an iterated tabu search heuristic for the single source capacitated facility location problem

A

AF

SQ1

D

a

ARRAA

KFHT

1

teittbomvw

dtTfi(prbsbti

h1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

ARTICLE IN PRESSG ModelSOC 2603 1–10

Applied Soft Computing xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l h o mepage: www.elsev ier .com/ locate /asoc

n iterated tabu search heuristic for the Single Source Capacitatedacility Location Problem

in C. Ho ∗

epartment of Economics and Business, Aarhus University, Fuglesangs Allé 4, 8210 Aarhus V, Denmark

r t i c l e i n f o

rticle history:eceived 16 May 2013eceived in revised form 29 August 2014

a b s t r a c t

This paper discusses the Single Source Capacitated Facility Location Problem (SSCFLP) where the problemconsists in determining a subset of capacitated facilities to be opened in order to satisfy the customers’demands such that total costs are minimized. The paper presents an iterated tabu search heuristic to

ccepted 5 November 2014vailable online xxx

eywords:acility locationeuristic

solve the SSCFLP. The iterated tabu search heuristic combines tabu search with perturbation operatorsto avoid getting stuck in local optima. Experimental results show that this simple heuristic is capableof generating high quality solutions using small computing times. Moreover, it is also competitive withother metaheuristic approaches for solving the SSCFLP.

© 2014 Elsevier B.V. All rights reserved.

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

abu search

. Introduction

One of the most critical decisions in logistics is to decide whereo locate the facilities (e.g., warehouses, plants, factories, retail-rs). This is a decision problem at the strategic level where themplemented decisions remain unchanged for a long period ofime. The decisions lead to long-term investments as companiesypically invest several millions or even billions of US dollars inuilding facilities in different regions or countries [1]. The problemf determining new locations for a set of facilities subject to costinimization is called the Facility Location Problem (FLP) [2–4]. A

ariant of the FLP is the Capacitated Facility Location Problem (CFLP),here the problem includes capacities for the facilities.

There are two levels of decisions involved with the CFLP, namelyetermining which facilities to open and allocating customers tohe opened facilities without violating the capacity constraints.he objective is to minimize the costs of opening the facilities (i.e.,xed costs) and the costs of supplying the customers with goodsi.e., variable costs). There are two versions of the CFLP – multi-le sourcing and single sourcing. The more popular one amongesearchers is the multiple sourcing version. When a customer cane supplied by more than one facility (i.e., multiple sourcing), theub-problem of the CFLP (i.e., the customer allocation problem)

Please cite this article in press as: S.C. Ho, An iterated tabu search heuAppl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.

ecomes a transportation problem and can be solved efficiently byhe transportation simplex algorithm. But if we restrict the possibil-ty of multiple sourcing the customer allocation problem becomes

∗ Tel.: +45 8716 4808.E-mail address: [email protected]

ttp://dx.doi.org/10.1016/j.asoc.2014.11.004568-4946/© 2014 Elsevier B.V. All rights reserved.

67

68

69

a generalized assignment problem which is an NP-hard problem.Hence, the CFLP with single sourcing is a more difficult problemto solve than the CFLP with multiple sourcing. In this paper, thefocus is on the Single Source Capacitated Facility Location Problem(SSCFLP). Despite its NP-hardness, the SSCFLP serves also as thebasis for other more complex problems, e.g., the location-routingproblem [5] and the bi-objective SSCFLP [6].

Different methods for solving the SSCFLP have been proposedin the literature. These include exact methods, lagrangian relax-ations, and heuristics. One of the first exact methods proposed tosolve the SSCFLP was developed by Neebe and Rao [7]. They for-mulated the problem as a set partitioning problem and solved it bya column-generating branch-and-bound procedure. As Lagrangianheuristics have shown to be very successful for solving Facility Loca-tion Problems, Holmberg et al. [8] combined a Lagrangian heuristicwith a repeated matching heuristic [9] in a branch-and-bound pro-cedure to speed up the bounding process. Few years later, Díazand Fernández [10] developed an effective branch-and-price algo-rithm to solve the SSCFLP. The most recent exact method is thecut-and-solve approach proposed by Yang et al. [11]. Their methodrelies on the lifted cover inequalities and Fenchel cutting planes tostrengthen the lower bound of the problem. Lagrangian relaxation-based techniques are very effective in solving the SSCFLP [12]. Thesetechniques usually differ in (i) which constraints are relaxed (i.e.,capacity or assignment constraints); and (ii) how a feasible solu-tion can be constructed from the solution of the relaxed problem. In

ristic for the Single Source Capacitated Facility Location Problem,004

Klincewicz and Luss [13], the capacity constraints are relaxed andthe relaxed sub-problem is solved by the dual-ascent algorithm.The Add heuristic is used to generate an initial feasible solution, anda final adjustment heuristic is used to improve the solution by the

70

71

72

73

dx.doi.org/10.1016/j.asoc.2014.11.004

dx.doi.org/10.1016/j.asoc.2014.11.004

http://www.sciencedirect.com/science/journal/15684946

www.elsevier.com/locate/asoc

mailto:[email protected]

dx.doi.org/10.1016/j.asoc.2014.11.004

ING ModelA

2 mput

rcrthscp

lsdamiutaGs[hrvbbACuat

sposscbOaifbhoseib

msmS

2

Jdftci

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

ARTICLESOC 2603 1–10

S.C. Ho / Applied Soft Co

elaxations. Hindi and Pienkosz [14] developed a heuristic whichombined Lagrangian relaxation (the assignment constraints areelaxed) with restricted neighborhood search. Cortinhal and Cap-ivo [15] also relaxed the assignment constraints. Their Lagrangianeuristic consists of two phases. The first phase finds solutions thatatisfy the assignment constraints (but not necessarily the capacityonstraints), and these solutions are further improved in the secondhase using local search and tabu search.

Delmaire et al. [16] proposed different heuristics based on evo-utionary algorithms, GRASP, and tabu search approaches. Tabuearch is embedded in a multi-start heuristic where a number ofifferent initial solutions are generated and the initial solutionsre improved using tabu search. Only neighborhoods which do notodify the set of the opened facilities are explored. Diversification

s ensured by selecting less-frequently made moves. In their follow-p paper, Delmaire et al. [17] proposed some improving algorithmshat are based on their previous work; a reactive GRASP heuristicnd some hybrid heuristics. These hybrid heuristics use reactiveRASP to construct a number of different initial solutions and theolutions are improved by tabu search (similar to the one in Ref.16]) and a local search procedure (which is a variable neighbor-ood descent procedure). These heuristics outperformed previousesults using small computation times. Ahuja et al. [18] proposed aery large-scale neighborhood search algorithm where the neigh-orhood structures are induced by customer multi-exchanges andy facility moves. Good results are obtained with their method.

scatter search heuristic for solving the SSCFLP was proposed byontreras and Díaz [19]. Their method generated good solutionssing a reasonable computing time. Chen and Ting [20] proposed

hybrid of Lagrangian heuristic and ant colony system for solvinghe SSCFLP. Near-optimal solutions were created with this hybrid.

In this paper, a different strategy is utilized to design the tabuearch heuristic. The tabu search procedure employs two sim-le neighborhood structures which re-allocate customers betweenpened and closed facilities. The re-allocation may lead to infeasibleolutions, which are penalized in the evaluating function. The bestolution from tabu search is then perturbed by one of the randomlyhosen neighborhood operators. The iterative procedure that com-ines tabu search and perturbation is named iterated tabu search.ne of the strengths of the proposed heuristic is its simplicity: thelgorithm is based on a simple search paradigm and relies on eas-ly reproducible mechanisms for constructing a solution, movingrom one solution to another, deciding whether a solution shoulde accepted or not, and terminating the algorithm. In addition, thiseuristic allows the search to be conducted in infeasible regionsf the search space. With the possibility of infeasible intermediateolutions, it is possible to obtain good solutions that may never bencountered if only feasible solutions were allowed. This heuristics also capable of identifying high quality solutions for two sets ofenchmark instances using small computing times.

The rest of the paper is organized as follows. A mathematical for-ulation of the SSCFLP is given in Section 2, while the iterated tabu

earch heuristic is described in Section 3. Computational experi-ents are provided in Section 4. Finally, a conclusion is drawn in

ection 5.

. Problem definition

Let there be a set I = {1, . . ., m} of potential locations, and a set = {1, . . ., n} of customers. Each customer j is associated with aemand, dj, that must be satisfied by a single facility. A fixed cost,

Please cite this article in press as: S.C. Ho, An iterated tabu search heuAppl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11

i, is incurred for opening a facility at location i. Also, let bi denotehe capacity of the facility located at site i. Moreover, let cij be theost of supplying the demand of customer j from a facility at site. The overall problem consists of opening a number of facilities

PRESSing xxx (2014) xxx–xxx

such that the customers’ demands are satisfied by these facilitiesat minimum cost.

For each location i ∈ I, we define the decision variable yi as

yi ={

1, if facility at locationiisopened;

0, otherwise.

For each location i ∈ I and customer j ∈ J, we define the decisionvariable xij as

xij ={

1, if customerjis assigned to a facility at location i;

0, otherwise.

The Single Source Capacitated Facility Location Problem can bestated mathematically as:

min∑i∈I

∑j∈J

cijxij +∑i∈I

fiyi (1)

s.t.∑i∈I

xij = 1 ∀j ∈ J (2)

∑j∈J

djxij ≤ biyi ∀i ∈ I (3)

yi ∈ {0, 1} ∀i ∈ I (4)

xij ∈ {0, 1} ∀i ∈ I, j ∈ J (5)

Constraint (2) ensures that every customer is served by exactlyone facility, while constraint (3) ensures that the total demand ofcustomers served by a facility does not exceed its capacity. Eqs.(4) and (5) are the integrality constraints. The objective (1) is tominimize the fixed costs and the costs of supplying the goods tothe customers.

3. Iterated tabu search

This section describes the Iterated Tabu Search (ITS) heuristicdeveloped for the SSCFLP. ITS can be seen as a special case of IteratedLocal Search (ILS) [21], in which the local search phase is replacedby a tabu search phase. At each iteration of the ITS heuristic, solu-tion s is perturbed resulting in solution s′, which is then improvedby tabu search to obtain solution s. If solution s satisfies an accep-tance criterion, the search continues with solution s, otherwise thesearch proceeds with solution s. The overall structure of the Iter-ated Tabu Search heuristic is depicted in Algorithm 1, where g(s)is a function for evaluating solution s, s∗ is the best known feasiblesolution encountered and g∗ is the value of s∗.

Algorithm 1. Iterated Tabu Search

1: s0 = InitialSolution()2: s = TabuSearch(s0)3: if s is feasible, set s∗ = s and g∗ = g(s∗); otherwise, set s∗ =∅ andg∗ =∞.4: for v = 1, . . ., � do5: s′ = Perturbation(s, �)6: s = TabuSearch(s′)7: if s is feasible and g(s) < g∗ then8: Set s∗ = s, g∗ = g(s) and � = 0.9: else10: Set � = � + 1.11: end if

ristic for the Single Source Capacitated Facility Location Problem,.004

12: If s is feasible and s satisfies the acceptance criterion, sets = s.13: end for14: return s∗

182

183

184

185

dx.doi.org/10.1016/j.asoc.2014.11.004

ING ModelA

mput

3

aattrm∑fcaa(

3

tfilno

3

nttq

aicavw

3

trcIcsta

cSw

3

�tmvTfsib

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299


S.C. Ho / Applied Soft Co

.1. Initial solution

The procedure for finding an initial solution is a two-stagepproach. In the first stage, a subset of I needs to be opened. Thellocation of customers to the opened facilities is determined inhe second stage. In order to determine which facility to open,he facilities are first sorted in an ascending order based on theatio fi/bi. Starting with the facility with lowest fi/bi we open asany facilities as necessary in order to satisfy total demand (i.e.,

j∈Jdj ≤∑

i∈Ibiyi). A customer is assigned to its nearest openedacility. This allocation might lead to some facilities violating theapacity constraints and/or some facilities with no customersssigned. An infeasible initial solution will not cause any problems,s the tabu search heuristic is designed to deal with infeasibilitysee Section 3.2.1).

.2. Tabu search

Tabu search [22] is an iterative process where at each itera-ion, the algorithm moves from solution s′ to the best solutionound in its neighborhood. In order to prevent the search from vis-ting previously visited solutions and to avoid getting stuck in aocal optimum, recently made moves are declared tabu for a givenumber of iterations. The tabu status of a certain move might beverridden when some aspiration criterion is satisfied.

.2.1. Infeasibility and cost functionsThe algorithm allows solutions to be infeasible during the

eighborhood search. A solution is infeasible if it violateshe capacity constraints of the facilities. The total viola-ion of capacity constraints in solution s′ is computed as(s′) =

∑i∈I max{0,

∑j∈Jdjxij − biyi}. Each solution s′ is evalu-

ted by an objective g(s′) = z(s′) + ˛q(s′). The cost function z(s′)s equal to

∑i∈Ifiyi +

∑i∈I

∑j∈Jcijxij of solution s′, while the

oefficient is a self-adjusting positive parameter that is modifiedt every iteration. Parameter is divided by 1 + � if there is noiolation of capacity constraints, otherwise it is multiplied by 1 + �,here � is a positive parameter.

.2.2. Neighborhood structuresThe neighborhood N1(s) for solution s consists of all solutions

hat can be transformed from s by relocating customer j from its cur-ent facility k to facility l, where k /= l. Facility l might be opened orlosed. If l is closed, it becomes opened after j has been assigned to l.n the case where k has only one customer assigned to it, it becomeslosed after j is assigned to l. This move is labeled (j, k, l). The tran-ition of j from k to l is forbidden if TABU(j, l) ≥ �, where TABU(j, l) ishe last iteration number in which this forbidden transition is stillctive and � is the current iteration number.

The neighborhood N2(s) for solution s is obtained by exchangingustomer j1 from facility k with customer j2 of facility l, with k /= l.uch a move is denoted by (j1, j2, k, l). The exchange of j1 from kith j2 from l is forbidden if TABU(j1, l) ≥ � and TABU(j2, k) ≥ �.

.2.3. Tabu tenures and aspiration criteriaOnce a move has been recorded tabu, it is declared tabu for

= 7 + m(a/u) iterations. The parameter a is the number of timeshe move from solution s to solution s has been made, and u is the

aximum value of a, for all moves made. This way of setting thealue of the tabu tenure is the same as in Cortinhal and Captivo [15].he purpose of recording the moves tabu is to prevent the search


rom visiting previously visited-solutions. However, this rule is nottrict as it can be overridden when some aspiration criterion is sat-sfied. Two aspiration criteria are used; one is g∗

1, the value of theest known feasible solution s∗, and the other is g∗

2, the value of

PRESSing xxx (2014) xxx–xxx 3

the best (feasible or infeasible) solution s∗. Both s∗ and s∗ refer tosolutions encountered in the current tabu search.

3.2.4. Acceleration of neighborhood searchIn order to accelerate the neighborhood search, only a fraction,

ˇ, of the neighbor solutions in N1(s) and N2(s) is evaluated. Thisis implemented as follows: Every solution s ∈ N1(s) ∪ N2(s) is ran-domly assigned a probability ps, and s is only evaluated if ps ≤ ˇ.

The resulting neighborhoods are denoted Nˇ1 (s) and Nˇ

2 (s), respec-tively. This also acts as a diversification mechanism in which lessexplored regions of the solution space might be sampled.

3.2.5. An algorithmic description of the tabu search procedureAn algorithmic description of the tabu search heuristic is shown

in Algorithm 2.

Algorithm 2. Tabu search

Require: Initial solution s01: Set s∗ = s0 and g∗

2 = g(s0). If s0 is feasible, set s∗ = s0 and g∗1 =

g(s0); otherwise, set s∗ = ∅ and g∗1 = ∞.

2: Set TABU(j, l) = −1 ∀ customers j and locations l.3: Set = 1.4: Set s = s0.5: for � = 1, . . ., � do6: Select solution s ∈ Nˇ

1 (s) ∪ Nˇ2 (s) that minimizes g(s) and is

non-tabu or satisfies the aspiration criterion (i.e., g(s) < g∗1 if fea-

sible or g(s) < g∗2 if infeasible).

7: If s is infeasible and ∃s ∈ Nˇ1 (s) ∪ Nˇ

2 (s) which is feasible andg(s) < g∗

1, set s∗ = s and g∗1 = g(s).

8: If g(s) < g∗2, set s∗ = s and g∗

2 = g(s).9: If s is feasible and g(s) < g∗

1, set s∗ = s and g∗1 = g(s).

10: Set the move from s to s tabu for � iterations (i.e., TABU(j,k) = � + � if (j, k, l) or TABU(j1, k) = � + � and TABU(j2, l) = � + � if (j1,j2, k, l)).11: Compute q(s) and update ˛.12: Set s = s.13: end for14: If s∗ /= ∅, return s∗; otherwise, return s∗.

3.3. Perturbation

Random perturbation mechanisms are applied to a solution toachieve search diversification. At iteration v of the ITS heuristic, oneof the following perturbation mechanisms is applied to solution s:

1. Close a facility A facility k with one customer (denoted as j)assigned to it is randomly chosen and closed down. The customerwill then be re-assigned to the cheapest opened facility l. LetIo = {i ∈ I : yi = 1}, then facility l is determined as arg min

i∈I0\{k}cij .

Facility k cannot be opened again in the next run of tabu search.2. Open a facility Randomly choose a closed facility l and open it.

Facility l cannot be closed down in the next run of tabu search.3. Close one facility and open one facility Randomly choose an

opened facility k and a closed facility l, and exchange their sta-tuses (i.e., close down facility k and open facility l). The exchangeshould not violate the total capacity constraint. Customers offacility k will then be re-assigned to facility l. Facility k cannotbe opened again during the tabu search and facility l cannot beclosed in the next run of tabu search.

4. Re-assign customers 1 Randomly re-assign all customers to theopened facilities. Note that the set of opened facilities does not


change with this perturbation operator.5. Re-assign customers 2 Let there be a set O2 consisting of �Io/2

randomly chosen opened facilities, and let set O1 denote theremaining opened facilities (i.e., O1 = Io\O2). Close the facilities

300

301

302

303

dx.doi.org/10.1016/j.asoc.2014.11.004

ARTICLE ING ModelASOC 2603 1–10

4 S.C. Ho / Applied Soft Comput

6

7

fiaHbo

iw

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

Fig. 1. The flowchart of the perturbation procedure.

in O2 and re-assign the customers in P = {j ∈ J : xij = 1 ∧ i ∈ O2}.Customer j ∈ P is re-assigned to facility l ∈ O1 that has enoughspare capacity to cover the demand dj, otherwise j is re-assignedto facility k = arg min i∈I\O1

fi and yk = 1.. Close a facility and open two facilities Let � =

∑j∈Jdj, =∑

i∈Ibiyi and Ic = I\Io.

If min{−fh + fr + fw : − bh + br + bw − �

≥0, h ∈ Io, r ∈ Ic, w ∈ Ic} < 0,

then facility k is closed down and facilities l1 and l2 are opened.The combination of opening and closing the facilities is deter-mined as follows.

(k, l1, l2) = arg minh∈Io,r∈Ic,w∈Ic

{−fh + fr + fw : − bh + br + bw − �≥0}.

Customers originally assigned to facility k will then be re-assigned to whichever facility (l1 or l2) is cheapest. Facility kcannot be opened, while facilities l1 and l2 cannot be closed inthe next run of tabu search.

. Open a facility and close two facilities

If min{fh − fr − fw : − br − bw + bh − �

≥0, h ∈ Ic, r ∈ Io, w ∈ Io} < 0,

then facility k is opened and facilities l1 and l2 are closed down.The combination of opening and closing the facilities is deter-mined as follows.

(k, l1, l2) = arg minh∈Ic,r∈Io,w∈Io

{fh − fr − fw : − br − bw + bh − �≥0}.

Customers originally assigned to facilities l1 and l2 will then bere-assigned to facility k. Facilities l1 and l2 cannot be opened,while facility k cannot be closed in the next run of tabu search.

Usually, perturbation is done by randomly applying one of therst five perturbation operators to a solution. However, these oper-tors might not always achieve the right diversification effect.ence, every consecutive iterations without improvement to theest known solution s∗, some stronger perturbation operators (i.e.,


perators 6 and 7) are used to perturb a solution (Fig. 1).It is vital that the perturbation applied does not get revoked

n the next run of the tabu search procedure. Hence, tabu searchill not consider any neighbor solutions obtained by reversing the


move applied by the perturbation operator (except for operators 4and 5) at iteration v of the ITS heuristic.

3.4. Acceptance criterion

A solution s in the ITS algorithm is accepted if s is feasible andg(s) < g∗(1 + ı) where ı ≥ 0. This acceptance criterion differs fromstandard acceptance criteria associated with ILS, where the com-parison is between the current solution s and its local optimum s.The implemented acceptance criterion is inspired from the Record-to-Record travel algorithm [23], where solution s is accepted if g(s)is not much worse than g∗.

4. Computational experiments

The ITS heuristic was coded in C++ and all computational exper-iments were carried out on a Dell notebook with an Intel Corei5-2520M CPU @ 2.5 GHz. Two sets of benchmark instances wereused to test the described iterated tabu search heuristic. The firstset includes the 57 instances available from http://www-eio.upc.edu/ elena/sscplp/. The number of customers in these instancesrange from 20 to 90, while the number of potential locations rangefrom 10 to 30. The second set of instances contains 71 instancesand is available for download from https://db.tt/PAwTKatL. Thenumber of customers in these instances range from 50 to 200 cus-tomers, and the number of potential locations range from 10 to30. The fundamental difference between the two sets of instancesis the relationship between the fixed costs (fi) and the assignmentcosts (cij). In the first set, fixed costs dominate the assignment costsresulting in solutions with a small set of opening facilities and notmuch slack for the capacity constraints. The situation in the secondset is completely different as the assignment costs dominate thefixed costs thus may lead to a larger set of opening facilities andmuch more slack for the capacity constraints.

4.1. Parameter settings

To tune the six parameters �, , ı, � , � and ˇ, a two-stageapproach is employed. First, the parameters �, and ı are jointlytested, and the values of the other parameters are fixed (i.e.,� = 100, � = 50 and = 1). Second, the remaining parameters � , �and are jointly tested, and the values of the other parametersare fixed from the first-stage tuning process. Table 1 shows theresults of the first-stage experiments of the first set of benchmarkinstances when � ∈ {0.1, 0.2, 0.3}, ∈ {10, 20} and ı ∈ {0.01, 0.02,0.03, 0.04, 0.05}. Every instance was run 20 times for each combi-nation of the three parameters. The column Best shows the averagedeviations of best solutions from the optimal solutions, while thecolumn Avg lists the average deviations of average solutions fromoptimal solutions. The table shows that when setting = 10 theresults are in general much better than when setting = 20. Thisparameter denotes how often operators 6 and 7 are applied inthe ITS heuristic. The usefulness of these operators is shown inTable 3. The average gaps do not vary much when = 10 which mayindicate that the heuristic may not be very sensitive to the testedvalues. However, it can be observed that setting � = 0.1, = 10 andı = 0.01 yields the best results. Hence, these parameters are fixedwith the chosen values for the second-stage tuning process.

In the second-stage tuning process, parameters � , � and arejointly tested. If � , the number of TS iterations, is set with a too lowvalue, then tabu search might not be capable of reaching promisingregions of the solution space. This is partly due to the infeasibility


of the solution caused by the perturbation operator in which feasi-bility might not be achieved if too few iterations of tabu search arerun. If feasibility is achieved, then tabu search needs some time inorder to reach good local minima. The parameter � denotes the

397

398

399

400

dx.doi.org/10.1016/j.asoc.2014.11.004

http://www-eio.upc.edu/~elena/sscplp/

http://www-eio.upc.edu/~elena/sscplp/

https://db.tt/PAwTKatL

ARTICLE IN PRESSG ModelASOC 2603 1–10

S.C. Ho / Applied Soft Computing xxx (2014) xxx–xxx 5

Table 1Results when varying �, and ı.

(�, ) ı = 0.01 ı = 0.02 ı = 0.03 ı = 0.04 ı = 0.05

Best Avg Best Avg Best Avg Best Avg Best Avg

(0.1,10) 0.085 0.464 0.090 0.471 0.093 0.474 0.092 0.508 0.090 0.473(0.2,10) 0.086 0.503 0.088 0.469 0.095 0.469 0.087 0.472 0.087 0.474(0.3,10) 0.092 0.473 0.086 0.480 0.097 0.494 0.084 0.492 0.092 0.475(0.1,20) 0.103 0.579 0.089 0.567 0.082 0.605 0.115 0.579 0.112 0.591(0.2,20) 0.113 0.581 0.078 0.562 0.105 0.557 0.102 0.585 0.077 0.574(0.3,20) 0.104 0.559 0.073 0.575 0.078 0.567 0.097 0.571 0.095 0.590

Table 2Results when varying �, � and ˇ.

(�, �) = 0.25 = 0.5 = 0.75 = 1

Best Avg CPU Best Avg CPU Best Avg CPU Best Avg CPU

(50,50) 0.364 1.721 1.46 0.249 1.403 2.14 0.215 1.213 2.81 0.245 1.277 3.24(100,50) 0.271 1.310 2.84 0.198 1.002 4.20 0.156 0.976 5.41 0.175 0.901 6.34(200,50) 0.252 1.065 5.48 0.114 0.748 8.23 0.080 0.620 10.74 0.123 0.652 12.41

4.18.1

16.0

npAtwtsotfnn

t1ttwfgvcFs((ehsr

TS

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

(50,100) 0.168 0.721 2.86 0.102 0.542

(100,100) 0.106 0.594 5.46 0.066 0.408

(200,100) 0.099 0.474 10.75 0.051 0.321

umber of ITS iterations. Obviously, the larger the value of thisarameter, the better the resulting solution quality of the heuristic.s a consequence, the computing time will soar. Hence, it is impor-

ant to find a balance between solution quality and computing timehen determining values for these parameters. The last parame-

er to impact both the solution quality and computing time is ˇ, theampling parameter. It might seem obvious that the larger the valuef ˇ, the better the resulting solution quality. However, computa-ional experiments show that better solution quality is achievedor = 0.75 than for = 1. This is due to the stochastic nature of theeighborhood sampling (where acts as a diversification mecha-ism) and the perturbation mechanism.

Table 2 shows the results of the computational experiments ofhe first set of benchmark instances when � ∈ {50, 100}, � ∈ {50,00, 200} and ∈ {0.25, 0.5, 0.75, 1}. The column CPU denoteshe average computing time (in seconds). As can be seen fromhe table better solution quality is achieved with = 0.75 thanith = 1, which indicates that proper randomness is beneficial

or diversification purposes when searching the neighborhood forood solutions. Table 2 also confirms that setting � with a lowalue and � with a large value does not necessarily outperform theases where � is set with larger values and � with smaller values.or example, setting � = 200 and � = 50 does not provide betterolution quality than for the heuristic using the combinations100,100) and (50,100). Employing combinations (200,50) and100,100) are approximately equivalent in terms of computationalffort. In contrast, setting � = 50 and � = 100 results in a fastereuristic than setting � = 200 and � = 50 but the former setting


till resulted in better solutions. The table also shows that the bestesults are obtained by setting � equal to 100, � equal to 100 and

able 3tatistics for the perturbation operators.

Operator Solution degradation (in %)

Close a facility 0.167Open a facility 0.149Close a facility, open a fcility 0.235Re-assign customers 1 0.098Re-assign customers 2 0.134Close one facility, open two facilities 0.224Open one facility, close two facilities 0.835

All 1.781

456

457

458

459

460

461

462

463

464

465

466

467

8 0.070 0.447 5.58 0.085 0.464 6.298 0.019 0.254 10.91 0.045 0.326 12.398 0.022 0.259 21.21 0.037 0.265 24.65

equal to 0.75. Hence, the computational results provided in theremainder of this paper are obtained using those values.

Table 3 shows the statistics on the different operators. The sec-ond column lists the average solution degradation of the solutionsfound from running the heuristic excluding the operator (listedat the same row) for all instances of the first set of benchmarkinstances. Every instance was run 20 times. This table shows thatoperator 7 (i.e., open one facility, close two facilities) is the mostuseful one. One reason for the usefulness could be that a “better” setof opened facilities is created when this operator is applied. As men-tioned earlier, fixed costs dominate the assignment costs in whichsolutions with a small set of opening facilities are preferred. Thisis achieved by operator 7. Operator 4 (i.e., re-assign customers 1)is the least useful operator among the seven operators. The reasonfor this could be that this operator does not lead to a different setof opened facilities, but it is still useful for diversification purposes.Perturbation is very useful for the iterated tabu search heuristic.When every operator is excluded from the heuristic, the averagedeviation degrades with 1.781%. As the tabu search procedure isnot complex, the ITS relies on the different perturbation operatorsfor diversification.

4.2. Computational results

The results of the first set of benchmark instances are depicted inseven tables (Tables A.1–A.7). In these tables, the first column indi-cates the problem, while the second and third columns denote thesize of the instance. The column Optimal lists the optimal solutionto each of the problems. The optimal solutions are obtained fromDíaz and Fernández [10]. The fifth column lists the best solution(out of 20 runs) obtained by the ITS heuristic, and the sixth columnis the deviation of the best solution from the optimal solution. TheAverage column is the average solution of 20 runs, while the follow-ing column is the average deviation of the average solution fromthe optimal solution. The last column is the average CPU time (inseconds). The results are obtained by setting � = 0.1, = 10, ı = 0.01,

= 0.75, � = 100 and � = 100 as good results were achieved usingreasonable computing time. For class C1 and class C2, the ITS heuris-tic obtained 17 optimal solutions out of 17 benchmark instances.


For the rest of the classes, the average deviations are less than 0.1.Table 4 provides a comparison of the published metaheuristics

applied on the 57 benchmark instances. GRASP is a multistart pro-cedure and RGRASP is a reactive GRASP. TS1 and TS2 are tabu search

468

469

470

471

dx.doi.org/10.1016/j.asoc.2014.11.004


6 S.C. Ho / Applied Soft Computing xxx (2014) xxx–xxx

Table 4Comparison of metaheuristics on the first set of benchmark instances.

Class GRASP RGRASP TS1 TS2 HB1 HB2 SS ITS

C1 0.235 0.071 0.144 0.000 0.000 0.000 0.019 0.0000.09 1.33 2.95 6.47 6.65 6.06 0.32 1.81

C2 0.482 0.406 0.441 0.394 0.000 0.008 0.210 0.0000.22 4.40 6.55 13.18 11.89 13.68 0.99 3.65

C3 0.293 0.124 0.078 0.024 0.000 0.004 0.064 0.0130.38 8.86 13.27 21.57 20.88 23.52 2.35 6.30

C4 0.500 0.345 0.295 0.046 0.026 0.038 0.390 0.0710.58 14.03 19.75 34.66 33.54 36.20 3.47 8.69

C5 n/a 0.059 n/a 0.007 0.004 0.007 0.051 0.00921.30 44.15 40.42 51.76 6.57 13.55

C6 n/a 0.182 n/a 0.044 0.030 0.017 0.228 0.02632.86 59.14 56.75 79.90 11.07 18.50

C7 n/a 0.132 n/a 0.098 0.094 0.093 0.152 0.01740.35 67.42 65.26 105.80 14.46 24.35

C1 − C7 0.396 0.204 0.264 0.107 0.022 0.024 0.167 0.019n/a n/a n/a n/a n/a n/a 5.18 10.91

N opt Computer 5 19 14 39 46 45 23 40

Computer VAX-200 system with an alpha server 2000 4/275 CPU Athlon2.1 GHz

Intel Core i52.5 GHz

Table 5Comparison of different heuristics on the second set of benchmark instances.

Class RM VLSN SS ALH ITS

C8 0.041 0.000 0.000 0.000 0.000C9 0.509 0.073 0.005 0.022 0.000C10 0.056 0.011 0.000 0.056 0.030C11 n/a 0.027 0.004 0.035 0.075C8 − C11 0.181 0.028 0.002 0.025 0.023Time 31.21 6.91 20.07 11.96 44.07N opt 40 53 68 60 61

pomiIopt(hotOepatecl

TR

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

Computer CPU DEC2000300AXP

Athlon1.2 GHz

rocedures, while HB1 and HB2 are hybrid heuristics. Descriptionsf these methods can be found in Delmaire et al. [16] and Del-aire et al. [17]. SS denotes the Scatter Search heuristic described

n Contreras and Díaz [19]. The last column lists the results of theTS method. Table 4 lists the average gaps of best solutions fromptimal solutions and average running times (in seconds) for everyroblem class as well as for all 57 instances. This table also giveshe number of optimal solutions found by the different heuristicsi.e., N opt). ITS generates overall better solutions than any of theeuristics, but HB1 and HB2 managed to identify 6 and 5 moreptimal solutions than the ITS heuristic. The ITS heuristic findshe optimal solution to 70% of the first set of problem instances.ptimal solutions to these problem instances are difficult to find,specially for the smaller-sized instances. The hardness of theseroblem instances is also indicated by Delmaire et al. [17, p. 220]nd Contreras and Díaz [19, p. 81]. Comparing running times ofhe different metaheuristics is not an easy task due to the differ-


nt computers, operating systems, compilers, data structures, andomputer languages used. It might seem that the ITS heuristic is aittle slower compared to the other methods. However, the reported

able A.1esults for class C1.

Problem m n Optimal Best

p1 10 20 2014 2014

p2 10 20 4251 4251

p3 10 20 6051 6051

p4 10 20 7168 7168

p5 10 20 4551 4551

p6 10 20 2269 2269

Average

Athlon2.1 GHz

Athlon XP1.83 GHz

Intel Core i52.5 GHz

computing times seem reasonable and acceptable for this type ofproblem.

Results for the second set of problem instances are listed infour tables (Tables A.8–A.11) and are obtained with the same setof parameter settings as described earlier. These tables have thesame format as Tables A.1–A.7. The optimal solutions are obtainedfrom Holmberg et al. [8] and Ahuja et al. [18]. For classes C8 and C9,the ITS heuristic obtained 40 optimal solutions out of 40 benchmarkinstances. For the remaining two classes, the average deviations are0.03 and 0.075, respectively.

Table 5 provides a comparison of different heuristics applied onthe second set of benchmark instances. RM is a repeated match-ing heuristic [9], VLSN denotes the very large-scale neighborhoodsearch heuristic [18], SS is the scatter search heuristic [19], ALH isthe hybrid heuristic of ant colony and lagrangian relaxation [20],and lastly ITS is our ITS heuristic. The table lists the average devi-ations of best solutions from optimal solutions for every problem


class as well as for the 71 instances. Time denotes the average com-puting times (in seconds) for the 71 benchmark instances and notfor each individual class. The ITS heuristic performs quite well for

Gap Average Gap CPU

0.000 2015.00 0.050 1.820.000 4257.00 0.141 1.780.000 6053.40 0.040 1.820.000 7179.70 0.163 1.880.000 4563.00 0.264 1.780.000 2271.40 0.106 1.75

0.000 0.127 1.81

509

510

511

dx.doi.org/10.1016/j.asoc.2014.11.004

Please cite this article in press as: S.C. Ho, An iterated tabu search heuristic for the Single Source Capacitated Facility Location Problem,Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.004



Table A.2Results for class C2.

Problem m n Optimal Best Gap Average Gap CPU

p7 15 30 4366 4366 0.000 4382.65 0.381 4.03p8 15 30 7926 7926 0.000 7952.20 0.331 3.72p9 15 30 2480 2480 0.000 2480.35 0.014 3.41p10 15 30 23,112 23,112 0.000 23,113.10 0.005 3.50p11 15 30 3447 3447 0.000 3469.35 0.648 3.40p12 15 30 3711 3711 0.000 3720.00 0.243 3.78p13 15 30 3760 3760 0.000 3770.05 0.267 3.63p14 15 30 5965 5965 0.000 6001.00 0.604 3.98p15 15 30 7816 7816 0.000 7827.00 0.141 3.35p16 15 30 11,543 11,543 0.000 11,558.70 0.136 3.35p17 15 30 9884 9884 0.000 9923.60 0.401 4.00

Average 0.000 0.288 3.65

Table A.3Results for class C3


p18 20 40 15,607 15,607 0.000 15,718.15 0.712 5.76p19 20 40 18,683 18,683 0.000 18,690.35 0.039 5.87p20 20 40 26,561 26,578 0.064 26,609.30 0.182 5.55p21 20 40 7295 7298 0.041 7341.60 0.639 5.48p22 20 40 3271 3271 0.000 3271.90 0.028 6.63p23 20 40 6036 6036 0.000 6089.80 0.891 6.85p24 20 40 6327 6327 0.000 6369.75 0.676 6.96p25 20 40 8947 8947 0.000 8954.05 0.079 7.25

Average 0.013 0.406 6.30



p26 20 50 4448 4448 0.000 4462.85 0.334 9.34p27 20 50 10,921 10,921 0.000 10,924.25 0.030 10.04p28 20 50 11,117 11,117 0.000 11,121.60 0.041 8.48p29 20 50 9832 9832 0.000 9838.55 0.067 9.13p30 20 50 10,816 10,868 0.481 10,903.55 0.809 7.40p31 20 50 4466 4467 0.022 4515.60 1.111 8.10p32 20 50 9881 9881 0.000 9923.25 0.428 9.63p33 20 50 39,463 39,489 0.066 39,583.00 0.304 7.42

Average 0.071 0.390 8.69



p34 30 60 4701 4701 0.000 4704.50 0.074 14.44p35 30 60 5456 5456 0.000 5467.70 0.214 14.13p36 30 60 16,781 16,789 0.048 16,814.55 0.200 13.23p37 30 60 14,668 14,668 0.000 14,705.25 0.254 14.05p38 30 60 47,249 47,249 0.000 47,259.80 0.023 11.61p39 30 60 41,007 41,016 0.022 41,043.30 0.089 12.10p40 30 60 61,633 61,636 0.005 61,679.20 0.075 12.03p41 30 60 17,246 17,246 0.000 17,246.00 0.000 16.80

Average 0.009 0.116 13.55



p42 30 75 7887 7887 0.000 7891.35 0.055 18.80p43 30 75 5114 5114 0.000 5150.45 0.713 20.22p44 30 75 36,022 36,064 0.117 36,185.95 0.455 16.02p45 30 75 17,676 17,676 0.000 17,701.35 0.143 18.72p46 30 75 48,701 48,729 0.057 48,761.45 0.124 17.93p47 30 75 66,230 66,233 0.005 66,252.50 0.034 18.42p48 30 75 58,964 58,971 0.012 59,041.30 0.131 20.14p49 30 75 79,614 79,631 0.021 79,707.10 0.117 17.74

Average 0.026 0.222 18.50

dx.doi.org/10.1016/j.asoc.2014.11.004


8 S.C. Ho / Applied Soft Computing xxx (2014) xxx–xxx



p50 30 90 5937 5937 0.000 5950.20 0.222 27.39p51 30 90 9060 9060 0.000 9098.90 0.429 25.15p52 30 90 34,652 34,661 0.026 34,680.50 0.082 23.07p53 30 90 30,038 30,043 0.017 30,053.50 0.052 22.75p54 30 90 43,853 43,854 0.002 43,863.35 0.024 25.44p55 30 90 69,610 69,674 0.092 70,059.95 0.646 21.84p56 30 90 64,474 64,474 0.000 64,479.90 0.009 24.42p57 30 90 49,791 49,791 0.000 49,791.90 0.002 24.75

Average 0.017 0.183 24.35



hp1 10 50 8848 8848 0.000 8851.75 0.042 7.05hp2 10 50 7913 7913 0.000 7914.35 0.017 6.64hp3 10 50 9314 9314 0.000 9324.35 0.111 7.07hp4 10 50 10,714 10,714 0.000 10,779.25 0.609 7.33hp5 10 50 8838 8838 0.000 8838.00 0.000 6.59hp6 10 50 7777 7777 0.000 7777.00 0.000 6.40hp7 10 50 9488 9488 0.000 9492.25 0.045 6.60hp8 10 50 11,088 11,088 0.000 11,088.00 0.000 6.66hp9 10 50 8462 8462 0.000 8462.75 0.009 7.11hp10 10 50 7617 7617 0.000 7617.90 0.012 6.55hp11 10 50 8932 8932 0.000 8941.50 0.106 7.19hp12 10 50 10,132 10,132 0.000 10,167.35 0.349 7.38hp13 20 50 8252 8252 0.000 8255.25 0.039 8.58hp14 20 50 7137 7137 0.000 7138.35 0.019 8.53hp15 20 50 8808 8808 0.000 8873.45 0.743 8.84hp16 20 50 10,408 10,408 0.000 10,425.00 0.163 8.28hp17 20 50 8227 8227 0.000 8230.35 0.041 8.65hp18 20 50 7125 7125 0.000 7137.60 0.177 8.74hp19 20 50 8886 8886 0.000 8893.85 0.088 8.97hp20 20 50 10,486 10,486 0.000 10,504.50 0.176 8.35hp21 20 50 8068 8068 0.000 8068.00 0.000 8.97hp22 20 50 7092 7092 0.000 7093.40 0.020 8.35hp23 20 50 8746 8746 0.000 8753.40 0.085 8.97hp24 20 50 10,273 10,273 0.000 10,328.00 0.535 9.26

titt

f

TR

512

513

514

515

516

517

518

519

Average

his set of instances, but it is slower than all the methods listedn Table 5. However, the ITS heuristic has found the optimal solu-


ion to 61 instances out of 71 instances using reasonable computingime.

As can be seen from Tables 4 and 5, the ITS heuristic’s per-ormance on the two sets of benchmark instances is quite good,

able A.9esults for class C9.

Problem m n Optimal Best

hp25 30 150 11,630 11,630

hp26 30 150 10,771 10,771

hp27 30 150 12,322 12,322

hp28 30 150 13,722 13,722

hp29 30 150 12,371 12,371

hp30 30 150 11,331 11,331

hp31 30 150 13,331 13,331

hp32 30 150 15,331 15,331

hp33 30 150 11,629 11,629

hp34 30 150 10,632 10,632

hp35 30 150 12,232 12,232

hp36 30 150 13,832 13,832

hp37 30 150 11,258 11,258

hp38 30 150 10,551 10,551

hp39 30 150 11,824 11,824

hp40 30 150 13,024 13,024

Average

0.000 0.141 7.82

especially on the first set where the ITS heuristic outperforms HB1and HB2. To the best of our knowledge, Contreras and Díaz [19] is


the only published paper which also considered the same two setsof benchmark instances. In that paper, the scatter search heuris-tic did extremely well on the second set of instances, while itsperformance on the first set of instances is only moderate.

Gap Average Gap CPU

0.000 11,683.75 0.462 59.020.000 10,774.70 0.034 58.660.000 12,340.55 0.151 61.200.000 13,763.85 0.305 62.480.000 12,433.15 0.502 54.710.000 11,361.10 0.266 55.580.000 13,348.30 0.130 56.960.000 15,337.30 0.041 57.290.000 11,705.40 0.657 59.730.000 10,691.65 0.561 59.470.000 12,260.65 0.234 60.380.000 13,873.75 0.302 61.090.000 11,262.90 0.044 66.240.000 10,584.10 0.314 65.780.000 11,827.50 0.030 66.290.000 13,080.90 0.437 67.55

0.000 0.279 60.78

520

521

522

523

dx.doi.org/10.1016/j.asoc.2014.11.004





hp41 10 90 6589 6589 0.000 6650.55 0.934 20.26hp42 20 80 5663 5665 0.035 5715.90 0.934 20.72hp43 30 70 5214 5214 0.000 5320.20 2.037 20.22hp44 10 90 7028 7028 0.000 7059.45 0.447 18.79hp45 20 80 6251 6251 0.000 6337.90 1.390 19.37hp46 30 70 5651 5669 0.319 5770.80 2.120 19.01hp47 10 90 6228 6228 0.000 6228.00 0.000 19.19hp48 20 80 5596 5598 0.036 5674.95 1.411 19.28hp49 30 70 5302 5305 0.057 5384.00 1.547 18.94hp50 10 100 8741 8741 0.000 8794.70 0.614 24.06hp51 20 100 7414 7414 0.000 7479.05 0.877 29.01hp52 10 100 9178 9178 0.000 9178.30 0.003 23.10hp53 20 100 8531 8531 0.000 8552.80 0.256 27.85hp54 10 100 8777 8777 0.000 8796.70 0.224 22.81hp55 20 100 7654 7654 0.000 7733.20 1.035 27.70

Average 0.030 0.922 22.02

Table A.11Results for class C11


hp56 30 200 21,103 21,144 0.194 21,302.80 0.947 94.36hp57 30 200 26,039 26,089 0.192 26,342.50 1.166 96.95hp58 30 200 37,239 37,288 0.132 37,498.85 0.698 95.43hp59 30 200 27,282 27,392 0.403 27,689.50 1.494 93.02hp60 30 200 20,534 20,534 0.000 20,633.30 0.484 100.53hp61 30 200 24,454 24,454 0.000 24,638.05 0.753 103.12hp62 30 200 32,643 32,730 0.267 32,812.20 0.518 103.26hp63 30 200 25,105 25,105 0.000 25,192.65 0.349 103.73hp64 30 200 20,530 20,530 0.000 20,557.30 0.133 105.14hp65 30 200 24,445 24,445 0.000 24,580.65 0.555 107.58hp66 30 200 31,415 31,415 0.000 31,603.25 0.599 114.00hp67 30 200 24,848 24,848 0.000 24,896.00 0.193 104.34hp68 30 200 20,538 20,538 0.000 20,664.60 0.616 105.36hp69 30 200 24,532 24,532 0.000 24,683.15 0.616 105.59hp70 30 200 32,321 32,323 0.006 32,755.00 1.343 106.51hp71 30 200 25,540 25,540 0.000 25,766.05 0.885 100.05

5

tmamTtdtosmoitt

A

2t

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

Average

. Conclusion

In this paper, we presented an iterated tabu search heuristic forhe Single Source Capacitated Facility Location Problem. The imple-

ented heuristic makes use of randomized neighborhood samplingnd perturbation to obtain diversification in the search. Experi-ents have been conducted on two sets of benchmark instances.

he heuristic finds the optimal solution in 40 out of 57 instances ofhe first set of benchmark instances and yields the lowest averageeviation (0.019%) among eight metaheuristic approaches. Further,he heuristic also manages to obtain the optimal solution in 61ut of 71 instances of the second set of benchmark instances. Thishows that the simple iterated tabu search heuristic produces opti-al and near-optimal solutions, and it is also competitive with

ther heuristics found in the literature. Future work will be onmproving the tabu search algorithm and on investigating whetherhe path-relinking procedure can be successfully combined withhe iterated tabu search heuristic.


cknowledgments

This work was partially supported by NordForsk under grant5900. This support is gratefully acknowledged. Thanks are also dueo the four anonymous reviewers for their constructive comments.

[

[

567

0.075 0.709 102.44

Results on the benchmark instances

References

[1] H. Timmons, Ford to increase its plant capacity in India, 2011 http://www.nytimes.com/2011/07/28/business/ford-to-increase-its-plant-capacity-in-india.html (accessed 15.05.13).

[2] A. Klose, A. Drexl, Facility location models for distribution system design, Eur.J. Oper. Res. 162 (1) (2005) 4–29.

[3] M.T. Melo, S. Nickel, F. Saldanha-da-Gama, Facility location and supply chainmanagement – a review, Eur. J. Oper. Res. 196 (2) (2009) 401–412.

[4] H.A. Eiselt, V. Marianov, Foundations of Location Analysis, Springer, 2011.[5] V.F. Yu, S.-W. Lin, Multi-start simulated annealing heuristic for the location

routing problem with simultaneous pickup and delivery, Appl. Soft Comput.24 (2014) 284–290.

[6] I. Harris, C.L. Mumford, M.M. Naim, A hybrid multi-objective approach to capac-itated facility location with flexible store allocation for green logistics modeling,Transp. Res. E 66 (2014) 1–22.

[7] A.W. Neebe, M.R. Rao, An algorithm for the fixed-charge assigning users tosources problem, J. Oper. Res. Soc. 34 (11) (1983) 1107–1113.

[8] K. Holmberg, M. Rönnqvist, D. Yuan, An exact algorithm for the capacitatedfacility location problems with single sourcing, Eur. J. Oper. Res. 113 (3) (1999)544–559.

[9] M. Rönnqvist, S. Tragantalerngsak, J. Holt, A repeated matching heuristic forthe single-source capacitated facility location problem, Eur. J. Oper. Res. 116(1) (1999) 51–68.


10] J.A. Díaz, E. Fernández, A branch-and-price algorithm for the single sourcecapacitated plant location problem, J. Oper. Res. Soc. 53 (7) (2002)728–740.

11] Z. Yang, F. Chu, H. Chen, A cut-and-solve based algorithm for the single-sourcecapacitated facility location problem, Eur. J. Oper. Res. 221 (3) (2012) 521–532.

568

569

570

571

572

dx.doi.org/10.1016/j.asoc.2014.11.004

http://www.nytimes.com/2011/07/28/business/ford-to-increase-its-plant-capacity-in-india.html



ING ModelA

1 mput

[

[

[

[

[

[

[

[

[

[

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596


0 S.C. Ho / Applied Soft Co

12] R.D. Galv ao, V. Marianov, Lagrangean relaxation-based techniques for solvingfacility location problems, in: H.A. Eiselt, V. Marianov (Eds.), Foundations ofLocation Analysis, Springer, 2011, pp. 391–420.

13] J.G. Klincewicz, H. Luss, A lagrangian relaxation heuristic for capacitated facilitylocation with single-source constraints, J. Oper. Res. Soc. 37 (5) (1986) 495–500.

14] K.S. Hindi, K. Pienkosz, Efficient solution of large scale, single-source, capaci-tated plant location problems, J. Oper. Res. Soc. 50 (3) (1999) 268–274.

15] M.J. Cortinhal, M.E. Captivo, Upper and lower bounds for the single sourcecapacitated location problem, Eur. J. Oper. Res. 151 (2) (2003) 333–351.

16] H. Delmaire, J.A. Díaz, E. Fernández, M. Ortega, Comparing new heuristics for the


pure integer capacitated plant location problem, Invest. Oper. 8 (1–3) (1999)217–242.

17] H. Delmaire, J.A. Díaz, E. Fernández, M. Ortega, Reactive GRASP and tabu searchbased heuristics for the single source capacitated plant location problem, INFOR37 (3) (1999) 194–225.

[

[


18] R.K. Ahuja, J.B. Orlin, S. Pallottino, M.P. Scaparra, M.G. Scutellà, A multi-exchange heuristic for the single-source capacitated facility location problem,Manage. Sci. 50 (6) (2004) 749–760.

19] I.A. Contreras, J.A. Díaz, Scatter search for the single source capacitated facilitylocation problem, Ann. Oper. Res. 157 (1) (2008) 73–89.

20] C.-H. Chen, C.-J. Ting, Combining lagrangian heuristic and ant colony system tosolve the single source capacitated facility location problem, Transp. Res. E 44(6) (2008) 1099–1122.

21] H.R. Lourenc o, O. Martin, T. Stützle, Iterated local search, in: F. Glover, G.Kochenberger (Eds.), Handbook of Metaheuristics, Kluwer Academic Publish-


ers, 2003, pp. 321–353.22] F. Glover, Future paths for integer programming and links to artificial intelli-

gence, Comput. Oper. Res. 13 (5) (1986) 533–549.23] G. Dueck, New optimization heuristics: the great deluge algorithm and the

record-to-record travel, J. Comput. Phys. 104 (1) (1993) 86–92.

597

598

599

600

601

dx.doi.org/10.1016/j.asoc.2014.11.004

an iterated tabu search heuristic for the single source capacitated facility location problem

Documents