7 Extensions and Concluding Remarks

In this paper, we have reviewed various models of manufacturing systems consisting of completely flexible machines for which a theory of optimal control has been developed and for which hierarchical controls that are asymptotically optimal have been constructed. We have looked at systems in which production capacity and/or demand are random, in which state constraints may or may not be present, and in which there are reentrant flows and multiple hierarchical decision levels. We have considered for different criteria. These are discounted costs, long-run average costs, risk-sensitive controls with discounted costs and risk-sensitive controls with average costs.

While asymptotic optimal controls have been constructed for these systems under fairly general conditions, many problems remain open. We shall now describe some of them.

All of the results on the hierarchical controls in this survey assume the costs of inventory/shortage and of production to be separable. Lehoczky, Sethi, Soner, and Taksar (1991) assume a nonseparable cost and prove that the value function of the original problem converges to the value function of the limiting problem. Controls are constructed and are only conjectured to be asymptotically optimal.

With regards to systems with state constraints such as flowshops and jobshops discussed in Section 3 and Section 5, respectively, only asymptotic optimal open-loop controls are constructed in general. Because of the absence of the Lipschitz property of the constructed feedback controls, their asymptotic optimality is much harder to establish. It has only been done in Sethi and Zhou (1996a), Sethi and Zhou (1996b), Fong and Zhou (1996), and Fong and Zhou (1997) for a two- machine flowshop with a specific cost structure defined in (3.14). Generalization of their results to jobshops and to general cost functions represents a challenging research problem.

When the Markov processes involved depend on control variables, as they do in Soner (1993), Sethi and Zhang (1994c), and Sethi and Zhang (1995a), no error bounds are available for constructed asymptotic optimal controls. Estimation of these errors and extensions of the results to Markov processes depending as well on the state variables remain open problems.

In the case of the the long-run average cost criterion, all of models dealing with hierarchical controls except Section 5.2 have constructed open-loop hierarchical controls in presence of an attrition rate. Extensions of these results to the systems without an attrition term remain open. No hierarchical feedback controls that are optimal have been constructed. In case of optimal control of these ergodic systems, verification theorems have been obtained, but the existence of optimal controls is proved only in the single product case of Section 4.1. In other cases, even the existence of optimal controls remains an open issue.

An important class of manufacturing systems consists of systems that have machines which are not completely flexible, and thus involve setup costs and/or setup times, when switching from production of one product to that of another. Such systems have been considered by Gershwin (1986), Gershwin, Caramanis, and Murray (1988), Sharifnia, Caramanis, and Gershwin (1991), Caramanis, Sharifnia, Hu, and Gershwin (1991), Connolly, Dallery, and Gershwin (1992), Hu and Caramanis (1992), and Srivatsan and Gershwin (1990). They have examined various possible heuristic policies and have carried out numerical computations and simulations. They have not studied their policies from the viewpoint of asymptotic optimality. Sethi and Zhang (1995b) have made some progress in this direction.

Based on the theoretical work on hierarchical control of stochastic manufacturing systems, Srivatsan, Bai, and Gershwin (1994) have developed a hierarchical framework and describe its experimental implementation in a semiconductor research laboratory at MIT. It is expected that such research would lead to the development of real-time decision making algorithms suitable for use in actual flexible manufacturing facilities; see also Caramanis, Sharifnia, Hu, and Gershwin (1991).

Finally, while error bounds have been obtained in some cases and not others, they do not provide information on how small $\varepsilon$, the rate of slow and fast rates, have to be for asymptotic hierarchical controls to be acceptable in practice. This issue can only be investigated computationally, and there is much work remaining to be done in this regard.