, Volume 13, Issue 4, pp 490–517 | Cite as

A Global Chance-Constraint for Stochastic Inventory Systems Under Service Level Constraints

  • Roberto Rossi
  • S. Armagan Tarim
  • Brahim Hnich
  • Steven Prestwich


We consider a class of production/inventory control problems that has a single product and a single stocking location, for which a stochastic demand with a known non-stationary probability distribution is given. Under the widely-known replenishment cycle policy the problem of computing policy parameters under service level constraints has been modeled using various techniques. Tarim and Kingsman introduced a modeling strategy that constitutes the state-of-the-art approach for solving this problem. In this paper we identify two sources of approximation in Tarim and Kingsman’s model and we propose an exact stochastic constraint programming approach. We build our approach on a novel concept, global chance-constraints, which we introduce in this paper. Solutions provided by our exact approach are employed to analyze the accuracy of the model developed by Tarim and Kingsman.


Global chance-constraints Stochastic inventory control Non-stationary (R,S) policy Uncertainty 


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  1. 1.
    Apt, K. (2003). Principles of constraint programming. New York, NY, USA: Cambridge University Press.Google Scholar
  2. 2.
    Askin, R. G. (1981). A procedure for production lot sizing with probabilistic dynamic demand. AIIE Transactions, 13, 132–137.MathSciNetGoogle Scholar
  3. 3.
    Balafoutis, T., & Stergiou, K. (2006). Algorithms for stochastic csps. In Proceedings of the 12th international conference on the principles and practice of constraint programming. Lecture notes in computer science (No. 4204, pp. 44–58). Springer Verlag.Google Scholar
  4. 4.
    Bellman, R. E. (2003). Dynamic programming. Dover Publications, Incorporated.Google Scholar
  5. 5.
    Berry, W. L. (1972). Lot sizing procedures for requirements planning systems: A framework for analysis. Production and Inventory Management Journal, 13, 19–34.MathSciNetGoogle Scholar
  6. 6.
    Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming. New York: Springer.MATHGoogle Scholar
  7. 7.
    Bookbinder, J. H., & Tan, J. Y. (1988). Strategies for the probabilistic lot-sizing problem with service-level constraints. Management Science, 34, 1096–1108.MATHMathSciNetGoogle Scholar
  8. 8.
    Brailsford, S. C., Potts, C. N., & Smith, B. M. (1999). Constraint satisfaction problems: Algorithms and applications. European Journal of Operational Research, 119, 557–581.MATHCrossRefGoogle Scholar
  9. 9.
    Charnes, A., & Cooper, W. W. (1959). Chance-constrainted programming. Management Science, 6(1), 73–79.MATHMathSciNetGoogle Scholar
  10. 10.
    Davis, T. (1993). Effective supply chain management. Sloan Management Review.Google Scholar
  11. 11.
    de Kok, A. G. (1991). Basics of inventory management: Part 2 the (R,S)-model. Research memorandum, FEW 521, 1991. Tilburg, The Netherlands: Department of Economics, Tilburg University.Google Scholar
  12. 12.
    de Kok, T., & Inderfurth, K. (1997). Nervousness in inventory management: Comparison of basic control rules. European Journal of Operational Research, 103, 55–82.MATHCrossRefGoogle Scholar
  13. 13.
    Devore, J. L. (1995). Probability and statistics for engineering and the sciences (4th ed.). Duxbury PressGoogle Scholar
  14. 14.
    Florian, M., Lenstra, J. K., & Rinooy Kan, A. H. G. (1980). Deterministic production planning: Algorithms and complexity. Management Science, 26(7), 669–679.MATHMathSciNetGoogle Scholar
  15. 15.
    Graves, S. C. (1999). A single-item inventory model for a non-stationary demand process. Manufacturing & Service Operations Management, 1, 50–61.Google Scholar
  16. 16.
    Heisig, G. (2002). Planning stability in material requirements planning systems. New York: Springer.MATHGoogle Scholar
  17. 17.
    Janssen, F., & de Kok, T. (1999). A two-supplier inventory model. International Journal of Production Economics, 59, 395–403.CrossRefGoogle Scholar
  18. 18.
    Jeffreys, H. (1961). Theory of probability. Oxford, UK: Clarendon Press.MATHGoogle Scholar
  19. 19.
    Laburthe, F., & the OCRE project team. (1994). Choco: Implementing a cp kernel. Technical report. France: Bouygues e-Lab.Google Scholar
  20. 20.
    Lustig, I. J., & Puget, J. F. (2001). Program does not equal program: Constraint programming and its relationship to mathematical programming. Interfaces, 31, 29–53.Google Scholar
  21. 21.
    Regin, J.-C. (1994). A filtering algorithm for constraints of difference in csps. In Proceedings of the national conference on artificial intelligence (AAAI-94) (pp. 362–367). Seattle, WA, USA.Google Scholar
  22. 22.
    Regin, J.-C. (2003). Global constraints and filtering algorithms. In M. Milano (Ed.), Constraints and integer programming combined. Kluwer.Google Scholar
  23. 23.
    Rossi, F., Petrie, C., & Dhar, V. (1990). On the equivalence of constraint satisfaction problems. In Proceedings of the 9th ECAI. European conference on artificial intelligence (pp. 550–556). Stockholm, Sweden: Pitman Publishing.Google Scholar
  24. 24.
    Scarf, H. (1960). The optimality of (s,S) policies in dynamic inventory problem. In K. J. Arrow, S. Karlin, & P. Suppes (Eds.), Mathematical methods in social sciences. Stanford Unversity Press.Google Scholar
  25. 25.
    Silver, E. A. (1978). Inventory control under a probabilistic time-varying demand pattern. AIIE Transactions, 10, 371–379.Google Scholar
  26. 26.
    Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory management and production planning and scheduling. New York: Wiley.Google Scholar
  27. 27.
    Smits, S. R., Wagner, M., & de Kok, T. G. (2004). Determination of an order-up-to policy in the stochastic economic lot scheduling model. International Journal of Production Economics, 90, 377–389.CrossRefGoogle Scholar
  28. 28.
    Tang, C. S. (2006). Perpectives in supply chain risk management. International Journal of Production Economics, 103, 451–488.CrossRefGoogle Scholar
  29. 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. 30.
    Tarim, S. A., & Smith, B. (2008). Constraint programming for computing non-stationary (R,S) Inventory Policies. European Journal of Operational Research (to appear).Google Scholar
  31. 31.
    Tarim, S. A., Hnich, B., Rossi, R., & Prestwich, S. (2007). Cost-based filtering for stochastic inventory control. In Recent advances in constraints joint ERCIM/CoLogNET international workshop on constraint solving and constraint logic programming, CSCLP 2006. Lecture Notes in Artificial Intelligence (No. 4651, pp. 169–183). Springer.Google Scholar
  32. 32.
    Tarim, S. A., Hnich, B., Rossi, R., & Prestwich, S. (2008). Cost-based filtering techniques for stochastic inventory control under service level constraints. Constraints (to appear).Google Scholar
  33. 33.
    Tarim, S. A., Manandhar, S., & Walsh, T. (2006). Stochastic constraint programming: A scenario-based approach. Constraints, 11(1), 53–80.MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Wagner, H. M., & Whitin, T. M. (1958). Dynamic version of the economic lot size model. Management Science, 5, 89–96.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Walsh, T. (2002). Stochastic constraint programming. In Proceedings of the 15th ECAI. European conference on artificial intelligence. IOS Press.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Roberto Rossi
    • 1
    • 2
  • S. Armagan Tarim
    • 3
  • Brahim Hnich
    • 4
  • Steven Prestwich
    • 2
  1. 1.Centre for Telecommunication Value—Chain Driven ResearchDublinIreland
  2. 2.Cork Constraint Computation CentreUniversity CollegeCorkIreland
  3. 3.Department of ManagementHacettepe UniversityAnkaraTurkey
  4. 4.Faculty of Computer ScienceIzmir University of EconomicsIzmirTurkey

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