Optimal control of reconfigurable capacity in manufacturing systems.

Abstract

Over-capacity has been a major problem in the world economy over the past decade. Reconfigurable capacity, and optimal capacity management policies, can contribute to increased economic stability. This research introduces a new approach to optimal capacity management for a firm faced with uncertainties and imperfect information of the market demand. It presents an optimal policy for the capacity management problem in a firm facing stochastic market demand, based on Markov decision theory. It is also assumed that the firm faces delay times between the times capacity changes are ordered and the times they are delivered. Optimal policies are presented as boundaries representing the optimal capacity expansion and reduction levels. It is shown that there exists an optimum region which is guaranteed to contain the optimal feasible policy. This reduces the numerical burden of finding the optimal policy. To improve robustness of the optimal policy to unexpected events, for the first time, the concept of feedback control is applied to address the capacity management problem. It is shown that feedback provides sub-optimal solutions for the capacity management problem, which are more robust under unexpected disturbances in the forecasts of market demand and unexpected events. The results are compared to the existing approaches and the advantages of feedback are demonstrated. Motivated by delays in the feedback capacity management problem, an analytical solution, based on Lambert functions, is presented to solve delay differential equations (DDE's). Stability analysis, free response and forced response for several cases of DDE's are presented in the paper based on this new solution approach. As an example, the new approach is applied to find an analytic solution to a linear machine tool chatter problem, and its stability lobes are obtained. The main advantage of the presented analytical approach is its ability to provide a closed form solution to systems of homogeneous linear delay differential equations in a compact form similar to systems of ordinary differential equations. The solution is in the form of an infinite series of modes, and it provides a tool to study the behavior of the individual modes of the equation.