Annals of Operations Research

, Volume 210, Issue 1, pp 245–272 | Cite as

Single-facility scheduling by logic-based Benders decomposition

Article

Abstract

Logic-based Benders decomposition can combine mixed integer programming and constraint programming to solve planning and scheduling problems much faster than either method alone. We find that a similar technique can be beneficial for solving pure scheduling problems as the problem size scales up. We solve single-facility non-preemptive scheduling problems with time windows and long time horizons. The Benders master problem assigns jobs to predefined segments of the time horizon, where the subproblem schedules them. In one version of the problem, jobs may not overlap the segment boundaries (which represent shutdown times, such as weekends), and in another version, there is no such restriction. The objective is to find feasible solutions, minimize makespan, or minimize total tardiness.

Keywords

Logic-based Benders decomposition Constraint programming Mixed integer programming Single-facility scheduling Makespan Tardiness Feasibility 

References

  1. Aggoun, A., & Vazacopoulos, A. (2004). Solving sports scheduling and timetabling problems with constraint programming. In S. Butenko, J. Gil-Lafuente, & P. M. Pardalos (Eds.), Economics, management and optimization in sports (pp. 243–264). New York: Springer. CrossRefGoogle Scholar
  2. Babonneau, F., Beltran, C., Haurie, A., Tadonki, C., & Vial, J. P. (2007). Proximal-ACCPM: A versatile oracle based optimization method. In E. J. Kontoghiorghes & C. Gatu (Eds.), Advances in computational management science: Vol. 9. Optimisation, econometric and financial analysis (pp. 69–92). New York: Springer. Google Scholar
  3. Baptiste, P., Le Pape, C., & Nuijten, W. (2001). Constraint-based scheduling: applying constraint programming to scheduling problems. Dordrecht: Kluwer. CrossRefGoogle Scholar
  4. Barlatta, A. Y., Cohn, A. M., & Gusikhinc, O. (2010). A hybridization of mathematical programming and dominance-driven enumeration for solving shift-selection and task-sequencing problems. Computers & Operations Research, 37, 1298–1307. CrossRefGoogle Scholar
  5. Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252. CrossRefGoogle Scholar
  6. Benini, L., Bertozzi, D., Guerri, A., & Milano, M. (2005). Allocation and scheduling for MPSoCs via decomposition and no-good generation. In Lecture notes in computer science: Vol. 3709. Principles and practice of constraint programming (CP 2005) (pp. 107–121). Berlin: Springer. CrossRefGoogle Scholar
  7. Bent, R., & Van Hentenryck, P. (2010). Spatial, temporal, and hybrid decompositions for large-scale vehicle routing with time windows. In D. Cohen (Ed.), Lecture notes in computer science: Vol. 6308. Principles and practice of constraint programming (CP 2010) (pp. 99–113). Berlin: Springer. CrossRefGoogle Scholar
  8. Cambazard, H., Hladik, P.-E., Déplanche, A.-M., Jussien, N., & Trinquet, Y. (2004). Decomposition and learning for a hard real time task allocation problem. In M. Wallace (Ed.), Lecture notes in computer science: Vol. 3258. Principles and practice of constraint programming (CP 2004) (pp. 153–167). Berlin: Springer. CrossRefGoogle Scholar
  9. Chu, Y., & Xia, Q. (2004). Generating Benders cuts for a class of integer programming problems. In J. C. Régin & M. Rueher (Eds.), Lecture notes in computer science: Vol. 3011. Integration of AI and OR techniques in constraint programming for combinatorial optimization problems (CPAIOR 2004) (pp. 127–141). Berlin: Springer. CrossRefGoogle Scholar
  10. Codato, G., & Fischetti, M. (2006). Combinatorial Benders’ cuts for mixed-integer linear programming. Operations Research, 54, 756–766. CrossRefGoogle Scholar
  11. Corréa, A. I., Langevin, A., & Rousseau, L. M. (2004). Dispatching and conflict-free routing of automated guided vehicles: a hybrid approach combining constraint programming and mixed integer programming. In J. C. Régin & M. Rueher (Eds.), Lecture notes in computer science: Vol. 3011. Integration of AI and OR techniques in constraint programming for combinatorial optimization problems (CPAIOR 2004) (pp. 370–378). Berlin: Springer. CrossRefGoogle Scholar
  12. Fazel-Zarandi, M. M., & Beck, J. C. (2009). Solving a location-allocation problem with logic-based Benders’ decomposition. In I. P. Gent (Ed.), Lecture notes in computer science: Vol. 5732. Principles and practice of constraint programming (CP 2009) (pp. 344–351). Berlin: Springer. CrossRefGoogle Scholar
  13. French, S. (1982). Sequencing and scheduling. New York: Wiley. Google Scholar
  14. Fulkerson, D. R. (1971). Blocking and anti-blocking pairs of polyhedra. Mathematical Programming, 1, 168–194. CrossRefGoogle Scholar
  15. Geoffrion, A. M. (1972). Generalized Benders decomposition. Journal of Optimization Theory and Applications, 10, 237–260. CrossRefGoogle Scholar
  16. Harjunkoski, I., & Grossmann, I. E. (2001). A decomposition approach for the scheduling of a steel plant production. Computers & Chemical Engineering, 25, 1647–1660. CrossRefGoogle Scholar
  17. Harjunkoski, I., & Grossmann, I. E. (2002). Decomposition techniques for multistage scheduling problems using mixed-integer and constraint programming methods. Computers & Chemical Engineering, 26, 1533–1552. CrossRefGoogle Scholar
  18. Hooker, J. N. (1995). Logic-based Benders decomposition. In INFORMS national meeting (INFORMS 1995). Google Scholar
  19. Hooker, J. N. (1996). Inference duality as a basis for sensitivity analysis. In E. C. Freuder (Ed.), Lecture notes in computer science: Vol. 1118. Principles and practice of constraint programming (CP 1996) (pp. 224–236). Berlin: Springer. CrossRefGoogle Scholar
  20. Hooker, J. N. (2000). Logic-based methods for optimization: combining optimization and constraint satisfaction. New York: Wiley. CrossRefGoogle Scholar
  21. Hooker, J. N. (2004). A hybrid method for planning and scheduling. In M. Wallace (Ed.), Lecture notes in computer science: Vol. 3258. Principles and practice of constraint programming (CP 2004) (pp. 305–316). Berlin: Springer. CrossRefGoogle Scholar
  22. Hooker, J. N. (2005a). A hybrid method for planning and scheduling. Constraints, 10, 385–401. CrossRefGoogle Scholar
  23. Hooker, J. N. (2005b). Planning and scheduling to minimize tardiness. In Lecture notes in computer science: Vol. 3709. Principles and practice of constraint programming (CP 2005) (pp. 314–327). Berlin: Springer. CrossRefGoogle Scholar
  24. Hooker, J. N. (2006). An integrated method for planning and scheduling to minimize tardiness. Constraints, 11, 139–157. CrossRefGoogle Scholar
  25. Hooker, J. N. (2007a). Integrated methods for optimization. Berlin: Springer. Google Scholar
  26. Hooker, J. N. (2007b). Planning and scheduling by logic-based Benders decomposition. Operations Research, 55, 588–602. CrossRefGoogle Scholar
  27. Hooker, J. N., & Ottosson, G. (2003). Logic-based Benders decomposition. Mathematical Programming, 96, 33–60. Google Scholar
  28. Hooker, J. N., & Yan, H. (1995). Logic circuit verification by Benders decomposition. In V. Saraswat & P. Van Hentenryck (Eds.), Principles and practice of constraint programming: the newport papers (pp. 267–288). Cambridge: MIT Press. Google Scholar
  29. Jain, V., & Grossmann, I. E. (2001). Algorithms for hybrid MILP/CP models for a class of optimization problems. INFORMS Journal on Computing, 13, 258–276. CrossRefGoogle Scholar
  30. Jeroslow, R. G. (1987). Representability in mixed integer programming, I: Characterization results. Discrete Applied Mathematics, 17, 223–243. CrossRefGoogle Scholar
  31. Keha, A. B., Khowala, K., & Fowler, J. W. (2009). Mixed integer programming formulations for single machine scheduling problems. Computers & Industrial Engineering, 56, 357–367. CrossRefGoogle Scholar
  32. Koulamas, C. (2010). The single-machine total tardiness scheduling problem: review and extensions. European Journal of Operational Research, 202, 1–7. CrossRefGoogle Scholar
  33. Maravelias, C. T. (2006). A decomposition framework for the scheduling of single- and multi-stage processes. Computers & Chemical Engineering, 30, 407–420. CrossRefGoogle Scholar
  34. Maravelias, C. T., & Grossmann, I. E. (2004a). A hybrid MILP/CP decomposition approach for the continuous time scheduling of multipurpose batch plants. Computers & Chemical Engineering, 28, 1921–1949. CrossRefGoogle Scholar
  35. Maravelias, C. T., & Grossmann, I. E. (2004b). Using MILP and CP for the scheduling of batch chemical processes. In J. C. Régin & M. Rueher (Eds.), Lecture notes in computer science: Vol. 3011. Integration of AI and OR techniques in constraint programming for combinatorial optimization problems (CPAIOR 2004) (pp. 1–20). Berlin: Springer. CrossRefGoogle Scholar
  36. Padberg, M. (1973). On the facial structure of set packing polyhedra. Mathematical Programming, 5, 199–215. CrossRefGoogle Scholar
  37. Pinedo, M. (1995). Scheduling: theory, algorithms, and systems. New York: Prentice Hall. Google Scholar
  38. Rasmussen, R. (2008). Scheduling a triple round robin tournament for the best Danish soccer league. European Journal of Operational Research, 185, 795–810. CrossRefGoogle Scholar
  39. Rasmussen, R., & Trick, M. A. (2007). A Benders approach to the constrained minimum break problem. European Journal of Operational Research, 177, 198–213. CrossRefGoogle Scholar
  40. Sadykov, R., & Wolsey, L. A. (2006). Integer programming and constraint programming in solving a multimachine assignment scheduling problem with deadlines and release dates. INFORMS Journal on Computing, 18, 209–217. CrossRefGoogle Scholar
  41. Tarim, S. A., & Miguel, I. (2006). A hybrid Benders’ decomposition method for solving stochastic constraint programs with linear recourse. In B. Hnich, M. Carlsson, F. Fages, & F. Rossi (Eds.), Lecture notes in computer science: Vol. 3978. Recent advances in constraints (CSCLP 2005) (pp. 133–148). Berlin: Springer. CrossRefGoogle Scholar
  42. Terekhov, D., Beck, J. C., & Brown, K. N. (2005). Solving a stochastic queueing design and control problem with constraint programming. In Proceedings of the 22nd national conference on artificial intelligence (AAAI 2005) (pp. 261–266). Google Scholar
  43. Thorsteinsson, E. (2001). Branch and check: a hybrid framework integrating mixed integer programming and constraint logic programming. In T. Walsh (Ed.), Lecture notes in computer science: Vol. 2239. Principles and practice of constraint programming (CP 2001) (pp. 16–30). Berlin: Springer. CrossRefGoogle Scholar
  44. Timpe, C. (2002). Solving planning and scheduling problems with combined integer and constraint programming. OR-Spektrum, 24, 431–448. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations