A Multi-item Inventory Model for Two-stage Production System with Imperfect Processes Using Differential Evolution and Credibility Measure

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1 Inernaonal Journal of Operaons Research Inernaonal Journal of Operaons Research Vol. 9, No., (0 A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly Debass Das, Arndam Roy and Samarj Kar 3,3 Deparmen of Mahemacs, Naonal Insue of echnology, Durgapur, W.B, Inda, Pn-7309 Deparmen of Compuer Scence, Prabha Kumar College, Cona, Purba- Mednpur, W.B, Inda, Pn-70 Receved Sepember 0; Revsed March 0; Acceped Aprl 0 Absrac A mul-em wo sage producon nvenory sysem wh mperfec producon process s formulaed. Here, a consran on he oal budge s mposed where he oal budge s mprecse n naure. Shorages are allowed and compleely backlogged. Sage I (raw maerals o sem-fnshed producs s an auomac process and hs process s reaed by machnes. Sage II (sem-fnshed producs o fnshed producs s also an auomac process and hs process s reaed by anoher machnes. I s assumed ha he me of ransporng ems from Sage I o Sage II s neglgble. he mperfec ems are reworked and assumed ha he nspecon me and rework me are very shor whch also can be negleced. he model has been formulaed as prof maxmzaon problem n sochasc and fuzzy-sochasc envronmens by consderng nvenory coss as mprecse n naure. Credbly heory has been used o ransform he fuzzy-sochasc model no an equvalen deermnsc one. o solve he problems, Dfferenal Evoluon (DE algorhm has been suably developed and appled. Fnally, o llusrae he model and o show he effecveness of he proposed approach a numercal example s presened. Keywords wo-sage sysem, nvenory, sock-dependen demand, mperfec qualy, rework, credbly measure, dfferenal evoluon. INRODUCION In mos of he classcal economc producon quany (EPQ model, s assumed ha ems produced are of perfec qualy, he qualy conrol of he produc generally s no consdered. However, n a producon sysem, s que naural ha a machne canno produce all ems perfecly durng whole producon perod. In mos of he producon sysem, a ceran poron of defecve ems are produced and wased as scraps snce hey have no recyclng or reworkng facly. Bu, n modern manufacurng companes four sysems of mprovng producon effcences are hghly apprecaed. hese are maerals requremen plannng (MRP, flexble manufacurng sysem (FMS, opmzed produc echnology (OP, and jus n me (JI. he adjusmen of producon rae wh varably n marke demand s a major componen n FMS. Goyal and Gunasekaran (995 have developed an negraed producon-nvenory-markeng model for deerorang he EPLS and EOQ for raw maerals n a mul-sage producon sysem. Bhuna and Ma (998 exend he EPLS model by consderng he fne producon rae dependng on on-hand nvenory and demand smulaneously. wo-sage producon sysems can be found n dfferen applcaons, lke processng and packagng food, exrudng and mllng plascs, shearng and punchng or rollng and cung meals [cf. Szendrovs (983]. Szendrovs (983 proposed wo-sage producon/nvenoy models n whch smaller los are produced a one sage and one larger lo s produced a he oher sage. Km s (999 consdered a wo-sage lo szng problems wh varous lo szng dependng on bach ransfer and producon raes beween sages. Hll (000 exended Km s (999 model provdng an alernave way of performng he analyss whch s easer o undersand. Darwsh and Ben-Daya (007 nvesgaed he effec of mperfec producon processes nvolvng varable he frequency of prevenve manenance. Recenly, Pearn e al. (00 also nvesgaed he effec of mperfec producon processes wh allowable shorages for wo sage producon sysem. Expensve producs, processed or assembled, are no usually scrapped. Consequenly, a procedure for recoverng he defecve ems would be benefcal o he company. For example, meal book-shelves and defecve flng cabnes are usually repared n shee meal ndusres, and defecve algnmen of seerng wheels s correced o fx he seerng column a a rgh-angle wh he seerng wheels n auomoble ndusres. Scrappng many such ems s an expensve proposal for any company. Hence, he rejeced ems are accumulaed for a ceran number of cycles and reworked whle a rework cos s Correspondng auhor s emal: kar_s_k@yahoo.com 83-73X Copyrgh 0 ORSW

2 88 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 assessed for no sasfyng he demand and oher resource consrans. Hayek and Salameh (00 derved an opmal operang polcy for he fne producon model under he effec of reworkng of mperfec qualy ems and assumng ha all he defecve ems are reparable. Chu (003 examned an EPQ model wh scrap ems and he reworkng of reparable ems. Konsanaras and Papachrsos (007 also exended he salameh and Jaber s (000 model o he case n whch whdrawng of defecve uns akes place a he end of he plannng horzon and mnmzed drecly he mean average cos nsead of maxmzng he mean average prof. ll now, algorhms have been developed for solvng he nvenory problems when nvenory parameers lke, oal floor space and oal budge allocaon for replenshmen, ec. are precsely known. Bu n real lfe suaon, hese parameers may be unceran n non-sochasc sense. In he compeve marke, s no possble o do he busness wh predefned fxed budgeary capal. Inally, a decson maker (DM may sar wh an amoun, bu, a a laer sage, o mee he sudden ncrease of demand or o aval he sudden fall n he prce of he commody, he / she s forced o augmen some more capal as per demand of he suaon. Hence, n hs case, budgeary allocaon s mprecse. Recenly, several researchers such as Roy and Ma (997, Kar e al. (000, Roy e al. (008, Das e al. (00, 0 ec. have developed several fuzzy nvenory models. he purpose of hs paper s o sudy a mul-em, wo-sage producon nvenory cum sale model havng mperfec producon process wh rework under budge consran. Here he producon sysem wh random mperfec ems s separaed no wo sages. In Sage I, sem-fnshed producs are produced by a se of machnes and n Sage II, fnshed producs are produced by anoher se of machnes. he model has been defned as a prof maxmzaon problem n sochasc and fuzzy-sochasc naure. In fuzzy-sochasc model, nvenory coss and he consran goal are mprecse n naure. A credbly measure and dfferenal evoluon (DE algorhm are used o solve he model. A numercal example s gven for llusraon of he heorecal resuls, and sensvy analyss for he prof funcon wh respec o some parameers are carred ou.. ASSUMPIONS AND NOAIONS A mul-em, wo-sage producon nvenory model wh rework sysem s developed on he bass of followng assumpons and noaons:. Assumpons he followng assumpons are made: ( Invenory sysem nvolves wo sage and mul-em and s a self producon sysem. ( he me horzon s nfne. ( Shorages are allowed and backlogged. (v Lead me s zero. (v Producon of Sage I and Sage II sars a same me. (v Machne breakdown does no occur a any producon sage and he handlng me beween processes s assumed o be zero. (v No defecve produc s scraped. (v No defecve ems are produced durng he rework. (x Se-up me s neglgble. (x I s well known ha he producon rae of Sage I s always hgher han Sage II. (x he producon sysem s mperfec, and he nspecon me and rework me of defecve producs are very shor, whch can be negleced. (x Inspecon cos s neglgble. (x ransporng me from Sage I o Sage II s gnored.. Noaons (for h em, =,,..., n he followng noaons are employed hrough hs paper as o develop he proposed model. N s he oal number of machnes for h em n Sage I. N s he oal number of machnes for h em n Sage II. P s he producon rae per machne for h em n Sage I.

3 89 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 P s he producon rae per machne for h em n Sage II, where W s he maxmum shorage amoun for h em. 0 N. P > N W s he maxmum nvenory level of sem-fnshed produc for h em n Sage I. W s he maxmum nvenory level of fnshed produc for h em n Sage II. q ( s he on hand nvenory of he em a me for h em n Sage I. D ( q s he sock-dependen demand rae. P = he producon cos per un em per un me n Sage I. c P c = he producon cos per un em per un me n Sage II. C = he holdng cos per un nvenory held per un of me n Sage I. h C h C 3 = he holdng cos per un nvenory held per un of me n Sage II. = he se-up cos per producon run. C = he shorage cos per un per un me. s C = Rework cos per defecve em n Sage I. r C = Rework cos per defecve em n Sage II. r s = Sellng prce per un em. d = Percenage of defecve sem-fnshed produc produced, a random varable. d = Percenage of defecve fnshed produc produced, a random varable.. P. fd ( = he probably densy funcon of d, ( j =,, unformly dsrbued wh p.d.f as j j fd (, 0d a a j 0, oherwse j j j m = Mark-up of he sellng prce per un em. = Producon sars a ha me. (decson varable 3 = me a whch sock of nvenory sars o accumulae of fnshed produc a Sage II. = me when he maxmum sock of nvenory of sem-fnshed producs occur a Sage I. (decson varable = me when he sock of sem-fnshed producs vansh a Sage I and sock of fnshed producs are maxmum a Sage II. = Duraon of he cycle. B = he oal budge. 3. MAHEMAICAL FORMULAION In hs model, we have consdered he demand rae s dependen on he on-hand nvenory.e. D( q q (,, >0 are consans. In he developmen of he wo sage producon model, we assume ha here exs allowable shorages and he shorages are backlogged and also he cycle sars wh shorage a me 0. he producon run begns a n boh he Sages bu producon and demand occur smulaneously n Sage II, back-orders are made up o Invenory ems n Sage I begn o accumulae up o W uns whou deeroraon. Afer 3. W uns and nvenory ems n Sage II begn o accumulae up o he producon n Sage I sops bu he producon run s connuous up o

4 90 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 n Sage II (cf. Fg.. A he end of producon, a he nvenory n Sage II would be depleed due o demand and vanshes a. hs cycle repeas agan and agan. Fgure. he graph of nvenory level durng me perod [ 0, ] For sem-fnshed produc n Sage I we have he followng resul: ( N P N P.( W. Also 3 N P.( W. 3 herefore N P.( ( N P N P.( 3 3 N P ( N P N P. N P 3 ( Now he change of nvenory level n Sage II wh respec o me can be descrbed by he followng dfferenal equaons:, 0 dq ( N P, d N P ( q (, ( q (, ( wh he boundary condons q (0 0, q ( W, q ( 0, q ( W, q ( 0. 0 hen he soluons of he dfferenal equaons ( are represened by, 0 ( N P.(, N P q e ( (, N P e e e ( ( (.., (3 A, q ( 0and from (3 we ge

5 9 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 N P N P. ( A, q ( 0 and from (3 we ge ( ( N P. e ln. (5 Now he oal holdng cos ( C durng he perod (0, HOL s gven by, C C C (6 HOL HOL HOL where C C ( N P N P.(.(. HOL h 3 And C C q ( d q ( d HOL h N P ( C h e d N P ( ( ( e. e d. e d N P C ( ( ( e h ( ( N P { e } ( e (. oal producon cos ( PC s gven by PC N P P ( N P P (. (7 c 3 c he sales revenue ( SR s gven by SR s N P ( m ( P P N P ( where c c m >. (8 he oal rework cos ( RC s gven by RC C d N P ( C d N P (. (9 r 3 r he oal shorage cos ( SHC s gven by SHC C q ( d q ( d s 0 C ( ( N P ( N P s (0 Hence he average prof ( AP durng he cycle (0, s gven by

6 9 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 AP SR PC C RC SHC C HOL 3 m ( P P N P( N PP ( c c c 3 N P P ( C ( N P N P.(.( c h 3 N P C e ( ( ( h ( ( N P { e } ( e ( C dnp( C dnp( r 3 r C ( ( ( s N P N P C 3. ( hen he expeced value of he average prof ( EAP s gven by EAP E SR PC C RC SHC C 3 HOL m ( P P N P( N PP ( c c c 3 N P P ( C ( N P N P.(.( c h 3 N P ( C ( ( e h ( ( N P { e } ( e ( C. Ed (. N P( C. Ed (. N P( r 3 r C ( ( N P ( N P s C. 3 ( Hence he oal expeced value of he average prof ( EAP s gven by n EAP EAP (3. CREDIBILIY MEASURE o consruc he nvenory model for wo-sage producon sysem n fuzzy envronmen, we shall frs nroduce some knowledge of credbly heory. Credbly heory was nalzed by Lu and Lu (00. If s a fuzzy varable wh membershp funcon ( x, hen for any se A of, he possbly measure of fuzzy even { A } s defned as Pos{ A} Sup ( x. xa he necessy of hs fuzzy even s defned as he mpossbly of he oppose even. ha s Nec{ A} Sup ( x. c xa he credbly measure of { A } s defned as he average of s possbly and necessy measure. herefore Cr{ A} Pos{ A} Nec{ A}, for any A, where s he power se of.

7 93 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 I s easy o check ha Cr sasfes he followng condons: ( Cr( 0 and Cr( ; ( Cr( Cr( whenever, and ; hus, Cr s also a fuzzy measure defned on (,. Besdes, Cr s self dual,.e., Cr( A Cr( c for any. Credbly measure s defned as he followng form: Cr( [ Pos( ( Nec(] [cf. Lu and Lu (00] for any and confdence level, 0. I also sasfes he above condons. Fgure. Membershp funcon of a rfn rapezodal Fuzzy Number: Le A s he rapezodal fuzzy number (rfn wh he membershp funcon ( x A, a connuous mappng : ( : [0,] x A 0 x a for x a for a x a a a ( x for a x a A a x 3 for a x a 3 a a 3 0 for a x f r a3 a r Pos( A r f a r a a a 3 0 f r a 3 a Nec( A r f a r a a a 0 r f r a f r a

8 9 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 he credbly measure for rfn can be defne as f r a a a ( r f a r a a a a a Cr( A r f a r a 3 ( a r f a r a a a 3 0 f r a 3 0 ( r a f a r a a a Cr( A r f a r a a a r( 3 3 f a r a 3 a a 3 f r a f r a Based on he credbly measure, Lu and Lu (00 presened he expeced value operaor of a fuzzy varable as follows: Le X be a normalzed fuzzy varable, hen he expeced value of he fuzzy varable X s defned by 0 0 E [ X ] Cr( X r dr Cr( X r dr. ( When he rgh sde of ( s of form, he expeced value s no defned. Also, he expeced value operaon has been proved o be lnear for bounded fuzzy varables,.e., for any wo bounded fuzzy varables X and Y, we have E [ ax by] ae[ X ] be [ Y ] for any real numbers a and b. he expeced value of rapezodal fuzzy varable X [ a, a, a, a ], 0 s defned as 3 EX [ ] ( ( aa ( a a PROBLEM FORMULAION 5. Sochasc Model So, he equvalen deermnsc form of he above sochasc model wh budge consran can be expressed as, Maxmze EAP(, subjec o, N P N P D( q n [ PC RC ] B. 3 (5 5. Fuzzy-Sochasc Model As n hs model, nvenory coss C, C, C and avalable budge B are mprecse, C, C and C n ( h h 3 h h 3 are replaced by C, C, C and B n consran of (5 s replaced by B and he expeced value of he average h h 3 prof s represened by EAP.

9 95 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 Credbly Approach (CrA In hs paper we consder C, C, C and B as rapezodal fuzzy number.e. C ( C, C, C, C, h h 3 h h h h 3 h C ( C, C, C, C, C ( C, C, C, C and B ( B, B, B, B. Snce opmzaon of a fuzzy h h h h3 h objecve s no well defned, so nsead of EAP one can opmze s equvalen opmsc and pessmsc reurn as saed n secon. So, he problem can be represened n followng way. When decson maker lkes o opmze he opmsc and pessmsc equvalen of EAP wh C, C, C and h h 3 B hen, he problem reduces o, Maxmze EAP (, subjec n o, N P N P D( q [ PC RC ] B, 3 (6 where EAP EAP. n EAP m ( P P N P ( N P P ( c c c 3 N PP ( {( ( C C ( C C }.( N P N P.(.( 3 ( ( C C h h N P ( C C. ( h3 h ( ( e ( ( N P { e } ( e ( c h h h3 h C. Ed (. N P( C. Ed (. N P( r 3 r C ( ( N P ( N P {( ( C C 3 3 ( C C 33 3 }, s (7 and B {( ( B B ( B B 3 }, SOLUION PROCEDURE 6.. Dfferenal Evoluon (DE Many heursc algorhms have been proposed for global opmzaon of nonlnear non-convex and non-dfferenable funcons. hese mehods are more flexble han classcal one as hey do no requre dfferenably, connuy or oher resrcve properes whch are usually requred for he objecve funcon o be opmzed. Some of such mehods are genec algorhm, evoluonary sraeges, colony opmzaon, parcle swarm opmzaon and dfferenal evoluon (DE. Dfferenal Evoluon (DE[cf. Sorn and Prce (997] s a novel populaon based sochasc drec search opmzaon algorhm ha s farly fas and reasonably robus. DE resembles he srucure of an evoluonary algorhm bu dffers from classcal evoluonary algorhms n s generaon of new canddae soluons and by s use of a `greedy' selecon scheme. he key dfference s ha muaon n DE algorhm s an arhmec combnaon of ndvduals whereas n radonal evoluonary algorhms, s he resul of small perurbaons o he genes of an ndvdual. Moreover, n DE, he ral

10 96 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 soluons are generaed by addng weghed dfference vecors o he arge vecor followed by a recombnaon (or crossover sep o produce an offsprng whch s only acceped f mproves he fness of he paren ndvdual. DE auomacally adaps he muaon ncremens (.e. search sep o he bes value based on he sage of he evoluonary process. In GA Muaon s caused by small aleraons of genes, whereas n DE Muaon s provded by arhmecal combnaons of ndvduals. he core of hs operaon s he formaon of a dfference vecor whch makes muae an ndvdual. he basc operaors of DE are descrbed n he followng secons: Inal Populaon An N dmenson parameer opmzaon problem can be represened as an N-dmensonal vecor x x x, x,..., x. as:,,,, j N If here s no prelmnary knowledge abou he opmzaon, he frs populaon soluons can be generaed randomly x x Rx [ x ],, G,( L,( H,( L where x and x are he lower and hgher boundares of he vecor x and R (0,, drawn unformly for each,( L,( H. Muaon Muaon operaor s employed o expand he search space. By he combnaon of vecors randomly chosen from he curren populaon a generaon G, a muan vecor v s generaed for each arge vecor x as, G, G v x Fx ( x, (8 G, r, G r, G r, G 3 where, r, r and r are recprocally dfferen random negers less han or equal o populaon sze of soluon vecors. 3 F (0, s a real consan posve weghng facor whch conrols he amplfcaon of he dfferenal varaon. Crossover o ncrease he dversy of he populaon, DE ulzes crossover operaon ha negraes successful soluons from he prevous generaon. he ral vecor ug, s found from s parens x and v G, G, usng he followng crossover rule: j j v f R C or j I, j G, R u G, (9 j j x f R C and j I G, R where,,..., N and C s crossover parameer. I s an neger randomly chosen wh replacemen from he R se I,.e., I I {,,..., N} ; he superscrp j represens he j h componen of respecve vecors; R (0,, drawn unformly for each j. j Selecon o decde wheher or no o nclude he ral vecor n he populaon of he nex generaon G, s compared wh he arge vecor usng greedy creron. If he value of he objecve funcon for he ral vecor ug, s beer han or equal o he value obaned for he arge vecor x ; he laer s replaced by he former oherwse he laer s reaned n he populaon of he nex G, generaon. u f f( u f( x G, G, G, x G, (0 x oherwse G, where,,... pop. sze, f : ( s assumed o be he feasble search space of he problem s a connuous

11 97 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 n real valued funcon and x s connuous varable vecor wh doman. DE Procedure: Sep-: Inalze conrol parameers: Se he values of he DE conrol parameers ( N, FC,. R Sep-: Deermne he nal populaon x x x, x, x,... x,, 3, N, where he componens of each pon x ( j,,..., N are floang pon numbers randomly chosen whn he, j range (0,. Sep-3: Generae he soluons of he nex populaon for = o N do Muaon phase: Generae a muan vecor vg, usng equaon (8. Crossover phase: Generae a ral vecor ug, by s parens x and v G, G, usng equaon (9. Acceped phase: Accepance of he oal expeced average prof (EAP funcon s occurred wh he equaon (0. Endfor Sep-: Populaon sascs deermne he populaon bes soluon f ermnaon condon s no sasfed hen go o sep 3. Reurn he models (5 and (6 are solved by usng dfferenal evoluon algorhm approach and credbly measure, dscussed n subsecon-6. and secon- respecvely. Our DE consss of some parameers, he sze N of he populaon, scalng parameer F n s muaon scheme (8 and he conrollng parameer C n he crossover scheme (9. Here we R consder N 00, C 0.5 and F NUMERICAL ILLUSRAION R In hs secon, boh he sochasc model and fuzzy-sochasc model are llusraed and solve by DE algorhm wh a numercal example. 7. Sochasc Model o llusrae he proposed wo sage sochasc producon nvenory model, le us consder he oal budge B and he npu daa for followng wo ems are shown n able. Iem C C C 3 h h able. Some daa for above nvenory model C N N s C C r r m P P P P c c Iem Iem able. Expeced value of defecve ems Iem Ed ( Ed ( Iem Iem he compuaonal resul s shown n able 3.

12 98 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 able 3. Opmal soluons for llusraed example wh allowable shorage 3 3 EAP Sensvy Analyss: For he gven numercal example menoned n secon 7., sensvy analyses are performed o sudy he effec of changes of dfferen values of he demand parameers,, and on maxmum expeced average prof of he sysem. I s observed ha for dfferen values of as ncreases when and are fxed, expeced value of he average prof ncreases and also for dfferen values of as ncreases when and are fxed, expeced value of he average prof also ncreases. All hese observaons agree wh he realy. ables and 5 show ha sensvy analyss of he demand parameer for dfferen values of when = 0 and = 0.33 and sensvy analyss of he demand parameer for dfferen values of when = 50 and = 0.35 respecvely. able. Sensvy analyss for able 5. Sensvy analyss for 3 EAP 3 EAP Fuzzy-Sochasc Model o llusrae he wo sage fuzzy-sochasc producon nvenory model numercally, he npu daa are aken as follows. C (.,.,.5,.75, C (.,.3,.5,.75, C (.5,.75, 3.5, 3.7, h h h C (.6,.8, 3.0, 3.5, C (0, 5, 9, 35 and C (0,, 9, 3, h 3 3 B ( 000, 5000, 50000, 5000 and he oher daa are same as n sochasc model. able 6 shows he resuls for dfferen values of he confdence level. able 6. Resuls usng credbly approach 3 3 EAP

13 99 Das, Roy and Kar: A Mul-em Invenory Model for wo-sage Producon Sysem wh Imperfec Processes Usng Dfferenal Evoluon and Credbly IJOR Vol. 9, No., (0 8. CONCLUSION AND FUURE SCOPE In hs sudy a wo-sage producon nvenory model for mul-em wh budgeary consran has been presened. Here we have analyzed an nvenory sysem where he demand can be sasfed by he producs of sage II producon, assumng ha mperfec producs are reworked. Here s also assumed ha defecve ems are produced n boh he sages n a random fashon. For fuzzy-sochasc model, holdng cos, seup cos and oal avalable budge are mprecse. For he frs me, random producon of defecve uns wh rework have consdered n a wo-sage producon sysem. Credbly heory approach has been nroduced for an mprecse nvenory sysem. A dfferenal evoluon (DE algorhm has been desgned for numercal llusraon of he proposed model. Fnally, a fuure sudy wll ncorporae more realsc assumpons n he proposed model, such as varable producon rae, unceran/mprecse naure of demand fne or random plannng horzon.in hs secon, boh he sochasc model and fuzzy-sochasc model are llusraed and solve by DE algorhm wh a numercal example. REFERENCES. Bhuna, A.K. and Ma, M. (998. A wo warehouses nvenory model for deerorang ems wh a lnear rend n demand and shorages. Journal of he Operaonal Research Socey, 9: Chu, Y.P. (003. Deermnng he opmal lo sze for he fne producon model wh random defecve rae, he rework process, and backloggng. Engneerng Opmzaon, 35(: Darwsh, M.A. and Ben-Daya, M. (007. Effec of nspecon errors and prevenve manenance on a wo-sage producon nvenory sysem. Inernaonal Journal of Producon Economcs, 07: Das, D., Roy, A. and Kar, S. (00. A producon-nvenory model for a deerorang em ncorporang learnng effec usng Genec Algorhm. Advances n Operaons Research, do: 0.55/00/ Das, D., Roy, A. and Kar, S. (0. Opmal paymen me for a realer under permed delay of paymen by he wholesaler wh dynamc demand and hybrd number cos parameers. OPSEARCH, 8(3: Davs, L. (99. Handbook of Genec Algorhms, Van Nosrand Renhold. 7. Goyal, S.K. and Gunasekaran, A. (995. An negraed producon-nvenory-markeng model for deerorang ems. Compuers and Indusral Engneerng, 8: Hayek, P. and Salameh, M.K. (00. Producon lo szng wh he reworkng of mperfec qualy ems produced. Producon Plannng and Conrol, (6: Hll, R.M. (000. On opmal wo-sage lo szng and nvenory bachng polces. Inernaonal Journal of Producon Economcs, 66: Kar, S., Roy,.K. and Ma, M. (000. A fuzzy deerorang mul-ems EOQ model wh prce dependen demand under budgeary consran. AMSE, France, 9: Km, D. (999. Opmal wo-sage lo szng and nvenory bachng polces. Inernaonal Journal of Producon Economcs, 58(3: -3.. Konsanaras, S. and Papachrsos, S. (007. Opmal polcy and holdng cos sably regons n a perodc revew nvenory sysem wh manufacurng and remanufacurng opons. European Journal of Operaonal Research, 78(: Lu, B. and Lu, Y.K. (00. Expeced value of fuzzy varable and fuzzy expeced value models. IEEE ransacons of Fuzzy Sysems, 0(: Pearn, W.L., Su, R.H. and Hsu, C.H. (00. Opmal producon run me for wo-sage producon sysem wh mperfec processes and allowable shorages. Cenral European Journal of Operaons Research, DOI: 0007/s x. 5. Roy,.K. and Ma, M. (997. A fuzzy EOQ model wh demand-dependen un cos under lmed sorage capacy. European Journal of Operaonal Research, 99: Roy, A., Kar, S. and Ma, M. (008. A deerorang mul-em nvenory model wh fuzzy coss and resources based on wo dfferen defuzzfcaon echnques. Appled Mahemacal Modellng, 3: Salameh, M.K. and Jaber, M.Y. (000. Economc producon quany model for ems wh mperfec qualy. Inernaonal Journal of Producon Economcs, 6: Sorn, R. and Prce, K. (997. Dfferenal evoluon a smple and effcen heursc for global opmzaon over connuous spaces. Journal of Global Opmzaon, : Szendrovs, A.Z. (983. Non-neger opmal lo sze raos n wo-sage producon/invenory sysems. Inernaonal Journal of Producon Research, (3: