OPTIMALITY OF STATE-DEPENDENT (s,S) POLICIES IN INVENTORY MODELS WITH MARKOV-MODULATED DEMAND AND LOST SALES

Markov-modulated processes have been used in stochastic inventory models with setup costs for modeling demand under the influence of uncertain environmental factors, such as fluctuating economic and market conditions. The analyses of these models have been carried out in the literature only under the assumption that unsatisfied demand is fully backlogged. The lost sales situation occurs in many retail establishments such as department stores and supermarkets. We use the analysis of the Markovian demand model with backlogging to analyze the lost sales case; in particular, we establish the optimality of an (s, S)-type policy under fairly general conditions.     

CAPACITY AND PRODUCTION DECISIONS IN STOCHASTIC MANUFACTURING SYSTEMS: AN ASYMPTOTIC OPTIMAL HIERARCHICAL APPROACH

We present a new paradigm of hierarchical decision making in production planning and capacity expansion problems under uncertainty. We show that under reasonable assumptions, the strategic level management can base the capacity decision on aggregated information from the shopfloor, and the operational level management, given this decision, can derive a production plan for the system, without too large a loss in optimality when compared to simultaneous determination of optimal capacity and production decisions. The results are obtained via an asymptotic analysis of a manufacturing system with convex costs, constant demand, and with machines subject to random breakdown and repair. The decision variables are purchase time of a new machine at a given fixed cost and production plans before and after the purchase. The objective is to minimize the discounted costs of investment, production, inventories, and backlogs. If the rate of change in machine states such as up and down is assumed to be much larger than the rate of discounting costs, one obtains a simpler limiting problem in which the random capacity is replaced by its mean. We develop methods for constructing asymptotically optimal decisions for the original problem from the optimal decisions for the limiting problem. We obtain error estimates for these constructed decisions.     

Exponential families of markov processes -part II: birth- and death processes

A common prior distribution and loss structure are set up to be appropriate for the sorting of batches using sampling inspection by variable and by attribute. Approxi¬mations to the exact optimal sampling plans are derived to gain a better understanding of BAYEsian sampling plans and to compare the plans using variable sampling and attribute sampling. It is assumed that the process quality distribution is normal with a known variance     

COMBINED PRODUCTION AND MAINTENANCE SCHEDULING FOR A MULTIPLE‐PRODUCT,SINGLE‐ MACHINE PRODUCTION SYSTEM

Traditionally, the problems of equipment maintenance scheduling and production scheduling in a multi‐product environment have been treated independently. In this paper, we develop a Markov decision process model that simultaneously determines maintenance and production schedules for a multiple‐product, single‐machine production system, accounting for the fact that equipment condition can affect the yield of different product types differently. The problem was motivated by an application in semiconductor manufacturing. After examining structural properties of the optimal policy, we compare the combined method to an approach often used in practice. In the nearly 6,000 test problems studied, the reward from the combined method was an average of more than 25 percent greater than the reward from the traditional method.     

INVENTORY RATIONING AND SHIPMENT FLEXIBILITY ALTERNATIVES FOR DIRECT MARKET FIRMS

This paper investigates inventory‐rationing policies of interest to firms operating in a direct market channel. We model a single product with two demand classes, where one class requests a lower order fulfillment lead time but pays a higher price. Demand for each class follows a Poisson process. Inventory is fed by a production system with exponentially distributed build times. We study rationing policies in which the firm either blocks or backlogs orders for the lower priority customers when inventory drops below a certain level. We compare the performance of these rationing policies with a pure first‐come, first‐serve policy under various scenarios for customer response to delay: lost sales, backlog, and a combination of lost sales and backlog.     

RETAIL INVENTORY CONTROL WITH LOST SALES,SERVICE CONSTRAINTS,AND FRACTIONAL LEAD TIMES

This paper describes a periodic review, fixed lead time, single-product, single-facility model with random demand, lost sales and service constraints that was developed for potential application at a Western Canadian retailer. The objective of this study was to determine optimal (s, S) policies for a large number of products and locations. To this end, we evaluate the long run average cost and service level for a fixed (s, S) policy and then used a search procedure to locate an optimal policy. The search procedure is based on an efficient updating scheme for the transition probability matrix of the underlying Markov chain, bounds on S and monotonicity assumptions on the cost and service level functions. A distinguishing feature of this model is that lead times are shorter than review periods so that the stationary analysis underlying computation of costs and service levels requires subtle analyses. We compared the computed policies to those currently in use on a test bed of 420 products and found that stores currently hold inventories that are 40% to 50% higher than those recommended by our model and estimate that implementing the proposed policies for the entire system would result in significant cost savings.