IPA Gradient Estimators for Production-Inventory Systems

Yao Zhao, PhD

Professor in

Supply Chain Management

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Description:

Online algorithms aim to adaptively control supply chains over time with the objective of minimizing the inventory on-hand but without compromising customer service metrics.  Infinitesimal Perturbation Analysis (IPA) is a research area that develops sensitivity estimators in terms of gradients of system performance metrics with respect to parameters of interest.  IPA can be used in the optimal design and online control of various systems, including production-inventory systems, via gradient-driven methods.

My research in this area focuses on developing unbiased IPA gradients for various production inventory systems (under either stochastic fluid model or discrete model).

 

Zhao, Y., B. Melamed (2007). IPA Gradients for Make-to-Stock Production-Inventory Systems with Lost SalesIEEE Transactions on Automatic Control 52: 1491-1495.

Abstract: This note applies the stochastic fluid model (SFM) paradigm to a class of single-stage, single-product make-to-stock (MTS) production-inventory systems with stochastic demand and random production capacity, where the finished-goods inventory is controlled by a continuous-time base-stock policy and unsatisfied demand is lost. This note derives formulas for infinitesimal perturbation analysis (IPA) derivatives of the sample-path time averages of the inventory level and lost sales with respect to the base-stock level and a parameter of the production rate process. These formulas are comprehensive in that they are exhibited for any initial inventory state, and include right and left derivatives (when they differ). The formulas are obtained via sample path analysis under very mild regularity assumptions, and are inherently nonparametric in the sense that no specific probability law need be postulated. It is further shown that all IPA derivatives under study are unbiased and fast to compute, thereby providing the theoretical basis for online adaptive control of MTS production-inventory systems.

Zhao, Y., B. Melamed (2006). IPA Gradients for Make-to-Stock Production-Inventory Systems with Backorders. Methodology and Computing in Applied Probability 8: 191-222.

Abstract: These two papers model a Make-to-Stock (MTS) production-inventory system as a stochastic fluid model (SFM), and derive formulas for IPA derivatives of the sample-path time averages of the inventory level, backorder level and lost sales with respect to control parameters (base-stock level and production rate).  The IPA derivatives are comprehensive because they are exhibited for any initial inventory state, and include right and left derivatives (when they differ). The IPA derivatives obtained can be applied to online control, since they are fast to compute and unbiased.  Furthermore, these derivatives are nonparametric in the sense that they do not assume any underlying probability law, but only require sample path information. This generality renders them suitable for simulation as well as real-life systems.

Fan, Y., B. Melamed, Y. Zhao, Y. Wardi (2009). IPA Derivatives for Make-to-Stock Production-Inventory Systems with Backorders under the (R,r) Policy. Methodology and Computing in Applied Probability 11: 159-179.

Abstract: This paper addresses Infinitesimal Perturbation Analysis (IPA) in the class of Make-to Stock (MTS)production-inventory systems with backorders under the continuous-review (R,r) policy, where R is the stock-up-to level and r is the reorder point.  We map an underlying discrete MTS system to a Stochastic Fluid Model(SFM) counterpart and derives closed-form IPA derivative formulas of the time-averaged inventory level and time-averaged backorder level with respect to the policy parameters,  R and r, and shows them to be unbiased.  The obtained formulas are comprehensive in the sense that they are computed for any initial inventory state and any time horizon, and are simple and fast to compute.  These properties hold the promise of utilizing IPA derivatives as an ingredient of offline design algorithms and online management and control of the class of systems under study.