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Annals of Operations Research

, Volume 195, Issue 1, pp 49–71 | Cite as

Constraint programming for stochastic inventory systems under shortage cost

  • Roberto RossiEmail author
  • S. Armagan Tarim
  • Brahim Hnich
  • Steven Prestwich
Open Access
Article

Abstract

One of the most important policies adopted in inventory control is the replenishment cycle policy. Such a policy provides an effective means of damping planning instability and coping with demand uncertainty. In this paper we develop a constraint programming approach able to compute optimal replenishment cycle policy parameters under non-stationary stochastic demand, ordering, holding and shortage costs. We show how in our model it is possible to exploit the convexity of the cost-function during the search to dynamically compute bounds and perform cost-based filtering. Our computational experience show the effectiveness of our approach. Furthermore, we use the optimal solutions to analyze the quality of the solutions provided by an existing approximate mixed integer programming approach that exploits a piecewise linear approximation for the cost function.

Keywords

Inventory control Constraint programming Decision making under uncertainty Replenishment cycle policy Non-stationary demand Shortage cost 

References

  1. Apt, K. (2003). Principles of constraint programming. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  2. Bayraktar, E., & Ludkovski, M. (2010). Inventory management with partially observed nonstationary demand. Annals of Operation Research, 176(1), 7–39. CrossRefGoogle Scholar
  3. Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer. Google Scholar
  4. Bookbinder, J. H., & Tan, J. Y. (1988). Strategies for the probabilistic lot-sizing problem with service-level constraints. Management Science, 34, 1096–1108. CrossRefGoogle Scholar
  5. de Kok, A. G. (1991). Basics of inventory management. Part 2. The (R,S)-model. Research memorandum, FEW 521. Department of Economics, Tilburg University, Tilburg, The Netherlands. Google Scholar
  6. de Kok, T., & Inderfurth, K. (1997). Nervousness in inventory management: comparison of basic control rules. European Journal of Operational Research, 103, 55–82. CrossRefGoogle Scholar
  7. Fahle, T., & Sellmann, M. (2002). Cost-based filtering for the constrained knapsack problem. Annals of Operation Research, 115, 73–93. CrossRefGoogle Scholar
  8. Focacci, F., Lodi, A., & Milano, M. (1999). Cost-based domain filtering. In Lecture notes in computer science: Vol. 1713. Proceedings of the 5th international conference on the principles and practice of constraint programming (pp. 189–203). Berlin: Springer. Google Scholar
  9. Focacci, F., Lodi, A., & Milano, M. (2002). Optimization-oriented global constraints. Constraints, 7(3–4), 351–365. CrossRefGoogle Scholar
  10. Focacci, F., & Milano, M. (2001). Connections and integrations of dynamic programming and constraint programming. In Proceedings of the international workshop on integration of AI and OR techniques in constraint programming for combinatorial optimization problems CP-AI-OR 2001. Google Scholar
  11. Fortuin, L. (1980). Five popular probability density functions: a comparison in the field of stock-control models. The Journal of the Operational Research Society, 31(10), 937–942. Google Scholar
  12. Graves, S. C. (1999). A single-item inventory model for a non-stationary demand process. Manufacturing & Service Operations Management, 1, 50–61. CrossRefGoogle Scholar
  13. Hadley, G., & Whitin, T. M. (1964). Analysis of Inventory Systems. New York: Prentice Hall. Google Scholar
  14. Heisig, G. (2002). Planning stability in material requirements planning systems. New York: Springer. Google Scholar
  15. Hnich, B., Rossi, R., Tarim, S. A., & Prestwich, S. D. (2009). Synthesizing filtering algorithms for global chance-constraints. In Lecture notes in computer science: Vol. 5732. Proceedings of principles and practice of constraint programming (CP 2009) (pp. 439–453). Berlin: Springer. CrossRefGoogle Scholar
  16. Hnich, B., Rossi, R., Tarim, S. A., & Prestwich, S. (2011). A survey on CP-AI-OR hybrids for decision making under uncertainty. In P. van Hentenryck & M. Milano (Eds.), Springer optimization and its applications: Vol. 45. Hybrid optimization (pp. 227–270). New York: Springer. Chap. 7. CrossRefGoogle Scholar
  17. F. Laburthe and the OCRE Project Team (1994). Choco: Implementing a CP kernel. Technical report, Bouygues e-Lab, France. Google Scholar
  18. Levi, R., Roundy, R. O., & Shmoys, D. B. (2006). Provably near-optimal sampling-based algorithms for stochastic inventory control models. In Proceedings of the thirty-eighth annual ACM symposium on theory of computing (STOC ’06) (pp. 739–748). New York: ACM Press. CrossRefGoogle Scholar
  19. Pujawan, I. N., & Silver, E. A. (2008). Augmenting the lot sizing order quantity when demand is probabilistic. European Journal of Operational Research, 127(3), 705–722. CrossRefGoogle Scholar
  20. Regin, J.-C. (1994). A filtering algorithm for constraints of difference in csps. In American Association for Artificial Intelligence (pp. 362–367). Seattle, Washington. Google Scholar
  21. Regin, J.-C. (2003). Global constraints and filtering algorithms. In M. Milano (Ed.), Constraints and integer programming combined. Dordrecht: Kluwer Academic. Google Scholar
  22. Rossi, F., van Beek, P., & Walsh, T. (2006). Handbook of constraint programming. Foundations of artificial intelligence. New York: Elsevier. Google Scholar
  23. Rossi, R., Tarim, S. A., Hnich, B., & Prestwich, S. (2007). Replenishment planning for stochastic inventory systems with shortage cost. In Lecture notes in computer science: Vol. 4510. Proceedings of the international conference on integration of AI and OR techniques in constraint programming for combinatorial optimization problems CP-AI-OR 2007 (pp. 229–243). Berlin: Springer. CrossRefGoogle Scholar
  24. Rossi, R., Tarim, S. A., Hnich, B., & Prestwich, S. D. (2008). A global chance-constraint for stochastic inventory systems under service level constraints. Constraints, 13(4), 490–517. CrossRefGoogle Scholar
  25. Rossi, R., Tarim, S. A., Hnich, B., & Prestwich, S. (2011). A state space augmentation algorithm for the replenishment cycle inventory policy. International Journal of Production Economics. 113(1), 377–384 CrossRefGoogle Scholar
  26. Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling. New York: Wiley. Google Scholar
  27. Tang, C. S. (2006). Perspectives in supply chain risk management. International Journal of Production Economics, 103, 451–488. CrossRefGoogle Scholar
  28. Tarim, S. A. (1996). Dynamic lotsizing models for stochastic demand in single and multi-echelon inventory systems. Ph.D. thesis, Lancaster University. Google Scholar
  29. Tarim, S. A., & Kingsman, B. G. (2004). The stochastic dynamic production/inventory lot-sizing problem with service-level constraints. International Journal of Production Economics, 88, 105–119. CrossRefGoogle Scholar
  30. Tarim, S. A., & Kingsman, B. G. (2006). Modelling and computing (R n,S n) policies for inventory systems with non-stationary stochastic demand. European Journal of Operational Research, 174, 581–599. CrossRefGoogle Scholar
  31. Tarim, S. A., & Smith, B. (2008). Constraint programming for computing non-stationary (R,S) inventory policies. European Journal of Operational Research, 189, 1004–1021. CrossRefGoogle Scholar
  32. Tarim, S. A., Hnich, B., Prestwich, S. D., & Rossi, R. (2009a). Finding reliable solution: event-driven probabilistic constraint programming. Annals of Operation Research, 171(1), 77–99. CrossRefGoogle Scholar
  33. Tarim, S. A., Hnich, B., Rossi, R., & Prestwich, S. D. (2009b). Cost-based filtering techniques for stochastic inventory control under service level constraints. Constraints, 14(2), 137–176. CrossRefGoogle Scholar
  34. Tempelmeier, H. (2007). On the stochastic uncapacitated dynamic single-item lotsizing problem with service level constraints. European Journal of Operational Research, 127(1), 184–194. CrossRefGoogle Scholar
  35. Wagner, H. M., & Whitin, T. M. (1958). Dynamic version of the economic lot size model. Management Science, 5, 89–96. CrossRefGoogle Scholar
  36. Walsh, T. (2002). Stochastic constraint programming. In Proceedings of European conference on artificial intelligence (ECAI’2002) (pp. 111–115). Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Roberto Rossi
    • 1
    Email author
  • S. Armagan Tarim
    • 2
  • Brahim Hnich
    • 3
  • Steven Prestwich
    • 4
  1. 1.Logistics, Decision and Information SciencesWageningen URWageningenThe Netherlands
  2. 2.Department of ManagementHacettepe UniversityAnkaraTurkey
  3. 3.Faculty of Computer ScienceIzmir University of EconomicsIzmirTurkey
  4. 4.Cork Constraint Computation CentreUniversity CollegeCorkIreland

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