Optimal feedback production planning in a stochastic two-machine flowshop

In this paper, the authors consider a production planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to non-negativity constraints on work-in-process. The machine capacities and demand processes are assumed to be finite state Markov chains. The problem is to choose the rate of production over time so as to minimize the expected discounted cost of production and inventory/backlog over an infinite horizon. The problem is formulated as a stochastic dynamic programming problem. It is shown that the value function of the problem is locally Lipschitz and is a viscosity solution to the dynamic programming equation together with certain boundary conditions. Optimal feedback control policies are obtained in terms of the ‘partial derivatives’ of the value function. It is shown that the derived control policy is feasible on the boundary of the state space. The results establish the theoretical framework within which further analyses and/or numerical solutions of the problem can be attempted.