Part I: DISCOUNTED-COST MODELS

One of the most important cost criteria is that of the total discounted cost. In this part we will discuss optimal and hierarchical controls of stochastic manufacturing systems with convex cost under the discounted cost criterion. In Section 2, we consider the theory of optimality for these systems. In Section 3, the theory will be used in deriving hierarchical controls. Moreover, the hierarchical controls will be proved to be nearly optimal.

  
2. Optimal Control under the Discounted Cost 
Criterion

The class of convex production planning models is an important paradigm in the operations management/operations research literature. The earliest formulation of a convex production planning problem in a discrete-time framework dates back to . They were interested in obtaining a production plan over a finite horizon in order to satisfy a deterministic demand and minimize the total discounted convex costs of production and inventory holding. Since then the model has been further studied and extended in both continuous-time and discrete-time frameworks by a number of researchers, including Johnson (1957), , Veinott (1964), Adiri and Ben-Israel (1966), Sprzeuzkouski (1967), Lieber (1973), and Hartl and Sethi (1984). A rigorous formulation of the problem along with a comprehensive discussion of the relevant literature appears in Bensoussan, Crouhy, and Proth (1983).

Extensions of the convex production planning problem to handle stochastic demand have been analyzed mostly in the discrete-time framework. A rigorous analysis of the stochastic problem has been carried out in Bensoussan, Crouhy, and Proth (1983). Continuous-time versions of the model that incorporate additive white noise terms in the dynamics of the inventory process were analyzed by and .

Earlier works that relate most closely to problems under consideration here include Kimemia and Gershwin (1983), Akella and Kumar (1986), Fleming, Sethi, and Soner (1987), Sethi, Soner, Zhang and Jiang (1992), and Lehoczky, Sethi, Soner, and Taksar (1991). These works incorporate piecewise deterministic processes (PDP) either in the dynamics or in the constraints of the model. Fleming, Sethi, and Soner (1987) consider the demand to be a finite state Markov process. In the models of Kimemia and Gershwin (1983), Akella and Kumar (1986), Sethi, Soner, Zhang, and Jiang (1992a) and Lehoczky, Sethi, Soner, and Taksar (1991), the production capacity rather than the demand for production is modeled as a stochastic process. In particular, the process of machine breakdown and repair is modeled as a birth-death process, thus making the production capacity over time a finite state Markov process.

Here we will discuss the optimality of single/parallel machine systems, N-machine flowshops, and general jobshops.