In this paper, we consider the parallel machine problems with sequence-dependent setup time and job arrivals in dynamics and the objective is to minimize the makespan.
For any specific problem, we characterize it as four features and using the modified parallel insertion algorithms to solve the problem. However, for the influence of jobs dynamic arriving, we develop the idle preventing mechanism to reduce the machine idle time caused by the stochastic inter-arrival time. Moreover, we apply the neural network and SVM to estimate the rescheduling criteria and values for parameters used in the modified parallel insertion algorithms and select an appropriate considered algorithm for a particular problem. The computational results show that our proposed approaches for choosing a proper algorithm and estimating the rescheduling criteria and values for parameters perform much better than deciding arbitrary. Applying neural network to estimate the rescheduling criteria and values for parameters obtains the even or better cases of 86.11% for all considered algorithms and SVM accomplishes the even or better cases of 92.90%. The results also display the performance of SVM is better than neural network in our testing problems, especially in the results of selecting the consider algorithm and getting the even or better cases of 88.89%. The proposed approaches may be applied to different types of scheduling problems and tested for various features of problems.
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Appendix
Table A 1. Summary of problem design
Job-type Tightness Setup Time Severity Setup Time Range Problem
No. 15 30 60 1975 1775 1708.3 4 2 1 2 1 0.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Job-type Tightness Setup Time Severity Setup Time Range
Table A 2. Real features of 81 problems
Problem No. T ST SR JR 44 322420.0645 0.9609 0.9718 30 45 322403.6161 1.0173 0.4899 30 46 358400.2723 3.9672 1.9553 30 47 358435.2037 3.9518 0.8198 30 48 358415.8067 3.9730 0.4702 30 49 358395.7453 1.8021 1.9745 30 50 358412.6671 1.9928 0.9262 30 51 358412.2981 2.0106 0.4830 30 52 358409.2526 0.8532 1.9782 30 53 358418.6830 1.0622 0.8445 30 54 358407.3905 0.9982 0.4784 30 55 310305.2614 4.0841 1.2622 60 56 310299.9887 4.0112 0.8511 60 57 310301.6925 3.9823 0.4544 60 58 310319.3757 2.0491 1.6770 60 59 310299.9080 2.2027 0.5535 60 60 310333.5706 1.9817 0.3464 60 61 310316.5424 1.0877 1.8578 60 62 310327.5719 1.0832 0.5967 60 63 310320.6761 1.0481 0.4827 60 64 322317.9139 3.7789 1.3252 60 65 322320.4340 3.9389 0.8498 60 66 322321.2408 4.0106 0.3874 60 67 322320.5954 1.8768 1.7241 60 68 322323.6802 2.1591 0.8503 60 69 322297.3168 1.9577 0.4381 60 70 322320.3534 0.9167 1.0152 60 71 322317.3492 0.8344 0.7242 60 72 322296.5812 1.0110 0.4360 60 73 358324.8904 3.8208 1.9011 60 74 358325.9488 3.8876 0.6983 60 75 358296.6618 4.0868 0.3199 60 76 358304.5353 2.0160 1.7431 60 77 358317.1878 2.0426 0.8427 60 78 358302.0152 1.9809 0.4035 60 79 358320.1113 0.8724 1.4782 60 80 358310.2114 0.9608 0.9495 60 81 358298.9399 1.0393 0.4181 60
Table A 3. The best rescheduling criterion and value for parameter for each algorithm
PIA-I PIA-II PIA-III PIA-IV
Table A 4. The makespan of each algorithm with the best rescheduling criteria and value for parameter and the best algorithm for each problem
Problem
No. PIA-I PIA-II PIA-III PIA-IV Best
Algorithm 1 112740.6072 122514.2697 112061.6115 122180.9924 PIA-III 2 109839.5459 118098.3491 108631.2987 116485.8495 PIA-III 3 116636.9068 118591.1993 111051.6052 125424.3713 PIA-III 4 88366.8958 96901.62309 89307.59095 92432.21312 PIA-I 5 93284.4870 96864.26208 94869.38649 101264.8968 PIA-I 6 98414.9048 98807.15599 95074.13531 99204.63112 PIA-III 7 89814.5885 89708.31368 87909.67825 88746.85128 PIA-III 8 86064.7380 87845.95273 87370.37218 87450.75238 PIA-I 9 88357.8945 91626.2354 88740.43535 89480.99432 PIA-I 10 100387.6688 111967.8208 100387.6688 108489.3183 PIA-I 11 109518.9357 114519.4534 100193.2667 114277.3759 PIA-III 12 112450.1267 121168.912 114549.8745 122513.229 PIA-I 13 98774.1072 100890.3783 96656.42491 102263.5047 PIA-III 14 95226.9439 95457.72776 90607.16086 102448.0536 PIA-III 15 102711.7323 108864.9743 103804.0783 110056.504 PIA-I 16 90331.4389 87701.39636 88624.09879 88043.77168 PIA-II 17 88077.2150 89557.51078 88077.21497 89557.51078 PIA-I 18 89761.0681 91909.53756 91535.61486 92823.84581 PIA-I 19 108959.1908 112943.0899 106279.0917 111257.9157 PIA-III 20 109808.9648 116252.2843 110837.9169 118780.5904 PIA-I 21 111108.5951 110358.7365 108205.7121 115297.4453 PIA-III 22 98363.0726 103281.755 97897.26305 103591.9883 PIA-III 23 98929.1437 105764.686 97877.70889 104077.5324 PIA-III 24 99212.6181 105884.3919 98282.76452 102100.1904 PIA-III 25 90641.1409 89658.35041 90187.11576 90560.84762 PIA-II 26 79142.9743 84162.25057 79650.58427 82476.72275 PIA-I 27 83192.9142 87267.49734 82539.74482 87888.7009 PIA-III 28 95421.4111 98119.20754 100405.8135 98180.32736 PIA-I 29 97716.5254 101701.2783 95691.09331 102185.7129 PIA-III 30 102505.2011 108219.3232 108910.75 107917.9137 PIA-I 31 85562.3423 90557.51296 87396.31865 83927.38296 PIA-IV 32 91704.7056 92063.10791 92165.21372 92920.14844 PIA-I 33 89619.9485 87910.33868 87215.32657 92288.84504 PIA-III 34 90071.8299 89378.45152 89106.63691 89134.33974 PIA-III 35 83606.8445 85143.00858 84496.23563 85755.5259 PIA-I 36 92607.8285 92387.61743 92451.02712 92748.50119 PIA-II 37 103260.5839 99142.80152 98886.84171 106156.9802 PIA-III 38 108788.5978 107678.0665 102544.6969 111218.8908 PIA-III 39 104342.4425 104696.8688 95430.54066 101220.9838 PIA-III 40 89594.7632 93585.1617 89537.94109 93289.36855 PIA-III
Problem
No. PIA-I PIA-II PIA-III PIA-IV Best
Algorithm 41 90616.3740 88892.12443 92156.43989 92457.91637 PIA-II 42 94606.6635 98040.40368 94761.46806 93414.12716 PIA-IV 43 86000.0793 86282.54365 87249.33977 86212.52229 PIA-I 44 81653.6883 84224.27522 82920.66797 81740.012 PIA-I 45 81591.0027 83398.52903 83093.37993 84093.65449 PIA-I 46 100904.1013 102604.9877 99812.79783 105258.6446 PIA-III 47 103314.2740 105289.372 107185.0767 110897.2789 PIA-I 48 111550.3743 113077.823 110106.1967 116149.8193 PIA-III 49 91205.1265 96337.56436 94871.16364 94778.35232 PIA-I 50 97516.1935 96460.79313 94612.60185 98335.30752 PIA-III 51 101366.6825 101138.58 97295.0429 102027.8805 PIA-III 52 86866.3709 85806.52795 86967.17457 86312.18491 PIA-II 53 88756.9071 87846.01942 92025.33803 86355.90305 PIA-IV 54 91160.1116 91160.1116 90917.26855 90682.5268 PIA-IV 55 95232.0633 98287.54894 94482.4952 97946.47661 PIA-III 56 104063.2170 106298.0642 106040.1797 104063.217 PIA-I 57 101184.1166 99053.48656 107430.2394 103138.0744 PIA-II 58 94944.3561 91200.10204 94736.66319 96068.70802 PIA-II 59 92435.8302 93385.82021 93385.82021 92450.3522 PIA-I 60 93808.9540 94878.923 96532.54317 92171.33746 PIA-IV 61 84589.9346 85133.39028 84618.06533 84618.06533 PIA-I 62 98549.5386 100000.4975 98549.53859 101152.801 PIA-I 63 91829.8356 92237.5216 91984.46392 91454.26666 PIA-IV 64 106131.5055 99318.8338 99762.77805 103922.846 PIA-II 65 99799.0007 100695.9995 96187.62584 92778.91546 PIA-IV 66 98042.3652 102386.6976 102137.1813 97529.21401 PIA-IV 67 96576.3577 97269.02674 94110.92992 94030.10507 PIA-IV 68 92126.9046 90006.26499 91166.65653 90433.79775 PIA-II 69 91915.3141 93327.64719 93723.44674 89520.0143 PIA-IV 70 87506.5192 86737.33932 86503.40365 86503.40365 PIA-III 71 88631.6868 88062.70139 88062.70139 88062.70139 PIA-III 72 93308.7476 93482.4465 93079.07779 93308.74756 PIA-III 73 103206.8625 101985.0213 98339.16515 98428.57852 PIA-III 74 103723.6989 102623.6727 107800.7126 95498.23371 PIA-IV 75 98759.2837 102769.3771 107916.6709 98759.28366 PIA-I 76 91865.6872 93139.32699 96035.58322 92982.18431 PIA-I 77 96459.7964 97299.3702 101840.8094 96459.79636 PIA-I 78 95726.1540 95379.01378 97793.35217 96205.15787 PIA-II 79 93420.1079 92152.04177 93838.48949 93992.04055 PIA-II 80 95905.7460 95540.93006 95672.10471 95761.44929 PIA-II 81 97603.4799 96126.98244 97481.77579 95460.53804 PIA-IV
Table A 5. The estimated rescheduling criterion and value for parameter by neural
PIA-I PIA-II PIA-III PIA-IV
Table A 6. The estimated rescheduling criterion and value for parameter by SVM
PIA-I PIA-II PIA-III PIA-IV
Table A 7. The makespan solved by each algorithm with estimated rescheduling criteria and values for parameters by neural network and SVM
PIA-I PIA-II PIA-III PIA-IV
Problem
No. NN SVM NN SVM NN SVM NN SVM
1 114759.78 112740.61 125551.28 122514.27 113645.82 112061.61 122226.35 122180.99 2 109839.55 109839.55 119474.75 119474.75 108631.30 108631.30 116485.85 116485.85 3 117894.92 116636.91 118591.20 118591.20 111051.61 127527.18 125424.37 125424.37 4 88366.90 88366.90 96901.62 96901.62 89307.59 89307.59 92432.21 100237.47 5 100591.80 93284.49 106979.65 106979.65 103473.02 103473.02 105711.99 105711.99 6 98414.90 98414.90 98807.16 98807.16 95074.14 95074.14 99204.63 99204.63 7 89814.59 89814.59 90416.41 89708.31 87909.68 87909.68 89700.44 90184.32 8 86064.74 86064.74 87845.95 87845.95 87370.37 87370.37 87450.75 87450.75 9 88357.89 88357.89 92701.43 92701.43 88740.44 88740.44 89480.99 89480.99 10 100387.67 100387.67 111967.82 111967.82 100387.67 100387.67 111735.12 111735.12 11 114078.86 109518.94 114519.45 114519.45 102100.43 100193.27 114277.38 114277.38 12 112450.13 112450.13 123302.31 123302.31 117608.44 114549.87 122513.23 122513.23 13 101933.70 98774.11 105674.54 105674.54 96854.59 96656.42 102263.50 107297.80 14 95226.94 95226.94 95457.73 95457.73 90607.16 90607.16 102448.05 102448.05 15 111817.91 102711.73 108864.97 108864.97 111011.01 103804.08 112781.93 110056.50 16 90331.44 90331.44 87701.40 87701.40 88624.10 88624.10 90767.72 88043.77 17 88077.21 88077.21 89557.51 89557.51 88077.21 88077.21 89557.51 89557.51 18 89761.07 89761.07 91909.54 91909.54 91535.61 91535.61 92823.85 92823.85 19 108959.19 108959.19 112943.09 112943.09 106279.09 106279.09 120576.80 111257.92 20 114639.10 109808.96 119988.44 116252.28 114920.16 110837.92 120599.08 118780.59 21 111108.60 111108.60 110358.74 110358.74 111262.51 108205.71 115297.45 115297.45 22 98363.07 98363.07 108187.42 103281.75 97897.26 97897.26 106580.92 103591.99 23 103880.19 98929.14 111211.15 111211.15 97877.71 106245.65 111211.15 104077.53 24 99288.54 99212.62 105884.39 105884.39 99416.68 98282.76 102100.19 102100.19 25 90641.14 90641.14 90775.89 89658.35 90187.12 90187.12 90737.30 90560.85 26 79142.97 79142.97 84162.25 84162.25 96412.48 79650.58 82476.72 82476.72 27 83192.91 83192.91 87267.50 87267.50 88318.26 82539.74 87888.70 87888.70 28 95421.41 95421.41 101178.89 107175.87 101939.87 100405.81 99794.62 99794.62 29 97716.53 97716.53 101701.28 101701.28 104402.30 104402.30 102185.71 102185.71 30 102505.20 102505.20 108219.32 108219.32 108944.65 108910.75 110389.94 107917.91 31 91274.96 85562.34 90803.96 90557.51 88089.90 87396.32 83927.38 83927.38 32 92683.19 91704.71 92063.11 92063.11 92165.21 92165.21 92920.15 92920.15 33 89619.95 89619.95 87910.34 87910.34 87215.33 87215.33 92288.85 92288.85 34 90071.83 90071.83 89946.11 89378.45 89106.64 89106.64 90954.43 90954.43 35 83606.84 83606.84 85708.31 85143.01 84496.24 84496.24 85755.53 85755.53 36 92607.83 92607.83 92387.62 92387.62 92451.03 92451.03 92748.50 92748.50 37 103260.58 103260.58 99806.59 99142.80 98886.84 105557.51 106156.98 106156.98 38 108788.60 108788.60 107678.07 107678.07 102544.70 102544.70 111218.89 111218.89 39 104342.44 104342.44 104696.87 104696.87 104765.16 95430.54 107300.07 101220.98 40 89594.76 89594.76 96246.21 93585.16 101697.92 89537.94 94293.93 94293.93
PIA-I PIA-II PIA-III PIA-IV Problem
No. NN SVM NN SVM NN SVM NN SVM
41 90616.37 90616.37 88892.12 88892.12 92156.44 92156.44 92457.92 92457.92 42 94606.66 94606.66 98040.40 98040.40 94761.47 94761.47 93414.13 93414.13 43 101596.52 86000.08 102385.89 101013.10 102385.89 102385.89 101035.53 86212.52 44 86211.49 81653.69 84224.28 84224.28 82920.67 82920.67 81740.01 81740.01 45 97733.82 81591.00 96930.14 97669.19 83093.38 83093.38 97733.82 97733.82 46 100904.10 100904.10 102604.99 102604.99 99812.80 99812.80 105258.64 105622.64 47 103314.27 103314.27 105289.37 105289.37 107185.08 107185.08 114298.91 114298.91 48 111550.37 111550.37 113077.82 113077.82 110106.20 110106.20 116149.82 116149.82 49 91205.13 91205.13 96337.56 97880.93 94871.16 94871.16 95866.83 94778.35 50 97516.19 97516.19 105045.93 96460.79 104803.91 94612.60 103985.56 98335.31 51 105699.48 101366.68 105996.37 105996.37 97295.04 97295.04 105898.70 102459.15 52 86866.37 86866.37 86993.22 85806.53 86967.17 86967.17 86910.91 86312.18 53 88756.91 88756.91 87846.02 96095.70 92025.34 92025.34 86355.90 86355.90 54 91160.11 91160.11 91160.11 91160.11 90917.27 90917.27 92767.52 92767.52 55 95232.06 95232.06 105545.96 105545.96 94482.50 94482.50 97946.48 97946.48 56 110627.28 104063.22 109591.12 115933.63 106040.18 106040.18 104063.22 104063.22 57 103618.53 101184.12 99053.49 99053.49 107430.24 107430.24 103618.53 103618.53 58 94944.36 94944.36 111345.90 91200.10 94736.66 108691.23 96068.71 96068.71 59 92435.83 92435.83 93385.82 93385.82 93385.82 93385.82 92450.35 92450.35 60 93808.95 93808.95 102536.26 94878.92 98796.63 96532.54 92231.40 92171.34 61 84589.93 84589.93 85133.39 85133.39 84618.07 84618.07 84618.07 84618.07 62 110893.86 98549.54 101152.80 111135.95 110893.86 110893.86 110893.86 101152.80 63 91829.84 91829.84 94059.90 92237.52 91984.46 91984.46 91829.84 91454.27 64 106131.51 106131.51 99318.83 99318.83 99762.78 99762.78 103922.85 103922.85 65 106189.72 99799.00 100696.00 100696.00 105668.57 105668.57 100501.33 100501.33 66 110210.70 98042.37 102386.70 102386.70 102137.18 102137.18 107015.98 107015.98 67 96576.36 96576.36 106346.92 97269.03 94110.93 106346.92 108634.25 94030.11 68 92126.90 92126.90 90006.26 90006.26 91166.66 91166.66 111548.29 90433.80 69 91915.31 91915.31 96938.75 93327.65 93723.45 93723.45 91915.31 91915.31 70 87506.52 87506.52 87387.16 86737.34 86503.40 86503.40 87506.52 86503.40 71 88631.69 88631.69 88062.70 88062.70 88062.70 88062.70 88062.70 88062.70 72 93308.75 93308.75 94514.20 93482.45 93079.08 93079.08 93308.75 93308.75 73 103206.86 103206.86 115733.27 101985.02 98339.17 98339.17 98428.58 98428.58 74 103723.70 103723.70 102623.67 102623.67 116244.96 107800.71 103554.31 95498.23 75 98759.28 98759.28 112955.48 102769.38 107916.67 107916.67 98759.28 98759.28 76 91865.69 91865.69 97048.59 93139.33 97937.17 96035.58 96119.94 95255.39 77 96459.80 96459.80 98639.40 97299.37 101840.81 101840.81 96459.80 96459.80 78 108659.27 95726.15 110689.08 95379.01 110939.18 97793.35 108659.27 108659.27 79 93420.11 93420.11 92152.04 92152.04 94928.41 93838.49 94068.99 93992.04 80 95905.75 95905.75 95741.09 95540.93 95672.10 95672.10 95892.82 95761.45 81 97603.48 97603.48 96126.98 96126.98 97481.78 97481.78 97603.48 95460.54
Table A 8. The estimated algorithm and makespan solved by the estimated algorithm with the estimated rescheduling criteria and values of parameters by neural network and
SVM
NN SVM
Problem
No. Estimated Algorithm Makespan Estimated Algorithm Makespan
1 3 113645.82 3 112061.61
NN SVM Problem
No. Estimated Algorithm Makespan Estimated Algorithm Makespan
40 3 101697.92 3 89537.94