Optimal Feedback Production Planning in a Stochastic N-machine Flowshop
Posted: 6 May 2020
Date Written: 1995
We consider a production planning problem in an N-machine flowshop subject to breakdown and repair of machines and 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 rates of production on the N machines over time to minimize the expected discounted cost of production and inventory/backlog over an infinite horizon. It is formulated as a stochastic dynamic programming problem. We show that the value function of the problem is locally Lipschitz and is a solution to a dynamic programming equation together with a certain boundary condition. We provide an interpretation of the boundary condition. We also prove a verification theorem and derive the optimal feedback control policy in terms of the directional derivatives of the value function. Finally, we obtain a deterministic optimal control problem that is equivalent to the stochastic production planning problem under consideration.
Keywords: Manufacturing systems; optimal control: stochastic jump processes; feedback control; piecewise-deterministic processes; dynamic programming: flowshops
JEL Classification: C61, M11, M20
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