﻿ Computing the Run Length Probability Distribution for CUSUM Charts - ASQ

# Computing the Run Length Probability Distribution for CUSUM Charts

It is widely recognized that the run length probability distribution of a cumulative sum (CUSUM) chart may be rather different from a geometric distribution. This is generally true for the left tail of the distribution, but, when the decision interval is large, it can apply to the whole distribution whether the process is in control or out of control. This paper provides a computer implementation of a fast and accurate algorithm to compute the run length probability distribution for CUSUM charts to monitor the process mean. The program may be used not only under the usual normality assumption but also for nonsymmetric and long-tailed continuous distributions.

Key Words: Cumulative Sum Control Charts, Statistical Process Control.

By ALBERTO LUCEÑO and JAIME PUIG-PEY, E. T. S. de Ingenieros de Caminos, University of Cantabria, 39005 Santander, Spain

INTRODUCTION

CUMULATIVE sum (CUSUM) charts are used in statistical process control (SPC) to detect small persistent shifts in a process mean. They have been receiving considerable attention in the statistical literature for many years. A recent review of this literature may be found in Hawkins and Olwell (1998). The cumulative sum approach has also led to some important generalizations such as the cumulative score (CUSCORE) charts, which are currently finding their place in the SPC tool kit (see, for example, Box and Luceño (1997, chaps. 10 and 11), Box and Ramírez (1992) , Luceño (1999), and Luceño and Box (2000)).

Suppose that samples of m independent observations of a quality characteristic Y are taken at equispaced times where D is the sam-pling interval. Let and be the mean and standard deviation of Y. CUSUM charts are designed to quickly detect small persistent deviations of with respect to a target value T. Suppose that is the sample average of the m observations taken at time , and let be the standardized value of so that the variance of is 1. The mean of is , and, when the process is on target, we have that µ = 0. Starting with S0 = 0 and using a reference value k, a one-sided CUSUM chart is obtained by plotting the cumulative sums versus . The chart proceeds until reaches a decision level h; at this time, it triggers an alarm because the mean µY might be larger than T, and the search for a special (and hopefully assignable) cause should start. Two-sided CUSUM charts to detect negative as well as positive deviations of µY with respect to T may be implemented running two symmetric one-sided charts simultaneously.

The performance of a one-sided CUSUM chart may be adequately described by the run length distribution ( RLD) for di erent values of µ - k and h. The mean of the RLD, which is usually called the average run length (ARL), provides a partial description of the chart performance that may be quite complete if the RLD is close to a geometric distribution (see Brook and Evans (1972) and Ewan and Kemp (1960)). However, the geometric distribution sometimes provides a very rough approximation to the RLD, so that the whole RLD should be computed in these cases (see, for example, Barnard (1959), Bissell (1969), Gan (1993), Woodall (1983), and Yashchin (1985)). The purpose of this paper is to provide an ANSI FORTRAN implementation of the numerical algorithm recently developed by Luce no and Puig-Pey (2000) to compute the RLD for different values of µ - k and h. The distribution of is often as-sumed to be normal, but the algorithm is valid for many other probability distributions including non-symmetric and long-tailed continuous distributions. 