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IMA Journal of Management Mathematics Advance Access originally published online on May 30, 2008
IMA Journal of Management Mathematics 2009 20(1):59-79; doi:10.1093/imaman/dpn010
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© The authors 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Stability and optimization of a production inventory system under prioritized base-stock control

Dong-Ping Song{dagger}

International Shipping & Logistics Group, The Business School, University of Plymouth, Plymouth, PL4 8AA, UK

{dagger} Email: dongping.song{at}plymouth.ac.uk

Received on 19 July 2007. Accepted on 1 May 2008.

A make-to-stock system producing two part-types with stochastic processing times and random demands is considered. A prioritized base-stock policy is used to control the production to meet exogenous Poisson demands, where the unmet high-priority demands are partially backordered due to their limited patience and the unmet low-priority demands are fully backordered. Based on the matrix analytical method, a necessary and sufficient condition for the stability of the system is provided. The explicit stationary distribution is derived using the spectral expansion approach. Interesting steady-state performance measures such as stock-out probabilities and lost-sale fraction are then calculated. By investigating the structural properties of the objective functions, simple procedures are presented to find the optimal base-stock levels either to minimize the expected cost or to satisfy the stock-out probability and the lost-sale fraction constraints. In addition, the optimization problem with respect to the maximum backlog level for part-type one and two base-stock levels is addressed and a solution procedure is presented. Numerical examples are given to demonstrate the results.

Keywords: production inventory system; prioritized base-stock control; stability; stationary distribution; optimization


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