TWO STAGE FACILITY LOCATION PROBLEM: LAGRANGIAN BASED HEURISTICS

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1 TWO STAGE FACILITY LOCATION PROBLEM: LAGRANGIAN BASED HEURISTICS Igor Ltvnchev Nuevo Leon State Unversty Monterrey, Méxco Mguel Mata Pérez Nuevo Leon State Unversty Monterrey, Méxco Edth Lucero Ozuna Espnosa Nuevo Leon State Unversty Monterrey, Méxco ABSTRACT In the two-stage capactated faclty locaton problem a sngle product s produced at some plants n order to satsfy customer demands. The product s transported from these plants to some depots and then to the customers. The capactes of the plants and depots are lmted. The am s to select cost mnmzng locatons from a set of potental plants and depots. Ths cost ncludes fxed cost assocated wth openng plants and depots, and varable cost assocated wth both transportaton stages. In ths work, several Lagrangan relaxatons are analyzed and compared, a Lagrangan heurstc producng feasble solutons s presented. The results of a computatonal study are reported. KEYWORDS. Two stage faclty locaton problem, Lagrangan Relaxaton, Lagrangan Heurstc, Subgradent Technque. Man area: Mathematcal Programmng. 3640

2 1. Introducton The two-stage capactated faclty locaton problem can be defned as follows: a sngle product s produced at plants and then transported to depots, both havng lmted capactes. From the depots the product s transported to customers to satsfy ther demands. The use of the plants/depots ncurs a fxed cost, whle transportaton from the plants to the customers through the depots results n a varable cost. We need to dentfy what plants and depots to use, as well as the product flows from the plants to the depots and then to the customers such that the demands are met at a mnmal cost. Faclty locaton problems have numerous applcatons and have been wdely studed n the lterature, see the revew publcatons by Daskn, Snyder & Berger (2003), Klose & Drexl (2004) and Melo, Mckel & Saldanha-da-Gama (2009) and the references theren. Varous applcatons of the faclty locaton n supply chan optmzaton and management are presented n Wang (2011) and Mns et al. (2011). Varous exact approaches have been proposed for the locaton problems. For example, Avella & Bocca (2007) presented a famly of mnmum knapsack nequaltes of a mxed type, contanng both bnary and contnuous (flow) varables for the capactated problem and developed a branch and cut and prce algorthm to deal wth large scale nstances. Klose & Drexl (2005) consdered a new lower bound for the capactated faclty locaton problem based on parttonng the plant set and employng column generaton. Approxmate approaches can be roughly dvded nto two large groups: metaheurstcs and Lagrangan based technques. Metaheurstc approaches to the problem lke tabu search, GRASP, are dscussed n Flho & Galvão (1998). An algorthm for large nstances s presented n Barahona & Chudack (2005), they used a heurstc procedure that produces a feasble nteger soluton and used a Lagrangan relaxaton to obtan a lower bound on the optmal value. A Lagrangan based heurstc for solvng the capactated plant locaton problem wth sde constrants was presented n Srdharan (1991). Approaches and relaxatons proposed n the lterature for the capactated faclty locaton problem are compared n Cornueols, Srdharan & Thzy (1991). A lnear programmng based heurstc s consdered n Klose (1999) for a twostage capactated problem wth sngle source constrants. Wollenweber (2008) proposed a greedy constructon heurstc and a Varable Neghborhood Descent and a Varable Neghborhood Search for the mult-stage faclty locaton problem wth starcase costs and splttng of commodtes. In Landete & Marín (2009) the asymmetry nherent to the problem n plants and depots s takng nto account to strengthenng the formulaton. Gendron & Semet (2009) presented two formulatons for the problem and compared the lnear relaxaton of each formulaton and the bnary relaxaton of the model. Several Lagrangan relaxaton approaches have been proposed for the two stage faclty locaton problem. For the uncapactated case Chardare, Lutton & Sutter (1999) studed the effectveness of the formulaton for the two level smple plant locaton problem ncorporatng polyhedral cuts and proposed an approach combnng a Lagrangan relaxaton method and a smulated annealng algorthm. Lu & Bostel (2005) proposed an algorthm based on Lagrangan heurstcs for a 0-1 mxed nteger model of a two level locaton problem wth three types of faclty to be located. In Marín (2007) a mxed nteger formulaton and several Lagrangan relaxatons to determne lower bounds for the two stage uncapactated faclty locaton problem are presented. The Lagrangan relaxaton for the capactated case was studed and numercally tested n Barros & Labbé (1994). Bloemhof et al. (1996) studed alternatve model formulatons of the capactated problem obtanng lower bounds by Lagrangan relaxatons of the flow-balancng constrants. They also developed heurstc procedures to obtan feasble solutons. In Marín & Pelegrín (1999) several Lagrangan relaxatons for two dfferent formulatons of the two-stage problem are computatonally compared. Tragantalerngsak et al. (1999) proposed a Lagrangan relaxaton-based branch and bound algorthm for the two-echelon, sngle source, capactated problem. A Lagrangan heurstc s proposed n Klose (2000) usng relaxaton of the capacty 3641

3 constrants for the problem wth a fxed number of plants. Feasble solutons are constructed from those of the Lagrangan sub problems by applyng smple reassgnment procedures. In many technques Lagrangan relaxaton s used n twofold: the optmal value of the Lagrangan (dual) problem s used as a dual bound, whle the Lagrangan soluton s used as a startng or reference pont to produce a feasble soluton and a correspondng prmal bound. Frequently a relaxaton s consdered as good f t produces a tght dual bound. Meanwhle, the qualty of the feasble Lagrangan based soluton has also to be taken nto account n evaluatng the relaxaton. There are often dfferent ways n whch a gven problem can be relaxed n a Lagrangan fashon. It s unlkely to hghlght a sngle relaxaton producng hgh qualty bounds of both types, prmal and dual. Moreover, f the qualty of the dual bound s bascally defned by the constrants relaxed, the qualty of the prmal bound depends also on the algorthm used to restore the feasblty of the Lagrangan soluton. In ths paper we consder a smple decomposable relaxaton and an algorthm to restore the feasblty of the correspondng Lagrangan soluton. Ths relaxaton produces a poor dual bound and never been consdered before as a promsng relaxaton. Meanwhle, the relaxaton results n a very tght feasble soluton typcally wthn % of the relatve sub optmalty. The rest of the paper s organzed as follows. The next secton presents a mathematcal formulaton for the two-stage faclty locaton problem and the Lagrangan bound. A heurstc procedure to get feasble solutons s presented n secton 3. Computatonal results are reported n secton 4, whle secton 5 concludes. 2. Problem formulaton and Lagrangan bound To formally descrbe the problem, let I = 1,,n be the ndex set of potental plants, J = 1,,m the ndex set of potental depots and K=1,,k the ndex set of clents. Then, the problem can be formulated as the followng mxed nteger lnear program: w = mín f y + g z + c x + d s (1) k k I J ( I, J ) ( J, k K ) s. t.: x b ; I, J x p ; J, (3) I s k qk ; k K, (4) J x s k ; J, (5) I k K x m y ; I, J, (6) s k l k z ; J, k K, (7) { } + x, s R, y, z 0,1 (8) k c and g are the fxed costs assocated wth the nstallaton of plant and depot ; d k are the costs of transportaton from plant to depot and from depot to clent k, Here f and respectvely; q k s the demand of clent k; whle b and (2) p are the capactes of the correspondng plant and depot. The varables n ths formulaton are y = 1 f plant s nstalled and y = 0 otherwse, z = 1 f depot s nstalled and z = 0 otherwse, x, s k are the transportaton flows between the correspondng unts. Constrants (2) and (3) represent capacty lmts for plants and depots, (4) s the demand constrant (for each customer, at least the demand must be met), (5) s the relaxed flow conservaton constrant (the product transported from the depot must at least be transported to t from the plants), constrants (6) and (7), together wth (8), assure that there s a flow only from plants and depots nstalled. Constants m, l represent the upper bounds for the respectve flows, k 3642

4 we may set, e.g., m = mn{ b, p } ; l k mn{ p, qk } =. Note that by mnmzng the obectve (1), constrants (5) are fulflled as equaltes for an optmal soluton of (1)-(8). Constrants (6), (7) can be stated n a more compact form yeldng an equvalent formulaton of the two stage locaton problem (formulaton B): w = mín f y + g z + c x + d s (9) k k I J ( I, J ) ( J, k K ) s. t.: s q ; k K, J k x s ; J, I k K k k x b y ; I, (12) J I x p z ; J, { } + x, s k R, y, z 0,1 (14) Constrants (12) together wth (14) assure the outflows only from plants opened, whle constrants (13) together wth (11) assure the flows only from and to depots opened. Formulatons A and B are equvalent n the sense that both result n the same optmal soluton. However, they have dfferent polyhedral structure of the feasble sets and thus we may expect that relaxng the same constrants may result n dfferent values for the correspondng Lagrangan bounds. Lagrangan bounds are wdely used as a core of many numercal technques, e.g. n branch and bound schemes for nteger and combnatoral problems. Most Lagrangan relaxaton approaches for the capactated faclty locaton problem are based ether on dualzng the demand constrants or the depot capacty constrants (Klose, 1999). For the formulaton A defned by (1)-(8) fve decomposable Lagrangan relaxatons are consdered, denoted as follows (all Lagrangan multplers are assumed to be nonnegatve): RA1: constrants (5), representng nterconnectons between the two stages, are dualzed gvng a Lagrangan problem that can be decomposed nto a sub problem for the frst stage (plants) n varables x, y and a sub problem correspondng to the second stage (depots) n s, z. RA2: constrants (6) are dualzed gvng the Lagrangan problem that can be decomposed nto I subproblems of the form w RA2 1 = mín{ y ( f um ), y {0,1}}, whch can be analyzed analytcally and a sub problem n varables x, s, z. RA3: constrants (7) are dualzed. Smlar to RA2 we get J sub problems n varables z and a sub problem n (x,s,y). RA4: constrants (2) and (4), bndng wth respect to ndex, are dualzed. The Lagrangan problem then decomposes nto J ndependent sub problems. RA5: constrants (3) and (5) are dualzed. Smlar to RA1 ths Lagrangan problem decomposes nto a sub problem correspondng to the frst stage (plants), and a sub problem correspondng to the second stage (depots). The Lagrangan relaxatons for the formulaton B defned by (9)-(14) are as follows: RB1: constrants (11) are dualzed gvng two Lagrangan sub problems: a sub problem n (x,y,z), and a sub problem n s. The latter problem s decomposed nto K ndependent contnuous onedmensonal knapsack problems whch can be analyzed analytcally. RB2: constrants (12) are dualzed, resultng n I ndependent sub problems of the form RB2 w1 = mín{ y ( f ub ), y {0,1}} wth only one bnary varable and a sub problem n (x; s; z). RB3: constrants (13) are relaxed gvng smlar to RB2 J ndependent sub problems n z and a sub problem n (x; s; y). (10) (11) (13) 3643

5 RB4: constrants (10) and (12) are dualzed. The Lagrangan problem then decomposes nto I ndependent sub problems of the form w RB4 1 = mín{ y ( f vb ), y {0,1}}, that can be analyzed analytcally, and a sub problem n (x; s; z). The latter decomposes nto J ndependent sub problems wth only one bnary varable and can be solved by nspecton (see, e.g., Wolsey, 1999), fxng z to 0 or 1 and then solvng the remanng problem wth contnuous varables (x,s). RB5: constrants (11) and (13) are dualzed, the Lagrangan problem decomposes nto three types of sub problems. We have J ndependent sub problems n z of the form RB5 w = mín{ z ( g v p ), z {0,1}}. We also have K ndependent contnuous one-dmensonal knapsack sub problems n s, and I ndependent sub problems n x, y. The latter problem has only one bnary varable and can be solved by nspecton. The problem of fndng the best,.e. bound maxmzng Lagrange multplers, s called the Lagrangan dual. To solve the Lagrangan dual problem one can apply a constrant generaton scheme (Benders method) transformng the dual problem nto a large-scale lnear programmng problem. The man advantage of usng Benders technque s that t generates two-sded estmatons for the dual bound n each teraton thus producng near-optmal dual bound wth guaranteed qualty. Meanwhle, the computatonal cost of ths scheme s typcally hgh. Another popular approach to solve the dual problem s by subgradent optmzaton. In contrast to the Benders method, the subgradent technque does not provde the value of the bounds wth the prescrbed accuracy. That s, termnatng teratons of the subgradent method usng some stoppng crtera we can expect only approxmate values of the bound. We do not consder here these two well-known approaches n detals, referrng the reader to Lasdon(1970), Wolsey (1999) and Coneo (2000) for the constrant generaton (Benders) technque, and to Martn (1999) and Gugnard (2003) for the subgradent scheme. 4. Gettng feasble solutons To get a feasble soluton from the Lagrangan one we use a smple algorthm to recover feasblty. In fact, ths approach can be appled to any nonfeasble soluton. Let x, s be a nonfeasble soluton k y = 0,, I =, I = I ; Do 1 0 z = 0,, J =, J = J. 1 0 x Step 0: Do J y, z b s k k K. p * Step 1: arg max{ y I0} =. * * Step 2: y * 1, I1 I1 { }, I0 I0 { } Step 3: If. b qk go to step 4 and do y 0, I0 I1 k K * Step 4: arg max{ z J0} =. * * Step 5: z * 1, J1 J1 { }, J0 J0 { }. =, otherwse, return to step 1. Step 6: If p qk go to step 7 and do z 0, J0 J1 k K =, otherwse, return to step

6 Step 7: Fx y and z n the orgnal problem and solve the correspondng lnear problem to obtan the flows. In ths algorthm we calculate for each plant a saturaton ndcator representng the relatve usage of ts capacty (step 0). Then the plant havng the hghest saturaton s opened (step 1). If the capacty s suffcent to satsfy the total customers' demand, the rest of the plants are closed, otherwse the plant havng the next hghest ndcator s opened too (steps 2 and 3). The depots are opened n a smlar way (steps 3, 5 and 6). Fxng the bnary varables obtaned by ths procedure, the flows are determned from the correspondng lnear problem. 5. Computatonal results A numercal study for the two-stage capactated faclty locaton problem was conducted to compare the bounds. The followng sets of nstances were generated accordng to the values (I; J; K): A(3; 5; 9); B(5; 7; 30); C(7; 10; 50); D(10; 10; 100); E(10;16;30); F(30;30;30); G(30;36;120); H(30;30;100); I (50;50;200). Every set contans 20 problem nstances. The data were random ntegers generated as follows: c, d U 10, 20, q U 1,10, [ ] [ ] k k J + K K b 10 + U [ 0,10 ], p 10 + U [ 0,10] I J Two dfferent ways to generate the fxed costs were mplemented. For the frst ten nstances n each class the fxed costs f,g were random ntegers generated ndependently on the number clents, plants and depots: f, g U [ 100, 200]. For the remanng ten nstances the fxed costs f for plants were proportonal to the number of depots and clents, whle the fxed costs g for depots were proportonal to the number of clents: K + J K f U [ 0,100 ] ; g U [ 0,100] I J The dual bound correspondng to the Lagrangan relaxaton was calculated by the subgradent technque. In each teraton of ths method the feasble soluton was obtaned by the Algorthm. The best (over all teratons) feasble soluton was stored. The current best feasble soluton was used to update the step sze. If after 5 consecutve teratons of the subgradent technque the dual bound was not mproved, the half of the step sze scalng parameter was used. The process stops f the step sze scalng parameter s less than 0:0001, or f the maxmum number (300) of teratons s reached. The procedure was mplemented n GAMS/CPLEX 11.2 usng a Sun Fre V440 termnal, connected to 4 processors Ultra SPARC III wth 1602 Hhz, 1 MB of CACHE, and 8 GB of memory. For all the nstances we have calculated: z the value of the optmal obectve of the two stage locaton problem. IP zl - the value of the best Lagrangan bound. z BF - the obectve value correspondng to the best feasble soluton. The relatve qualty of the Lagrangan bound and of the best feasble soluton was measured by ε L zip zl zbf zip = 100% and ε BF = 100%, z z IP BF correspondngly. The smlar proxmty ndcators are used to measure the qualty of the bounds and feasble solutons derved from other relaxatons and other feasble solutons. The results obtaned for the Type 1 nstances are presented n Table

7 Table 1 Type 1 nstances. Lagrangan relaxatons Feasble solutons In the top 3 (%) Best bound (%) In the top 3 (%) Best bound (%) Sze RB1 RB2 RB3 RB1 RB2 RB3 RA2 RA3 RB4 RA2 RA3 RB4 A B C D The frst group of columns n Table 1 represents (n %) how many tmes the correspondng dual bound appeared among the best 3 bounds. The second group of columns shows (n %) how many tmes the correspondng dual bound was the best. The ndcators are presented only for the dual bounds correspondng to RB1, RB2 and RB3 snce they were most frequently among the best. The last columns present ndcators for the best feasble solutons obtaned by the Algorthm n the course of solvng the dual problem. The thrd group of columns n Table 1 represents (n %) how many tmes the correspondng feasble soluton appeared among the best 3 solutons. The last group of columns shows (n %) how many tmes the correspondng feasble soluton was the best. The ndcators are presented only for the feasble solutons derved from RA2, RA3, and RB4 snce they were most frequently among the best. Table 2 presents smlar results for the Type 2 nstances. Table 2 Type 2 nstances. Lagrangan relaxatons Feasble solutons In the top 3 (%) Best bound (%) In the top 3 (%) Best bound (%) Sze RB1 RB2 RB3 RB1 RB2 RB3 RA2 RA3 RB4 RA2 RA3 RB4 A B C D As can be seen from Tables 1 and 2, the bound correspondng to RB3 appears frequently among the best Lagrangan bounds. Consderng the qualty of the feasble solutons we may hghlght RB4 whch s frequently among the best and s easer to calculate than RA3. We note that relaxatons gvng the best dual bounds were never among those producng the best feasble solutons. In the Lagrangan problem correspondng to RB4 the demand and the capacty plant constrants are relaxed. The problem decomposes nto the followng sub problems: RB4: constrants (10) and (12) are dualzed: RB4 w = mín f y + g z + cx + d ks k + u k q k s k + v x b y I J ( I, J ) ( J, k K ) k K J I J x s ; J, I k K I k x p z ; J, { } + x, s R, y, z 0,1 k ( ) ( ) The Lagrangan problem then decomposes nto I ndependent sub problems of the form RB4 w1 = mín{ y ( f vb ), y {0,1}}, that can be analyzed analytcally, and a sub problem n (x; s; z): 3646

8 w = mín g z + ( c + v ) x + ( d u ) s RB4 2 k k k J ( I, J ) ( J, k K ) x s ; J, I k K I k x p z ; J, { } + x, s R, z 0,1. k The latter decomposes nto J ndependent sub problems of the form: w = mín g z + ( c + v ) x + ( d u ) s RB4 k k k I k K x I k K I x s p z + x, s k R, z 0,1. Ths problem has only one bnary varable and can be solved by nspecton (see, e.g., Wolsey, 1999), fxng z to 0 or 1 and then solvng the remanng problem wth contnuous varables (x,s). Thus we may conclude that the computatonal cost to solve the Lagrangan problem correspondng to the relaxaton RB4 s very low, n fact no nteger problem s nvolved. Table 3 present the results obtaned for the frst way to generate data for the RB4 relaxaton, whle Table 4 gves the results for the second way to generate data for the same relaxaton. The results are shown for 5 dfferent nstances for each problem sze. The frst two columns present the proxmty ndcators for the correspondng dual bound and for the best (over all teratons) feasble soluton obtaned by the Algorthm. The last two columns gve the proxmty ndcator for the feasble soluton correspondng to the last and the frst teraton of the subgradent technque. The number n the parenthess ndcates the number of the teraton correspondng to the bound value. Table 3 Results for RB4 type 1. k { } ε L (%) ε BF (%) ε LF (%) ε FF (%) Sze A (14) 5.07 (135) A (31) 0.00 (89) A (5) 4.21 (138) 4.21 A (9) 0.00 (128) A (4) 3.79 (130) 3.79 B (27) 0.99 (124) 3.47 B (17) 0.00 (135) 2.35 B (16) 4.73 (132) 2.66 B (15) 4.91 (147) 4.18 B (28) 0.18 (102) 8.37 C (36) 1.78 (118) 3.39 C (9) 2.81 (116) 8.22 C (8) 2.28 (151) 7.59 C (28) 6.57 (116) 8.49 C (28) 2.43 (118) 6.91 D (16) 4.26 (113) 5.30 D (24) 3.14 (129) 7.44 D (96) 5.55 (131) 9.71 D (1) 2.16 (130)

9 D (105) 1.75 (153) 3.79 ε L (%) Table 4 Results for RB4 type 2. ε BF (%) ε LF (%) ε FF (%) Sze A (9) 4.72 (214) A (29) 0.00 (166) A (2) (158) A (28) (188) A (1) 0.00 (157) 0.00 B (137) 0.79 (212) 2.09 B (58) 1.62 (250) 1.44 B (22) 9.79 (250) 8.63 B (61) 6.01 (243) 6.01 B (21) 0.11 (252) C (62) 8.28 (223) 9.95 C (181) 2.11 (270) 5.06 C (76) 2.28 (258) 4.64 C (27) 4.80 (237) 5.28 C (47) 1.49 (250) 4.34 D (62) 2.12 (284) 6.79 D (237) 4.34 (256) 4.48 D (67) 1.62 (277) 5.64 D (18) 8.18 (250) 8.45 D (153) 3.58 (276) 2.19 Along wth the set nstances A D we have used larger nstances generated accordng to the values (I; J; K): E(10; 16; 30); F(30; 30; 30); G(30; 60; 120); H(30; 30; 100); I(50; 50; 200). Table 5 Results for RB4 type 1. Sze ε L (%) ε BF (%) ε LF (%) ε FF (%) E (59) 0.10 (225) E (92) 0.75 (217) 7.79 E (49) 5.67 (231) E (134) 2.05 (256) 9.14 E (191) 6.07 (256) 9.10 F (109) F (53) 0.89 (235) F (58) 1.17 (198) F (264) F (60) 1.66 (266) G (240) G (88) G (249) G (100)

10 G (194) H (284) H (168) H (191) H (284) H (147) I (199) I (130) I (196) I (202) I (94) Table 6 Results for RB4 type 2. Sze ε L (%) ε BF (%) ε LF (%) ε FF (%) E (44) 9.06 (227) E (71) 0.79 (205) 4.93 E (51) 6.83 (293) E (108) 1.69 (233) 7.35 E (56) 8.03 (277) 8.79 F (133) 1.31 (257) F (60) 1.14 (261) F (56) 2.93 (250) F (51) 2.06 (261) F (32) 5.42 (277) G (194) G (128) G (120) G (260) G (87) H (206) H (214) H (240) H (127) H (78) I (211) I (207) I (221) I (168) I (146) As we can see from the Tables for both ways to generate the data the approach provdes very tghts feasble solutons, typcally wthn % of the relatve proxmty. So we may expect that the populaton of the Lagrangan solutons generated by the subgradent technque n the course of solvng the Lagrangan dual s suffcent for the Algorthm to generate hgh qualty feasble solutons. The qualty of the dual bund s poor, mprovng for larger nstances. The feasble soluton derved from the soluton of the Lagrangan dual and correspondng to the last teraton of the subgradent technque not necessarly s the best feasble soluton. 3649

11 Moreover, typcally ε LF > ε BF and the best feasble soluton s obtaned on the early teratons of the subgradent method. Thus we may conclude that t s mportant to generate feasble solutons n all teratons of the subgradent technque. 5. Conclusons Lagragan bound was presented for the two stage capactated faclty locaton problem. Two ndcators were consdered: the qualty of the dual bound and the proxmty of the Lagrangan based feasble soluton. It turned out that relaxng the demand and the capacty plant constrants provdes a rather poor dual bound, but the Lagrangan based feasble solutons are good, typcally wthn % of the relatve suboptmalty. Relaxng the demand and the capacty plant constrants result n a decomposable Lagrangan problem wth all sub problems analyzed by nspecton. Thus ths low cost relaxaton seems to be promsng to form the core of the Lagrangan based heurstcs. Solvng the dual problem by the subgradent technque we compared two approaches to generate a feasble Lagrangan based soluton. One s to get a feasble one by the soluton of the dual problem,.e. at the last teraton of the subgradent technque. Another approach s to generate feasble solutons n all teratons of the subgradent method and then choose the tghtest. It turned out that the best (over all teratons) feasble soluton never was obtaned at the last teraton. That s, smply solvng the Lagrangan dual and gettng a correspondng Lagrangan based feasble soluton s not suffcent to produce a tght feasble soluton. On the contrary, the populaton of the Lagrangan solutons generated by the subgradent technque n the course of solvng the Lagrangan dual s suffcent to generate hgh qualty feasble solutons. An nterestng drecton for the future research s mprovng the heurstc used to derve the feasble solutons. Some complements n ths drecton are n course. References Aardal K. (1998). Capactated faclty locaton: Separaton algorthms and computatonal experence. Mathematcal Programmng, 81, Avella, P. & Bocca, M. (2007). A cuttng plane algorthm for the capactated faclty locaton problem. Computatonal Optmzaton and Applcatons, 43, Barahona F. & Chudak F. (2005). Near-optmal solutons to large-scale faclty locaton problems. Dscrete Optmzaton Barros A. & Labbé M. (1994). A general model for the uncapactated faclty and depot locaton problem. Locaton scence, 2, Bloemhof-Ruwaard J.M., Salomon M. & Van Wassenhove L.N. (1996). The capactated dstrbuton and waste dsposal problem. European Journal of Operatonal Research, 88, Caserta M & Quñonez E. (2009). A cross entropy-based metaheurstc algorthm for largescale capactated faclty locaton problems. Journal of Operaton Research Socety, 60, Chardare P., Lutton J.L. & Sutter A. (1999). Upper and lower bounds for the two-level smple plant locaton problem. Annals of Operatons Research, 86, Coneo, A.J., Castllo, E., Mínguez,R. & García-Bertrand, R. (2006). Decomposton Technques n Mathematcal Programmng. Sprnger-Verlag. Cornueols G., Srdharan R. & Thzy J.M. (1991). A comparson of heurstcs and relaxatons for the Capactated Plant Locaton Problem. European Journal of Operatonal Research, 50, Daskn, M., Snyder, L. & Berger, R. (2003). Logstcs Systems: Desgn and Optmzaton. Sprnger. Flho V.J. & Galvão R.D. (1998). A tabu search heurstc for the concentrator locaton problem. Locaton Scences, 6, Fsher M.L. (1985). An applcaton orented gude to Lagrangan relaxaton. Interfaces, 15,

12 Gendron B. & Semet F. (2009). Formulatons and relaxatons for a mult-echelon capactated locaton-dstrbuton problem. Computers and Operatons Research, 36, Gugnard M. (2003). Lagrangan Relaxaton, TOP, 11, Klose A. (2000). A Lagrangan relax and cut approach for the two-stage capactated faclty locaton problem. European Journal of Operatonal Research, 126, Klose A. (1999). An LP-based heurstc for two-stage capactated faclty locaton problems. Journal of the Operatonal Research Socety, 50, Klose A. & Drexl A. (2004). Faclty locaton models for dstrbuton system desgn. European Journal of Operaton Research, 162, Klose A. & Drexl A. (2005). Lower bounds for the capactated faclty locaton problem based on column generaton. Management Scence, 51, Landete M. & Marín A. (2009). New facets for the two-stage uncapactated faclty locaton polytope. Computatonal Optmzaton and Applcatons, 44, Ltvnchev, I & Ozuna, E.L. (2012). Lagrangan bounds and a heurstc for the two-stage capactated faclty locaton problem. Internatonal Journal of Energy Optmzaton and Engneerng, v 1, No.1, Ltvnchev, I., Mata, M. & Ozuna, E.L. (2012). Lagrangan heurstc for the two-stage capactated faclty locaton problem. Appled and Computatonal Mathematcs, v 11, No.1, Ltvnchev, I., Mata, M., Ozuna, E.L., Rangel, S. & Saucedo, J. (2012, to appear) Two stage capactated faclty locaton problem: Lagrangan based heurstcs. Meta-Heurstcs Optmzaton Algorthms n Engneerng, Busness, Economcs, and Fnance. A book edted by Dr. Pandan Vasant. IGI global USA. Lasdon L.S. (2002). Optmzaton Theory for Large Systems. Boston, MA: Dover Publcatons. Lu Z. & Bostel N. (2005). A faclty locaton model for logstcs systems ncludng reverse flows: The case of remanufacturng actvtes. Computer and Operatons Research, 34, Marín A. (2007). Lower bounds for the two-stage uncapactated faclty locaton problem. European Journal of Operatonal Research, 179, Marín A. & Pelegrín B. (1999). Applyng Lagrangan relaxaton to the soluton of two-stage locaton problems. Annals of Operaton Research, 86, Martn, R. (1999). Large Scale Lnear and Integer Optmzaton: A Unted Approach. Boston, MA: Kluwer Academc Publshers. Melo M.T, Nckel S. & Saldanha-da-Gama F. (2009). Faclty locaton and supply chan management A revew. European Journal of Operatonal Research, 196, Mns I., Zempeks V., Dounas G. & Ampazs N. (2011). Supply chan management: Advances and ntellgent methods. Greece. IGI global. Osoro M. & Sánchez A. (2008). A preprocessng procedure for fxng the bnary varables n the capactated faclty locaton problem through parng and surrogate constrant analyss. WSEAS Transactons on Mathematcs, 8, Ramos M.T. & Sáenz J. (2005). Solvng capactated faclty locaton problems by Fenchel cuttng planes. Journal of the Operatonal Research Socety, 56, Srdharan R. (1991). A Lagrangan heurstc for the capactated plant locaton problem wth sde constrants. Journal of the Operatonal Research Socety, 42, Sun M., Ducat E. & Armentano V. (2007). Solvng capactated faclty locaton problem usng tabu search. Proceedngs of ICM Tragantalerngsak S., Holt J. & Ronnqvst M. (2000). An exact method for the two-echelon, sngle-source, capactated faclty locaton problem. European Journal of Operatonal Research, 123, Wang, J. (2011). Supply Chan Optmzaton, Management and Integraton: Emergng Applcatons. New Jersey, USA. IGI Global. Wollenweber J. (2008). A mult-stage faclty locaton problem wth starcase and splttng of commodtes: model, heurstc approach and applcaton. OR Spectrum, 30, Wolsey L.A. (1999). Integer programmng. New York, NY: John Wley and Sons. 3651