Skip to main content
SpringerLink
Log in
Book cover

Just-in-Time Systems pp 21–40Cite as

An Exact Algorithm for the Single-Machine Earliness–Tardiness Scheduling Problem

  • Chapter
  • First Online:

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 60))

Abstract

This paper introduces our exact algorithm for the single-machine total weighted earliness–tardiness scheduling problem, which is based on the Successive Sublimation Dynamic Programming (SSDP) method. This algorithm starts from a Lagrangian relaxation of the original problem and then constraints are successively added to it until the gap between lower and upper bounds becomes zero. The relaxations are solved by dynamic programming, and unnecessary dynamic programming states are eliminated in the course of the algorithm to suppress the increase of states caused by the addition of constraints. This paper explains the methods employed in our algorithm to construct the Lagrangian relaxations, to eliminate states and to compute an upper bound together with some other improvements. Then, numerical results for known benchmark instances are given to show the effectiveness of our algorithm.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    When α i are chosen as zero, the total weighted earliness-tardiness problem is reduced to the total weighted tardiness problem.

  2. 2.

    To be more precise, the algorithm can be terminated when the gap becomes less than one because the objective function is integral.

References

  1. Abdul-Razaq, T.S., Potts, C.N.: Dynamic programming state-space relaxation for single-machine scheduling. Journal of the Operational Research Society 39, 141–152 (1988)

    MATH  Google Scholar 

  2. Bülbül, K., Kaminsky, P., Yano, C.: Preemption in single machine earliness/tardiness scheduling. Journal of Scheduling 10, 271–292 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chang, P.C.: A branch and bound approach for single machine scheduling with earliness and tardiness penalties. Computers and Mathematics with Applications 37, 133–144 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Christofides, N., Mingozzi A, Toth P.: State-space relaxation procedures for the computation of bounds to routing problems. Networks 11, 145–164 (1981)

    Google Scholar 

  5. Congram, R.K., Potts, C.N., van de Velde, S.L.: An iterated dynasearch algorithm for the single machine total weighted tardiness scheduling problem. INFORMS Journal on Computing 14, 52–67 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davis, J.S., Kanet, J.J.: Single-machine scheduling with early and tardy completion costs. Naval Research Logistics 40, 85–101 (1993)

    Article  MATH  Google Scholar 

  7. Detienne, B., Pinson, É., Rivreau., D.: Lagrangian domain reductions for the single machine earliness-tardiness problem with release dates, European Journal of Operational Research 201, 45–54 (2010)

    Google Scholar 

  8. Dyer, M.E, Wolsey, L.A.: Formulating the single-machine sequencing problem with release dates as a mixed integer problem. Discrete Applied Mathematics 26, 255–270 (1990)

    Google Scholar 

  9. Fisher, M.L.: Optimal solution of scheduling problems using Lagrange multipliers: Part I. Operations Research 21 1114–27 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fry, T.D., Armstrong, R.D., Darby-Dowman, K., Philipoom, P.R.: A branch and bound procedure to minimize mean absolute lateness on a single processor. Computers & Operations Research 23, 171–182 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics 5, 287–326 (1979)

    Google Scholar 

  12. Grosso, A., Della Croce, F., Tadei, R.: An enhanced dynasearch neighborhood for the single machine total weighted tardiness scheduling problem. Operations Research Letters 32, 68–72 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hoogeveen, J.A., van de Velde, S.L.: A branch-and-bound algorithm for single-machine earliness-tardiness scheduling with idle time. INFORMS Journal on Computing 8, 402–412 (1996)

    Article  MATH  Google Scholar 

  14. Ibaraki, T.: Enumerative approaches to combinatorial optimization. Annals of Operations Research 10 and 11 (1987)

    Google Scholar 

  15. Ibaraki, T., Nakamura, Y.: A dynamic programming method for single machine scheduling, European Journal of Operational Research 76, 72–82 (1994)

    Article  MATH  Google Scholar 

  16. Kim, Y.-D., Yano, C.A.: Minimizing mean tardiness and earliness in single-machine scheduling problems with unequal due dates. Naval Research Logistics 41, 913–933 (1994)

    Article  MATH  Google Scholar 

  17. Lawler, E.L.: A “pseudopolynomial” algorithm for sequencing jobs to minimize total tardiness, Annals of Discrete Mathematics 1, 331–342 (1977)

    Article  MathSciNet  Google Scholar 

  18. Lenstra, J.K, Rinnooy Kan, A.H.G. and Brucker, P.: Complexity of machine scheduling problems. Annals of Discrete Mathematics 1, 343–362 (1977)

    Google Scholar 

  19. Péridy, L., Pinson, É., Rivreau, D.: Using short-term memory to minimize the weighted number of late jobs on a single machine. European Journal of Operational Research 148, 591–603 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Potts, C.N., Van Wassenhove, L.N.: A branch and bound algorithm for the total weighted tardiness problem. Operations Research 33, 363–377 (1985)

    Article  MATH  Google Scholar 

  21. Pritsker, A.A.B., Watters, L.J., Wolfe, P.M.: Multiproject scheduling with limited resources: A zero-one programming approach. Management Science 16, 93–108 (1969)

    Article  Google Scholar 

  22. Sourd, F., Kedad-Sidhoum, S.: The one-machine problem with earliness and tardiness penalties. Journal of Scheduling 6 533–549 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Sourd, F.: Optimal timing of a sequence of tasks with general completion cost. European Journal of Operational Research 165, 82–96 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sourd, F.: Dynasearch for the earliness-tardiness scheduling problem with release dates and setup constraints. Operations Research Letters 34, 591–598 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sourd, F., Kedad-Sidhoum, S.: A faster branch-and-bound algorithm for the earliness-tardiness scheduling problem. Journal of Scheduling 11, 49–58 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sourd, F.: New exact algorithms for one-machine earliness-tardiness scheduling. INFORMS Journal on Computing 21, 167–175 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sousa, J.P., Wolsey, L.A.: A time indexed formulation of non-preemptive single machine scheduling problems. Mathematical Programming 54, 353–367 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. Tanaka, S., Fujikuma, S., Araki, M.: An exact algorithm for single-machine scheduling without machine idle time. Journal of Scheduling 12, 575–593 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Tanaka, S., Fujikuma, S.: An efficient exact algorithm for general single-machine scheduling with machine idle time. 4th IEEE Conference on Automation Science and Engineering (IEEE CASE 2008), 371–376 (2008)

    Google Scholar 

  30. Tanaka, S., Fujikuma, S.: A dynamic-programming-based exact algorithm for single-machine scheduling with machine idle time. Journal of Scheduling, available online. DOI: 10.1007/s10951-011-0242-0

    Google Scholar 

  31. van den Akker, J.M., van Hoesel, C.P.M., Savelsbergh, M.W.P.: A polyhedral approach to single-machine scheduling problems. Mathematical Programming 85 541–572 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. van den Akker, J.M., Hurkens, C.A.J., Savelsbergh, M.W.P.: Time-indexed formulations for machine scheduling problems: column generation. INFORMS Journal on Computing 12, 111–124 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  33. Yano, C.A., Kim, Y.-D.: Algorithms for a class of single-machine weighted tardiness and earliness problems. European Journal of Operational Research 52, 167–178 (1991)

    Article  MATH  Google Scholar 

  34. Yau, H., Pan, Y., Shi, L.: New solution approaches to the general single machine earliness-tardiness problem. IEEE Transactions on Automation Science and Engineering 5, 349–360 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

This work is partially supported by Grant-in-Aid for Young Scientists (B) 19760273, from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) Japan.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto, 615-8510, Japan

    Shunji Tanaka

Authors
  1. Shunji Tanaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shunji Tanaka .

Editor information

Editors and Affiliations

  1. , Graduate Program in Systems Engineering, Universidad Autónoma de Nuevo León, San Nicolás de los Garza, 66450, Nuevo León, Mexico

    Roger Z. Ríos-Mercado

  2. , Graduate Program in Systems Engineering, Universidad Autónoma de Nuevo León, San Nicolás de los Garza, 66450, Nuevo León, Mexico

    Yasmín A. Ríos-Solís

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Tanaka, S. (2012). An Exact Algorithm for the Single-Machine Earliness–Tardiness Scheduling Problem. In: Ríos-Mercado, R., Ríos-Solís, Y. (eds) Just-in-Time Systems. Springer Optimization and Its Applications(), vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1123-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation