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

  • * Corresponding author: Yuanguo Zhu

    * Corresponding author: Yuanguo Zhu
Abstract Full Text(HTML) Figure(3) / Table(5) Related Papers Cited by
  • 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.


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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

    $i=1, 2, ..., n$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, ...
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    Table 2.  Results for small scale

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    Table 3.  Results for mesoscale

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    Table 4.  Results for large scale

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    Table 5.  Average relative percentage errors of GA and HGA

    $20$ $0.06266$ $0.05713$
    $50$ $0.03316$ $0.02685$
    $100$ $0.01295$ $0.007372$
    $200$ $0.005723$ $0.004993$
    Average$0.028623$ $0.024086$
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