IPA derivatives for a discrete model of make-to-stock production-inventory systems with backorders |
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Authors: | Benjamin Melamed Yihong Fan Yao Zhao Yorai Wardi |
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Institution: | 1.Rutgers Business School—Newark and New Brunswick, Department of Supply Chain Management and Marketing Sciences,Rutgers University,Piscataway,USA;2.Rutgers Business School—Newark and New Brunswick, Department of Management Science and Information Systems,Rutgers University,Newark,USA;3.Rutgers Business School—Newark and New Brunswick, Department of Supply Chain Management and Marketing Sciences,Rutgers University,Newark,USA;4.School of Electrical and Computer Engineering,Georgia Institute of Technology,Atlanta,USA |
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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. |
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