Regulated quality diffusion under quadratic costs

We study how to regulate quality processes so as to attain a desirable target. The effect of the regulation varies in a random fashion due to disturbances. The interaction between regulatory actions and outcomes is termed quality diffusion. In this paper, we address a specific version of such diffusion known as a mean-reverting process (also termed Ornstein Uhlenbeck process). The total production cost includes both a quadratic cost of quality deviations, and a linear cost of production control. Assuming that both regulation and quality processes are continuously reviewed, an optimal quality regulation policy is sought to minimize the total production cost over a finite horizon. The optimal regulation is found to be a surge-discharge control that can be characterized by a triplet including a discharge time, a stopping time, and a discharge function. The surge-discharge control is easy to be implemented for a variety of applications. Computational algorithms for the optimal control are developed.