Recent Researches in Applied Mathematics

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1 Solvig the Stochastic Capacitated Locatio-Allocatio Proble by Usig a New Hybrid Algorith Esaeil Mehdizadeh*, Mohaad Reza Tavarroth, Seyed ohse ousavi Departet of Idustrial Egieerig, Faculty of Idustrial ad Mechaical Egieerig, Islaic Azad Uiversity (IAU), Qazvi Brach,Qazvi, Ira *eehdi@qiau.ac.ir Abstract: The ai of this study is to preset a hybrid algorith based o two eta-heuristic algoriths aed Vibratio Dapig Optiizatio (VDO) algorith ad Geetic Algorith (GA) for solvig the stochastic capacitated locatio-allocatio proble. For solvig the odel ore efficietly, the siplex algorith, stochastic siulatio ad the proposed hybrid algorith are itegrated to desig a powerful hybrid itelliget algorith. Fially, to illustratio the efficiecy of the proposed algorith, a few uerical exaples are selected fro literature ad it is foud out that the proposed algorith is able to obtai good results. Key-Words: - Stochastic siulatio, Locatio-allocatio proble, Geetic algorith, Vibratio dapig optiizatio.. Itroductio Locatio-allocatio (LA) proble is to locate a set of ew facilities such that the trasportatio cost fro facilities to custoers is iiized [8]. This proble has bee studied by ay researchers i the past years. Sice LA proble was proposed by Cooper [] for the first tie, uch work has bee doe o this proble. LA proble was studied i detail i Ge ad Cheg [, 3], where all kids of cases were discussed. Also ay odels were preseted. Logedra ad Terrell firstly itroduced the stochastic ucapacitated FLA odel [4]. Zhou proposed the expected value odel, chacecostraied prograig ad depedet-chace prograig for ucapacitated LA proble with stochastic deads [5]. For solvig LA odels, uerous algoriths have bee desiged. For exaple Kuee ad Solad used a brach-ad-boud algorith [6]. Sherali ad Nordai have show that capacitated LA proble is NP-hard eve if all the custoers are located o a straight lie [7]. The capacitated cotiuous locatio allocatio proble is also called capacitated ulti source Weber proble. Aras ad Altel proposed a ew heuristic ethod for the capacitated ulti-facility Weber proble. The capacitated ulti-facility Weber proble is cocered with locatig facilities i the Euclidea plae ad allocatig their capacities to custoers at iiu total cost [8]. Duras et al. preseted the discrete approxiatio heuristics for the capacitated cotiuous locatio allocatio proble with probabilistic custoer locatios [9]. Also uerous eta-heuristic algoriths have bee desiged ivolvig siulated aealig [0], tabu search [] ad so o. Soe hybrid algoriths have bee also suggested, such as oe of the is the cobiatio of siulated aealig ad rado descet ethod [] ad the other oe is the cobiatio of the Lagrage relaxatio ethod ad geetic algorith [3]. Zhou ad Liu proposed the expected value odel, chace-costraied prograig ad depedet-chace prograig for capacitated LA proble with stochastic deads. For solvig these stochastic odels efficietly, the etwork siplex algorith, stochastic siulatio ad geetic algorith are itegrated to produce a hybrid itelliget algorith [4]. Vibratio dapig optiizatio (VDO) algorith proposed by Mehdizadeh ad Tavakkoli- Moghadda is a eta-heuristic algorith [5]. The ai of this paper is to preset a ew hybrid algorith, aed vibratio dapig optiizatio algorith ad geetic algorith (VDO & GA) for solvig stochastic capacitated locatio-allocatio proble to iiizig the total cost. The siplex algorith, stochastic siulatio ad the proposed hybrid algorith are itegrated to produce a hybrid itelliget algorith. The paper is orgaized as follows. I Sectio the LA proble is itroduced. The capacitated LA proble as the expected value odel, chace-costraied prograig forulates i Sectio 3, VDO algorith ad GA is described i sectio 4. The itegrated algorith preseted i Sectio 5. Fially, Sectio 6 provides a few uerical exaples to illustrate the perforace ad the effectiveess of the proposed algoriths. The proposed algorith copared with the preseted algorith by Zhou ad Liu [4].. Locatio-allocatio proble I order to odel capacitated stochastic LA proble, the followig idices, paraeters, ad decisio variables are used [4] : i =,,, : idex of facilities; =,,, : idex of custoers; ( ): locatio of custoer ; ; ξ : stochastic dead of custoer ; ; : capacity of facility i; i ; ISBN:

2 ( ): decisio variable represetig the locatio of facility i; i ; : quatity supplied to custoer by facility i after the stochastic deads are realized, i ;. Here we will cosider the capacitated LA proble with assuptios that the path betwee ay custoer ad facility is coected ad uit trasportatio cost is proportioate of the quatity supplied ad the travel distace. Ad facility i is assued to be located withi a certai regio R i ={( ) ( 0}; i =,,,; respectively. We suppose that the stochastic dead vector ξ= ( ξ, ξ,, ξ ) is defied o the probability space (Ω, A, Pr). For coveiece, we also write x y x y LL x y z z L z z ( XY, ) = z z Z L = L L L L z z L z I a deteriistic LA proble, the allocatio z is a decisio variable which is costat all the tie. However, i a stochastic LA proble, the decisio z will be ade every period after the deads of custoers are realized. For each ωєω; ξ(ω) is a realizatio of stochastic vector ξ.a allocatio z is said to be feasible if z 0, i =,,...,, =,,..., z = ξω (), =,,..., i= z si, i=,,..., = () We deote the feasible allocatio set by z 0, i=,,...,, =,,..., Z( ω) = z z = ξ( ω), =,,..., i= z si, i=,,..., = () Note that Z(ω) ay be a epty set for soe ω. The for each ωєω; the iial trasportatio cost is the oe associated with the best allocatio z; i.e. ω z z ( ω ) i i C ( x, y ) = i z i = = ( x a ) + ( y b ) (3) whose optial solutio is called the optial allocatio. If Z(ω) = Ø; the deads of soe custoers are ipossible to be et, ad the right-had side of Eq. (3) becoes eaigless. As a pealty, we defie C ( x, y ω ) = ax ξ ( ω ) i = ( x a ) + ( y b ) (4) i i I the followig sectios, stochastic prograig odels will be provided for the capacitated LA proble with stochastic deads. 3. Proble odels 3.. Expected value odel (EVM) The expected value odel has bee described by Zhou ad Liu [4]. The first type of stochastic prograig is the so-called expected value odel (EVM). The essetial idea of EVM is to optiize the expected values of obective fuctios subect to soe expected costraits. Whe the deads of custoers are stochastic, the trasportatio cost C(x,ylω) becoes stochastic, too. The we cosider iiizig the expected value of C(x, ylω). Though the stochastic capacitated LA proble has bee studied by ay researchers, a geeral EVM has ot bee give due to the coplexity. This sectio provides a ew idea to forulate the LA proble by EVM as follows, i pr{ ω Ω C( x, y ω) r} dr x, y 0 subect. to : g ( x, y ) o, i =,..., where gi( xi, y i) 0; i =,,, are the potetial regios of locatios of ew facilities, C(x,y ω) is defied b y E qs. () (4), ad is the expected value of C(x, y ω). The odel is differet fro traditioal stochastic prograig odels because there is a sub-optial proble i it, i.e. i z ( xi a ) + ( yi b ) subect to i= = z = ξ ( ω ), =,..., i = = z s, i =,..., i Z 0, i =,..., =,..., i i i (6) (5) ISBN:

3 Note that i Eq. (6) the paraeters x i, y i ad ξ (ω) are fixed real ubers for each ωєω; i =,,, ; =,,,; respectively. It is clearly a liear prograig proble. 3.. Chace costraied prograig (CCP) As the secod stochastic prograig, chacecostraied prograig (CCP) offers a powerful approach of odelig stochastic decisio systes. CCP was iitialized (Chares & Cooper, 96) ad subsequetly developed by ay researchers. Here we will itroduce the latest versio (Liu, 999a,b). Sice ix,y C(x,ylω) is eaigless if C(x,ylω) is stochastic, a atural idea is to provide a cofidece level b at which it is desired that C(x,ylω) ; where the cofidece level β is provided as a appropriate safety argi by the decisio-aker. So the obective is to iiize istead of C(x,ylω) with a chace costrait as follows, { (, ω) f } pr C x y β (7) where is referred to as the β-optiistic value to C(x,y)lω) (Liu, 999a,b). A poit (x, y) is called feasible if ad oly if the probability easure of the evet C(x, ylω) is at least β. I other words, the costrait will be violated at ost ( - β) of tie. Hece we have the followig LA CCP odel, i f subect to xy, pr{ ω Ω C( x, y ω) f } β, gi( xi, yi) 0 i =,..., (8) where i is called the β-optiistic trasportatio cost ad C(x,y lω ) is defied by Eqs.()-(4)., 4. GA ad VDO algorith LA is a NP-hard proble [7]. Oe way to solve NP-hard probles is to use eta-heuristic algoriths. I this work, two eta-heuristic algoriths, GA ad VDO are used. 4.. Geetic algorith Geetic algorith (GA) was first itroduced by Joh Hollad i the 970s. It is a search techique based o the cocept of the atural selectio ad evolutio. Geetic algorith, starts with a iitial set of rado solutios called a populatio. Each idividual i the populatio is referred to a chroosoe, represetig a solutio to the proble at had. Chroosoes evolve through successive iteratios, aed geeratios. Durig each geeratio, chroosoes are evaluated by usig soe easures of fitess. To create the ext geeratio, ew chroosoes, referred to offsprig, are fored by either () ergig two chroosoes fro the curret geeratio by usig a crossover operator, or () odifyig a chroosoe by usig a utatio operator. A ew geeratio is fored by () selectig, accordig to the fitess value, soe of parets ad offsprig, ad () reectig others so as to keep the populatio size costat. After several geeratios, the algorith coverges to the best chroosoe, which hopefully represets the optial or sub-optial solutio to the give proble. 4.. Vibratio dapig optiizatio algorith Vibratio dapig optiizatio (VDO) algorith was itroduced by Mehdizadeh ad Tavakkoli-Moghadda [5] as a ethod to solve parallel achie schedulig proble. It is a stochastic search ethod based o the cocept of the vibratio dapig i echaical vibratio. VDO is a eighborhood search techique that has produced good results for cobiatorial probles. VDO begis with a iitial solutio (X), iitial aplitude (A), ad a iteratio uber (L). Aplitude (A) cotrols the possibility of the acceptace of a deterioratig solutio, ad the iteratio uber (L) deteries the uber of repetitios util a solutio reaches a stable state uder the aplitude. The A ay have the followig iplicit eaig of a flexibility idex. At high aplitude (early i the search), there is soe flexibility to ove to a worse solutio; but at lower aplitude (later i the search) less of this flexibility exists. A ew eighborhood solutio (Y) is geerated based o these A ad L through a heuristic perturbatio o the existig solutios. The eighborhood solutio (Y) becoes a ew solutio if the chage of a obective fuctio ( Δ =E(Y) - E(X)) is iproved (i.e., Δ 0 for a iiizatio proble). Eve though it is ot iproved, the eighborhood solutio becoes a ew solutio with a appropriate probability based σ o. This leaves the possibility of fidig a global optial solutio out of a local optiu. The algorith teriates if there is o chage after L repetitios. Otherwise, the iteratio cotiues with ew a plitude (A). The pseudo code of VDO algorith is as follow: e A Begi; INITIALIZE (X; A; L ad t); INPUT γ adσ (γ is dapig coefficiet adσ is Rayleigh distributio Costat); Repeat For i = to L do Y=PERTURB(X); {geerate ew eighborhood solutio} Δ =E(Y) - E(X); A σ e If Δ 0 or ( > RANDOM (0, )) The X = Y; {accept the oveet} ISBN:

4 Edif Edfor; UPDATE (A, L ad t, t = t + ) Util (Stop-Criterio) Ed. γt A= Ae, L = L 5. Hybrid itelliget algorith + & Geerally speakig, ucertai prograig odels are difficult to solve by traditioal ethods. A good way is to desig soe hybrid itelliget algoriths for solvig the. I this sectio, we itegrate the siplex algorith, stochastic siulatios ad the proposed hybrid algorith (GA&VDO) to produce a hybrid itelliget algorith for solvig geeral stochastic prograig odels of LA proble. 5.. Coputig optial allocatios Assue that the locatio (x,y) of the facilities has bee give. For each ωєω,we have the realized deads ξ(ω). I order to iiize the trasportatio cost C(x,ylω), we * should deterie the optial allocatios(z ). I other words, we ust solve the liear prograig (6), which is a liear trasportatio proble virtually. So we use the siplex algorith to solve it. 5.. Coputig ucertai fuctios By Ucertai fuctios we ea the fuctios with stochastic paraeters. Due to the coplexity, we desig soe stochastic siulatios to calculate ucertai fuctios. The first type of ucertai fuctio is: U :( x, y) pr{ ω Ω ( x, y ω) rdr } (9) 0 I order to copute it, we desig a stochastic siulatio as follows, Step. Set U(x,y) = 0. Step. Geerate ω fro Ω accordig to the probability easure Pr. Step 3. Solve the liear prograig (6) by the siplex algorith ad deote its optial obective value by c. Step 4. U(x,y) U(x, y) + c. Step 5. Repeat the secod to fourth steps for N ties, where N is a sufficietly large uber. Step 6. Retur U(x,y)/N. The secod type ucertai fuctio is U :( x, y ) i{ f pr { ω Ω ( x, y ω )) f } β } (0) which ay be estiated by the followig procedure, Step. Geerate ω, ω,..., ωn fro Ω accordig to the probability easure Pr, where N is a sufficietly large uber. Step. For eachω k ; solve the liear prograig (6) by the siplex algorith ad deote the optial obective value by ck; k =,,,N; respectively. Step 3. Set as the iteger part of βn. Step 4. Retur the th least eleet i {,,, } Hybrid itelliget algorith We describe the proposed hybrid algorith as the followig procedure: Step. Iitialize pop_size chroosoe sv k = ( ) = (,,,,,,,, ), k =,,,pop_size potetial regio {(x,y) gi( x,y i i) 0; i =,,,} uiforly. Subect to α % of chroosoes i the GA iitial populatio are produced by VDO algorith. Step. Calculate the obective values ( =,) for all chroosoes Vk; k =,,,pop_size by stochastic siulatios, respectively, where the siplex algorith is used to solve the liear prograig (6). Step 3. Copute the fitess of all chroosoesv k ; k =,,,pop_size. The rak-based evaluatio fuctio is defied as Eval( )=α, pop_size, 0 < () is a paraeter i the geetic syste. Step4. Perfor the selectio process based o spii g the roulette wheel characterized by the fitess for ( pop_size) ties, ad each tie select a sigle chroosoe where the paraeter is the probability of crossover. The group the selected parets,, ito the pairs ( Without loss of geerality, let us illustrate the crossover operator o each pair by (.At first, we geerate a rado uber λ fro the ope iterval (0,), the the crossover operator o ad will produce two childre X ad Y as follows: x = λ. v + ( λ ). v () y = ( λ ). v + λ. v If both childre are i the regio {(x, y) ( 0; i =,,, }; the we replace the parets with the. If ot, we keep the feasible oe if it exists, ad the redo the crossover operator by regeerati g aother real uber fro (0, ) util two feasible childre are obtaied. I this case, we oly replace the parets with the feasible childre. Fially, ew pop_size chroosoes are obtaied by crossover operator. Step 5. procure pop_size - pop_size) chroosoe by reproductio operator. Fially, ew pop_size chroosoes ; k =,,,pop_size are obtaied. Step6. Geerate the pop_size rado iteger uber fro the iterval, the select chroosoes that rival with the geerated uber, ISBN:

5 where the paraeter is the probability of utatio. For each selected paret V = ( x, x,..., x, y, y,..., y),we utate it i the followig way. At first, we choose a utatio directio d i radoly. If (3) is ot feasible for the regio costraits, the we set M as a rado uber betwee 0 ad M util it is feasible, whe re M is a appropriate large positive uber. If the above process caot fid a feasible solutio i a predeteried uber of iteratios, the we set M = 0.Ayway, we replace the paret with its child X.The pop_size ew chroosoes ay be geerated, ad we still deote the by ; k =,,,pop_size. Step 7. Repeat the secod to sixth steps for a give uber of cycles. Step 8. Report the best chroosoe = ( ) as the optial locatios. 6. Nuerical exaples I this sectio, the exaples preseted by Zhou ad Liu [4] are used for illustratio the efficiecy of the proposed itelliget algorith. I the exaples four facilities are located, where each facility is assued to be located withi a certai regio R i = {(, ) 0 }; i =,,3,4; respectively. Assue that there are 0 custoers whose deads ξ are orally distributed variable N(5,) for =,,,0; ad are uiforly distributed variable U(8,) for =,,, 0. The locatios ( a, b ), =,,,0 of custoers are give i Table. The capacities (i =,, 3, 4) of the four facilities are 40, 50, 60 ad 70, respectively. The proposed algoriths are coded i MATLAB ad ru o a PC with itel core Duo cpu.00ghz, 4GB of RAM. We use the Liprog fuctio i MATLAB to solve the the liear prograig (6) durig ruig progra. The paraeters of VDO algorith are the followig: A 0 =6, T=50, N=0, σ=, =0.05; Ad also α is percet of the GA iitial populatio chroosoes that produce by VDO algorith. We obtai the coputatioal results for EVM ad CCP odels show i Table ad 3, where cost is the iial expected cost. I the Table 4, the proposed hybrid itelliget algorith is copared with the preseted algorith by Zhou ad Liu [4] accordig to the cost. Nuerical results i Table 4 show that the proposed hybrid itelliget algorith is ore efficiet tha hybrid itelliget algorith preseted by Zhou ad Liu ad gives better solutios particularly i the CCP odel. 7. Coclusios I this paper, a hybrid itelliget algorith based o a ew hybrid algorith, aed vibratio dapig optiizatio algorith ad geetic algorith (VDO & GA) is proposed for solvig stochastic capacitated locatio-allocatio proble to iiize the total cost. I the proposed hybrid algorith, α % of chroosoes i the GA iitial populatio are produced by VDO algorith. The for solvig the preseted stochastic odels efficietly, the siplex algorith, stochastic siulatio ad the proposed hybrid algorith is itegrated to desig a powerful hybrid itelliget algorith. we coputatioally evaluated the perforace of the proposed hybrid itelliget algorith ad copared it with hybrid itelliget algorith preseted by Zhou ad Liu[4] ad it is foud out that our proposed algorith is able to obtai better results. Refereces: [] Cooper, L. (963). Locatio-allocatio probles. Operatios Research,, [] Ge, M., & Cheg, R. (997). Geetic algoriths ad egieerig desig. New York: Wiley. [3] Ge, M., & Cheg, R. (000). Geetic algoriths ad egieerig optiizatio. New York: Wiley. [4] Logedra, R., & Terrell, M. P. (988). Ucapacitated plat locatio-allocatio probles with price sesitive stochastic deads. Coputers ad Operatios Research, 5, [5] Zhou, J. (000). Ucapacitated facility layout proble with stochastic deads. Proceedigs of the Sixth Natioal Coferece of Operatios Research Society of Chia, [6] Kuee, R. E., & Solad, R. M. (97). Exact ad approxiate solutios to the ultisource Weber proble. Matheatical Prograig, 3, [7] Sherali, H.D., Nordai, F.L., 988. NP-hard, capacitated, balaced p-edia probles o a chai graph with a cotiuu of lik deads. Matheatics of Operatios Research 3, [8]Aras N, Altel IK, OrbayM. New heuristic ethods for the capacitated ulti- facility Weber proble.naval Research Logistics 007;54: 3. [9] Duraz E, Aras N, Altel IK. Discrete approxiatio heuristics for the capacitated cotiuous locatio allocatio proble with probabilistic cu stoer locatios. 009;36: [0] Murray, A. T., & Church, R. L. (996). Applyig siulated aealig to locatio-plaig odels. Joural of Heuristics,, [] Ohleuller, M. (997). Tabu search for large locatioallocatio probles. Joural of the Operatioal Research Society, 48, [] Erst, A. T., & Krishaoorthy, M. (999). Solutio algoriths for the capacitated sigle allocatio hub locatio proble. Aals of Operatios Research, 86, ISBN:

6 [3] Gog, D., Ge, M., Yaazaki, G., & Xu, W. (997). Hybrid evolutioary ethod for capacitated locatioallocatio proble. Coputers ad Idustrial Egieerig, 33, [4] Zhou J, Liu B. New stochastic odels for capacitated locatio- allocatio proble003;45:-6. [5] Mehdizadeh, E., ad Tavakkoli-Moghadda R,. Vibratio dapig optiizatio, Proceedig of the Iteratioal Coferece Operatios Research 008 OR &Global Busiess, Geray, Septeber 3-5, 008. Table: Locatios of 0 custoers Table: EVM odel Probles popsize p c p α optial locatios cost (63.70,6.56) (36.80,87.7) (77.5,8.4) (8.60,5.95) (37.75,86.96) (3.7,5.8) (3.76,5.43) (74.89,8.39) (6.88,79.3) (78.3,9.60) (6.38,55.8) (6.69,49.67) (6.8,58.56) (7.5,8.0) (5.80,5.65) (78.6,.70) (6.0,56.98) (34.58,89.56) (78.4,0.04) (9.,46.47) (9.97,3.4) (60.5,60.3) (78.80,9.4) (9.37,56.7) (.76,8.67) (64.0,55.9) (6.45,50.78) (77.30,.4) (60.88,59.9) (3.44,79.43) (77.53,.70) (6.78,48.4) (6.74,58.7) (37.44,85.03) (75.9,0.96) (9.54,50.0) (37.6,86.59) (6.97,58.4) (6.3,50.57) (76.96,.36) Table3: CCP odel Probles popsize p c p α optial locatios cost (5.94,34.50) (7.9,57.00) (80.68,7.64) (5.07,78.4) (58.09,7.5) (60.30,.6) (6.36,75.6) (6.77,46.9) (39.09,86.0) (58.0,59.05) (83.00,9.4) (7.5,49.34) (57.8,56.6) (6.36,74.68) (3.09,50.99) (70.54,.60) (38.8,.77) (57.7,74.65) (84.54,5.58) (8.6,5.4) (36.68,7.8) (3.7,45.89) (6.57,63.90) (78.67,.59) (4.9,.7) (8.55,.3) (5.9,56.5) (57.39,7.6) (60.,50.7) (8.37,.7) (5.46,48.9) (39.65,86.0) (5.80,3.73) (33.57,86.9) (76.57,5.9) (34.50,50.95) (60.37,56.83) (3.83,76.) (8.60,45.04) (74.69,.04) 05 Table 4: Copariso betwee the algoriths Probles Z&L GA (EVM) VDO&GA(EVM) Z&L GA(CCP) VDO&GA(CCP) ISBN: