Average Run Length on CUSUM Control Chart for Seasonal and Non-Seasonal Moving Average Processes with Exogenous Variables

Rapin Sunthornwat,Yupaporn Areepong

doi:10.3390/sym12010173

Abstract

The aim of this study was to derive explicit formulas of the average run length (ARL) of a cumulative sum (CUSUM) control chart for seasonal and non-seasonal moving average processes with exogenous variables, and then evaluate it against the numerical integral equation (NIE) method. Both methods had similarly excellent agreement, with an absolute percentage error of less than 0.50%. When compared to other methods, the explicit formula method is extremely useful for finding optimal parameters when other methods cannot. In this work, the procedure for obtaining optimal parameters—which are the reference value ( a ) and control limit ( h )—for designing a CUSUM chart with a minimum out-of-control ARL is presented. In addition, the explicit formulas for the CUSUM control chart were applied with the practical data of a stock price from the stock exchange of Thailand, and the resulting performance efficiency is compared with an exponentially weighted moving average (EWMA) control chart. This comparison showed that the CUSUM control chart efficiently detected a small shift size in the process, whereas the EWMA control chart was more efficient for moderate to large shift sizes.

Highlights

Statistical process control (SPC) has been widely used to monitor processes and services, so as to avoid any instabilities and inconsistencies
The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts were developed from the Shewhart control chart [3], which is suitable for processes with a large shift size in the parameters of interest when the observations follow a normal distribution
The average run length (ARL) that was calculated from the explicit formula was in excellent agreement with the ARL obtained from the numerical integral equation (NIE) method with the percentage error at less than 0.50%

Summary

Introduction

Statistical process control (SPC) has been widely used to monitor processes and services, so as to avoid any instabilities and inconsistencies. The CUSUM and EWMA control charts were developed from the Shewhart control chart [3], which is suitable for processes with a large shift size in the parameters of interest (the mean or variance) when the observations follow a normal distribution. The CUSUM and EWMA control chart can detect small shift sizes in the parameters of interest and they are suitable for observations following more complex patterns, such as auto-correlated observations, trending and seasonal observations, and changing point observations [4,5,6]. SPC has been adopted for monitoring production and service processes in several fields, such as medical sciences, industrial manufacturing, network analysis, mechanical trading on securities, and healthcare. A systematic review of the researches on the limitations and benefits of SPC for the quality improvement of healthcare systems can be found in [7,8], and a comparison of SPC to several

Objectives

Results

Conclusion