Abstract
Homogeneously weighted moving average () charts have recently achieved popularity for monitoring small changes in process parameters (location and/or dispersion). Furthermore, the (double ) and (triple ) are the advanced versions of the charts. The chart for the process dispersion is designed to detect only the upward (one-sided) shift (i.e., process deterioration). Employing a two-sided chart for concurrently detecting both process improvement and process deterioration is an important aspect of statistical process monitoring. By taking this point as motivation, one and two-sided charts (symbolized as the ) are proposed for monitoring the process dispersion. The Monte Carlo simulations are performed to investigate the performance behavior of the charts in terms of certain performance indicators, including , , , , and . The comparison among the versus existing charts (, , , , and ) indicates that the charts outperform the existing charts. Finally, a dataset is also analyzed to illustrate the implementation of the charts.
1. Introduction
Every manufacturing and production process involves two kinds of variations: one common cause and other special cause variations. The common cause variations are natural and essential parts of any stable process. A process that operates with the common cause variations is known to be statistically IC (in-control). On the other hand, the special cause variations are unnatural and deteriorate the process stability. A process that functions with the special cause variations is called statistically OC (out-of-control). The special cause variations demand special attention for quick detection and removal to get the process back on track (i.e., statistically IC).
Control charts are the most well-known SPM (statistical process monitoring) tools for detecting special cause variations that trigger shifts in the process parameters (location and/or dispersion). Shewhart [1] introduced the basic chart, which is used to monitor the process parameters and decide whether the process is IC or OC. The Shewhart chart is used to monitor the mean level of the process, whereas the Shewhart , , and charts are implemented to detect the changes in the process dispersion. The Shewhart charts are also referred to as memoryless-type control as they only use the current process information; consequently, the Shewhart charts are less sensitive to small changes in the process parameters. The memory-type charts, on the other hand, incorporate both current and previous process information and are more sensitive than the Shewhart charts. The memory-type charts include the (cumulative sum) proposed by Page [2], (exponentially weighted moving average) designed by Roberts [3], (moving average) introduced by Roberts [3], and (double moving average) suggested by Khoo and Wong [4] and Alevizakos et al. [5].
The classical chart has become the most powerful tool for researchers to detect the small shifts that ruin the manufacturing and production process stability. Numerous authors have improved the performance of the classical chart by integrating different techniques, emphasizing the process mean monitoring (see [6–10]). It is important to note that, in general, many special cause variations can potentially affect the process dispersion (variance or standard deviation). Furthermore, dispersion monitoring is of utmost importance because the mean structure is based on it [11]. Moreover, an increase in process dispersion leads to process deterioration, while a decrease in process dispersion improves process performance [12]. Various authors have investigated the classical -type charts in the context of process dispersion monitoring. For example, Crowder and Hamilton [13] developed the chart, which uses the log transformation to the sampling variance and detects changes in process standard deviation. Following Crowder and Hamilton [13], Shu and Jiang [14] introduced the chart to monitor the process dispersion, which has a better detection ability than the Crowder and Hamilton [13] chart. In the same way, Huwang et al. [15] designed the charts for process dispersion, and they demonstrated that their charts uniformly outperformed the Crowder and Hamilton [13] and Shu and Jiang [14] charts. Similarly, Castagliola [16] proposed the bilateral chart using a three-parameter log transformation to , which efficiently monitors the process dispersion. Likewise, Ali and Haq [17] proposed the (generally weighted moving) and charts to detect the process dispersion shifts. In addition, Chatterjee et al. [18] used the transformation of Castagliola [16] and introduced the (triple ) chart for monitoring the process dispersion.
Abbas [19] introduced the (homogeneously weighted moving average) chart and showed that the chart outperformed the classical chart. Later, Abid et al. [20] proposed the (double ) chart for the process mean that outperformed the chart. Similarly, Riaz et al. [21] proposed the (triple ) chart for the process mean, which uniformly outperformed the and charts. Other -based studies for mean process monitoring are offered by Rasheed et al. [22], Rasheed et al. [23], and Zhang et al. [24]. After that, Knoth et al. [25] highlighted a few concerns about the chart when compared to the chart in the case of the steady-state comparison scenario. However, Riaz et al. [26] reinvestigated the zero-state as well as the steady-state performances of the chart for various shift sizes and smoothing parameters. They performed a comprehensive comparative analysis of the run-length profiles of the and charts for several values of the design parameters. The results indicate that the concerns of Knoth et al. [25] are not always true, i.e., the chart outperforms the chart under a zero state for numerous regions of shifts, and that it can maintain its dominance over the chart despite varied delays in process shifts.
Recently, Riaz et al. [27] developed the chart for effective process dispersion monitoring. They showed that the dispersion chart outperformed the dispersion charts proposed by Crowder and Hamilton [13], Shu and Jiang [14], and Huwang et al. [15], respectively. It should be noted, the chart has a significant disadvantage, i.e., it is a one-sided chart and detects only the upward shifts in the process, i.e., process deterioration. However, in practice, both the process deterioration and the process improvement are important, which necessitates the use of a single two-sided chart that monitors the process deterioration and the process improvement. Also, as mentioned earlier, Riaz et al. [21] improved the performance of the -type charts by introducing the chart for efficient mean process monitoring. So, there is a clear research gap to further enhance the performance of existing charts by extending the idea of Riaz et al. [21] and designing a two-sided chart for monitoring the upward and downward process dispersion shits at the same time. Capitalizing the aforementioned research gap, this paper aims to present the design of one and two-sided charts with time-varying control limits, which effectively monitor the process dispersion. These charts are labeled as charts. The upper-sided chart is formulated that identifies the upward changes (process deterioration), while the two-sided chart is developed to monitor both upward and downward changes (process deterioration and process improvement) in the dispersion parameter. The Monte Carlo simulations are performed to compute the approximate run-length indicators, such as (average run length), (standard deviation run length), (extra quadratic loss), (relative average run length), and (performance comparison index). Based on these run-length indicators, the one and two-sided charts are compared to the one and two-sided competing charts denoted as , , , , and charts. The comparison demonstrates that the charts achieve better performance against the competing charts in monitoring small and large shifts in the process dispersion. Finally, to illustrate the practical implementation of the charts, the wind farm data analysis is also provided.
The structure for the rest of the paper is given as follows. Section 2 offers the theory and background of some competing charts for dispersion monitoring. Similarly, the methodologies of the charts are specified in Section 3. Likewise, Section 4 presents the design of the charts. In addition, Section 5 includes a comprehensive comparative study of the charts. Furthermore, the application of the chart to the wind farm dataset is given in Section 6. The final section addresses the summary, conclusions, and recommendations of the study.
2. Competing Methods
This section provides the concepts and background of some charts that monitor the process dispersion shifts. Section 2.1 introduces the process variable and transformation. Similarly, Sections 2.2–2.5 present the methodologies and formulation of the one and two-sided , , , and charts.
2.1. Process Variable and Transformation
Suppose that are independent identically normal process random variables, i.e., , where and are the IC process mean and dispersion, respectively, is the subgroup number, and refers to the amount of shift in process dispersion. As the only concern is to monitor the shifts in process dispersion, the process mean is considered to be zero, i.e., . Here, the shifts can be defined as , where is the OC process dispersion. So, for the IC process, , and for the OC process with decreasing shifts, , while for increasing shifts, .
Assuming the sample variance computed from the th subgroup having size , where is the sample mean, then it is well known that for the IC process, has a chi-square distribution with degrees of freedom equal to , i.e., . As the sample variance has a skewed distribution, it is not an appropriate statistic for the design of certain memory charts. To overcome this limitation, Quesenberry [28] suggested the transformation for the process dispersion, defined as follows:where denotes the CDF (cumulative distribution function) of chi-square distribution at degrees of freedom and denotes the inverse standard normal CDF. If the underlying process is IC, then the statistic follows the standard normal distribution, i.e., .
2.2. Chart
The chart for the process dispersion, based on the statistic , is proposed by Huwang et al. [15] and denoted by the chart. Hereafter this chart is labeled by chart. If denotes the plotting statistic of the chart, then it can be defined by the relation given as follows:where is the smoothing parameter, satisfying . The initial value of the statistic is set to zero, i.e., . The expected value and variance of the statistic are, respectively, given as follows:
If (upper control limit) and (lower control limit) denote the upper and lower control limits that monitor the two-sided shifts in process dispersion, then its time-varying form for the chart is given as follows:where is the width coefficient of the chart, and it is determined so that the (in-control ) is equal to its desired value. The two-sided chart is constructed by plotting the statistic against the subgroup number and the process is declared to be OC if or ; otherwise, the process is considered to be IC.
Similarly, to detect the upward changes in the process dispersion with the chart, only the upper limit is considered, i.e.,
The statistic is plotted against the subgroup number and the upper-sided chart will detect the process to be OC if ; otherwise, the process remains IC.
2.3. Chart
The chart for process dispersion (hereafter labeled by chart) is based on the statistic and can be designed by combining the features of the two charts. If the plotting statistic for the chart is denoted by , then it can be defined as follows:
The starting value of is equal to zero, i.e., . The mean and variance of for the IC process are, respectively, given as follows:
Based on and in equation (7), the control limits, and , of the chart are defined as follows:where is the chart width coefficient. To diagnose the upward or downward changes in the process dispersion, is plotted against the subgroup number and if or , then the underlying process is considered to be OC; otherwise, it is considered to be IC.
Similarly, the control limit for the upper-sided chart is given as follows:
To detect the increasing shifts in process dispersion, is plotted against the subgroup number and the process is said to be OC when ; otherwise, the process is said to be IC.
2.4. Chart
The chart based on the three-parameter log transformation is constructed by Chatterjee et al. [18], which effectively monitors the process dispersion. Similarly, the chart based on transformation , i.e., chart, can also be formulated to monitor the process dispersion changes. The plotting statistic of the chart, i.e., can be defined as follows:
The initial values of are set as zero, i.e., . The mean of is given as , whereas its variance is defined as follows:where . The limits and of the two-sided chart can be defined as follows:where denotes the width of the chart. To detect the increasing or decreasing changes in the dispersion parameter, the two-sided chart is constructed, and the statistic is plotted against the subgroup number . If the statistic falls outsides the control limits, i.e., or , then the process is deemed to be OC; otherwise, it is IC. On the other hand, to detect the upward changes in the process, however, only in equation (12) is used. In this case, the process is said to be OC when ; otherwise, the process is said to be IC.
2.5. Chart
The chart, based on the statistic , can be used to detect the process dispersion shifts. This chart is referred to as the chart. If denotes the plotting statistic of the chart, then it can be given as follows:where is the mean of previous observations and () is the smoothing constant. The initial values and are zero, i.e., . The IC mean and variance of , from Appendix A.1, are given as follows:
The two-sided time-varying control limits, and for the chart are given as follows:where is called the width coefficient and it is chosen to obtain close to prespecified value. To monitor an increase or decrease in process dispersion, the two-sided chart is constructed, and is plotted versus the subgroup number . The underlying process is considered to be OC when or ; otherwise, it is considered to be IC. Likewise, based on equation (14), the time-varying upper control limit, , of the chart is given as follows:
To identify the upward changes in the process dispersion, the statistic and if then the process is considered to be OC; otherwise, the process is regarded to be IC.
2.6. Chart
The chart is based on symbolized by chart, can be used to monitor the process dispersion, and detects both the downward and upward shifts. The statistic is and it can be given as follows:
It is assumed that . The statistic in equation (17) can be written as follows:
The IC mean and variance of , from Appendix A.2, are provided as follows:
The control limits, i.e., and , for the two-sided chart are given as follows:where denotes the width coefficient of the chart, and its value is chosen so that the is approximately equal to a specified value. To detect an increase or decrease in the process dispersion, the two-sided chart is constructed, and is plotted against . The process is stated to be OC when or ; otherwise, the process is stated to be IC.
Similarly, the control limit, , of the chart is given as follows:
To diagnose the upward changes, the statistic is plotted against the subgroup number and if , the process is stated as OC; otherwise, the process is considered IC.
3. Chart
This section describes the methodology of the chart, referred to as the chart, that efficiently monitors the process dispersion shifts. To construct the chart, the charting statistic for the chart is given as follows:where is the charting statistic of the chart, specified by equation (17), and is a design parameter. Here, . Equation (22), alternatively, can be written as follows:
The mean and variance of the statistic , from Appendix A.3, are given as follows:
The two-sided time-varying control limits, and , of the chart, using the defined mean and variance in equation (24), are given as follows:where denotes the width coefficient of the chart, and its value is chosen so that is approximately equal to the desired value. For monitoring an increase or a decrease in the process, the two-sided chart is constructed, and is plotted against . The two-sided chart declares the process to be OC when or ; otherwise, it declares that the process is IC.
Similarly, the upper-sided time-varying limit, , of the chart is given as follows:
To detect the upward shifts in the process dispersion, is plotted against . If , then the process is considered OC; otherwise, it is considered IC.
4. Design and Implementation of Charts
This section provides the performance evaluation measures in Section 4.1. Similarly, Section 4.2 offers the overall performance measures. Likewise, Section 4.3 provides the run-length distribution of the chart. Also, Section 4.4 explains the IC design of the chart. Moreover, the OC performance of the chart is also provided in Section 4.4.
4.1. Performance Measures
The and are the most popular and frequently used measures to evaluate the performance of the chart at a specified shift. is defined as the expected number of sample points plotted on the chart before the chart indicates an OC signal [29]. are of two types, i.e., (IC ) and (OC ). The should be high enough to prevent many false alarms if the process operates in an IC state, whereas the should be lower to rapidly identify process shifts [30]. To evaluate the effectiveness of two or more charts, it is necessary to set the common for all of them. For a certain shift, the chart with the lower value is considered sensitive and may identify a shift quicker than the other charts.
4.2. Overall Performance Measures
Other performance indicators that assess the overall performance of a control chart over the entire range of shifts are known as (extra quadratic loss), (relative average run length), and (performance comparison index). The is the weighted over the entire range of shifts, i.e., , under the square of the shift as a weight. Symbolically, can be defined as follows:where is the of a specific chart at a shift . The presents the ratio between the of a particular chart and the benchmark chart . Symbolically, the can be defined as follows:
A benchmark chart is often recognized as the chart with the lowest or as one of the existing standard charts. The is the ratio of the of a chart and the of the benchmark chart. Mathematically, it can be defined as follows:
The benchmark chart has , and for the remaining charts, [31].
4.3. Run-Length Distribution of Chart
To compute the approximate run-length distribution of the chart, the Monte Carlo simulation technique is used. A simulation algorithm is developed using the statistical software package, which can be explained in the steps given as follows:(i)Choose sample size , smoothing parameter , and parametric values, i.e., in the case of IC process .(ii)Generate random observations , for , from .(iii)Compute the statistic in equation (1).(iv)Using the statistic , calculate the statistic in equation (13).(v)Using as input, compute the statistic, , in equation (17).(vi)Using the statistic as input, compute the statistic, , in equation (22).(vii)Select for desired and compute and in equation (25).(viii)Plot the statistic versus the subgroup number . If or , record sequence order, known as run length ().(ix)Repeat steps (ii)–(viii) for times and record and hence compute the approximate by and approximate by The above simulation algorithm is used to construct the two-sided (i.e., and ) chart; however, only one of the two control limits in equation (25) is required to design the one-sided (i.e., or ) chart. In addition, the and values of the competing , , , , and charts are computed similarly.
4.4. IC Design and OC Performance of Charts
The IC designs of the one and two-sided charts are based on smoothing parameters and the width coefficient, . The combinations () are chosen such that the is close to the desired value. Also, time-varying control limits are considered for the one and two-sided charts. The smoothing parameter values are set as follows: , 0.1, 0.2, and 0.3, for each of the values of 5, 10, and 20, respectively, to execute the simulation study. Each combination of and is then used to compute the values of the THWMAV chart coefficient, i.e., , to make sure that is roughly equal to the desired value. Table 1 lists the values computed for the one and two-sided charts to get close to 200, 370, and 500. Furthermore, Table 1 also presents the (IC ) for one and two-sided charts.
To address the OC performance of the one and two-sided charts, different combinations are used for each , 10, 15, as listed in Table 1. As mentioned above, these combinations provide the which is approximately equal to 370. For the IC process, it is assumed that the shift in process dispersion is , while for the OC process, , 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 0.2 for the upward shifts, and for the downward shifts, , 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, and 0.5 are chosen. The and (given in the parentheses) values for the one and two-sided chart, along with competing charts, are presented in Tables 1–6. For each , the smallest values are shown in bold font for each shift. The main findings from these results are as follows:(i)As rises, the values of for the charts also increase to achieve desired value. For instance, with , the value of for upper-sided chart is 0.205, 0.279, 0.636, and 1.305 when , , , and , respectively, to obtain close to 370 (see Table 1).(ii)Similarly, for the two-sided chart, if , then , if then , if , then , and if , then to achieve which is approximately 370 (see Table 1).(iii)As increases, the values for the charts decrease. For instance, with , , for the upper-sided chart, if , then , if then , if then and if then (see Table 1). Similarly, for the upper-sided chart, the value is 2129.33 when , 1747.29 when , 473.58 when , and 234.46 when (see Table 1).(iv)For a specified , as increases, the OC performance of the one and two-sided charts deteriorates. For example, for and , the upper-sided THWMAV chart has ARL1 and SDRL1 (OC SDRL) values are 3.92 and 7.25 when ; 4.01 and 7.49 when ; 6.66 and 12.16 when ; and 13.96 and 20.45 when ; respectively (see Table 2).(v)Similarly, for the two-sided chart, and = 17.15 when , and = 18.14 at , and = 27.45 for , and and = 36.52 when (see Table 2).(vi)For a given , the OC performances for the charts improve as increases. For example, with and , the upper-sided THWMAV chart has ARL1 values are 3.49, 2.32, and 1.92 and SDRL1 values are 4.50, 2.29, and 1.67 when , 10, and 15, respectively (see Tables 2, 4, and 6). Similarly, for the two-sided chart, , 4.64, 3.59 and = 8.78, 4.30, 2.95 when , 10, 15, respectively (see Tables 3, 5, and 7).(vii)As increases, the values also increase; as a result, the overall performance of the one and two-sided charts deteriorates. For example, with , the values of for upper-sided chart are 13.59, 13.64, 14.54, and 16.74 for , , , and , respectively (see Table 2). Similarly, for the two-sided chart, when , when , when and when (see Table 3).
5. Comparative Study
The current section compares the upper and two-sided charts against the upper and two-sided competing charts, such as , , , , and charts. The charts are compared with common , using , 0.1, 0.2, 0.3 and , 10, 15. The comparisons are given in the following subsections.
5.1. versus Charts
The charts achieve supreme performance against the charts in terms of smaller and for single shifts and a certain range of shifts. For example, for a single shift, at , , and , the upper-sided chart yields , 2.38, 1.89, … and , 2.91, 1.87, …, while the upper-sided chart provides , 10.83, 6.21, … and , 10.40, 5.73, … (see Table 2 and Figure 1), respectively. Similarly, the two-sided chart produces , 5.36, 3.78, … and , 6.30, 3.74, …, whereas the upper-sided delivers , 15.17, 8.33, … and , 13.46, 7.08, … (see Table 2 and Figure 2) for the same sizes of shift. Similarly, the charts attain better overall performance over the charts for a certain range of shifts. For instance, when and , the upper-sided chart’s , , and values are 13.59, 1.00, and 1.00, while the upper-sided chart’s , , and values are 20.37, 1.50, and 2.75 (see Table 2), respectively. Moreover, the two-sided chart has the , , and values of 15.91, 1.00, and 1.00, respectively; however, the two-sided chart achieves larger , , and values which are 25.53, 1.60, and 2.13 (see Table 3), respectively.
5.2. versus Charts
The upper and two-sided charts reveal outstanding performance over the upper and two-sided charts, respectively, in the terms of the least and . For example, at , , and , the and values for the upper and two-sided charts are 2.85, 4.01, and 7.09, 9.28, whereas the and values for the upper- and two-sided charts are; 16.07, 15.99, and 22.96, 21.22, respectively (see Tables 4 and 5). Likewise, the one and two-sided charts achieve superior overall performance against the one and two-sided chart, respectively, in terms of minimum , , and values. For example, with and , the one- and two-sided THWMAV chart's EQL, RARL, and PCI values are 12.89, 1.00, and 10.00 and 13.97, 1.00, and 1.00, respectively, which are smaller than the one- and two-sided DEWMAV chart's EQL, RARL, and PCI values of 16.08, 1.25, and 2.35 and 18.51, 1.32, and 1.70 (see Tables 4 and 5), respectively.
5.3. versus Charts
The upper-sided chart shows excellent performance against the upper-sided chart by keeping the lower and values. Similarly, the two-sided chart achieves dominant performance against the two-sided chart. In detail, when , , and , then the upper-sided chart bears the and values of 2.31, 1.51, 1.25, 1.13, 1.07 and 2.78, 1.21, 0.76, 0.53, 0.37, respectively, while, the upper-sided chart achieves and values of 9.44, 3.38, 1.96, 1.44, 1.22 and 11.53, 3.92, 1.89, 1.08, 0.68, respectively (see Table 6). Similarly, for the same , , and values, the and for the two-sided chart are 5.56, 2.93, 2.04, 1.61, 1.37 and 6.33, 2.46, 1.52, 1.11, 0.85, respectively, whereas the and values of the two-sided chart are larger, i.e., 14.70, 5.07, 2.72, 1.84, 1.45, and 15.14, 5.20, 2.59, 1.52, 0.98, respectively (see Table 7). Besides, at and , the , , and values of the upper-sided chart indicate a better overall detection ability as compared to the upper-sided chart because , , and of the upper-sided chart are larger. Correspondingly, the two-sided chart’s , , and are smaller relative to the two-sided chart’s , , and , respectively (see Table 6).
5.4. versus Charts
The upper-sided chart achieves superior performance against the chart in terms of minimum and . For example, for , , and , the upper-sided chart provides and , whereas the upper-sided chart provides and (see Table 4). Similarly, the two-sided chart also gets excellent detection ability against the two-sided chart in terms of smaller and . For instance, at , , and , the upper-sided chart’s and are smaller as compared to the two-sided chart’s and values (see Table 5). Moreover, the upper-sided chart has a superior overall performance to the upper-sided chart. In detail, at and , the upper-sided chart has , , and values, whereas the upper-sided chart’s , , and are larger in comparison (see Table 4). Likewise, the two-sided chart reveals improved detection ability against the two-sided chart, as it offers smaller , , and values of 15.10, 1.00, 1.00, as compared to the , , and values for the upper-sided chart, which are: 21.20, 1.40, 1.88 (see Table 5).
5.5. versus Charts
At , , and , the charts achieve better detection ability than the charts. For instance, the upper and two-sided THWMAV charts ARL1 and SDRL1 values are equal to 2.46 and 5.51, and 3.00 and 6.61, respectively, while upper and two-sided DHWMAV charts ARL1 and SDRL1 values are 3.69 and 8.13, and 4.65 and 9.27, respectively (see Tables 2 and 3 and Figures 1 and 2). It indicates the inferior performance for the upper and two-sided charts. In addition, with , , the , , and values of the upper and two-sided charts are 13.64, 1.00, 1.00, and 16.07, 1.00, 1.00, respectively, which suggests the better overall detection ability for the upper and two-sided charts against the upper and two-sided charts that have the , , and values of 14.79, 1.08, 1.29, and 18.93, 1.18, 1.32, respectively (see Tables 2 and 3).
5.6. Main Findings of the Study
Some essential findings of this study can be given as follows:(i)The one and two-sided charts undoubtedly boost the efficiency of the process dispersion monitoring.(ii)The one and two-sided charts perform better against the competing , , , , and charts for all parametric choices.(iii)The one and two-sided charts reveal better overall detection ability against the competing , , , , and charts.(iv)The performance for the one and two-sided charts deteriorates as the value of increases.(v)The OC performance for the one and two-sided charts improves as the sample size increases.(vi)The overall performance for the one and two-sided charts degrades as the value of increases.
6. Application of Charts
This section analyzes the dataset of the wind station, Dhahran (, , Saudi Arabia), to demonstrate the practical application of the one and two-sided charts. This dataset is considered from the study of Riaz et al. [21], which addresses the daily power generated by the wind station during the winter (15 Nov. to 29 Feb. 2020). Table 8 presents the dataset consisting of 21 subgroups, each of size . The (Anderson–Darling) test determines normal distribution as a goodness of fit to the dataset. The test provides the value as 0.353, which suggests that the underlying process of the daily power generating at the wind station follows the normal distribution having a mean of and standard deviation of , i.e., . This means that this dataset can be regarded as random samples from the IC process having a standard deviation of 1.1, i.e., . To implement the upper-sided chart, following Riaz et al. [21], the shift of moderate size, i.e., , is introduced artificially in the dataset at subgroup number 16 so that OC process standard deviation is equal to . At , the design parameters for the upper-sided , , , , and charts are chosen such as , , , , , and , respectively. Using these design parameters, respectively, the upper-sided , , , , , and charts are designed, and the statistics , , , , and , respectively, are plotted against the subgroup number . The upper-sided , , , , , and charts are depicted in Figure 3, which indicates that the upper-sided chart detects OC signal at the 18th subgroup number, while the upper-sided , , and charts detect the OC signal at subgroup number 21, 21, 19, respectively, and the and charts do not detect any OC signal (see Table 8 and Figure 3). This indicates that the upper-sided chart is more sensitive than the upper-sided competing charts.
Similarly, to construct the two-sided chart, the artificial shift of size 1.25 is introduced in the dataset at the start of the process. In this case, the OC process dispersion is equal to 1.375. At , the plotting statistics , , , , and for the two-sided charts are constructed using the design parameters for the two-sided , , , , , and charts, i.e., , , , , and , respectively. The two-sided charts are depicted in Figure 4, which demonstrate that the two-sided chart triggers the OC signal at the 8th subgroup number, while the two-sided , , , , and charts trigger the OC signal at the 8th, 11th, 15th, 12th, and 10th subgroup numbers, respectively (see Table 8 and Figure 4). In addition, the two-sided chart overall triggers 14 OC signals, while the two-sided , , , , and charts overall detect 14, 8, 7, 10, and 10 OC signals, respectively. This reveals that the two-sided chart is more sensitive than the two-sided competing charts.
7. Summary, Conclusions, and Recommendations
This paper introduced the one and two-sided triple homogenously weighted moving average charts, denoted as charts, to detect the process dispersion shifts. The upper-sided chart monitors the upward shifts in the process dispersion. In contrast, the two-sided chart monitors both upward and downward shifts in the process dispersion parameter. Extensive Monte Carlo simulations are used to compute approximate run-length indicators, such as and for the one and two-sided charts. Also, to investigate the overall performance of the charts, , , and measures are computed. A comprehensive comparative study is performed, and the one and two-sided charts are compared to the competing one and two-sided , , , , and charts. The comparison demonstrated that the charts have better detection ability than the competing charts. Finally, the dataset of the wind farm is also analyzed to support the one and two-sided charts, which also reveals that the charts are more sensitive in detecting process dispersion shifts. Finally, several issues are recommended for future studies of the charts. For example, the concepts of the charts can be used with the idea of neutrosophic statistics [32, 33], in the high-yield processes [34], in multivariate process cases [35], etc.
Appendix
A. Process Variable and Transformation
Let (, ) denote the process variable having independently identically normal distribution with mean and dispersion , where denotes the amount of shift in process dispersion. When the process is IC then ; otherwise, for downward shifts and for upward shifts. Let be the sample mean and denote the sample variance for the th subgroup having size ; then, for the IC process, has a chi-square distribution with degrees of freedom equal to , i.e., . In this case, according to Quesenberry [28], the transformation is known as the independently identically standard normal random variable, i.e., for . So,
For the IC process, the expected value of is given as
Similarly, the variance of is given as
Likewise, the covariance of and , i.e., , is given as
A1. Mean and Variance of Chart
The plotting statistic of the chart is defined as
For the stable process, the mean of is given as(cf. equations A.1 and A.2).
Similarly, the variance of is given as(cf. equations A.1, A.3, and A.4)
A2. Mean and Variance of Chart
The statistic is given as(cf. equation A.5)
For the IC process, the mean of is given as(cf. equations A.1 and A.2).
In the same way, the variance of is given as(cf. equations A.1, A.3, and A.4)
A3. Mean and Variance of Chart
The plotting statistic for the chart is given as(cf. equation A.9)
For the IC process, the mean of is given as(cf. equations A.1 and A.2).
Similarly, the IC variance of is defined as(cf. equations A.1, A.3, and A.4)
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.