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Table 5 Experimental results for ADVRP instances with \(D_{{{{{\text{max}}}}}(2)}=0.9 \times LT(1)\)

From: A hybrid metaheuristic for solving asymmetric distance-constrained vehicle routing problem

# vertices

# vehicles

p

OPT (M5SBB)

VNS

HVT

Best.Sol

\(\hbox {Gap}_1\)

Time

Best.Sol

Aver.Sol

\(\hbox {Gap}_1\)

Gap Best-to-Best

Time

40

2

1

inf

inf

inf

inf

157

157

inf

inf

0.45

40

2

2

inf

inf

inf

inf

214

214

inf

inf

0.44

40

2

3

(180)

188

4.44

100

176

176

2.22

− 6.38

0.47

40

2

4

187

202

8.02

100

187

187

0.00

− 7.43

0.47

40

4

1

(171)

177

3.51

100

169

169

1.17

− 4.52

0.43

40

4

2

224

228

1.79

100

224

224

0.00

− 1.75

0.45

40

4

4

inf

inf

inf

inf

223

223

inf

inf

0.46

60

3

2

222

241

8.56

100.05

230

230

3.60

− 4.56

1.54

60

3

3

192

213

10.94

100

198

198

3.13

− 7.04

1.46

60

3

4

205

218

6.34

100.03

205

205

0.00

− 5.96

1.46

60

6

1

203

217

6.9

100.02

215

215

5.91

− 0.92

1.53

60

6

2

(254)

265

4.33

100.06

252

252

0.79

− 4.91

1.57

60

6

3

(228)

234

2.63

100.05

221

221

3.07

− 5.56

1.6

60

6

4

238

254

6.72

100.02

238

238

0.00

− 6.30

1.45

80

4

1

198

228

15.15

100.05

219

219

10.61

− 3.95

4.56

80

4

2

237

260

9.7

100.02

249

249

5.06

− 4.23

4.59

80

4

3

211

233

10.43

100.03

211

211

0.00

− 9.44

4.45

80

4

4

215

250

16.28

100.03

215

215

0.00

− 14.00

4.29

80

8

1

231

252

9.09

100.03

239

239

3.46

− 5.16

4.35

80

8

2

(281)

302

7.47

100.05

276

276

1.78

− 8.61

4.66

80

8

3

242

255

5.37

100.2

261

261

7.85

2.35

4.32

80

8

4

238

251

5.46

100.19

238

238

0.00

− 5.18

4.62

100

5

2

256

287

12.11

100.17

279

279

8.98

− 2.79

17.6

100

10

1

232

259

11.64

100.08

238

238

2.59

− 8.11

16.36

100

10

2

309

331

7.12

100.08

315

315

1.94

− 4.83

16.53

100

10

3

270

292

8.15

100.23

271

271

0.37

− 7.19

16.29

100

10

4

255

276

8.24

100.22

279

279

9.41

1.09

16.27

120

12

1

269

292

8.55

100.56

282

282

4.83

− 3.42

73.66

120

12

2

329

inf

inf

inf

331

331

0.61

inf

73.93

120

12

3

277

309

11.55

100.06

291

291

5.05

− 5.83

71.98

120

12

4

308

339

10.06

100.02

339

339

10.06

0.00

70.8

140

14

1

280

317

13.21

100.38

288

288.6

2.86

− 9.15

84.4

140

14

2

334

360

7.78

100.11

342

342.8

2.40

− 5.00

88.34

140

14

3

283

324

14.49

100.17

307

307.3

8.48

− 5.25

85.73

140

14

4

326

346

6.13

100.31

346

346.7

6.13

0.00

83.88

160

16

1

309

354

14.56

100.23

347

348.9

12.30

− 1.98

97.72

160

16

2

(365)

inf

inf

inf

346

347.6

5.21

inf

97.76

160

16

3

310

345

11.29

100.19

324

324.9

4.52

− 6.09

99.32

180

18

2

(392)

inf

inf

inf

369

374.6

5.87

inf

110.53

180

18

4

347

386

11.24

100.27

386

368.6

11.24

0.00

110.82

200

10

2

322

376

16.77

101.25

376

376.8

16.77

0.00

121.22

200

20

3

358

401

12.01

100.41

401

409.8

12.01

0.00

116.4