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

• * Corresponding author: Yuanguo Zhu
• 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: • • 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

 notations definitions $i=1, 2, ..., n$ the index of job $j=1, 2, ...$ the index of position $b=1, 2, ...$ the index of batch $l=1, 2, ...$, K the 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, ...

Table 2.  Results for small scale

 No GA HGA min max time(s) min max time(s) 1 3079 3361 356 3125 3349 301 2 2953 3415 338 2837 3437 312 3 2556 2889 313 2398 2735 263 4 3582 3764 359 3619 3923 329 5 2046 2358 271 2098 2491 254

Table 3.  Results for mesoscale

 No GA HGA min max time(s) min max time(s) 1 8466 8893 1603 8501 8697 1354 2 9320 9671 1754 9455 9612 1340 3 8323 8569 1625 8361 8538 1385 4 8736 9028 1701 8798 9110 1313 5 7954 8077 1590 7839 8154 1397

Table 4.  Results for large scale

 No GA HGA min max time(s) min max time(s) 1 21542 22634 8655 21696 21873 6320 2 19556 21391 8367 19374 20065 5966 3 21406 23767 8961 21250 22034 6257 4 21973 23891 9058 21630 22741 6299 5 20592 22378 8537 20063 20097 6035

Table 5.  Average relative percentage errors of GA and HGA

 $n$ GA 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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