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On the Objective Function Behavior in ( s , S ) Inventory Models

Author

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  • Izzet Sahin

    (University of Wisconsin, Milwaukee, Wisconsin)

Abstract

The most common measure of effectiveness used in determining the optimal ( s , S ) inventory policies is the total cost function per unit time, E ( s , Δ), Δ = S − s . In stationary analysis, this function is constructed through the limiting distribution of on-hand inventory, and it involves some renewal-theoretic elements. For Δ ≥ 0 given, E ( s , Δ) turns out to be convex in s , so that the corresponding optimal reorder point, s 1 (Δ), can be characterized easily. However, E ( s 1 (Δ), Δ) is not in general unimodal on Δ ≥ 0. This requires the use of complicated search routines in computations, as there is no guarantee that a local minimum is global.Both for periodic and continuous review systems with constant lead times, full backlogging and linear holding and shortage costs, we prove in this paper that E ′( s 1 (Δ), Δ) = 0, Δ ≥ 0, is both necessary and sufficient for a global minimum ( E ( s 1 (Δ), Δ) is pseudoconvex on Δ ≥ 0) if the underlying renewal function is concave. The optimal stationary policy can then be computed efficiently by a one-dimensional search routine. The renewal function in question is that of the renewal process of periodic demands in the periodic review model and of demand .sizes in the continuous review model.

Suggested Citation

  • Izzet Sahin, 1982. "On the Objective Function Behavior in ( s , S ) Inventory Models," Operations Research, INFORMS, vol. 30(4), pages 709-724, August.
    • Handle: RePEc:inm:oropre:v:30:y:1982:i:4:p:709-724
    DOI: 10.1287/opre.30.4.709
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    Cited by:

    1. Nasr, Walid W. & Maddah, Bacel, 2015. "Continuous (s, S) policy with MMPP correlated demand," European Journal of Operational Research, Elsevier, vol. 246(3), pages 874-885.
    2. Zhang, Zhe George & Kim, Ilhyung & Springer, Mark & Cai, Gangshu (George) & Yu, Yugang, 2013. "Dynamic pooling of make-to-stock and make-to-order operations," International Journal of Production Economics, Elsevier, vol. 144(1), pages 44-56.
    3. Tovey C. Bachman & Pamela J. Williams & Kristen M. Cheman & Jeffrey Curtis & Robert Carroll, 2016. "PNG: Effective Inventory Control for Items with Highly Variable Demand," Interfaces, INFORMS, vol. 46(1), pages 18-32, February.
    4. Tamjidzad, Shahrzad & Mirmohammadi, S. Hamid, 2015. "An optimal (r, Q) policy in a stochastic inventory system with all-units quantity discount and limited sharable resource," European Journal of Operational Research, Elsevier, vol. 247(1), pages 93-100.
    5. 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—Continuous‐time case," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 154-169, January.
    6. 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.
    7. 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.
    8. Carl R. Schultz, 1990. "On the optimality of the (S — 1,S) policy," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(5), pages 715-723, October.
    9. Kleijnen, Jack P.C. & Mehdad, E. & van Beers, W.C.M., 2012. "Convex and monotonic bootstrapped kriging," Other publications TiSEM 972e079d-0209-45bf-b25e-a, Tilburg University, School of Economics and Management.
    10. 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.
    11. Tamer Boyacı & Guillermo Gallego, 2002. "Managing waiting times of backordered demands in single‐stage (Q, r) inventory systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(6), pages 557-573, September.

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