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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