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An Efficient Algorithm for Computing Optimal ( s , S ) Policies

Author

Listed:
  • Awi Federgruen

    (Columbia University, New York, New York)

  • Paul Zipkin

    (Columbia University, New York, New York)

Abstract

This paper presents an algorithm to compute an optimal ( s , S ) policy under standard assumptions (stationary data, well-behaved one-period costs, discrete demand, full backlogging, and the average-cost criterion). The method is iterative, starting with an arbitrary, given ( s , S ) policy and converging to an optimal policy in a finite number of iterations. Any of the available approximations can thus be used as an initial solution. Each iteration requires only modest computations. Also, a lower bound on the true optimal cost can be computed and used in a termination test. Empirical testing suggests very fast convergence.

Suggested Citation

  • Awi Federgruen & Paul Zipkin, 1984. "An Efficient Algorithm for Computing Optimal ( s , S ) Policies," Operations Research, INFORMS, vol. 32(6), pages 1268-1285, December.
    • Handle: RePEc:inm:oropre:v:32:y:1984:i:6:p:1268-1285
    DOI: 10.1287/opre.32.6.1268
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    Citations

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    Cited by:

    1. Pisch, Frank, 2020. "Managing global production: theory and evidence from just-in-time supply chains," LSE Research Online Documents on Economics 108488, London School of Economics and Political Science, LSE Library.
    2. Tunc, Huseyin & Kilic, Onur A. & Tarim, S. Armagan & Eksioglu, Burak, 2011. "The cost of using stationary inventory policies when demand is non-stationary," Omega, Elsevier, vol. 39(4), pages 410-415, August.
    3. D. Beyer & S. P. Sethi, 1999. "The Classical Average-Cost Inventory Models of Iglehart and Veinott–Wagner Revisited," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 523-555, June.
    4. Lee, Jun-Yeon & Ren, Louie, 2011. "Vendor-managed inventory in a global environment with exchange rate uncertainty," International Journal of Production Economics, Elsevier, vol. 130(2), pages 169-174, April.
    5. Chou, Mabel & Sim, Chee-Khian & Yuan, Xue-Ming, 2013. "Optimal policies for inventory systems with two types of product sharing common hardware platforms: Single period and finite horizon," European Journal of Operational Research, Elsevier, vol. 224(2), pages 283-292.
    6. Gah-Yi Ban, 2020. "Confidence Intervals for Data-Driven Inventory Policies with Demand Censoring," Operations Research, INFORMS, vol. 68(2), pages 309-326, March.
    7. Sandun C. Perera & Suresh P. Sethi, 2023. "A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Discrete‐time case," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 131-153, January.
    8. Y. Feng & J. Sun, 2001. "Computing the Optimal Replenishment Policy for Inventory Systems with Random Discount Opportunities," Operations Research, INFORMS, vol. 49(5), pages 790-795, October.
    9. Hao Yuan & Qi Luo & Cong Shi, 2021. "Marrying Stochastic Gradient Descent with Bandits: Learning Algorithms for Inventory Systems with Fixed Costs," Management Science, INFORMS, vol. 67(10), pages 6089-6115, October.
    10. Chan, Gin Hor & Song, Yuyue, 2003. "A dynamic analysis of the single-item periodic stochastic inventory system with order capacity," European Journal of Operational Research, Elsevier, vol. 146(3), pages 529-542, May.
    11. D. Beyer & S. P. Sethi, 1997. "Average Cost Optimality in Inventory Models with Markovian Demands," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 497-526, March.
    12. Kleijnen, J.P.C. & Wan, J., 2007. "Optimization of simulated systems : OptQuest and alternatives [also see “Simulation for the optimization of (s, S) inventory system with random lead times and a service level constraint by using Arena," Other publications TiSEM ffaee312-9f6a-4452-9ccc-9, Tilburg University, School of Economics and Management.
    13. Tarim, S. Armagan & Smith, Barbara M., 2008. "Constraint programming for computing non-stationary (R, S) inventory policies," European Journal of Operational Research, Elsevier, vol. 189(3), pages 1004-1021, September.
    14. Xie, Xiaolan, 1998. "Stability analysis and optimization of an inventory system with bounded orders," European Journal of Operational Research, Elsevier, vol. 110(1), pages 126-149, October.
    15. Cong Shi & Huanan Zhang & Xiuli Chao & Retsef Levi, 2014. "Approximation algorithms for capacitated stochastic inventory systems with setup costs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(4), pages 304-319, June.
    16. B S Maddah & M Y Jaber & N E Abboud, 2004. "Periodic review (s, S) inventory model with permissible delay in payments," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(2), pages 147-159, February.
    17. Bijvank, Marco & Vis, Iris F.A., 2011. "Lost-sales inventory theory: A review," European Journal of Operational Research, Elsevier, vol. 215(1), pages 1-13, November.
    18. Nir Halman & Diego Klabjan & Mohamed Mostagir & Jim Orlin & David Simchi-Levi, 2009. "A Fully Polynomial-Time Approximation Scheme for Single-Item Stochastic Inventory Control with Discrete Demand," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 674-685, August.
    19. Retsef Levi & Cong Shi, 2013. "Approximation Algorithms for the Stochastic Lot-Sizing Problem with Order Lead Times," Operations Research, INFORMS, vol. 61(3), pages 593-602, June.
    20. Xiang, Mengyuan & Rossi, Roberto & Martin-Barragan, Belen & Tarim, S. Armagan, 2018. "Computing non-stationary (s, S) policies using mixed integer linear programming," European Journal of Operational Research, Elsevier, vol. 271(2), pages 490-500.
    21. Chen, Zhen & Rossi, Roberto, 2021. "A dynamic ordering policy for a stochastic inventory problem with cash constraints," Omega, Elsevier, vol. 102(C).
    22. Dural-Selcuk, Gozdem & Rossi, Roberto & Kilic, Onur A. & Tarim, S. Armagan, 2020. "The benefit of receding horizon control: Near-optimal policies for stochastic inventory control," Omega, Elsevier, vol. 97(C).

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