Lower Bounding Procedures for the Single Allocation Hub Location Problem

Transcription

1 Lower Boundng Procedures for the Snge Aocaton Hub Locaton Probem Borzou Rostam 1,2 Chrstoph Buchhem 1,4 Fautät für Mathemat, TU Dortmund, Germany J. Faban Meer 1,3 Uwe Causen 1 Insttute of Transport Logstcs, TU Dortmund, Germany Abstract Ths paper proposes a new ower boundng procedure for the Uncapactated Snge Aocaton p-hub Medan Probem based on Lagrangean reaxaton. For sovng the resutng Lagrangean subprobem, the gven probem structure s expoted: t can be decomposed nto smaer subprobems that can be soved effcenty by combnatora agorthms. Our computatona experments for some benchmar nstances demonstrate the strength of the new approach. Keywords: Hub Locaton, Lagrangan reaxaton, Lower bounds. 1 Ths research has been funded by the German Research Foundaton (DFG) wthn the project Lenung des Güterfusses n durch Gateways geoppeten Logst-Servce- Netzweren mttes quadratscher Optmerung (CL 318/14 and BU 2313/2) 2 Ema: brostam@mathemat.tu-dortmund.de 3 Ema: meer@t.tu-dortmund.de 4 Ema: chrstoph.buchhem@math.tu-dortmund.de

2 1 Introducton Consder a compete graph G = (V, E), where V = {1, 2,..., n} corresponds to orgns, destnatons and possbe hub ocatons, and E s the edge set. Let b j be the transport cost per unt of fow from node to node j, and W j be the amount of fow from node to node j. The cost per unt of fow for each path Pj from an orgn node to a destnaton node j whch passes hubs and respectvey, s β 1 b + αb + β 2 b j, where β 1, α, and β 2 are the coecton, transfer and dstrbuton costs respectvey. The Uncapactated Snge Aocaton p-hub Medan Probem (USApHMP) conssts of seectng p nodes as hubs and assgnng the remanng nodes to these p hubs such that each non-hub node s assgned to exacty one hub node wth the mnmum overa cost. The quadratc bnary programmng formuaton for the (USApHMP) s: mn b (β 1 W j + β 2 W j )x + αb W j x x j j j s.t. x = 1 (1) x x, (2) x = p (3) x {0, 1},, (4) where the bnary varabe x ndcates the aocaton of node to the hub ocated at node. Constrants (1) ndcate that non-hub node s aocated to precsey one hub node. Constrants (2) enforce that node s aocated to a hub node at ony f a hub s ocated at node. Constrant (3) restrcts the number of seected hubs to p. To ease the argumentaton, we defne C = b (β 1 j W j + β 2 j W j) and Q j = αb W j. Ths aows us to wrte down the objectve functon n a more condensed form: C x + Q j x x j. j The USApHMP was frst ntroduced n [9] as a quadratc bnary program. Snce then, many exact and heurstc agorthms have been proposed n the terature, e.g., by Campbe [3], Ernst and Krshnamoorthy [5], Sorn-Kapov et a. [10], and Ić et a. [8]. Due to the quadratc nature of the probem,

3 many attempts have been made to nearze the objectve functon so that the resutng ower bound s strong enough to be used n a branch-and-bound agorthm. Sorn-Kapov et a. [10] and Ernst and Krshnamoorthy [5] proposed Mxed Integer Lnear Programmng (MILP) formuatons wth O ( n 4) and O ( n 3) varabes, repectvey. The ower bound obtaned from the contnuous reaxaton of the four ndex MILP formuaton of Sorn-Kapov et a. [10] s tghter than the one obtaned usng the three ndex MILP formuaton of Ernst and Krshnamoorthy [5]. However, t requres consderaby more runnng tme to be computed. In ths paper we consder two new ower bounds for the USApHMP. The frst bound trvay buds a new p-medan probem from the quadratc cost matrx and soves the resutng probem to obtan a bound. Ths boundng procedure s then shown to be equvaent to the contnuous reaxaton of an MILP. Due to the arge duaty gap mped by ths approach, we deveop a new MILP formuaton and show how to hande t va a Lagrangan reaxaton approach to obtan a Lagrangan functon wth boc-dagona structure. 2 A reaxed Gmore-Lawer type bound The Gmore-Lawer procedure, shorty denoted by GL, s one of the most popuar approaches to fnd a ower bound for the Quadratc Assgnment Probem. The new bound we consder s derved from a smpe observaton on the structure of the USApHMP, n a smar sprt to the GL procedure: If we defne C TOT = C + j, Q jx j, ths aows us to rewrte USApHMP as { } z = mn C TOT x : (1) (4). (5) A ower bound for the probem can, therefore, be obtaned f we repace each C TOT wth ts mnmum vaue over the set of possbe feasbe soutons whch contan the assgnment of node to node. In other words, for each arc (, ), potentay n the souton, we consder the best cumuaton provdng the mnmum nteracton cost wth (, ). Ths can be done by sovng a set of subprobems wth a near objectve functon, one for each possbe assgnment. Let P represent such a subprobem for a gven arc (, ) E: P : { } mn Q j x j : (1) (4), x = 1. (6) j

4 The dea s thus to sove, for each P, a p-medan probem whch contans arc (, ), usng the -th coumn of the quadratc cost matrx as the cost vector. Ths yeds a ower bound for the vaue of C TOT n any feasbe souton contanng (, ). However, probem P s a p-medan probem whch s wenown to be NP-hard [6]. Therefore, we consder a reaxaton of P caed P whch ony requres the aocaton of each non-hub node j to precsey one hub node,.e., { } P : z = mn Q j x j : (1), (4), x = 1. (7) j The probem P s a sem-assgnment probem and can be soved n O( n 2) tme. The vaue of z combned wth the near cost C of arc (, ) yeds a ower bound for C TOT, whch can then be ntegrated nto (5), resutng n { } P 1 : z1 = mn (C + z )x : (1) (3). (8) Note that we do not requre (4) here, as otherwse sovng P 1 woud be NPhard agan. We obtan Theorem 2.1 Sovng P 1 yeds a ower bound for USApHMP,.e., z1 z. Athough the man part of the boundng procedure that we just descrbed s combnatora, the same bounds can be obtaned by sovng a near program. More precsey, we ntroduce the non-negatve contnuous varabes y j for a,, j, V, and a set of constrants as y j = x, j, ; j. (9) Now we consder the foowng MILP formuaton: { P1: mn C x + } Q j y j : (1) (4), (9), y 0. j For ths MILP, we can show the foowng resut. The proof s omtted due to space restrctons. Theorem 2.2 The optma objectve vaue for the contnuous reaxaton of probem P1 agrees wth z1.

5 3 New formuaton and Lagrangan reaxaton In order to mprove the bound presented n the prevous secton we foow the dea proposed n [4]. We consder probem P1 wth separate varabes y j and y j for a (, ), (j, ) E and ntroduce an addtona set of constrants: y j = y j (, ), (j, ), < j. (10) We refer to ths new formuaton as probem P2. We can prove that probems P2 and USApHMP are equvaent n the sense that for any x feasbe for USApHMP there exsts a y such that (x, y) s feasbe for P2, and conversey, for any (x, y) feasbe for P2, x s feasbe for USApHMP wth the same objectve vaue. Consder the contnuous reaxaton of P2. Due to the arge number of varabes and constrants, and aso degeneracy of the probem, sovng ths reaxaton n order to obtan a ower bound for USApHMP s too tme consumng. Therefore we consder the Lagrangan dua obtaned from reaxng constrants (1) and (10), usng a set of Lagrangan mutpers µ, V, and λ j for a (,, j, ), < j. For convenence we assume that λ j = λ j for a (,, j, ), < j. The resutng Lagrangan functon s as foows: P(µ, λ): mn (C µ )x + (Q j λ j )y j j s.t. (2) (3), (9) and x, y 0. Note that P (µ, λ) s the contnuous reaxaton of P1 where constrants (1) have been reaxed nto the objectve functon. Therefore, we have: Theorem 3.1 For any gven vaues of µ and λ, an optma souton (x, y ) to P (µ, λ) s gven by x = ˆx and y j = ŷ jˆx (,, j, ), where ŷ s the optma souton of subprobem P (wth Q j = Q j λ j ) whe ˆx s the optma souton of the foowng probem: { } MP : mn (C µ )x : (2) (3). (11) Observe that MP can be soved by a smpe nspecton: Let C = C µ for a,, and defne d = C + mn{0, C }. Then Probem MP s reduced to seectng the p hubs wth smaest cost d.

6 4 Computatona resuts In ths secton we present computatona experments on the CAB nstances from OR Lbrary [2]. We consder the bggest nstances, havng n = 20, 25 nodes, and dfferent dscount factors α {0.2, 0.4, 0.6, 0.8, 1} whe aways eepng β 1 = β 2 = 1. In order to deveop a procedure to fnd the optma (or near-optma) dua mutpers of the Lagrangan dua probem, we use the subgradent method [7] wth nta step sze of 1 and maxmum number of aowed teratons of Tabe 1 presents the reatve gap between the ower bounds and the optma objectve vaues, and CPU executon tmes (n seconds) of usng dfferent approaches. The frst four coumns of the tabe ndcate the probem sze (n), the number of desred hubs (P ), the dscount factor (α), and the optma objectve vaues obtaned from [5]. The next three coumns, from eft to rght, gve the reatve gaps obtaned usng the reaxed GL procedure descrbed n Secton 2 (RGL), the subgradent mpementaton for P2 (SubP2), and the near programmng reaxaton obtaned wth the three ndex formuaton of Ernst and Krshnamoorthy [5] (LPEK). The CPU executon tmes of each approach are gven n the ast three coumns of the tabes. The formua we used to compute the reatve gaps s 100 (Opt Lb)/Lb, where Opt and Lb stand for the optma vaue and the vaue of the ower bound, respectvey. As we can observe from the tabe for a probems tested, the Lagrangan dua procedure sgnfcanty outperforms the LPEK n terms of the strength of the bounds. In fact, n 21 out of the 30 nstances even optmaty was proven usng our Lagrangan dua. For the remanng nstances, the reatve gap of the Lagrangan dua s ess than 1%, except for the snge case n = 20, p = 3, α = 0.8, for whch the gap s 1.37%. These resuts confrm the strength of the four ndex MILP formuatons as shown aready by Sorn-Kapov et a. [10]. However, our new four ndex MILP formuaton presented n ths paper s favorabe, snce ts contnuous reaxaton possesses a speca structure wth ends tsef to decomposton technques. Even f the tme needed to compute our bounds s consderaby onger than for computng the LPEK bound, the sgnfcanty stronger bounds w ey ead to compettve or even faster runnng tmes when sovng USApHMP to optmaty. Moreover, wthn a branch-and-bound scheme, the number of teratons of the subgradent method can be reduced sgnfcanty by ntazng t wth the optma mutpers of the parent node, as aready observed for other Lagrangan reaxaton based branch-and-bound methods, e.g., [1].

7 Tabe 1 Comparson of dfferent ower boundng approaches on TestSet CAB for probem sze n = 20, 25. Instance Gap(%) CPU Tme n P α Opt. RGL SubP2 LPEK RGL SubP2 LPEK

8 5 Concuson We presented a new MIP formuaton of the Uncapactated Snge Aocaton p-hub Medan Probem. Due to the sze of the resutng mode, we deveoped a Lagrangean reaxaton approach to compute a strong ower bound. For sovng the Lagrangan subprobem, we used a smpe observaton on the structure of the USApHMP combned wth the Gmore-Lawer procedure. Our future wor w concentrate on the ntegraton of our ower bounds nto a branchand-bound scheme and on extendng our deas to the Capactated SApHMP. References [1] Baumann, F., C. Buchhem and A. Iyna, Lagrangean decomposton for meanvarance combnatora optmzaton, n: Proceedngs of ISCO 2014, pp [2] Beasey, J. E., OR Lbrary (2012). URL [3] Campbe, J. F., Integer programmng formuatons of dscrete hub ocaton probems, European Journa of Operatona Research 72 (1994), pp [4] Caprara, A., Constraned 0-1 quadratc programmng: Basc approaches and extensons, European Journa of Operatona Research 187 (2008), pp [5] Ernst, A. T. and M. Krshnamoorthy, Effcent agorthms for the uncapactated snge aocaton p-hub medan probem, Locaton Scence 4 (1996), pp [6] Garey, M. R. and D. S. Johnson, Computers and Intractabty; A Gude to the Theory of NP-Competeness, W. H. Freeman & Co., New Yor, NY, [7] Hed, M. and R. Karp, The traveng-saesman probem and mnmum spannng trees: Part II, Mathematca Programmng 1 (1971), pp [8] Ić, A., D. Uroševć, J. Brmberg and N. Madenovć, A genera varabe neghborhood search for sovng the uncapactated snge aocaton p-hub medan probem, European Journa of Operatona Research 206 (2010), pp [9] O Key, M. E., A quadratc nteger program for the ocaton of nteractng hub factes, European Journa of Operatona Research 32 (1987), pp [10] Sorn-Kapov, D., J. Sorn-Kapov and M. O Key, Tght near programmng reaxatons of uncapactated p-hub medan probems, European Journa of Operatona Research 94 (1996), pp