Stability of re-entrant flow lines∗

The paper addresses the issue of stability for re-entrant flow lines, producing multiple products, with capacitated machines, random demand and random yield. The policies of interest are multi-echelon base stock policies, combined with a set of static and dynamic management rules of the available capacity. We introduce three classes of capacitated echelon base stock policies: Π0, the pure multi-echelon base stock; Π1, like Π0 with a possibly finite upper bound on the admission of raw materials; and Π2, like Π1 with a possibly finite upper bound on the utilization of intermediate inventories. The order of business is: establishing conditions for the stability of the shortfall echelon process when demands are stationary and ergodic; examining the regenerative structure of the shortfall process when demands are given by an i.i.d. sequence. The regenerative properties are valuable in establishing the convergence of costs and also simulation estimators, which enables the utilization of Infinitesimal Perturbation Analysis to optimize the policy parameters. We use a coupling argument for shortfalls while establishing the stability conditions, which will, by itself, render the Harris ergodicity of the shortfall process. We show that the stability condition suffices to ensure that the shortfall process possesses the regenerative structure of a Harris ergodic Markov chain. Under a stronger condition, we establish that the vector of shortfalls returns to the origin infinitely often, with probability one. We show that the necessary stability condition is also sufficient for any sort of re-entrant system, in the presence of random yield, provided the control policy is in Π2. Carlos F. Bispo Instituto Superior Tcnico, Av. Rovisco Pais, 1096-001 Lisboa, Portugal, e-mail: [email protected] ∗ This work was supported by the FCT Project PEst-OE/EEI/LA0009/2011.