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An uncertain programming model for single machine scheduling problem with batch delivery

  • * Corresponding author: Yuanguo Zhu

    * Corresponding author: Yuanguo Zhu
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  • A single machine scheduling problem with batch delivery is studied in this paper. The objective is to minimize the total cost which comprises earliness penalties, tardiness penalties, holding and transportation costs. An integer programming model is proposed and two dominance properties are obtained. However, sometimes due to the lack of historical data, the worker evaluates the processing time of a job according to his past experience. A pessimistic value model of the single machine scheduling problem with batch delivery under an uncertain environment is presented. Since the objective function is non-monotonic with respect to uncertain variables, a hybrid algorithm based on uncertain simulation and a g#enetic algorithm (GA) is designed to solve the model. In addition, two dominance properties under the uncertain environment are also obtained. Finally, computational experiments are presented to illustrate the modeling idea and the effectiveness of the algorithm.

    Mathematics Subject Classification: Primary: 90B99; Secondary: 90C10.

    Citation:

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  • Figure 1.  An example of crossover

    Figure 2.  An example of mutation

    Figure 3.  The sensitivity of the solution with respect to the confidence level

    Table 1.  List of notations

    notationsdefinitions
    $i=1, 2, ..., n$the index of job
    $j=1, 2, ...$the index of position
    $b=1, 2, ...$the index of batch
    $l=1, 2, ...$, Kthe index of customer
    $n_{l}$the number of jobs of customer $l$, l=1, 2, ..., K
    $p_i$processing time of job $i$, i=1, 2, ..., n
    $[d^{s}_{i}, d^{t}_{i}]$due window of job $i$, i=1, 2, ..., n
    $\alpha_i$unit earliness penalty cost of job $i$, i=1, 2, ..., n
    $\beta_i$unit tardiness penalty cost of job $i$, i=1, 2, ..., n
    $h_i$unit holding cost of job $i$, i=1, 2, ..., n
    $D_l$transportation cost of customer $l$, l=1, 2, ..., K
    $C_i$completion time of job $i$, i=1, 2, ..., n
    $T_j$completion time of the $j$th job on machine}, j=1, 2, ...
    $R_i$delivery date of job $i$, i=1, 2, ..., n
    $R^{'}_{lb}$delivery date of the $b$th batch of customer $l$, l=1, 2, ..., K; b=1, 2, ...
     | Show Table
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    Table 2.  Results for small scale

    NoGAHGA
    minmaxtime(s)minmaxtime(s)
    13079336135631253349301
    22953341533828373437312
    32556288931323982735263
    43582376435936193923329
    52046235827120982491254
     | Show Table
    DownLoad: CSV

    Table 3.  Results for mesoscale

    NoGAHGA
    minmaxtime(s)minmaxtime(s)
    1846688931603850186971354
    2932096711754945596121340
    3832385691625836185381385
    4873690281701879891101313
    5795480771590783981541397
     | Show Table
    DownLoad: CSV

    Table 4.  Results for large scale

    NoGAHGA
    minmaxtime(s)minmaxtime(s)
    12154222634865521696218736320
    21955621391836719374200655966
    32140623767896121250220346257
    42197323891905821630227416299
    52059222378853720063200976035
     | Show Table
    DownLoad: CSV

    Table 5.  Average relative percentage errors of GA and HGA

    $n$GAHGA
    $20$ $0.06266$ $0.05713$
    $50$ $0.03316$ $0.02685$
    $100$ $0.01295$ $0.007372$
    $200$ $0.005723$ $0.004993$
    Average$0.028623$ $0.024086$
     | Show Table
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