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Minimizing total completion time in a two-machine no-wait flowshop with uncertain and bounded setup times

  • 1Corresponding Author: Muberra Allahverdi

    1Corresponding Author: Muberra Allahverdi 
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  • We address a two-machine no-wait flowshop scheduling problem with respect to the performance measure of total completion time. Minimizing total completion time is important when inventory cost is of concern. Setup times are treated separately from processing times. Furthermore, setup times are uncertain with unknown distributions and are within some lower and upper bounds. We develop a dominance relation and propose eight algorithms to solve the problem. The proposed algorithms, which assign different weights to the processing and setup times on both machines, convert the two-machine problem into a single-machine one for which an optimal solution is known. We conduct computational experiments to evaluate the proposed algorithms. Computational experiments reveal that one of the proposed algorithms, which assigns the same weight to setup and processing times, is superior to the rest of the algorithms. The results are statistically verified by constructing confidence intervals and test of hypothesis.

    Mathematics Subject Classification: Primary: 90; Secondary: 90B36.

    Citation:

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  • Figure 1.  Flowchart of the algorithms

    Figure 2.  The distributions used for generating $ s_{i, k} $ within $ Ls_{i, k} $ and $ Us_{i, k} $

    Figure 3.  Overall Avg. Error of Algorithms

    Figure 4.  Avg. Std. of Algorithms

    Figure 5.  Overall Avg. Error of Algorithms with respect to $H$

    Figure 6.  Overall Avg. Error of Algorithms with respect to $H$

    Figure 7.  Overall Avg. Error of Algorithm $ALG-6$ with respect to $H$

    Figure 8.  Overall Avg. Error of Algorithms with respect to distributions

    Table 1.  Error of Algorithms for Positive Linear Distribution

    $n$
    Algorithm $H$ 100 200 300 400 500 Avg.
    $ALG-1$ 20 9.08 9.11 9.17 9.17 9.27 9.16
    $ALG-2$ 9.38 9.37 9.28 9.34 9.3 9.33
    $ALG-3$ 3.05 2.98 2.95 2.96 2.97 2.98
    $ALG-4$ 4.99 4.93 4.88 4.9 4.9 4.92
    $ALG-5$ 4.85 4.81 4.76 4.81 4.83 4.79
    $ALG-6$ 0.1 0.02 0 0 0 0.02
    $ALG-7$ 4.23 4.1 4.02 4 3.95 4.06
    $ALG-8$ 2.44 2.35 2.44 2.44 2.44 2.42
    $ALG-1$ 30 9.17 9.33 9.42 9.4 9.4 9.34
    $ALG-2$ 9.5 9.56 9.48 9.47 9.47 9.50
    $ALG-3$ 3.04 3.1 3.03 3.03 3 3.04
    $ALG-4$ 5.04 5.07 4.97 4.98 4.98 5.01
    $ALG-5$ 4.81 4.93 4.95 4.95 4.95 4.92
    $ALG-6$ 0.08 0.01 0 0 0 0.02
    $ALG-7$ 4.2 4.12 4.02 4.04 3.96 4.07
    $ALG-8$ 2.48 2.45 2.46 2.49 2.49 2.47
    $ALG-1$ 40 9.37 9.47 9.49 9.54 9.56 9.49
    $ALG-2$ 9.65 9.69 9.64 9.63 9.67 9.66
    $ALG-3$ 3.01 3.07 3.06 3.05 3.08 3.05
    $ALG-4$ 5.06 5.06 5.06 5.05 5.06 5.06
    $ALG-5$ 4.93 5.01 4.96 5 5.05 4.99
    $ALG-6$ 0.09 0.02 0 0 0 0.02
    $ALG-7$ 4.24 4.17 4.03 4.1 4.12 4.13
    $ALG-8$ 2.47 2.51 2.53 2.51 2.57 2.52
    Avg. 4.80 4.80 4.78 4.79 4.79
     | Show Table
    DownLoad: CSV

    Table 2.  Error of Algorithms for Negative Linear Distribution

    $n$
    Algorithm $H$ 100 200 300 400 500 Avg.
    $ALG-1$ 20 9.08 9.13 9.22 9.26 9.27 9.19
    $ALG-2$ 9.38 9.26 9.33 9.35 9.27 9.32
    $ALG-3$ 3.05 2.97 3.04 3.02 2.99 3.01
    $ALG-4$ 4.99 4.85 4.92 4.9 4.87 4.91
    $ALG-5$ 4.75 4.73 4.89 4.92 4.86 4.83
    $ALG-6$ 0.1 0.02 0 0 0 0.02
    $ALG-7$ 4.23 4.1 4.06 4.07 3.95 4.08
    $ALG-8$ 2.44 2.41 2.44 2.5 2.43 2.44
    $ALG-1$ 30 9.17 9.37 9.37 9.37 9.41 9.34
    $ALG-2$ 9.5 9.51 9.53 9.5 9.51 9.51
    $ALG-3$ 3.04 3.05 3.04 3.04 3.05 3.04
    $ALG-4$ 5.04 5 5.03 5 4.99 5.01
    $ALG-5$ 4.81 4.92 4.9 4.94 4.93 4.90
    $ALG-6$ 0.08 0.01 0 0 0 0.02
    $ALG-7$ 4.2 4.15 4.1 4.05 4.01 4.10
    $ALG-8$ 2.48 2.54 2.46 2.55 2.53 2.51
    $ALG-1$ 40 9.37 9.37 9.5 9.51 9.55 9.46
    $ALG-2$ 9.65 9.54 9.59 9.64 9.62 9.61
    $ALG-3$ 3.01 2.96 3.04 3.09 3.07 3.03
    $ALG-4$ 5.06 4.97 5.01 5.06 5.05 5.03
    $ALG-5$ 4.93 4.85 4.95 4.97 5.01 4.94
    $ALG-6$ 0.09 0.01 0 0 0 0.02
    $ALG-7$ 4.24 4.12 4.08 4.07 4.08 4.12
    $ALG-8$ 2.47 2.44 2.58 2.54 2.57 2.52
    Avg. 4.80 4.76 4.80 4.81 4.79
     | Show Table
    DownLoad: CSV

    Table 3.  Error of Algorithm for Uniform Distribution

    $n$
    Algorithm $H$ 100 200 300 400 500 Avg.
    $ALG-1$ 20 9.05 9.32 9.34 9.37 9.41 9.30
    $ALG-2$ 9.39 9.45 9.43 9.49 9.43 9.44
    $ALG-3$ 3.09 3.02 2.97 3.04 3.04 3.03
    $ALG-4$ 4.98 4.96 4.92 4.98 4.96 4.96
    $ALG-5$ 4.86 4.88 4.87 4.94 4.95 4.90
    $ALG-6$ 0.1 0.02 0 0 0 0.02
    $ALG-7$ 4.24 4.08 4.09 4.09 4.05 4.11
    $ALG-8$ 2.42 4.08 4.09 4.09 4.05 4.11
    $ALG-1$ 30 9.09 9.47 9.52 9.64 9.56 9.46
    $ALG-2$ 9.59 9.61 9.63 9.67 9.59 9.63
    $ALG-3$ 3.1 3.04 3.04 3.09 3.03 3.06
    $ALG-4$ 5.13 5.03 5.02 5.06 5 5.05
    $ALG-5$ 4.86 4.94 4.96 5.05 5.02 4.97
    $ALG-6$ 0.1 0.02 0 0 0 0.02
    $ALG-7$ 4.17 4.17 4.08 4.13 4.06 4.12
    $ALG-8$ 2.44 2.49 2.53 2.59 2.56 2.52
    $ALG-1$ 40 9.32 9.68 9.68 9.7 9.79 9.63
    $ALG-2$ 9.67 9.89 9.85 9.87 9.89 9.83
    $ALG-3$ 3.05 3.06 3.06 3.08 3.1 3.07
    $ALG-4$ 5.07 5.14 5.13 5.13 5.13 5.12
    $ALG-5$ 4.86 5.09 5.04 5.07 5.09 5.03
    $ALG-6$ 0.09 0.01 0 0 0 0.02
    $ALG-7$ 4.31 4.28 4.23 4.19 4.21 4.24
    $ALG-8$ 2.43 2.61 2.64 2.68 2.68 2.61
    Avg. 4.81 4.86 4.86 4.89 4.88
     | Show Table
    DownLoad: CSV

    Table 4.  Error of Algorithm for Normal Distribution

    $n$
    Algorithm $H$ 100 200 300 400 500 Avg.
    $ALG-1$ 20 9.04 9.3 9.32 9.35 9.38 9.28
    $ALG-2$ 9.5 9.46 9.45 9.46 9.5 9.47
    $ALG-3$ 3.03 3 3 3.03 3.06 3.02
    $ALG-4$ 5.01 4.97 4.92 4.96 4.95 4.90
    $ALG-5$ 4.81 4.88 4.9 4.94 4.95 4.90
    $ALG-6$ 0.08 0.01 0 0 0 0.02
    $ALG-7$ 4.27 4.09 4.13 4.08 4.06 4.13
    $ALG-8$ 2.47 2.46 2.45 2.5 2.55 2.49
    $ALG-1$ 30 9.43 9.43 9.54 9.57 9.71 9.54
    $ALG-2$ 9.8 9.71 9.66 9.7 9.72 9.72
    $ALG-3$ 3.11 3 3.09 3.08 3.12 3.08
    $ALG-4$ 5.16 5.06 5.06 5.05 5.11 5.09
    $ALG-5$ 4.99 4.92 5 4.99 5.08 5.00
    $ALG-6$ 0.1 0.02 0 0 0 0.02
    $ALG-7$ 4.38 4.15 4.17 4.17 4.17 4.21
    $ALG-8$ 2.52 2.52 2.59 2.57 2.62 2.56
    $ALG-1$ 40 9.57 9.7 9.77 9.78 9.86 9.74
    $ALG-2$ 10.04 9.96 9.9 9.86 9.86 9.92
    $ALG-3$ 3.12 3.03 3.05 3.07 3.05 3.06
    $ALG-4$ 5.25 5.17 5.14 5.11 5.12 5.16
    $ALG-5$ 5.01 5.03 5.05 5.11 5.1 5.06
    $ALG-6$ 0.08 0.01 0 0 0 0.02
    $ALG-7$ 4.47 4.2 4.21 4.24 4.19 4.26
    $ALG-8$ 2.69 2.6 2.73 2.71 2.7 2.69
    Avg. 4.91 4.86 4.88 4.89 4.91
     | Show Table
    DownLoad: CSV

    Table 5.  Error of Algoithms with respect to $ n $

    n
    Algorithm 100 200 300 400 500 Avg.
    $ALG-1$ 9.27 9.39 9.45 9.47 9.51 9.42
    $ALG-2$ 9.64 9.58 9.56 9.58 9.57 9.59
    $ALG-3$ 3.07 3.02 3.03 3.05 3.05 3.04
    $ALG-4$ 5.09 5.02 5.01 5.02 5.01 5.03
    $ALG-5$ 4.88 4.92 4.94 4.97 4.99 4.94
    $ALG-6$ 0.09 0.02 0.00 0.00 0.00 0.02
    $ALG-7$ 4.28 4.14 4.10 4.10 4.07 4.14
    $ALG-8$ 2.49 2.49 2.53 2.55 2.56 2.52
     | Show Table
    DownLoad: CSV

    Table 6.  Error of Algorithms with respect to $ H $

    $H$
    Algorithm 20 30 40 Avg.
    $ALG-1$ 9.24 9.42 9.59 9.42
    $ALG-2$ 9.40 9.60 9.77 9.59
    $ALG-3$ 3.01 3.06 3.06 3.04
    $ALG-4$ 4.94 5.04 5.10 5.03
    $ALG-5$ 4.86 4.94 5.01 4.94
    $ALG-6$ 0.02 0.02 0.02 0.02
    $ALG-7$ 4.10 4.13 4.19 4.14
    $ALG-8$ 2.46 2.52 2.59 2.52
     | Show Table
    DownLoad: CSV

    Table 7.  Confidence Intervals for Average Errors

    Algorithm 95$\%$ Confidence Interval on the Avg. Error
    $ALG-1$ (09.33-9.51)
    $ALG-2$ (9.50-9.68)
    $ALG-3$ (2.98-3.11)
    $ALG-4$ (4.95-5.10)
    $ALG-5$ (4.86-5.01)
    $ALG-6$ (0.02-0.03)
    $ALG-7$ (4.07-4.21)
    $ALG-8$ (2.45-2.59)
     | Show Table
    DownLoad: CSV
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