# American Institute of Mathematical Sciences

December  2017, 7(4): 465-480. doi: 10.3934/naco.2017029

## A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine flow shop problem

 1 Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran 2 Department of Industrial and Systems Engineering, Isfahan University of Technology, 84156-83111, Isfahan, Iran

* Corresponding author

Received  December 2016 Revised  September 2017 Published  October 2017

Fund Project: The reviewing process of this paper was handled by Associate Editors A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey, at the occasion of 12th International Conference on Industrial Engineering (ICIE 2016), which was held in Tehran, Iran during 25-26 January, 2016.

In this paper, first, the problem of dynamic two-machine flow shop scheduling with the objective of minimizing the number of tardy jobs is investigated. Second, a hybrid algorithm based on genetic algorithm and parallel simulated annealing algorithm is presented. In order to solve large scale instances of the problem and to generate the initial population for the hybrid approach, a heuristic algorithm is also presented. To evaluate the efficiency of the proposed algorithms, they are compared with an optimal branch-and-bound algorithm which has been already developed in the literature. Computational experiments demonstrate that the proposed hybrid algorithm can solve the entire small-sized problems and more than 95% of medium-sized problems optimally.

Citation: Mostafa Abouei Ardakan, A. Kourank Beheshti, S. Hamid Mirmohammadi, Hamed Davari Ardakani. A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine flow shop problem. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 465-480. doi: 10.3934/naco.2017029
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The flowchart of the proposed hybrid meta-heuristic algorithm
An example for R-K method
Crossover operator
The differences between two groups of generated problems
 Influential Parameter High setting Low setting $r_j$ Jobs are available in a longer time interval after the beginning of scheduling Jobs are available in a short time interval which is closer to the close to the beginning of scheduling $D_j$ Tight due date intervals Loose due date intervals $D_j$ Different due dates Similar due dates Number of tardy jobs More tardy jobs Less tardy jobs
 Influential Parameter High setting Low setting $r_j$ Jobs are available in a longer time interval after the beginning of scheduling Jobs are available in a short time interval which is closer to the close to the beginning of scheduling $D_j$ Tight due date intervals Loose due date intervals $D_j$ Different due dates Similar due dates Number of tardy jobs More tardy jobs Less tardy jobs
Performance of PSA-GA and HEU1 algorithms in set High (compared with B & B)
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 n min avg max OPT# OPT(%) avg max avg max 5 0.00 0.09 0.17 20 100% 1.00 1.00 1.06 1.50 6 0.05 0.16 0.28 20 100% 1.00 1.00 1.11 1.33 7 0.02 0.20 0.37 20 100% 1.00 1.00 1.09 1.40 8 0.02 0.18 0.37 20 100% 1.00 1.00 1.10 1.20 9 0.02 0.21 0.45 20 100% 1.00 1.00 1.06 1.17 10 0.08 0.16 0.41 20 100% 1.00 1.00 1.12 1.29 12 0.02 0.06 0.50 20 $100\%$ 1.00 1.00 1.01 1.10 14 0.23 0.41 0.86 20 $100\%$ 1.00 1.00 1.12 1.25 16 0.33 0.70 1.15 19 $95\%$ 1.00 1.07 1.11 1.25 18 0.62 1.34 2.11 19 $95\%$ 1.01 1.07 1.04 1.13 20 0.44 0.90 1.67 19 $95\%$ 1.01 1.08 1.05 1.12
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 n min avg max OPT# OPT(%) avg max avg max 5 0.00 0.09 0.17 20 100% 1.00 1.00 1.06 1.50 6 0.05 0.16 0.28 20 100% 1.00 1.00 1.11 1.33 7 0.02 0.20 0.37 20 100% 1.00 1.00 1.09 1.40 8 0.02 0.18 0.37 20 100% 1.00 1.00 1.10 1.20 9 0.02 0.21 0.45 20 100% 1.00 1.00 1.06 1.17 10 0.08 0.16 0.41 20 100% 1.00 1.00 1.12 1.29 12 0.02 0.06 0.50 20 $100\%$ 1.00 1.00 1.01 1.10 14 0.23 0.41 0.86 20 $100\%$ 1.00 1.00 1.12 1.25 16 0.33 0.70 1.15 19 $95\%$ 1.00 1.07 1.11 1.25 18 0.62 1.34 2.11 19 $95\%$ 1.01 1.07 1.04 1.13 20 0.44 0.90 1.67 19 $95\%$ 1.01 1.08 1.05 1.12
Performance of PSA-GA and HEU1 algorithms in set Low (compared with B & B)
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 n min avg max OPT# OPT(%) avg max avg max 5 0.02 0.09 0.16 20 100% 1.00 1.00 1.10 1.33 6 0.06 0.15 0.31 20 100% 1.00 1.00 1.10 1.25 7 0.14 0.27 0.38 20 100% 1.00 1.00 1.05 1.20 8 0.11 0.20 0.34 20 100% 1.00 1.00 1.07 1.40 9 0.23 0.41 0.55 20 100% 1.00 1.00 1.06 1.33 10 0.11 0.29 0.55 20 100% 1.00 1.00 1.09 1.60 12 0.22 0.33 0.61 20 $100\%$ 1.00 1.00 1.07 1.43 14 0.22 0.48 0.67 20 $100\%$ 1.00 1.00 1.08 1.20 16 0.37 0.70 1.08 20 $100\%$ 1.00 1.00 1.09 1.17 18 0.70 1.14 1.62 19 $95\%$ 1.01 1.03 1.06 1.15 20 0.78 1.12 2.29 19 $95\%$ 1.01 1.03 1.07 1.12
 CPU time (Sec.) for PSA-GA Number of problems optimally solved* $Z^{opt}/ Z^{heu}$ PSA-GA $Z^{opt}/ Z^{heu}$ HEU1 n min avg max OPT# OPT(%) avg max avg max 5 0.02 0.09 0.16 20 100% 1.00 1.00 1.10 1.33 6 0.06 0.15 0.31 20 100% 1.00 1.00 1.10 1.25 7 0.14 0.27 0.38 20 100% 1.00 1.00 1.05 1.20 8 0.11 0.20 0.34 20 100% 1.00 1.00 1.07 1.40 9 0.23 0.41 0.55 20 100% 1.00 1.00 1.06 1.33 10 0.11 0.29 0.55 20 100% 1.00 1.00 1.09 1.60 12 0.22 0.33 0.61 20 $100\%$ 1.00 1.00 1.07 1.43 14 0.22 0.48 0.67 20 $100\%$ 1.00 1.00 1.08 1.20 16 0.37 0.70 1.08 20 $100\%$ 1.00 1.00 1.09 1.17 18 0.70 1.14 1.62 19 $95\%$ 1.01 1.03 1.06 1.15 20 0.78 1.12 2.29 19 $95\%$ 1.01 1.03 1.07 1.12
Performance of algorithms HEU1 and PSA-GA in large problems; set High
 CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ n min avg max min avg max avg max 30 1.06 2.16 3.45 0.00 0.00 0.01 1.16 1.26 40 1.44 2.01 2.87 0.01 0.01 0.01 1.13 1.31 50 2.79 4.90 6.72 0.01 0.02 0.03 1.11 1.20 60 5.01 7.96 11.06 0.00 0.01 0.04 1.11 1.21 70 6.44 10.63 14.45 0.01 0.02 0.04 1.11 1.25 80 11.29 18.65 26.46 0.03 0.04 0.06 1.08 1.22 90 10.42 23.23 36.60 0.02 0.04 0.07 1.07 1.14 100 13.29 19.39 24.73 0.03 0.05 0.09 1.06 1.10
 CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ n min avg max min avg max avg max 30 1.06 2.16 3.45 0.00 0.00 0.01 1.16 1.26 40 1.44 2.01 2.87 0.01 0.01 0.01 1.13 1.31 50 2.79 4.90 6.72 0.01 0.02 0.03 1.11 1.20 60 5.01 7.96 11.06 0.00 0.01 0.04 1.11 1.21 70 6.44 10.63 14.45 0.01 0.02 0.04 1.11 1.25 80 11.29 18.65 26.46 0.03 0.04 0.06 1.08 1.22 90 10.42 23.23 36.60 0.02 0.04 0.07 1.07 1.14 100 13.29 19.39 24.73 0.03 0.05 0.09 1.06 1.10
Performance of algorithms HEU1 and PSA-GA in large problems; set Low
 CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ n min avg max min avg max avg max 30 1.53 2.52 3.89 0.00 0.00 0.01 1.06 1.13 40 2.78 4.16 6.07 0.01 0.01 0.01 1.05 1.10 50 4.54 6.83 9.19 0.01 0.01 0.03 1.05 1.10 60 6.25 9.65 12.78 0.00 0.02 0.03 1.03 1.06 70 8.42 12.07 19.63 0.02 0.03 0.04 1.03 1.09 80 13.34 17.40 28.78 0.01 0.04 0.08 1.04 1.08 90 15.34 22.24 28.78 0.03 0.06 0.09 1.03 1.07 100 17.82 38.95 58.25 0.05 0.10 0.14 1.03 1.07
 CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$ n min avg max min avg max avg max 30 1.53 2.52 3.89 0.00 0.00 0.01 1.06 1.13 40 2.78 4.16 6.07 0.01 0.01 0.01 1.05 1.10 50 4.54 6.83 9.19 0.01 0.01 0.03 1.05 1.10 60 6.25 9.65 12.78 0.00 0.02 0.03 1.03 1.06 70 8.42 12.07 19.63 0.02 0.03 0.04 1.03 1.09 80 13.34 17.40 28.78 0.01 0.04 0.08 1.04 1.08 90 15.34 22.24 28.78 0.03 0.06 0.09 1.03 1.07 100 17.82 38.95 58.25 0.05 0.10 0.14 1.03 1.07
Results of the HEU1 algorithm; large-sized problem instances
 CPU time (Sec.) set High CPU times(Sec.) set Low n min avg max min avg max 200 0.08 0.13 0.17 0.22 0.25 0.30 400 0.70 1.05 1.33 1.64 1.85 2.06 600 2.31 3.03 3.64 5.60 6.13 6.55 60 6.25 9.65 12.78 0.00 0.02 0.03 800 5.53 7.10 8.64 12.97 14.46 19.42 1000 11.16 13.98 16.37 25.55 32.08 43.46
 CPU time (Sec.) set High CPU times(Sec.) set Low n min avg max min avg max 200 0.08 0.13 0.17 0.22 0.25 0.30 400 0.70 1.05 1.33 1.64 1.85 2.06 600 2.31 3.03 3.64 5.60 6.13 6.55 60 6.25 9.65 12.78 0.00 0.02 0.03 800 5.53 7.10 8.64 12.97 14.46 19.42 1000 11.16 13.98 16.37 25.55 32.08 43.46
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