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 *S*_{0} = 0 and using a *reference
value k*, a one-sided CUSUM chart is obtained by plotting
the cumulative sums

versus

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.

Read Full Article (PDF, 346 KB)