222222222214 Journal of Uncertan Sytem Vol.7, No.1, pp.22-35, 2013 Onlne at: www.u.org.uk Robut Capactated Faclty Locaton Problem: Optmzaton Model and Soluton Algorthm Ragheb Rahmanan *, Mohammad Sad-Mehrabad, Hoat Ahour Department of Indutral Engneerng, Iran Unverty of Scence and Technology, P.C. 1684613114, Narmak, Tehran, Iran Receved 6 July 2011; Reved 13 November 2012 Abtract In th artcle, we propoe an extenon of the capactated faclty locaton problem under uncertanty, where uncertanty may appear n the model key parameter uch a demand and cot. In th model, t aumed that faclte have hard contrant on the amount of demand they can erve and, a a reult, ome cutomer may not be fully atfed. Unfortunately, tradtonal model gnore th tuaton and f faclte do not erve all demand, the model become nfeable. Accordngly, we develop the mathematcal formulaton n order to allow partal atfacton by ntroducng penalty cot for unatfed demand. In general, th model optmze locaton for predefned number of capactated faclte n uch a way that mnmze total expected cot of tranportaton, contructon, and penalty cot of uncovered demand, whle relatve regret n each cenaro mut be no greater than a potve number ( p 0 ). The developed model NP-hard and very challengng to olve. Therefore, an effcent heurtc oluton algorthm baed on the varable neghborhood earch developed to olve the problem. The algorthm effcency compared wth the mulated annealng algorthm and CPLEX olver by olvng varety of tet problem.computatonal experment how that the propoed algorthm more effectve and effcent n term of CPU tme and oluton qualty. 2013 World Academc Pre, UK. All rght reerved. Keyword: capactated faclty locaton problem, uncertanty, robut optmzaton, partal atfacton, varable neghborhood earch, mulated annealng 1 Introducton Locaton problem are crtcal manageral decon wth a large body of lterature and numerou applcaton n the real-world applcaton [5, 9, 10, 24, 26]. In fact, faclty locaton decon are long-term trategcal decon and almot mpoble to revere. Thee trategc decon have a great mpact on the network flow and cutomer atfacton. Eentally, thee problem deal wth locatng a number of faclte to upply a et of cutomer at the mnmum cot wth repect to everal contrant and nvolve varou obectve whch naturally are n conflct [27]. For example, n ome locaton problem there mght be hard contrant on amount of avalable budget, number of faclte, faclte capacty, coverage dtance, and o forth, whle the obectve functon maxmze total covered demand and mnmze total contructon and tranportaton cot. In lne wth th ubect, numerou paper have been publhed n whch the Capactated Faclty Locaton Problem (CFLP) one of the mot mportant and bac model. Th problem ha been ntenvely condered n the lterature, nce t nclude more realtc aumpton n compare wth, for example, uncapactated locaton problem. In the CFLP, a number of capactated faclte, P, are to be located among J poble te n order to atfy demand of I cutomer by mnmzng total cot of tranportaton and fxed charge of etablhng faclte. The CFLP NP-hard problem whch generalzed from the mple plant locaton problem. The CFLP ha been effectvely mplemented to olve real-world applcaton uch a plant locaton, power taton locaton, warehoue locaton, to ut name a few. Though th problem and t varaton have been ntenvely tuded n the lterature, th problem ha been manly nvetgated wth determntc data and wth th aumpton that faclte capacty can meet all cutomer demand. * Correpondng author. Emal: ragheb.rahmanan@gmal.com (R. Rahmanan).
Journal of Uncertan Sytem, Vol.7, No.1, pp.22-35, 2013 23 However, n the realty, thee aumpton are rarely atfed. Utlzed data n manageral decon are encountered wthn complete factor uch a noy and erroneou defect. Meanwhle, n many cae due to lack of htorcal nformaton, t almot mpoble to obtan a relable and certan etmaton for the model key parameter. On the other hand, locaton decon are long-term trategc decon and very entve to the change n demand and other parameter. Therefore, gnorng uncertanty n parameter yeld n neffectve decon. A a reult, optmzaton under uncertanty ha receved ncreang attenton n the locaton theory durng the lat few decade [2, 13, 32, 30, 35] and, varou approache are propoed to face and reolve the uncertanty. Thee approache can be generally categorzed nto robut and tochatc optmzaton approache. In the robut approach, the uncertan parameter are etmated by dcrete or nterval data and the obectve functon typcally mnmze mnmax cot or mnmax regret. The regret n each cenaro defned a dtance between the current cot of that cenaro and t optmal cot. Dcrete cenaro are ued when probablty dtrbuton for uncertan parameter not known. Robut oluton are naturally too conervatve decon and mpoe unneceary long-term cot. On the other hand, tochatc programmng model partton decon n twotage n whch obectve functon mnmze the um cot of the frt-tage and the expected cot of the econdtage. In general, the typcal tochatc programmng may yeld nexpenve oluton n the long run but perform poorly under certan realzaton of the random data. Bede, the robut problem due to the mnmax tructure are much harder to olve. Baed on thee weaknee, Snyder and Dakn [32] ntroduced the tochatcp-robut optmzaton approach whch come through combnng man concept of thee approache. One of the man goal of th approach to degn a more robut ytem wth a lttle ncreae n the expected cot. In th tudy, th approach appled to optmze the CFLP under uncertanty. Conderng uncertanty ncreae relablty of oluton. Moreover, tradtonal CFLP model become nfeable f total opened capacty lower than um of cutomer demand. Wherea, n the realty, t lkely that faclte due to ther lmted capacty leave ome of thee demand unatfed. For ntance, power taton have hard contrant on the amount of generated electrcty and they cannot provde unbounded amount of electrcty to the cutomer. Moreover, due to the huge nvetment cot of etablhng power taton, manager cannot contruct more than a lmted number of power taton. Hence, wth a lmted number of capactated power taton, logcally, we wll not be able to upply all cutomer ncreang demand. Th tuaton can be oberved n thrd-world and deprved area by far. Unfortunately, tradtonal model of CFLP cannot fnd even a feable oluton for uch problem. Therefore, we alo develop the mathematcal formulaton of CFLP wth th aumpton that our faclte may not be able to upply all demand and any proporton of cutomer demand may reman unatfed. On the other hand, long any proporton of demand mpoe a great deal of cot to the company, nce they loe the market hare. A a reult, the obectve functon mnmze penalty cot of unatfed demand, a well. Generally peakng, the man goal of th tudy to develop the mathematcal formulaton of CFLP under uncertanty by takng thee fact nto account that each cutomer may not be fully uppled and ther demand are uncertan and characterzed by a gven et of cenaro. Moreover, we aocate a penalty cot to any proporton of unatfed demand. Therefore, the obectve functon mnmze total cot of etablhng new faclte, expected cot of tranportaton, and expected penalty cot of unatfed demand. The developed model wll be a two-tage model n whch the frt-tage optmze faclte locaton and the econd-tage optmze cutomer agnment. Each cutomer may be uppled by everal faclte (.e. multple agnment property) and any proporton of t demand may be atfed (.e. ay 60% of t total demand). Th formulzaton wll be referred to u a the Stochatc p-robut Capactated Faclty Locaton Problem, or p-scflp a an abbrevaton. In addton, nce the developed model formulaton hard and cannot be effectvely olved ung ordnary optmzaton method, we propoe oluton algorthm baed on Varable Neghborhood Search (VNS) and Smulated Annealng (SA) to olve the developed model. We tet the effcency of thee algorthm on a varety tet ntance and compare ther reult wth the CPLEX olver n term of optmalty gap and CPU tme requrement. The remnder of th paper organzed a follow: n the next Secton the related lterature revewed. In Secton 3 the mathematcal formulaton of the p-scflp preented. Subequently, oluton approache baed on the VNS and SA are outlned n Secton 4. Lkewe, our numercal experment are ummarzed n the Secton 5 and fnally, our concluon and future reearch drecton are dcued n the Secton 6. 2 Related Lterature The CFLP a clacal faclty locaton problem wth large body of the lterature. The CFLP ha been extenvely nvetgate don the both de of model formulaton and olvng algorthm. Varou algorthm that have been mplemented to olve the Uncapactated Faclty Locaton Problem (UFLP) are uually generalzed to olve the
24 R. Rahmanan et al.: Robut Capactated Faclty Locaton Problem CFLP. For example, Kuehn and Hamburger [22] propoed an algorthm to olve the UFLP and then Jacoben [17] extended th algorthm for the CFLP. However, n th ecton, we are gong to addre the mot relevant paper to the condered problem and the ntereted reader are referred to [1, 26, 33] and reference theren. Lagrangan Relaxaton (LR) ha been wdely condered a an effcent oluton algorthm to olve the CFLP. Context preented by Cornueol et al. [7] provd an excellent theoretcal analy of all poble Lagrangan relaxaton and the lnear programmng relaxaton for the CFLP. A Lagrangan Heurtc (LH) framework preented by the Bealey [4] to olve dfferent faclty locaton problem. In the propoed method for the CFLP, allocaton contrant and capacty contrant are ncorporated nto obectve functon by ung Lagrangan multpler. In addton, LH method for the both UFLP and the CFLP alo propoed by other author [3, 23]. Barahona and Chudak [3] ntally provded the lnear programmng relaxaton of the CFLP and then propoed the Lagrangan relaxaton to olve the lnear problem. Heurtc and meta-heurtc algorthm are alo appled to olve the CFLP n a wde range. Sun [34] appled the Tabu Search (TS) to olve the CFLP and compared t wth the Lagrangan and the urrogate/lagrangan heurtc method. He ued long term memory baed on prmogentary lnked quad tree to tore vted oluton and prohbt them from beng vted agan. Cortnhal and Captvo [8] propoed upper and lower bound for the ngle ource capactated locaton problem. In that artcle, Lagrangan relaxaton ued to obtan lower bound for th problem and upper bound are obtaned by Lagrangan heurtc followed by earch method (e.g. TS). A mple local earch heurtc for the capactated faclty locaton problem preented by Korupolu et al. [20]. Hnd and Penkoz [15] preented a heurtc that combne Lagrangan relaxaton wth retrcted neghborhood earch for the capactated ngle ource locaton problem. In th paper, a heurtc procedure wth three phae ued to fnd feable oluton. An effcent VNS heurtc for the capactated P-medan problem nvetgated by Flezar and Hnd [12]. They alo propoed everal ntermedate heurtc earch algorthm. Thee algorthm were able to fnd very good oluton wthn much horter computaton tme than the full VNS algorthm.smlarly, exact method have been nvetgated to olve the CFLP. Bacally, the propoed exact method are baed on branch and bound and et parttonng technque wth dfference n tratege to ncreae the lower bound and type of relaxaton. For ntance, Sa [28] relaxed the CFLP to obtan tranportaton problem but Nau [25] relaxed capacty contrant and added et of urrogate contrant to obtan tghter bound. Neverthele, ntereted reader are referred to the context by Srdharan [33] to get comprehenve revew on the oluton algorthm for the CFLP. However, the publhed work related to CFLP have motly concentrated on oluton algorthm, and extenon of mathematcal formulaton are uually ntroduced a newer work uch a capactated hub locaton, capactated maxmal coverng, capactated faclty locaton and network degn, and o forth. Faclty locaton problem are ntenvely tuded under uncertanty [2, 13, 16, 32, 30, 35]. For ntance, Ghezavat et al. [13] propoed a robut approach to locaton-allocaton problem under uncertanty. Snyder and Dakn [32] propoed a new approach for optmzng faclty locaton problem under uncertanty. They combned the mnmum-expected-cot and p-robutne meaure together n order to ntroduce the tochatc p-robut optmzaton model. Th approach mplemented on the clacal model of the UFLP and P-medan problem n order to reolve the uncertanty n demand and dtance. They ntended to fnd oluton that ha mnmum-expected-cot whle the obtaned oluton p-robut;.e., cot under each cenaro for each feable oluton mut be no greater than 100(1+p) % of the optmal cot of that cenaro, where p a non-negatve value known a robutne coeffcent. They olved the propoed model by ung Lagrangan decompoton and reducng them to the multple-choce knapack problem. Addtonally, they dcued a mechanm for detectng nfeablty. The nteret reader are referred to the [5, 21, 26, 29, 31] to get more nformaton about locaton and locaton under uncertanty. 3 Problem Decrpton and Formulaton Conder a producton-dtrbuton network that ha I maor center of cutomer and J potental faclty te. A company want to locate P faclte n th network n order to ervce thee cutomer. Thee faclte cannot upply unbounded amount of demand for cutomer, and conequently, ome cutomer may not be fully atfed or even may reman completely or partally unatfed. On the other hand, nablty n atfyng cutomer demand ncur huge cot for the company n conequence of long the market hare. A a reult, we defne penalty cot for unatfed demand a a functon of unmet-demand. Therefore, f X, denote fracton of demand of cutomer whch atfed by faclty at node under cenaro, we wll have 1 X, percentage of t demand unatfed n cenaro. Where, X, J J ndcate total fracton of demand of cutomer whch atfed by faclte through the network. Accordngly, we defne a varable n order to capture the proporton of demand of cutomer under
Journal of Uncertan Sytem, Vol.7, No.1, pp.22-35, 2013 25 cenaro whch unuppled,.e., Z. By keepng th n mnd that um of total fracton of atfed and unatfed demand for each cutomer n each cenaro hould be equal to one or, one hundred percent of t demand, we have equaton (1). X, Z 1 I, S. (1) J Logcally, each cutomer may be agned to everal faclte and may be partally uppled (ay 80 percentage of t total demand). Th more cloe to the realty becaue n the clacal model f faclte could not upply all demand, the model become nfeable, whle there tll a uboptmal oluton to the problem. To mplfy the preentaton of the mathematcal formulaton n th paper, the followng notaton are defned.the ndex et and model parameter are decrbed n Table 1. Table 1: Parameter and ndex et Symbol Indexed by Decrpton 1,2,..., J Set of potental faclty te; J I 1,2,..., I S 1,2,..., S Parameter C W f d * Q a Set of cotumer; Set of cenaro; Decrpton Faclte capacty Probablty of that cenaro occur Dered robutne coeffcent Demand at node under cenaro The fxed charge of openng a faclty on node Number of faclte to be open Travel dtance between node and under cenaro Optmal cot of CFLP under data from cenaro A contant number * where Q the optmal obectve value of the determntc CFLP problem wth partal atfacton (ee equaton (15-21)) under data from cenaro that can be computed by CPLEX or any exact method. The decon of the tochatc p-robut capactated faclty locaton nclude decon about locatng faclte and agnng cotumer to thee faclte n uch a way that total expected cot beng mnmzed. Thee decon are made n two tage and are defned a follow. 1 f one faclty located at node y 0 otherwe X fracton of demand of node under cenaro that atfed by faclty located at node, 0 Z 0 fracton of demand of node under cenaro that notatfed. Note that, here the locaton decon varable (y ), unlke the agnment varable ( X, and Z ), are ndependent on the ndex n order to reflect the two-tage nature of the problem. Fnally, we aume that uncertanty aocated wth demand and dtance. Ung th notaton and aumpton, the propoed two-tage mxed-nteger model for the problem n hand formulated a follow.
26 R. Rahmanan et al.: Robut Capactated Faclty Locaton Problem Mn SCFLP : f y W d X, a W Z J S I J I Subect to: (2) X, Z 1 I, S (3) J I W X C y J S,, f y W d X, a W Z (1 ) Q S (5) * J I J I J y P Z 0 I, S (7) X, 0 I, J, S (8) y 0,1 J. (9) In the propoed model, equaton (2) repreent the obectve functon. Frt term n th equaton mnmze the fxed cot of openng faclte. Second term mnmze total expected tranportaton cot. Fnally, the thrd term mnmze total expected penalty cot of unatfed demand. Penalty cot for each cutomer calculated a a functon of lot demand ( W * Z ) multple to a contant number. Here, we have aumed that there lnear relaton between lot demand and cot mpoed to the company, whle, n the realty, t lkely that penalty cot be a power functon of lot demand a equaton (10), b a* W * Z. (10) However, for the ake of mplcty n olvng the model, we have et b=1. Otherwe, the model would become a nonlnear mxed-nteger formulaton whch much more challengng to olve. Moreover, n the obectve functon, agn weght to each cenaro whch ndcate mportance of that cenaro n decon makng, where 1. S Contrant (3) guarantee that um of total atfed and unatfed demand for cutomer n each cenaro do not exceed than t total demand. Equaton (4) repreent the capacty contrant n whch we enure that the total allocated demand to the faclty at node n cenaro (.e. W * X, ) doe not exceed t capacty. Addtonally, th contran ndcate that I cutomer hould be only agned to the faclte,.e., X, y. Equaton (5) enforce the p-robutne crteron. Accordngly, the obectve functon of each cenaro hould not be greater than 1 % of optmal cot of that cenaro. Robutne coeffcent can be et dfferently n each cenaro and denote by, n order to account for the fact that dfferent cenaro may have dfferent mportance level n our decon. However, for the ake of mplcty, we conder the ame value for each cenaro. Addtonally, th contrant ndcate that we are wllng to make addtonal nvetment n the nfratructure n order protect agant future poble drupton. Note that, th model would be equal to the clacal capactated P-medan problem f (robutne contrant wll be nactve), S 1, a=0, and contrant (3) replaced by the followng equaton X, 1 I, S. (11) J The mnmax regret formulaton whch a typcal obectve functon of robut model generally ued where the cenaro probablty not known. Th obectve functon mnmze maxmum regret over all poble cenaro. We can ealy change the propoed model n order to obtan th formulaton. To do o, we mply replace (4) (6)
Journal of Uncertan Sytem, Vol.7, No.1, pp.22-35, 2013 27 the obectve functon wth the robutne coeffcent whch yeld n the followng formulaton. We call th formulaton Robut Capactated Faclty Locaton Problem or, RCFLP. Mn RCFLP : (12) Subect to: 0 (13) (3), (4), (5), (6), (7), (8), (9). (14) Snce locatng large number of faclte not derable, equaton (6) lmt number of faclte that can be opened. Fnally, equaton (7 and 9) declare type of agnng varable and locaton varable repectvely. The determntc form of the propoed model formulaton whch would be ued to determne optmal cot of each cenaro a follow. Mn DCFLP : f y W d X, a W Z (15) J I J I Subect to: J I J X, Z 1 I, W X C y J y,, P (16) (17) (18) 4 Soluton Approach Z 0 I, (19) X, 0 I, J, (20) y 0,1 J. (21) For realtcally zed ntance t very challengng to olve mot of the p-scflp problem to optmalty wthn a reaonable computer CPU tme and memory by the well-known optmzaton olver uch a GAMS/CPLEX or Lngo. Therefore, n th part, two heurtc algorthm baed on the VNS and SA are developed to olve the p-scflp. Varable neghborhood earch a mple and powerful oluton framework to fnd near to optmal oluton to large cale and complex problem n a reaonable computer tme. It wa frtly propoed by Brmberg and Mladenovć [6] to olve contnue locaton allocaton problem. Comprehenve revew on the method and applcaton for th metaheurtc prepared by Hanen and Mladenovć [14]. Lkewe, SA that mmc coolng proce of materal another effcent oluton algorthm to olve hghly complex problem [36]. Broadly peakng, the p-scflp nvolve two man decon: (Ι) decon regardng faclte locaton, (.e., y ) and, (ΙΙ) decon about agnng cutomer to the located faclte (.e. X, and Z ). In the propoed model, f faclte locaton wa gven, an optmal oluton for the allocaton ub-problem eay to obtan. In other word, the problem can be ealy olved under any gven vector for the y. Therefore, the propoed algorthm n each teraton determne faclte' locaton, y, and then ue a local earch to determne the local optma for the allocaton ubproblem. Th local earch baed on an exact optmzaton algorthm that ue Branch & Bound and cuttng plane method n t framework. Addtonally, nce tetng a move everal tme not computatonally effcent, n thee algorthm, a long-term memory appled to tore vted move and prevent them from beng vted agan. In order to how a move vector, a J -dgt bnary vector wth equal length to the total potental faclte te ued. The th cell on the chromoome vector ndcate that at node faclty located (1) or not (0) n the current oluton. For example, Fg. 1 repreent the move vector of a network that nclude 10 potental faclte te and node 1 and 7 are elected n current oluton to be open and other node to be cloe.
28 R. Rahmanan et al.: Robut Capactated Faclty Locaton Problem Fgure 1: A chromoome for oluton repreentaton Moreover, the GAP value n percentage (relatve dtance between oluton and bet poble oluton) calculated by mean of equaton (22). In th equaton, Ffnal ndcate fnal oluton obtaned wth a heurtc method, and Fbet denote the lower bound obtaned by CPLEX. Ffnal Fbet % GAP 100. (22) Ffnal At the ubequent ubecton the propoed algorthm are dcued n detal and fnally, the flowchart of propoed VNS and outlne of SA are preented. 4.1 Contructon of an Intal Soluton Wth no doubt, ntal oluton ha a great mpact on every oluton algorthm. More mportantly, the VNS algorthm ue the bet recorded oluton a the ncumbent oluton to generate next move. Therefore, a cloe ntal oluton to the optmal oluton ha an enormou mpact on a ucceful VNS mplementaton. Our computatonal tet ndcated that optmal oluton of each cenaro ha a cloe gap wth the optmal oluton of the propoed model. A a reult, the ntal oluton obtaned from electng the optmal oluton of that cenaro that ha greater probablty of occurrence. 4.2 Local Search Local earch a method to return the local optma for the generated oluton. In th tudy oftware that make an nterface between MATLAB and GAMS ued to obtan the local optmal from CPLEX. Th oftware wa developed by Ferr [11]. In better word, algorthm are coded n MATLAB oftware and t generate the move (ee Fg. 2 and 3) and pa them to GAMS oftware and then the CPLEX olver wll olve the problem whle the locaton varable (.e. y ) are fxed. Th method mple and rather effcent. 4.3 Man Procedure of VNS Algorthm The man procedure of the algorthm llutrated n Fg. 2. The algorthm tart wth an ntal oluton for the locaton of faclte and then determne t local optma by CPLEX olver.in the mprovement phae, everal parameter uch a, pre-elected neghborhood tructure (.e. K max ) and et of oluton n the generated k th neghborhood (.e. N k (x)) have extreme mpact on the algorthm performance and need to be carefully tenured. A a reult, we examned everal tratege and the bet combnaton dcued here. For a gven neghborhood tructure, k, VNS randomly elect k bac node from the ncumbent oluton and replace them wth k non-bac node n order to generate a new oluton. Our computatonal tet ndcated that mot of mprovement move are acheved when k fxed at one. Conequently, we change the neghborhood tructure after a predefned number of teraton wthout mprovement ( U ). In the clacal VNS, U et to one; that mean, k k after any non-mprovement teraton neghborhood tructure ncreae by one. Note that, th algorthm et k to 1 f the obectve functon mproved or k k. max Algorthm take an ntal oluton a the ncumbent oluton, denoted by q bet, and fnd the local optma for th oluton (.e. F ) and whenever a move could mprove the obectve functon they wll be updated. Accordngly, t bet generate k th neghborhood from the bet recorded move, denoted by q, and f th oluton ha not been teted prevouly, t wll be checked to ee f t mprove the obectve functon. After applyng the local earch for th oluton, denoted by F current, followng tep are appled n order to decde whether we can change the neghborhood tructure or not, and the algorthm wll be repeated untl one of the toppng crtera perform.
Journal of Uncertan Sytem, Vol.7, No.1, pp.22-35, 2013 29 If F current < F bet then let { q bet = q, k=1, and UI=0} Ele f UI<=U k, then let {UI=UI+1, k =k, and q bet = q bet } Otherwe let {k =k+1, UI=0, and q bet = q bet } where, UI a counter for teraton and, whenever k k t wll be et to zero. Fnally, the algorthm top after max elapng the condered CPU tme, havng 3* J teraton wthout mprovement, or achevng the optmal oluton f t wa known beforehand. 4.4 Man Procedure of SA Algorthm Smulated Annealng (SA) a probabltc meta-heurtc that ha wdely been ued to fnd reaonable oluton n a lmted amount of tme (ee, [18, 19]). SA, unlke VNS, accept wore move wth a mall probablty n order to ecape from trappng n local optma. Th probablty calculated by the Boltzmann functon whch ue equaton (23). T Pr e r (23) where, C change n the evaluaton functon (.e. Fcurrent Fbet ), a contant, and T current temperature. If Pr wa greater than a random number, r, n nterval [0, 1], t accept wore move. Fgure 2: Man flow chart of the VNS algorthm The peudo-code of propoed SA algorthm to olve the model outlned at Fg. 3. To begn wth, an ntal oluton obtaned, then, the algorthm fnd t local optma, denoted by F bet. At the tep (2), the algorthm generate
30 R. Rahmanan et al.: Robut Capactated Faclty Locaton Problem a new oluton from X by randomly electng one bac node and replacng t wth one non-bac node. Accordngly, f the new oluton mproved the obectve functon both F bet and X wll be updated. Moreover, f the obectve functon dd not mprove, we can tll update X and accept wore move wth a mall probablty. Fnally, at each teraton temperature (.e. T) updated by ung the coolng rate parameter. Th parameter uually between 0.8 and 0.99. In th way, a tme elape the probablty of acceptng wore move reduce. Moreover, we record the bet acheved move and after a predefned number of teraton (ay 100), the algorthm retart from that oluton f obectve functon dd not mproved. Fnally, the algorthm termnated n four cae: (1) the temperature to be le than TF, (2) the obectve functon not mproved durng 3* J teraton, (3) the relatve gap wth the bet poble oluton become approxmately zero and (4) maxmum condered CPU run tme elape. Intalze nput parameter {model,, T0, TF, toppng crtera} Step (1): Calculate an ntal oluton: X Fnd a local optma for th oluton X: F bet Repeat followng tep untl one of the toppng crtera perform Step (2): Generate a new oluton Y from X baed on k=1 Step (3): Apply the local earch to fnd the local optma of Y: F current Step (4): Calculate change n the obectve functon: Fcurrent Fbet If 0then let { F bet F Current and X Y } Otherwe {r = random(0, 1)} If exp( ) r then let { X Y } T Step (5): T T 0 0 5 Numercal Experence 0 Fgure 3: Outlne of the propoed SA algorthm In th part, expermental reult are preented. The propoed problem wa mplemented on a range of tet problem and were olved wth tandard mathematcal programmng oftware GAMS 23.3.3, namely wth the branch-andbound algorthm of CPLEX 12.1, and algorthm were coded n MATLAB 7.6 and run on a Core 2 Dual @2.22GHz DELL Notebook wth 2GB RAM. In all data et, each node erve a both a cutomer and a potental faclty te (.e., I = J ). 5.1 Data Generaton To tet the algorthm performance, we appled numercal experment on both randomly generated and tandard tet problem. The tandard tet problem were obtaned from that publhed by Snyder and Dakn [32]. Thee tet problem have 49, 55, 88, and 150 node wth repectvely 9, 5, 9, and 9cenaro. We have ummarzed computatonal reult for thee ntance n Table 3 and 5. In the generated tet problem, the locaton of the cotumer are generated randomly and unformly dtrbuted over an100 100 area. In each data et for the generated tet problem, demand and fxed cot for cenaro-1 were drawn unformly from [0,10000] and [4000, 8000] repectvely and then rounded to the nearet nteger. Addtonal cenaro are dentfed by multplyng cenaro-1 to a random number drawn unformly from nterval [0.5, 1.5]. Travel dtance or travel tme between faclte and cutomer for cenaro-1 were equal to Eucldean dtance, and other cenaro are obtaned by multplyng random number drawn unformly from [0.8, 1.3] to the cenaro-1 data. Moreover, probablty of that cenaro occur calculated from equaton (24). I W S I W S. Condered capacty for each tet problem computed by mean of equaton (25). In other word, maxmum demand n all poble cenaro determned and then dvded to number of faclte. (24)
Journal of Uncertan Sytem, Vol.7, No.1, pp.22-35, 2013 31 max C S P I W. (25) 5.2 Computatonal Reult The propoed p-scflp ha not been condered formerly n the lterature. Conequently, we cannot provde comparon wth other computatonal tet. Accordngly, we tet the effcency of algorthm by olvng eghteen tet problem wth varou dmenon and comparng obtaned reult wth CPLEX. Followng parameter n th experment were ued. Maxmum condered CPU tme for all tet problem wa fxed at 4000 econd. Ten and ffteen percent of network node are condered a the number of faclte, or P. In addton, parameter of the propoed VNS algorthm were et to K max 3, B1 15, B2 10, B3 5 and the SA parameter were 0.97, T0 100, TF 0.1. 4 Regardng the propoed model, the requred parameter a and b were repectvely et to10 and 1whch yeld n a lnear mxed-nteger formulaton. To carryng the expermental tet, n th ecton, we frt olve the mnmax regret formulaton, RCFLP, and then preent computatonal reult for the propoed mn-expected-cot formulaton, p- SCFLP. In Table 2 through 5, thoe column under Tet Problem" whch are labeled by "TP", "N, "P", and "S" ndcate number of tet problem, number of network node, number of allowable faclte to be open, and number of cenaro, repectvely. Lkewe, thoe column whch are labeled by "L.B","GAP%", and"cpu Tme()" repreent lower bound, gap or relatve error (ee equaton (22)), and elaped CPU tme n econd, repectvely. In addton, the average CPU tme requrement n econd and average gap are lted n the lat row of Table 2-5. Moreover, n thee table bet performance n olvng each tet ntance boldfaced for the better algorthm. Table 2: Computatonal reult for the generated tet ntance: RCFLP 5.3 Numercal Experment for the RCFLP Note from Table 2 that, for TP1-3, on the average, VNS and SA found the optmal oluton n le than 1.66 and 7.48 econd, repectvely, whle CPLEX requred 18.28 econd n order to acheve to the optmal oluton. Lkewe, for larger ntance (TP4 and 5), the heurtc repectvely by conumng 271.63 and 566.36 econd alo outperformed CPLEX whch elaped 2917.21 econd. All n all, VNS, SA, and CPLEX repectvely by elapng 109.12, 219.68, and 1177.85 econd obtaned oluton that have 2.47, 3.73, and 3.81 percentage gap wth the lower bound. A a reult, the propoed VNS algorthm ha been able to outperform both SA and CPLEX n term of
32 R. Rahmanan et al.: Robut Capactated Faclty Locaton Problem optmalty gap and CPU tme requrement. More nteretngly that n one cae the VNS wa able to mprove CPLEX oluton by more than 10 percentage. The propoed algorthmhave been alo appled to olve well-known tet problem taken from lterature. Our heurtc olved mot of thee ntance to optmalty wthn a mall fracton of CPLEX CPU tme. On the average, VNS wa able to mprove oluton optmalty gap by 40.04% and reduce CPU tme requrement by 30.47% n compare wth CPLEX. On the other hand, thee value for SA algorthm are repectvely 38.67 and 10.37 percentage. Accordngly, the propoed VNS outperform agan both SA and CPLEX. Table 3: Computatonal reult for the tandard tet ntance: RCFLP 5.4 Numercal Experment for the p-scflp In th part, we preent computatonal reult for the propoed model wth mn-expected-cot obectve functon (equaton (2 to 9)). To carryng th experment, the p value for each ntance wa et to t lower cae taken from Table 2 and 3. In general, nce the mn-expected-cot formulaton much le complex than mnmax formulaton, CPLEX could fnd optmal oluton almot for all ntance wthn a mall fracton of the condered CPU tme. A a reult, the algorthm were not able to mprove CPLEX reult too much. The next two conecutve table ummarze the computatonal reult of th experment. Table 4: Computatonal reult for the generated tet ntance: p-scflp
Journal of Uncertan Sytem, Vol.7, No.1, pp.22-35, 2013 33 Table 5: Computatonal reult for the tandard tet ntance: p-scflp Table 4 ummarze performance comparon between algorthm and CPLEX for the generated tet ntance. A hown n Table 4, VNS and SA, unlke CPLEX, faled to obtan the optmal oluton for 2 and 1 ntance, repectvely. Moreover, on the average, CPLEX, VNS, and SA for the ntance whch are olved optmally requred 63.61, 21.20, and 16.52 econd, repectvely. However, takng all ntance together, the propoed VNS outperform both SA and CPLEX n term of CPU tme requrement and oluton qualty. Table 5 reveal computatonal reult on the tandard tet problem. For the maorty of ntance, all approache found the optmal oluton takng a few mnute. However, the bet oluton qualte and fatet performance wa acheved by VNS. Note that, the propoed VNS n two cae, TP9, n addton to reducng the CPLEX CPU tme, mproved optmalty gap a well. Thee expermental tude clearly demontrate the advantage of ung the propoed VNS n olvng the p-scflp. In order to nvetgate the convergence peed of propoed oluton algorthm, we took TP5, the tet problem wth 80 node and 8 faclte, a a repreentatve and oberved algorthm and CPLEX behavor n reducng optmalty gap over tme. A a reult, whenever the obectve functon mproved, t gap wth the lower bound reported n Table 2 and t tme were recorded and the reult are depcted n Fg. 4. A th plot how, the propoed ntal oluton ha le optmalty gap n compare wth the frt feable oluton of the CPLEX, roughly 34%. Clearly, the propoed VNS much fater than SA or CPLEX converge to the bet oluton. Though after approxmately 600 econd the VNS could not mprove the oluton any further, t ha acheved to the cloet oluton to the lower bound. Fgure 4: Convergence comparon between CPLEX, VNS and SA to olve TP5 over tme
34 R. Rahmanan et al.: Robut Capactated Faclty Locaton Problem 6 Concluon and Future Reearch Drecton In th paper an extenon of the capactated faclty locaton problem under uncertanty wa nvetgated. The mathematcal formulaton wa developed wth thee aumpton that demand and tranportaton cot are uncertan. In addton, nce faclte have hard contrant on the amount of demand they can upply, t lkely that they cannot completely erve all demand. A a reult, we developed the model formulaton to allow partal atfacton by ntroducng a penalty cot a a functon of unuppled demand. In th model, the obectve functon mnmzed expected cot whle relatve regret n each cenaro wa retrcted.in addton, we dcued the mnmax formulaton n whch the wort cae mnmzed. The propoed model formulaton a NP-Hard problem and very challengng to olve. A a reult, two fx-andoptmze heurtc baed on varable neghborhood earch and mulated annealng were developed to olve the model. In lne wth th ubect, the problem wa plt nto two ub-problem, locaton ub-problem and agnment ub-problem. The propoed algorthm teratvely determne the faclte locaton and then the agnment ubproblem olved by CPLEX. Algorthm performance n term of optmalty gap and CPU tme requrement wa teted by ung a varety of tet problem up to 150 node. We preented numercal experment for both mnmax regret formulaton and mn-expected-cot obectve functon. The reult demontrated that the algorthm bede of mplcty outperform CPLEX n term of oluton qualty and computer tme requrement. In general, the bet performance wa acheved by the VNS. Further attenton alo requred n the future reearche to nclude addtonal real aumpton uch a drupton n whch faclte may randomly fal to ervce cutomer or any practcal aumpton whch can be helpfully condered. Another nteretng reearch avenue to extend the model to take nto account acceblty to dfferent clae of faclte wth varou level of herarchy. Another lne of reearch would be developng other alternatve oluton algorthm to olve larger ntance. Moreover, developng an effcent local earch algorthm mght be an approprate future reearche drecton. Acknowledgement We would lke to thank two anonymou referee for ther contructve comment and uggeton whch mproved both the nght and preentaton of the paper. Reference [1] Aardal, K., Capactated faclty locaton: eparaton algorthm and computatonal experence, Mathematcal Programmng, vol.81, pp.149 175, 1998. [2] Averback, I., The mnmax relatve regret medan problem on network, INFORMS Journal on Computng, vol.17, no.4, pp.451 461, 2005. [3] Barahona, F., and F.A. Chudak, Near-optmal oluton to large-cale faclty locaton problem, Dcrete Optmzaton, vol.2, pp.35 50, 2005. [4] Bealey, J.E., Lagrangan heurtc for locaton problem, European Journal of Operatonal Reearch, vol.65, pp.383 399, 1993. [5] Brandeau, M.L., and S.S. Chu, An overvew of repreentatve problem n locaton reearch, Management Scence, vol.35, no.6, pp.645 674, 1989. [6] Brmberg, J., and N. Mladenovć, A varable neghborhood algorthm for olvng the contnuou locaton-allocaton problem, Stude n Locatonal Analy, vol.10, pp.1 12, 1996. [7] Cornueol, G., R. Srdharan, and J.M. Thzy, A comparon of heurtc and relaxaton for the capactated plant locaton problem, European Journal of Operatonal Reearch, vol.50, pp.280 297, 1991. [8] Cortnhal, M.J., and M.E. Captvo, Upper and lower bound for the ngle ource capactated locaton problem, European Journal of Operatonal Reearch, vol.151, pp.333 351, 2003. [9] Dakn, M.S., Network and Dcrete Locaton: Model, Algorthm, and Applcaton, Wley, New York, 1995. [10] Drezner, Z., and H.W. Hamacher, Faclty Locaton: Theory and Algorthm, Sprnger, 2001. [11] Ferr, M.C., MATLAB and GAMS: Interfacng Optmzaton and Vualzaton Software, Unverty of Wconn, 2005.
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