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IPA Derivatives for Make-to-Stock Production-Inventory Systems with Backorders

Author

Listed:
  • Yao Zhao

    (Rutgers University)

  • Benjamin Melamed

    (Rutgers University)

Abstract

A single-stage Make-to-Stock (MTS) production-inventory system consists of a production facility coupled to an inventory facility, and is subject to a policy that aims to maintain a prescribed inventory level (called base stock) by modulating production capacity. This paper considers a class of single-stage, single-product MTS systems with backorders, driven by random demand and production capacity, and subject to a continuous-review base-stock policy. A model from this class is formulated as a stochastic fluid model (SFM), where all flows are described by stochastic rate processes with piecewise constant sample paths, subject to very mild regularity assumptions that merely preclude accumulation points of jumps with probability 1. Other than that, the MTS model in SFM setting is nonparametric in that it assumes no specific form for the underlying probability law, and as such is quite general. The paper proceeds to derive formulas for the (stochastic) IPA (Infinitesimal Perturbation Analysis) derivatives of the sample-path time averages of the inventory level and backorders level with respect to the base-stock level and a parameter of the production rate. These formulas are comprehensive in that they are exhibited for any initial condition of the system, and include right and left derivatives (when they do not coincide). The derivatives derived are then shown to be unbiased and their formulas are seen to be amenable to fast computation. The generality of the model and comprehensiveness of the IPA derivative formulas hold out the promise of gradient-based applications. More specifically, since the base-stock level and production rate are the key control parameters of MTS systems, the results provide the theoretical underpinnings for optimizing the design of MTS systems and for devising prospective on-line adaptive control algorithms that employ IPA derivatives. The paper concludes with a discussion of those issues.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:8:y:2006:i:2:d:10.1007_s11009-006-8548-7
    DOI: 10.1007/s11009-006-8548-7
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    References listed on IDEAS

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    1. Uday S. Karmarkar, 1987. "Lot Sizes, Lead Times and In-Process Inventories," Management Science, INFORMS, vol. 33(3), pages 409-418, March.
    2. 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.
    3. 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.
    4. Lawrence M. Wein, 1992. "Dynamic Scheduling of a Multiclass Make-to-Stock Queue," Operations Research, INFORMS, vol. 40(4), pages 724-735, August.
    5. 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.
    6. Michael C. Fu, 1994. "Sample Path Derivatives for (s, S) Inventory Systems," Operations Research, INFORMS, vol. 42(2), pages 351-364, April.
    7. Paul H. Zipkin, 1986. "Models for Design and Control of Stochastic, Multi-Item Batch Production Systems," Operations Research, INFORMS, vol. 34(1), pages 91-104, February.
    8. Paul Glasserman & Sridhar Tayur, 1995. "Sensitivity Analysis for Base-Stock Levels in Multiechelon Production-Inventory Systems," Management Science, INFORMS, vol. 41(2), pages 263-281, February.
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