Hybridized Evolutionary Local Search Algorithm for the Team

Hybridized Evolutionary Local Search Algorithm for the Team

Journal of Heuristics manuscript No.
(will be inserted by the editor)
Hybridized Evolutionary Local Search Algorithm for the Team Orienteering
Problem with Time Windows - Appendix
Nacima Labadie · Jan Melechovský · Roberto Wolfler Calvo
List of Tables
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Results of GRASP-ELS on Solomon’s instances with m = 1. . . . . . . . . . . . .
Results of GRASP-ELS on Solomon’s instances with m = 2. . . . . . . . . . . . .
Results of GRASP-ELS on Solomon’s instances with m = 3. . . . . . . . . . . . .
Results of GRASP-ELS on Solomon’s instances with m = 4. . . . . . . . . . . . .
Results of GRASP-ELS on Cordeau’s instances with m = 1. . . . . . . . . . . . .
Results of GRASP-ELS on Cordeau’s instances with m = 2. . . . . . . . . . . . .
Results of GRASP-ELS on Cordeau’s instances with m = 3. . . . . . . . . . . . .
Results of GRASP-ELS on Cordeau’s instances with m = 4. . . . . . . . . . . . .
Comparison of results for the new data set of Solomon’s instances. . . . . . . . . .
Results of GRASP-ELS and ILS for the new data set of Solomon’s instances. . . .
Results of GRASP-ELS, VNS and ILS for the new data set of Cordeau’s instances.
Comparison of results for Solomon’s instances with n = 50 and m = 1. . . . . . . .
Comparison of results for Solomon’s instances with n = 50 and m = 2. . . . . . . .
Comparison of results for Cordeau’s instances with m = 1 . . . . . . . . . . . . . .
Comparison of results for Solomon’s instances with n = 100 and m = 1. . . . . . .
Comparison of results for Cordeau’s instances with m = 2 . . . . . . . . . . . . . .
Comparison of results for Solomon’s instances with n = 100 and m = 2. . . . . . .
N. Labadie · J. Melechovský (B)
Institute Charles Delaunay, University of Technology of Troyes,
BP 2060, 10010 Troyes Cedex, France
E-mail: {nacima.labadi;jan.melechovsky}@utt.fr
R. Wolfler Calvo
Laboratoire d’Informatique de Paris-Nord, University Paris 13
99, Avenue J.-B. Clement, 93430 Villetaneuse, France
E-mail: [email protected]
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
3
4
5
6
6
7
7
8
9
9
10
11
11
12
12
13
Table 7 Results of GRASP-ELS on Solomon’s instances with m = 1.
2
Instance
BK
ACS
ILS
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c101
c102
c103
c104
c105
c106
c107
c108
c109
320
360
400
420
340
340
370
370
380
320
360
400
420
340
340
370
370
380
320
360
390
400
340
340
360
370
380
320
360
400
420
340
340
370
370
380
320.0
360.0
400.0
420.0
340.0
340.0
370.0
370.0
380.0
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
198
286
293
303
247
293
299
308
277
284
297
298
198
286
293
303
247
293
299
308
277
284
297
298
182
286
286
297
247
293
288
297
276
281
295
295
198
286
290
302
247
293
296
304
277
282
297
297
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
219
266
266
301
244
252
277
298
219
266
266
301
244
252
277
298
219
259
265
297
221
239
274
288
219
259
265
297
244
252
277
298
Time(s)
Instance
BK
ACS
ILS
Min
Avg
Max
320
360
400
420
340
340
370
370
380
320
360
400
420
340
340
370
370
380
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.1
15.6
27.2
73.2
2.3
3.5
3.5
6.1
23.4
1.2
16.5
38.5
105.2
2.3
3.8
3.7
6.6
25.5
198.0
286.0
290.4
302.8
247.0
293.0
296.4
306.8
277.0
283.6
297.0
297.2
198
286
291
303
247
293
297
308
277
284
297
298
198
286
291
303
247
293
297
308
277
284
297
298
0.0
0.0
0.9
0.1
0.0
0.0
0.9
0.4
0.0
0.1
0.0
0.3
0.8
2.2
4.0
4.3
1.2
2.8
3.7
4.4
2.6
2.5
4.0
4.1
219.0
259.0
265.2
300.2
244.0
252.0
277.0
298.0
219
259
266
301
244
252
277
298
219
259
266
301
244
252
277
298
0.0
2.6
0.3
0.3
0.0
0.0
0.0
0.0
0.5
1.9
2.5
2.3
1.3
1.1
1.7
2.5
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
1.3
17.8
52.5
179.8
2.4
4.4
3.9
7.1
27.1
c201
c202
c203
c204
c205
c206
c207
c208
870
930
960
970
910
920
920
950
870
930
960
970
910
920
920
950
840
910
940
950
900
910
910
930
870
910
960
970
900
920
930
940
870.0
910.0
960.0
970.0
906.0
926.0
930.0
942.0
0.9
2.3
4.8
5.1
1.2
3.3
4.8
5.0
2.7
2.8
4.6
4.6
0.9
2.6
5.3
5.7
1.2
4.1
5.8
5.8
2.8
3.0
5.4
5.0
r201
r202
r203
r204
r205
r206
r207
r208
r209
r210
r211
797
903
993
1073
949
1008
1038
1071
938
970
1023
797
903
993
1053
949
1008
1035
1071
938
970
1016
788
880
980
1073
931
996
1038
1069
926
958
1023
784
900
994
1065
911
1020
1042
1095
934
958
1024
0.6
2.1
2.7
3.1
1.4
1.2
1.9
2.9
0.7
2.2
3.0
3.5
1.5
1.3
2.1
3.9
rc201
rc202
rc203
rc204
rc205
rc206
rc207
rc208
795
932
979
1117
855
888
967
1040
795
932
979
1107
855
888
967
1040
780
882
960
1117
840
860
926
1037
777
919
970
1090
839
865
960
993
Time(s)
Min
Avg
Max
870
910
960
970
910
930
930
950
870
910
960
970
910
930
930
950
0.0
2.2
0.0
0.0
0.4
-0.7
-1.1
0.8
2.5
17.4
41.4
122.4
6.9
11.0
16.1
16.7
2.5
20.4
45.2
130.3
10.6
13.4
17.3
17.7
2.6
24.2
51.7
135.9
21.5
16.1
19.7
19.0
788.2
909.8
1001.4
1071.8
927.2
1023.6
1049.2
1100.2
937.4
969.0
1028.2
792
925
1010
1077
953
1029
1061
1103
943
987
1036
792
929
1021
1086
953
1029
1061
1109
949
987
1046
1.1
-0.8
-0.8
0.1
2.3
-1.5
-1.1
-2.7
0.1
0.1
-0.5
4.8
9.2
12.3
14.4
6.7
9.4
11.8
13.6
9.6
9.1
9.5
5.3
9.6
12.6
15.2
7.8
10.7
13.5
15.9
10.3
10.8
11.3
6.3
10.0
13.2
16.2
8.6
11.6
14.5
16.8
10.7
12.3
12.7
786.4
922.8
978.2
1096.0
844.6
886.8
965.4
1007.0
795
924
986
1103
853
895
970
1024
795
934
995
1132
859
895
977
1041
1.1
1.0
0.1
1.9
1.2
0.1
0.2
3.2
3.4
8.5
9.1
11.7
6.5
5.7
6.6
10.2
4.2
8.9
9.6
11.9
6.8
6.1
7.5
10.7
4.6
9.2
10.2
12.0
7.0
6.4
8.3
11.1
Table 8 Results of GRASP-ELS on Solomon’s instances with m = 2.
3
Instance
BK
ACS
ILS
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c101
c102
c103
c104
c105
c106
c107
c108
c109
590
660
710
760
640
620
670
680
720
590
660
710
760
640
620
670
680
720
590
650
700
750
640
620
670
670
710
590
650
720
760
640
620
670
680
720
590.0
656.0
720.0
760.0
640.0
620.0
670.0
680.0
720.0
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
349
508
520
544
453
529
529
556
506
525
538
543
349
508
520
544
453
529
529
556
506
525
538
543
330
508
513
539
430
529
529
549
498
515
535
515
344
500
510
536
447
501
524
551
504
503
529
525
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
427
505
519
575
480
481
534
550
427
505
516
575
480
481
534
550
427
494
519
565
459
458
515
546
412
477
509
549
473
467
521
546
Time(s)
Instance
BK
ACS
ILS
Min
Avg
Max
590
660
720
760
640
620
670
680
720
590
660
720
760
640
620
670
680
720
0.0
0.6
-1.4
0.0
0.0
0.0
0.0
0.0
0.0
2.8
27.9
170.6
276.9
4.3
9.7
8.1
15.0
43.7
3.0
34.1
190.9
319.7
4.6
10.2
8.9
17.7
49.4
3.2
39.1
207.2
373.3
5.0
10.8
9.7
20.5
53.4
346.0
505.2
512.2
543.8
447.0
512.4
526.8
553.4
504.8
507.4
531.0
528.4
349
508
518
546
447
524
530
555
506
515
534
532
349
508
519
546
447
524
533
557
506
515
534
534
0.9
0.6
1.5
0.0
1.3
3.1
0.4
0.5
0.2
3.4
1.3
2.7
1.4
5.3
9.7
10.6
2.3
7.0
9.4
10.7
4.3
6.1
7.4
7.9
1.6
6.5
11.1
12.1
2.5
7.5
10.0
12.8
5.3
6.4
10.5
9.3
414.0
484.6
510.4
555.8
475.8
472.0
527.0
550.2
416
494
513
559
480
481
529
552
417
504
517
574
480
483
529
554
3.0
4.0
1.7
3.3
0.9
1.9
1.3
0.0
1.3
3.8
6.4
6.4
2.6
3.5
3.6
5.8
1.5
4.4
6.8
7.1
2.9
3.7
4.6
6.3
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c201
c202
c203
c204
c205
c206
c207
c208
1460
1460
1460
1460
1460
1460
1460
1470
1460
1460
1460
1440
1460
1460
1460
1470
1400
1430
1430
1460
1450
1440
1450
1460
1450
1470
1470
1480
1470
1480
1480
1480
1450.0
1470.0
1472.0
1480.0
1470.0
1480.0
1482.0
1486.0
1.9
7.1
13.1
13.1
2.7
7.9
10.8
16.5
6.0
6.9
14.9
10.9
r201
r202
r203
r204
r205
r206
r207
r208
r209
r210
r211
1239
1310
1377
1440
1346
1401
1428
1458
1361
1365
1422
1239
1310
1358
1404
1346
1381
1400
1433
1361
1360
1411
1231
1270
1377
1440
1338
1401
1428
1458
1345
1365
1422
1227
1334
1407
1449
1363
1421
1456
1458
1375
1403
1454
1.6
5.1
7.5
7.7
3.5
3.9
5.4
7.1
rc201
rc202
rc203
rc204
rc205
rc206
rc207
rc208
1376
1472
1573
1656
1428
1514
1544
1646
1376
1472
1573
1622
1428
1514
1544
1646
1305
1461
1573
1656
1381
1495
1531
1606
1366
1498
1579
1696
1440
1503
1564
1662
Time(s)
Min
Avg
Max
1450
1470
1480
1480
1470
1480
1490
1490
1450
1470
1480
1480
1470
1480
1490
1490
0.7
-0.7
-0.8
-1.4
-0.7
-1.4
-1.5
-1.1
4.0
16.4
40.3
83.0
10.6
15.9
18.1
21.3
4.1
17.7
45.0
94.4
10.9
17.1
19.9
25.0
4.3
20.3
51.4
104.0
11.0
20.5
23.1
28.7
1236.8
1339.2
1409.6
1456.2
1365.0
1428.0
1457.2
1458.0
1384.4
1408.8
1455.2
1247
1347
1413
1458
1368
1432
1458
1458
1397
1412
1457
1248
1347
1414
1458
1379
1440
1458
1458
1400
1416
1458
0.2
-2.2
-2.4
-1.1
-1.4
-1.9
-2.0
0.0
-1.7
-3.2
-2.3
13.8
20.9
22.5
0.9
15.2
23.4
1.6
0.1
22.4
18.8
20.7
14.4
21.4
23.4
9.6
18.0
24.0
16.4
0.1
23.4
20.4
22.3
15.3
21.9
24.5
23.2
19.0
24.7
21.5
0.1
24.2
23.4
26.6
1370.0
1502.4
1606.4
1701.2
1446.6
1511.4
1569.6
1667.4
1374
1506
1624
1707
1454
1527
1579
1669
1377
1509
1632
1713
1455
1538
1587
1680
0.4
-2.1
-2.1
-2.7
-1.3
0.2
-1.7
-1.3
10.2
13.3
16.0
19.5
13.4
15.5
16.0
19.5
11.1
14.9
19.5
22.1
14.6
16.0
18.2
20.7
11.7
16.7
20.8
23.5
15.8
16.6
19.1
22.0
Table 9 Results of GRASP-ELS on Solomon’s instances with m = 3.
Instance
4
BK
ACS
ILS
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c101
c102
c103
c104
c105
c106
c107
c108
c109
810
920
980
1020
870
870
910
920
970
810
920
980
1020
870
870
910
920
970
790
890
960
1010
840
840
900
900
950
810
920
980
1020
850
860
910
900
960
810.0
920.0
980.0
1028.0
854.0
864.0
910.0
910.0
968.0
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
481
691
736
773
620
722
757
790
710
737
770
769
481
691
736
773
620
722
757
790
710
737
770
769
481
685
720
765
609
719
747
790
699
711
764
758
476
676
725
767
610
710
749
781
697
724
760
754
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
621
710
747
823
682
695
755
783
621
710
747
823
682
695
755
783
604
698
747
822
654
678
745
757
597
693
745
786
644
684
751
766
Time(s)
Instance
BK
ACS
ILS
Min
Avg
Max
810
920
980
1030
860
870
910
920
970
810
920
980
1030
860
870
910
920
970
0.0
0.0
0.0
-0.8
1.8
0.7
0.0
1.1
0.2
4.3
27.5
141.5
322.9
5.9
11.0
11.3
19.4
55.9
4.6
33.1
161.3
462.9
6.6
13.5
13.2
24.2
61.3
5.1
36.2
199.4
516.5
7.7
15.3
15.1
26.4
68.4
478.0
680.8
728.0
768.8
613.6
713.6
751.0
784.0
700.4
727.0
762.6
760.6
481
685
730
771
620
723
753
787
703
731
770
772
481
691
735
774
620
723
758
793
703
736
772
772
0.6
1.5
1.1
0.5
1.0
1.2
0.8
0.8
1.4
1.4
1.0
1.1
2.4
7.8
12.3
22.5
5.5
11.9
14.6
21.9
8.1
10.3
13.9
14.6
2.7
9.9
14.5
25.3
5.8
12.9
15.9
24.2
9.1
12.2
15.9
17.9
605.8
695.2
748.6
799.8
659.6
686.2
753.4
773.4
614
699
752
817
673
689
758
778
614
714
764
832
680
698
768
782
2.4
2.1
-0.2
2.8
3.3
1.3
0.2
1.2
3.0
7.0
10.5
12.3
4.8
6.6
8.2
9.0
3.4
7.8
12.1
12.9
5.4
6.9
9.2
11.5
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c201
c202
c203
c204
c205
c206
c207
c208
1810
1790
1760
1780
1800
1800
1810
1810
1810
1790
1760
1770
1800
1800
1790
1800
1750
1750
1760
1780
1770
1770
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810.0
1810.0
1810.0
1810.0
1810.0
1810.0
1810.0
1810.0
3.1
11.0
17.0
29.0
6.1
14.6
17.0
28.6
11.2
14.9
18.6
20.6
r201
r202
r203
r204
r205
r206
r207
r208
r209
r210
r211
1432
1449
1458
1458
1458
1458
1458
1458
1458
1458
1458
1432
1449
1456
1458
1458
1458
1458
1458
1458
1458
1458
1408
1443
1458
1458
1458
1458
1458
1458
1458
1458
1458
1435
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
3.7
8.2
14.1
13.9
5.8
7.5
10.3
14.1
rc201
rc202
rc203
rc204
rc205
rc206
rc207
rc208
1681
1706
1724
1724
1698
1722
1722
1724
1681
1706
1724
1724
1698
1722
1722
1724
1625
1686
1724
1724
1659
1708
1713
1724
1682
1714
1724
1724
1700
1724
1724
1724
Time(s)
Min
Avg
Max
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
0.0
-1.1
-2.8
-1.7
-0.6
-0.6
0.0
0.0
0.0
0.1
0.2
0.2
0.0
0.0
0.0
0.0
0.0
0.1
0.2
0.2
0.0
0.0
0.0
0.0
0.0
0.1
0.3
0.3
0.0
0.0
0.0
0.0
1438.4
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1441
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1441
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
-0.4
-0.6
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
16.4
1.8
0.2
0.1
0.0
0.1
0.0
0.0
0.0
0.0
0.0
19.0
7.9
0.2
0.1
0.0
0.1
0.0
0.0
0.0
0.1
0.0
20.2
15.7
0.3
0.1
0.0
0.1
0.0
0.0
0.0
0.1
0.0
1686.8
1715.4
1724.0
1724.0
1703.2
1724.0
1724.0
1724.0
1695
1719
1724
1724
1709
1724
1724
1724
1698
1724
1724
1724
1709
1724
1724
1724
-0.3
-0.6
0.0
0.0
-0.3
-0.1
-0.1
0.0
17.9
22.0
0.6
0.1
19.4
0.5
0.3
0.0
19.0
22.6
1.0
0.1
22.5
0.7
0.8
0.0
19.9
22.9
2.1
0.1
24.1
0.9
1.8
0.0
Table 10 Results of GRASP-ELS on Solomon’s instances with m = 4.
Instance
5
BK
ACS
ILS
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c101
c102
c103
c104
c105
c106
c107
c108
c109
1020
1150
1190
1240
1060
1070
1120
1120
1190
1020
1150
1190
1240
1060
1070
1120
1120
1190
1000
1090
1150
1220
1030
1040
1100
1100
1180
1020
1140
1190
1250
1050
1060
1100
1110
1180
1020.0
1146.0
1194.0
1254.0
1050.0
1062.0
1106.0
1114.0
1180.0
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
608
836
909
957
771
893
937
994
879
908
944
954
608
836
909
957
771
893
937
994
879
908
944
954
601
807
878
941
735
870
927
982
866
870
935
939
597
826
914
954
759
883
935
976
853
881
938
945
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
808
903
948
1052
875
908
964
1007
808
903
948
1052
875
908
964
1007
794
881
947
1019
841
874
951
998
795
879
940
1026
848
875
954
993
Time(s)
Instance
BK
ACS
ILS
Min
Avg
Max
1020
1150
1200
1260
1050
1070
1110
1120
1180
1020
1150
1200
1260
1060
1070
1110
1120
1180
0.0
0.3
-0.3
-1.1
0.9
0.7
1.3
0.5
0.8
4.9
33.5
120.4
402.6
7.5
15.3
12.3
30.4
55.2
5.5
36.2
132.3
450.8
8.5
16.2
14.9
31.3
65.5
6.3
38.6
140.7
469.3
9.4
17.3
17.1
32.3
73.7
600.2
829.2
917.6
957.6
766.4
888.2
938.6
980.0
865.8
889.2
942.8
953.4
608
834
923
962
773
891
945
983
879
895
947
963
608
837
924
967
776
901
945
990
879
903
949
963
1.3
0.8
-0.9
-0.1
0.6
0.5
-0.2
1.4
1.5
2.1
0.1
0.1
4.1
12.7
21.6
36.4
7.9
16.8
23.8
35.0
11.5
17.5
23.3
27.6
4.4
14.3
24.6
50.8
9.0
19.3
29.6
42.3
13.4
20.9
24.6
36.9
798.0
885.0
950.0
1033.0
854.4
887.0
962.6
1002.6
801
894
957
1042
858
898
968
1011
808
908
967
1059
875
906
971
1023
1.2
2.0
-0.2
1.8
2.4
2.3
0.1
0.4
5.0
9.9
14.0
19.6
9.0
7.6
11.6
17.3
5.2
10.2
16.3
21.9
10.3
9.2
14.1
19.6
GRASP-ELS
Best
Gap(%)
Min
Avg
Max
c201
c202
c203
c204
c205
c206
c207
c208
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810.0
1810.0
1810.0
1810.0
1810.0
1810.0
1810.0
1810.0
4.8
16.0
27.8
92.5
9.5
21.5
39.5
45.7
16.0
23.5
26.7
63.3
r201
r202
r203
r204
r205
r206
r207
r208
r209
r210
r211
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
5.4
10.7
18.7
24.2
11.0
10.3
18.9
22.5
rc201
rc202
rc203
rc204
rc205
rc206
rc207
rc208
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
Time(s)
Min
Avg
Max
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
1810
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458.0
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
1458
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.2
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.3
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1724.0
1724.0
1724.0
1724.0
1724.0
1724.0
1724.0
1724.0
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
1724
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.1
0.0
Table 11 Results of GRASP-ELS on Cordeau’s instances with m = 1.
Instance
BK
ACS
ILS
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
308
404
394
489
595
567
298
463
493
591
308
404
394
485
577
567
298
463
482
591
304
385
384
447
576
538
291
463
461
539
GRASP-ELS
Min
Avg
Max
308
399
394
469
575
582
293
462
466
568
308.0
402.6
394.0
474.6
581.0
583.4
294.4
462.8
477.8
574.2
308
404
394
480
585
585
298
463
491
579
Best
Gap(%)
308
404
394
489
594
590
298
463
493
583
0.0
0.3
0.0
2.9
2.4
-2.9
1.2
0.0
3.1
2.8
Time(s)
Min
Avg
Max
1.1
2.6
2.9
5.0
8.7
6.7
1.4
2.6
6.0
6.6
1.2
3.0
3.1
5.3
9.1
8.6
1.6
3.3
7.0
8.1
1.4
3.2
3.3
5.7
9.6
9.9
1.7
3.6
7.6
9.2
Instance
BK
ACS
ILS
pr11
pr12
pr13
pr14
pr15
pr16
pr17
pr18
pr19
pr20
330
437
450
501
638
559
358
504
499
570
328
437
442
501
528
525
358
504
480
556
330
431
450
482
638
559
346
479
499
570
Instance
BK
ACS
ILS
547
768
816
952
1126
1119
652
907
955
1122
547
768
816
952
1120
1119
652
907
950
1122
542
727
757
925
1126
1110
624
877
955
1056
GRASP-ELS
Min
Avg
Max
327
442
452
538
653
624
351
505
528
653
329.2
442.0
456.6
541.2
665.0
652.6
356.2
517.6
537.8
656.6
330
442
461
546
674
668
359
529
555
661
Best
Gap(%)
330
442
461
567
685
672
359
535
558
667
0.2
-1.1
-1.5
-8.0
-4.2
-16.7
0.5
-2.7
-7.8
-15.2
Best
Gap(%)
545
774
826
996
1216
1218
652
934
1008
1223
0.8
1.6
0.2
-0.4
-4.8
-2.6
1.5
-1.0
-3.8
-6.3
Time(s)
Min
Avg
Max
1.9
3.6
4.5
7.5
11.9
14.5
2.3
4.5
7.3
12.9
2.0
4.0
4.9
8.8
13.1
16.2
2.4
5.0
8.7
13.9
2.0
4.5
5.2
10.2
15.2
17.7
2.6
5.3
9.7
14.5
Table 12 Results of GRASP-ELS on Cordeau’s instances with m = 2.
Instance
6
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
BK
ACS
ILS
502
714
740
899
1034
997
566
819
880
1078
502
714
740
899
1034
995
566
819
880
1078
471
660
714
863
1011
997
552
796
867
1004
GRASP-ELS
Min
Avg
Max
502
697
722
900
1060
1046
554
819
844
1060
502.0
701.0
732.0
905.0
1069.4
1058.8
560.2
826.2
858.6
1072.8
502
705
742
917
1 084
1069
566
832
874
1095
Best
Gap(%)
502
712
742
921
1089
1076
566
832
878
1124
0.0
1.8
1.1
-0.7
-3.4
-6.2
1.0
-0.9
2.4
0.5
Time(s)
Min
Avg
Max
3.5
9.0
13.9
20.0
32.3
24.5
6.2
11.7
26.4
28.5
3.8
10.1
14.3
21.4
35.5
29.1
6.5
13.2
27.0
33.7
4.1
11.4
14.6
22.2
37.7
34.4
6.7
14.2
27.3
36.5
pr11
pr12
pr13
pr14
pr15
pr16
pr17
pr18
pr19
pr20
GRASP-ELS
Min
Avg
Max
541
747
806
947
1162
1127
638
904
981
1186
542.8
755.8
814.4
955.6
1179.6
1148.6
642.2
915.8
991.4
1193.2
545
769
822
964
1203
1182
647
925
1000
1198
Time(s)
Min
Avg
Max
4.8
14.1
17.5
29.3
43.6
64.1
7.9
18.6
25.6
38.1
5.2
14.8
19.9
34.7
50.1
67.0
8.2
19.2
27.8
40.8
5.7
15.5
21.4
37.5
54.4
69.1
8.6
19.6
29.7
43.1
Table 13 Results of GRASP-ELS on Cordeau’s instances with m = 3.
Instance
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
BK
ACS
ILS
619
942
999
1243
1417
1376
744
1118
1227
1492
619
942
999
1243
1417
1370
744
1118
1227
1492
598
899
946
1195
1356
1376
713
1082
1144
1473
GRASP-ELS
Min
Avg
Max
619
926
995
1255
1442
1464
733
1116
1212
1517
619.6
929
999
1260.4
1450.8
1478.4
736.4
1118.6
1220.2
1529
622
936
1005
1267
1464
1490
741
1120
1229
1549
Best
Gap(%)
622
940
1010
1284
1482
1514
744
1138
1248
1549
-0.1
1.4
0.0
-1.4
-2.4
-7.4
1.0
-0.1
0.6
-2.5
Time(s)
Instance
Min
Avg
Max
6.6
16.7
22.8
37.0
59.0
57.8
11.7
25.9
46.8
75.5
7.0
19.1
27
42.7
68.2
66.5
12.4
26.3
49.7
86.6
7.2
21
29.8
47.0
72.8
69.9
13.1
27.0
52.7
90.1
pr11
pr12
pr13
pr14
pr15
pr16
pr17
pr18
pr19
pr20
BK
ACS
ILS
649
985
1101
1263
1509
1516
832
1229
1320
1514
649
985
1101
1263
1509
1516
832
1229
1320
1505
632
902
1046
1197
1488
1478
808
1165
1238
1514
GRASP-ELS
Min
Avg
Max
649
974
1071
1308
1544
1625
815
1229
1310
1626
652.0
976.6
1092.8
1324.6
1563.0
1631.0
823.2
1234.8
1330.4
1646.6
654
981
1110
1339
1580
1640
829
1243
1362
1667
Best
Gap(%)
654
1002
1129
1368
1622
1667
832
1276
1385
1678
-0.5
0.9
0.7
-4.9
-3.6
-7.6
1.1
-0.5
-0.8
-8.8
Time(s)
Min
Avg
Max
6.0
20.1
21.0
37.3
68.1
80.2
8.9
21.5
52.0
64.5
6.6
23.1
24.5
43.9
72.7
92.2
10.5
25.3
54.2
76.5
6.9
27.2
27.0
42.1
77.9
99.6
11.3
28.5
56.0
84.3
Table 14 Results of GRASP-ELS on Cordeau’s instances with m = 4.
Instance
7
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
BK
ACS
ILS
657
1072
1222
1515
1740
1740
872
1376
1561
1827
657
1072
1222
1515
1740
1740
872
1376
1561
1827
644
1014
1162
1452
1665
1696
840
1267
1460
1782
GRASP-ELS
Min
Avg
Max
657
1059
1196
1512
1755
1816
851
1345
1554
1847
657
1062.8
1200
1524.6
1774.8
1820.8
858
1349.8
1562.4
1866.4
657
1073
1203
1533
1790
1828
864
1359
1576
1889
Best
Gap(%)
657
1079
1216
1557
1833
1854
871
1382
1592
1933
0.0
0.9
1.8
-0.6
-2.0
-4.6
1.6
1.9
-0.1
-2.2
Time(s)
Instance
Min
Avg
Max
0.3
18.3
24.3
43.8
68.1
76.2
9.9
26.8
56.3
88.2
1.2
21.0
28.0
49.7
77.2
85.1
11.3
30.2
59.2
94.6
3.6
21.9
31.3
53.1
82.8
90.6
12.0
33.2
65.0
99.1
pr11
pr12
pr13
pr14
pr15
pr16
pr17
pr18
pr19
pr20
BK
ACS
ILS
657
1118
1329
1568
1854
1889
925
1470
1596
1846
657
1118
1329
1568
1854
1887
925
1470
1596
1841
654
1041
1263
1528
1818
1889
889
1352
1560
1846
GRASP-ELS
Min
Avg
Max
657
1114
1312
1607
1904
1969
923
1471
1615
1970
657.0
1121.0
1314.8
1623.4
1917.6
1976.6
927.2
1490.6
1632.0
1993.8
657
1128
1319
1 641
1 942
1 986
932
1 521
1 655
2012
Best
Gap(%)
657
1130
1346
1660
1954
2021
933
1521
1676
2037
0.0
-0.3
1.1
-3.5
-3.4
-4.6
-0.2
-1.4
-2.3
-8.0
Time(s)
Min
Avg
Max
0.0
23.2
38.3
57.5
100.0
141.8
13.2
35.9
69.7
111.3
0.1
26.7
41.3
72.2
111.9
149.6
15.2
38.1
77.0
121.2
0.1
28.9
43.7
79.8
122.9
154.2
16.0
39.8
85.5
127.3
Table 15 Comparison of results for the new data set of Solomon’s instances.
Instance
m
BK
GRASP-ELS
VNS
ILS
Avg
Gap(%)
Time(s)
Avg
Gap(%)
Time(s)
z
Gap(%)
Time(s)
c101
c102
c103
c104
c105
c106
c107
c108
c109
10
10
10
10
10
10
10
10
10
1 810
1 810
1 810
1 810
1 810
1 810
1 810
1 810
1 810
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
8.9
0.3
0.4
0.4
0.3
0.1
0.1
0.2
0.0
1 809.0
1 810.0
1 810.0
1 810.0
1 810.0
1 808.0
1 810.0
1 810.0
1 810.0
0.1
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
21.7
19.8
18.8
16.8
22.7
23.5
21.9
19.6
16.1
1 720
1 790
1 810
1 810
1 770
1 750
1 790
1 810
1 810
5.0
1.1
0.0
0.0
2.2
3.3
1.1
0.0
0.0
4.1
4.2
3.0
1.8
2.8
3.8
3.1
2.5
2.0
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
19
17
13
9
14
12
10
9
11
10
10
9
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 454.4
1 451.8
1 455.0
1 445.0
1 449.6
1 453.8
1 450.6
1 455.0
1 439.8
1 434.2
1 438.6
1 440.4
0.2
0.4
0.2
0.9
0.6
0.3
0.5
0.2
1.2
1.6
1.3
1.2
60.9
146.5
178.6
103.9
74.4
89.1
96.6
116.8
78.6
102.6
88.6
127.3
1 455.8
1 451.1
1 453.3
1 443.5
1 450.1
1 451.9
1 449.7
1 453.2
1 448.3
1 450.5
1 449.2
1 451.9
0.2
0.5
0.3
1.0
0.5
0.4
0.6
0.3
0.7
0.5
0.6
0.4
2.5
3.1
2.0
2.3
4.1
3.1
3.3
2.7
2.5
4.4
3.0
2.4
1 441
1 450
1 450
1 402
1 435
1 441
1 431
1 430
1 432
1 419
1 410
1 418
1.2
0.5
0.5
3.8
1.6
1.2
1.9
1.9
1.8
2.7
3.3
2.7
2.5
3.1
2.0
2.3
4.1
3.1
3.3
2.7
2.5
4.4
3.0
2.4
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
14
12
11
10
13
11
11
10
1 724
1 724
1 724
1 724
1 724
1 724
1 724
1 724
1 705.2
1 693.8
1 716.4
1 724.0
1 694.4
1 691.8
1 724.0
1 718.8
1.1
1.8
0.4
0.0
1.7
1.9
0.0
0.3
65.4
74.6
129.3
4.0
90.4
77.8
66.4
121.7
1 704.5
1 696.0
1 715.8
1 722.7
1 698.2
1 694.8
1 721.0
1 721.8
1.1
1.6
0.5
0.1
1.5
1.7
0.2
0.1
4.3
2.9
3.4
3.2
3.9
4.8
2.4
5.6
1 686
1 659
1 689
1 719
1 691
1 665
1 701
1 698
2.2
3.8
2.0
0.3
1.9
3.4
1.3
1.5
4.3
2.9
3.4
3.2
3.9
4.8
2.4
5.6
0.5
65.7
0.4
8.5
1.8
3.2
Average
8
Table 16 Results of GRASP-ELS and ILS for the new data set of Solomon’s instances.
Instance
m
BK
GRASP-ELS
Avg
Gap(%)
Time(s)
z
Gap(%)
Time(s)
c201
c202
c203
c204
c205
c206
c207
c208
4
4
4
4
4
4
4
4
1 810
1 810
1 810
1 810
1 810
1 810
1 810
1 810
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
1 810.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1 810
1 810
1 810
1 810
1 810
1 810
1 810
1 810
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.5
1.1
1.0
1.0
1.0
1.0
1.0
0.9
r201
r202
r203
r204
r205
r206
r207
r208
r209
r210
r211
4
3
3
2
3
3
2
2
3
3
2
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458
1 458.0
1 456.2
1 458.0
1 458.0
1 458.0
1 458.0
1 456.0
1 458.0
1 458.0
1 458.0
1 456.0
0.0
0.1
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.1
0.2
19.2
0.0
6.1
0.0
0.0
19.9
0.1
0.0
0.1
26.5
1 458
1 443
1 458
1 440
1 458
1 458
1 428
1 458
1 458
1 458
1 422
0.0
1.0
0.0
1.2
0.0
0.0
2.1
0.0
0.0
0.0
2.5
1.3
2.7
1.5
3.3
1.1
1.0
1.6
1.6
1.1
1.2
1.9
rc201
rc202
rc203
rc204
rc205
rc206
rc207
rc208
4
3
3
3
4
3
3
3
1 724
1 724
1 724
1 724
1 724
1 724
1 724
1 724
1 724.0
1 719.8
1 724.0
1 724.0
1 724.0
1 724.0
1 724.0
1 724.0
0.0
0.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
20.3
0.7
0.0
0.1
0.7
1.1
0.0
1 724
1 686
1 724
1 724
1 724
1 708
1 713
1 724
0.0
2.2
0.0
0.0
0.0
0.9
0.6
0.0
2.2
1.7
2.8
1.0
2.1
1.3
1.4
1.0
0.0
3.5
0.4
1.5
Average
ILS
Table 17 Results of GRASP-ELS, VNS and ILS for the new data set of Cordeau’s instances.
Instance
m
BK
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
3
6
9
12
15
18
5
10
15
20
657
1 220
1 788
2 477
3 351
3 671
948
2 006
2 736
3 850
Average
GRASP-ELS
VNS
ILS
Avg
Gap(%)
Time(s)
Avg
Gap(%)
Time(s)
z
Gap(%)
Time(s)
619.0
1 202.2
1 771.0
2 475.6
3 351.0
3 671.0
938.8
2 006.0
2 736.0
3 850.0
5.8
1.5
1.0
0.1
0.0
0.0
1.0
0.0
0.0
0.0
10.1
81.9
171.0
341.2
3.5
31.5
35.4
3.8
33.9
2.5
612.9
1 195.3
1 759.0
2 466.4
3 349.5
3 670.1
928.3
2 005.0
2 735.6
3 850.0
6.7
2.0
1.6
0.4
0.0
0.0
2.1
0.0
0.0
0.0
69.6
85.0
86.0
81.2
71.7
83.5
69.5
63.2
54.6
52.4
608
1 180
1 738
2 428
3 297
3 650
909
1 984
2 729
3 850
7.5
3.3
2.8
2.0
1.6
0.6
4.1
1.1
0.3
0.0
0.7
4.5
11.3
45.4
37.3
106.1
1.5
12.0
33.0
52.0
0.9
42.6
1.3
71.7
2.3
30.4
9
Table 18 Comparison of results for Solomon’s instances with n = 50 and m = 1.
Inst.
BK
GRASP-ELS
VNS
ILS
Avg Gap(%) Time(s)
Avg Gap(%) Time(s)
z
ACS
Gap(%) Time(s)
Avg Gap(%) Time(s)
c101_50
c102_50
c103_50
c104_50
c105_50
c106_50
c107_50
c108_50
c109_50
270
300
320
340
300
280
310
320
340
270.0
300.0
320.0
340.0
300.0
280.0
310.0
320.0
340.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.7
2.9
14.2
33.1
0.9
1.0
1.6
3.0
5.7
270.0
300.0
320.0
340.0
300.0
280.0
310.0
320.0
340.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
44.5
65.4
69.0
80.7
45.2
39.7
53.8
42.7
41.7
270
300
320
330
300
280
310
320
340
0.0
0.0
0.0
2.9
0.0
0.0
0.0
0.0
0.0
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.3
0.2
270
300
320
340
300
280
310
320
340
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.8
0.0
0.1
0.1
0.1
0.7
1.0
r101_50
r102_50
r103_50
r104_50
r105_50
r106_50
r107_50
r108_50
r109_50
r110_50
r111_50
r112_50
126
198
214
227
159
208
220
227
192
208
223
226
126.0
195.0
214.0
227.0
159.0
201.8
217.4
227.0
192.0
208.0
223.0
226.0
0.0
1.5
0.0
0.0
0.0
3.0
1.2
0.0
0.0
0.0
0.0
0.0
0.4
0.6
0.8
1.0
0.7
1.3
1.1
0.9
1.0
0.8
1.5
1.0
126.0
198.0
214.0
227.0
159.0
208.0
219.9
227.0
192.0
208.0
223.0
226.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
13.6
24.5
26.5
29.3
16.1
24.8
32.5
28.2
20.1
23.4
25.2
26.0
126
195
210
227
159
203
220
223
192
208
223
226
0.0
1.5
1.9
0.0
0.0
2.4
0.0
1.8
0.0
0.0
0.0
0.0
0.1
0.2
0.2
0.3
0.1
0.2
0.3
0.2
0.2
0.2
0.2
0.2
126
198
214
227
159
208
220
227
192
208
223
226
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
2.2
2.6
0.0
2.0
9.3
1.5
0.0
0.0
5.9
2.8
rc101_50
rc102_50
rc103_50
rc104_50
rc105_50
rc106_50
rc107_50
rc108_50
180
230
240
270
210
210
240
250
180.0
230.0
240.0
268.0
210.0
210.0
230.0
240.0
0.0
0.0
0.0
0.7
0.0
0.0
4.2
4.0
0.4
0.7
1.1
1.5
0.7
0.7
1.0
1.2
180.0
230.0
240.0
266.0
210.0
210.0
240.0
250.0
0.0
0.0
0.0
1.5
0.0
0.0
0.0
0.0
30.8
32.8
32.5
38.3
22.2
33.9
31.1
33.8
180
230
240
270
200
200
230
240
0.0
0.0
0.0
0.0
4.8
4.8
4.2
4.0
0.1
0.2
0.2
0.2
0.2
0.2
0.1
0.2
180
230
240
270
210
210
240
250
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.7
9.4
0.6
0.0
3.5
8.6
0.5
2.8
0.1
35.5
1.0
0.2
0.0
1.8
Average
10
Table 19 Comparison of results for Solomon’s instances with n = 50 and m = 2.
Inst.
BK
GRASP-ELS
c101_50
c102_50
c103_50
c104_50
c105_50
c106_50
c107_50
c108_50
c109_50
490
530
570
590
520
500
540
540
570
486.0
530.0
570.0
590.0
510.0
500.0
540.0
540.0
570.0
0.8
0.0
0.0
0.0
1.9
0.0
0.0
0.0
0.0
1.3
4.9
15.4
49.8
2.0
1.7
3.3
5.3
8.7
490.0
530.0
570.0
590.0
519.0
500.0
540.0
540.0
570.0
0.0
0.0
0.0
0.0
0.2
0.0
0.0
0.0
0.0
53.4
85.9
72.6
55.6
63.4
34.4
61.8
72.3
47.1
480
520
560
590
520
490
530
540
570
2.0
1.9
1.8
0.0
0.0
2.0
1.9
0.0
0.0
0.8
0.9
0.7
0.8
0.7
0.7
0.7
0.9
1.0
r101_50
r102_50
r103_50
r104_50
r105_50
r106_50
r107_50
r108_50
r109_50
r110_50
r111_50
r112_50
246
357
383
412
292
376
398
417
364
393
406
428
246.0
356.4
379.4
409.4
290.4
371.4
395.4
413.8
352.0
388.2
404.8
417.0
0.0
0.2
0.9
0.6
0.5
1.2
0.7
0.8
3.3
1.2
0.3
2.6
0.7
1.9
2.4
4.9
1.3
2.0
2.6
4.4
1.9
2.8
2.7
2.7
246.0
355.7
379.2
411.5
292.0
373.9
392.1
416.4
364.0
392.0
399.5
424.2
0.0
0.4
1.0
0.1
0.0
0.6
1.5
0.1
0.0
0.3
1.6
0.9
10.2
17.5
19.6
28.7
14.9
22.1
25.8
27.6
18.2
22.9
26.6
23.4
244
343
374
410
282
373
385
412
350
384
397
417
0.8
3.9
2.3
0.5
3.4
0.8
3.3
1.2
3.8
2.3
2.2
2.6
0.3
0.5
0.6
0.7
0.3
0.5
0.6
0.7
0.5
0.5
0.5
0.6
rc101_50
rc102_50
rc103_50
rc104_50
rc105_50
rc106_50
rc107_50
rc108_50
360
430
460
520
400
410
470
490
360.0
430.0
450.0
520.0
394.0
390.0
440.0
472.0
0.0
0.0
2.2
0.0
1.5
4.9
6.4
3.7
0.9
1.7
2.5
3.9
1.8
1.6
2.9
3.3
360.0
426.0
455.0
499.0
400.0
409.0
459.0
488.0
0.0
0.9
1.1
4.0
0.0
0.2
2.3
0.4
36.4
35.9
27.2
32.7
37.8
30.6
24.5
24.9
360
430
450
500
370
400
470
480
0.0
0.0
2.2
3.8
7.5
2.4
0.0
2.0
0.4
0.7
0.5
0.6
0.5
0.5
0.5
0.5
1.2
4.9
0.5
36.3
1.9
0.6
Avg
VNS
Gap(%) Time(s)
Average
Avg
ILS
Gap(%) Time(s)
z
Gap(%) Time(s)
Table 20 Comparison of results for Cordeau’s instances with m = 1
Inst.
BK
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
308
404
394
489
595
567
298
463
493
591
Average
GRASP-ELS
VNS
ILS
Avg Gap(%) Time(s)
Avg Gap(%) Time(s)
308.0 0.0
402.6 0.3
394.0 0.0
474.6 2.9
581.0 2.4
583.4 -2.9
294.4 1.2
462.8 0.0
477.8 3.1
574.2 2.8
1.2
3.0
3.1
5.3
9.1
8.6
1.6
3.3
7.0
8.1
308.0 0.0
404.0 0.0
391.4 0.7
486.0 0.6
589.6 0.9
589.0 -3.9
298.0 0.0
459.5 0.8
482.0 2.2
572.9 3.1
77.9
244.6
387.2
541.6
1 455.2
1 633.8
131.0
514.1
920.8
1 534.1
1.0
5.0
0.4
744.0
11
z
304
385
384
447
576
538
291
463
461
539
ACS
Gap(%) Time(s)
1.3
4.7
2.5
8.6
3.2
5.1
2.3
0.0
6.5
8.8
0.5
0.6
1.0
1.9
4.6
2.5
0.4
1.0
1.4
3.6
4.3
1.8
Avg
308.0
403.8
394.0
482.0
576.8
564.6
298.0
462.6
481.8
588.4
Gap(%) Time(s)
0.0
0.0
0.0
1.4
3.1
0.4
0.0
0.1
2.3
0.4
256.2
1 147.8
2 024.3
1 404.7
2 075.7
2 199.8
20.1
2 476.0
2 318.2
2 343.5
0.8
1 626.6
Table 21 Comparison of results for Solomon’s instances with n = 100 and m = 1.
Inst.
BK
GRASP-ELS
VNS
ILS
Avg Gap(%) Time(s)
Avg Gap(%) Time(s)
z
ACS
Gap(%) Time(s)
Avg Gap(%) Time(s)
c101
c102
c103
c104
c105
c106
c107
c108
c109
320
360
400
420
340
340
370
370
380
320.0
360.0
400.0
420.0
340.0
340.0
370.0
370.0
380.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.2
16.5
38.5
105.2
2.3
3.8
3.7
6.6
25.5
320.0
360.0
397.0
418.0
340.0
340.0
370.0
370.0
380.0
0.0
0.0
0.8
0.5
0.0
0.0
0.0
0.0
0.0
77.9
114.7
111.2
108.1
84.6
90.2
82.0
95.6
113.3
320
360
390
400
340
340
360
370
380
0.0
0.0
2.5
4.8
0.0
0.0
2.7
0.0
0.0
0.4
0.3
0.5
0.3
0.3
0.3
0.3
0.3
0.3
320.0
360.0
400.0
420.0
340.0
340.0
370.0
370.0
380.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.5
0.7
16.9
33.5
0.9
1.0
2.1
0.8
0.8
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
198
286
293
303
247
293
299
308
277
284
297
298
198.0
286.0
290.4
302.8
247.0
293.0
296.4
306.8
277.0
283.6
297.0
297.2
0.0
0.0
0.9
0.1
0.0
0.0
0.9
0.4
0.0
0.1
0.0
0.3
0.9
2.3
4.8
5.1
1.2
3.3
4.8
5.0
2.7
2.8
4.6
4.6
198.0
285.2
293.0
303.0
247.0
292.2
298.8
307.8
277.0
284.0
297.0
298.0
0.0
0.3
0.0
0.0
0.0
0.3
0.1
0.1
0.0
0.0
0.0
0.0
47.8
89.8
101.0
110.6
80.1
97.5
96.6
97.4
77.6
97.2
104.0
107.8
182
286
286
297
247
293
288
297
276
281
295
295
8.1
0.0
2.4
2.0
0.0
0.0
3.7
3.6
0.4
1.1
0.7
1.0
0.1
0.2
0.2
0.2
0.1
0.2
0.2
0.2
0.2
0.3
0.2
0.2
198.0
286.0
292.6
303.0
247.0
293.0
294.6
306.0
277.0
283.2
296.6
297.4
0.0
0.0
0.1
0.0
0.0
0.0
1.5
0.6
0.0
0.3
0.1
0.2
0.1
11.1
640.6
164.0
3.0
86.3
922.6
696.1
28.0
617.6
484.4
947.1
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
219
266
266
301
244
252
277
298
219.0
259.0
265.2
300.2
244.0
252.0
277.0
298.0
0.0
2.6
0.3
0.3
0.0
0.0
0.0
0.0
0.6
2.1
2.7
3.1
1.4
1.2
1.9
2.9
219.0
266.0
266.0
301.0
243.2
249.4
277.0
298.0
0.0
0.0
0.0
0.0
0.3
1.0
0.0
0.0
61.2
54.7
69.2
62.5
65.2
67.4
72.4
71.5
219
259
265
297
221
239
274
288
0.0
2.6
0.4
1.3
9.4
5.2
1.1
3.4
0.2
0.2
0.3
0.3
0.2
0.2
0.2
0.2
219.0
266.0
266.0
301.0
244.0
252.0
277.0
298.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.2
30.9
57.2
29.4
9.7
308.6
502.1
207.5
0.2
9.0
0.1
86.5
1.9
0.2
0.1
200.1
Average
Table 22 Comparison of results for Cordeau’s instances with m = 2
Inst.
pr01
pr02
pr03
pr04
pr05
pr06
pr07
pr08
pr09
pr10
Average
BK
502
714
740
899
1 034
997
566
819
880
1 078
GRASP-ELS
VNS
ILS
ACS
Avg
Gap(%)
Time(s)
Avg
Gap(%)
Time(s)
z
Gap(%)
Time(s)
Avg
Gap(%)
Time(s)
502.0
701.0
732.0
905.0
1 069.4
1 058.8
560.2
826.2
858.6
1 072.8
0.0
1.8
1.1
-0.7
-3.4
-6.2
1.0
-0.9
2.4
0.5
3.8
10.1
14.3
21.4
35.5
29.1
6.5
13.2
27.0
33.7
484.4
705.3
721.1
906.2
1 062.4
999.6
549.7
797.9
850.9
1 073.2
3.5
1.2
2.6
-0.8
-2.7
-0.3
2.9
2.6
3.3
0.4
79.8
211.4
363.2
562.0
963.0
914.4
122.8
333.5
578.7
1 060.0
471
660
714
863
1 011
997
552
796
867
1 004
6.2
7.6
3.5
4.0
2.2
0.0
2.5
2.8
1.5
6.9
0.5
1.2
3.3
4.1
7.1
9.8
1.0
5.1
5.2
10.0
502.0
705.2
731.4
883.6
1 021.8
987.6
566.0
813.0
862.0
1 064.6
0.0
1.2
1.2
1.7
1.2
0.9
0.0
0.7
2.0
1.2
602.2
1 017.6
2 083.0
2 439.3
2 278.5
1 724.1
972.3
1 730.0
3 131.0
2 918.6
-0.4
19.5
1.3
518.9
3.7
4.7
1.0
1889.7
12
Table 23 Comparison of results for Solomon’s instances with n = 100 and m = 2.
Inst.
BK
Avg Gap(%) Time(s)
Avg Gap(%) Time(s)
z
c101
c102
c103
c104
c105
c106
c107
c108
c109
590
660
710
760
640
620
670
680
720
590.0 0.0
656.0 0.6
720.0 -1.4
760.0 0.0
640.0 0.0
620.0 0.0
670.0 0.0
680.0 0.0
720.0 0.0
3.0
34.1
190.9
319.7
4.6
10.2
8.9
17.7
49.4
590.0 0.0
652.0 1.2
719.0 -1.3
759.0 0.1
640.0 0.0
620.0 0.0
667.0 0.4
680.0 0.0
719.0 0.1
70.1
83.3
73.2
83.6
70.6
83.4
91.9
88.3
72.4
590
650
700
750
640
620
670
670
710
0.0
1.5
1.4
1.3
0.0
0.0
0.0
1.5
1.4
1.4
0.9
1.2
1.5
0.8
0.8
1.4
0.8
0.9
588.0
660.0
710.0
754.0
640.0
620.0
670.0
680.0
720.0
0.3
0.0
0.0
0.8
0.0
0.0
0.0
0.0
0.0
110.8
1 427.4
938.9
1 355.0
1 436.6
104.6
76.0
95.3
1 817.3
r101
r102
r103
r104
r105
r106
r107
r108
r109
r110
r111
r112
349
508
520
544
453
529
529
556
506
525
538
543
346.0
505.2
512.2
543.8
447.0
512.4
526.8
553.4
504.8
507.4
531.0
528.4
0.9
0.6
1.5
0.0
1.3
3.1
0.4
0.5
0.2
3.4
1.3
2.7
1.6
6.5
11.1
12.1
2.5
7.5
10.0
12.8
5.3
6.4
10.5
9.3
349.0 0.0
495.0 2.6
519.5 0.1
542.4 0.3
453.0 0.0
511.3 3.3
533.3 -0.8
548.8 1.3
502.4 0.7
509.3 3.0
538.6 -0.1
532.8 1.9
35.4
64.9
58.8
67.9
68.3
76.6
71.0
73.3
71.5
62.7
61.7
82.5
330
508
513
539
430
529
529
549
498
515
535
515
5.4
0.0
1.3
0.9
5.1
0.0
0.0
1.3
1.6
1.9
0.6
5.2
0.4
0.9
0.9
1.5
0.8
0.9
1.0
1.4
0.5
1.0
0.6
0.5
349.0
506.6
518.8
540.0
453.0
523.8
527.4
553.6
505.4
523.4
535.6
541.6
0.0
0.3
0.2
0.7
0.0
1.0
0.3
0.4
0.1
0.3
0.4
0.3
36.5
2 006.6
1 095.1
2 291.8
1 037.3
2 034.7
2 357.6
1 906.2
1 386.6
1 180.2
1 726.2
1 653.6
rc101
rc102
rc103
rc104
rc105
rc106
rc107
rc108
427
505
519
575
480
481
534
550
414.0
484.6
510.4
555.8
475.8
472.0
527.0
550.2
3.0
4.0
1.7
3.3
0.9
1.9
1.3
0.0
1.5
4.4
6.8
7.1
2.9
3.7
4.6
6.3
419.0
504.0
516.7
570.8
480.0
479.9
521.5
541.3
1.9
0.2
0.4
0.7
0.0
0.2
2.3
1.6
46.0
53.8
61.9
57.9
44.7
57.0
58.7
65.3
427
494
519
565
459
458
515
546
0.0
2.2
0.0
1.7
4.4
4.8
3.6
0.7
0.6
0.8
1.1
0.7
0.8
0.6
0.5
0.6
427.0
497.2
510.2
568.8
478.4
480.2
530.2
541.6
0.0
1.5
1.7
1.1
0.3
0.2
0.7
1.5
27.9
1 496.9
1 965.9
2 381.3
1 676.7
1 865.1
771.5
821.0
1.1
26.6
0.7
67.5
1.6
0.9
0.4
1 278.6
Average
GRASP-ELS
VNS
ILS
13
ACS
Gap(%) Time(s)
Avg Gap(%) Time(s)