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IPA derivatives for a discrete model of make-to-stock production-inventory systems with backorders

Author

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  • Benjamin Melamed
  • Yihong Fan
  • Yao Zhao
  • Yorai Wardi

Abstract

We consider a class of single-stage, single-product Make-to-Stock production-inventory system (MTS system) with backorders. The system employs a continuous-review base-stock policy which strives to maintain a prescribed base-stock level of inventory. In a previous paper of Zhao and Melamed (Methodology and Computing in Applied Probability 8:191–222, 2006 ), the Infinitesimal Perturbation Analysis (IPA) derivatives of inventory and backorders time averages with respect to the base-stock level and a parameter of the production-rate process were computed in Stochastic Fluid Model (SFM) setting, where the demand stream at the inventory facility and its replenishment stream from the production facility are modeled by stochastic rate processes. The advantage of the SFM abstraction is that the aforementioned IPA derivatives can be shown to be unbiased. However, its disadvantages are twofold: (1) on the modeling side, the highly abstracted SFM formulation does not maintain the identity of transactions (individual demands, orders and replenishments) and has no notion of lead times, and (2) on the applications side, the aforementioned IPA derivatives are brittle in that they contain instantaneous rates at certain hitting times which are rarely known, and consequently, need to be estimated. In this paper, we remedy both disadvantages by using a discrete setting, where transaction identity is maintained, and order fulfillment from inventory following demand arrivals and inventory restocking following replenishment arrivals are modeled as discrete jumps in the inventory level. We then compute the aforementioned IPA derivatives with respect to the base-stock level and a parameter of the lead-time process in the discrete setting under any initial system state. The formulas derived are shown to be unbiased and directly computable from sample path observables, and their computation is both simple and computationally robust. Copyright Springer Science+Business Media, LLC 2010

Suggested Citation

  • Benjamin Melamed & Yihong Fan & Yao Zhao & Yorai Wardi, 2010. "IPA derivatives for a discrete model of make-to-stock production-inventory systems with backorders," Annals of Operations Research, Springer, vol. 181(1), pages 1-19, December.
    • Handle: RePEc:spr:annopr:v:181:y:2010:i:1:p:1-19:10.1007/s10479-009-0662-9
    DOI: 10.1007/s10479-009-0662-9
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    References listed on IDEAS

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    1. Ioannis Paschalidis & Yong Liu & Christos Cassandras & Christos Panayiotou, 2004. "Inventory Control for Supply Chains with Service Level Constraints: A Synergy between Large Deviations and Perturbation Analysis," Annals of Operations Research, Springer, vol. 126(1), pages 231-258, February.
    2. Y. Wardi & B. Melamed & C.G. Cassandras & C.G. Panayiòtou, 2002. "Online IPA Gradient Estimators in Stochastic Continuous Fluid Models," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 369-405, November.
    3. Philip Heidelberger & Xi-Ren Cao & Michael A. Zazanis & Rajan Suri, 1988. "Convergence Properties of Infinitesimal Perturbation Analysis Estimates," Management Science, INFORMS, vol. 34(11), pages 1281-1302, November.
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