An Explicit Analytical Solution of the Average Run Length of an Exponentially Weighted Moving Average Control Chart using an Autoregressive Model

Abstract

The aim of this paper is to derive explicit formulas of Average Run Length (ARL) using a Fredholm integral equation of the second kind for an Exponentially Weighted Moving Average (EWMA) control chart using an Autoregressive Model. A common characteristic used for comparing the performance of control charts is Average Run Length (ARL), the expected number of observations taken from an in-control process until the control chart falsely signals out-of-control is denoted by ARL0. An ARL0 will be regarded as acceptable if it is large enough to keep the level of false alarms at an acceptable level. A second common characteristic is the expected number of observations taken from an out-of-control process until the control chart signals that the process is out-of-control is denoted by ARL1. Explicit formulas for the ARL of an AR(p) process with exponential white noise were derived. To check the accuracy, the results obtained were compared with those from explicit formulas using numerical integral equations based on the Gauss-Legendre rule. There was an excellent agreement between the explicit formulas and the numerical solutions. The computational time for the explicit formulas was approximately one second; much less than that required for the numerical approximations. The explicit analytical formulas for evaluating ARL0 and ARL1 can produce a set of optimal parameters which depend on the smoothing parameter (l) and the width of control limit (h), for designing an EWMA chart with a minimum ARL1.