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

  • Shunji TanakaEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 60)


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 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.


  1. 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)zbMATHGoogle Scholar
  2. 2.
    Bülbül, K., Kaminsky, P., Yano, C.: Preemption in single machine earliness/tardiness scheduling. Journal of Scheduling 10, 271–292 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Davis, J.S., Kanet, J.J.: Single-machine scheduling with early and tardy completion costs. Naval Research Logistics 40, 85–101 (1993)zbMATHCrossRefGoogle Scholar
  7. 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. 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. 9.
    Fisher, M.L.: Optimal solution of scheduling problems using Lagrange multipliers: Part I. Operations Research 21 1114–27 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 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. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 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)zbMATHCrossRefGoogle Scholar
  14. 14.
    Ibaraki, T.: Enumerative approaches to combinatorial optimization. Annals of Operations Research 10 and 11 (1987)Google Scholar
  15. 15.
    Ibaraki, T., Nakamura, Y.: A dynamic programming method for single machine scheduling, European Journal of Operational Research 76, 72–82 (1994)zbMATHCrossRefGoogle Scholar
  16. 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)zbMATHCrossRefGoogle Scholar
  17. 17.
    Lawler, E.L.: A “pseudopolynomial” algorithm for sequencing jobs to minimize total tardiness, Annals of Discrete Mathematics 1, 331–342 (1977)CrossRefMathSciNetGoogle Scholar
  18. 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. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 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)zbMATHCrossRefGoogle Scholar
  21. 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)CrossRefGoogle Scholar
  22. 22.
    Sourd, F., Kedad-Sidhoum, S.: The one-machine problem with earliness and tardiness penalties. Journal of Scheduling 6 533–549 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sourd, F.: Optimal timing of a sequence of tasks with general completion cost. European Journal of Operational Research 165, 82–96 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sourd, F.: Dynasearch for the earliness-tardiness scheduling problem with release dates and setup constraints. Operations Research Letters 34, 591–598 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sourd, F.: New exact algorithms for one-machine earliness-tardiness scheduling. INFORMS Journal on Computing 21, 167–175 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sousa, J.P., Wolsey, L.A.: A time indexed formulation of non-preemptive single machine scheduling problems. Mathematical Programming 54, 353–367 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 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. 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/s10951-011-0242-0Google Scholar
  31. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 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)zbMATHCrossRefGoogle Scholar
  34. 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)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Electrical EngineeringKyoto UniversityNishikyo-kuJapan

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