Mathematical Programming

, Volume 82, Issue 1–2, pp 255–271 | Cite as

Approximation algorithms for two-machine flow shop scheduling with batch setup times

  • Bo Chen
  • Chris N. Potts
  • Vitaly A. Strusevich


In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.


Scheduling Flow shop Batch setup times Approximation algorithm Performance analysis 


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  1. [1]
    E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, Sequencing and scheduling: Algorithms and complexity, in: S.C. Graves, A.H.G. Rinnooy Kan, P. Zipkin, (Eds.), Handbooks in Operations Research and Management Science, vol. 4: Logistics of Production and Inventory, North-Holland, Amsterdam, 1993, pp. 445–522.Google Scholar
  2. [2]
    C.L. Monma, C.N. Potts, On the complexity of scheduling with batch setup times, Operations Research 37 (1989) 798–804.Google Scholar
  3. [3]
    C.N. Potts, L.N. Van Wassenhove, Integrating scheduling with batching and lot-sizing: A review of algorithms and complexity, Journal of the Operational Research Society 43 (1992) 395–406.Google Scholar
  4. [4]
    S.M. Johnson, Optimal two- and three-stage production schedules with setup times included, Naval Research Logistics Quarterly 1 (1954) 61–68.Google Scholar
  5. [5]
    T. Yoshida, K. Hitomi, Optimal two-stage production scheduling with setup times separated, AIIE Transactions 11 (1979) 261–263.Google Scholar
  6. [6]
    U. Kleinau, Two-machine shop scheduling problems with batch processing, Mathematical and Computer Modelling 17 (1993) 55–66.Google Scholar
  7. [7]
    T. Gonzalez, S. Sahni, Open shop scheduling to minimize finish time, Journal of the Association for Computing Machinery 23 (1976) 665–679.Google Scholar
  8. [8]
    S. Zdrzałka, Analysis of approximation algorithms for single-machine scheduling with delivery times and sequence independent batch setup times, European Journal of Operational Research 80 (1995) 371–380.Google Scholar
  9. [9]
    C.L. Monma, C.N. Potts, Analysis of heuristics for preemptive parallel machine scheduling with batch setup times, Operations Research 41 (1993) 981–993.Google Scholar
  10. [10]
    B. Chen, A better heuristic for preemptive parallel machine scheduling with batch setup times, SIAM Journal on Computing 22 (1993) 1303–1318.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1998

Authors and Affiliations

  • Bo Chen
    • 1
  • Chris N. Potts
    • 2
  • Vitaly A. Strusevich
    • 3
  1. 1.Warwick Business SchoolUniversity of WarwickCoventryUK
  2. 2.Faculty of Mathematical StudiesUniversity of SouthamptonSouthamptonUK
  3. 3.School of Computing and Mathematical SciencesUniversity of GreenwichLondonUK

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