Stochastic Approximation Process Adjustment For Asymmetric Cost Functions

Optimizing adjustment rule for asymmetric cost functions

Abstract

This paper presents a feedback control rulefor the machine start-up adjustment problem when the cost function of themachining process is not symmetric around its target. In particular, thepresence of a bias term in the control rule permits the process qualitycharacteristic to converge to a steady-state target from the lower cost side,thus reducing the process quality losses incurred during the transient phase ofadjustment. A machining application is used to demonstrate the savingsgenerated by the biased linear feedback adjustment rule compared to anadjustment rule due to Grubbs (1954, 1983) and to an integral (or EWMA)controller. The performance of the different adjustment schemes is studied froma small-sample point of view, showing that the advantage of the proposed ruleis significant especially for expensive parts which are usually produced insmall lots. In this paper, two asymmetric cost functions {constant and quadratic{are considered. Optimal biased control rules for both cost functions arederived.