Research Article
Exploring Further Advantages in an Alternative Formulation for the Set Covering Problem
Table 4
RPD metrics, the average total execution time, and average number of evaluations obtained during the first study for ACO algorithm.
| | Instance | | d-ACO | n-ACO | | | (%) | (%) | (%) | | | (%) | (%) | (%) |
| | 4_1 | 429 | 1880 | 845 | 0.23 | 0.23 | 0.23 | 1350 | 109 | 0.00 | 0.00 | 0.00 | | 4_2 | 512 | 3950 | 900 | 0.20 | 0.00 | 0.59 | 2700 | 224 | 0.00 | 0.00 | 0.00 | | 4_3 | 516 | 1600 | 916 | 0.97 | 0.19 | 1.16 | 4200 | 704 | 0.19 | 0.00 | 0.78 | | 4_4 | 494 | 4850 | 880 | 0.40 | 0.20 | 0.81 | 2100 | 829 | 0.20 | 0.20 | 0.20 | | 4_5 | 512 | 1950 | 680 | 0.10 | 0.00 | 0.39 | 600 | 340 | 0.00 | 0.00 | 0.39 | | 4_6 | 560 | 3780 | 574 | 0.00 | 0.00 | 0.36 | 2625 | 240 | 0.00 | 0.00 | 0.00 | | 4_7 | 430 | 1430 | 858 | 0.70 | 0.00 | 1.16 | 1425 | 118 | 0.00 | 0.00 | 0.00 | | 4_8 | 492 | 2440 | 415 | 0.00 | 0.00 | 0.20 | 1350 | 97 | 0.00 | 0.00 | 0.00 | | 4_9 | 641 | 5120 | 932 | 1.25 | 0.31 | 2.34 | 5550 | 848 | 0.31 | 0.00 | 0.78 | | 4_10 | 514 | 1390 | 115 | 0.00 | 0.00 | 0.00 | 750 | 60 | 0.00 | 0.00 | 0.00 |
| | 5_1 | 253 | 4860 | 891 | 0.40 | 0.00 | 1.58 | 3000 | 526 | 0.00 | 0.00 | 1.19 | | 5_2 | 302 | 4970 | 945 | 1.49 | 0.66 | 2.32 | 2700 | 870 | 1.32 | 0.33 | 1.99 | | 5_3 | 226 | 2080 | 753 | 0.22 | 0.00 | 1.33 | 1725 | 125 | 0.00 | 0.00 | 0.00 | | 5_4 | 242 | 2780 | 245 | 0.00 | 0.00 | 0.00 | 1500 | 110 | 0.00 | 0.00 | 0.00 | | 5_5 | 211 | 1160 | 857 | 0.47 | 0.47 | 0.47 | 750 | 223 | 0.00 | 0.00 | 0.47 | | 5_6 | 213 | 640 | 59 | 0.00 | 0.00 | 0.00 | 300 | 25 | 0.00 | 0.00 | 0.00 | | 5_7 | 293 | 5490 | 901 | 0.51 | 0.34 | 1.02 | 1575 | 837 | 0.34 | 0.34 | 0.34 | | 5_8 | 288 | 3820 | 653 | 0.35 | 0.00 | 0.69 | 1650 | 122 | 0.00 | 0.00 | 0.00 | | 5_9 | 279 | 20 | 823 | 0.00 | 0.00 | 0.36 | 975 | 78 | 0.00 | 0.00 | 0.00 | | 5_10 | 265 | 2370 | 617 | 0.38 | 0.00 | 0.75 | 1350 | 112 | 0.00 | 0.00 | 0.00 |
| | 6_1 | 138 | 3050 | 802 | 1.45 | 0.00 | 1.45 | 2175 | 332 | 0.00 | 0.00 | 1.45 | | 6_2 | 146 | 1180 | 122 | 0.00 | 0.00 | 0.00 | 2025 | 164 | 0.00 | 0.00 | 0.00 | | 6_3 | 145 | 1530 | 128 | 0.00 | 0.00 | 0.00 | 900 | 70 | 0.00 | 0.00 | 0.00 | | 6_4 | 131 | 930 | 70 | 0.00 | 0.00 | 0.00 | 300 | 22 | 0.00 | 0.00 | 0.00 | | 6_5 | 161 | 1130 | 831 | 1.24 | 0.00 | 2.48 | 2025 | 378 | 0.00 | 0.00 | 1.86 |
| | a_1 | 253 | 2725 | 904 | 0.79 | 0.79 | 0.79 | 4875 | 976 | 0.40 | 0.40 | 0.40 | | a_2 | 252 | 4700 | 908 | 1.19 | 1.19 | 1.19 | 3000 | 967 | 0.40 | 0.00 | 0.79 | | a_3 | 232 | 2925 | 903 | 0.43 | 0.43 | 0.43 | 5700 | 971 | 0.43 | 0.43 | 0.43 | | a_4 | 234 | 1575 | 895 | 0.43 | 0.43 | 0.43 | 3150 | 444 | 0.00 | 0.00 | 0.43 | | a_5 | 236 | 1815 | 902 | 0.42 | 0.42 | 0.42 | 3675 | 478 | 0.00 | 0.00 | 0.85 |
| | b_1 | 69 | 360 | 45 | 0.00 | 0.00 | 0.00 | 450 | 45 | 0.00 | 0.00 | 0.00 | | b_2 | 76 | 370 | 37 | 0.00 | 0.00 | 0.00 | 900 | 93 | 0.00 | 0.00 | 0.00 | | b_3 | 80 | 320 | 34 | 0.00 | 0.00 | 0.00 | 300 | 33 | 0.00 | 0.00 | 0.00 | | b_4 | 79 | 380 | 38 | 0.00 | 0.00 | 0.00 | 450 | 44 | 0.00 | 0.00 | 0.00 | | b_5 | 72 | 10 | 1 | 0.00 | 0.00 | 0.00 | 150 | 16 | 0.00 | 0.00 | 0.00 |
| | c_1 | 227 | 2690 | 973 | 1.32 | 0.88 | 1.76 | 6225 | 1034 | 0.88 | 0.88 | 0.88 | | c_2 | 219 | 2780 | 979 | 0.91 | 0.91 | 0.91 | 3975 | 1056 | 0.91 | 0.91 | 0.91 | | c_3 | 243 | 3305 | 993 | 1.65 | 0.82 | 2.47 | 5775 | 1067 | 0.82 | 0.41 | 1.23 | | c_4 | 219 | 2400 | 235 | 0.00 | 0.00 | 0.00 | 3900 | 380 | 0.00 | 0.00 | 0.00 | | c_5 | 215 | 1305 | 979 | 0.47 | 0.00 | 0.93 | 2925 | 1052 | 0.47 | 0.47 | 0.47 |
| | d_1 | 60 | 415 | 1174 | 1.67 | 0.00 | 1.67 | 2025 | 473 | 0.00 | 0.00 | 0.00 | | d_2 | 66 | 1675 | 236 | 0.00 | 0.00 | 0.00 | 1800 | 350 | 0.00 | 0.00 | 0.00 | | d_3 | 72 | 775 | 1198 | 1.39 | 1.39 | 1.39 | 975 | 1019 | 1.39 | 0.00 | 1.39 | | d_4 | 62 | 235 | 25 | 0.00 | 0.00 | 0.00 | 150 | 20 | 0.00 | 0.00 | 0.00 | | d_5 | 61 | 600 | 63 | 0.00 | 0.00 | 0.00 | 675 | 75 | 0.00 | 0.00 | 0.00 |
| | nre_1 | 29 | 100 | 29 | 0.00 | 0.00 | 0.00 | 50 | 18 | 0.00 | 0.00 | 0.00 | | nre_2 | 30 | 1650 | 902 | 0.00 | 0.00 | 3.33 | 1400 | 423 | 0.00 | 0.00 | 0.00 | | nre_3 | 27 | 900 | 216 | 0.00 | 0.00 | 0.00 | 700 | 170 | 0.00 | 0.00 | 0.00 | | nre_4 | 28 | 1200 | 452 | 0.00 | 0.00 | 3.57 | 1000 | 260 | 0.00 | 0.00 | 0.00 | | nre_5 | 28 | 200 | 49 | 0.00 | 0.00 | 0.00 | 100 | 31 | 0.00 | 0.00 | 0.00 |
| | nrf_1 | 14 | 200 | 118 | 0.00 | 0.00 | 0.00 | 200 | 112 | 0.00 | 0.00 | 0.00 | | nrf_2 | 15 | 100 | 64 | 0.00 | 0.00 | 0.00 | 75 | 40 | 0.00 | 0.00 | 0.00 | | nrf_3 | 14 | 100 | 5464 | 7.14 | 0.00 | 7.14 | 50 | 4966 | 7.14 | 0.00 | 7.14 | | nrf_4 | 14 | 500 | 271 | 0.00 | 0.00 | 0.00 | 200 | 122 | 0.00 | 0.00 | 0.00 | | nrf_5 | 13 | 100 | 5511 | 7.69 | 7.69 | 7.69 | 100 | 5614 | 7.69 | 7.69 | 7.69 |
| | nrg_1 | 176 | 5750 | 1905 | 0.28 | 0.00 | 2.27 | 5350 | 1250 | 0.00 | 0.00 | 0.00 | | nrg_2 | 154 | 4800 | 2166 | 1.95 | 1.95 | 1.95 | 3425 | 2171 | 1.30 | 0.65 | 1.95 | | nrg_3 | 166 | 4350 | 2253 | 3.61 | 3.01 | 4.82 | 2775 | 2253 | 3.01 | 2.41 | 4.22 | | nrg_4 | 168 | 5250 | 2207 | 2.68 | 1.19 | 3.57 | 4100 | 2217 | 2.38 | 1.19 | 2.98 | | nrg_5 | 168 | 6450 | 2258 | 0.60 | 0.60 | 0.60 | 5500 | 1830 | 0.00 | 0.00 | 0.60 |
| | nrh_1 | 63 | 3650 | 5499 | 3.17 | 1.59 | 4.76 | 3525 | 5768 | 3.17 | 3.17 | 3.17 | | nrh_2 | 63 | 3300 | 5437 | 3.17 | 3.17 | 3.17 | 2100 | 5706 | 3.17 | 3.17 | 3.17 | | nrh_3 | 59 | 4200 | 5511 | 3.39 | 3.39 | 3.39 | 2750 | 5776 | 3.39 | 3.39 | 3.39 | | nrh_4 | 58 | 3600 | 5434 | 3.45 | 1.72 | 3.45 | 2200 | 5664 | 3.45 | 1.72 | 3.45 | | nrh_5 | 55 | 4250 | 2329 | 0.00 | 0.00 | 0.00 | 2975 | 1708 | 0.00 | 0.00 | 0.00 |
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