Existence of Optimal Feedback Production Plans in Stochastic Flowshops with Limited Buffers

This paper extends the work PSZ, who only required the work-in-process to be nonnegative. Inclusion of lower and upper bound constraints on the work-inprocess and upper bound on the finished good surplus represents an important feature that is usually present in real life. There has been a substantial number of works related to the problems considered here. In view of the literature review in PSZ and Sethi and Zhang [8], we choose not to discuss the earlier literature in this paper. Instead, we discuss the relevant research that has appeared subsequent to PSZ. Fong and Zhou [2] have treated a two-machine flowshop with limited buffers in the context of hierarchical controls. While they are not able to show the local Lipschitz continuity of the value function as in PSZ for an n-machine flowshop with unlimited buffers or in Sethi, Zhang and Zhou [9] for a 2-machine flowshop with 2 limited internal buffer, they prove a weaker property that is sufficient for their analysis of hierachical controls. But when it comes to optimal controls, local Lipschitz property of the value function is one of the most important things to establish. We do not know how to extend the construction procedure used in PSZ and Sethi, Zhang and Zhou [9] to allow for upper bounds on the buffer sizes. In this paper, therefore, we develop a new methodology that allows us to prove that the value function of a general N-machine flowshop is locally Lipschitz continuous. While useful in the present context, we believe that the methodology would find applications in other contexts. It therefore represents a main contribution of this paper. The plan of the paper is as follows. In Section 2, we give a formulation of the problem and state the result on the existence and uniqueness of the optimal feedback contr...