Supplementary Materials — On Local Search for Bi-objective Knapsack Problems Arnaud Liefooghe
[email protected] LIFL, Universit´e Lille 1, UMR CNRS 8022, 59655 Villeneuve d’Ascq cedex, France
Lu´ıs Paquete
[email protected] CISUC, Department of Informatics Engineering, Faculty of Science and Technology, University of Coimbra, 3030-290 Coimbra, Portugal
Jos´e Rui Figueira
[email protected]
CEG-IST, Instituto Superior T´ecnico, 1040-001 Lisboa, Portugal (Associate Researcher at LORIA Laboratory, Nancy, France)
Abstract This document reports the whole set of experimental results obtained for the connectedness analysis and the local search analysis for the bi-objective unconstrained knapsack problem (BUKP), the bi-objective knapsack problem with bounded cardinality (BKP-BC), and the bi-objective knapsack problem with fixed cardinality (BKP-FC).
size of the efficient set
350000
ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
300000 250000 200000 150000 100000 50000 0 0
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instance size (n)
Figure 1: Size if the efficient set for BUKP with respect to the instance size for different data correlation values.
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14000 12000 10000 8000 6000 4000 2000 0
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ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
35000 30000 25000
size of the efficient set
ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
16000
size of the efficient set
size of the efficient set
18000
20000 15000 10000 5000 0
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ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
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instance size (n)
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instance size (n)
Figure 2: Size if the efficient set for BKP-BC with respect to the instance size for k = n/10 (left), k = n/5 (middle), and k = n/2 (right).
60000
15000 10000 5000 0
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ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
50000 40000
size of the efficient set
ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
20000
size of the efficient set
size of the efficient set
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ρ = 0.8 ρ = 0.4 ρ = 0.0 ρ = -0.4 ρ = -0.8
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Figure 3: Size if the efficient set for BKP-FC with respect to the instance size for k = n/10 (left), k = n/5 (middle), and k = n/2 (right).
2
Table 1: Results of the connected analysis for BUKP instances with respect to the 1flip neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
1518 5320 11083 18713 27903 38380 50415 63546 78478 94035
100.0 96.6 96.6 100.0 93.3 93.3 90.0 90.0 76.6 83.3
100 200 300 400 500 600 700 800 900 1000
1798 6188 12637 21254 31601 43186 56940 72280 88331 106405
100.0 100.0 100.0 96.6 100.0 93.3 93.3 93.3 96.6 93.3
100 200 300 400 500 600 700 800 900 1000
2214 7685 15482 25977 38210 53478 69711 88658 109420 131184
100.0 100.0 100.0 90.0 96.6 96.6 96.6 93.3 83.3 86.6
100 200 300 400 500 600 700 800 900 1000
2790 10102 21090 35203 52537 73010 94548 119336 146931 177283
100.0 100.0 100.0 86.6 90.0 100.0 90.0 90.0 83.3 83.3
100 200 300 400 500 600 700 800 900 1000
4884 17666 36044 60678 89851 125859 162028 207343 257282 310945
100.0 100.0 90.0 93.3 96.6 86.6 86.6 80.0 80.0 76.6
proportion of the largest connected component ρ = −0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = −0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
3
number of unconnected solutions 0 1 1 0 1 1 1 1 1 2 0 0 0 1 0 1 1 1 1 3 0 0 0 1 1 1 1 1 2 1 0 0 0 1 1 0 1 1 2 2 0 0 1 1 3 1 3 2 2 1
Table 2: Results of the connected analysis for BKP-BC instances (k = n/10) with respect to the 1-flip neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
76 283 559 998 1551 2113 2782 3593 4376 5341
6.6 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
92 315 621 1013 1586 2113 2800 3471 4290 5091
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
101 294 646 1034 1599 2096 2771 3474 4240 5015
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
125 424 842 1386 2136 2949 3876 4799 5927 7240
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
241 847 1675 2939 4488 6207 8245 9919 12348 15074
0 0 0 0 0 0 0 0 0 0
proportion of the largest connected component ρ = −0.8 93.2 90.4 89.7 88.7 88.9 89.1 88.5 89.7 89.1 89.0 ρ = −0.4 73.6 72.5 71.9 71.5 71.4 71.8 72.1 72.7 72.1 70.7 ρ = 0.0 66.6 58.9 58.8 57.1 58.6 56.4 56.6 57.0 56.4 55.8 ρ = 0.4 57.7 55.0 55.1 54.8 55.3 54.9 55.4 55.5 54.8 55.3 ρ = 0.8 58.2 59.5 60.2 61.9 63.3 62.5 63.8 63.0 63.0 63.7
4
number of unconnected solutions 23 81 157 262 346 369 555 557 767 946 57 214 447 658 1031 1190 1467 1698 2096 2757 120 320 650 1060 1698 2104 2662 3551 4095 5041 150 457 853 1464 2487 3021 4035 4815 5949 7537 286 908 1424 2565 3236 4978 5569 6842 8252 10389
Table 3: Results of the connected analysis for BKP-BC instances (k = n/5) with respect to the 1-flip neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
215 771 1584 2608 4066 5629 7237 9192 11363 13606
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
248 811 1614 2686 4139 5673 7292 9172 11241 13568
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
273 895 1745 2977 4357 5998 8055 9961 12195 14626
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
361 1100 2296 3710 5557 7865 10006 12406 15358 18237
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
642 1960 4022 6763 10480 14372 18469 22955 27603 34219
0 0 0 0 0 0 0 0 0 0
proportion of the largest connected component ρ = −0.8 92.3 90.5 90.8 89.8 90.9 91.1 90.1 90.3 90.6 90.3 ρ = −0.4 75.5 75.2 75.1 74.0 75.7 75.9 73.6 74.5 74.3 74.0 ρ = 0.0 66.9 63.9 63.5 61.4 61.2 61.1 62.7 61.1 61.2 61.3 ρ = 0.4 60.1 55.9 56.1 54.1 54.1 54.3 53.9 53.3 53.7 52.9 ρ = 0.8 58.1 54.1 53.9 54.7 55.7 54.6 55.3 54.7 54.0 55.0
5
number of unconnected solutions 40 168 280 640 672 965 1303 1337 1580 2006 142 409 818 1606 2024 2595 3194 4604 4902 6007 270 780 1608 2685 3992 4755 6186 8373 8987 11909 444 1125 2498 3870 5942 7973 10022 12586 15817 18678 809 2188 4043 6671 9673 15112 18096 21088 26504 29719
Table 4: Results of the connected analysis for BKP-BC instances (k = n/2) with respect to the 1-flip neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
780 2745 5645 9473 14114 19340 25384 32040 39392 47114
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
938 3127 6439 10769 16132 21724 28492 36065 44223 53387
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700 800 900 1000
1145 3999 7840 13357 19495 27350 35541 44447 54842 66374
0 0 0 0 0 0 0 0 0 0
100 200 300 400 500 600 700
1505 5168 10646 17873 26370 37043 47697
0 0 0 0 0 0 0
100 200 300 400 500 600 700
2645 9169 18415 31370 45185 64794 81700
0 0 0 0 0 0 0
proportion of the largest connected component ρ = −0.8 97.3 97.0 96.7 96.5 96.6 96.6 96.3 96.4 96.2 96.3 ρ = −0.4 91.2 90.6 90.5 89.9 90.4 90.2 89.5 89.5 89.4 89.7 ρ = 0.0 86.1 84.4 85.6 84.5 85.0 84.5 84.5 84.0 83.9 84.6 ρ = 0.4 83.1 80.5 80.2 80.6 79.7 80.8 79.5 ρ = 0.8 80.4 77.7 77.0 77.7 76.3 78.0 76.1
6
number of unconnected solutions 49 241 390 552 846 1228 1411 1673 3489 2950 210 671 1057 1963 2423 3466 4898 5813 9859 8811 303 1111 1914 3751 4946 6688 8917 12024 13635 15489 604 2220 3889 6158 9524 11692 17144 920 5125 8059 11725 19235 24240 35859
Table 5: Results of the connected analysis for BKP-FC instances (k = n/10) with respect to the 1-exchange neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
20 55 102 175 268 337 449 530 656 807
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
57 179 331 525 839 1080 1417 1703 2152 2654
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
92 317 625 1067 1573 2199 2904 3563 4360 5281
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
167 568 1181 2003 2993 4170 5358 6800 8194 9976
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
369 1270 2611 4549 6692 9295 12010 14971 18233 21850
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
proportion of the largest connected component ρ = −0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = −0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
7
number of unconnected solutions 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 0 0 0 0 0 0 0 0 0 0 0 0 0
Table 6: Results of the connected analysis for BKP-FC instances (k = n/5) with respect to the 1-exchange neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
41 123 236 404 561 753 1048 1313 1562 1899
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
128 375 758 1308 1824 2475 3535 4193 5209 6265
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
218 779 1442 2659 3829 5232 6731 8599 10588 12866
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
400 1309 2753 4665 7005 10145 12589 16103 19481 23649
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
857 2916 6068 10072 15205 21657 27086 34608 41577 50717
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
proportion of the largest connected component ρ = −0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = −0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
8
number of unconnected solutions 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 0 0 0 0 0 0 0 0 0 0 0 0 0
Table 7: Results of the connected analysis for BKP-FC instances (k = n/2) with respect to the 1-exchange neighborhood. Rounded average values are reported. n
size of the efficient set
proportion of connected instances
100 200 300 400 500 600 700 800 900 1000
72 223 477 757 1121 1484 2010 2574 3207 3735
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
226 733 1535 2525 3698 4960 6585 8663 10449 12302
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
442 1546 2959 5043 7268 10411 13403 17798 22241 25453
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800 900 1000
737 2745 5456 9134 13849 19118 24872 31545 38984 46989
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
100 200 300 400 500 600 700 800
1578 5862 11564 19436 29002 40313 52594 67298
100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
proportion of the largest connected component ρ = −0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = −0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 ρ = 0.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
9
number of unconnected solutions 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 0 0 0 0 0 0 0 0 0 0 0
100 DP PLS
10 1 0.1 0.01 0.001
DP PLS
10
CPU time (seconds)
CPU time (seconds)
100
1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
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1000
0
200
instance size (n)
400
600
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1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 100 DP PLS
CPU time (seconds)
10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 100 DP PLS
10
CPU time (seconds)
CPU time (seconds)
100
1 0.1 0.01 0.001 0.0001
DP PLS
10 1 0.1 0.01 0.001 0.0001
0
200
400
600
800
1000
0
instance size (n)
200
400
600
800
1000
instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 4: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BUKP and PLS–flip for BUKP instances.
10
Table 8: Results obtained by PLS–flip for BUKP instances. Rounded average values are reported. n
size of the output set
100 200 300 400 500 600 700 800 900 1000
1518 5311 11045 18628 27673 37958 49748 62576 76900 92043
100 200 300 400 500 600 700 800 900 1000
1796 6175 12580 21136 31296 42619 56035 70810 86209 103338
100 200 300 400 500 600 700 800 900 1000
2211 7649 15377 25713 37647 52412 68081 85986 105427 125456
100 200 300 400 500 600 700 800 900 1000
2782 10038 20825 34550 51143 70210 90335 112318 136762 162908
100 200 300 400 500 600 700 800 900 1000
4858 17280 34797 56803 82255 111259 139832 172739 206914 242689
proportion of efficient solutions found ρ = −0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = −0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
11
number of missing efficient solutions
epsilon value
0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.2 0.2 0.2
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.1 0.0 0.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.1 0.1 0.1
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 0.0 0.1 0.1 0.0 0.1 0.1 0.3 0.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
0.0 0.0 0.2 0.0 0.1 0.1 0.2 0.3 0.2 0.3
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1000 DP PLS-flip-ex PLS-flip
10 1 0.1 0.01 0.001
DP PLS-flip-ex PLS-flip
100
CPU time (seconds)
CPU time (seconds)
100
10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
200
instance size (n)
400
600
800
1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 1000 DP PLS-flip-ex PLS-flip
CPU time (seconds)
100 10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 10000 DP PLS-flip-ex PLS-flip
100
DP PLS-flip-ex PLS-flip
1000 CPU time (seconds)
CPU time (seconds)
1000
10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
instance size (n)
200
400
600
800
1000
instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 5: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BKP-BC, PLS–flip and PLS–flip-exchange for BKP-BC instances (k = n/10).
12
1000 DP PLS-flip-ex PLS-flip
100 10 1 0.1 0.01 0.001
DP PLS-flip-ex PLS-flip
100
CPU time (seconds)
CPU time (seconds)
1000
10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
200
instance size (n)
400
600
800
1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 10000 DP PLS-flip-ex PLS-flip
CPU time (seconds)
1000 100 10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 10000
10000 DP PLS-flip-ex PLS-flip
100
DP PLS-flip-ex PLS-flip
1000 CPU time (seconds)
CPU time (seconds)
1000 10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
instance size (n)
200
400
600
800
1000
instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 6: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BKP-BC, PLS–flip and PLS–flip-exchange for BKP-BC instances (k = n/5).
13
10000 DP PLS-flip-ex PLS-flip
100
DP PLS-flip-ex PLS-flip
1000 CPU time (seconds)
CPU time (seconds)
1000
10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
200
instance size (n)
400
600
800
1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 10000 DP PLS-flip-ex PLS-flip
CPU time (seconds)
1000 100 10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 10000
10000 DP PLS-flip-ex PLS-flip
100
DP PLS-flip-ex PLS-flip
1000 CPU time (seconds)
CPU time (seconds)
1000 10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
100 200 300 400 500 600 700 800 instance size (n)
0
100 200 300 400 500 600 700 800 instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 7: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BKP-BC, PLS–flip and PLS–flip-exchange for BKP-BC instances (k = n/2).
14
Table 9: Results obtained by PLS–flip-exchange and PLS–flip for BKP-BC instances (k = n/10). Rounded average values are reported. n
PLS–flip-exchange size of the proportion of efficient output set solutions found
100 200 300 400 500 600 700 800 900 1000
82 312 623 1121 1711 2338 3114 3963 4831 5902
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
125 436 867 1420 2203 2919 3862 4735 5873 7100
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
154 504 1098 1808 2733 3689 4864 6042 7441 8887
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
216 764 1522 2515 3846 5322 6909 8507 10582 12785
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
410 1416 2754 4698 6955 9641 12505 15115 18713 22420
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
PLS–flip size of the proportion of efficient number of missing output set solutions found efficient solutions ρ = −0.8 76 0.9 6.2 282 0.9 30.0 558 0.9 65.2 993 0.9 128.1 1521 0.9 190.0 2084 0.9 253.8 2759 0.9 354.5 3554 0.9 409.4 4306 0.9 525.1 5253 0.9 648.8 ρ = −0.4 92 0.7 32.7 315 0.7 120.7 620 0.7 246.3 1012 0.7 408.0 1572 0.7 630.2 2097 0.7 821.2 2782 0.7 1079.9 3443 0.7 1292.2 4234 0.7 1638.8 5024 0.7 2076.4 ρ = 0.0 98 0.7 55.9 291 0.6 213.1 640 0.6 458.0 1030 0.6 778.0 1593 0.6 1139.7 2077 0.6 1612.1 2752 0.6 2112.3 3432 0.6 2610.0 4186 0.6 3254.1 4950 0.6 3937.2 ρ = 0.4 114 0.5 102.0 376 0.5 388.6 739 0.5 783.2 1170 0.5 1345.3 1767 0.5 2078.9 2417 0.5 2905.4 3107 0.5 3801.9 3834 0.5 4673.6 4832 0.5 5750.2 5783 0.5 7001.7 ρ = 0.8 185 0.5 225.4 584 0.4 832.3 1112 0.4 1641.3 1821 0.4 2876.6 2607 0.4 4347.4 3704 0.4 5936.8 4650 0.4 7855.3 5694 0.4 9421.1 7094 0.4 11619.5 8371 0.4 14048.7
15
epsilon value 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4
Table 10: Results obtained by PLS–flip-exchange and PLS–flip for BKP-BC instances (k = n/5). Rounded average values are reported. n
PLS–flip-exchange size of the proportion of efficient output set solutions found
100 200 300 400 500 600 700 800 900 1000
233 851 1739 2897 4425 6106 7954 10058 12322 14817
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
328 1079 2148 3634 5432 7411 9826 12148 14908 17999
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
410 1406 2751 4831 7074 9729 12684 16039 19540 23328
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
602 1966 4090 6816 10131 14257 18194 22662 27634 33165
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
1103 3603 7329 12080 18100 25029 31433 38945 47068 56139
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
PLS–flip size of the proportion of efficient number of missing output set solutions found efficient solutions ρ = −0.8 215 0.9 18.3 769 0.9 81.7 1580 0.9 159.7 2600 0.9 296.7 4021 0.9 403.7 5562 0.9 543.4 7169 0.9 784.8 9086 0.9 972.2 11166 0.9 1155.9 13381 0.9 1435.4 ρ = −0.4 248 0.8 80.2 810 0.8 268.2 1607 0.8 540.1 2680 0.7 953.7 4110 0.8 1321.7 5614 0.8 1796.9 7232 0.7 2593.5 9062 0.7 3086.2 11068 0.7 3839.8 13328 0.7 4670.4 ρ = 0.0 273 0.7 137.1 896 0.6 509.7 1745 0.6 1005.8 2956 0.6 1875.2 4328 0.6 2746.7 5947 0.6 3781.7 7966 0.6 4718.2 9848 0.6 6190.5 12001 0.6 7539.1 14344 0.6 8983.7 ρ = 0.4 345 0.6 257.5 1106 0.6 859.9 2291 0.6 1798.5 3639 0.5 3177.1 5260 0.5 4870.5 7489 0.5 6768.4 9593 0.5 8600.6 12015 0.5 10647.3 14801 0.5 12832.6 17501 0.5 15663.6 ρ = 0.8 578 0.5 524.3 1777 0.5 1826.4 3598 0.5 3730.6 5678 0.5 6401.9 8204 0.5 9895.4 11784 0.5 13245.3 14511 0.5 16922.8 18262 0.5 20682.9 22236 0.5 24831.5 26077 0.5 30061.9
16
epsilon value 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.4 1.4 1.4 1.4 1.4 1.4
Table 11: Results obtained by PLS–flip-exchange and PLS–flip for BKP-BC instances (k = n/2). Rounded average values are reported. n
PLS–flip-exchange size of the proportion of efficient output set solutions found
100 200 300 400 500 600 700 800 900 1000
801 2828 5820 9767 14483 19802 26020 32753 40203 47953
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
1029 3449 7092 11916 17686 23768 31344 39527 48280 57775
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700 800 900 1000
1329 4714 9121 15651 22626 31665 41113 51422 62923 75109
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700
1811 6382 13137 21761 32283 44178 57525
1.0 1.0 1.0 1.0 1.0 1.0 1.0
100 200 300 400 500 600 700
3275 11561 23135 37945 54597 73824 93470
1.0 1.0 1.0 1.0 1.0 1.0 1.0
PLS–flip size of the proportion of efficient number of missing output set solutions found efficient solutions ρ = −0.8 780 1.0 21.7 2742 1.0 86.0 5628 1.0 191.3 9426 1.0 340.4 13992 1.0 490.9 19119 1.0 683.1 25049 1.0 970.6 31559 1.0 1194.9 38660 1.0 1542.9 46165 1.0 1788.0 ρ = −0.4 938 0.9 91.6 3123 0.9 325.9 6412 0.9 680.6 10711 0.9 1205.3 15982 0.9 1704.1 21448 0.9 2319.6 28082 0.9 3262.5 35385 0.9 4142.0 43218 0.9 5061.5 51859 0.9 5915.4 ρ = 0.0 1149 0.9 180.6 3996 0.8 717.6 7839 0.9 1281.2 13270 0.8 2381.4 19278 0.9 3347.7 26825 0.8 4839.5 34813 0.8 6299.8 43332 0.8 8090.1 53021 0.8 9901.3 63665 0.8 11444.0 ρ = 0.4 1525 0.8 286.3 5190 0.8 1192.8 10653 0.8 2484.1 17714 0.8 4046.7 25926 0.8 6357.4 35974 0.8 8203.4 46129 0.8 11395.7 ρ = 0.8 2699 0.8 576.3 9208 0.8 2352.9 18192 0.8 4942.1 29925 0.8 8019.2 42307 0.8 12289.5 58121 0.8 15702.7 72056 0.8 21414.3
17
epsilon value 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.2 1.2
100 DP PLS
10 1 0.1 0.01 0.001
DP PLS
10
CPU time (seconds)
CPU time (seconds)
100
1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
200
instance size (n)
400
600
800
1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 1000 DP PLS
CPU time (seconds)
100 10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 10000 DP PLS
100
DP PLS
1000 CPU time (seconds)
CPU time (seconds)
1000
10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
instance size (n)
200
400
600
800
1000
instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 8: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BKP-FC and PLS–exchange for BKP-FC instances (k = n/10).
18
1000 DP PLS
100 10 1 0.1 0.01 0.001
DP PLS
100
CPU time (seconds)
CPU time (seconds)
1000
10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
200
instance size (n)
400
600
800
1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 10000 DP PLS
CPU time (seconds)
1000 100 10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 10000
10000 DP PLS
DP PLS
1000 CPU time (seconds)
CPU time (seconds)
1000 100 10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
instance size (n)
200
400
600
800
1000
instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 9: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BKP-FC and PLS–exchange for BKP-FC instances (k = n/5).
19
10000 DP PLS
100
DP PLS
1000 CPU time (seconds)
CPU time (seconds)
1000
10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
200
instance size (n)
400
600
800
1000
instance size (n)
(a) ρ = -0.8
(b) ρ = -0.4 10000 DP PLS
CPU time (seconds)
1000 100 10 1 0.1 0.01 0.001 0.0001 0
200
400
600
800
1000
instance size (n)
(c) ρ = 0.0 10000
10000 DP PLS
DP PLS
1000 CPU time (seconds)
CPU time (seconds)
1000 100 10 1 0.1 0.01 0.001
100 10 1 0.1 0.01 0.001
0.0001
0.0001 0
200
400
600
800
1000
0
instance size (n)
200
400
600
800
1000
instance size (n)
(d) ρ = 0.4
(e) ρ = 0.8
Figure 10: CPU time in seconds (average value and deviation, given in log-scale) of MDP-BKP-FC and PLS–exchange for BKP-FC instances (k = n/2).
20
Table 12: Results obtained by PLS–exchange for BKP-FC instances (k = n/10). Rounded average values are reported. n
size of the output set
100 200 300 400 500 600 700 800 900 1000
20 55 102 174 264 332 444 523 646 793
100 200 300 400 500 600 700 800 900 1000
56 179 329 523 833 1070 1407 1686 2114 2609
100 200 300 400 500 600 700 800 900 1000
92 317 624 1061 1565 2180 2876 3518 4301 5205
100 200 300 400 500 600 700 800 900 1000
167 566 1176 1989 2969 4122 5286 6662 8008 9693
100 200 300 400 500 600 700 800 900 1000
368 1265 2587 4468 6520 8978 11500 14228 17177 20461
proportion of efficient solutions found ρ = −0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = −0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
21
number of missing efficient solutions
epsilon value
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 0.0 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 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 0.0 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 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 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Table 13: Results obtained by PLS–exchange for BKP-FC instances (k = n/5). Rounded average values are reported. n
size of the output set
100 200 300 400 500 600 700 800 900 1000
41 123 234 403 557 746 1038 1297 1538 1863
100 200 300 400 500 600 700 800 900 1000
128 374 753 1302 1812 2456 3496 4118 5127 6136
100 200 300 400 500 600 700 800 900 1000
218 778 1439 2641 3784 5165 6633 8436 10346 12520
100 200 300 400 500 600 700 800 900 1000
398 1305 2736 4625 6910 9914 12304 15581 18775 22555
100 200 300 400 500 600 700 800 900 1000
855 2896 5948 9792 14626 20430 25334 31762 37785 45138
proportion of efficient solutions found ρ = −0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = −0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
22
number of missing efficient solutions
epsilon value
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 0.0 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 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 0.0 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 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 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Table 14: Results obtained by PLS–exchange for BKP-FC instances (k = n/2). Rounded average values are reported. n
size of the output set
100 200 300 400 500 600 700 800 900 1000
72 222 476 753 1116 1469 1981 2537 3140 3658
100 200 300 400 500 600 700 800 900 1000
226 732 1528 2512 3666 4902 6474 8493 10167 11982
100 200 300 400 500 600 700 800 900 1000
442 1540 2928 5000 7170 10215 13118 17248 21259 24423
100 200 300 400 500 600 700 800 900 1000
733 2726 5407 8989 13520 18478 23888 29925 36464 43494
100 200 300 400 500 600 700 800
1571 5776 11209 18526 27064 36432 46532 58005
proportion of efficient solutions found ρ = −0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = −0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.4 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 ρ = 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
23
number of missing efficient solutions
epsilon value
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 0.0 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 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 0.0 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 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 0.0 0.0 0.0 0.0
