Skip to main content
SpringerLink
Log in

MP or not MP: that is the question

  • Published:
Journal of Scheduling Aims and scope Submit manuscript

Abstract

It is well known that in the twentieth century, mathematical programming (MP) modeling and particularly linear programming (LP) modeling, even though strongly applied to combinatorial optimization, were not too successful when directed to scheduling problems. The purpose of this paper is to show that the field of successful applications of LP/MP modeling is still growing and includes also scheduling topics. We first focus on single machine scheduling. We consider a single machine scheduling model where a quadratic programming (QP) formulation handled by means of a QP solver is shown to be competitive with the state of the art approaches. Also, we discuss a single machine bicriterion scheduling problem and show that a standard LP formulation based on positional completion times performs reasonably well when handled by means of a LP solver. Then, we show how LP can be used to tighten bounds for approximation results in sequencing problems. Finally, we show how to enhance the complexity bounds of branch-and-reduce exact exponential algorithms by means of the so-called measure-and-conquer paradigm requiring always the solution of a specific MP model.

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

Access this article

Log in via an institution

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

  • Azizoglu, M., Koksalan, M., & Koksalan, S. K. (2003). Scheduling to minimize maximum earliness and number of tardy jobs where machine idle time is allowed. Journal of the Operational Research Society, 54, 661–664.

    Article  Google Scholar 

  • Baker, K. R., & Keller, B. (2010). Solving the single-machine sequencing problem using integer programming. Computers and Industrial Engineering, 59, 730–735.

    Article  Google Scholar 

  • Baptiste, Ph, Della Croce, F., Grosso, A., & Tkindt, V. (2010). Sequencing a single machine with due dates and deadlines: An ILP-based approach to solve very large instances. Journal of Scheduling, 13, 39–47.

    Article  Google Scholar 

  • Blazewicz, J., Dror, M., & Weglarz, J. (1991). Mathematical programming formulations for machine scheduling: A survey. European Journal of Operational Research, 51, 283–300.

    Article  Google Scholar 

  • Boria, N., & Della Croce, F. (2014). Re-optimization in machine scheduling. Theoretical Computer Science, 540–541, 13–26.

    Article  Google Scholar 

  • Bourgeois, N., Escoffier, B., Paschos, V Th, & van Rooij, J. M. M. (2012). Fast algorithms for max independent set. Algorithmica, 62, 382–415.

    Article  Google Scholar 

  • Christofides, N. (1976). Worst-case analysis of a new heuristic for the traveling salesman problem, Technical report, GSIA, Carnegie-Mellon University, Pittsburgh.

  • Della Croce, F., Salassa, F., & T’kindt, V. (2014). A hybrid heuristic approach for single machine scheduling with release times. Computers and Operations Research, 45, 7–11.

    Article  Google Scholar 

  • Della Croce, F., & T’kindt, V. (2003). Improving the preemptive bound for the one-machine dynamic total completion time scheduling problem. Operations Research Letters, 31, 142–148.

    Article  Google Scholar 

  • Engels, D. W., Karger, D. R., Kolliopoulos, S. G., Sengupta, S., Uma, R. N., & Wein, J. (2003). Techniques for scheduling with rejection. Journal of Algorithms, 49, 175–191.

    Article  Google Scholar 

  • Eppstein, D. (2001). Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction, In Proceedings of Symposium on Discrete Algorithms, SODA01 (pp. 329–337).

  • Ergun, O., & Orlin, J. B. (2006). Fast neighborhood search for the single machine total weighted tardiness problem. Operations Research Letters, 34, 41–45.

    Article  Google Scholar 

  • Fomin, F. V., Grandoni, F., & Kratsch, D. (2009). A measure and conquer approach for the analysis of exact algorithms. Journal of the ACM, 56, 1–32.

  • Lasserre, J. B., & Queyranne, M. (1992). Generic scheduling polyhedral and a new mixed integer formulation for single machine scheduling. In Proceedings of the IPCO Conference (pp. 136–149).

  • Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1985). The traveling salesman problem: A guided tour of combinatorial optimization. New York: Wiley. (Ed.).

    Google Scholar 

  • Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., & Shmoys, D. B. (1993). Sequencing and scheduling: Algorithms and complexity. In S. C. Graves, A. H. G. Rinnooy Kan, & P. H. Zipkin (Eds.), Logistics of Production and Inventory (pp. 445–522). Amsterdam: North Holland.

    Chapter  Google Scholar 

  • Lee, C. Y., & Vairaktarakis, G. L. (1993). Complexity of single machine hierarchical scheduling: A survey. In P. M. Pardalos (Ed.), Complexity in numerical optimization (pp. 269–298). Singapore: World Scientific.

    Chapter  Google Scholar 

  • MHallah, R., & Bulfin, R. L. (2003). Minimizing the weighted number of tardy jobs on a single machine. European Journal of Operational Research, 145, 45–56.

    Article  Google Scholar 

  • Moghaddam, A., Amodeo, L., Yalaoui, F., & Karimi, B. (2012). Single machine scheduling with rejection: Minimizing total weighted completion time and rejection cost. International Journal of Applied Evolutionary Computation, 3, 42–61.

    Article  Google Scholar 

  • Molaee, E., Moslehi, G., & Reisi, M. (2010). Minimizing maximum earliness and number of tardy jobs in the single machine scheduling problem. Computers and Mathematics with Applications, 60, 2909–2919.

    Article  Google Scholar 

  • Papadimitriou, C. H., & Yannakakis, M. (1993). The traveling salesman problem with distances one and two. Mathematics of Operations Research, 18, 1–11.

    Article  Google Scholar 

  • Plastria, F. (2002). Formulating logical implications in combinatorial optimisation. European Journal of Operational Research, 140, 338–353.

    Article  Google Scholar 

  • Potts, C. N. P., & Van de Velde, S. L. (1995). Dynasearch: Iterative local improvement by dynamic programming: Part i, the traveling salesman problem, faculty of mathematical studies. Southampton: University of Southampton.

    Google Scholar 

  • Smith, S. S. (1994). Reactive scheduling systems. In D. E. Brown & W. T. Scherer (Eds.), Intelligent scheduling systems (pp. 155–192). Boston: Kluwer Academic Publishers.

    Google Scholar 

  • T’Kindt, V., & Billaut, J.-C. (2002). Multicriteria scheduling: Theory, models and algorithms. Heidelberg: Springer.

    Book  Google Scholar 

  • Williams, H. P. (1990). Model building in mathematical programming. New York: Wiley.

    Google Scholar 

  • Woeginger, G. J. (2003). Exact algorithms for NP-hard problems: A survey. In M. Juenger, G. Reinelt, & G. Rinaldi (Eds.), Combinatorial Optimization: Eureka! You shrink!, volume 2570 of Lecture Notes in Computer Science (pp. 185–207). New York: Springer.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. D.A.I. Politecnico di Torino, Torino, Italy

    Federico Della Croce

  2. CNR, IEIIT, Torino, Italy

    Federico Della Croce

Authors
  1. Federico Della Croce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Federico Della Croce.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Della Croce, F. MP or not MP: that is the question. J Sched 19, 33–42 (2016). https://doi.org/10.1007/s10951-015-0459-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10951-015-0459-4

Keywords

Search

Navigation