### G-2001-44

# Decomposition/Aggregation Based Dynamic Programming Optimization of Partially Homogeneous Unreliable Transfer Lines

## and BibTeX reference

In this paper, the problem of optimally controlling production in a single part unreliable, manufacturing flow line, subjected to a constant rate of demand for parts is considered. A performance measure while combines storage and production backlog costs is used. For Markovian, two-state machines, an analytic solution of the associated Hamilton-Jacobi-Bellman equation is beyond reach. Instead, the focus here is on a suboptimal class of decentralized hedging policies parameterized by a set of *critical inventory levels*, one for each machine in the transfer line. Thus, each machine strives to achieve as quickly as possible a given critical level of processed parts in the associated storage bin, which, once reached, it will attempt to maintain by producing exactly at the current rate of demand for parts until failure or starvation occurs. Once starvation ceases or the machine is repaired, it will resume the same production strategy. Our objective is to optimize the choice of the processed parts critical levels. For doing so, we first recall a decomposition technique based on two decoupling approximations; the *machine decoupling approximation* aims at simplifying the world *upstream* of a given machine; the *demand averaging principle* aims at effectively shielding the machine from the precise instantaneous behavior of the machines *downstream*. The decomposition technique yields particularly simple models in so-called partially homogeneous flow lines, i.e. lines such that mean repair rates for machines, after failure, are all identical. Under the partially homogeneous assumption, an efficient dynamic programming solution to the optimization problem is developed. Existence of an optimum distribution of critical levels is established for the finite transfer line case. In addition, for a discretized finite state version of the homogeneous infinite transfer line problem, existence of an optimal feedback policy which is stationary is established. Finally, under specific structural assumptions, qualitative properties of optimal critical levels profile for the homogeneous (continuous states) but finite transfer line are derived. From these properties, homogeneous infinite (continuous state) transfer line behavior is inferred. Results of numerical experiments, and comparisons with optimal control performance control performance are reported.

Published **November 2001**
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30 pages