Abstract
We study the two-machine flow shop problem with minimum delays. The problem is known to be strongly NP-hard even in the case of unit processing times and to be approximable within a factor of 2 of the length of an optimal schedule in the general case. The question whether there exists a polynomial-time algorithm with a better approximation ratio has been posed by several researchers but still remains open. In this paper, we improve the above bound to 3/2 for the special case of the problem when both operations of each job have equal processing times (this case of flow shop is known as the proportionate flow shop). Our analysis of the algorithm relies upon a nontrivial generalization of the lower bound established by W. Yu for the case of unit processing times.
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References
A. A. Ageev and A. E. Baburin, “Approximation Algorithms for UET Scheduling Problems with Exact Delays,” Oper. Res. Lett. 35(4), 533–540 (2007).
A. A. Ageev and A. V. Kononov, “Approximation Algorithms for Scheduling Problems with Exact Delays,” in: Lecture Notes in Computer Sciences, Vol. 4368: Approximation and Online Algorithms. 4th International Workshop, WAOA 2006 (Zurich, September 14–15, 2006) (Springer, Berlin, 2006), pp. 1–14.
M. Dell’Amico, “Shop Problems with Two Machines and Time Lags,” Oper. Res. 44(4), 777–787 (1996).
M. Dell’Amico and R. J. M. Vaessens, “Flow and Open Shop Scheduling on Two Machines with Transportation Times and Machine-Independent Processing Times is NP-Hard,” in Materiali di discussione 141 (Dipartimento di Economia Politica, Università diModena, 1996).
R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, “Optimization and Approximation in Deterministic Sequencing and Scheduling: A Survey,” Ann. of Discrete Math. 5, 287–326 (1979).
S. M. Johnson, “Optimal Two-and Three-Stage Production Schedules with Setup Times Included,” Naval Res. Logist. Quart. 1, 61–68 (1954).
S.M. Johnson, “Discussion: Sequencing n Jobs on Two Machines with Arbitrary Time Lags,” Management Sci. 5, 299–303 (1958).
W. Kern and W. M. Nawijn, “Scheduling Multi-Operation Jobs with Time Lags on a Single Machine,” in: Proceedings of the 2nd Workshop on Graphs and Combinatorial Optimization (Enschede, 1991).
J. K. Lenstra (private communication, 1991).
L. G. Mitten, “Sequencing n Jobs on Two Machines with Arbitrary Time Lags,” Management Sci. 5, 293–298 (1958).
V. A. Strusevich, “A Heuristic for the Two-Machine Open-Shop Scheduling with Transportation Times,” Discrete Appl. Math. 93(2–3), 287–304 (1999).
W. Yu, The Two-Machine Shop Problem with Delays and the One-Machine Total Tardiness Problem, PhD Thesis (Depart. Math. Comput. Sci., Technische Universiteit Eindhoven, 1996).
W. Yu, H. Hoogeveen and J. K. Lenstra, “Minimizing Makespan in a Two-Machine Flow Shop with Delays and Unit-Time Operations is NP-Hard,” J. Sched. 7(5), 333–348 (2004).
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Original Russian Text © A.A. Ageev, 2007, published in Diskretnyi Analiz i Issledovanie Operatsii, Ser. 1, 2007, Vol. 14, No. 4, pp. 3–15.
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Ageev, A.A. A 3/2-Approximation for the proportionate two-machine flow shop scheduling with minimum delays. J. Appl. Ind. Math. 2, 447–454 (2008). https://doi.org/10.1134/S1990478908040017
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DOI: https://doi.org/10.1134/S1990478908040017