Part II: AVERAGE-COST MODELS

In this part we review another important cost criterion, namely, the long-run average cost criterion.

The criterion presents a stark contrast to the discounted-cost criterion. The discounted-cost criterion considers near-term costs to be more important than costs occuring in the long term. In fact, the discounted-cost criterion all but ignores the distant future costs. The long-run cost criterion on the other hand considers only the long-term costs to be important.

In Section 4, we consider the theory of optimal control of stochastic manufacturing problems with the long-run average cost criterion. We use the vanishing discount approach for the purpose of this paper. Section 5 is concerned with hierarchical controls for long-run average cost problems.

  
4 Optimal Control with the Long-Run Average Cost Criterion

Beginning with Bielecki and Kumar (1988), there has been a considerable interest in studying the problem of convex production planning in stochastic manufacturing systems with the objective of minimizing long-run average cost. Bielecki and Kumar (1988) dealt with a single machine (with two states: up and down), single product problem with linear holding and backlog costs. Because of the simple structure of their problem, they were able to obtain an explicit solution for the problem, and thus verify the optimality of the resulting policy. They showed that the so-called hedging point policy is optimal in their simple model. It is a policy to produce at full capacity if the surplus is smaller than a threshold level, produce nothing if the surplus is higher than the threshold level, and produce as much as possible but no more than the demand rate when the surplus is at the threshold level.

Sharifnia (1988) dealt with an extension of the Bielecki-Kumar model with more than two machine states. Liberopoulos and Caramanis (1995) showed that Sharifnia's method for evaluating hedging point policies applies even when the transition rates of the machine states depend on the production rate. Liberopoulos and Hu (1995) obtained monotonicity of the threshold levels corresponding to different machine states. Srivastsan and Dallery (1998) generalized the Bielecki-Kumar problem to allow for two products. They limited their focus to only the class of hedging point policies and attempted to partially characterize an optimal solution within that class. Bai and Gershwin (1990) and Bai (1991) use heuristic argument to obtain suboptimal controls in two-machine flowshops. In addition to nonnegative constraints on the inventory levels in the internal buffer, they also consider inventory level in this buffer to be bounded above by the size of the buffer. Moreover, Srivatsan, Bai, and Gershwin (1994) apply their results to semiconductor manufacturing (jobshop). All of these papers, however, are heuristic in nature, since they do not rigorously prove the optimality of the policies for their extensions of the Bielecki-Kumar model.

Presman, Sethi, Zhang, and Zhang (1998a) extend the hedging point policy to the problem of two part types by using the potential function related to the dynamic programming equation of the two part type problem. Using a verification theorem for the two part type problem, they prove the optimality of the hedging point policy. Furthermore, by the Bielecki and Kumar result, Sethi and Zhang (1999) prove the optimality of the hedging point policy for a general n-product problem under a special class of cost functions. For the deterministic model, Sethi, Zhang and Zhang (1996) provide optimal control policies and the optimal value.

The difficulty in proving the optimality for cases rather than these special cases lies in the fact that when the problem is generalized to include convex cost and multiple machine capacity levels, explicit solutions are no longer possible. One then needs to develop appropriate dynamic programming equations, existence of their solutions, and verification theorems for optimality. In this section, we will review some works related to this.