of exponenal and unform dsrbuons Produc λ /30 /30 /60 /60 /30 /30 /60 /60 Mn Max able 3: he parameers and he bes combnaon of he GA mehod he bes combnaon for Unform Exponenal Parameer Alernaves dsrbuon dsrbuon P c P m N able 4: he bes resul for Produc srbuon Z(,, h) Unform Exponenal Obecve funcon value Generaon number Fg. 4: he convergence pah of he bes resul n unform example Obecve fucnon value Generaon number Fg. 5: he convergence pah of he bes resul n exponenal example CONCLUSION AN ECOMMENAIONS FO FUUE ESEACH In hs research, a sochasc replenshmen mulproduc nvenory model wh dscoun and fuzzy purchasng prce and holdng cos was nvesgaed. wo mahemacal modelng for wo cases of unform and exponenal dsrbuon of he me beween wo replenshmens n case of ncremenal dscoun have been developed and shown o be fuzzy mxed negernonlnear programmng problems. hen, a hybrd nellgen algorhm (fuzzy smulaon+ga) has been proposed o solve he fuzzy neger non-lnear problems. Some recommendaons for fuure works are () consderng demands as fuzzy or random varables, () employng a oal dscoun polcy and (3) applyng some oher mea-heursc algorhms. EFEENCES. Chang, C., 003. Opmal replenshmen for a perodc revew nvenory sysem wh wo supply modes. Eur. J. Opera. es., 49: Mohebb, E. and M.J.M. Posner, 00. Mulple replenshmen orders n connuous-revew nvenory sysem wh los sales. Opera. es. Le., 30: Feng, K. and U.S. ao, 007. Echelon-sock (n) conrol n wo sage seral sochasc nvenory sysem. Opera. es. Le., 35: Ouyang, L.Y. and B.. Chuang, 000. A perodc revew nvenory model nvolvng varable leadme wh a servce level consran. In. J. Sys. Sc., 3: Chang, C., 006. Opmal orderng polces for perodc-revew sysems wh replenshmen cycles. Eur. J. Opera. es., 70: Qu,.., J.H. Bookbnder and P. Iyogun, 999. An negraed nvenory-ransporaon sysem wh modfed perodc polcy for mulple producs. Eur. J. Opera. es., 5: Eynan, A. and. Kropp, 007. Effecve and smple EOQ-lke soluons for sochasc demand perodc revew sysems. Eur. J. Opera. es., 80: Bylka, S., 005. urnpke polces for perodc revew nvenory model wh emergency orders. In. J. Prod. Econ., 93-94: Mohebb, E., 004. A replenshmen model for he supply-uncerany problem. In. J. Econ., 87: 5-37.

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