## 1. INTRODUCTION

Control chart is an important tool of statistical process control (SPC) used for the improvement of products and services. The main objective of the control chart is to help process engineers, in the visual monitoring of unusual changes in a process on which corrective action should be taken to bring back the process into the in-control state. The Shewhart control charts introduced by Walter Shewhart in 1920s has the ability to detect a large shifts in the process but become less efficient in detecting process shifts of small magnitude, because the Shewhart chart uses only the information based on the most recent observation. The Exponentially Weighted Moving Average (EWMA) (Roberts, 1959), Cumulative Sum (CUSUM) (Page, 1954) and Moving Average (MA) (Wong *et al*., 2004) control charts are memory-type control charts proposed as alternatives to the Shewhart-type (memoryless) control charts when the detection of small process shifts are of interest. The memory-type control charts use information contained in the entire process observations rather than the current observations. The EWMA uses a weighted moving average as chart statistics while the MA is based on simple moving averages that are easily computed.

In recent times, few researchers focused on the MA charts in order to improve the Shewhart-type control chart for monitoring and detecting small shifts in process parameters. Wong *et al*. (2004) proposed design procedures and step-by-step guidelines for the MA and combined moving average-Shewhart schemes for use by process personnel. Khoo and Yap (2004) proposed the MA charts for a joint monitoring of the process mean and variance to detect both increases and decreases in the mean and/or variance. Khoo (2004a) proposed MA chart for monitoring fraction non-conforming units as a better alternative to the traditional p chart. Khoo (2004b) proposed a Poisson moving average control chart for the number of nonconforming items as an alternative to the standard c chart. Sparks (2004) proposed the use of weighted moving average control charts for the early detection of shifts from target when the shift size is known. The development of MA chart from an economic viewpoint was introduced by Yu and Chen (2005). Ghute and Shirke (2013) developed a multivariate MA control chart for monitoring mean vec-tor of a process. Ghute and Shirke (2014) proposed a MA control chart for monitoring the variability of a multivariate process. Pawar and Shirke (2014) developed a nonparametric control chart for monitoring process variability. Ghute and Rajmanya (2014) developed a control chart for monitoring process variability based on Downton estimator and Gini’s mean difference. Khan *et al*. (2016) proposed a EWMA control chart for exponential quality characteristic using the moving average. Adeoti and Olaomi (2016) proposed the moving average control chart based on the sample standard deviation for detecting small shifts in process variability because the S-chart is deficient for small sudden shifts in process parameters and also because the sampling distribution of S is not symmetric when sampling is from the normal distribution. Khoo and Wong (2008) introduced the double moving average (DMA) control chart as an alternative to the moving average chart for early detection of small to moderate shifts in process mean. Using simulation to compute the values of the average run length, they showed that the efficiency of the standard MA control chart is enhanced by the use of double moving average statistic in the detection of small and moderate shifts in the process mean. The DMA control chart is designed by assuming that the quality characteristics follow the normal distribution. The use of double moving average together with EWMA control chart when the quality characteristic follow the exponential distribution was studied by Aslam *et al*. (2017). More information on moving average technique for process monitoring can be found in Zhang *et al*. (2004), Khoo and Wong (2008), Montgomery (2009) and Khan *et al*. (2016).

Though, the MA control chart is less sensitive than the EWMA and CUSUM control charts in the detection of small process mean shifts. Research articles dealing with the use of double MA in quality control for the detection of small process shifts in variability and performance comparisons of the double MA, EWMA and CUSUM control chart for monitoring process variability has not been given adequate attention in the literature, even as they are all known as memory control charts and have been shown to be superior to the Shewhart charts. Therefore, research on the use of double MA control chart, in particular, in the case that more superior EWMA and CUSUM charts have been enhanced for small shift detection is for increased sensitivity of control charts for the monitoring of small and moderate shifts in process variability, because numerous extensions on EWMA and CUSUM charts that has been undertaken by different authors towards increasing the sensitivity of the chart is with regard to small process mean shift (see Riaz *et al*., 2011;Maravelakis *et al*., 2017). So, research on the use of double MA control chart for detecting process shift in process variability is desirable.

This paper is then focused on developing an efficient alternative to the moving average S control chart using double moving average technique that is expected to perform better than some of the existing control charts in the detection of small and moderate shift in process variability. The performance of the DMA control chart based on the sample standard deviation (DMA-S) is compared with some existing control charts in the literature for the detection of small and moderate shifts in process variability.

The article is organised as follows: A brief review of existing control charts used in this study is given in Section 2. Section 3 presents the design of the proposed double moving average control chart. The formula to compute the average run length (ARL) of the DMA-S chart and the performance measures of the proposed control chart is presented in Section 4. Section 5 compares the performance of the proposed DMA-S control chart with the existing control charts based on the ARL values. The implementation of the proposed chart is illustrated with a simulated and real-life data in section 6. Section 7 gives the concluding remarks.

## 2. BRIEF REVIEW OF EXISTING CONTROL CHARTS

### 2.1 S Control Chart

Let ${X}_{1},\hspace{0.33em}{X}_{2},\hspace{0.33em}\dots ,\hspace{0.33em}{X}_{n}$ represent random sample of size *n* from a normal distribution with mean, *μ* and standard deviation, *σ*, that is, ${X}_{i}~N(\mu ,\hspace{0.33em}{\sigma}^{2})$. The S control chart limits are computed based on the 3 *σ* approach i.e ${\mu}_{S}\pm 3{\sigma}_{S}$ where *μ _{S}* and

*σ*are the mean and standard deviation of the process. In order to construct the control limits for the S chart, the value of the true standard deviation

_{S}*σ*ought to be known or estimated. For a known value of the standard deviation, the control limits are defined as (see Montgomery, 2009)

where ${B}_{6}={C}_{4}+3\sqrt{1-{c}_{4}^{2}}\hspace{0.33em}\text{and}\hspace{0.33em}{B}_{5}={C}_{4}-3\sqrt{1-{c}_{4}^{2}}$

For an unknown but estimated value of the standard deviation, the control limits are defined as

where ${B}_{4}=1+\left(\frac{3}{{C}_{4}}\right)\sqrt{1-{c}_{4}^{2}},\hspace{0.33em}{B}_{3}=\text{}1-\left(\frac{3}{{C}_{4}}\right)\sqrt{1-{c}_{4}^{2}}\hspace{0.33em}\text{and}\hspace{0.33em}\overline{S}=\frac{1}{m}{\displaystyle {\sum}_{i=1}^{m}{S}_{i}}$

*B*_{3}, *B*_{4}, *B*_{5}, *B*_{6} are the control chart constants obtained in Montgomery (2009). *m* is the number of subgroups and *S _{i}* denotes the standard deviation of the

*i*-th subgroup defined as

If a sample points plot outside the control limit it is an indication that the process is unstable.

### 2.2 The Moving Average-S Control Chart

The moving average S (MA-S) control chart is a time weighted control chart based on the sample standard deviation. Suppose *S*_{1}, *S*_{2}, … be the standard deviations from a series of subgroups ${X}_{i}=({X}_{i1},\hspace{0.33em}{X}_{i2},\hspace{0.33em}{X}_{i3}\hspace{0.33em}\dots )$ with normal distribution. Here, it is assumed that ${X}_{i}~N(\mu ,\hspace{0.33em}{\sigma}^{2})$ and *S _{i}* is as defined in equation (3).

The moving average s control chart of span *w* at time *i* is defined as (see Adeoti and Olaomi, 2016)

For periods *i* < *w*, the average of all the standard deviation up to period *i* defines the moving average at time *i*, that is,

The mean and variance of the moving average-S control chart when the process is in-control for *i* ≥ *w* are given as

For periods *i*<*w*,

The upper and lower control limits (UCL and LCL) for the moving average S control chart for periods *i* ≥ *w* when the parameters are unknown is given as

For period *i* < *w*, the upper and lower control limits are given as

where $\begin{array}{l}{D}_{5}^{\text{*}}=1-\frac{3}{{C}_{4}}\sqrt{\frac{1-{C}_{4}^{2}}{i}}\hspace{0.33em}\text{and}\hspace{0.33em}{D}_{6}^{\text{*}}=1+\frac{3}{{C}_{4}}\sqrt{\frac{1-{C}_{4}^{2}}{i}}\\ {D}_{7}^{*}=1-\frac{3}{{C}_{4}}\sqrt{\frac{1-{C}_{4}^{2}}{w}}\hspace{0.33em}\text{and}\hspace{0.33em}{D}_{8}^{\text{*}}=1+\frac{3}{{C}_{4}}\sqrt{\frac{1-{C}_{4}^{2}}{w}}\end{array}$

The Values of ${D}_{5}^{*},\hspace{0.33em}\hspace{0.33em}{D}_{6}^{*},\hspace{0.33em}\hspace{0.33em}{D}_{7}^{*}\hspace{0.33em}\text{and}\hspace{0.33em}{D}_{8}^{*}$ for sample size *n* and span *w* = 2, 3, 4 and more details about MA-S are given in Adeoti and Olaomi (2016).

### 2.3 EWMA Control Chart

The control chart based on the EWMA statistic was proposed by Roberts (1959) primarily to detect shift of small and moderate size quickly. The EWMA control chart utilizes past and present information for process monitoring as the most recent observations are assigned a larger weight, while the less recent observations are assigned smaller weights. The EWMA control chart is based on the statistic

Where *X _{i}* is the mean of the

*i*th subgroup,

*λ*is the smoothing constant such that 0 <

*λ*≤ 1 and

*i*is the sample number. The starting value for

*Z*denoted

_{i}*Z*

_{0}is taken equal to the target mean or the mean of given data when the target mean is not available. The upper control limit (UCL), center line (CL) and lower control limit (LCL) of the EWMA control chart based on the statistic

*Z*are given as

_{i}

where *μ*_{0} is the target mean, *L* is the control limit coefficient and *λ* is the smoothing constant that determines the decline of the weight. The process is said to be out-ofcontrol when the EWMA statistic *Z _{i}* plot beyond the control limits i.e.

*P*(

_{i}*Z*>

_{i}*UCL*or

*Z*<

_{i}*UCL*) . Otherwise, it is in-control.

### 2.4 Moving Average Chart Based on Downton and Gini Mean Difference

The moving average control chart based on Downton estimator and Gini’s mean difference (hereafter denoted MA-D and MA-G charts) was proposed by Ghute and Rajmanya (2014). The MA-D and MA-G control charts uses additional information from recent history of process for monitoring changes in the process dispersion and are developed as an improvement to the Shewhart control chart. The moving average control chart based on Downton estimator and Gini’s mean difference are defined respectively as

where

and

where

The chart is declared to be out-of-control if *MD _{i}* >

*UCL or MD*<

_{i}*LCL*and

*MG*>

_{i}*UCL or MG*<

_{i}*LCL*. The control limits are obtained using simulation to satisfy a specified in-control average run length value.

## 3. DESIGN OF DOUBLE MOVING AVERAGE- S CONTROL CHART

The design of the double moving average S (DMA-S) control chart is based on computing the moving average of the subgroup standard deviation twice. The double moving average S control chart of span *w* at time *i* is defined as

Note that the moving average of the subgroup standard deviation, *MA _{i}* of span

*w*at time

*i*is computed using equation (4) for

*i*≥

*w*, For period

*i*<

*w*, the double moving average S statistic is computed to be the average of all moving average standard deviation up to period

*i*. That is,

The mean of the double moving average S statistic based on an in-control process where the underlying assumption follows a normal distribution, $N(\mu ,\hspace{0.33em}{\sigma}^{2})$ for period *i* ≥ *w* is given as

The variance of the double moving average S statistic is given as follows for *w* ≥ 2

Therefore, the usual 3σ upper and lower control limits of the proposed control chart for known process variability is

However, in most cases the value of the process variability, *σ* is unknown and as a result, it is estimated from past data when the process is assumed to be incontrol. Montgomery (2009) gave an unbiased estimator of the process variable that can be used to estimate the process standard deviation, *σ* in the design of the S and MA-S control chart as $\widehat{\sigma}=\frac{S}{{c}_{A}}$, where *S* is the average subgroup standard deviation estimated from the number of subgroups in a preliminary dataset assumed to be incontrol. Thus, the 3*σ* upper and lower control limits of the proposed control chart when *σ* is unknown becomes

Equations (22a) and (22b) can be re-written as

where

The values of ${B}_{5}^{*},\hspace{0.33em}\hspace{0.33em}{B}_{6}^{*},\hspace{0.33em}\hspace{0.33em}{B}_{9}^{*},\hspace{0.33em}\hspace{0.33em}{B}_{10}^{*}$ have been computed for values of span *w* = 2, 3, 4 and 5 and sample size *n* and presented in Table 1.

The control limits of the double moving average S control chart for *w* ≥ 2 are computed based on Equation (23a). The double moving average S control chart is constructed by plotting the double moving average S statistic computed from Equation (17) on the chart against the sample number *i* with control limits given in Equation (23a). The process is declared as in-control whenever *LCL*≤*DMA _{i}*≤

*UCL*. Otherwise, the process is declared as out-of-control.

## 3. THE PERFORMANCE EVALUATION OF DOUBLE MOVING AVERAGE-S CONTROL CHART

The performance of any control chart is determined on the basis of certain criteria. One of such criteria is the average run length (ARL). The average run length is the average number of samples that signals an out-of-control situation. Zhang *et al*. (2004) provided some useful discussion on the ARLs of a moving average control chart and showed that some of the formula for computing the ARLs can be misleading, because the moving average control chart is a memory-type control chart in which successive statistics are dependent upon each other. Therefore, the run length distribution of the moving average control chart does not become a geometric random variable. The same reason is also given here for the double moving average control chart. Therefore, for the double moving average S control chart we derive the formula for computing the in-control and out-of-control average run lengths. The formula of the average run length for the double moving average S-control chart is given in equation (24) following the work of Areepong and Sukparungee (2011). The detail of the ARL computation is provided in the appendix.

When the process is in-control, a large ARL value is desirable; however, when there is a shift in the process variability a small ARL is desired. The in-control average run length when the process is assumed to be normally distributed with mean *μ* = 0 and standard deviation ${\sigma}_{1}=\delta {\sigma}_{0}=1$ is set at 370, *δ* = 1.00 is an indication that there is no shift in the process variability. The numerical results for in-control and out-of-control average run length of the DMA-S control chart has been calculated from equation (24) for sample sizes *n*, spans *w*, and different shift sizes *δ*. The ARL profiles of the double moving average S control chart for various shift in the process standard deviation and sample sizes *n* = 5 and 10 are presented in Tables 2 and 3.

## 4. COMPARISON OF CONTROL CHARTS

In this section, we compare the performance of the proposed DMA-S control chart with some of the existing control charts for monitoring process variability available in the literature. The out-of-control ARL values are used to evaluate the detection ability of the control charts. The ARL values of the proposed and existing control charts are presented in Tables 4 and 5.

### 5.1 Proposed Chart Versus S Chart

From Table 4, it is interesting to note that the double moving average S-control chart perform better than the standard S-chart for small and moderate shifts in process variability. Also, the ARL values of the proposed control chart decreases rapidly as the span *w* increases. Figures 1 and 2 provide a clear picture about the efficiency of DMA-S chart over the S-chart.

### 5.2 Proposed Chart Versus MA-S Chart

By comparing DMA-S with MA-S chart, we observed that the DMA-S control chart outperforms MA-S control chart in detecting small shifts in process variability for span *w* and all shift sizes as it consistently gives a lower ARL values. For example, in Table 4 when *δ* = 1.25, the ARL value for the proposed control chart is 5.64 and 3.45 for *w* = 3 and 4 while that of the moving average S-chart is 14.35 and 11.65 respectively (see Figures 1 and 2).

### 5.3 Proposed Chart Versus EWMA Chart

The comparison of the proposed control chart with EWMA control chart reveal that the DMA-S chart is superior to the EWMA chart for detecting small shift (*δ* ≤ 1.25 ) for span *w* = 2, 3 and 4 because it has smaller ARL values but it is less efficient for moderate shift (*δ* ≥ 1.5) in the process variability (see Figures 1 and 2).

### 5.4 Proposed Chart Versus MA-D Chart

From Table 5, we see that the ARL values of DMAS chart are smaller than the MA-D chart proposed by Ghute and Rajmanya (2014) for span *w* ≥ 3, but inefficient for span *w* = 2. This means that the increase in span w leads to early detection of shifts in process variability by the DMA-S control chart. Figures 3 and 4 show the comparison of the proposed chart and MA-D chart

In summary, the results of Tables 4 and 5 and Figures 1-4 show that the proposed control chart is superior to the existing control charts in detecting small shifts in process variability. The fact that the S control chart is less efficient in detecting process shifts is obvious from Figures 1 and 2 because the ARL values of S-chart is greater than that of the proposed and existing control charts in this study.

## 6. EXAMPLE

### 6.1 Simulated Data

The application of the proposed control chart is illustrated based on the simulated data for twenty subgroups. The first ten subgroup is assumed to follow a normal distribution with mean *μ* =10 and standard deviation *σ* = 1 i.e. *Normal*(10, 1) and the next 10 subgroup are generated from a normal distribution with mean *μ* =10 and standard deviation*σ* = 2. i.e. *Normal*(10, 2) for sample size n = 5. The data are given in Adeoti and Olaomi (2016). The sample standard deviation, the moving average and double moving average is presented in Table 6. The control limits of the double moving average S chart are com puted using Equation (15). Note that the value of the control chart constant used to compute the control limits is obtained in Table 1. The values of the control chart limits are given as (*LCL*,*UCL*) = (0.6899, 2.3387) for span *w* = 2. The double moving average statistic in Table 6 plotted against the control limits for span *w* = 2 is shown in Figure 5. Figure 6 is the moving average statistics plotted against the control limits (*LCL*,*UCL*) = (0.3452, 2.6800). The figures show that the DMA-S chart detects an out-ofcontrol signal for measurements 19 and 20 while the moving average S (MA-S) control chart fails to detect the shift for the same span *w* = 2.

### 6.2 Real-Life Data

This section illustrates how the proposed control chart is used to construct control limits using real -life data taken from Montgomery (2009). The data is the inside diameter measurements for automobile engine piston rings. Twenty-five samples taken from (Table 6.1, Montgomery, 2009) are for phase I monitoring of the process. The sample standard deviation, moving average and double moving average statistics are summarized in Table 7.

The upper and lower control limits of the double moving average control chart for spans *w* = 2, 3 and 4 are given as (LCL,UCL) = (0.0043, 0.0145), (0.0063, 0.0128) and (0.0068, 0.0120) while control limits for the moving average s chart are given as (*LCL*,*UCL*) = (0.0022, 0.0166), (0.0035, 0.0153) and (0.0043, 0.0145) Figure 7 shows the proposed double moving average S control chart with the plotted statistics using real-life data for span *w* = 3. Here, it can be seen that the proposed control chart signals an out-of-control condition at the 13^{th} sample. The value of the moving average S is plotted against control chart limits in Figure 8. From Figure 8, the control chart was unable to signal an out-of-control condition for same span *w* = 3. Therefore, it is observed that decision on the use of S-control chart and moving average control for process monitoring will be ineffective when used for phase II monitoring of the process because of their failure to detect that the process is not stable but which was detected by the double moving average control chart.

## 7. CONCLUSION

This paper propose the double moving average S control chart as an improvement to the existing control charts for detecting small and moderate shift in process variability. The DMA-S chart is based on computing the moving average of the subgroup twice, since successive statistics are dependent upon each other. Control limit factors for the proposed chart are constructed for sample sizes n and span *w* = 2, 3, 4 and 5. The performance of the double moving average S control chart has been evaluated using the average run length derived in this paper. The proposed DMA-S chart is found to perform better than the Shewhart-S, the MA-S, MA-D and EWMA control charts for monitoring small shift in process variability but compete favourably well with the EWMA control chart for moderate shifts in process variability. It is observed that the proposed control chart can serve as a better alternative to the existing control charts for quick detection of small and moderate shift in process variability.

## Appendix

Let ARL = n, then the ARL with *w* ≥ 2 is computed as

Then

Therefore,