Annals of Operations Research

, Volume 210, Issue 1, pp 191–211 | Cite as

Multicut Benders decomposition algorithm for process supply chain planning under uncertainty

Article

Abstract

In this paper, we present a multicut version of the Benders decomposition method for solving two-stage stochastic linear programming problems, including stochastic mixed-integer programs with only continuous recourse (two-stage) variables. The main idea is to add one cut per realization of uncertainty to the master problem in each iteration, that is, as many Benders cuts as the number of scenarios added to the master problem in each iteration. Two examples are presented to illustrate the application of the proposed algorithm. One involves production-transportation planning under demand uncertainty, and the other one involves multiperiod planning of global, multiproduct chemical supply chains under demand and freight rate uncertainty. Computational studies show that while both the standard and the multicut versions of the Benders decomposition method can solve large-scale stochastic programming problems with reasonable computational effort, significant savings in CPU time can be achieved by using the proposed multicut algorithm.

Keywords

Benders decomposition Stochastic programming Planning Supply chain 

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References

  1. Archibald, T. W., Buchanan, C. S., McKinnon, K. I. M., & Thomas, L. C. (1999). Nested Benders decomposition and dynamic programming for reservoir optimisation. The Journal of the Operational Research Society, 50, 468–479. Google Scholar
  2. Bahn, O., Dumerle, O., Goffin, J. L., & Vial, J. P. (1995). A cutting plane method from analytic centers for stochastic-programming. Mathematical Programming, 69, 45–73. Google Scholar
  3. Benders, J. F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4, 238–252. CrossRefGoogle Scholar
  4. Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer. Google Scholar
  5. Birge, J. R., & Louveaux, F. V. (1988). A multicut algorithm for two-stage stochastic linear programs. European Journal of Operational Research, 34, 384–392. CrossRefGoogle Scholar
  6. Contreras, I., Cordeau, J. F., & Laporte, G. (2010). Benders decomposition for large-scale uncapacitated hub location. Operations Research, in press. Google Scholar
  7. Escudero, L. F., Garín, A., Merino, M., & Pérez, G. (2007). A two-stage stochastic integer programming approach as a mixture of branch-and-fix coordination and Benders decomposition schemes. Annals of Operations Research, 152, 395–420. CrossRefGoogle Scholar
  8. Fragniere, E., Gondzio, J., & Vial, J. P. (2000). Building and solving large-scale stochastic programs on an affordable distributed computing system. Annals of Operations Research, 99, 167–187. CrossRefGoogle Scholar
  9. Higle, J. L., & Sen, S. (1991). Stochastic decomposition—an algorithm for 2-stage linear-programs with recourse. Mathematics of Operations Research, 16, 650–669. CrossRefGoogle Scholar
  10. Infanger, G. (1993). Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs. Annals of Operations Research, 39, 69–95. CrossRefGoogle Scholar
  11. Infanger, G. (1994). Planning under uncertainty: solving large-scale stochastic linear programs. Danvers: Boyd and Fraser. Google Scholar
  12. Latorre, J. M., Cerisola, S., Ramos, A., & Palacios, R. (2009). Analysis of stochastic problem decomposition algorithms in computational grids. Annals of Operations Research, 166(1), 355–373. CrossRefGoogle Scholar
  13. Linderoth, J., Shapiro, A., & Wright, S. (2006). The empirical behavior of sampling methods for stochastic programming. Annals of Operations Research, 142, 215–241. CrossRefGoogle Scholar
  14. Linderoth, J., & Wright, S. (2003). Decomposition algorithms for stochastic programming on a computational grid. Computational Optimization and Applications, 24, 207–250. CrossRefGoogle Scholar
  15. Miller, N., & Ruszczyński, A. (2010). Risk-averse two-stage stochastic linear programming: modeling and decomposition. Operations Research, doi:10.1287/opre.1100.0847. Google Scholar
  16. Mulvey, J. M., & Ruszczynski, A. J. (1995). A new scenario decomposition method for large-scale stochastic optimization. Operations Research, 43, 477–490. CrossRefGoogle Scholar
  17. Ntaimo, L. (2010). Disjunctive decomposition for two-stage stochastic mixed-binary programs with random recourse. Operations Research, 58, 229–243. CrossRefGoogle Scholar
  18. Rosenthal, R. E. (2010). GAMS—a user’s manual. Washington: GAMS Development Corp. Google Scholar
  19. Ruszczynski, A. (1993). Parallel decomposition of multistage stochastic-programming problems. Mathematical Programming, 58, 201–228. CrossRefGoogle Scholar
  20. Ruszczyński, A. (1997). Decomposition methods in stochastic programming. Mathematical Programming, 79, 333–353. Google Scholar
  21. Saharidis, G. K. D., Boile, M., & Theofanis, S. (2011). Initialization of the Benders master problem using valid inequalities applied to fixed-charge network problems. Expert Systems with Applications, 38, 6627–6636. CrossRefGoogle Scholar
  22. Saharidis, G. K. D., & Ierapetritou, M. G. (2010). Improving Benders decomposition using maximum feasible subsystem (MFS) cut generation strategy. Computers & Chemical Engineering, 34, 1237. CrossRefGoogle Scholar
  23. Saharidis, G. K. D., Minoux, M., & Ierapetritou, M. G. (2010). Accelerating Bender’s method using covering cut bundle generation. International Transactions in Operational Research, 17, 221. CrossRefGoogle Scholar
  24. Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. European Journal of Operational Research, 167, 96–115. CrossRefGoogle Scholar
  25. Sen, S. (1993). Subgradient decomposition and differentiability of the recourse function of a 2-stage stochastic linear program. Operations Research Letters, 13, 143–148. CrossRefGoogle Scholar
  26. Sen, S., Zhou, Z., & Huang, K. (2009). Enhancements of two-stage stochastic decomposition. Computers & Operations Research, 36, 2434–2439. CrossRefGoogle Scholar
  27. Shapiro, A. (2000). Stochastic programming by Monte Carlo simulation methods. Stochastic Programming E-Prints Series, 03. Google Scholar
  28. Shapiro, A. (2008). Stochastic programming approach to optimization under uncertainty. Mathematical Programming, 112, 183–220. CrossRefGoogle Scholar
  29. Shapiro, A., & Homem-de-Mello, T. (1998). A simulation-based approach to two-stage stochastic programming with recourse. Mathematical Programming, 81, 301–325. Google Scholar
  30. Trukhanov, S., Ntaimo, L., & Schaefer, A. (2010). Adaptive multicut aggregation for two-stage stochastic linear programs with recourse. European Journal of Operational Research, 206, 395–406. CrossRefGoogle Scholar
  31. Van Slyke, R. M., & Wets, R. (1969). L-shaped linear programs with applications to optimal control and stochastic programming. SIAM Journal on Applied Mathematics, 17, 638–663. CrossRefGoogle Scholar
  32. Wassick, J. M. (2009). Enterprise-wide optimization in an integrated chemical complex. Computers & Chemical Engineering, 33, 1950–1963. CrossRefGoogle Scholar
  33. You, F., Wassick, J. M., & Grossmann, I. E. (2009). Risk management for global supply chain planning under uncertainty: models and algorithms. AIChE Journal, 55, 931–946. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Argonne National LaboratoryArgonneUSA
  2. 2.Northwestern UniversityEvanstonUSA
  3. 3.Carnegie Mellon UniversityPittsburghUSA

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