Part III: RISK-SENSITIVE COST MODELS
 In this part we consider robust production plans with risk sensitive cost criteria. This consideration is motivated by the following observations. First, since most manufacturing systems are large and complex, it is difficult to establish accurate mathematical models to describe these systems. Modeling errors are inevitable. Second, in practice, an optimal policy for a subdivision of a big corporation is usually not an optimal policy for the whole corporation. Optimal solutions with the usual cost criteria may not be desirable in many real situations. An alternative approach is to consider robust controls. In some manufacturing systems, it is more desirable to consider controls that are robust enough to attenuate uncertain disturbances, such as modeling errors, and therefore achieve the system stability. Robust control design is particularly important in manufacturing systems with unfavorable disturbances. There are two kinds of system disturbances in the systems under consideration: (1) unfavorable internal disturbances - usually associated with unfavorable machine capacity fluctuations; (2) unfavorable external disturbances such as fluctuations in demand.

  
6 Risk-Sensitive Hierarchical Controls

The basic idea of the risk-sensitive control is to consider a risk-sensitive cost function that penalizes heavily on costs associated with large values of state and control variables. Typically an exponential-of-integral cost criterion is considered. Such cost functions penalize heavily, state trajectories and controls which give large values to the exponent. Related literature on risk-sensitive and robust controls can be found in Whittle (1990), Fleming and McEneaney (1995), Barron and Jensen (1989), and references therein. For details of models discussed in this section, see Zhang (1995).

As the rate of fluctuation of the production capacity process goes to infinity, we show that the risk-sensitive control problem can be approximated by a limiting problem in which the stochastic capacity process can be averaged out and replaced by its average. We also show that the value function of the limiting problem satisfies the Isaacs equation of a zero-sum, two-player differential game. Then, we use a near optimal control of the limiting problem to construct a nearly optimal control for the original risk-sensitive control problem.

In Section 6.1 we consider risk-sensitive hierarchical controls with discounted costs. In Section 6.2 we study risk-sensitive hierarchical controls with average costs.