# Table 4 Cluster-based decomposition in small instances 1-5. Solutions the successful combinations of Γ, k and η: number of sub-problems (#subp); percentage of jobs for which a solution is provided over the sub-problems (%sol); sum of OF, mM, mF, OVT, TT over the sub-problems with a solution; sum of the computational times over them (CT); maximum percent optimality gap over them (Gapmax)

Instance Input Overall output
(roverall) Γ k η #subp %sol OF mM mF OVT TT CT Gapmax
1 (0.464) I(2) I(1) 1 2 100% 316002123.25 50 321 24 55.25 3017 3.0⋅ 10− 6%
II(11) I(1) 1 8 97% 440002157.5 66 323 20 64.5 1037 2.1⋅ 10− 6%
I(2) II(9) 1 1 100% 232002285.25 37 337 49 57.25 TL 1.8⋅ 10− 5%
II(11) II(3) 1 5 93% 639001669.75 96 241 34 59.75 196 0.0%
I(2) I(1) 0.9 2 100% 316002121.75 53 317 19 53.75 3016 2.5⋅ 10− 6%
II(11) I(1) 0.9 8 98% 560002247.75 79 335 4 70.75 1108 1.1⋅ 10− 6%
I(2) II(9) 0.9 1 100% 232002285.25 37 337 49 57.25 TL 1.8⋅ 10− 5%
II(11) II(3) 0.9 5 93% 639001669.75 96 241 34 59.75 196 0.0%
2 (0.548) I(2) I(1) 1 1 100% 120002512.75 15 348 22 72.75 TL 1.1⋅ 10− 5%
II(10) I(1) 1 4 48% 1162.5 0 174 18 31.5 13 0.0%
I(2) II(9) 1 1 100% 120002512.75 15 348 22 72.75 TL 1.1⋅ 10− 5%
II(10) II(3) 1 5 100% 184002992.5 23 410 20 72.5 377 0.0%
I(2) I(1) 0.9 1 100% 120002512.75 15 348 22 72.75 TL 1.1⋅ 10− 5%
II(10) I(1) 0.9 7 100% 292002743 38 385 19 80 1264 2.1⋅ 10− 6%
I(2) II(9) 0.9 1 100% 120002512.75 15 348 22 72.75 TL 1.1⋅ 10− 5%
II(10) II(3) 0.9 6 100% 225002631.25 33 362 34 79.25 110 0.0%
3 (0.636) I(2) I(1) 1 2 100% 229003065 34 420 39 69 2693 2.6⋅ 10− 5%
II(10) I(1) 1 4 100% 267002942 38 420 18 75 2475 1.8⋅ 10− 5%
I(2) II(7) 1 1 100% 219002612.25 32 343 62 76.25 TL 1.6⋅ 10− 5%
II(10) II(5) 1 4 95% 279003171.5 37 416 51 66.5 2680 4.3⋅ 10− 4%
I(2) I(1) 0.9 2 100% 229003065 34 420 39 69 2693 2.6⋅ 10− 5%
II(10) I(1) 0.9 5 100% 267002957.75 38 414 43 70.75 2263 9.2⋅ 10− 3%
I(2) II(7) 0.9 2 95% 191003051.5 25 402 96 98.5 3314 7.4⋅ 10− 5%
II(10) II(5) 0.9 4 100% 272003285.25 36 455 63 64.25 2686 3.6%
4 (0.657) I(3) I(1) 1 3 100% 384002578.25 49 358 23 81.25 36 0.0%
I(3) II(5) 1 3 100% 495002511.50 70 357 11 72.50 131 0.0%
I(3) I(1) 0.9 3 100% 399002575 52 380 27 68 38 0.0%
I(3) II(5) 0.9 3 100% 495002511.50 70 357 11 72.50 129 0.0%
5 (0.666) I(2) I(1) 1 1 100% 138003514.50 20 487 7 69.50 TL 1.7⋅ 10− 5%
II(10) I(1) 1 4 100% 448002886.5 67 398 35 68.5 1850 1.2⋅ 10− 5%
I(2) II(9) 1 1 100% 138003514.50 20 487 7 69.50 TL 1.7⋅ 10− 5%
II(10) II(5) 1 4 100% 318003566.25 50 477 12 58.25 2004 1.8⋅ 10− 6%
I(2) I(1) 0.9 1 100% 138003514.50 20 487 7 69.50 TL 1.7⋅ 10− 5%
I(2) II(9) 0.9 1 100% 138003514.50 20 487 7 69.50 TL 1.7⋅ 10− 5%
II(10) II(5) 0.9 6 100% 615002806.5 89 380 19 84.5 1665 3.5⋅ 10− 6%
1. TL denotes the time limit of 3600 seconds, and the best solution among those with %sol = 100% is highlighted in bold