Adaptive Memory Programming for the Robust Capacitated International Sourcing Problem

Transcription

1 Adaptve Memory Programmng for the Robut Capactated Internatonal Sourcng Problem Joé Lu González Velarde Centro de Stema de Manufactura, ITESM Monterrey, Méxco Rafael Martí Departamento de Etadítca e Invetgacón Operatva, Unverdad de Valenca, Span. Rafael.Mart@uv.e Latet revon: October 7, 2005 Abtract The Internatonal Sourcng Problem cont of electng a ubet from an avalable et of potental uppler nternatonally located. The elected uppler mut meet the demand for tem of a et of plant, whch are alo located worldwde. Snce the cot are affected by macroeconomc condton n the countre where the uppler and the plant are located, the formulaton conder the uncertanty aocated wth change n thee condton. We formulate the robut capactated nternatonal ourcng problem by mean of a cenaro-optmzaton approach. In th paper we propoe a contructve method baed on memory tructure to olve th problem. The method coupled wth a local earch procedure followed by a path relnkng for mproved outcome. We propoe nnovatve mechanm to acheve a good balance between ntenfcaton and dverfcaton n the earch proce. Moreover, our path relnkng mplementaton ue contructve neghborhood for extrapolated relnkng. The computatonal expermentaton favor th method when compared wth a recent tabu earch approach. Key Word: Memory tructure, Path Relnkng

2 Adaptve Memory Programmng for ROCIS / 2 1. Introducton The Robut Capactated Internatonal Sourcng Problem (ROCIS) cont of electng a et of uppler to meet the demand for tem at everal plant located n dfferent countre. Dfferent veron of the problem have been propoed dependng on the aumpton. Table 1 ummarze the more relevant work n th and related problem. Author Decrpton Year Jucker and Carlon Solve a ngle product, ngle perod problem, wth prce 1976 and demand uncertanty Hodder and Jucker Preent a determntc ngle perod, ngle product 1982 model Hodder and Jucker Optmally olve a ngle perod, ngle product model 1985 wth quantty ettng frm Haug Addree the determntc problem wth a ngle 1985 product and multple perod wth dcount factor Louveaux and Solve a cenaro-baed problem n whch capacty a 1992 Peter frt tage decon Guterrez and Explore the generaton of cenaro to model prce 1995 Kouvel uncertanty and olve mple plant locaton problem Kouvel and You Propoe an un-capactated veron robutne approach 1997 baed on a mnmax regret crteron González-Velarde and Laguna Propoe a tabu earch method for the robut capactated veron wth exchange rate uncertanty 2004 Table 1. Relevant prevou work In th paper we conder the varant ntroduced n González-Velarde and Laguna (2004), whch deal wth a ngle tem n a ngle perod and model uncertanty n the demand and the exchange rate va a et S of cenaro. A frt formulaton of the problem follow: Mn S p M e f y + M j N e c j x j ubject to: M j N x x j j d j b y j N, S M, S. The control decon varable x j reflect the hppng from uppler ( M={1,2,...,m}) to plant j (j N={1,2,...,n}) n cenaro. The degn varable y take the value 1 f uppler contracted, and 0 otherwe. The frt et of contrant nclude the condton that for each cenaro, the demand at plant j, d j, mut be atfed. The econd et conder that for each cenaro, the capacty of uppler, b, cannot be exceeded. The objectve functon reflect the fact that the total unt cot for delverng component from uppler to plant j, c j, known, however, the exchange rate at uppler locaton n cenaro, e, make cot data uncertanty under dfferent cenaro (where p the probablty of occurrence of cenaro ). Fnally, t alo ncorporate the fxed cot f of development of uppler. González-Velarde and Laguna (2004) mprove th formulaton, replacng the objectve functon above wth the followng expreon:

3 Adaptve Memory Programmng for ROCIS / 3 F( y) = p e f y + z + ω S M + S p ( z + S E( z)) p 2 where z the optmal objectve value aocated wth the tranportaton problem obtaned when fxed the problem above for a partcular cenaro, and: S + = { : z E( z) = S E( z) 0} p z Th objectve functon penalze only the potve devaton from the expected value (thoe tuaton n whch the objectve value n a gven cenaro exceed the expected cot) and nclude a normalzng term to account for the fact that not all the term n the probablty dtrbuton are beng added. The value of ω a factor that the decon-maker can adjut to gve more or le mportance to the rk component of the objectve functon. The author develop a tabu earch algorthm to obtan effcent oluton for th non-lnear nteger program. The propoed oluton method can be vewed a a heurtc baed on the paradgm of Bender decompoton. An ntal et of value agned to the y bnary varable, whch make the remanng problem lnear. Th problem n turn may be decompoed nto S maller lnear ub-problem, one for each cenaro. The optmal dual oluton for each ub-problem ued to fnd a new et of value for the bnary varable. Intead of generatng vald cut for an nteger problem a n Bender method, thee dual varable are combned to form neghborhood of promng oluton and a earch conducted n the generated neghborhood. The earch method baed on the hort-term memory tratege of tabu earch (Glover and Laguna, 1997). A new et of value elected for the bnary varable and the procedure contnue untl ome termnaton crteron reached. In th paper we propoe an alternatve oluton method for the robut capactated nternatonal ourcng problem. Secton 2 decrbe our olvng methodology baed on contructve memory tructure and path relnkng, to obtan hgh qualty oluton for th problem. Secton 3 devoted to the computatonal experment. It how that our adaptve memory programmng method outperform the prevou tabu earch approach a well a a generc catter earch code. The paper fnhe wth the aocated concluon. 2. Soluton Method Our oluton method for the ROCIS problem cont of three tage. The frt one a contructve procedure that ncorporate memory tructure for dverfcaton purpoe. The econd one a local earch method, whch electvely appled to mprove prevouly generated oluton. Here, the meanng of electvely not lmted to the objectve functon evaluaton, but alo nclude the concept of nfluence aocated wth the oluton tructure. The thrd tage create path connectng mproved oluton baed on the path relnkng methodology. Mot of the tabu earch applcaton (Glover and Laguna, 1997) mplement tranton neghborhood n the context of local earch method. Contructve neghborhood have been rarely ued wth memory tructure, although they were ntroduced from the very begnnng of the methodology (Glover, 1989). In th paper we preent a contructve method baed on a greedy functon modfed wth frequency baed memory. In common wth other evolutonary method, Path Relnkng operate wth a populaton of oluton, rather than wth a ngle oluton at a tme, and employ procedure for combnng thee oluton to create new one. From a patal orentaton, the proce of generatng lnear combnaton of a et of reference oluton may be characterzed a generatng path between and beyond thee oluton, where oluton on uch path alo erve a ource for generatng addtonal path. Th lead to a broader concepton of the meanng of creatng combnaton of oluton. By natural extenon, uch

4 Adaptve Memory Programmng for ROCIS / 4 combnaton may be conceved to are by generatng path between and beyond elected oluton n neghborhood pace, rather than n Eucldean pace. In th paper we propoe a path relnkng method that create path connectng all the par of oluton obtaned from the electve applcaton of the local earch algorthm. In the followng ubecton we decrbe the mplementaton of thee three tage, a adapted n the context of the ROCIS problem. 2.1 Contructve Method The frt tage of our oluton procedure partcularly mportant, gven the goal of developng a method that balance dverfcaton and ntenfcaton n the earch. We mplement a contructve method ung frequency-baed memory, a propoed n tabu earch. Th method baed on modfyng a greedy meaure of attractvene by ung a frequency counter that dcourage the electon of uppler frequently elected n prevou oluton generaton. The attractvene of electng uppler gven by the greedy functon G() addng both, the fxed cot aocated wth th uppler f, and the um of the hppng unt cot from th uppler to all the plant, relatve to the uppler capacty b. The hppng cot multpled by the probablty of each cenaro, makng G() a meaure of expected attractvene. G( ) = f + p b n S j= 1 c j e A frequency counter Freq mantaned to record the number of tme uppler ha been elected n prevou oluton. Th frequency counter ued to penalze the attractvene of an element, and therefore, nducng dverfcaton wth repect to the oluton already generated. We modfy the value of G() to reflect prevou electon of element, a follow: MaxG G ( ) = G( ) + β Freq, MaxFreq where MaxFreq the maxmum Freq value for all, and MaxG the maxmum G() value for all. Each contructon tart by creatng a lt of unagned uppler U, whch at the begnnng cont of all the uppler n the problem (.e., ntally U = m). Then, we retrct th canddate lt conderng the et U' wth the k mot attractve uppler, accordng to the G -value. In each contructon tep, the next uppler randomly elected from the et U', then U updated (U = U {}) and U recalculated. The method fnhe when the um of the capacte of the elected uppler at leat a large a D. D = max d S j N Note that we top the contructon when the elected plant can atfy the demand n any cenaro. However, t poble that an optmal oluton have more elected plant than th mnmum number. Snce we do not have any nformaton about the bet number of elected plant, we do not mplement here any mechanm to ncreae t n the current oluton under contructon becaue t would be abolutely artfcal. In ubecton 2.3 we decrbe a mechanm to explore varaton on the cardnalty of th et n the combnaton element of the path relnkng method. To avod ntal bae, the frequency mechanm actvated after the frt IntIter contructon, and before th, electon are made wth the G-value. Let P be the et of oluton generated wth th method. j

5 Adaptve Memory Programmng for ROCIS / 5 It mportant to pont out that although our contructon method employ a canddate and a retrcted canddate lt, the evaluaton functon G () not adaptve nce t value reman contant thorough the contructon of a oluton (and change from one contructon to the next one accordng to the frequence). Therefore, trctly peakng, th not a GRASP contructon (Feo and Reende, 1995) and can be better clafed a a memory-baed contructon. Each generated oluton evaluated, by olvng the cenaro ub-problem and calculatng F(y). We alo ue a hah functon to codfy each oluton a follow: H ( y) = y M 2. The value of F(y) and H(y) are tored to avod evaluatng a oluton that ha already been generated. The ratonal for th that the objectve functon evaluaton can be conderably expenve n term of computatonal tme a the number of cenaro ncreae. It alo more convenent to tore the hah value than the entre bnary trng n order to effcently earch for the memberhp of the oluton n the lt. We apply th evaluaton flter baed on the hah functon n the entre procedure, ncludng the mprovement and path relnkng method decrbed below. 2.2 Improvement Method We partton the et P of generated oluton nto clae accordng to the cardnalty of the elected uppler n the oluton. Specfcally, let A k be the et of oluton wth k elected uppler: A k = y P / y = 1,.., n = k Note that P=A 1 A 2.. A n where ome A k mght be empty. In order to have a dvere et of oluton we want to have good repreentaton of each cardnalty et. Snce the objectve functon evaluaton computatonally expenve, the local earch only appled to a percentage pr of the bet oluton n each et A k. The local earch cont of exchange of uppler: a elected uppler replaced by a non-elected one n the current oluton. We examne the uppler y from =1 to n-1, and conder the mot attractve unelected uppler for exchange, where attractvene now meaured wth G (j). The value G(j) modfed addng the term V(j) whch meaure the relatve contrbuton of uppler j to the qualty of the generated oluton (we cale th term to be n the ame range than G(j)). We defne S j P a the et of generated oluton n whch uppler j elected: S j ={y P y j =1}. Then, we compute V(j) a the average value of the oluton n S j. G' '( MaxG j) = G( j) + V ( j) MaxV, V ( j) = y S j F( y) S j Where MaxV repreent the maxmum of the V(j) value for j=1,,m, and, a n prevou expreon, MaxG the maxmum of the G-value. The local earch procedure examne, for each uppler y, the bet alternatve uppler for exchange. Non elected uppler y j are then canned n the order gven by the G -value (where the uppler wth the lowet G -value examned frt). For each uppler y j we tet whether th exchange feable n term of the capacty (we only admt thoe oluton where the upply exceed the demand for every cenaro). The frt feable exchange that reult n a oluton wth a lower objectve value performed. The algorthm fnhe when no further mprovement poble.

6 Adaptve Memory Programmng for ROCIS / 6 The local earch appled to the bet oluton n each et A k a determned by the parameter pr. Let A k be the et of mproved oluton obtaned wth the applcaton of the local earch to the elected oluton n A k. In the followng ub-ecton we combne the oluton wthn each A k and between par of them. 2.3 Path Relnkng Path Relnkng, PR, can be condered an extenon of the clacal combnaton mechanm of other evolutonary method. Intead of drectly producng a new oluton when combnng two or more orgnal oluton, PR generate path between and beyond the elected oluton n the neghborhood pace. The character of uch path ealy pecfed by reference to oluton attrbute that are added, dropped or otherwe modfed by the move executed. Example of uch attrbute nclude edge and node of a graph, equence poton n a chedule, vector contaned n lnear programmng bac oluton, and value of varable and functon of varable. To generate the dered path, t only neceary to elect move that perform the followng role: upon tartng from an ntatng oluton, the move mut progrevely ntroduce attrbute contrbuted by a gudng oluton (or reduce the dtance between attrbute of the ntatng and gudng oluton). Then, conder the creaton of a path that jon two elected oluton y and y, retrctng attenton to the part of the path that le between the oluton, producng a oluton equence y = y(l), y(2),, y(r) = y. Path Relnkng tart from a gven et of elte oluton obtaned durng a earch proce. Followng the termnology gven n Laguna and Martí (2003), we wll let RefSet (hort for Reference Set ), refer to th et of b oluton that have been elected durng the applcaton of the prevou earch method. It ha been well documented n the Scatter Search context (Glover 1998), that the ze of the RefSet the 10% of the ze of the et P. Then, we wll conder th rato n our oluton procedure and apply the PR method to all par of the bet 10% mproved oluton obtaned after the applcaton of the local earch procedure. In mathematcal term: RefSet =A 1 A 2.. A n. For each par of oluton y and y n A j we frt compute two new oluton, y and y. In the former we elect the uppler preent n both oluton and the latter contan thoe uppler preent n at leat one of them. In mathematcal term: y = mn (y, y ), y = max (y, y ) Then, ntead of creatng a path between y and y, we create a path between thee two new pont y and y. Let y be the ntatng oluton and y.be the gudng oluton n the path (whch contan y and y a hown n Fgure 1). Let S be the et of uppler elected n y but non-elected n y. Symmetrcally, let S be the et of uppler non-elected n y and elected n y. S = { uppler j y j =1 and y j =0 }, S ={ uppler j y j =0 and y j =1 } Startng wth y, ntermedate oluton n the frt part of the path are generated by addng a uppler from S to the current oluton. Gven an ntermedate oluton, non-elected uppler are ordered accordng to ther relatve mert a meaured by the G value. Th value wa ntroduced n the mprovement phae; however, we redefne S j RefSet a the et of mproved oluton n the RefSet n whch uppler j elected: S j ={y RefSet y j =1}. Then, we compute G wth the expreon gven n the prevou ubecton. Gven an ntermedate oluton y n the frt part of the path (ntally y= y ), non elected uppler n S are ordered accordng to ther G value. Then, the uppler j* n S wth the lowet G value elected (y j* =1), thu obtanng the next oluton n the path. Once y ha been reached (after S electon), we alternate between addng and deletng uppler from the current oluton to reach y. Specfcally, uppler to be deleted are thoe n S, whle uppler to be added are thoe n S. Even teraton correpond to addton and odd correpond to deleton. Gven an ntermedate oluton y, the non elected uppler n S wth lowet G value, j*, elected n an even teraton for ncluon n y (y j* =1). Smlarly, n an odd teraton, the non-elected uppler n S wth larget G value, j+, elected for deleton (y j+ =0). Fnally, once y ha been reached, n the thrd part of the path, we add the uppler n S to obtan y. A n the frt part of the path, at each teraton we add the non elected uppler n S wth lowet G value. The relnkng fnhe when the ntatng oluton matche the gudng oluton (after 3 S + S ntermedate oluton have been generated).

7 Adaptve Memory Programmng for ROCIS / 7 A t done n prevou path relnkng mplementaton (Laguna and Martí, 1999), we have alo condered the ncluon of a extenve exploraton at certan pont of the relnkng proce. Specfcally, an expanded neghborhood from ome of the feable oluton along the path examned. It cont of exchange of uppler n whch a elected uppler replaced by a non-elected one untl no more mprovement can be made. Th the ame exchange mechanm ued n the mprovement phae. Once the expanded neghborhood ha been explored, the relnkng contnue from the oluton before the exchange were made. number of uppler Feable regon y y Expanded neghborhood y Unfeable regon y Fgure 1. Path relnkng llutraton Note that two conecutve oluton after a relnkng tep dffer only n the electon of one uppler. Therefore, t not effcent to apply the expanded neghborhood exploraton (.e., the exchange mechanm) at every tep of the relnkng proce. A recommended n Laguna and Martí (1999), the exchange mechanm appled every 10 tep of the relnkng proce. Note that y a well a the frt oluton n the path can eventually be nfeable wth repect to the capacty. However, once the feablty attaned n an ntermedate oluton, we try to keep the feablty n the remanng oluton n the path. Th why we alternate between addng and deletng uppler n the econd part of the path nce ucceve deleton would caue unfeablty. Note that we cannot guarantee that every oluton n the path wll be feable but feablty wll be retored wth the propoed mechanm. Once the path ha been travered n the drecton defned from y' to y'', the procedure appled n revere drecton (form y'' to y') gven that a dfferent path generated. The Path Relnkng procedure termnate when all par of oluton have been examned n both drecton. 3. Computatonal Experment For our computatonal tetng we frt ue the et of 90 ntance reported n González-Velarde and Laguna (2004). In th et the number of cenaro fxed to 27, the number of plant to 10 and the number of uppler to 10, 15 and 20, thee number defne three group of 30 ntance each. Wthn thee three ze categore, x ubgroup of ze 5 were formed by varyng the parameter to defne the relatonhp between demand at the plant and the capacty of the uppler, a well a the parameter to defne the relatonhp between fxed and varable cot. A n th prevou work, we ue a value of ω =2 n the robut objectve functon, whch the one that penalze potve devaton from the expected cot. Note that the computatonal experment n mlar tude (e.g., Kouvel and Yu, 1997), deal wth problem ntance of comparable ze. However, the cenaro ub-problem n uch tude are trvally olved, gven that they aume an nfnte capacty for each uppler. Addtonally to thee 90 ntance,

8 Adaptve Memory Programmng for ROCIS / 8 we have generated 30 new larger ntance wth 20 plant and 40 uppler. Thee ntance have been generated wth the ame procedure reported n González-Velarde and Laguna (2004). Laguna and Martí (2003) propoe a generc catter earch for bnary problem. Ther method wa orgnally degned to olve a knapack problem; however t can be ealy adapted to our problem. Bacally we only need to ue the evaluaton functon decrbed n the ntroducton and modfy the knapack capacty contrant to control the feablty. We have ncluded th generc olver n our comparon a a baelne to meaure the contrbuton of the pecfc olver uch a the prevou tabu earch method by González-Velarde and Laguna (2004) and our current mplementaton. In our prelmnary expermentaton the value of IntIter wa et to 10 and we have condered the key earch parameter and 15 ntance wth 10 plant and 20 uppler. In the frt experment we undertake to meaure the value β. For each value of β (0.3, 0.5 and 0.7) Table 2 how the average of the bet objectve value found wth the contructve method, a well a the number of optma and average runnng tme. Table 2. Prevou experment. β Devaton 12.8% 12.2% 15.3% Num. of Opt CPU ec Table 2 how that the bet oluton obtaned, on average, wth the contructve method wth a β value of 0.5. In the econd experment we meaure the contrbuton of the percentage parameter pr n the qualty acheved by the local earch method. Note that the local earch only appled to the bet pr% oluton n each et A k. We tet three value for th parameter: 25%, 50% and 75% and ue the ame 15 ntance that n the prevou experment. We do not produce table for th experment, nce thee three value provde the ame oluton n the local earch procedure. However, a expected, run tme ncreae a pr ncreae. Therefore we et pr=25% n our oluton method. In the next experment, we employ the 90 problem ntance reported n González-Velarde and Laguna (2004). A mentoned, thee ntance have 10 plant and are grouped n three categore accordng to the number of uppler (10, 15 and 20). Table 3, 4 and 5 report for each group of 30 ntance the average objectve value, the average devaton from the optmal oluton, the number of optma acheved, and the average CPU econd of the dfferent method under conderaton. We compare the performance of the tabu earch method (TS, González-Velarde and Laguna 2004), the generc catter earch method (SS), the contructve procedure decrbed n ecton 2.1 (Cont), the contructve procedure followed by the local earch decrbed n ecton 2.2 (Cont+LS) and, the path relnkng method (PR). Table 3. n=10, m=10. TS SS Cont Cont+LS PR Value Devaton 0.00% 6.49% 1.33% 0.35% 0.00% Num. of Opt CPU ec Table 4. n=10, m=15. TS SS Cont Cont+LS PR Value Devaton 2.74% 9.29% 5.27% 0.11% 0.00% Num. of Opt CPU ec

9 Adaptve Memory Programmng for ROCIS / 9 Table 5. n=10, m=20. TS SS Cont Cont+LS PR Value Devaton 7.42% 15.57% 9.95% 0.62% 0.44% Num. of Opt CPU ec Thee table how that the bet oluton qualty obtaned by the path relnkng method (PR), whch able to match a larger number of optmal oluton than the other method. Th epecally true n the ntance wth 20 uppler n whch PR matche 21 optmal oluton, Cont+LS 19, and none of them the other method. Conderng the 90 ntance n Table 3, 4 and 5 together, TS matche 34, SS 3, Cont 16, Cont+LS 74 and PR 81. However, although n th problem run tme not a crtcal factor, t hould be noted that the PR method conume a runnng tme 26 tme hgher than the mple contructon method (Cont). Thee table alo how that the performance of the SS method clearly nferor wth a gnfcantly lower number of optmal oluton than thoe acheved by the other approache. However, t a generc method and t reult are qute acceptable conderng t wde applcablty to any 0-1 optmzaton problem. Regardng the relatve devaton from optmalty, Table 3 how that both, the TS and the PR method preent a 0.00% devaton on average, whle SS, Cont and Cont+LS preent 6.49%, 1.33% and 0.35% repectvely. However, a hown n Table 4 and 5, n larger graph (relatve to the number of uppler) the method quckly deterorate preentng larger devaton from optmalty. The rankng of the method accordng to the average percentage devaton value acro the 90 ntance SS (10.45%), Cont (5.51%), TS (3.39%), Cont+LS (0.36%) and PR (0.15%). In our lat experment we undertake to compare the performance of our propoed procedure ung relatvely larger graph (a compared to thoe n the frt experment). In pecfc, we generate 30 addtonal ntance wth 20 plant and 40 uppler. We cannot obtan the optmal oluton for thee large ntance. The BEST column repreent the mnmum value of the objectve functon for each ntance after runnng all procedure durng the experment. (We cannot ae how cloe the BEST value are from the optmal oluton, and we are only ung thee value a a way of comparng the method.) Table 6. n=20, m=40. Bet TS Cont Cont+LS PR Value Devaton 0.00% 2.93% 19.41% 0.11% 0.00% Num. of Opt CPU ec Table 6 clearly how that the propoed procedure outperform the prevou tabu earch approach nce t able to obtan all the bet known oluton n thee large ntance. The prevou tabu earch mplementaton alo perform well nce t preent an average devaton from the bet oluton known of 2.93%. It hould be alo noted that the contructon wth local earch (wthout the path relnkng phae) perform remarkably well n a relatve hort computatonal tme (t acheve a 0.11% of average relatve devaton and matche 27 out of 30 bet known oluton n about 3 mnute of CPU tme). Concluon We have developed a heurtc procedure baed on the Tabu Search methodology to provde hgh qualty oluton to the Robut Capactated Sourcng Problem (Roc). Our oluton method cont of three tage. A contructve procedure that ncorporate memory tructure for dverfcaton purpoe, a local earch method, whch electvely appled to mprove prevouly generated oluton wth an aocated

10 Adaptve Memory Programmng for ROCIS / 10 low computatonal effort, and a path relnkng mplementaton to create path connectng mproved oluton. The propoed procedure wa hown compettve n a et of problem ntance for whch the optmal oluton are known. For a et of larger ntance, the propoed contructon wth t local earch and path relnkng varant performed remarkably well (outperformng the bet procedure reported n the lterature). Acknowledgment Reearch by Joé Lu González-Velarde partally upported by the ITESM Reearch Char n Indutral Engneerng. Reearch by Rafael Martí partally upported by the Mntero de Educacón y Cenca (ref. TIN E and TIC2003-C05-01) and by the Agenca Valencana de Cènca Tecnologa (ref. GRUPOS03 /189). Reference Feo, T. and M. G. C. Reende (1995) Greedy Randomzed Adaptve Search Procedure, Journal of Global Optmzaton, Vol. 2, pp Glover, F. (1989) Tabu Search-Part I, ORSA Journal on Computng, Vol. 1, pp Glover, F. and M. Laguna (1997) Tabu Search, Kluwer Academc Publher. Glover, F., A Template for Scatter Search and Path Relnkng. In: Hao, J.-K., Lutton, E., Ronald, E., Schoenauer, M., Snyer, D. (Ed.), Artfcal Evoluton, Lecture Note n Computer Scence 1363, Sprnger, pp González-Velarde, J.L. and M. Laguna (2004), "A Bender-baed heurtc for the robut capactated nternatonal ourcng problem", IIE Tranacton, vol. 36, pp Gutérrez, G.J. and P. Kouvel (1995), A Robutne Approach to Internatonal Sourcng, Annal of Operaton Reearch, vol. 59, pp Haug, P. (1985), A Multple-Perod, Mxed-Integer-Programmng Model for Multnatonal Faclty Locaton, Journal of Management, vol. 11, no. 3, pp Hodder, J.E. and J. V. Jucker (1985), A Smple Plant-Locaton Model for Quantty-Settng Frm ubject to Prce Uncertanty, European Journal of Operatonal Reearch, vol. 21, pp Hodder, J.E. and J.V. Jucker (1982), Plant Locaton Modelng for the Multnatonal Frm, Proceedng of the Academy of Internatonal Bune Conference on the Aa-Pacfc Dmenon of Internatonal Bune, Honolulu, December, 1982, pp Jucker, J.V. and R.C. Carlon (1976) The Smple Plant-Locaton Problem under Uncertanty, Operaton Reearch, vol. 24, no. 6, pp Kouvel, P. and G. Yu (1997) Robut Dcrete Optmzaton and t Applcaton Kluwer Academc Publher, Dordrecht. Laguna, M. and R. Martí (1999) GRASP and Path Relnkng for 2-Layer Straght Lne Crong Mnmzaton, INFORMS Journal on Computng, Vol. 11, No. 1, pp Laguna, M., Martí, R., Scatter Search Methodology and Implementaton n C, Kluwer Academc Publher, Boton. Louveaux F.V. and D. Peeter. (1992) A dual-baed procedure for tochatc faclty locaton. Operaton Reearch, vol. 40 no. 3, pp