Replenishment Planning for Stochastic Inventory Systems with Shortage Cost

  • Roberto Rossi
  • S. Armagan Tarim
  • Brahim Hnich
  • Steven Prestwich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4510)


One of the most important policies adopted in inventory control is the (R,S) policy (also known as the “replenishment cycle” policy). Under the non-stationary demand assumption the (R,S) policy takes the form (R n ,S n ) where R n denotes the length of the n th replenishment cycle, and S n the corresponding order-up-to-level. Such a policy provides an effective means of damping planning instability and coping with demand uncertainty. In this paper we develop a CP approach able to compute optimal (R n ,S n ) policy parameters under stochastic demand, ordering, holding and shortage costs. The convexity of the cost-function is exploited during the search to compute bounds. We use the optimal solutions to analyze the quality of the solutions provided by an approximate MIP approach that exploits a piecewise linear approximation for the cost function.


Mixed Integer Programming Inventory Level Piecewise Linear Approximation Mixed Integer Programming Model Shortage Cost 
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  1. 1.
    Tarim, S.A., Hnich, B., Rossi, R., Prestwich, S.D.: Cost-Based Filtering for Stochastic Inventory Control. In: Azevedo, F., Barahona, P., Fages, F., Rossi, F. (eds.) CSCLP. LNCS (LNAI), vol. 4651, pp. 169–183. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Tarim, S.A., Smith, B.: Constraint Programming for Computing Non-Stationary (R,S) Inventory Policies. European Journal of Operational Research (to appear)Google Scholar
  3. 3.
    Tarim, S.A., Kingsman, B.G.: The Stochastic Dynamic Production/Inventory Lot-Sizing Problem With Service-Level Constraints. International Journal of Production Economics 88, 105–119 (2004)CrossRefGoogle Scholar
  4. 4.
    Tarim, S.A., Kingsman, B.G.: Modelling and Computing (R n,S n) Policies for Inventory Systems with Non-Stationary Stochastic Demand. European Journal of Operational Research 174, 581–599 (2006)zbMATHCrossRefGoogle Scholar
  5. 5.
    Tarim, S.A.: Dynamic Lotsizing Models for Stochastic Demand in Single and Multi-Echelon Inventory Systems. PhD Thesis, Lancaster University (1996)Google Scholar
  6. 6.
    Bookbinder, J.H., Tan, J.Y.: Strategies for the Probabilistic Lot-Sizing Problem With Service-Level Constraints. Management Science 34, 1096–1108 (1988)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Wagner, H.M., Whitin, T.M.: Dynamic Version of the Economic Lot Size Model. Management Science 5, 89–96 (1958)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Silver, E.A., Pyke, D.F., Peterson, R.: Inventory Management and Production Planning and Scheduling. John Wiley and Sons, New York (1998)Google Scholar
  9. 9.
    Porteus, E.L.: Foundations of Stochastic Inventory Theory. Stanford University Press, Stanford (2002)Google Scholar
  10. 10.
    Apt, K.: Principles of Constraint Programming. Cambridge University Press, Cambridge (2003)Google Scholar
  11. 11.
    Charnes, A., Cooper, W.W.: Chance-Constrainted Programming. Management Science 6(1), 73–79 (1959)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fortuin, L.: Five Popular Probability Density Functions: a Comparison in the Field of Stock-Control Models. Journal of the Operational Research Society 31(10), 937–942 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lustig, I.J., Puget, J.-F.: Program Does Not Equal Program: Constraint Programming and its Relationship to Mathematical Programming. Interfaces 31, 29–53 (2001)Google Scholar
  14. 14.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)zbMATHGoogle Scholar
  15. 15.
    Focacci, F., Lodi, A., Milano, M.: Cost-Based Domain Filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (1999)Google Scholar
  16. 16.
    Laburthe, F., OCRE project team: Choco: Implementing a CP Kernel. Bouygues e-Lab, FranceGoogle Scholar
  17. 17.
    Hadley, G., Whitin, T.M.: Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs (1964)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Roberto Rossi
    • 1
  • S. Armagan Tarim
    • 2
  • Brahim Hnich
    • 3
  • Steven Prestwich
    • 1
  1. 1.Cork Constraint Computation Centre, University College, CorkIreland
  2. 2.Department of Management, Hacettepe UniversityTurkey
  3. 3.Faculty of Computer Science, Izmir University of EconomicsTurkey

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