PADE APPROXIMANTS FOR THE TRANSIENT OPTIMIZATION OF HEDGING CONTROL POLICIES IN MANUFACTURING

Part production is considered over a finite horizon in a single-part m ultiple-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 charac terized by a set of constant critical machine-mode dependent inventory levels that must be aimed at and maintained whenever possible, For th e finite-horizon (transient) case, the critical levels still exist, bu t they are now time-varying and in general very difficult to character ize, Thus, in an attempt to render the problem tractable, transient pr oduction optimization is sought within the (suboptimal) class of time- invariant hedging control policies, A renewal equation is developed fo r the cost functional over finite horizon under an arbitrary time-inva riant hedging control policy, The kernel of that renewal equation is a first-return time probability density function which satisfies an aux iliary system of Kolmogorov-type partial differential equations (PDE), The renewal equation and the auxiliary PDE system are used to generat e the terms in an infinite Laurent series expansion of the Laplace tra nsform of the finite-horizon cost functional viewed as a function of t he length of that horizon T. The terms in the infinite series expansio n are generated recursively, and their calculation is based on the sol ution of a system of piecewise smooth coupled linear differential equa tions, the associated Jordan canonical form of which is explicitly con structed, In the two-state machine case, this shows immediately that t he Bielecki-Kumar infinite-horizon cost is approached via a term that decays to zero as 1/T and that can be computed exactly, Furthermore, P ade approximants to the resulting infinite series expansion yield a ge neric (and quite accurate) approximate expression of the cost function al in terms of T and z, the arbitrary hedging level, In the multistate case, Pade approximants yield excellent numerical approximations to t he cost functional as a function of T for given choices of hedging lev els, This is subsequently used as part of an optimization scheme, wher eby hedging levels which are optimal for a given finite-horizon length are efficiently computed, The algorithms presented here can also be a pplied to the finite-horizon optimization for multipart failure-prone manufacturing systems, provided that only the partwise decoupled hedgi ng control policies of Caramanis and Sharifnia are considered.