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An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm

  • Corresponding author: Sadoullah Ebrahimnejad*

    Corresponding author: Sadoullah Ebrahimnejad* 
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  • The aim of this research is to study the dynamic facility layout and job-shop scheduling problems, simultaneously. In fact, this paper intends to measure the synergy between these two problems. In this paper, a multi-objective mixed integer nonlinear programming model has been proposed where areas of departments are unequal. Using a new approach, this paper calculates the farness rating scores of departments beside their closeness rating scores. Another feature of this paper is the consideration of input and output points for each department, which is crucial for the establishment of practical facility layouts in the real world. In the scheduling problem, transportation delay between departments and machines' setup time are considered that affect the dynamic facility layout problem. This integrated problem is solved using a hybrid two-phase algorithm. In the first phase, this hybrid algorithm incorporates the non-dominated sorting genetic algorithm. The second phase also applies two local search algorithms. To increase the efficacy of the first phase, we have tuned the parameters of this phase using the Taguchi experimental design method. Then, we have randomly generated 20 instances of different sizes. The numerical results show that the second phase of the hybrid algorithm improves its first phase significantly. The results also demonstrate that the simultaneous optimization of those two problems decreases the mean flow time of jobs by about 10% as compared to their separate optimization.

    Mathematics Subject Classification: Primary: 90B50; Secondary: 90B85, 90B35.

    Citation:

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  • Figure 1.  Two possible cases that could happen to determine the start time of a job

    Figure 2.  An illustrative example of a solution with 12 departments and 20 jobs

    Figure 3.  The candidate locations and departments arrangements for an example with 12 departments

    Figure 4.  An illustrative example of the calculation of PUS

    Figure 5.  The possible movements, and rotations in the local search for layout

    Figure 6.  The flowchart of the second phase of the hybrid algorithm (local search algorithms)

    Figure 7.  Illustrative examples of the violation of departments

    Figure 8.  Illustration of the solution found for the discrete facility layout of Instance 15

    Figure 9.  Illustration of the solution found for the continuous facility layout of Instance 15

    Figure 10.  Illustration of the solution found for the scheduling of Instance 15 at period 1 (Phase 1)

    Figure 11.  Illustration of the solution found for the scheduling of Instance 15 at period 1 (Phase 2)

    Figure 12.  The distribution and the interval estimation of the assessment metrics for separate optimization and simultaneous optimization

    Table 1.  The features and objectives studied in the literature

    Problem Rows Features Rows Objectives
    FLP [F1] Inequality of departments [O1] Material handling cost
    [F2] Input and output for departments [O2] Rearrangement cost of departments
    [F3] Multiple periods [O3] Desirability of closeness rating scores
    [F4] Continuous Optimization [O4] PUS
    [O5] Work in process
    JSS [F5] Setup time [O6] Makespan
    [F6] Transportation delay time [O7] Mean Flow Time (MFT)
    [F7] Multiple periods [O8] Earliness
    [F8] Due date of jobs [O9] Lateness
    [F9] Machine breakdown
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    Table 2.  A summary of the features for a number of studies published recently

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    Table 3.  Specifications of randomly generated instances

    Size of Instance No. of No. of No. of
    instances (No. of periods) departments machines jobs
    Small 1 (2), 11 (3) 3 3 3
    2 (2), 12 (3) 4 5 5
    3 (2), 13 (3) 5 7 7
    Medium 4 (2), 14 (3) 6 9 9
    5 (2), 15 (3) 8 11 11
    6 (2), 16 (3) 10 13 13
    Large-scale 7 (2), 17 (3) 12 16 16
    8 (2), 18 (3) 14 19 19
    9 (2), 19 (3) 16 21 21
    10 (2), 20 (3) 18 23 23
     | Show Table
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    Table 4.  The demand for products over different periods ([7])

    1 (*10) 2 (*10) 3 (*10)
    1 T(250,280,300) T(40, 50, 60) T(40, 50, 60)
    2 T(70, 75, 90) T(350,400,430) T(110,125,135)
    3 N(5, 56) N(2, 55) N(20,550)
    4 N(4, 40) N(4, 50) N(4, 70)
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    Table 5.  The levels of parameters defined for experiments

    Parameter Level of parameters
    Small size Medium size Large-scale
    I II III I II III I II III
    Iteration 60 80 100 80 100 120 100 150 200
    Initial population 10 20 30 30 40 50 80 100 120
    $ C_p $ 0.7 0.8 0.9 0.7 0.8 0.9 0.7 0.8 0.9
    $ M_p $ 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3
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    Table 6.  The optimal setting for the parameters of NSGA-II

    Parameter Size of instances
    Small Medium Large-scale
    Iteration 60 100 150
    Initial population 20 30 100
    $ C_p $ 0.7 0.7 0.8
    $ M_p $ 0.2 0.3 0.3
     | Show Table
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    Table 7.  The comparison of the traditional and proposed method of the PUS

    4 7 10 13 16 19
    Traditional method $ (\%) $ 35.2 48.8 57.3 34 55.6 51.6
    Proposed method $ (\%) $ 38.9 41.1 49.8 35.3 42 37.6
    Gap $ (\%) $ -10.6 15.9 13.1 -3.8 24.3 27
     | Show Table
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    Table 8.  Pareto solutions found by the Baron solver and the hybrid algorithm for Instance 1

    Row Baron solver Hybrid algorithm
    Obj. 1 Obj. 2 Obj. 3 (%) Obj. 4 Obj. 1 Obj. 2 Obj. 3 (%) Obj. 4
    1 596,320.6 0.2777 0.68 23.3751 223,500 0.58 24.9 22.851
    2 441,669.9 0 3.33 22.1017 224,616.6 0.57 24.7 22.798
    3 482,948.3 0.2148 1.70 22.2769 236,688.6 0.555 24.2 22.764
    4 227,695.7 0.2838 20.44 21.5836 249,431.5 0.54 23.1 22.693
    5 269,847.4 0.2532 20.93 21.3166 223,500 0.6 25.7 21.854
    6 227,756.6 0.28528 20.44 21.5832 375,100 0.3 25.3 22.903
    7 351,791.2 0.40051 9.68 21.9565 223,500 0.6 20.3 22.379
    8 268,928.5 0.25386 20.76 21.3141 375,100 0.45 20.6 22.903
    9 228,763.6 0.2868 20.45 21.5826 300,388.8 0.494 20.6 21.854
    10 228,571.3 0.2871 20.38 21.5841 379,873.8 0 20.6 21.64
    11 360,249.7 0.1717 40.14 22.5674 223,500 0.786 20 21.645
     | Show Table
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    Table 9.  The comparison of the solutions' quality for both the separate optimization and simultaneous optimization

    Instance QM MID DM SM
    Sep. Sim. $ \bar{d}_1 $ Sep. Sim. $ \bar{d}_2 $ Sep. Sim. $ \bar{d}_3 $ Sep. Sim. $ \bar{d}_4 $
    1 0.600 1 0.400 0.975 0.781 0.194 1.310 1.933 0.623 0.655 1.151 -0.496
    2 0.600 1 0.400 1.010 0.586 0.424 1.906 0.739 -1.167 1.454 1.730 -0.276
    3 0.500 0.750 0.250 0.994 1.269 -0.275 1.213 1.967 0.754 0.464 0.548 -0.084
    4 0.555 0.888 0.333 1.241 1.141 0.100 1.337 1.479 0.142 0.610 0.861 -0.251
    5 0.500 1 0.500 1.902 1.432 0.470 1.666 1.479 -0.187 1.037 1.524 -0.487
    6 0.428 0.714 0.286 1.342 1.286 0.056 1.555 1.294 -0.261 0.504 0.677 -0.173
    7 0.875 0.375 -0.500 1.107 1.287 -0.180 1.576 1.461 -0.115 0.415 0.985 -0.570
    8 0.500 1 0.500 0.893 0.624 0.269 1.324 1.636 0.312 0.602 0.854 -0.252
    9 0.600 1 0.400 1.365 1.017 0.348 1.521 1.241 -0.280 0.439 0.950 -0.511
    10 0.555 0.875 0.320 1.698 1.205 0.493 1.722 1.625 -0.097 0.520 0.991 -0.471
    11 0.666 1 0.334 1.031 0.743 0.288 0.883 1.397 0.514 0.999 1.986 -0.987
    12 1 1 0 0.863 1.041 -0.178 1.068 0.883 -0.185 0.080 1.278 -1.198
    13 0.500 1 0.500 2.631 2.115 0.516 1.536 1.625 0.089 0.268 0.790 -0.522
    14 0.666 1 0.334 1.656 1.328 0.328 1.658 1.031 -0.627 0.771 0.790 -0.019
    15 0.666 0.666 0 1.246 1.101 0.145 1.521 1.677 0.156 0.950 0.721 0.229
    16 0.666 0.500 -0.166 1.462 1.482 -0.020 1.409 1.324 -0.085 0.537 0.746 -0.209
    17 0.666 0.666 0 0.877 1.077 -0.200 1.624 1.359 -0.265 0.357 0.472 -0.115
    18 0.500 0.777 0.277 0.645 0.639 0.006 1.446 1.365 -0.081 0.698 0.685 0.013
    19 0.833 1 0.167 1.791 1.450 0.341 1.552 1.701 0.149 0.578 0.773 -0.195
    20 0.666 0.888 0.222 1.308 0.812 0.496 1.291 1.105 -0.186 0.661 0.808 -0.147
    Average 0.626 0.854 0.227 1.301 1.120 0.181 1.455 1.415 -0.039 0.629 0.960 -0.336
    Gap (%) 36.42 13.91 -2.74 -52.62
    Sep: Separate Optimization and Sim: Simultaneous Optimization
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    Table 10.  The comparison of the average unit of the MFT for both the separate optimization and simultaneous optimization

    1 2 3 4 5 6 7 8 9 10
    Separate 23.2 42.2 80.1 94.7 124 185 238.3 253.8 295.7 317.6
    Simultaneous 22.3 40.6 74.2 90.3 118.9 161.8 211.1 212.7 250.1 265.6
    Gap (%) 3.7 3.9 7.4 4.6 4.1 12.5 11.4 16.2 15.4 16.4
    11 12 13 14 15 16 17 18 19 20
    Separate 23.1 42.3 73 94.6 126.6 197.4 245.6 266.4 299.9 324.5
    Simultaneous 21.8 41 70.6 70.6 118.8 163.6 212.9 223.7 254.7 285.6
    Gap (%) 5.5 3.1 3.3 25.4 6.1 17.1 13.3 16 15 11.98
     | Show Table
    DownLoad: CSV
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