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

  • * Corresponding author

    * Corresponding author 
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.
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  • 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.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  The flowchart of the proposed hybrid meta-heuristic algorithm

    Figure 2.  An example for R-K method

    Figure 3.  Crossover operator

    Table 1.  The differences between two groups of generated problems

    Influential ParameterHigh settingLow setting
    $r_j$Jobs are available in a longer time interval after the beginning of schedulingJobs are available in a short time interval which is closer to the close to the beginning of scheduling
    $D_j$Tight due date intervalsLoose due date intervals
    $D_j$Different due datesSimilar due dates
    Number of tardy jobsMore tardy jobsLess tardy jobs
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    Table 2.  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
    nminavgmaxOPT#OPT(%)avgmaxavgmax
    50.000.090.1720100%1.001.001.061.50
    60.050.160.2820100%1.001.001.111.33
    70.020.200.3720100%1.001.001.091.40
    80.020.180.3720100%1.001.001.101.20
    90.020.210.4520100%1.001.001.061.17
    100.080.160.4120100%1.001.001.121.29
    120.020.060.5020 $100\%$1.001.001.011.10
    140.230.410.8620 $100\%$1.001.001.121.25
    160.330.701.1519 $95\%$1.001.071.111.25
    180.621.342.1119$95\%$1.011.071.041.13
    200.440.901.6719 $95\%$1.011.081.051.12
     | Show Table
    DownLoad: CSV

    Table 3.  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
    nminavgmaxOPT#OPT(%)avgmaxavgmax
    50.020.090.1620100%1.001.001.101.33
    60.060.150.3120100%1.001.001.101.25
    70.140.270.3820100%1.001.001.051.20
    80.110.200.3420100%1.001.001.071.40
    90.230.410.5520100%1.001.001.061.33
    100.110.290.5520100%1.001.001.091.60
    120.220.330.6120 $100\%$1.001.001.071.43
    140.220.480.6720 $100\%$1.001.001.081.20
    160.370.701.0820 $100\%$1.001.001.091.17
    180.701.141.6219$95\%$1.011.031.061.15
    200.781.122.2919 $95\%$1.011.031.071.12
     | Show Table
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    Table 4.  Performance of algorithms HEU1 and PSA-GA in large problems; set High

    CPU time (Sec.) for PSA-GACPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$
    nminavgmaxminavgmaxavgmax
    301.062.163.450.000.000.011.161.26
    401.442.012.870.010.010.011.131.31
    502.794.906.720.010.020.031.111.20
    605.017.9611.060.000.010.041.111.21
    706.4410.6314.450.010.020.041.111.25
    8011.2918.6526.460.030.040.061.081.22
    9010.4223.2336.600.020.040.071.071.14
    10013.2919.3924.730.030.050.091.061.10
     | Show Table
    DownLoad: CSV

    Table 5.  Performance of algorithms HEU1 and PSA-GA in large problems; set Low

    CPU time (Sec.) for PSA-GA CPU times(Sec.) for HEU1 $Z^{PSA-GA}/ Z^{HEU1}$
    nminavgmaxminavgmaxavgmax
    301.532.523.890.000.000.011.061.13
    402.784.166.070.010.010.011.051.10
    504.546.839.190.010.010.031.051.10
    606.259.6512.780.000.020.031.031.06
    708.4212.0719.630.020.030.041.031.09
    8013.3417.4028.780.010.040.081.041.08
    9015.3422.2428.780.030.060.091.031.07
    10017.8238.9558.250.050.100.141.031.07
     | Show Table
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    Table 6.  Results of the HEU1 algorithm; large-sized problem instances

    CPU time (Sec.) set High CPU times(Sec.) set Low
    nminavgmaxminavgmax
    2000.080.130.170.220.250.30
    4000.701.051.331.641.852.06
    6002.313.033.645.606.136.55
    606.259.6512.780.000.020.03
    8005.537.108.6412.9714.4619.42
    100011.1613.9816.3725.5532.0843.46
     | Show Table
    DownLoad: CSV
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