Abstract
The paper deals with the problem of minimizing the expected makespan in a two-machine flow shop with blocking and random job processing times. It is well known that it reduces to an instance of the traveling salesman problem (TSP). Assuming that the job processing times can be stochastically ordered on both machines, we show that the problem under study is equivalent to TSP on a permuted Monge matrix. This allows us to prove that it is NP-hard for the independently and exponentially distributed job processing times, and identify a new class of efficiently solvable special cases.
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References
Allahverdi, A., “Stochastically minimizing total flowtime in flowshops with no waiting space”, European Journal of Operational Research, 113, 101–112 (1999).
Burkard, R. E., V. G. Deineko, R. Van Dal, J. A. A. Van der Veen and G. J. Woeginger, “Well-solvable special cases of the traveling salesman problem: a survey”, SIAM Rev. 40, 496–546 (1998).
Chang, C. S. and D. D. Yao, “Rearrangement, majorization and stochastic scheduling”, Mathematics of Operations Research, 18, 658–684 (1993).
Gilmore, P. C. and R. E. Gomory, “Sequencing a one state variable machine: a solvable case of the traveling salesman problem”, Operations Research, 12, 655–679 (1964).
Gilmore, P. C., E. L. Lawler and D. B. Shmoys, “Well-solved special cases,” In E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy-Kan and D. B. Shmoys (eds.), Chapter 4 in The Traveling Salesman Problem: a Guided Tour of Combinatorial Optimization. Wiley: Chichester, 1985, pp. 87–144.
Gourgand, M., N. Grangeon and S. Norre, “A contribution to the stochastic flow shop scheduling problem”, European Journal of Operational Research, 151, 415–433 (2003).
Hall, N. G. and C. Sriskandarajah, “A survey of machine scheduling problems with blocking and no-wait in process”, Operations Research, 44, 510–515 (1996).
Jia, C., “Minimizing variation in stochastic flow shop”, Operations Research Letters, 23, 109–111 (1998).
Kabadi, S. N., “Generalization of the Gilmore-Gomory traveling salesman problem and the Gilmore-Gomory scheme: a survey”, International Game Theory Review, 3, 213–235 (2001).
Park, J. K., “A special case of the n-vertex traveling salesman problem that can be solved in O(n) time”, Information Processing Letters, 40, 247–254 (1991).
Pinedo, M., “Minimizing the expected makespan is stochastic flow shops”, Operations Research, 30, 148–162 (1982).
Pinedo, M., Scheduling: Theory, Algorithms, and Systems. Prentice Hall: Upper Saddle, N.J., 2002.
Reddi, S.S. and C. V. Ramamoorthy, “On the flowshop sequencing problem with no-wait in process”, Operational Research Quarterly, 23, 323–331 (1972).
Sarvanov, V. I., “On the complexity of minimizing a linear form on a set of cyclic permutations,” Doklady AN SSSR, 253, 533–535 (1980) (in Russian) Soviet Math. Dokl., 22, 118–120 (in English).
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Kalczynski, P.J., Kamburowski, J. Two-Machine Stochastic Flow Shops With Blocking and the Traveling Salesman Problem. J Sched 8, 529–536 (2005). https://doi.org/10.1007/s10951-005-4782-z
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DOI: https://doi.org/10.1007/s10951-005-4782-z