|Computer Modeling in Engineering & Sciences|
New Hybrid EWMA Charts for Efficient Process Dispersion Monitoring with Application in Automobile Industry
1Faculty of Economics, Taiyuan Normal University, Taiyuan, 030619, China
2Department of Mathematics and Statistics, Riphah International University, Islamabad, 44000, Pakistan
3Department of Mathematics, Women University of Azad Jammu and Kashmir, Bagh, 12500, Pakistan
4Department of Statistics, University of Azad Jammu and Kashmir, Muzaffarabad, 13100, Pakistan
5School of Statistics, Shanxi University of Finance and Economics, Taiyuan, 030619, China
*Corresponding Author: Syed Masroor Anwar. Email: email@example.com
Received: 08 September 2021; Accepted: 09 November 2021
Abstract: The EWMA charts are the well-known memory-type charts used for monitoring the small-to-intermediate shifts in the process parameters (location and/or dispersion). The hybrid EWMA (HEWMA) charts are enhanced version of the EWMA charts, which effectively monitor the process parameters. This paper aims to develop two new upper-sided HEWMA charts for monitoring shifts in process variance, i.e., HEWMA1 and HEWMA2 charts. The design structures of the proposed HEWMA1 and HEWMA2 charts are based on the concept of integrating the features of two EWMA charts. The HEWMA1 and HEWMA2 charts plotting statistics are developed using one EWMA statistic as input for the other EWMA statistic. A Monte Carlo simulations method is used as a computational technique to determine the numerical results for the performance characteristics, such as average run length (ARL), median run length, and standard deviation run length (SDRL) for assessing the performance of the proposed HEWMA1 and HEWMA2 charts. In addition, to evaluate the overall performance of the proposed HEWMA1 and HEWMA2 charts, other numerical measures consisting of the extra quadratic loss (EQL), relative average run length (RARL), and performance comparison index (PCI) are also computed. The proposed HEWMA1 and HEWMA2 charts are compared to some existing charts, such as CH, CEWMA, HEWMA, AEWMA HHW1, HHW2, AIB-EWMA-I, and AIB-EWMA-II charts, on the basis aforementioned numerical measures. The comparison reveals that the proposed HEWMA1 and HEWMA2 charts achieve better detection ability against the existing charts. In the end, a real-life data application is also provided to enhance the implementation of the proposed HEWMA1 and HEWMA2 charts practically.
Keywords: Average run length; extra quadratic loss; memory-type charts; Monte Carlo simulations; smoothing parameter
Control charts are the essential tools of the statistical process monitoring (SPM) toolkit, used to detect the shifts in manufacturing and production processes parameter(s). The control charts are generally classified into memory-type and memoryless-type charts . The memoryless-type charts are used only for current information of the process, while the memory-type charts are based on both current and previous information of the process. The basic memoryless-type charts are the Shewhart charts, like the Shewhart , , and charts, etc. . The Shewhart charts are simple and easy to apply; however, they are only efficient for the cases where large shifts occur in the process parameter(s). On the contrary, the memory-type charts, such as the exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts are sensitive in monitoring small-to-intermediate shifts in the process parameter(s).
Roberts  was the first to introduce the classical EWMA chart for monitoring the mean level of the process. Later, Hunter et al. [4–7] further investigated the various EWMA-type charts in order to facilitate the application of the classical EWMA chart in mean process monitoring. Recent studies have shown that EWMA-type charts are the most useful tools for researchers. For example, Haq , Abbas et al. , Ali et al. , Tang et al. , Haq , Rasheed et al. , Rasheed et al. , etc., are the few recent references in this regard.
In general, most manufacturing and production processes have a shift in the mean level; however, the process variance (or standard deviation) may be shifted from the target in many practical situations. Domangue et al.  suggested that monitoring an increase in process variance is more important for the processes. Although the EWMA-types charts are primarily used in mean process monitoring; however, few works address the variance monitoring via these charts. For example, Crowder et al.  used the logarithmic transformation to the sample variance in order to develop the EWMA chart (also known as the CH chart) for monitoring the process standard deviation. Similarly, Shu et al.  suggested the EWMA chart, denoted as the NEWMA chart, efficiently monitors process variance compared to the CH chart. Correspondingly, Huwang et al.  developed the EWMA-type charts for detecting shifts in the process variance and demonstrated that their control charts outperform the CH and NEWMA charts. Equally, Castagliola  used the three parameters logarithmic transformation of and proposed the bilateral EWMA chart to monitor the process variance shifts. Besides, Chang et al.  designed the optimal EWMA chart in order to monitor the process variance shifts. In addition, Razmy  offered the EWMA chart the monitors the standardized process variance. Furthermore, Haq  proposed two auxiliary information-based charts, symbolized by AIBEWMA1 and AIBEWMA2 charts, that monitor the process variance efficiently. Also, Ali et al.  suggested the generally weighted moving average (GWMA) and hybrid EWMA (HEWMA) chart to monitor process variance changes. Both GWMA and HEWMA perform better than classical memory charts. Other studies based on the EWMA-type charts for tracking the process variance are provided by Saghir et al. , Zaman et al. , Riaz et al.  and Chatterjee et al. , etc.
The use of hybrid charts enhances the efficiency of traditional charts. For example, Haq  and Haq  proposed the HEWMA chart to monitor the mean level of the process. Later on, numerous authors used the HEWMA charts in different process monitoring schemes. The HEWMA charts are more efficient than the classical EWMA and CUSUM charts in terms of small to moderate shifts monitoring. For example, Aslam et al.  introduced the HEWMA chart to monitor the mean level of the process under repetitive sampling. Similarly, Aslam et al.  monitored the COM-Poisson process by designing the HEWMA chart. Equally, Aslam et al.  developed the mixed chart, named the HEWMA-CUSUM chart, for the Weibull process monitoring. Correspondingly, Noor-ul-Amin et al.  recommended the HEWMA chart for Phase-II mean monitoring, based on the auxiliary information. Besides, Aslam et al.  suggested the HEWMA-p chart to monitor the variance of the non-normal process. Also, Noor et al.  constructed the Bayesian HEWMA chart, using two loss functions, to monitor the mean level of the normal process. The other studies about the chart are offered by Asif et al. , Noor-Ul-Amin et al. , etc.
The majority of manufacturing and service processes are affected by the gradual increase in process variance. The increase in the process variance indicates a deterioration in the process performance. This study's first and most important goal is to propose efficient charts with effective shifts detection ability in monitoring an increase in the process variances. So, motivated by Crowder et al. , Castagliola , and Haq , this study proposes two new HEWMA charts to monitor the increasing shifts in the process variance. The proposed charts are known as HEWMA1 and HEWMA2 charts. The design structure of the HEWMA1 chart uses the CH statistic as the input for the HEWMA1 statistic. In the same lines, the CEWMA statistic is considered an input to the HEWMA2 statistic to formulate the HEWMA2 chart. The Monte Carlo simulations are employed to compute the numerical results associated with average run length (), standard deviation run length (), extra quadratic loss (), relative average run length (), and performance comparison index () for the proposed HEWMA1 and HEWMA2 charts. Based on these measures, the proposed HEWMA1 and HEWMA2 charts are compared to the existing CH, CEWMA, HEWMA, AEWMA, HHW1, HHW2, AIBEWMA1, and AIBEWMA2 charts. The comparison shows that the proposed HEWMA1 and HEWMA2 charts have better detection ability to detect the shift in the process variance. Finally, two applications of the proposed HEWMA1 and HEWMA2 charts are provided, one with simulated data and the other with real-life data, to aid in the comparison of the proposed charts.
The remainder of the article is set out in the following way: Section 2 presents the existing methods. Likewise, Section 3 lays out the methodologies and formulation of the proposed HEWMA1 and HEWMA2 charts. In addition, the performance evaluation measures and simulation study are included in Section 4. Furthermore, Section 5 consists of the comparison and performance analysis of the proposed HEWMA1 and HEWMA2 charts against some existing charts. Section 6 offers a real-life data application to enhance the performance of the proposed HEWMA1 and HEWMA2 charts. The last section addresses the concluding remarks.
2 Existing Schemes
This section defines the process variable in Subsection 2.1. Similarly, Subsection 2.2 explains the details about the transformations to the sample variance. In addition, Subsections 2.3 to 2.6 provide the design and formulation of the CH, CEWMA, HHW2, and HEWMA charts for monitoring process variance, respectively.
2.1 Process Variable
Assuming, at the time , there are ; i.e., independent identically normal random variables with mean and variance and , i.e., for . As the only concern is to detect the increasing changes in , so the mean level of the process is assumed to be IC, i.e., . Suppose the underlying process variance remains IC for a particular time, i.e., for , then the process goes in OOC state, i.e., for . Let denotes the size of shifts in then it can be defined as a ratio IC process standard deviation to OOC process standard deviation, i.e., . So, whenever the underlying process is IC, and in the case of the OOC process. If be the sample mean and denotes the sample variance of the th subgroup, respectively, then for IC process follows a chi-square distribution with degrees of freedom, i.e., .
In order to implement the EWMA-type charts, the assumption of normality is required for the plotting statistic of the charts. However, because the sample variance has a chi-square distribution, so it is not an acceptable statistic for the design structure of the EWMA-type charts. In order to cope with this issue, a few transformations are available in the literature, given as follows.
Let denote a log transformation of , defined as
where the ratio is follows the gamma distribution with parameter and . According to Lawless , is a log-gamma random variable and it has a approximate normal distribution, i.e., , where and .
Castagliola  recommended the three-parameter logarithmic transformation of into a new variable, given as
where , and are the three constants of the transformation, defined as; , , , respectively. The values of , , , and based on the sample size are given by Castagliola . The statistic , in this case, is approximately normally distributed with and variance , i.e., .
In order to detect the shifts in the process variance, Quesenberry  suggested another transformation defined as
where , is chi-square variable with degrees of freedom, i.e., . The is the distribution function (DF) of the chi-square variable, while denotes the inverse DF of the standard normal variable. In this case, the statistic follows a standard normal distribution, i.e., .
2.3 CH Chart
Crowder et al.  introduced the CH chart, which monitored the process variance shifts. Let for be the sequence of IID random variable, defined on the sequence , where is defined by Eq. (1), then using the recurrence relationship, the CH plotting statistic can be given as
where is known as the smoothing parameter. The CH statistic mean and variance are, respectively, given as
In the case of large , the variance of is reduced to; . In order to detect the gradual rise in the process variance, let be the charting statistic for the CH chart is given by
The initial value of and is set on 0, i.e., . The upper control limit for the CH chart is denoted by and can be defined by
where is the CH chart width coefficients and can be computed so that IC \; is approximately equal to the desired value. The CH chart detects the upward shifts in the process whenever .
2.4 CEWMA Chart
Castagliola  proposed the EWMA chart (also known as -EWMA chart) to monitor the shifts in the process variance. Hereafter the -EWMA chart is labeled as the CEWMA chart. The CEWMA chart used the three parameters logarithmic transformation to to obtain the approximate normality for the plotting statistic. Let be the CEWMA sequence, based on another sequence for , where is defined by Eq. (2), then the charting statistic of the CEWMA chart is given by
The initial value of is denoted by and can be is defined as
The values for various can be taken from . The values are close to 0, so one can replace them with 0. Here, the plotting statistic is an approximate normal distributed variable having mean and variance , i.e., . However, for a large value of , the factor tends to 1, and in this case . If denotes the upper control limit for the CEWMA chart, then it can be defined as
where is called the width coefficient of the CEWMA chart. The CEWMA chart detects OOC signals with increasing shift whenever .
2.5 HHW2 Chart
Huwang et al.  proposed the EWMA chart to monitor the process variance, denoted as the HHW2 chart. Let , for be the HHW2 sequence that based on the IID sequence of random variable , where is defined in Eq. (3), then the HHW2 statistic based on the sequence is defined as
The initial value of and is set on 0, i.e., . The plotting statistic has a normal distribution with mean zero, and variance , i.e., . The time-dependent control limits, for the HHW2 chart can be defined as
In case of large values, the upper control limit for of the HHW2 chart is denoted as and can be given as
The is the HHW2 chart width coefficients. The HHW2 chart detects OOC signals whenever fall above the control limits specified in Eq. (12).
2.6 HEWMA Chart
Ali et al.  followed the idea of  and designed the HEWMA chart for process variance. Let the IID for is known as the HEWMA sequence, which is based on the , then the statistic for the HEWMA chart is given as
where is defined by Eq. (10) and is also a smoothing constant, such that . The initial values of are set to 0, i.e., . The mean and the variance of is given as
The control limit for the HEWMA chart, based on Eq. (14) is defined by
When gets larger, the control limits defined by Eq. (15) is reduced to and given as
where is the width coefficient for the HEWMA chart. The HEWMA chart triggers OOC signals whenever .
3 Proposed Methods
Haq  suggested the HEWMA chart to monitor the process mean. Similarly, Ali et al.  presented the HEWMA chart's design structure for tracking the process variance. Following Haq , the HEWMA1 and HEWMA2 charts can be developed using Transformations I and II, respectively. These charts detect increasing shifts in process variance. The methodologies and construction of the HEWMA1 and HEWMA2 charts are, respectively, presented in Subsections 3.1 and 3.2.
3.1 HEWMA1 Chart
The design structure for the HEWMA1 chart can be constructed using the CH statistic as input for the HEWMA1 statistic. Let defined the sequence of IID random variable, say for , based on the CH sequence then the HEWMA1 statistic can be defined by the relation given as
where is the CH statistic defined by Eq. (4). The starting value of is equal to 0, i.e., . The mean of is and its variance, for the case of very large , is defined as
In order to monitor the increasing shift in the process, the HEWMA1 statistic is defined as
The initial value of is set on , i.e., . The control limit for upper sided HEWMA1 is given as
where is the width coefficient for the HEWMA1 chart. The HEWMA1 chart detects OOC signals whenever . Similarly, for monitoring the gradual decrease in the variance, the HEWMA1 statistic is given by
The initial value of is denoted by set on , i.e., . The control limit for lower sided HEWMA1 is given as
The lower-sided HEWMA1 chart detects OOC signals whenever . The control limits defined in Eqs. (20) and (22) are called the HEWMA1 upper and lower control limits, respectively. However, the HEWMA1 two-sided control limits are given as
The two-sided HEWMA1 chart detects OOC signals whenever or .
3.2 HEWMA2 Chart
In order to formulate the design of the HEWMA2 chart, the charting statistic of the CEWMA chart can be used as an input for the HEWMA2 statistic. Let for be the HEWMA2 sequence, then the HEWMA2 chart statistic is , and it can be defined by
where is the HHW2 plotting statistic defined by Eq. (7). The starting values of is equal to defined by Eq. (8). The expected value for the statistic is and its variance is given as
The HEWMA2 control limits and are given as
The control limits defined in Eq. (26) are known as the two-sided time-dependent control limits; however, in the case of large values, the two-sided fixed HEWMA2 control limits are defined as
In this case, The HEWMA2 chart triggers OOC signals whenever or . The control limits specified in Eq. (27) are the two-sided control limits; however, the HEWMA2 upper control limit is defined as
The is known as the chart constants. The HEWMA2 chart detects OOC signals with increasing shift whenever .
4 Performance Analysis and Simulation Study
This section defines the performance evaluation measures, such as average run length in Subsection 4.1 and overall performance measures in Subsection 4.2. Similarly, the simulation study for the proposed HEWMA1 and HEWMA2 charts is designed in Subsection 4.3. Likewise, Subsection 4.4 provides the choices design parameters for the proposed HEWMA1 and HEWMA2 charts.
4.1 Average Run Length
The most popular and commonly used performance evaluation measures are the and measures. The can be defined as the average number of sample points until a chart indicates the OOC signal . The is further categorized as IC (, and OOC (. When a process is working in an IC state, the should be as large enough to prevent the frequent false alarms, while the should be smaller so that the shift is detected quickly . A chart with smaller is preferred over the competing charts at a prespecified .
4.2 Overall Performance Evaluation Measures
The and measures evaluate the performance of the charts on a single specified shift. However, sometimes the researcher may want to investigate the charts performances for the entire range of shifts, i.e., . For this purpose, the other performance measures, such as extra quadratic loss (), relative average run length (), and performance comparison index (), are used. The details on the EQL, RARL, and PCI measures are provided in the following subsections.
4.2.1 Extra Quadratic Loss
The can be considered as a weighted average of defined over the range of shifts to , using as a weight. Symbolically, the can be defined as
where is the at specific shift and and are the minimum and maximum shift values, respectively. A chart with a low value is considered to have a better overall detection ability .
4.2.2 Relative Average Run Length
Like the measure, the also evaluates the overall performance of the charts. The mathematically can be defined as
where is the value for benchmark chart at shift . A chart with a smaller at specific is known as a benchmark chart. The value for the benchmark chart is always equal to 1. The benchmark chart is considered superior to the competing chart if .
4.2.3 Performance Comparison Index
The also assesses the overall performance of the best chart. Ou et al.  defined the as a ratio of of the best chart to the of the benchmark chart. Mathematically, it can be given by the expression given as
The PCI value for the benchmark chart is equal to be 1, and for the rest of the charts, .
4.3 Monte Carlo Simulations
The random sample of size , i.e., for , is generated from a normal distribution under different parameter settings. Domangue et al.  suggested that monitoring a gradual rise (process deterioration) in the process variance is more important; therefore, an upward shift is considered in the process variance. The shift is reflected in the process standard deviation, i.e., , where 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0. The Monte Carlo simulation approach is utilized as a computational methodology for the numerical results by designing an algorithm in the statistical package . At each shift size , the simulations are performed with 20,000 replicates. The simulation algorithm for the HEWMA1 and HEWMA2 charts include the following steps:
i) Specify sample size , smoothing parameters and parameters of the process distribution, i.e., for IC process and for OOC process , where .
ii) Generate random observations for from .
iii) Compute the statistics, and in Eqs. (1) and (2), respectively.
iv) Using from , compute the EWMA statistics, and in Eqs. (4) and (7), respectively.
v) Using and , compute the HEWMA1 statistic and HEWMA2 statistic in Eqs. (19) and (24), respectively.
vi) Selected and for desired and compute and in Eqs. (20) and (28), respectively.
vii) Plot the statistic against the and the statistic against the . If and then record sequence order called the run length for the HEWMA1 and HEWMA2 charts, respectively.
viii) Repeat Steps (ii)–(vii) times and record run lengths and hence compute the approximate by and approximate by .
ix) For values generate random observations for from and repeat the Steps (iii)–(viii).
4.4 Choices of Design Parameters
The design parameters for the proposed HEWMA1 and HEWMA2 charts are the smoothing constants , and the chart width coefficient , which have a certain effect on the chart performance. Therefore, the different settings of the design parameters are used in computing and measures. The various combination of smoothing parameters are chosen as 0.05, , 0.20), ( 0.1, , 0.20), ( 0.2, , 0.200001), ( 0.3, , 0.2) to determine the values of and , so that . The numerical results regarding the proposed HEWMA1 and HEWMA2 charts are displayed in Tables 1–4.
5 Comparative Study
This section addresses the detailed comparative study of the proposed charts to the existing charts. The ARL values in Tables 1–4 reveal that the HEWMA2 chart outperforms the HEWMA1 chart; therefore, the HEWMA2 chart is recommended to compare with the existing charts for better detection performance. Thus the HEWMA2 charts is compared against the existing CH , CEWMA , HEWMA , AEWMA , AIBEWMA1, and AIBEWMA2  charts. Table 5 presents the ARL values for comparison, while Table 6 contains the overall performance values. The following Subsections offer further details about the comparisons.
5.1 Proposed vs. CH Chart
The proposed HEWMA2 chart achieves better performance against the CH chart. For example, at , with , 0.2, and , the proposed HEWMA2 charts provide the values 25.40 and 26.91, respectively, whereas the CH chart produces the 44.26, 46.63, respectively (see Table 5 & Fig. 1). Similarly, the proposed HEWMA2 chart indicates improved overall performance against the CH chart. As for and the , , and values of the proposed HEWMA2 charts are 19.5141, 1.0000, and 1.0000, which are less than the , , and values of the CH chart; 27.7505, 1.4221, and 1.8851 (see Table 6).
5.2 Proposed vs. CEWMA Chart
The proposed HEWMA2 chart shows lower values when it is compared to the CEWMA chart. For instance, assuming , with , 0.2, and the proposed HEWMA2 chart has values of 25.40 and 26.91, respectively, whereas the CEWMA chart owns the values of 31.80 and 37.80 (see Table 5 & Fig. 1). Similarly, in terms of overall performance (see Table 6), the proposed HEWMA2 charts attained smaller the , , and values, i.e., 20.1797, 1.0000, and 1.0000 against the CEWMA chart the , , and values, i.e., 23.5747, 1.1682, and 1.2818, respectively, when 0.2, and .
5.3 Proposed vs. HEWMA Chart
The proposed HEWMA2 chart achieves superior performance over the HEWMA chart. For instance, at , , and the proposed HEWMA2 chart gives the values 9.94, 5.73, 3.87, 2.87, whereas the HEWMA chart yields the 10.11, 5.75, 3.92, 2.96 (see Table 5 & Fig. 2). Similarly, the proposed HEWMA2 chart indicates improved overall performance against the HEWMA chart. For example, the proposed charts deliver the , , and values as 19.5141, 1.0000, 1.0000, respectively; however, the HEWMA chart provides the , , and values as 20.6981, 1.0607, and 1.1126, respectively (see Table 6).
5.4 Proposed vs. AEWMA Chart
Haq  developed the adaptive EWMA (AEWMA) chart for monitoring the process variance. The proposed HEWMA2 chart is compared to the AEWMA chart at and the results indicate that the proposed HEWMA1 and HEWMA2 chart has better detection ability against the AEWMA chart for the small shift, i.e., . For instance, with , , and , the proposed HEWMA-2 charts deliver the values as 25.40, 9.94, while the AEWMA chart has the values equal to 26.04, 10.35 (see Table 5 & Fig. 2). Likewise, when and , the overall performance of the proposed HEWMA2 chart is superior to the AEWMA chart as the proposed chart has a smaller than the AEWMA chart (see Table 6).
5.5 Proposed vs. AIBEWMA1 and AIBEWMA2 Charts
Haq  proposed the AIBEWMA1 and AIBEWMA2 charts for monitoring the process variance. The comparison of the HEWMA2 chart against the AIBEWMA1 and AIBEWMA2 charts demonstrates that the proposed HEWMA2 chart is more efficient than the AIBEWMA1 and AIBEWMA2 charts. For example, with chart properties, i.e., , , , and the values for the proposed HEWMA2 charts is observed as 25.40, while the values for the AIBEWMA1 and AIBEWMA2 charts are reported as 30.11 and 30.79 (see Table 5 & Fig. 3). Likewise, the proposed charts' , , and values also show the edge in the overall detection ability of the HEWMA2 chart over the AIBEWMA1 and AIBEWMA2 charts. As the , , and values for the proposed HEWMA2 charts are 19.5141, 1.0000, and 1.0000, respectively, where , , and values for the AIBEWMA1 and AIBEWMA2 charts are 20.8605, 1.0690 and 1.1185, and 21.4587, 1.0996 and 1.1919, respectively (see Table 6).
6 Important Points of the Study
A few important points related to the HEWMA1 and HEWMA2 charts can be listed as:
i) The HEWMA statistics undoubtedly boost the efficiency of the proposed HEWMA1 and HEWMA2 charts.
ii) The proposed HEWMA2 chart has better detection performance than the proposed HEWMA1 chart (see Tables 1–4).
iii) At different parametric settings, the values for the proposed HEWMA1 and HEWMA2 charts are less than the values of the CH, CEWMA, HEWMA, AEWMA, AIBEWMA1, and AIBEWMA2 charts (see Section 5).
iv) The overall performance reveals the dominance of HEWMA1 and HEWMA2 charts over the CH, CEWMA, HEWMA, AEWMA, AIBEWMA1, and AIBEWMA2 charts (see Section 5).
v) The proposed HEWMA1 and HEWMA2 charts have better performance for smaller and (see Tables 1–4).
vi) The control limit coefficient for the suggested HEWMA1 and HEWMA2 charts increases as and increases.
7 Real-life Application of the Proposed Charts
This subsection explains the application of the proposed HEWMA1 and HEWMA2 charts to real-life data. For this purpose, the real-life data are considered, representing the inside diameter of the cylinder bores in an engine block. These real-life data are used by [46,47] in their studies. The data comprise 30 samples, each size given in Table 7. In order to implement the proposed HEWMA1 and HEWMA2 control along with CH and CEWMA charts, following [23,48], an upward shift of size is introduced artificially after sample number 16 [49,50]. At , the smoothing parameter values , are used, which provides the width of the the HEWMA1, and HEWMA2, CH and CEWMA charts, respectively, given as 1.365, 1.399, 1.303 and 2.198. Using the aforementioned parameters, the charting statistics for the HEWMA1, and HEWMA2, CH and CEWMA charts and their corresponding upper control limits are computed. The charting statistics for HEWMA1, and HEWMA2, CH and CEWMA charts are given in Table 7, while their corresponding upper control limits are given as 0.1528, 0.1314, 0.2547, and 0.4876, respectively. Figs. 4–7 display the charting statistics of the CH, CEWMA, HEWMA1, and HEWMA2 charts against sample number. The results show that the proposed HEWMA1 chart outperforms the CH chart as the proposed HEWMA1 chart trigger the first OOC point after sample number 25, while the CH chart detects OOC point after sample number 28. Overall, the proposed HEWMA1 chart declares 7 OOC points, while the CH chart detects 2 OOC signals. Similarly, the HEWMA2 chart gains better detection ability relative to the CEWMA chart, as as the proposed HEWMA2 chart identifies the first OOC signal at sample number 26, while the CH chart diagnoses OOC signal at sample number 29. This indicates that the proposed HEWMA1 is more efficient than the CH chart, and the HEWMA2 chart achieves better detection ability than the CHWMA chart.
8 Concluding Remarks
This paper proposes two new hybrid EWMA charts to monitor the shifts in the process variance. The proposed charts are called HEWMA1 and HEWMA2 charts. The HEWMA1 chart is designed using the CH statistic as the input for the HEWMA1 statistic, while in the same lines, CEWMA statistic is used as the input for the HEWMA2 statistic to construct the proposed HEWMA2 chart. In order to evaluate the performance of the proposed HEWMA1 and HEWMA2 charts, the extensive Monte Carlo simulation approach is used to approximate the run length properties, including the average run length and standard deviation run length. Similarly, to assess the overall performances of the proposed HEWMA1 and HEWMA2 charts, the extra quadratic loss, relative average run length, and performance comparison index are computed. The proposed HEWMA1 and HEWMA2 charts are compared to existing CH, CEWMA, HEWMA, AEWMA, HHW1, HHW2, AIBEWMA1, and AIBEWMA2 charts, and the comparison indicates that the proposed HEWMA1 and HEWMA2 charts outperform the existing charts. In the end, real-life data are analyzed to enhance the efficiency of the proposed HEWMA1 and HEWMA2 charts.
Funding Statement: 2019 Shanxi Province Soft Science Research Program Project ``Research on Sustainable Development Capacity and Classification Construction of Shanxi Development Zone'' (Project No. 2019041005-2).
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
|This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.|