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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 \(\mathrm{LPS}(r,o,n)>t\).

References

  • Adams, J., Balas, E., & Zawack, D. (1988). The shifting bottleneck procedure for job shop scheduling. Management Science, 34(3), 391–401.

    Article  Google Scholar 

  • Amin, J., Shafia, M. A., & Tavakkoli-Moghaddam, R. (2011). A hybrid algorithm based on particle swarm optimization and simulated annealing for a periodic job shop scheduling problem. The International Journal of Advanced Manufacturing Technology, 54(1–4), 309–322.

    Google Scholar 

  • Beasley, J. E. (1990). Or-library, http://people.brunel.ac.uk/~mastjjb/jeb/info.html.

  • Bertsimas, D., & Gamarnik, D. (1999). Asymptotically optimal algorithm for job shop scheduling and packet routing. Journal of Algorithms, 33(2), 296–318.

    Article  Google Scholar 

  • Bertsimas, D., Gamarnik, D., & Sethuraman, J. (2003). From fluid relaxations to practical algorithms for job shop scheduling: The holding cost objective. Operational Research, 51(5), 798–813.

    Article  Google Scholar 

  • Bertsimas, D., & Sethuraman, J. (2002). From fluid relaxations to practical algorithms for job shop scheduling: The makespan objective. Mathematical Programming, 92(1), 61–102.

    Article  Google Scholar 

  • Blackstone, J. H., Phillips, D. T., & Hogg, G. L. (1982). A state-of-the-art survey of dispatching rules for manufacturing job shop operations. International Journal of Production Research, 20(1), 27–45.

    Article  Google Scholar 

  • Boudoukh, T., Penn, M., & Weiss, G. (2001). Scheduling job shop with some identical or similar jobs. Journal of Scheduling, 4, 177–199.

    Article  Google Scholar 

  • Brauner, N., Crama, Y., Grigoriev, A., & van de Klundert, J. (2005). A framework for the complexity of high-multiplicity scheduling problems. Journal of Combinatorial Optimization, 9(3), 313–323.

    Article  Google Scholar 

  • Chekuri, C., & Khanna, S. (2004). Handbook of sheduling: Algorithms, models, and performance analysis. Approximation algorithms for minimizing average weighted completion time (Vol. 33431, pp. 11-1–11-30). Boca Raton, FL: CRC Press.

    Google Scholar 

  • Correa, J. R., Wagner, M. R. (2005). Lp-based online scheduling: From single to parallel machines. In M. Jnger & V. Kaibel (Eds.), Integer programming and combinatorial optimization. Lecture Notes in Computer Science, vol. 3509 (pp. 196–209). Berlin Heidelberg: Springer

  • Dai, J. G., & Weiss, G. (1996). Stability and instability of fluid models for certain re-entrant lines. Mathematics of Operations Research, 21(1), 115–134.

    Article  Google Scholar 

  • Dai, J. G., & Weiss, G. (2002). A fluid heuristic for minimizing makespan in job-shops. Operations Research, 50(4), 692–707.

    Article  Google Scholar 

  • Goemans, M., Queyranne, M., Schulz, A. S., Skutella, M., & Wang, Y. (2002). Single machine scheduling with release dates. SIAM Journal on Discrete Mathematics, 15(2), 165–192.

    Article  Google Scholar 

  • Goldberg, L. A., Mike, P., Aravind S., Elizabeth S. (1997). Better approximation guarantees for job-shop scheduling (pp. 599–608). In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: SODA ’97, Society for Industrial and Applied Mathematics.

  • Hall, N. G., Lee, T. E., & Posner, M. E. (2002). The complexity of cyclic shop scheduling problems. Journal of Scheduling, 5(4), 307–327.

    Article  Google Scholar 

  • Kechadi, M-Tahar, & Low, Kok Seng. (2013). Recurrent neural network approach for cyclic job shop scheduling problem. Journal of Manufacturing Systems, 32(4), 689–699.

    Article  Google Scholar 

  • Kimbrel, T., & Sviridenko, M. (2008). High-multiplicity cyclic job shop scheduling. Operations Research Letters, 36(5), 574–578.

    Article  Google Scholar 

  • Lee, T. E., & Posner, M. E. (1997). Performance measures and schedule patterns in periodic job shops. Operational Research, 45(1), 72–91.

    Article  Google Scholar 

  • Lenstra, J. K., & Rinnooy-Kan, A. H. G. (1979). Computational complexity of discrete optimization problems. Annals of Discrete Mathematics, 4(8), 121–140.

    Article  Google Scholar 

  • Leung, J. M. Y., Zhang, G., Yang, X., Mak, R., & Lam, K. (2004). Optimal cyclic multi-hoist scheduling: A mixed integer programming approach. Operations Research, 52(6), 965976.

    Article  Google Scholar 

  • Nowicki, E., & Smutnicki, C. (1996). A fast taboo search algorithm for the job shop problem. Management Science, 42(6), 797–813.

    Article  Google Scholar 

  • Savelsbergh, Martin W. P., Uma, R. N., Joel W. (1998). An experimental study of Lp-based approximation algorithms for scheduling problems (pp. 453–462). In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: SODA ’98, Society for Industrial and Applied Mathematics.

  • Sevast’janov, S. V. (1994). On some geometric methods in scheduling theory: A survey. Discrete Applied Mathematics, 55(1), 59–82.

    Article  Google Scholar 

  • Shmoys, D. B., Stein, C., & Wein, J. (1994). Improved approximation algorithms for shop scheduling problems. SIAM Journal Computing, 23(3), 617–632.

    Article  Google Scholar 

  • Skutella, M. (2006). List scheduling in order of \(\alpha \)-points on a single machine. Lecture Notes in Computer Science, 3484 250291.

  • Sotskov, Y. N., & Shakhlevich, N. V. (1995). Np-hardness of shop-scheduling problems with three jobs. Discrete Applied Mathematics, 59(3), 237–266.

    Article  Google Scholar 

  • Weiss, G. (2008). A simplex based algorithm to solve separated continuous linear programs. Mathematical Programming Series A, 115(1), 151–198.

    Article  Google Scholar 

  • Williamson, D. P., Hall, L. A., Hoogeveen, J. A., Hurkens, C. A. J., Lenstra, J. K., Sevast’janov, S. V., et al. (1997). Short shop schedules. Operations Research, 45(2), 288–294.

    Article  Google Scholar 

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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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  • DOI: https://doi.org/10.1007/s10951-014-0376-y

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