The Performance of S Control Charts for the Lognormal Distribution with Estimated Parameters

Abstract

Control charts, one of the powerful tools in statistical process control (SPC), are widely used to monitor and detect out-of-control processes in the manufacturing industry. Many researchers have pointed out the effects of using estimated parameters on the average run length (ARL) performance metric. Most of the previous literature has studied the expected value of the average run length (AARL) and the standard deviation of the average run length (SDARL) to evaluate the performance of control charts. In this article, we study the performance of three S control charts, the Shewhart S-chart, the median absolute deviation (MAD) control chart, and the lognormal S control chart, for a lognormal distribution in terms of the AARL and SDARL. Simulation results indicate the sample size to reach the specified in-control ARL value is very large. The lognormal S control chart has a smaller SDARL value than the other two S-charts.

1. Introduction

In statistical process control (SPC), the control chart is quite a popular tool for monitoring and improving a process through statistical analysis. In general, there is a two-phase procedure for the implementation of a control chart. For the Phase I procedure, a set of historical data was collected and used to estimate the process parameters and construct the control limits. As for the Phase II procedure, the control chart obtained from Phase I is used to quickly detect whether the parameters of the process have changed from the estimated value in the Phase I study.

The performance of a control chart is typically evaluated by studying the run length distribution of the chart. The run length (RL) of a control chart is defined as the number of samples taken before there is an out-of-control signal, and the average run length (ARL) is the expected value of the run length. When the process is in control, a control chart should have a large ARL, which indicates the chart should signal infrequently. On the contrary, when the process is out of control, a chart should have a small ARL, which denotes it should detect an out-of-control signal quickly. For a given shift parameter, the control chart that has a smaller out-of-control ARL (ARL${}_1$) is considered to outperform. Many studies have pointed out the effect of parameter estimation on the performance of control charts. For example, see Chen [1], Quesenberry [2], Jones et al. [3], Bischak and Trietsch [4], and Castagliola et al. [5]. A comprehensive overview of the current literature on the impact of parameter estimation on the performance of control charts is given by Jensen et al. [6] and Psarakis et al. [7].

Most researchers have determined the required size of the Phase I data so that the average of the in-control ARL values (AARL) is close to the desired value [1,2,3,4,5]. Because using different Phase I data will result in different parameter estimates to construct control charts, the performance of Phase II control chart will vary with the use of different Phase I data. Thus, there is sampling distribution variation in the in-control ARL values. One can average over the distribution of the parameter estimators to obtain the AARL values, which reflects marginal performance. Moreover, some other researchers have applied the standard deviation of the ARL (SDARL) as a criterion to evaluate the conditional effect of various amounts of Phase I data on the Phase II performance of control charts with estimated parameters. Any given control chart with a smaller SDARL indicates that it has a better ARL performance. Therefore, practitioners can not only determine the required size of the Phase I data needed to satisfy the in-control AARL value close to the desired value but also require a sufficiently small SDARL value.

Jones and Steiner [8] are the first to propose the use of the SDARL metric to assess the variation in a risk-adjusted cumulative sum (CUSUM) control chart performance due to parameter estimation. Zhang et al. [9,10] investigated the effect of parameter estimation on a one-sided lower exponential CUSUM chart. Lee et al. [11] focus on the effect of parameter estimation for the upper-sided Bernoulli cumulative sum chart. Saleh et al. [12] and Aly et al. [13] presented the adaptive EWMA control chart with estimated parameters. Saleh et al. [14] investigate the effect of different standard deviation estimators on the $\overline{X}$-chart’s performance.

Goedhart et al. [15] evaluate the guaranteed in-control performance of the Shewhart X control charts. Faraz et al. [16] and Guo and Wang [17] considered the variation in the conditional ARL for the $S^2$ control chart with estimated parameters.

Recently, Huang [18] used the SDARL metric to study the performance of the lognormal $\overline{X}$ control charts with estimated parameters. In this article, we adapt the methodology developed by Huang [19] to further study the AARL and SDARL performance of the S-charts for a lognormal distribution. Several studies for the S-charts of a lognormal distribution have been developed. Abu-Shawiesh [20] proposed a simple, robust control chart based on the median absolute deviation from the sample median for monitoring the process variability. Other studies, such as Figueiredo and Gomes [21] and Adekeye and Azubuike [22], have designed robust control charts for the non-normal processes. Huang et al. [19] proposed a lognormal S control chart for monitoring the lognormal standard deviation and concluded the proposed S-chart has a better performance. Shaheen et al. [23] developed a control chart for monitoring the lognormal process variation using repetitive sampling. Akhtar et al. [24] designed an exponentially weighted moving average (EWMA) chart for the lognormal process. In this study, we discuss three S-charts, which include the Shewhart S-chart, the MAD control chart, and the lognormal S control chart, on the AARL and SDARL performance. Although Huang et al. [19] indicated that the lognormal S control chart has better performance, they only focused on the ARL performance. Therefore, we are motivated to study the effects of using estimated parameters on the AARL and SDARL performance.

The rest of the paper is organized as follows. In Section 2, we introduce three S-charts for monitoring the lognormal standard deviation. The AARL and SDARL performance of three S-charts is evaluated in Section 3. Finally, a conclusion is given in Section 4.

2. The $\mathbf{S}$ Control Chart

Three S-charts for monitoring the lognormal standard deviation are introduced in this section. Assume that there are m samples with subgroup size n, denoted by $X_{i1},X_{i2},\dots ,X_{in}$, $i=1,2,\dots ,m$, from a lognormal distribution

$$f\left(x\right)=\frac{1}{x\sqrt{2\pi }\sigma }{e}^{-\frac{{(logx-\mu )}^2}{2{\sigma }^2}}.$$

Further, we denote the lognormal mean and the lognormal standard deviation by $\theta$ and $\xi$, respectively. That is, $\theta =e^{\mu +{\sigma }^2/2}$ and $\xi =\sqrt{(e^{{\sigma }^2}-1)e^{2\mu +{\sigma }^2}}$.

2.1. The Shewhart S-Chart

The Shewhart S-chart is designed based on the sample standard deviation S, and under the normal process, $S/c_4$ is an unbiased estimator of the population standard deviation, where $c_4={(2/(n-1))}^{1/2}[\mathsf{\Gamma }(n/2)/\mathsf{\Gamma }((n-1)/2)]$ (Montgomery [25]). Further, as for the lognormal process, the standard deviation of S is $\xi \sqrt{1-c_4^2}$. If $\xi$ is known, the control limits of the Shewhart S-chart are given by

$$\left\{\begin{array}{cc}\mathrm{UCL}\hfill & =c_4\xi +K\xi \sqrt{1-c_4^2}\hfill \\ \mathrm{LCL}\hfill & =c_4\xi -K\xi \sqrt{1-c_4^2},\hfill \end{array}\right.$$

where K is a multiplier that is chosen to achieve a specific in-control chart performance.

If $\xi$ is unknown, one can use $\widehat{\xi }$ to estimate it from the Phase I data. Let $S_X\left(i\right)$ be the ith sample standard deviation, $i=1,2,\dots ,m$. The average of the m standard deviations is ${\overline{S}}_X=\frac{1}{m}{\sum }_{i=1}^m{S}_X\left(i\right)$. Thus, the parameter $\xi$ is estimated by ${\widehat{\xi }}_{SW}={\overline{S}}_X/c_4$. The control limits of the Shewhart S-chart can be estimated by

$$\left\{\begin{array}{cc}{\mathrm{UCL}}_{SW}\hfill & ={\overline{S}}_X+L_{SW}\frac{{\overline{S}}_X}{c_4}\sqrt{1-c_4^2}\hfill \\ {\mathrm{LCL}}_{SW}\hfill & ={\overline{S}}_X-L_{SW}\frac{{\overline{S}}_X}{c_4}\sqrt{1-c_4^2},\hfill \end{array}\right.$$

where $L_{SW}$ is a multiplier that is chosen to satisfy a specific in-control ARL (ARL${}_0$).

For Phase II monitoring, independent samples of size n are repeatedly taken from the process. For each sample, one computes the sample standard deviation $S_X$. The chart signals when $S_X$ is below ${\mathrm{LCL}}_{SW}$ or above ${\mathrm{UCL}}_{SW}$.

2.2. The Median Absolute Deviation (MAD) Chart

The median absolute deviation (MAD) for a sample, $X_1,X_2,\dots ,X_n$, is defined as

$$\mathrm{MAD}=1.4826Median\left\{\right|X_i-\mathrm{MD}\left|\right\},$$

where $\mathrm{MD}$ is the sample median of $X_1,X_2,\dots ,X_n$ (Hampel [26]). Rousseeuw and Croux [27] show that the statistic $b_n\mathrm{MAD}$ is an unbiased estimator of the standard deviation, where $b_n$ can be found in Abu-Shawiesh [20].

Let $\overline{\mathrm{MAD}}={\sum }_{i=1}^m{\mathrm{MAD}}_i/m$ be the average median absolute deviation, where MAD${}_i$ is the median absolute deviation of the ith sample, $i=1,2,\dots ,m$. One can use ${\widehat{\xi }}_{\mathit{MAD}}=b_n\overline{\mathrm{MAD}}$ to estimate $\xi$, and the control limits of the MAD-chart can be given by

$$\left\{\begin{array}{cc}{\mathrm{UCL}}_{MAD}\hfill & =c_4{b}_n\overline{\mathrm{MAD}}+L_{MAD}{b}_n\overline{\mathrm{MAD}}\sqrt{1-c_4^2}\hfill \\ {\mathrm{LCL}}_{MAD}\hfill & =c_4{b}_n\overline{\mathrm{MAD}}-L_{MAD}{b}_n\overline{\mathrm{MAD}}\sqrt{1-c_4^2},\hfill \end{array}\right.$$

where $L_{MAD}$ is a multiplier that depends on n and the desired ARL${}_0$.

For Phase II monitoring, independent samples of size n are repeatedly taken from the process. For each sample, $b_n\mathrm{MAD}$ is computed, and an out-of-control signal is detected as soon as $b_n\mathrm{MAD}$ is below ${\mathrm{LCL}}_{MAD}$ or above ${\mathrm{UCL}}_{MAD}$.

2.3. The Lognormal S-Chart

Huang et al. [19] proposed a lognormal S-chart, which we will call the $LS$-chart thereafter, for monitoring the lognormal standard deviation for two cases when $\sigma$ is less than 1 and greater than 1. Assume that $Y_{ij}=log\left(X_{ij}\right)$, $i=1,2,\dots ,m$, $j=1,2,\dots ,n$, which follow a normal distribution with the mean, $\mu$, and the variance, ${\sigma }^2$. Note that the variance of a lognormal distribution is ${\xi }^2=e^{2\mu +{\sigma }^2}(e^{{\sigma }^2}-1)$.

2.3.1. Case I: Assuming $\sigma <1$

According to Tang and Yeh [28], if $\sigma <1$, one can ignore the higher-order terms in the Taylor’s expansion of $e^{{\sigma }^2}-1=({\sigma }^2+\frac{{\left({\sigma }^2\right)}^2}{2!}+\frac{{\left({\sigma }^2\right)}^3}{3!}+\cdots )$, and then ${\xi }^2$ can be approximated by

$${\xi }^2\cong e^{2\mu +{\sigma }^2}\times {\sigma }^2.$$

Therefore, the standard deviation, $\xi$, can be approximated by $e^{\mu +{\sigma }^2/2}\times \sigma$, and, further, $log\left(\xi \right)$ can be approximated by $\mu +\frac{{\sigma }^2}{2}+log\sigma .$ A naive point estimator of $log\left(\xi \right)$ is $(\overline{Y}+S^2/2+logS)$ where $\overline{Y}$ and $S^2$ are, respectively, the sample mean and the sample variance, and the approximated variance of $\overline{Y}+S^2/2+logS$ is given by

$$\frac{S^2}{n}+\frac{S^4}{2(n+1)}+\frac{(n-4)}{2{(n-3)}^2}+\frac{S^2}{(n-1)}.$$

Note that a control chart for monitoring $\xi$ is equivalent to monitoring $log\left(\xi \right)$. We denote the sample mean of $Y_{i1},Y_{i2},\dots ,Y_{in}$ by ${\overline{Y}}_i$ and the sample standard deviation of $Y_{i1},Y_{i2},\dots ,Y_{in}$ by $S_Y\left(i\right)$ for the ith sample, $i=1,2,\dots ,m$. Thus, the grand sample mean of ${\overline{Y}}_1,{\overline{Y}}_2,\dots ,{\overline{Y}}_m$ is $\overline{\overline{Y}}={\sum }_{i=1}^m{\overline{Y}}_i/m$ and the average of the m sample standard deviations is ${\overline{S}}_Y={\sum }_{i=1}^m{S}_Y\left(i\right)/m$. Therefore, for the case when $\sigma <1$, the control limits of the $LS$-chart are estimated by

$$\left\{\begin{array}{cc}{\mathrm{UCL}}_S\hfill & =\left(\overline{\overline{Y}}+\frac{{\overline{S}}_Y^2}{2}+log{\overline{S}}_Y\right)+L_S\sqrt{\frac{{\overline{S}}_Y^2}{n}+\frac{{\overline{S}}_Y^4}{2(n+1)}+\frac{(n-4)}{2{(n-3)}^2}+\frac{{\overline{S}}_Y^2}{(n-1)}}\hfill \\ {\mathrm{LCL}}_S\hfill & =\left(\overline{\overline{Y}}+\frac{{\overline{S}}_Y^2}{2}+log{\overline{S}}_Y\right)-L_S\sqrt{\frac{{\overline{S}}_Y^2}{n}+\frac{{\overline{S}}_Y^4}{2(n+1)}+\frac{(n-4)}{2{(n-3)}^2}+\frac{{\overline{S}}_Y^2}{(n-1)}},\hfill \end{array}\right.$$

(1)

where $L_S$ is a multiplier that is chosen to satisfy the desired ARL${}_0$.

As for the Phase II monitoring, independent samples of size n each are repeatedly taken from the process. For each sample, one computes the plotting statistic $\overline{Y}+S_Y^2/2+log\left(S_Y\right)$. An out-of-control signal is detected if the plotting statistic falls below ${\mathrm{LCL}}_S$ or above ${\mathrm{UCL}}_S$.

2.3.2. Case II: Assuming $\sigma >1$

According to Huang et al. [19], if $\sigma >1$, ${\xi }^2$ can be approximated by

$${\xi }^2\cong e^{2\mu +{\sigma }^2}\times e^{{\sigma }^2}=e^{2(\mu +{\sigma }^2)}.$$

Thus, the standard deviation, $\xi$, can be approximated by $e^{\mu +{\sigma }^2}$ and the $log\left(\xi \right)$ can be approximated by $\mu +{\sigma }^2$. A naive point estimator of $(\mu +{\sigma }^2)$ is $(\overline{Y}+S^2)$ and an unbiased estimator of $\mathrm{Var}(\overline{Y}+S^2)$ is $\frac{S^2}{n}+\frac{2S^4}{n+1}.$ Therefore, for the case when $\sigma >1$, the control limits of the $LS$-chart are given by

$$\left\{\begin{array}{cc}{\mathrm{UCL}}_S\hfill & =\left(\overline{\overline{Y}}+{\overline{S}}_Y^2\right)+L_S\sqrt{\frac{{\overline{S}}_Y^2}{n}+\frac{2{\overline{S}}_Y^4}{n+1}}\hfill \\ {\mathrm{LCL}}_S\hfill & =\left(\overline{\overline{Y}}+{\overline{S}}_Y^2\right)-L_S\sqrt{\frac{{\overline{S}}_Y^2}{n}+\frac{2{\overline{S}}_Y^4}{n+1}}.\hfill \end{array}\right.$$

(2)

where $L_S$ is a multiplier that is chosen to satisfy the desired ARL${}_0$.

As for the Phase II monitoring, independent samples of size n are each repeatedly taken from the process. For each sample, one computes the plotting statistic $\overline{Y}+S_Y^2$. An out-of-control signal is detected if the plotting statistic falls below ${\mathrm{LCL}}_S$ or above ${\mathrm{UCL}}_S$.

3. The AARL and SDARL Metrics

In this section, we compare the performance of the three S-charts in terms of the AARL and SDARL metrics.

3.1. Simulated Settings

The in-control process parameter is set equal to ${\theta }_0=1$ and ${\xi }_0=\sqrt{e^{{\sigma }_0^2}-1}$. We set various value of ${\sigma }_0$, ranging from 0.2 to 3.0, for both cases of standard deviation ($\sigma <1$ and $\sigma >1$), and consider the different values of $m=20$, 50, 100, 500, 800, 1200, 3000, and 5000, with a fixed subgroup size $n=5$ and 10.

Table 1. The multipliers L of three S-charts for different values of ${\sigma }_0$ when subgroup sample sizes $n=$ 5 and 10, and ARL${}_0=$ 200.

	$\mathit{n}=5$			$\mathit{n}=10$
	S-Chart	MAD-Chart	LS-Chart	S-Chart	MAD-Chart	LS-Chart
${{\sigma }}_{\mathbf{0}}$	$\left({\mathit{L}}_{\mathit{SW}}\right)$	$\left({\mathit{L}}_{\mathit{MAD}}\right)$	$\left({\mathit{L}}_{\mathit{S}}\right)$	$\left({\mathit{L}}_{\mathit{SW}}\right)$	$\left({\mathit{L}}_{\mathit{MAD}}\right)$	$\left({\mathit{L}}_{\mathit{S}}\right)$
0.2	3.539	3.150	3.864	3.462	3.470	3.139
0.4	5.018	3.591	3.552	5.162	3.514	3.005
0.6	7.252	4.336	3.315	7.768	3.611	2.947
0.8	10.220	5.354	3.253	11.316	3.816	2.949
1.0	13.880	6.644	3.290	15.916	4.612	2.991
1.2	18.524	8.170	4.764	21.906	5.754	3.745
1.4	24.138	9.987	4.945	28.651	7.079	3.830
1.6	30.134	12.168	5.052	36.664	8.619	3.883
1.8	37.129	14.553	5.118	46.068	10.366	3.946
2.0	44.226	17.192	5.215	55.968	12.371	3.971
2.5	60.068	24.961	5.315	79.736	18.205	4.017
3.5	57.649	42.227	5.424	87.391	33.975	4.100

presents the multipliers of the three S-charts based on a desired ARL${}_0=$ 200.

The algorithm for simulating the AARL and SDARL value of an S-chart is as follows:

• Step 1: Given ${\sigma }_0$, generate m samples of size n from a lognormal distribution. Compute the control limits of the S-chart with its own multipliers obtained from Table 1.
• Step 2: Calculate the in-control ARL for the S-chart when the future observations are repeatedly generated from a lognormal distribution.
• Step 3: Repeat Steps 1 and 2 10,000 times.
• Step 4: Calculate the value of the AARL and the SDARL for the S-chart.

3.2. Discussion of Results

The AARL and SDARL values of three S-charts for the different values of m and subgroup size $n=5$ and $n=10$ are summarized in

Table 2. The AARL and SDARL of the three S-charts for different sample m with subgroup sample size $n=5$ under various values of ${\sigma }_0$, and ARL${}_0=$ 200.

			S-Chart		MAD-Chart		$\mathit{LS}$-Chart				S-Chart		MAD-Chart		$\mathit{LS}$-Chart
n	${\mathsf{\sigma }}_0$	m	AARL	SDARL	AARL	SDARL	AARL	SDARL	${\mathsf{\sigma }}_0$	m	AARL	SDARL	AARL	SDARL	AARL	SDARL
5	0.2	20	295.94	322.63	518.11	920.13	210.53	62.76	1.4	20	272.35	613.37	276.89	412.62	281.77	404.82
		50	237.20	143.83	277.09	280.46	203.93	38.55		50	227.87	249.92	219.20	129.02	222.10	127.08
		100	218.96	88.33	235.37	141.13	201.75	27.44		100	213.50	144.52	208.36	83.76	207.42	82.63
		500	206.44	37.23	209.70	51.38	200.98	14.52		500	203.44	37.76	202.31	34.34	203.51	33.52
		800	205.61	29.64	207.69	40.02	200.96	12.72		800	202.35	29.40	201.52	27.78	202.03	26.88
		1200	205.21	24.72	206.42	32.92	200.79	11.57		1200	208.80	24.98	201.10	23.12	201.50	23.04
		3000	204.14	16.86	204.68	21.43	200.71	10.00		3000	201.50	17.31	200.07	16.07	200.77	16.01
		5000	203.82	14.36	204.34	17.71	200.66	9.66		5000	201.26	14.26	200.24	13.63	200.76	13.52
		∞	200.07	0	200.05	0	200.13	0		∞	200.07	0	200.05	0	200.06	0
	0.4	20	253.11	223.66	367.17	811.88	211.13	38.81	1.8	20	291.16	1070.03	294.78	914.87	307.15	487.58
		50	218.32	105.50	248.04	192.87	201.38	27.82		50	243.25	826.47	230.92	176.97	240.52	164.67
		100	210.00	67.84	221.80	111.08	201.18	19.75		100	222.87	564.53	215.18	96.45	220.10	94.94
		500	204.80	30.15	204.71	44.19	200.77	12.65		500	204.20	59.11	202.55	38.67	210.61	36.13
		800	204.52	23.67	203.52	34.23	200.57	11.16		800	201.93	42.44	201.79	31.08	202.60	30.20
		1200	203.31	19.54	203.44	28.46	200.31	10.45		1200	201.87	34.56	200.96	25.60	201.29	24.11
		3000	202.01	14.66	202.62	19.29	200.59	9.58		3000	200.50	22.55	200.63	17.59	200.72	16.90
		5000	202.26	12.69	202.05	15.79	200.52	9.33		5000	200.58	18.44	200.34	14.72	200.66	15.17
		∞	200.04	0	200.06	0	200.07	0		∞	200.07	0	200.05	0	200.06	0
	0.6	20	247.19	210.03	311.50	446.18	212.90	41.04	2.0	20	311.39	1923.42	288.71	904.12	291.29	402.79
		50	217.68	100.34	240.86	168.64	202.30	24.97		50	245.67	667.85	223.95	176.99	230.27	158.51
		100	209.47	65.09	216.95	96.20	201.81	17.98		100	224.20	330.99	212.06	106.12	213.23	92.72
		500	204.04	28.95	204.42	39.60	200.35	11.36		500	203.86	69.24	203.62	41.07	203.13	37.13
		800	203.27	23.11	204.14	31.11	200.40	10.36		800	203.55	57.87	202.26	31.65	202.00	29.83
		1200	202.73	19.77	203.18	25.96	200.60	10.10		1200	202.87	54.46	201.20	26.02	201.67	24.60
		3000	202.36	14.24	202.31	17.90	200.83	9.61		3000	201.70	27.67	200.75	17.86	201.10	16.72
		5000	202.38	12.56	202.10	15.05	200.96	9.35		5000	200.73	22.81	200.81	14.76	200.77	14.24
		∞	200.04	0	200.05	0	200.05	0		∞	200.07	0	200.05	0	200.06	0
	0.8	20	251.46	301.02	291.95	363.77	213.37	81.04	2.5	20	310.23	2183.47	342.07	1882.81	391.09	729.62
		50	218.85	99.90	233.36	146.06	202.68	52.61		50	264.77	1712.68	234.38	251.63	251.63	209.04
		100	211.13	66.83	217.22	89.10	201.70	38.27		100	251.06	1375.28	222.00	244.31	225.30	115.93
		500	204.04	29.13	204.85	37.13	200.38	19.10		500	216.35	481.38	203.92	53.53	203.43	43.84
		800	203.47	23.46	203.63	29.70	200.91	16.45		800	210.62	416.46	201.79	38.66	202.16	34.67
		1200	203.57	19.99	202.70	24.72	200.79	14.30		1200	203.14	103.20	201.68	33.34	201.51	28.70
		3000	202.64	14.40	202.44	17.02	200.70	11.46		3000	201.93	74.07	200.43	21.21	200.97	19.23
		5000	202.68	12.45	202.39	14.46	200.10	10.45		5000	201.82	54.87	200.56	17.37	200.15	16.08
		∞	200.07	0	200.05	0	200.06	0		∞	200.08	0	200.06	0	200.03	0
	1.0	20	244.11	272.83	281.26	324.93	229.61	153.38	3.0	20	305.25	2696.07	349.44	1858.75	393.80	723.92
		50	216.33	126.64	227.28	133.09	214.77	92.55		50	265.53	1812.02	268.38	914.72	260.23	222.45
		100	206.99	69.77	214.96	85.32	207.83	63.81		100	247.79	1560.98	232.82	277.69	229.70	119.44
		500	204.40	29.81	204.63	35.90	203.15	29.60		500	238.97	1383.79	208.81	75.40	206.53	45.64
		800	203.75	24.30	202.90	28.59	201.22	23.73		800	231.69	1223.72	207.03	59.11	203.96	34.57
		1200	203.59	20.30	202.91	23.89	200.51	20.14		1200	217.10	468.96	207.19	50.21	203.00	29.22
		3000	202.32	14.42	201.94	16.52	200.42	14.47		3000	209.31	352.91	204.49	36.27	201.67	19.58
		5000	202.81	12.55	200.98	14.21	200.07	12.64		5000	206.88	302.39	204.07	22.13	200.37	15.97
		∞	200.07	0	200.05	0	200.06	0		∞	200.09	0	200.06	0	200.03	0

Note: The bolded values indicate the case in which an SDARL is within 10% of the ARL₀.

and

Table 3. The AARL and SDARL of the three S-charts for different sample m with subgroup sample size $n=10$ under various values of ${\sigma }_0$, and ARL${}_0=$ 200.

			S-Chart		MAD-Chart		$\mathit{LS}$-Chart				S-Chart		MAD-Chart		$\mathit{LS}$-Chart
n	${\mathsf{\sigma }}_0$	m	AARL	SDARL	AARL	SDARL	AARL	SDARL	${\mathsf{\sigma }}_0$	m	AARL	SDARL	AARL	SDARL	AARL	SDARL
10	0.2	20	263.59	244.67	448.64	1109.33	212.18	71.27	1.4	20	253.88	1051.31	256.83	442.99	327.23	433.59
		50	222.89	112.55	243.48	250.47	206.08	48.51		50	221.48	432.53	220.59	133.84	240.45	125.33
		100	211.21	73.92	221.34	130.34	203.67	35.24		100	208.80	78.57	209.80	77.65	222.24	75.91
		500	201.74	31.19	203.94	45.78	201.61	17.75		500	202.01	31.62	202.45	31.20	201.21	28.61
		800	201.64	25.11	201.88	34.13	201.08	14.91		800	201.50	25.20	201.69	25.49	200.52	20.28
		1200	200.63	21.33	200.63	27.58	201.29	13.38		1200	201.21	21.48	201.12	21.40	200.88	15.33
		3000	200.28	15.18	200.31	19.69	200.97	10.81		3000	200.99	15.08	200.66	15.12	200.71	13.45
		5000	200.76	12.98	200.75	15.31	200.13	10.12		5000	200.80	13.15	200.48	12.90	200.67	14.24
		∞	200.07	0	200.05	0	200.13	0		∞	200.07	0	200.05	0	200.06	0
	0.4	20	235.11	160.76	351.11	705.11	201.58	36.89	1.8	20	267.52	1212.25	265.17	888.91	330.00	409.65
		50	213.17	87.11	243.14	181.37	201.66	27.17		50	227.10	302.14	222.08	190.61	246.09	160.93
		100	206.41	57.89	216.43	105.31	201.19	21.39		100	212.24	189.14	211.78	98.28	223.90	96.65
		500	200.96	25.96	201.84	42.51	200.19	12.93		500	202.08	44.20	202.58	42.37	202.76	38.96
		800	200.86	21.08	200.73	31.32	200.90	11.61		800	201.73	35.95	202.01	35.89	201.73	31.15
		1200	200.40	18.05	200.24	25.34	200.24	10.86		1200	201.83	31.28	201.49	31.91	200.67	24.32
		3000	200.24	13.25	200.01	17.10	200.10	9.70		3000	200.96	19.46	201.12	19.34	200.02	17.15
		5000	200.16	11.86	200.12	13.95	200.16	9.28		5000	200.59	15.79	201.28	15.18	200.91	14.99
		∞	200.04	0	200.06	0	200.07	0		∞	200.07	0	200.05	0	200.06	0
	0.6	20	234.79	156.85	293.45	373.88	202.42	37.67	2.0	20	292.15	1336.94	267.73	884.76	332.91	590.27
		50	212.82	81.17	228.36	150.33	202.58	23.37		50	233.61	400.40	221.41	189.87	241.40	161.67
		100	206.93	54.53	212.05	84.13	201.19	16.63		100	217.83	198.32	211.77	117.67	219.13	94.46
		500	202.03	24.72	201.79	37.50	201.74	10.54		500	205.33	61.41	203.20	43.42	204.12	38.79
		800	201.60	20.60	201.15	29.84	200.22	10.07		800	204.65	48.78	202.22	36.59	202.61	30.77
		1200	200.85	17.16	200.41	23.36	200.38	9.58		1200	203.80	38.03	201.80	32.09	201.97	25.20
		3000	200.84	12.93	200.97	15.66	200.58	9.15		3000	203.55	24.14	201.21	19.65	200.94	17.39
		5000	200.78	11.54	200.80	13.14	200.81	9.08		5000	202.75	19.40	200.05	17.30	200.72	14.56
		∞	200.04	0	200.05	0	200.05	0		∞	200.07	0	200.05	0	200.06	0
	0.8	20	231.09	158.96	271.43	296.48	203.94	60.01	2.5	20	295.83	2075.37	300.07	533.41	321.09	457.00
		50	212.53	84.92	230.87	139.55	202.34	41.54		50	275.65	1719.93	227.05	240.11	237.66	161.18
		100	205.78	54.55	215.67	80.21	201.77	30.72		100	241.99	1066.25	213.74	218.36	218.97	93.05
		500	201.64	24.90	202.29	32.72	200.06	16.32		500	220.18	647.07	203.90	56.20	203.50	37.79
		800	201.15	20.56	201.69	25.22	200.37	13.94		800	205.31	232.33	201.52	38.73	202.86	29.81
		1200	201.30	17.24	200.12	21.64	200.61	12.59		1200	204.72	147.06	201.54	34.11	202.21	25.17
		3000	200.91	12.99	200.23	15.40	200.98	10.52		3000	204.93	121.22	200.27	19.85	200.94	17.14
		5000	200.86	11.50	200.01	12.87	200.93	9.87		5000	200.34	34.12	200.17	16.99	200.75	14.39
		∞	200.07	0	200.05	0	200.06	0		∞	200.08	0	200.06	0	200.03	0
	1.0	20	231.50	191.15	269.15	272.03	203.62	92.45	3.0	20	305.25	2696.07	349.44	1858.75	393.80	723.92
		50	211.45	87.40	220.88	117.74	202.36	64.59		50	265.53	1812.02	268.38	914.72	260.23	222.45
		100	206.21	59.14	208.37	74.90	201.66	47.89		100	247.79	1560.98	232.82	277.69	229.70	119.44
		500	202.45	26.07	202.82	31.76	201.44	23.60		500	238.97	1383.79	208.81	75.40	206.53	45.64
		800	201.31	20.78	201.68	25.25	201.01	19.67		800	231.69	1223.72	207.03	59.11	203.96	34.57
		1200	201.52	17.82	201.15	21.29	200.18	16.64		1200	217.10	468.96	207.19	50.21	203.00	29.22
		3000	200.08	13.22	200.68	14.95	200.01	12.49		3000	209.31	352.91	204.49	36.27	201.67	19.58
		5000	200.00	11.64	200.49	12.90	200.16	11.39		5000	206.88	302.39	204.07	22.13	200.37	15.97
		∞	200.07	0	200.05	0	200.06	0		∞	200.09	0	200.06	0	200.03	0

Note: The bolded values indicate the case in which an SDARL is within 10% of the ARL₀.

, respectively, under various values of ${\sigma }_0$. Under each table, the numbers at the bottom row (entitled $m=\infty$) are for the case when the process parameters are known. In order to achieve in-control ARL performance when the process parameters are estimated, the required amount of Phase I data should yield an in-control AARL value close to ARL${}_0$ and an SDARL value that is sufficiently small. From Table 2 and Table 3, the sample size for the three S-charts to reach the desired in-control ARL value of $AR{L}_0=$ 200 requires vary large values of m. In addition, the $LS$-chart has the smallest SDARL value among these three S-charts. It indicates that the $LS$-chart outperforms the other two S-charts.

Zhang et al. [10] recommended that an SDARL within 10% of the ARL${}_0$ is reasonable In our case, the 10% of the desired in-control ARL value of $AR{L}_0=$ 200 is 20; therefore, the bolded values in Table 2 and Table 3 indicate an SDARL is within 10% of the ARL${}_0$. Consequently, from Table 2 and Table 3, one needs about 100 or 500 samples for the $LS$-chart, 1200 or 3000 samples for the Shewhart S-chart, and 3000 or 5000 samples for the MAD-chart to obtain SDARL values of no more than 20 in the case when ${\sigma }_0<1$. When $\sigma$ is greater than 1, one need at least 3000 samples for these three S-charts. In addition, when $\sigma =$ 2.5 and 3.0, the SDARL of the Shewhart S-chart is greater than 20 and becomes very large even when m is large. It indicates that, in the case when ${\sigma }_0>1$, the Shewhart S-chart has the worst SDARL performance among the three S-charts. Therefore, for the in-control performance, the $LS$-chart has better performance than the Shewhart S-chart and the MAD control chart.

As for the out-of-control performance, we denoted the out-of-control process parameters by ${\xi }_1=\lambda {\xi }_0$, $\lambda =1.5$ and $2.0$. The AARL${}_1$ and SDARL${}_1$ for the three S-charts for various value of ${\sigma }_0$ and the different values for m samples with subgroup size $n=5$ and 10 are summarized in

Table 4. The AARL${}_1$ and SDARL${}_1$ of the three S-charts for different sample m with subgroup sample size $n=5$ under various values of ${\sigma }_0$ and $\lambda$.

				S-Chart		MAD-Chart		$\mathit{LS}$-Chart					S-Chart		MAD-Chart		$\mathit{LS}$-Chart
n	${\mathsf{\sigma }}_0$	$\mathsf{\lambda }$	m	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$	${\mathsf{\sigma }}_0$	$\mathsf{\lambda }$	m	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$
5	0.2	1.5	20	10.27	4.69	21.18	15.90	183.87	149.72	1.4	1.5	20	112.61	191.34	145.11	309.93	61.70	72.78
			100	9.47	1.79	17.75	5.19	133.31	114.43			100	98.56	37.95	100.77	78.68	47.13	16.06
			500	9.34	0.82	17.17	2.23	124.21	98.88			500	95.78	16.99	91.02	40.37	45.05	6.93
			1200	9.32	0.56	17.05	1.52	122.93	93.90			1200	95.59	12.85	91.27	33.40	44.79	5.07
			3000	9.32	0.41	17.04	1.09	121.68	95.40			3000	95.34	10.94	91.21	29.95	44.66	3.88
			∞	9.27	0	16.95	0	110.50	0			∞	86.40	0	81.25	0	40.59	0
		2.0	20	3.51	0.88	7.44	3.41	46.58	29.69		2.0	20	77.48	124.91	138.31	267.13	28.02	19.98
			100	3.40	0.37	6.85	1.29	40.14	10.62			100	70.83	22.16	101.54	79.63	23.79	6.35
			500	3.38	0.17	6.74	0.57	39.26	5.16			500	69.83	10.81	87.92	42.72	23.08	2.84
			1200	3.38	0.12	6.73	0.39	39.08	3.78			1200	69.35	7.96	86.52	36.11	23.00	2.00
			3000	3.38	0.08	6.72	0.28	39.02	3.03			3000	69.44	6.83	86.18	32.62	22.96	1.53
			∞	3.36	0	6.65	0	37.97	0			∞	67.49	0	76.27	0	19.96	0
	0.4	1.5	20	16.98	6.97	34.12	25.82	83.80	42.28	1.8	1.5	20	167.79	349.01	190.59	497.94	101.13	130.69
			100	15.97	2.87	28.57	8.43	78.13	20.37			100	143.44	155.65	168.32	132.57	72.33	28.45
			500	15.81	1.37	27.78	3.75	76.94	10.85			500	136.27	35.61	155.88	62.54	68.19	12.36
			1200	15.78	0.99	27.71	2.65	76.81	8.65			1200	134.45	24.95	154.26	51.88	67.39	8.97
			3000	15.76	0.78	27.61	1.99	76.73	7.61			3000	134.77	20.15	154.20	45.77	67.29	7.05
			∞	15.27	0	25.95	0	72.50	0			∞	112.40	0	121.25	0	60.59	0
		2.0	20	6.81	1.88	15.65	8.25	17.98	7.97		2.0	20	134.59	277.80	168.09	608.42	52.33	59.84
			100	6.63	0.82	14.32	3.10	16.70	3.25			100	115.70	61.24	132.83	156.18	41.10	13.74
			500	6.60	0.38	14.04	1.41	16.45	1.49			500	112.45	26.59	129.91	79.64	39.30	5.94
			1200	6.60	0.27	14.01	0.98	16.42	1.07			1200	111.74	19.57	127.24	65.49	39.04	4.32
			3000	6.59	0.21	14.00	0.73	16.41	0.84			3000	111.64	15.42	125.66	61.01	38.93	3.22
			∞	6.36	0	13.65	0	15.97	0			∞	101.49	0	116.27	0	33.96	0
	0.6	1.5	20	27.37	12.24	57.97	44.04	43.68	22.11	2.0	1.5	20	190.75	459.65	192.87	584.87	129.15	236.88
			100	25.87	4.93	49.69	14.78	40.12	9.38			100	161.28	141.09	178.37	151.88	86.93	36.67
			500	25.59	2.40	48.12	6.91	39.52	4.68			500	152.83	47.68	168.95	70.69	81.53	15.82
			1200	25.54	1.84	47.95	5.10	39.40	3.51			1200	152.28	39.45	167.90	57.08	80.92	11.59
			3000	25.54	1.52	47.83	4.08	39.39	2.95			3000	152.16	26.84	166.82	50.55	80.60	9.23
			∞	23.27	0	42.95	0	37.50	0			∞	132.40	0	141.25	0	70.59	0
		2.0	20	12.72	3.99	34.81	19.82	5.44	1.60		2.0	20	162.56	387.98	179.72	743.45	69.16	74.07
			100	12.37	1.75	31.37	7.43	5.24	0.68			100	138.84	103.34	158.23	185.30	52.40	18.96
			500	12.29	0.84	30.90	3.54	5.20	0.31			500	133.33	38.96	145.12	93.16	49.69	8.27
			1200	12.31	0.65	30.85	2.64	5.20	0.22			1200	132.25	27.90	140.67	75.70	49.25	5.92
			3000	12.30	0.51	30.77	2.13	5.19	0.16			3000	132.36	21.69	140.65	68.94	49.03	4.50
			∞	11.36	0	27.65	0	4.97	0			∞	117.49	0	120.34	0	44.92	0
	0.8	1.5	20	41.53	20.82	94.21	79.71	10.96	4.49	2.5	1.5	20	193.63	604.43	194.99	795.34	190.34	407.39
			100	39.40	8.46	80.91	25.30	10.32	1.79			100	188.98	429.63	192.99	235.72	116.37	54.47
			500	38.81	4.21	78.56	12.40	10.19	0.83			500	182.53	236.37	188.01	93.97	107.32	23.28
			1200	38.77	3.32	78.43	9.48	10.17	0.59			1200	180.25	179.48	185.41	70.26	106.04	16.74
			3000	38.73	2.76	78.29	7.96	10.16	0.43			3000	175.46	73.19	180.01	61.42	105.64	13.35
			∞	36.27	0	74.95	0	9.50	0			∞	142.40	0	151.25	0	90.59	0
		2.0	20	22.21	8.52	69.14	43.43	5.76	1.57		2.0	20	192.30	528.13	193.28	963.99	115.51	218.99
			100	21.37	3.59	62.25	16.32	5.57	0.66			100	187.89	321.01	190.92	408.73	78.80	32.90
			500	21.18	1.77	60.87	8.15	5.53	0.31			500	180.28	129.39	185.17	132.47	73.76	13.92
			1200	21.20	1.36	60.71	6.39	5.53	0.22			1200	177.42	122.67	180.30	104.38	72.98	10.14
			3000	21.17	1.14	60.70	5.44	5.52	0.17			3000	176.47	68.77	179.79	87.89	72.67	7.98
			∞	20.36	0	56.65	0	4.97	0			∞	107.49	0	110.23	0	65.96	0
	1.0	1.5	20	60.27	44.41	140.40	117.10	33.71	24.63	3.0	1.5	20	192.78	579.82	195.69	927.06	167.18	494.14
			100	55.43	13.58	119.85	39.21	28.37	7.76			100	188.46	504.68	190.77	364.14	140.28	67.92
			500	54.53	6.84	116.18	20.27	27.53	3.51			500	185.85	345.98	192.02	142.11	129.99	30.10
			1200	54.54	5.33	115.69	16.06	27.38	2.50			1200	185.69	289.20	191.27	88.78	128.16	22.03
			3000	54.54	4.57	115.50	13.91	27.33	1.89			3000	185.61	114.54	191.21	69.33	127.88	18.02
			∞	51.27	0	107.95	0	26.50	0			∞	142.40	0	151.25	0	112.59	0
		2.0	20	35.12	17.29	118.26	88.62	14.01	7.02		2.0	20	177.48	492.15	188.31	974.28	163.99	281.91
			100	33.72	6.73	105.99	30.95	12.74	2.53			100	170.83	384.17	181.54	519.58	104.75	47.78
			500	33.41	3.38	103.54	16.47	12.51	1.14			500	169.83	280.06	177.92	207.50	97.14	20.42
			1200	33.30	2.62	103.39	13.24	12.48	0.80			1200	169.35	235.63	176.52	162.55	96.08	14.66
			3000	33.31	2.25	103.17	11.65	12.46	0.59			3000	169.44	161.68	176.18	105.39	95.84	12.03
			∞	30.36	0	96.65	0	11.97	0			∞	137.49	0	146.27	0	79.96	0

and

Table 5. The AARL${}_1$ and SDARL${}_1$ of the three S-charts for different sample m with subgroup sample size $n=10$ under various values of ${\sigma }_0$ and $\lambda$.

				S-Chart		MAD-Chart		$\mathit{LS}$-Chart					S-Chart		MAD-Chart		$\mathit{LS}$-Chart
n	${\mathsf{\sigma }}_0$	$\mathsf{\lambda }$	m	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$	${\mathsf{\sigma }}_0$	$\mathsf{\lambda }$	m	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$	AARL${}_1$	SDARL${}_1$
10	0.2	1.5	20	5.27	1.65	40.68	26.41	28.01	16.82	1.4	1.5	20	91.91	98.85	185.52	402.66	11.54	5.53
			100	5.05	0.69	35.19	10.29	24.55	6.18			100	84.04	25.92	179.88	104.59	10.60	2.04
			500	4.99	0.32	34.09	4.77	23.84	2.89			500	82.19	12.30	172.05	60.48	10.40	0.91
			1200	4.98	0.22	33.94	3.38	23.73	2.01			1200	82.19	9.86	170.47	53.13	10.39	0.64
			3000	4.98	0.16	33.90	2.60	23.72	1.56			3000	82.14	8.57	169.17	48.78	10.37	0.42
			∞	4.96	0	32.85	0	22.62	0			∞	75.91	0	158.53	0	10.35	0
		2.0	20	1.91	0.28	8.30	3.81	3.83	1.07		2.0	20	59.80	45.05	169.73	305.59	6.63	2.25
			100	1.88	0.12	7.68	1.44	3.68	0.44			100	57.01	14.73	88.91	101.54	6.33	0.93
			500	1.87	0.06	7.56	0.64	3.65	0.20			500	56.41	7.45	75.43	54.70	6.26	0.43
			1200	1.87	0.04	7.55	0.44	3.65	0.14			1200	56.26	5.67	74.32	43.98	6.25	0.29
			3000	1.87	0.03	7.54	0.31	3.65	0.10			3000	56.18	4.87	72.38	39.59	6.25	0.22
			∞	1.86	0	7.52	0	3.64	0			∞	54.10	0	64.14	0	6.23	0
	0.4	1.5	20	10.42	3.50	49.36	25.70	20.55	10.48	1.8	1.5	20	145.28	267.45	190.41	639.46	13.73	7.12
			100	10.08	1.49	45.94	11.87	18.84	4.13			100	129.01	119.97	187.65	184.11	12.42	2.59
			500	9.99	0.69	44.90	5.80	18.40	1.86			500	123.56	27.49	168.03	96.88	12.17	1.19
			1200	9.97	0.50	44.75	4.40	18.33	1.34			1200	122.93	20.32	165.53	83.01	12.12	0.82
			3000	9.97	0.39	44.76	3.60	18.33	1.05			3000	122.82	16.57	165.27	77.35	12.12	0.61
			∞	9.96	0	43.55	0	18.31	0			∞	112.88	0	141.66	0	12.08	0
		2.0	20	3.97	0.82	18.94	8.90	5.06	1.46		2.0	20	111.13	157.40	166.57	545.98	8.86	3.63
			100	3.92	0.35	17.41	3.55	4.86	0.60			100	100.60	57.85	146.44	122.91	8.25	1.43
			500	3.92	0.17	17.17	1.67	4.82	0.28			500	97.81	20.00	140.14	87.88	8.12	0.66
			1200	3.91	0.12	17.12	1.20	4.81	0.19			1200	97.36	14.45	137.23	62.50	8.10	0.45
			3000	3.91	0.09	17.09	0.91	4.81	0.14			3000	97.43	11.76	134.38	56.16	8.10	0.33
			∞	3.90	0	17.05	0	4.80	0			∞	87.14	0	114.42	0	8.04	0
	0.6	1.5	20	18.91	6.88	59.58	24.37	21.48	9.94	2.0	1.5	20	173.62	349.41	191.44	690.56	14.88	8.16
			100	18.17	2.85	57.92	12.78	19.83	4.10			100	149.89	113.75	188.37	195.57	13.21	2.90
			500	18.06	1.42	57.41	6.95	19.50	1.94			500	143.76	44.86	158.93	105.83	12.94	1.30
			1200	18.03	1.09	57.28	5.46	19.44	1.39			1200	142.70	28.22	157.40	93.26	12.90	0.90
			3000	18.02	0.90	57.27	4.87	19.43	1.09			3000	142.41	22.66	156.37	86.02	12.87	0.67
			∞	18.00	0	56.23	0	18.45	0			∞	122.07	0	131.28	0	12.80	0
		2.0	20	8.13	2.03	35.89	12.77	2.37	0.46		2.0	20	137.74	237.34	172.94	772.23	9.98	4.45
			100	8.00	0.89	35.25	6.43	2.33	0.19			100	125.47	130.36	141.70	397.61	9.18	1.68
			500	7.96	0.44	34.99	3.44	2.31	0.09			500	119.79	32.34	133.95	214.79	9.03	0.77
			1200	7.95	0.33	34.97	2.70	2.31	0.06			1200	119.35	22.56	133.34	195.06	9.01	0.54
			3000	7.95	0.26	34.97	2.36	2.31	0.04			3000	119.23	20.45	128.45	173.10	9.01	0.39
			∞	7.94	0	33.86	0	2.30	0			∞	107.94	0	119.34	0	9.00	0
	0.8	1.5	20	31.22	13.69	82.15	38.28	5.66	1.89	2.5	1.5	20	194.67	501.40	195.99	834.26	132.05	169.72
			100	29.57	5.26	78.69	18.79	5.37	0.76			100	189.56	329.95	193.99	244.11	97.44	38.20
			500	29.44	2.71	77.93	10.58	5.33	0.35			500	181.41	169.04	187.01	121.40	91.68	16.97
			1200	29.37	2.12	77.51	8.54	5.32	0.24			1200	176.90	124.67	180.41	105.47	91.37	12.84
			3000	29.35	1.78	77.50	7.49	5.32	0.17			3000	173.48	65.55	175.01	92.74	91.04	10.56
			∞	29.15	0	74.55	0	5.30	0			∞	152.40	0	161.25	0	80.59	0
		2.0	20	15.11	4.63	63.87	20.38	3.24	0.66		2.0	20	187.00	483.29	193.12	905.24	78.28	74.19
			100	14.80	2.03	62.18	11.32	3.17	0.29			100	178.21	318.91	180.39	496.72	61.71	21.46
			500	14.74	1.00	61.96	6.76	3.15	0.13			500	164.71	102.96	175.07	252.17	58.84	9.65
			1200	14.72	0.78	61.97	5.72	3.15	0.09			1200	162.80	64.95	170.52	224.04	58.30	6.93
			3000	14.71	0.65	61.02	5.21	3.15	0.07			3000	161.77	66.86	169.55	206.11	58.14	5.58
			∞	14.70	0	58.34	0	3.14	0			∞	127.49	0	132.23	0	53.76	0
	1.0	1.5	20	47.40	25.29	160.86	129.84	9.56	3.80	3.0	1.5	20	192.17	601.33	195.22	943.75	181.20	276.41
			100	44.73	9.34	138.64	43.16	8.93	1.48			100	188.86	427.29	190.42	273.83	125.38	53.56
			500	44.31	4.79	134.33	23.09	8.81	0.69			500	185.62	286.52	192.04	126.04	117.78	24.17
			1200	44.22	3.82	133.27	19.10	8.80	0.48			1200	185.05	237.33	191.59	105.07	116.87	18.13
			3000	44.22	3.27	133.50	17.26	8.80	0.35			3000	185.95	119.69	191.61	90.32	116.37	15.12
			∞	41.27	0	117.95	0	8.78	0			∞	152.40	0	159.25	0	102.59	0
		2.0	20	25.86	9.92	155.88	114.25	4.88	1.30		2.0	20	177.01	603.54	188.23	974.97	117.32	132.69
			100	25.04	4.12	125.71	41.70	4.71	0.53			100	170.20	453.18	181.47	529.11	87.78	33.44
			500	24.93	2.13	123.24	23.67	4.68	0.25			500	169.44	291.97	177.13	256.52	83.02	15.08
			1200	24.91	1.69	122.55	20.16	4.68	0.18			1200	169.49	192.31	176.46	215.97	82.29	11.25
			3000	24.91	1.40	122.28	18.49	4.67	0.13			3000	169.12	133.00	176.36	201.52	82.11	9.22
			∞	23.82	0	106.65	0	4.57	0			∞	135.88	0	141.02	0	74.01	0

, respectively. For the case when ${\sigma }_0<1$, the AARL${}_1$ and SDARL${}_1$ values of the $LS$-chart are generally smaller than those of the other S-charts when ${\sigma }_0\ge 0.6$ and $\lambda \ge 2.0$. In the case when ${\sigma }_0>1$, the $LS$-chart has the smallest AARL${}_1$ and SDARL${}_1$ values for all the cases considered. In general, the $LS$-chart has a better out-of-control performance among the three S-charts in most cases.

4. Conclusions

In this study, we extended the work of Huang [18] to the S control chart for a lognormal distribution. Huang [19] showed that the lognormal S-chart has better performance than the Shewhart S-chart and the MAD control chart. However, they only focus on the in-control and out-of-control performance in terms of ARL. They did not study the effect of parameter estimation on the ARL performance. Therefore, we aim to further study the AARL and SDARL performance of three S-charts to monitor the standard deviation of a lognormal distribution. We evaluate the in-control performance of the S-chart with estimated parameters using SDARL metrics. The results indicated the sample size to reach the specified in-control ARL value is very large. The $LS$-chart has the smallest SDARL value among these three S-charts, showing that the $LS$-chart has better performance than the Shewhart S-chart and MAD-chart on the in-control performance. In addition, we also investigate the out-of-control performance of the three S-charts. The AARL${}_1$ and SDARL${}_1$ values of the $LS$-chart are generally smaller than those of the other S-charts when ${\sigma }_0\ge 0.6$. The $LS$-chart has a better out-of-control performance than the other two S-charts.