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The Nonstationary Infinite Horizon Inventory Problem

Author

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  • Thomas E. Morton

    (Carnegie-Mellon University)

Abstract

By allowing disposal in each period, a sequence of upper and lower bounds on the infinite horizon nonstationary periodic review inventory problem is obtained. The nth bounds depend only on knowledge of the demand distributions in the first n periods, giving planning horizon results. In general, calculation of any bound requires solution of an ordinary n period problem, although for the proportional cost case the solution for n = 1 and good approximations for n = 2 may easily be given. The upper and lower relative cost and policy bounds are monotonic in the horizon length and, under mild conditions, converge to a unique infinite horizon solution. An analytic expression is given for the asymptotic convergence rate, which is geometric even for the undiscounted case. The formulation given here is a concrete example of the general planning horizon formulation for infinite horizon dynamic programming models set forth in [Morton, T. 1975. Infinite horizon dynamic programming models--a planning horizon formulation. Carnegie-Mellon University Management Sciences Research Report No. 372, August (to appear in Operations Res.)].

Suggested Citation

  • Thomas E. Morton, 1978. "The Nonstationary Infinite Horizon Inventory Problem," Management Science, INFORMS, vol. 24(14), pages 1474-1482, October.
    • Handle: RePEc:inm:ormnsc:v:24:y:1978:i:14:p:1474-1482
    DOI: 10.1287/mnsc.24.14.1474
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    Cited by:

    1. Ehrenthal, J.C.F. & Honhon, D. & Van Woensel, T., 2014. "Demand seasonality in retail inventory management," European Journal of Operational Research, Elsevier, vol. 238(2), pages 527-539.
    2. Iida, Tetsuo, 2002. "A non-stationary periodic review production-inventory model with uncertain production capacity and uncertain demand," European Journal of Operational Research, Elsevier, vol. 140(3), pages 670-683, August.
    3. Alexandar Angelus, 2011. "A Multiechelon Inventory Problem with Secondary Market Sales," Management Science, INFORMS, vol. 57(12), pages 2145-2162, December.
    4. Iida, Tetsuo, 1999. "The infinite horizon non-stationary stochastic inventory problem: Near myopic policies and weak ergodicity," European Journal of Operational Research, Elsevier, vol. 116(2), pages 405-422, July.
    5. Cheaitou, Ali & van Delft, Christian, 2013. "Finite horizon stochastic inventory problem with dual sourcing: Near myopic and heuristics bounds," International Journal of Production Economics, Elsevier, vol. 143(2), pages 371-378.
    6. Torpong Cheevaprawatdomrong & Robert L. Smith, 2004. "Infinite Horizon Production Scheduling in Time-Varying Systems Under Stochastic Demand," Operations Research, INFORMS, vol. 52(1), pages 105-115, February.
    7. Van-Anh Truong, 2014. "Approximation Algorithm for the Stochastic Multiperiod Inventory Problem via a Look-Ahead Optimization Approach," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1039-1056, November.
    8. Xiangwen Lu & Jing-Sheng Song & Amelia Regan, 2006. "Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds," Operations Research, INFORMS, vol. 54(6), pages 1079-1097, December.
    9. Satya S. Malladi & Alan L. Erera & Chelsea C. White, 2023. "Inventory control with modulated demand and a partially observed modulation process," Annals of Operations Research, Springer, vol. 321(1), pages 343-369, February.
    10. Iida, Tetsuo, 2001. "The infinite horizon non-stationary stochastic multi-echelon inventory problem and near-myopic policies," European Journal of Operational Research, Elsevier, vol. 134(3), pages 525-539, November.
    11. Chaithanya Bandi & Eojin Han & Omid Nohadani, 2019. "Sustainable Inventory with Robust Periodic-Affine Policies and Application to Medical Supply Chains," Management Science, INFORMS, vol. 65(10), pages 4636-4655, October.
    12. Jing‐Sheng Song & Paul H. Zipkin, 2012. "Newsvendor problems with sequentially revealed demand information," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(8), pages 601-612, December.
    13. Srinivas Bollapragada & Thomas E. Morton, 1999. "Myopic Heuristics for the Random Yield Problem," Operations Research, INFORMS, vol. 47(5), pages 713-722, October.
    14. Robert L. Smith & Rachel Q. Zhang, 1998. "Infinite Horizon Production Planning in Time-Varying Systems with Convex Production and Inventory Costs," Management Science, INFORMS, vol. 44(9), pages 1313-1320, September.
    15. Zhaotong Lian & Liming Liu & Stuart X. Zhu, 2010. "Rolling‐horizon replenishment: Policies and performance analysis," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(6), pages 489-502, September.
    16. Tetsuo Iida & Paul H. Zipkin, 2006. "Approximate Solutions of a Dynamic Forecast-Inventory Model," Manufacturing & Service Operations Management, INFORMS, vol. 8(4), pages 407-425, October.

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