Skip to main content
Log in

On some properties of optimal schedules in the job shop problem with preemption and an arbitrary regular criterion

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Optimal schedules in the job shop problem with preemption and with the objective of minimizing an arbitrary regular function of operation completion times are studied. It is shown that for any instance of the problem there always exists an optimal schedule that meets several remarkable properties. Firstly, each changeover date coincides with the completion time of some operation, and so, the number of changeover dates is not greater than the total number of operations, while the total number of interruptions of the operations is no more than the number of operations minus the number of jobs. Secondly, every changeover date is “super-integral”, which means that it is equal to the total processing time of some subset of operations. And thirdly, the optimal schedule with these properties can be found by a simple greedy algorithm under properly defined priorities of operations on machines. It is also shown that for any instance of the job shop problem with preemption allowed there exists a finite set of its feasible schedules which contains at least one optimal schedule for any regular objective function (from the continuum set of regular functions).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akers, S. B. (1956). A graphical approach to production scheduling problems. Operations Research, 4, 244–245.

    Article  Google Scholar 

  • Akers, S. B., & Friedman, J. (1955). A non-numerical approach to production scheduling problems. Operations Research, 3, 429–442.

    Google Scholar 

  • Bansal, N., Kimbrel, T., & Sviridenko, M. (2006). Job shop scheduling with unit processing times. Mathematics of Operations Research, 31, 381–389.

    Article  Google Scholar 

  • Baptiste, Ph., Carlier, J., Kononov, A., Queyranne, M., Sevastyanov, S., & Sviridenko, M. (2009). Structural properties of optimal solutions in preemptive scheduling. Diskretnyj Analiz i Issledovanie Operacij, 16(1), 3–36 (in Russian). English translation in: Journal of Applied and Industrial Mathematics, 4(4), 455–474 (2010).

    Google Scholar 

  • Baptiste, Ph., Carlier, J., Kononov, A., Queyranne, M., Sevastyanov, S., & Sviridenko, M. (2011). Integrality property in preemptive shop scheduling. Discrete Applied Mathematics, 159(5), 272–280.

    Article  Google Scholar 

  • Brucker, P., Kravchenko, S. A., & Sotskov, Y. N. (1997). Osnabrücker Schriften zur Mathematik: Heft 184. Preemptive job-shop scheduling problems with a fixed number of jobs. Osnabrück: Universität Osnabrück.

    Google Scholar 

  • Chemisova, D. (2009). On some properties of optimal schedules for the flow shop problem with preemption and an arbitrary regular criterion. Diskretnyj Analiz i Issledovanie Operacij, 16(3), 73–96 (in Russian).

    Google Scholar 

  • Czumaj, A., & Scheideler, C. (2000). A new algorithm approach to the general Lovasz local lemma with applications to scheduling and satisfiability problems (extended abstract). In Proceedings of the 32nd annual ACM symposium on theory of computing (STOC) (pp. 38–47).

    Google Scholar 

  • Feige, U., & Scheideler, C. (2002). Improved bounds for acyclic job shop scheduling. Combinatorica, 22(3), 361–399.

    Article  Google Scholar 

  • Glebov, N. I. (1968a). An algorithm for scheduling problem with two jobs. Upravlyaemye Sistemy, 1(1), 14–20 (in Russian).

    Google Scholar 

  • Glebov, N. I. (1968b). On an upper bound on schedule length for the case of two jobs. Problemy Kibernetiki, 20, 225–229 (in Russian).

    Google Scholar 

  • Glebov, N. I., & Perepelitsa, V. A. (1970). On the lower and upper bounds for a scheduling problem. In Issledovanija po Kibernetike (pp. 11–17). Moscow: Sovetskoe Radio (in Russian).

    Google Scholar 

  • Goldberg, L. A., Paterson, M., Srinivasan, A., & Sweedyk, E. (2001). Better approximation guarantees for job shop scheduling. SIAM Journal on Discrete Mathematics, 14(1), 67–92.

    Article  Google Scholar 

  • Hefetz, N., & Adiri, I. (1982). An efficient optimal algorithm for the two-machines, unit-time, job shop, schedule-length problem. Mathematics of Operations Research, 7, 354–360.

    Article  Google Scholar 

  • Henkin, V. E. (1966). On one question in scheduling theory (2×M sequencing problem). Kibernetika, 6(6), 67–71 (in Russian).

    Google Scholar 

  • Jackson, J. R. (1956). An extension of Johnson’s result on job lot scheduling. Naval Research Logistics Quarterly, 3, 201–203.

    Article  Google Scholar 

  • Jansen, K., Solis-Oba, R., & Sviridenko, M. (2003). Makespan minimization in job shops: a linear time approximation scheme. SIAM Journal on Discrete Mathematics, 16, 288–300.

    Article  Google Scholar 

  • Kononov, A., Sevastyanov, S., & Sviridenko, M. (2012). A complete 4-parametric complexity classification of short shop scheduling problems. Journal of Scheduling, 15(4), 427–446.

    Article  Google Scholar 

  • Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1993). Sequencing and scheduling: algorithms and complexity. In Handbooks in operations research and management science (Vol. 4, pp. 445–522). Amsterdam: North-Holland.

    Google Scholar 

  • Leighton, T., Maggs, B., & Rao, S. (1994). Packet routing and job shop scheduling in O(Congestion+Dilation) steps. Combinatorica, 14, 167–186.

    Article  Google Scholar 

  • Leighton, T., Maggs, B., & Richa, A. (1999). Fast algorithms for finding O(Congestion+Dilation) packet routing schedules. Combinatorica, 19(3), 375–401.

    Article  Google Scholar 

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

    Google Scholar 

  • Mastrolilli, M., & Svensson, O. (2008). (Acyclic) job shops are hard to approximate. In 49th annual IEEE symposium on foundations of computer science (pp. 583–592).

    Google Scholar 

  • Nagarajan, V., & Sviridenko, M. (2008). Tight bounds for permutation flow shop scheduling. In Proceedings of the 13th conference on integer programming and combinatorial optimization (IPCO) (pp. 154–168).

    Chapter  Google Scholar 

  • Potts, C., Shmoys, D., & Williamson, D. (1991). Permutation vs. nonpermutation flow shop schedules. Operations Research Letters, 10, 281–284.

    Article  Google Scholar 

  • Sevast’janov, S. V. (1986). An algorithm with an estimate for a problem with routings of parts of arbitrary shape and alternative executors. Cybernetics, 22, 773–781.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Sevastianov, S. (1998). Nonstrict vector summation in multi-operation scheduling. Annals of Operations Research, 83, 179–211.

    Article  Google Scholar 

  • Sevastianov, S. V., & Woeginger, G. J. (1998). Makespan minimization in preemptive two machine job shops. Computing, 60(1), 73–79.

    Article  Google Scholar 

  • Sevastyanov, S., Chemisova, D., & Chernykh, I. (2006). On some properties of optimal schedules for the Johnson problem with preemption. Diskretnyj Analiz i Issledovanie Operacij. Seriya 1, 13(3), 83–102 (in Russian).

    Google Scholar 

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

    Article  Google Scholar 

  • Tanaev, V. S., Sotskov, Yu. N., & Strusevich, V. A. (1994). Scheduling theory. Multi-stage systems. Dordrecht: Kluwer Academic.

    Book  Google Scholar 

  • Timkovsky, V. G. (2004). Reducibility among scheduling classes, Chap. 8. In J. Y.-T. Leung (Ed.), Chapman & Hall/CRC Computer and Information Science Series. Handbook of scheduling: algorithms, models, and performance analysis. Boca Raton: CRC Press.

    Google Scholar 

  • Williamson, D., Hall, L., Hoogeveen, J., Hurkens, C., Lenstra, J. K., Sevastianov, S., & Shmoys, D. (1997). Short shop schedules. Operations Research, 45(2), 288–294.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. V. Sevastyanov.

Additional information

Supported by the Russian Foundation for Basic Research (grants no. 08-01-00370 and 08-06-92000-HHC) and by Federal Target Grant “Scientific and educational personnel of innovation Russia” for 2009–2013 (government contract No. 14.740.11.0362).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sevastyanov, S.V., Chemisova, D.A. & Chernykh, I.D. On some properties of optimal schedules in the job shop problem with preemption and an arbitrary regular criterion. Ann Oper Res 213, 253–270 (2014). https://doi.org/10.1007/s10479-012-1290-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1290-3

Keywords

Navigation