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Multiple-stage multiple-machine capacitated lot-sizing and scheduling with sequence-dependent setup: A case study in the wheel industry

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  • This paper studies a real-world problem of simultaneous lot-sizing and scheduling in a capacitated flow shop. The problem combines two significant characteristics in production which are multiple-stage production with heterogeneous multiple machines and sequence-dependent setup time. Setup time does not hold the triangle inequality, thus there may be a setup for a product without actual production. Consequently, a novel mixed integer programming (MIP) formulation is proposed and tested on real data sets of wheel production. Exact approaches cannot find a feasible solution for the model in a reasonable time, so MIP-based heuristics are developed to solve the model more quickly. Test results show that the formulation is able to contain the problem requirements and the heuristics are computationally effective. Moreover, the obtained solution can improve on a real practice at the plant.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Production process flow

    Figure 2.  Example of bill of materials from one type of first-stage product

    Figure 3.  A disconnected subtour and a main sequence

    Figure 4.  A subtour connected to a main sequence at the beginning of period

    Figure 5.  Relax and fix heuristic on multi-stage and over the periods

    Figure 6.  Comparison of total setup time between the company planning and our model

    Figure 7.  Comparison of total inventory level between the company planning and our model

    Figure 8.  Comparison of total overtime between the company planning and our model

    Table 1.  Average objective values in detailed

    q Setup time (sec) Inventory level (pieces) Overtime (sec)
    W=1000 W=100 W=10 W=1000 W=100 W=10 W=1000 W=100 W=10
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    Table 2.  Numerical results of small problems

    Problem size($N \times M \times T$) $<$1000 1000—4000 4000—6000
    MIP Heu. MIP Heu. MIP Heu.
    Avg. Time (sec)8716958352261090816461774
    Avg. Gap (%)3.945.675.448.636.719.22
    StDev. Gap1.814.061.886.192.733.36
     | Show Table
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    Table 3.  Numerical results of real problems by our heuristics

    Avg. Time(sec)Avg. LBDev(%)
    High variant of products family833018.54
    Low variant of products family27561.47
     | Show Table
    DownLoad: CSV

    Table 4.  Total objective value between the company solutions and our model solutions

     | Show Table
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    [2] B. Almada-loboD. KlabjanM. Antnia carravilla and J. F. Oliveira, Single machine multi-product capacitated lot sizing with sequence-dependent setups, International Journal of Production Research, 45 (2007), 4873-4894.  doi: 10.1080/00207540601094465.
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