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Linear programming-based algorithms for the minimum makespan high multiplicity jobshop problem

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Abstract

We study a generalized version of the minimum makespan jobshop problem in which multiple instances of each job are to be processed. The system starts with specified inventory levels in all buffers and finishes with some desired inventory levels of the buffers at the end of the planning horizon. A schedule that minimizes the completion time of all the operations is sought. We develop a polynomial time asymptotic approximation procedure for the problem. That is, the ratio between the value of the delivered solution and the optimal one converge into one, as the multiplicity of the problem increases. Our algorithm uses the solution of the linear relaxation of a time-indexed Mixed-Integer formulation of the problem. In addition, a heuristic method inspired by this approximation algorithm is presented and is numerically shown to out-perform known methods for a large set of standard test problems of moderate job multiplicity.

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Notes

  1. We present the results of Bertsimas and Sethuraman (2002) using our notation here for the sake of comparison. In their notation, \(U_\mathrm{max}\) is bounded from above by the number of job classes (\(I\)) times the maximal processing time \(P_\mathrm{max}\) and \(J_\mathrm{max} \equiv O_\mathrm{max}\).

  2. This may happen since, if a machine is ready at time \(t\), LPSA dispatches an available operation instance with the smallest \(\mathrm{LPS}(r,o,n)\) even if \(\mathrm{LPS}(r,o,n)>t\).

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Authors and Affiliations

  1. IBM Research – Haifa, Mount Carmel, 31905 , Haifa, Israel

    Michael Masin

  2. Department of Industrial Engineering, Tel Aviv University, 69978 , Tel Aviv, Israel

    Tal Raviv

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  1. Michael Masin
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  2. Tal Raviv
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Correspondence to Tal Raviv.

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Masin, M., Raviv, T. Linear programming-based algorithms for the minimum makespan high multiplicity jobshop problem. J Sched 17, 321–338 (2014). https://doi.org/10.1007/s10951-014-0376-y

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