Article Contents
Article Contents

# Effective heuristics for makespan minimization in parallel batch machines with non-identical capacities and job release times

• * Corresponding author: Zhao-Hong Jia
• We consider the problem of scheduling a set of $n$ jobs with arbitrary job sizes, processing times and release times on a set of $m$ parallel batch machines with non-identical capacities; the objective is to minimize the makespan. We first present an algorithm to compute a lower bound for the optimal makespan. Based on different rules of batching the jobs and assigning the batches to the machines, several heuristics are proposed to solve the problem. The performance of the proposed heuristics is evaluated by computational experiments. The proposed heuristics are compared against the lower bound and against each other. Our results show that the one of the proposed algorithms outperforms all the other heuristics.

Mathematics Subject Classification: Primary: 68T20, 68T05; Secondary: 05-04.

 Citation:

• Figure 1.  Comparison of solution quality and robustness on the problems with different numbers of jobs.

Table 1.  Parameters setting

 factors categories and values machine capacities $S_1=10, S_2=25, S_3=65$ machine numbers $m_1=5, m_2=3, m_3=2$ job numbers $n$ $n=\{90,108,126,144,162,180\}$ processing times of jobs $p_j$ U[5, 15] job arrival times $r_0=0,r_1=20,r_3=40,r_4=60$

Table 2.  Results for instances with 90 jobs.

 Ins. No. LB $R_{FF}$ $R_{FB}$ $R_{BF}$ $R_{BB}$ 1 75 1.33 1.33 1.33 0.00 2 75 5.33 4.00 5.33 0.00 3 75 4.00 2.67 4.00 0.00 4 75 4.00 4.00 4.00 2.67 5 75 6.67 4.00 5.33 2.67 6 75 13.33 10.67 13.33 8.00 7 75 10.67 8.00 10.67 1.33 8 75 10.67 10.67 10.67 6.67 9 75 1.33 1.33 4.00 0.00 10 75 8.00 8.00 8.00 4.00 Average 75 6.53 5.47 6.67 2.53

Table 3.  Results for instances with 108 jobs.

 Ins. No. LB $R_{FF}$ $R_{FB}$ $R_{BF}$ $R_{BB}$ 1 75 13.33 9.33 14.67 6.67 2 75 10.67 6.67 10.67 4.00 3 75 14.67 10.67 12.00 6.67 4 75 21.33 13.33 20.00 8.00 5 75 14.67 10.67 17.33 6.67 6 75 14.67 12.00 14.67 13.33 7 75 10.67 8.00 10.67 10.67 8 75 21.33 12.00 8.00 13.33 9 75 20.00 16.00 21.33 13.33 10 75 17.33 14.67 17.33 10.67 Average 75 15.87 11.33 14.67 9.33

Table 4.  Results for instances with 126 jobs.

 Ins. No. LB $R_{FF}$ $R_{FB}$ $R_{BF}$ $R_{BB}$ 1 80 21.25 16.25 20.00 12.50 2 75 34.67 30.67 32.00 20.00 3 75 28.00 24.00 24.00 16.00 4 75 36.00 32.00 34.67 22.67 5 83 21.69 18.07 20.48 16.87 6 75 29.33 28.00 28.00 21.33 7 78 24.36 20.51 23.08 14.10 8 79 27.85 22.79 26.58 16.46 9 77 27.27 18.18 25.97 11.69 10 77 22.08 20.78 22.08 19.48 Average 77.4 27.25 23.13 25.69 17.11

Table 5.  Results for instances with 144 jobs.

 Ins. No. LB $R_{FF}$ $R_{FB}$ $R_{BF}$ $R_{BB}$ 1 86 38.37 36.05 38.37 30.23 2 89 23.60 20.23 20.23 14.61 3 86 32.56 29.07 31.40 26.74 4 76 35.53 30.26 34.21 27.63 5 79 40.51 32.91 37.98 24.05 6 75 34.67 29.33 36.00 32.00 7 83 30.12 24.10 28.92 21.69 8 82 25.61 19.51 23.17 20.73 9 88 25.00 23.86 25.00 17.05 10 78 41.03 37.18 41.03 32.05 Average 82.2 32.70 28.25 31.63 24.68

Table 6.  Results for instances with 162 jobs.

 Ins. No. LB $R_{FF}$ $R_{FB}$ $R_{BF}$ $R_{BB}$ 1 82 37.81 32.93 37.81 28.05 2 91 39.56 32.97 39.56 31.87 3 87 34.48 29.89 31.03 27.59 4 88 27.27 25.00 27.27 21.59 5 88 31.82 30.68 32.96 25.00 6 80 38.75 33.75 37.50 31.25 7 79 27.85 27.85 27.85 24.05 8 83 33.74 33.74 36.15 31.33 9 81 39.51 34.57 37.04 29.63 10 90 30.00 26.67 27.78 21.11 Average 84.9 34.08 30.80 33.49 27.15

Table 7.  Results for instances with 180 jobs.

 Ins. No. LB $R_{FF}$ $R_{FB}$ $R_{BF}$ $R_{BB}$ 1 88 48.86 45.46 44.32 36.36 2 86 41.86 40.70 40.70 37.21 3 84 35.71 32.14 35.71 29.76 4 90 42.22 38.89 42.22 35.56 5 90 38.89 36.67 37.78 34.44 6 83 44.58 42.17 39.76 39.76 7 88 36.36 31.82 35.23 34.09 8 86 40.70 36.05 40.70 30.23 9 93 46.24 41.94 43.01 36.56 10 90 42.22 37.78 38.89 32.22 Average 87.8 41.77 38.36 39.83 34.62
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