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Sample Path Derivatives for (s, S) Inventory Systems

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  • Michael C. Fu

    (University of Maryland, College Park, Maryland)

Abstract

for ( s , S ) inventory systems, we derive sample path derivatives of performance measures with respect to the two parameters s and S . These derivatives yield derivative estimators which can be estimated from a single sample path or simulation of the inventory system, in some cases not even requiring actual knowledge of the underlying demand distribution. Such derivative estimates would be useful in sensitivity analysis or in gradient-based optimization techniques. We consider the nondiscounted periodic review system with general independent and identically distributed (i.i.d.) continuous demands, full backlogging, and general holding and shortage costs. For the infinite horizon model, consistency proofs are given for some special cases, although we argue why the estimators should be correct for the more general case.

Suggested Citation

  • Michael C. Fu, 1994. "Sample Path Derivatives for (s, S) Inventory Systems," Operations Research, INFORMS, vol. 42(2), pages 351-364, April.
    • Handle: RePEc:inm:oropre:v:42:y:1994:i:2:p:351-364
    DOI: 10.1287/opre.42.2.351
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    Cited by:

    1. Assaf Avrahami & Yale T. Herer & Retsef Levi, 2014. "Matching Supply and Demand: Delayed Two-Phase Distribution at Yedioth Group—Models, Algorithms, and Information Technology," Interfaces, INFORMS, vol. 44(5), pages 445-460, October.
    2. Zhenyu Cui & Michael C. Fu & Jian-Qiang Hu & Yanchu Liu & Yijie Peng & Lingjiong Zhu, 2020. "On the Variance of Single-Run Unbiased Stochastic Derivative Estimators," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 390-407, April.
    3. Özdemir, Deniz & Yücesan, Enver & Herer, Yale T., 2013. "Multi-location transshipment problem with capacitated production," European Journal of Operational Research, Elsevier, vol. 226(3), pages 425-435.
    4. 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.
    5. Ozdemir, Deniz & Yucesan, Enver & Herer, Yale T., 2006. "Multi-location transshipment problem with capacitated transportation," European Journal of Operational Research, Elsevier, vol. 175(1), pages 602-621, November.
    6. Foreest, N. D. van & Wijngaard, J., 2010. "On the Optimal Policy for the Single-product Inventory Problem with Set-up Cost and a Restricted Production Capacity," Research Report 10005, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    7. Michael C. Fu, 2008. "What you should know about simulation and derivatives," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(8), pages 723-736, December.
    8. 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.
    9. Sumit Kunnumkal & Huseyin Topaloglu, 2009. "A stochastic approximation method for the single-leg revenue management problem with discrete demand distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(3), pages 477-504, December.
    10. Sumit Kunnumkal & Huseyin Topaloglu, 2011. "A stochastic approximation algorithm to compute bid prices for joint capacity allocation and overbooking over an airline network," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(4), pages 323-343, June.
    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. 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.
    13. Yongqiang Wang & Michael C. Fu & Steven I. Marcus, 2012. "A New Stochastic Derivative Estimator for Discontinuous Payoff Functions with Application to Financial Derivatives," Operations Research, INFORMS, vol. 60(2), pages 447-460, April.
    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. repec:dgr:rugsom:10005 is not listed on IDEAS
    16. Dimitris Bertsimas & Aurélie Thiele, 2006. "A Robust Optimization Approach to Inventory Theory," Operations Research, INFORMS, vol. 54(1), pages 150-168, February.
    17. Huseyin Topaloglu & Warren Powell, 2007. "Incorporating Pricing Decisions into the Stochastic Dynamic Fleet Management Problem," Transportation Science, INFORMS, vol. 41(3), pages 281-301, August.
    18. Yao Zhao & Benjamin Melamed, 2006. "IPA Derivatives for Make-to-Stock Production-Inventory Systems with Backorders," Methodology and Computing in Applied Probability, Springer, vol. 8(2), pages 191-222, June.
    19. Sumit Kunnumkal & Huseyin Topaloglu, 2008. "Using Stochastic Approximation Methods to Compute Optimal Base-Stock Levels in Inventory Control Problems," Operations Research, INFORMS, vol. 56(3), pages 646-664, June.
    20. Nicky D. Van Foreest & Jacob Wijngaard, 2014. "On Optimal Policies for Production-Inventory Systems with Compound Poisson Demand and Setup Costs," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 517-532, May.
    21. Yihong Fan & Benjamin Melamed & Yao Zhao & Yorai Wardi, 2009. "IPA Derivatives for Make-to-Stock Production-Inventory Systems With Backorders Under the (R,r) Policy," Methodology and Computing in Applied Probability, Springer, vol. 11(2), pages 159-179, June.
    22. Huseyin Topaloglu, 2008. "A Stochastic Approximation Method to Compute Bid Prices in Network Revenue Management Problems," INFORMS Journal on Computing, INFORMS, vol. 20(4), pages 596-610, November.
    23. Gong, Y. & Yucesan, E., 2006. "The Multi-Location Transshipment Problem with Positive Replenishment Lead Times," ERIM Report Series Research in Management ERS-2006-048-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.

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