A Stochastic Programming Approach toManufacturing Flow Control

This paper proposes and tests an approximation of the solution of a class of piecewise deterministic control problems, typically used in the modeling of manufacturing ow processes. This approximation uses a stochastic programming approach on a suitably discretized and sampled system. The method proceeds through two stages: (i) the Hamilton-Jacobi-Bellman (HJB) dynamic programming equations for the nite horizon continuous time stochastic control problem are dis-cretized over a set of sampled times; this deenes an associated discrete time stochastic control problem which, due to the niteness of the sample path set for the Markov disturbance process, can be written as a stochastic programming problem. (ii) The very large event tree representing the sample path set is replaced with a reduced tree obtained by randomly sampling over the set of all possible paths. It is shown that the solution of the stochastic program deened on the randomly sampled tree converges toward the solution of the discrete time control problem when the sample size increases to innnity. The discrete time control problem solution converges to the solution of the ow control problem when the discretization mesh tends to 0. A comparison with a direct numerical solution of the dynamic programming equations is made for a single part manufacturing ow control model in order to illustrate the convergence properties. Applications to larger models aaected by the curse of dimensionality in a standard dynamic programming techniques show the possible advantages of the method.