IEEE Transactions on Automatic Control, Vol.42, No.4, 440-457, 1997
Pade Approximants for the Transient Optimization of Hedging Control Policies in Manufacturing
Part production is considered over a finite horizon in a single-part multiple-failure mode manufacturing system, When the rate of demand for parts is constant, for Markovian machine-mode dynamics and for convex running cost functions associated with part inventories or backlogs, it is known that optimal part-production policies are of the so-called hedging type. For the infinite-horizon case, such policies are characterized by a set of constant critical machine-mode dependent inventory levels that must be aimed at and maintained whenever possible, For the finite-horizon (transient) case, the critical levels still exist, but they are now time-varying and in general very difficult to characterize, Thus, in an attempt to render the problem tractable, transient production optimization is sought within the (suboptimal) class of time-invariant hedging control policies, A renewal equation is developed for the cost functional over finite horizon under an arbitrary time-invariant hedging control policy, The kernel of that renewal equation is a first-return time probability density function which satisfies an auxiliary system of Kolmogorov-type partial differential equations (PDE), The renewal equation and the auxiliary PDE system are used to generate the terms in an infinite Laurent series expansion of the Laplace transform of the finite-horizon cost functional viewed as a function of the length of that horizon T. The terms in the infinite series expansion are generated recursively, and their calculation is based on the solution of a system of piecewise smooth coupled linear differential equations, the associated Jordan canonical form of which is explicitly constructed, In the two-state machine case, this shows immediately that the Bielecki-Kumar infinite-horizon cost is approached via a term that decays to zero as 1/T and that can be computed exactly, Furthermore, Pade approximants to the resulting infinite series expansion yield a generic (and quite accurate) approximate expression of the cost functional in terms of T and z, the arbitrary hedging level, In the multistate case, Pade approximants yield excellent numerical approximations to the cost functional as a function of T for given choices of hedging levels, This is subsequently used as part of an optimization scheme, whereby hedging levels which are optimal for a given finite-horizon length are efficiently computed, The algorithms presented here can also be applied to the finite-horizon optimization for multipart failure-prone manufacturing systems, provided that only the partwise decoupled hedging control policies of Caramanis and Sharifnia are considered.