﻿ Evaluating and Improving the Synthetic Control Chart - ASQ

# Evaluating and Improving the Synthetic Control Chart

Wu and Spedding (2000a) proposed the synthetic control chart, which is designed to detect shifts in the process mean. Our paper shows that the synthetic control chart can be represented as a runs rule chart with a head start feature. We present a Markov chain model of the synthetic control chart and use it to evaluate the zero-state and steady-state average run length (ARL) performance. We also alter the synthetic chart to achieve better ARL performance. We show, however, that synthetic control charts do not perform as well as cumulative sum and exponentially weighted moving average charts.

Key Words: Average Run Length, Cumulative Sum Control Charts, Exponentially Weighted Moving Average Control Charts, Runs Rules, Statistical Process Control.

By ROBERT B. DAVIS, Miami University, Hamilton, OH 45011
WILLIAM H. WOODALL, Virginia Tech, Blacksburg, VA 24061-0439

INTRODUCTION

THE synthetic control chart was introduced by Wu and Spedding (2000a) as a statistical process control technique that can be used to detect shifts in the process mean. The rst step in developing a synthetic control chart is to establish upper and lower control limits of a Shewhart control chart. However, unlike the standard Shewhart charts used in statistical process control, a signal is not generated based upon any single point falling outside of the limits. Instead, when any sample produces a value outside of the control limits, the practitioner determines how many samples have been taken since the last time that a point fell outside the limits, or since the first sample if there have been no previous points outside the limits. If this number of samples is su ciently small, then a signal is generated. Wu and Spedding (2000b) provide a computer program that will compute upper and lower control limits and other chart parameters when the practitioner is interested in a specified shift size.

More formally, control limits are established at the constant values

where µ is the in-control process mean, k > 0, is the standard deviation of individual measurements, and n is the sample size. When any sample mean is outside of these limits, the conforming run length (CRL) is determined. The CRL is the number of samples since the most recent previous sample mean was outside the control limits or since sampling began if there has been no point outside the control limits. If we have CRL > L , where L is a speci ed positive integer, the process is still considered in-control. Otherwise, the process is considered out-of-control and a signal is generated. It should be noted that as L increases, the synthetic control chart behaves more and more like an ordinary Shewhart chart.

Wu and Spedding (2000a) present a variety of (k, L) pairs that lead to an in-control average run length (ARL) of roughly 370.4 when the sample statistic being monitored follows a normal distribution. This value is chosen to give the synthetic control chart an in-control average run length (ARL) identical to that of the traditional 3-sigma Shewhart chart under the assumption of known in-control parameter values.

Calzada and Scariano (2001) investigated the robustness of the synthetic control chart to non-normality. Wu, Yeo, and Spedding (2001) and Wu and Yeo (2001) proposed synthetic control charts for detecting fraction nonconforming increases with attribute data; however, the performance of these charts will not be investigated here.