Economia Basica

Annals of Operations Research 126, 21–41, 2004  2004 Kluwer Academic Publishers. Manufactured in The Netherlands.

Optimal Policies for Production/Inventory Systems with Finite Capacity and Markov-Modulated Demand and Supply Processes ∗
GUILLERMO GALLEGO and HAICHAO HU ggallego@ieor.columbia.edu; HH190@columbia.edu Department of Industrial Engineering and Operations Research ColumbiaUniversity, New York, NY 10027-6699, USA

Abstract. In many production/inventory systems, not only is the production/inventory capacity finite, but the systems are also subject to random production yields that are influenced by factors such as breakdowns, repairs, maintenance, learning, and the introduction of new technologies. In this paper, we consider a singleitem, single-location, periodic-reviewmodel with finite capacity and Markov modulated demand and supply processes. When demand and supply processes are driven by two independent, discrete-time, finite-state, time-homogeneous Markov chains, we show that a modified, state-dependent, inflated base-stock policy is optimal for both the finite and infinite horizon planning problems. We also show that the finite-horizon solution converges to theinfinite-horizon solution. Keywords: production/inventory system, finite-capacity, Markov-modulated demand and Markovmodulated yield, modified state-dependent inflated base-stock policy

Introduction In this paper, we analyze a discrete-time, single-item, single-location, periodic-review production/inventory system with finite production capacity where the demand and supply processes are driven by twoindependent, discrete-time, finite-state, timehomogeneous Markov chains. Markov chain modulated demand models have been used by many authors to capture the influence on demand by factors such as weather, product age, economic conditions, and price competition. We feel that a similar argument can be made to model the supply process. In many instances, not only is the production/inventory capacity finite,but the system is also subject to random production yields that are influenced by factors such as breakdowns, repairs, maintenance, learning, and the introduction of new technologies. Consider, for example, a machine that has just been repaired and is operating at an ideal state. Over time, the machine may start producing a larger percentage of defective products, with this percentage increasing overtime unless preventive maintenance or repairs are made. As a second example, consider the yields in semiconductor manufacturing. Production yields increase as the manufacturing process is fine tuned and drop again when new, more complex, products are introduced to replace older products. The influence of these factors on random yields
∗ The research for this paper was supported by NSF grantDMI-0075627.

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can be effectively modeled by a Markov chain driven process. Parlar, Wang, and Gerchak (1995), for example, present a model where the supply process can be in one of two states and these two states are governed by a Markov chain. The problem considered here brings together three literature streams (random yields, finite capacity, and Markov driven models) into arealistic problem faced frequently in industry by semiconductor manufacturers producing complex products with short life cycles, assemblers of personal computers with unreliable suppliers, and airlines managing spare parts which are reconditioned in congested job shops. Under the above assumptions on the demand and supply processes and some other mild conditions, we show that, given the demandstate and the yield state, the optimal production/ordering policy is a modified state-dependent inflated base-stock policy. This means that the optimal production/ordering quantity for each period is decreasing1 with respect to the initial level and the optimal order-up-to level is decreasing with respect to the initial level. The term “inflated base-stock policy” was coined by Zipkin, see (Zipkin,...