Optimal input flows for a pde-ode model of supply chains

Abstract

In this paper we deal with a continuous model for supply chains, consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. We discuss the optimal control problem stated as the minimization of the queues and the quadratic difference between the effective outflow and a desired one. Here the input flow is the control and is assumed to have uniformly bounded variation. Introducing generalized tangent vectors to piecewise constant controls, representing shifts of discontinuities, we analyze the dependence of the solution on the control function. Then existence of an optimal control for the original problem is obtained. Finally we study the sensitivity of the cost functional J as function of controlled inflow, providing an estimate of the derivative of J with respect to switching times. © 2012 International Press.