The Design of the ARL-Unbiased S ² Chart When the In-Control Variance Is Estimated

Abstract

The S 2 chart has been known as a powerful tool to monitor the variability of the normal process. When the variance of the process is unknown, it needs to be estimated by Phase I samples. It is well known that there are serious effects of parameter estimation on the performance of the S 2 chart based on known parameter assumption. If the effects of parameter estimation are not considered, it can lead to an increase in the number of false alarms and a reduction in the ability of the chart to detect process changes except for very small shifts in the variance. Based on the criterion of average run length (ARL) unbiased, a S 2 control chart is developed when the in-control variance is estimated. The performance of the proposed control chart is also evaluated in terms of the ARL and standard deviation of the run length. Finally, an example is used to illustrate the proposed control chart.

Optimal design of a synthetic chart for monitoring process dispersion with unknown in-control variance

“…Notice that Tðr 0 Þ ¼ qTðr 1 Þ and Tðr 0 Þ is an increasing function of q, thus Tðr 0 Þ has been proposed as a monitoring statistic (Montgomery, 2005;Guo & Wang, 2014;Zhang et al, 2005).…”

“…When in-control variance is unknown and is estimated by the first 30 samples, for m ¼ 30; n ¼ 5 and ARL 0 ¼ 370:37, from Table 1 given by Guo and Wang (2014), we have UCL 30;5 ¼ 21:1220 and LCL 30;5 ¼ 0:1340 of the ARL-unbiased S 2 control chart with adjusted control limits. Sincer 2 0;30 ¼ 1:0795 based on the first 30 samples, the sample values of the monitoring statistic Tðr 0;30 Þ ¼ ðn À 1ÞS 2 =r 2 0;30 ¼ ðn À 1ÞS 2 =1:0795 for the last 30 samples are also given in Table 6.…”

“…One is that the in-control variance is known, another is that the in-control variance is unknown. For comparison purpose, we also give the performance of the ARL-unbiased S 2 chart (Zhang et al, 2005;Guo & Wang, 2014).…”

“…3 also Table 3 The ARLs of the proposed synthetic S 2 control charts with estimated parameter when ARL0 ¼ 370:37 and n ¼ 4. Table 4 Sets of ðLm;n; UCLm;n; LCLm;nÞ and ARLsyn;m;nðqÞ for the proposed synthetic S 2 charts when ARL0=370.37 and n ¼ 4. provides the ARL curves of the synthetic S 2 charts with estimated parameter (denoted as ''estimated parameter case'') and with known parameter (denoted as ''known parameter case'') and the ARL-unbiased S 2 charts with adjusted control limits (denoted as ''ARL-unbiased S 2 chart'') proposed by Guo and Wang (2014).…”

“…Next, in Section 4, we propose a procedure to design the new synthetic S 2 chart with the desired in-control ARL when the in-control variance is estimated. The optimal design parameters are obtained and the performance comparison is conducted between the new synthetic S 2 chart and the ARL-unbiased S 2 chart with adjusted control limits developed by Guo and Wang (2014). Finally, we use an example to illustrate the operation of the proposed synthetic S 2 chart.…”