Optimizing Multiproduct Multiconstraint Inventory Control Systems with Stochastic Period Length and Emergency Order

Transcription

1 Joual of Uceta Systes Vol.7, No.1, pp.58-71, 013 Ole at: Optzg Multpoduct Multcostat Ivetoy Cotol Systes wth Stochastc Peod Legth ad egecy Ode Ata Allah Talezadeh 1, Seyed Tagh Akhava Nak, *, Gholaeza Jalal Na 3 1 Depatet of Idustal geeg, College of geeg, Uvesty of Teha, Teha Ia Depatet of Idustal geeg, Shaf Uvesty of Techology, P.O. Box Azad Ave Teha Ia 3 Depatet of Idustal geeg, Ia Uvesty of Scece ad Techology, Teha Ia Receved 4 Octobe 011; Revsed 6 May 01 Abstact Whle the usual assuptos ultpeodc vetoy cotol pobles ae that the odes ae placed ethe at the begg of each peod (peodc evew) o depedg o the vetoy level they ca happe at ay te (cotuous evew), ths pape we assue the peods betwee two epleshets of seveal poducts ae detcal ad depedet ado vaables. Fo the poble at had, the ode quattes (decso vaables) ae of tegetype ad thee ae two kds of space ad sevce level costats fo each poduct. A odel of the poble s fst developed whch a cobato of backode ad eegecy odes s cosdeed fo the shotages, ad the costs ae shotage, holdg, puchasg, ad taspotato. The, we show the odel s of a tege-oleapogag type ad to solve t, two eta-heustc algoths of geetc algoth ad sulated aealg ae eployed. At the ed, a uecal exaple s gve to deostate the applcablty of the poposed ethodology. 013 Wold Acadec Pess, UK. All ghts eseved. Keywods: vetoy cotol, stochastc epleshet, patal back-odeg, tege olea pogag, sulated aealg, geetc algoth 1 Itoducto ad Lteatue Revew Whle ultpeodc vetoy cotol odels the cotuous evew ad the peodc evew ae the two ao polces, the udelyg assuptos of the poposed odels estct the coect utlzato eal-wold evoets. I cotuous evew polcy, the use has the feedo to act at ayte ad eplesh odes based upo the avalable vetoy level. Howeve, the peodc evew, the use s allowed to eplesh the odes oly pedeteed tes. Chag [8] cosdeed a peodc evew odel wth dscouts, whch the peod was patly log ad the costs wee puchasg, holdg, ad fxed odeg. The potat aspect of hs study was to toduce eegecy odes to pevet shotages. Bylka [5] vestgated a odel costaed o the aouts of odes ad backodes, whch the lead-te was costat ad dead was stochastc. By aalyzg the chages the lead-te ad the odeg cost, they ted to deve the optzed odeg te. togal ad Rah [10] aalyzed a ultpeod vetoy poble wth depedetly ad detcally dstbuted (..d.) epleshet tevals. I ths close to ealty schee, a supple vsts a etale wth ado te-aval tes ad the etale epleshes hs vetoes based o a eplesh-up-to-level vetoy cotol polcy. They also assued that oly a ceta facto of the uet dead was backodeed ad the est was lost. I ths settg ude geeal dstbuto betwee epleshet epochs, they showed the cocavty of the expected poft fucto ad gave the codto that would hold fo the optal eplesh-up-to-level. Late, Chag [9] exteded a peodc evew vetoy syste whch the peod legth was ot costat ad followed a pobablty dstbuto. They assued the supple s vst-tevals wee..d. ado vaables. * Coespodg autho. al: Nak@Shaf.edu (S.T.A. Nak).

2 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, The peodc vetoy cotol pobles wee vestgated depth dffeet eseach woks. Chew at al. [7] exteded a peodc evew polcy fo a peshable poduct wth a pedeteed lfete whch the pce ad the vetoy allocato wee otly deteed. They assued a pce-sestve dead whee the pce would cease as the te goes o. Lee ad Schwaz [0] developed a peodc vetoy cotol syste wth oe poduct ad stochastc lead-te. They assued the ode would be delveed edately o oe peod late. Recetly, Aadua ad Uthayakua [4] cosdeed a peodc evew syste whch the decso vaables wee the peod legth, the axu vetoy level, ad the facto of the shotage that was lost. The a of the eseach was to ze the lost-sale ate. Futhe, Yu ad Xaobo [31] developed a peodc evew vetoy syste sevg ultple dead classes that wee dffeetated by a teatet fo shotages. They assued shotages of soe classes wee teated accodg to lost sales. The a of the eseach was to detee both the vetoy epleshet decsos ad the allocato fo all classes. Che ad Yag [6] cosdeed a peodc-evew sglepoduct vetoy syste wth stochastc dead whch the pcg ad the odeg decsos wee ade sultaeously ove a fte hozo. Ray et al. [4] studed two peodc evew vetoy odels that dffeed tes of how backodeg cost was chaged. I the fst odel, the backodeg cost was chaged pe ut backodeed ad was depedet of the legth of te. I the secod odel howeve, the backodeg cost was chaged based o the ube of backodes that wee te-depedet. Teute et al. [30] poposed a ew ethod fo deteg ode-up-to levels of tes a peodc evew syste whch the lead-te deads followed a copoud boal patte. Assug the odes to be placed ethe oe o at least as uch as a u ode quatty, Kesulle et al. [18] poposed a peodc evew sgle-te sgle-stage vetoy syste wth stochastc dead. Slve ad Bschak [6] exteded the peodc evew base stock syste ude exact fll ate ad oally stochastc dead. Talezadeh et al. [7] developed a ultpoduct ultcostat stochastc vetoy cotol odel whch the peod legth was a ado vaable. Late, Talezadeh et al. [8] exteded the pevous wok by cosdeg fuzzy cost facto ad the fuzzy stochastc peod legth [9]. Ths pape povdes a exteso of a odel of vetoy cotol toduced togal ad Rah [10] to ultple vetoy tes, tegal decso vaables, ultple costats, ad eegecy odes. As the deved optzato odels ae hghly olea, we popose to use ethe sulated aealg o geetc algoths to solve fo a ea optal soluto. Fou a specfcatos of the poposed odel of ths eseach that akes t close to eal-lfe vetoy evoets ad have led to ts ovelty ae 1) to odel both ultpoducts ad ultcostats pobles, ) the fact that the decso vaables ae tege, 3) the exstece of eegecy odes, ad 4) copoatg the taspotato cost. By deployg these codtos sultaeously, the developed odel s dffeet fo othe odels the peodc evew lteatue. Futhe, the odel s helpful stuatos whch a supple vsts a etale wth ado te-aval tes ad the etale epleshes hs vetoes based o a eplesh-up-to-level vetoy cotol polcy. Moeove, due to soe ltatos o the poducto capacty, the supply of the aw ateals, ad the lke, the peod legth ay be uceta ad the goods ay ot be delveed o te. As a exaple, whe dead ceases ad the poducto capacty s lted, case of beakdows o late ecepts of poted aw ateals (as they ae delayed at custos) the lead-te ad hece the cycle legth ae ceased. The stochastc atue of these factos causes the peod legth to be stochastc. The est of the pape s ogazed as follows. I Secto, the poble alog wth ts assuptos s defed. I Secto 3, we odel the defed poble of Secto. To do ths we fst toduce the paaetes ad the vaables of the poble. The, a sgle poduct poble s odeled, ad fally the ultpoduct poble s foulated. I the fouth secto of the pape, we expla the ways to solve the odel at had ad aalyze t ude specal codtos. Icopoatg a uecal exaple, the soluto ethods ae vestgated Secto 5. The cocluso ad ecoedatos fo futue eseach coe Secto 6. Poble Defto Cosde a peodc vetoy cotol poble fo oe povde whch the tes equed to ode seveal poducts ae stochastc atue. Let the te-peods betwee two epleshets of the poducts be detcal ad depedet ado vaables. The deads of the poducts ae costats ad case of shotage, a facto s cosdeed backode ad a facto lost-sale. I ths case, the pecetage of the ube of custoes that wat to eceve the odes s kow. To avod lost sales, eegecy odes ca be placed. Whle the puchasg cost pe ut of a eegecy ode s oe tha the puchasg cost pe ut of a oday ode, t s less tha the shotage cost pe ut of lost sales. Fo oday odes, the syste costs ae holdg, backode, ad puchase. Howeve, a taspotato cost s added to the puchasg cost of a eegecy ode. Futhe, thee s o dffeece the sellg pce pe ut of both oday ad eegecy odes ad the lead-tes of the eegecy odes ae zeo. Whle all

3 60 A.A. Talezadeh et al.: Optzg Multpoduct Multcostat Ivetoy Cotol Systes the puchased poducts ae sold, the waehouse space ad the sevce level of each poduct ae cosdeed costats of the poble. Moeove, the decso vaables ae teges. We eed to detfy the vetoy levels each cycle such that the expected poft s axzed. 3 Modelg Fo the poble at had, sce the te-peods betwee two epleshets ae depedet ado vaables, ode to axze the expected poft of the plag hozo we eed to cosde oly oe peod. Futheoe, sce we assued the costs assocated wth the vetoy cotol syste ae holdg ad shotage (cludg eegecy ode ad taspotato costs), we eed to calculate the expected vetoy level ad the expected equed stoage space each peod. Befoe odelg, we fst defe the paaetes ad the vaables of the odel. 3.1 The Paaetes ad the Vaables of the Model Fo 1,,...,, defe the paaetes ad the vaables of the odel as : The axu vetoy level of the th poduct T : A ado vaable deotg the te-peod betwee two epleshets (cycle legth) of the th poduct f ( t ): The Pobablty desty fucto of T T h : The holdg cost pe ut vetoy of the th poduct each peod : The backode cost pe ut dead of the th poduct c : The puchasg cost pe ut of the th poduct c : The puchasg cost pe ut of a eegecy ode of the th poduct k : The taspotato cost pe ut of a ode of the th poduct v : The sellg pce pe ut of the th poduct d : The costat dead ate of the th poduct t D : The te at whch the vetoy level of the th poduct eaches zeo : The pecetage of usatsfed deads of the th poduct that s backodeed I : The expected aout of the th poduct vetoy pe cycle l : The expected aout of the th poduct lost-sale each cycle b : The expected aout of the th poduct backode each cycle q : The expected aout of the th poduct ode each cycle q : The expected aout of the th poduct eegecy ode each cycle ( q = L ) : The lowe lt of the sevce level fo the th poduct f : The equed waehouse space pe ut of the th poduct F : Total avalable waehouse space : Nube of shpets A : The costat cost of each shpet C : The expected puchase cost of a eegecy ode of the th poduct CH : The expected holdg cost pe cycle of the th poduct CB : The expected shotage cost backode state of the th poduct CL : The expected shotage cost lost-sale state of the th poduct CP : The expected puchase cost of oday odes of the th poduct CT : The expected taspotato cost of the th poduct R : The expected eveue obtaed fo sales Z : The expected poft obtaed each cycle Fo sake of splcty, Secto 3.3 we fst cosde a sgle-poduct poble. The, we exted the odelg to the ult-poduct odelg Secto 3.4. Howeve, we fst toduce the pctoal epesetato of the sglepoduct poble Secto 3..

4 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, Ivetoy Daga Accodg to togal ad Rah [10] ad cosdeg the fact that the te-peods betwee epleshets ae stochastc vaables, two cases ay occu. I the fst case, the te-peod betwee epleshets s less tha the aout of te equed fo the vetoy level to each zeo (see Fgue 1). I the secod case, t s geate (see Fgue ). Fgue 3 depcts the shotages both cases. 3.3 Devg the Costs ad the Poft of a Sgle-Poduct Model I ode to obta the expected poft each cycle, we eed to evaluate all of the tes quato (1) [10]: Z R CP CH CB CL v q c q h I b ( v c ) l. (1) Based o Fgue 3, the ado lost sale quatty s (1 )( dt ), whee ts expected value s obtaed usg quato () TMax l (1 ) ( dt ) ft ( t ). dt td T TMax () td Futhe, the ado backodeed quatty s ( dt ) wth a expected value of TMax b ( dt ) ft ( t ). dt td T TMax (3) td Moeove, sce the ado vetoy whe the vetoy level s postve s T dt stock out peods, the expected vetoy level s obtaed usg quato (4) td T Max dt T T T M t d D I T f ( t ) dt f ( t ) dt., ad ( d ) dug (4) Fgue 1: Pesetg the vetoy cycle whe TM T t D Fgue : Pesetg the vetoy cycle whe t D T T Max

5 6 A.A. Talezadeh et al.: Optzg Multpoduct Multcostat Ivetoy Cotol Systes Fgue 3: Pesetg shotages two cases of copact back ode ad lost sales I ode to calculate the expected ode quatty, oe eeds to detee the equed quattes dug both stock ad stock-out peods. Sce they ae espectvely dt ad dt, the expected ode quatty wll be td TMax ( ) ( ). (5) q dt f t dt dt f t dt T T TM t D I ths pape, to avod lost sales of quato () we put a eegecy ode of sze q l. Sce the puchasg cost pe ut of a eegecy ode s c ad that the sellg pce pe ut s P, the cost of a eegecy ode s calculated as: T Max C ( ) 1 c q Pq c v ( dt ) ft ( t ) t dt D (6) whee c v. The taspotato cost s calculated based o quato (7), whch fq s the equed space to shp the ode fo the supple. Ak 0 ˆ q fq f A kq ˆ ˆ f fq f CT (7) A k ( 1) ˆ ˆ q f fq f By toducg bay vaables Y, 1,,...,, the taspotato cost ca be copoated wth the atheatcal odel of the poble as CT k q AY 1 f q fy ˆ 0 fy ˆ ˆ fq fy ( 1) fy ˆ ˆ fq fy Y1 Y Y 1 Y 0,1. As the total avalable waehouse space s F, the space equed fo each ut of poduct s f, ad the uppe lt fo the vetoy s, the space costat wll be f F. (9) 1 (8)

6 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, Sce the shotages oly occu whe the cycle te s oe tha t D ad that the lowe lt fo the sevce level s, the TMax PT ( ) td ( ) 1. f T t dt (10) d I shot, the coplete atheatcal odel of the sgle poduct vetoy s Max Z R CP CH CB C CT TMax d ( ) ( ) v c dt ft t ( ) ( ) dt T dt ft t dt M d s.. t f F TMax T Max ( dt ) ft ( t) ( ) 1 ( ) ( ) dt c v dt ft t dt td t D T Max d ( ) T ( ) ( ) T T M d 1 k dt f t dt dt f t dt AY (11) TMax T d f ( t ) dt 1 0 fq ˆ fy1 fy ˆ ˆ fq fy ( 1) fy ˆ f q fy ˆ Y1 Y Y 1 Y 0,1 0, Itege. 3.4 The Multpoduct Model The sgle-poduct vetoy odel of Secto 3.3 ca be easly exteded to a ultple-poduct odel as follows: [( ) ( ) ] 1 1 Max Z v c q h I b c v q k q AY st.. f F fq fy ˆ 1 ( 1) fy ˆ f q fy ˆ,3,..., 1 PT ( t ) 1 1,,..., 1 D Y 0,1 1,,..., 0, Itege 1,,...,. Y 1 I what follows, we cosde two pobablty desty fuctos fo T ad hece we develop two odels T Follows a Ufo Dstbuto I ths case the pobablty desty fucto of T s ft ( t) 1 ( tax t ). Accodgly, (1) wll chage to (1)

7 64 A.A. Talezadeh et al.: Optzg Multpoduct Multcostat Ivetoy Cotol Systes h ( c 3 c k)(1 ) ht ax Max Z 1 6 d ( tax t ) 1 d ( ax ) t t ( c c k)(1 ) tax ht t ax 1 tax t 3 ht d 3( )( ax ) 3 ax ( ( )(1 )) v c k t t d t d c v 6( tax t ) s.. t f F (13) dt Max 1 1,,..., d ( t t ) Max M ( 1) ( dt (1 )) ( t t ) d 0 ( ) fy ˆ Max Max M f 1 d( tmax t ) M ( 1) ( dt (1 )) ( t t ) d ( 1) fy ˆ f ( ) fy,,3,..., 1 Y 1 Max Max M ˆ 1 d( tmax t ) M Y 0,1 1,,..., 0, Itege 1,,...,. 1 AY 3.4. T Follows a xpoetal Dstbuto If T follows a expoetal dstbuto wth paaete, the the pobablty desty fucto of T wll be t f ( t ) e. I ths case, the odel becoes: 1 1 d hd d Max Z [ d(1 )( c k c ) d] e d( v c k) h (1 e ) 1 1 d e 1 1,,..., d d 0 f ( (( 1) 1)) ˆ e fy (14) 1 1 ( 1) ˆ d d fy ( (( 1) 1)) ˆ f e fy,3,..., 1 Y 1 1 s.. t f F Y 0,1 1,,..., 0, Itege 1,,...,. I the ext secto, we wll toduce two eta-heustc algoths to solve the poble. 4 The Soluto Algoths Sce the odels (13) ad (14) ae tege-olea atue, eachg a aalytcal soluto (f ay) s dffcult [11]. May eseaches have used eta-heustc algoths to solve coplcated optzato pobles ay

8 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, felds of scetfc ad egeeg dscples. Soe of these eta-heustc algoths ae sulatg aealg [7], Tabu seach [15], geetc algoths [, 5], patcle swa optzato [3, 13 ad 16], eual etwoks [], haoy seach [14, 17] ad at coloy [3]. As a esult, ths secto two teatve-seach algoth of sulated aealg ad populato-base geetc algoth ae eployed to solve the odels. 4.1 Sulated Aealg To solve coplex optzato pobles, Aats ad Kost [1] poposed a local seach algoth aed sulated aealg (SA) that was sped by physcal aealg pocesses. SA s a effcet ad effectve ethod that poduces good suboptal solutos ad has bee used ay cobatoal optzato pobles of dffeet aeas of sceces [19]. A SA algoth follows seach dectos that pove the obectve fucto value. Whle explog soluto space, SA offes the possblty of acceptg wose eghbo solutos a cotolled ae ode to escape fo local a. The a steps a SA algoth ae: (1) geeatg eghbo, () evaluatg the obectve fucto, (3) assgg a tal tepeatue, (4) chagg the tepeatue, (5) coolg schee, ad (6) stoppg. The eghbo geeato s a potat copoet of SA. I ths pape, the tal solutos ae geeated two dffeet ways. I the fst way, they ae adoly selected aog a feasble soluto space ad the secod, they ae geeated usg the best solutos obtaed by the geetc algoth descbed Secto 4.. Whe a soluto s geeated, t should be evaluated by ts obectve fucto value. I the axzato odels of ths eseach, f the obectve fucto of the ew soluto () becoes bgge tha the obectve fucto of the pevous soluto (), the () wll be eplaced by (). Othewse, by geeatg a ado ube the bette soluto s selected. Oe of the potat paaetes of the SA algoth s ts tal tepeatue. The tal tepeatue has a sgfcat effect o the possblty of selectg a bad soluto. O the oe had, f a hgh value assued fo the tal tepeatue, a soluto wth a bad obectve fucto value has a hgh chace of beg accepted. O the othe had, low value of the tal tepeatue akes the pobablty of the soluto to be a local optu hgh. I ths pape, dffeet lage values of 1000, 1500, ad 000 ae chose fo the tal tepeatues. The age of tepeatue chages a SA algoth s also oe of the pay aspects of the aealg pocess. I ths pape, we chage the tepeatue of the SA algoth based o a geoetc fucto gve quato (15) wth 0.9, 0.95, ad T T 1 ; 1,, (15) Aalyzg the equlbu state afte a couple of etece a specfc tepeatue of a SA algoth s potat ad ecessay as well. Ths step should be pefoed to ake sue f the aealg pocess eeds to cotue ts cuet tepeatue o t should be stopped ad tasfeed to the ext tepeatue. I ths eseach, eachg to the pe-defed fal tepeatue T F s used as the stoppg cteo. Futheoe, dffeet values of 50, 100, ad 00 ae eployed fo Nt ()(ube of teato each tepeatue). I shot, the steps volved the poposed SA algoth ae show Fgue Choosg a tal soluto fo the goup of feasble solutos S. Choosg the tal tepeatue T Selectg the ube of teatos Nt () at each tepeatue 4. Selectg the fal tepeatue T F 5. Deteg the pocess of the tepeatue educto utl t eachest F 6. Settg the tepeatue exchage coute to zeo fo each tepeatue 7. Ceatg the soluto at the eghbohood of the soluto 8. valuatg the obectve fucto at ay tepeatue ad calculate z( ) z( ) 9. Acceptg the soluto, f 0. lse, geeatg a ado ube RN ~ U [0,1]. T0 If RN e the select soluto 10. Settg 1. If s equal to Nt () the go to 1. Othewse, go to Reducg the tepeatue. If t eaches T F the stop. Othewse, go to 6 Fgue 4: The steps of the poposed SA algoth

9 66 A.A. Talezadeh et al.: Optzg Multpoduct Multcostat Ivetoy Cotol Systes 4. Geetc Algoth Hollad [1] was the fst who toduced the fudaetal pcpal of geetc algoths (GA). GA, as a populatobased eta-heustc algoth, was sped by the cocept of suvval of the fttest choosoes. A choosoe s a stg of gees that ae cosdeed the coded fgue of a possble soluto. I a optzato applcato of the GA, a vaable s cosdeed a gee ad a soluto vecto cotag seveal ges s a choosoe. I ths pape, the choosoes ae stgs of the vetoy levels of the poducts ( ). A GA opeates though a sple cycle of stages cludg 1) ceato of a populato of stgs, ) evaluato of the stgs, 3) selecto of the best stgs, ad 4) geetc apulato to ceate ew populato of stgs. A goup of choosoes s called populato. Oe of the a chaactestcs of a GA s wokg o a set of choosoes (solutos), stead of focusg o a sgle soluto (o oe choosoe). The ube of populato a geeato s the populato sze ad s deoted by N. Ceato of a populato s usually pefoed by ado geeato ove feasble o feasble soluto spaces of the o had poble. Moeove, soe hts o choosg a pope populato sze ae gve by Ma et al. [1]. I ths eseach, the feasble soluto space s cosdeed to geeate populatos of dffeet szes of 10, 100, ad A soluto s evaluated based o ts obectve fucto value. I the axzato poble of ths eseach, the choosoes wth hghe obectve fucto values ae accepted the best kow. Futhe, the ftess popotoal selecto assgs a selecto pobablty to each soluto. At the ed, the ceato of the ew populato s pefoed by e-cobatos of two types; utato ad cossove. The pobabltes of the cossove ( P c ) ad utato ( P ) ae the paaetes of the geetc algoths. I ths eseach, we test the sgle pot, the two pots, ad the ufo cossoves show Fgues 5 to 7 wth the cossove pobabltes of 0.85, 0.90, ad 0.95, whee shows the choosoe cotag the vetoy levels of the poducts. Futhe, the utato opeato of ths eseach, we ceate a ado ube RN betwee (0,1) fo each gee. If RN s less tha a pedeteed utato pobablty P, the the utato, pefoed based o the ufo fucto ove the specfc age of the vaable, occus the gee. Othewse, the utato opeato s ot pefoed that gee. Fgue 8 depcts a utato opeato whch P s chose Fgue 5: The sgle-pot cossove opeato wth M= Fgue 6: The two-pots cossove opeato wth M=3 ad Fgue 7: The ufo cossove opeato wth M=3 ad RN Fgue 8: A saple of the utato opeato

10 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, I ths pape, 0.078, 0.088, ad 0.1 ae eployed as dffeet values of the P paaete. Futhe, the steps of the GA used ths pape ae show Fgue 9. Fgue 9: The steps of the GA I ode to deostate the poposed SA ad GA algoths ad to evaluate the pefoaces, the ext secto we bg a uecal exaple used togal ad Rah [10] wth soe odfcatos. I ths exaple, two cases of the ufo ad the expoetal dstbutos fo the te-peod betwee two epleshets ae vestgated. 5 Nuecal xaples Cosde a ultpoduct vetoy cotol poble volvg eght poducts ad geeal data gve Table (1). Tables () ad (3) show the paaetes of the ufo ad the expoetal dstbutos used fo the te-peod betwee two epleshets, espectvely. The total avalable waehouse space s 18,000 ad the avalable space fo each shpet s 5,000 wth a costat cost of 500 pe shpet. Tables (4) ad (5) show dffeet values of the paaetes of the SA ad the GA ethods, espectvely. I ths eseach all the possble cobatos of the paaetes SA ( Nt (), Tad 0 ) ad GA ( Pc, Pad N ) ethods ae eployed ad usg the ax(ax) cteo the best cobato of the paaetes has bee selected. Futheoe, the sgle-pot cossove had bette pefoaces tha both the two-pots ad ufo cossove opeatos. Table (6) shows the best esult. The best cobatos of the SA ad the GA algoths ae show Tables (7) ad (8), espectvely. Moeove, the covegece paths of the obectve fucto values of the SA algoth ufo ad expoetal dstbutos ae show Fgues 10 ad 11. These gaphs fo the GA ethod ae show Fgues 1 ad 13. Fo the esults, we see that the best soluto of the GA ethod s bette tha the oe obtaed by the SA algoth. Table 1: Geeal data Poduct h v c c k f d Table : Data fo ufo dstbuto Poduct t M t Max 1. Settg the paaetes P c, P ad N. Italzg the populato adoly 3. valuatg the obectve fucto fo all choosoes based o obectve fucto 4. Selectg dvdual fo atg pool 5. Applyg the cossove opeato fo each pa of choosoes wth pobablty P c 6. Applyg utato opeato fo each choosoe wth pobablty P 7. Replacg the cuet populato by the esultg atg pool 8. valuatg the obectve fucto 9. If stoppg cteo s et, the stop. Othewse, go to step 5

11 68 A.A. Talezadeh et al.: Optzg Multpoduct Multcostat Ivetoy Cotol Systes Table 3: Data fo expoetal dstbuto Poduct /30 1/30 1/60 1/60 1/30 1/30 1/60 1/60 Dstbuto Ufo xpoetal Table 4: The paaetes of the SA algoth N t T P c Table 5: The paaetes of the GA ethod P Table 6: The best esult fo Appoach Poduct Z GA SA GA SA Table 7: The best cobato of the SA paaetes Nuecal xaple wth N t T 0 Ufo Dstbuto xpoetal Dstbuto Table 8: The best cobato of the GA paaetes Nuecal xaple wth P c P N Ufo Dstbuto xpoetal Dstbuto N Iteato Nube Obectve Fucto Fgue 10: The covegece path of the best esult ufo exaple of SA

12 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, Iteato Nube Obectve Fucto x 10 4 Fgue 11: The Covegece path of the best esult expoetal exaple of SA Obectve Fucto Geeato Nube Fgue 1: The covegece paths of the best esult ufo exaple of GA 6.6 x Obectve Fucto Iteato Nube Fgue 13: The covegece path of the best esult expoetal exaple of GA

13 70 A.A. Talezadeh et al.: Optzg Multpoduct Multcostat Ivetoy Cotol Systes 6 Cocluso ad Recoedato fo Futue Reseach I ths pape, a stochastc epleshet ultpoduct vetoy odel wth patal backodeg ad eegecy ode ude the sevce level ad space costats was vestgated. Two atheatcal odelg fo two cases of ufo ad expoetal dstbuto of the te betwee two epleshets have bee developed ad show to be tege-olea pogag. The, two eta-heustc soluto algoths of SA ad GA wee poposed to solve the odels. Fally, based upo the esults of two uecal exaples t was show that the best soluto of the GA algoth was bette tha the oe the SA algoth. Fuzzy paaetes, soe othe pobablty dstbuto fuctos fo the peod legth, ad deteoato ate fo the stock vetoy ca be cosdeed futue woks. Refeeces [1] Aats,.H.L., ad J.H.M. Kost, Sulated Aealg ad Boltza Mache, A Stochastc Appoach to Coputg, Joh Wley ad Sos, Chcheste, [] Abbas, B., ad H. Mahloo, Ipovg espose suface ethodology by usg atfcal eual etwok ad sulated aealg, xpet Systes wth Applcatos, vol.39, o.3, pp , 01. [3] Afsha, M.H., Coloy-utated at syste fo ppe etwok optzato, Iaa Joual of Scece ad Techology, Tasacto B: geeg, vol.35, o., pp.17 3, 011. [4] Aadua K., ad R. Uthayakua, Reducg lost-sales ate (T,R,L) vetoy odel wth cotollable lead te, Appled Matheatcal Modellg, vol.34, o.11, pp , 010. [5] Bylka, S., Tupke polces fo peodc evew vetoy odel wth eegecy odes, Iteatoal Joual of Poducto coocs, vols.93-94, pp , 005. [6] Che, Z., ad Y. Yag, Optalty of (s, S, p) polcy a geeal vetoy-pcg odel wth ufo deads, Opeatos Reseach Lettes, vol.38, o.4, pp.56 60, 010. [7] Chew,.P., L. Lee, ad R. Lu, Jot vetoy allocato ad pcg decsos fo peshable poducts, Iteatoal Joual of Poducto coocs, vol.10, o.1, pp , 009. [8] Chag, C., Optal epleshet fo a peodc evew vetoy syste wth two supply odes, uopea Joual of Opeatoal Reseach, vol.149, o.1, pp.9 44, 003. [9] Chag, C., Peodc evew vetoy odels wth stochastc supple s vst tevals, Iteatoal Joual of Poducto coocs, vol.115, o., pp , 008. [10] togal, K., ad M.A. Rah, Replesh-up-to vetoy cotol polcy wth ado epleshet teval, Iteatoal Joual of Poducto coocs, vols.93-94, pp , 005. [11] Ge, M., Geetc Algoth ad geeg Desg, Joh Wley & Sos, New Yok, [1] Hollad, J.H., Adopto Neual ad Atfcal Systes, The Uvesty of Mchga Pess, A Abo, USA, [13] Hosse, S.V., H. Moghadas, A.H. Noo, ad M.B. Roya, Newsboy poble wth two obectves, fuzzy costs ad total dscout stategy, Joual of Appled Sceces, vol.9, o.10, pp , 009. [14] Jabepou, M., ad. Khoa, A ew haoy seach algoth fo solvg xed-dscete egeeg optzato pobles, geeg Optzato, vol.43, o.5, pp , 011. [15] Joo, S.J., ad J.Y. Bog, Costucto of exact D-optal desgs by tabu seach, Coputatoal Statstc ad Data Aalyss, vol.1, o., pp , [16] Kaveh, A., ad K. Lakead, A ovel hybd chage syste seach ad patcle swa optzato ethod fo ultobectve optzato, xpet Systes wth Applcatos, vol.38, o.1, pp , 011. [17] Kaveh, A., ad M. Ahagaa, Dscete cost optzato of coposte floo syste usg socal haoy seach odel, Appled Soft Coputg Joual, vol.1, o.1, pp , 01. [18] Kesulle, G.P., A.G.D. Kok, ad S. Daba, Sgle te vetoy cotol ude peodc evew ad a u ode quatty, Iteatoal Joual of Poducto coocs, vol.133, o.1, pp.80 85, 011. [19] Kkpatck, S., C.D. Gelatt, ad M.P. Vecch, Optzato by sulated aealg, Scece, vol.0, o.4598, pp , [0] Lee, J.Y., ad L.B. Schwaz, Leadte aageet a peodc-evew vetoy syste: a state-depedet base-stock polcy, uopea Joual of Opeatoal Reseach, vol.199, o.1, pp.1 19, 009.

14 Joual of Uceta Systes, Vol.7, No.1, pp.58-71, [1] Ma, K.F., K.S. Tag, S. Kwog, ad W.A. Halag, Geetc Algoths fo Cotol ad Sgal Pocessg, Spge Velag, Lodo, UK, [] Passaddeh, S.H.R., ad S.T.A. Nak, ad J. Aya Yegaeh, A paaete-tued geetc algoth fo ult-poduct ecooc poducto quatty odel wth space costat, dscete delvey odes ad shotages, Advaces geeg Softwae, vol.41, o., pp , 010. [3] Rah-Vahed, A.R., S.M. Mghoba, ad M. Rabba, A hybd ultobectve patcle swa algoth fo a xedodel assebly le sequecg poble, geeg Optzato, vol.39, o.8, pp , 007. [4] Ray, S., Y. Sog, ad M. Vea, Copaso of two peodc evew odels fo stochastc ad pce-sestve dead evoet, Iteatoal Joual of Poducto coocs, vol.18, o.1, pp.09, 010. [5] Shahsava, M., S.T.A. Nak, ad A.A. Naaf, A effcet geetc algoth to axze et peset value of poect payets ude flato ad bous pealty polcy esouce vestet poble, Advaces geeg Softwae, vol.41, os.7-8, pp , 010. [6] Slve,.A., ad D.P. Bschak, The exact fll ate a peodc evew base stock syste ude oally dstbuted dead, Oega, vol.39, o.3, pp , 011. [7] Talezadeh, A.A., M.B. Ayaezhad, ad S.T.A. Nak, Optzg ult-poducts ult-costats vetoy cotol systes wth stochastc epleshets, Joual of Appled Scece, vol.6, o.1, pp.1 1, 008. [8] Talezadeh, A.A., S.T. Nak, ad M.B. Ayaezhad, Mult-poduct ult-costat vetoy cotol systes wth stochastc epleshet ad dscout ude fuzzy puchasg pce ad holdg costs, Aeca Joual of Appled Scece, vol.8, o.7, pp , 008. [9] Talezadeh, A.A., S.T.A. Nak, ad M.B. Ayaezhad, A hybd ethod of Paeto, TOPSIS ad geetc algoth to optze ult-poduct ult-costat vetoy cotol systes wth ado fuzzy epleshets, Matheatcal ad Copute Modelg, vol.49, os.5-6, pp , 009. [30] Teute, R.H., A.A. Sytetos, ad M.Z. Baba, Deteg ode-up-to levels ude peodc evew fo copoud boal (tettet) dead, uopea Joual of Opeatoal Reseach, vol.03, o.3, pp , 010. [31] Zhou, Y., ad X. Zhao, Optal polces of a vetoy syste wth ultple dead classes, Tsghus Scece ad Techology, vol.15, o.5, pp , 010.