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Efficient Auxiliary Information–Based Control Charting Schemes for the Process Dispersion with Application of Glass Manufacturing Industry
The hybrid exponentially weighted moving average (HEWMA) control chart is an enhanced version of the EWMA control chart that monitors the process parameters effectively. Similarly, the auxiliary information-based (AIB) EWMA control charts are very efficient for monitoring process parameters. The purpose of this paper is to propose two new control charts for the improved monitoring of process dispersion referred to as and control charts. A simulation study is carried out to assess the performance of the proposed and control charts. Average run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the performance of the proposed control charts against the existing counterparts. The comparisons reveal the superiority of the proposed control charts against other competing control charts, particularly for small to moderate shifts in the process dispersion. Finally, a real-life data set from the glass industry is used to demonstrate the practical implementation of the proposed control charts.
There are two types of variations in manufacturing and service processes; common (random) cause variations and special (assignable) cause variations. The common cause variations are an inherent part of every process and cannot be removed entirely. However, the special cause variations are harmful and may distract the processes from their target which results a shifts in the process parameter(s) (location and/or scale). As a result, the practitioner needs to identify and eliminate the assignable cause variations in the process. The statistical control chart is a primary tool in the statistical process control (SPC) toolkit that identifies and rectifies the special cause variations in the process. Memory-less control charts introduced by Shewhart  are used to monitor large shifts in the process. On the other hand, the classical memory control charts like cumulative sum (CUSUM) control chart designed by Page  and exponentially weighted moving average (EWMA) control chart offered by Roberts  are used to monitor small to moderate shifts in the process.
Generally, the classical EWMA control chart has been used to detect small shifts in the process mean. However, in many practical situations, the shifts may also occur in the process variance (or standard deviation); when the process variance increases, the productivity and capability of the process may be damaged. If the process variance decreases, more units will be closer to their target value, resulting in improved process functionality. If these changes in the process dispersion are not rectified quickly, unnecessary losses may occur. Numerous authors have constructed different EWMA control charts for the process variance. For example, Crowder and Hamilton  used a suitable log transformation to , and designed the EWMA control chart for monitoring the process standard deviation, where is the in-control (IC) process variance. Following the same lines, Shu and Jiang  suggested the new EWMA control chart in which truncated to its IC mean on every occasion whenever it is less than zero. Similarly, Chang and Gan  constructed a one-sided optimal EWMA for monitoring the process variance. Likewise, Khoo  discussed the double sampling variance control chart features for monitoring process variability. Similarly, Castagliola  used three parameters logarithmic transformation to to improve normality and hence the proposed bilateral EWMA control chart for monitoring the process dispersion. Later, Castagliola, et al.  constructed a new -EWMA control chart for flexible sampling intervals. Likewise, Eyvazian  proposed the EWMSV control chart for process dispersion when the sample size equals 1. In the same direction, Huwang, et al.  suggested the EWMA control charts named HHW1 and HHW2 for the process dispersion. They demonstrated that their control charts perform uniformly better against the control charts by Crowder and Hamilton  and Shu and Jiang . Razmy and Peiris  designed an standardized EWMA control chart for monitoring of process dispersion. Later, Yang and Arnold  constructed an unbiased EWMA-p control chart for monitoring the process dispersion. Currently, Castagliola, et al.  proposed the double sampling -chart for the process variance and elaborated various features of the proposed control chart. Further related work on dispersion control charts, see Abbasi and Miller , Ali and Haq  and Saghir, et al. , etc.
Combining the features of the different memory control charts enhances the performance of the ultimate control charts. For example, Haq  has constructed a new hybrid exponentially weighted moving average (HEWMA) control chart using one EWMA statistic as an input for another EWMA statistic. Subsequently, several other authors discussed some of the innovations in the HEWMA control chart. For instance, Azam, et al.  presented a HEWMA control chart for the process mean under repetitive sampling. Similarly, Aslam, et al.  proposed a HEWMA control chart for the COM-Poisson distribution. Likewise, Ali and Haq  suggested the generally weighted moving average and the HEWMA control charts for process dispersion. Recently, Chan, et al.  suggested improved double EWMA and homogeneously weighted moving average lepage schemes with real life applications. For a more detailed study, see Aslam, et al. , Noor-ul-Amin, et al. , Mukherjee, et al. , Song, et al. , and references therein.
Several auxiliary information-based (AIB) control charts have been suggested for efficient monitoring of process parameters. For example, Abbas, et al.  introduced AIB EWMA for the process mean, named as control chart. Likewise, Haq  recommended two new AIB EWMA control charts named as - and - control charts that efficiently monitor the process dispersion. Similarly, Haq  provided AIB maximum EWMA control chart for process location and dispersion. Hussain, et al.  suggested EWMA control chart based on dual auxiliary information-based estimator for the monitoring of process location. Similarly, Abbasi and Riaz  provides the control chart using dual auxiliary information under different ranked set sampling schemes. On the same lines, Riaz, et al.  suggested variability control chart using dual auxiliary information-based estimators under different ranked set sampling techniques and different runs rules. Besides, Noor‐ul‐Amin, et al.  suggested the control chart for the simultaneous monitoring of the process mean and coefficient of variation. Recently, Anwar, et al.  introduced an AIB combined mixed EWMA-CUSUM control chart for joint monitoring of process paramters.
As mentioned before, Ali and Haq  proposed the HEWMA control chart for the process dispersion which is more sensitive than the classical EWMA control chart. Sometimes, researchers, engineers, and practitioners are interested in utilizing the features of HEWMA control charts when the original variable carries other information, such as an auxiliary variable, to improve the process’s effectiveness. In this case, the HEWMA control chart will remain inefficient. So, to address this deficiency, this study introduces two auxiliary information-based HEWMA, symbolized as and control charts to monitor the small shifts in the process dispersion. To evaluate the performance of the proposed and control charts against other control charts, specific performance evaluation measures such as average run length (), extra quadratic loss (), performance comparison index (), and relative () measures are considered. Besides, an algorithm is designed in R using the Monte Carlo simulations method to calculate the performance evaluation measures. Existing control charts such as HEWMA, adaptive EWMA (AEWMA), HHW1, HHW2, -, and - control charts are considered for comparison. Moreover, the proposed control charts are also implemented with real-life applications to show the significance for practical importance.
The article’s remainder is organized as follows: variable of interest, auxiliary information, transformation based on auxiliary information, and the existing HEWMA control chart are highlighted in Section 2. Section 3 presents the design structure of the proposed and control charts. Besides, Section 4 highlights the performance evaluation measures. Furthermore, Section 5 consists of the performance comparison of the proposed and control charts against the existing control charts. Similarly, the real-life application of the proposed control charts is provided in Section 6. The last section presents an overall summary and conclusions.
2. Existing Method
2.1. Variable of Interest and Transformation
Suppose be normally distributed process variable, that is, . It is assumed that over a certain period, the underlying process remains IC with variance , but afterwards, it becomes out-of-control (OOC) with variance . Let be the amount of shift in process standard deviation . In the case of the IC process, and OOC process, . Also, represents the the sample mean and denotes sample variance of the process variable .
Suppose, be an auxiliary information variable of variable, then and follow a bivariate normal distribution. Suppose , for be a random sample of size . Let and be the sample mean and the sample variance of , respectively. According to Garcia and Cebrian , the unbiased regression estimator of say , is given byWhere is the correlation coefficient. The mean and variance of are given as , .
Similarly, Haq  suggested the difference estimator for process dispersion given aswhere , and represents the sample number. Also, is the cumulative distribution function (CDF) of chi-square distribution at degrees of freedom, and denotes the inverse CDF of the standard normal distribution. The is the correlation between and . The mean and variance of given by and .
2.3. HEWMA Control Chart for Process Dispersion
Ali and Haq  proposed the HEWMA control chart for the monitoring of process dispersion. Let for be the sequence of independentely and identically distributed (IID) observations based on the other sequence , then the plotting statistic , for the HEWMA control chart is defined as:where , and and are smoothing constants. The mean and variance of are, respectively, given as and The lower and upper control limits of the HEWMA control chart at the time , are presented aswhere control chart coefficient is used to adjust the IC of the HEWMA control chart at a pre-specified desired level. The HEWMA control chart triggers an OOC signal whenever fall outside of the control limits .
3. Proposed Methods
This section contains the methodology of the proposed and control charts for monitoring the process dispersion. Subsection 3.1 covers the design structure of the proposed control chart, whereas, the control chart are given in Subsection 3.2.
3.1. Control Chart
Let for be the sequence of IID random variables, then the plotting statistic , for control chart using the recurrence formula, given by
Here and are smoothing constants. The mean and variance of the can be given by the expression as and , respectively. The control limits of the proposed control chart are given bywhere is the coefficient for the control chart at a pre-specified false alarm rate. The statistic is plotted against the and . The process is considered to be OOC when or ; otherwise, it is IC.
3.2. Control Chart
Let for be the sequence of IID random variables, based on , we defined a new sequence , using the recurrence formula, given by
Here and are smoothing constants. The mean and variance of can be given by the expression as and . The control limits for the control chart are given by where is the control chart coefficient for control chart at a pre-specified false alarm rate. The statistic is plotted against the and , the process is considered to be OOC when or ; otherwise, IC.
4. Performance Evaluation Measures
Quality experts use different performance evaluation measures to evaluate the control charts’ performance. is mostly used for a single shift, while , , and are used to assess the overall performance of control charts. An algorithm is developed in software, and the Monte Carlo simulation technique is used to compute the numerical results. Monte Carlo simulation with 20000 iterations is performed at each shift , where 0.5, 0.6, 0.7, 0.8 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0. The details of these performance measures are given in the following subsections.
The is defined as the average number of sample points plotted until the OOC signal is detected. The is categorized as IC and OOC (). If the process is IC state, the needed to be large enough to avoid frequent false alarms. However, the should be small enough that it quickly detects the shift(s) in the process parameters. It is necessary for the better performance of the control chart that it should have a smaller with fixed at the desired level.
4.2. Overall Performance Measures
The , , and performance evaluation measures evaluate a control chart’s overall effectiveness by comparison method. The evaluates the overall performance of control charts over a specific range of shifts (Raza, et al. ). It is based on the loss function and is defined as:where is the of a particular control chart at shift . The is a weighted average over the entire shift domain using the square of shift as weight. A control chart with a minimum value is preferred over other control charts (Anwar, et al. ).
The is the average of the ratios among the of a particular control chart with the of a benchmark control chart for all desired shifts.where and symbolize the of a particular control chart and a benchmark control chart for the desired shift, respectively. The benchmark control chart is the control chart with the least . The value for the benchmark control chart is one, and for the other control charts, it is greater than 1.
The evaluates the performance of the best control chart. It is defined as the ratio between the of a control chart and the of the benchmark control chart.where is the of the best-performing control chart. The for the benchmark control chart is 1, while the other control chart’s is greater than 1.
4.3. Choices of Parameters
The parameters of the proposed and control charts are , and . Several settings of these parameters are used, and hence corresponding , and standard deviation of RL () are computed. The values of are set as (0.05,0.05), (0.05,0.1), (0.05,0.2), (0.1,0.05), (0.1,0.1), (0.1,0.2), (0.2,0.05), (0.2,0.1), (0.2,0.2), (0.3,0.05), (0.3,0.1), and (0.3,0.2) and the values of and are determined to obtain . The different settings of and are taken from the Haq  given as (0.25, 0.0563898), (0.50, 0.2293317), (0.75, 0.5313626), (0.90, 0.7870992), (0.95, 0.8880799). The numerical results of the proposed and control charts are presented in Tables 1–8.
5. Evaluation and Performance Comparison
This section provides extensive comparisons of the proposed and control charts with the HEWMA (Ali and Haq ), AEWMA (Haq ), HHW1 and HHW2 (Huwang, et al. ), and - and - (Haq ) control charts.
5.1. Proposed versus HEWMA Control Chart
The proposed and control charts are compared with the HEWMA control chart. The proposed control charts outperform the HEWMA control chart. For instance, at , , , and the proposed and control charts provides the values are (23.69, 9.05) and (25.05, 9.75), respectively, whereas the HEWMA control chart has the values equal to (25.79, 10.15) (see Tables 3, 7 versus 9). Similarly, the proposed control charts’ superiority over the HEWMA control chart can also be visualized in Figure 1. Besides, the overall performance measures confirm the better performance of the proposed and control charts than the HEWMA control chart. For example, the proposed and control charts have , , and values are 13.905,1,1 and 13.918, 1.001,0.989, respectively, whereas the HEWMA control chart has , , and values are 19.752, 1.421,2.273 (see Table 10).
5.2. Proposed versus AEWMA Control Chart
The proposed and control charts are more efficient than the AEWMA control chart. For example, with , , , and , the proposed and control charts hold the values 21.21 and 22.88, respectively, while the value with the AEWMA control chart is 26.04 (see Tables 4, 7 versus 9, and Figure 1). Likewise, the and control charts have smaller , , and values than the AEWMA control chart. For instance, the proposed and control charts provide , , and values are 13.905, 1.000, 1.000 and 13.918, 1.001, 0.989, respectively, while the AEWMA control chart produces , , and values are 19.609, 1.41, 2.234 (see Table 10).
5.3. Proposed versus HHW1 and HHW2 Control Charts
The proposed and control charts provide better performance against the HHW1 and HHW2 control charts. For example, at , , , and , the proposed and control charts attained the values as (23.69, 9.05) and (25.05, 9.75), respectively, while the HHW1 and HHW2 control charts have the values as (34.52, 14.09) and (32.11, 12.72), respectively (see Tables 3, 7 versus 9). Furthermore, Figure 2 also shows the superiority of the proposed control charts over the HHW1 and HHW2 control charts. In terms of overall effectiveness (see Table 10), the and control charts are superior to the HHW1 and HHW2 charts. For instance, the proposed and control charts have 13.905 and 13.918, whereas the HHW1 and HHW2 control charts have values 23.413 and 21.954, respectively (see Table 10).
5.4. Proposed versus - and - Control Charts
The proposed and control charts attain outstanding performance against the - and - control charts. For example, when , and , the values of proposed and control charts are 7.57 and 8.52, whereas, the values of - and - control charts are 9.73 and 10.96, respectively (see Tables 4, 7 versus 9). Additionally, the dominance of the proposed and control charts over the - and - control charts can be seen in Figure 3. For overall performance, the proposed and control charts have smaller , , and values than the , , and values of the - and - control charts, respectively (see Table 10).
5.5. Main Outcomes of the Study
Some interesting outcomes of the proposed and control charts are listed as follows:(i)The use of Hybrid EWMA statistic certainly boosts the detection ability of the proposed and control charts.(ii)The performance of the proposed and control charts improve with the induction of suitable auxiliary information in the model.(iii)The values of the proposed and control charts are smaller than HEWMA, AEWMA, HHW1, HHW2, -, and - control charts.(iv)The overall performance evaluation measures show the dominance of the and control charts against other control charts (see Subsections 5.1–5.4).(v)The proposed and control charts provide the best performance for larger values of (see Figures 4 and 5).(vi)The performance of the proposed and control charts are increased for smaller values of and (see Tables 1–8).
6. Real-Life Application
To demonstrate the practical implementation of the proposed and control charts, a real-life data set of glass thickness (), and its impact on the stress strength () of glass bottles is considered from Asadzadeh and Kiadaliry . This data set contains 40 samples, each of size 5, of stress strength (), thickness (cm). The proposed control charts are constructed under the assumption of known parameters. However, in real-life data application of the proposed control chart, the population parameters are not available. Therefore, for the practical implementation of the control charts, the estimated parameters are used for the empirical quantification of the quantities required to show the proposed control charts’ implementation. The estimates of process parameters are given as: , , , , and . In the data set, the first 20 samples are treated as IC, while the rest of the 20 samples are considered OOC. Following Anwar, et al. , the is multiplied by 1.3 for the OOC scenario.
The parameters of proposed and control charts are set on , , , and with . Similarly, the control chart parameters of the existing - and - control charts are at and .
The -, -, proposed , and proposed control charts detect the first OOC signal at sample number 29 (see Figures 6–9). Overall, the existing - control chart detects 2 OOC signals while the control chart detects 8 OOC signals. In the same manner, the - control chart detects a total of 2 OOC signals, and the control chart detects 5 OOC signals. This indicates that the proposed and control charts are more efficient than the existing - and - control charts.
7. Concluding remarks
This study presented the two new auxiliary information-based hybrid EWMA control charts, named and control charts for process dispersion. The control chart used the auxiliary information through the regression estimator for the population variance, whereas the control chart used the auxiliary information through the difference estimator. The and control charts are constructed by combining the features of the AIB dispersion estimators with the HEWMA control chart. The proposed control charts’ performance based on average run length, extra quadratic loss, relative average run length, and performance comparison index measures reveal the superiority over the competitive control charts. It is worth mentioning that the proposed and control charts performed very well to monitor small to moderate shifts in process dispersion, especially for large correlation coefficient values. Besides, a real-life application is also provided for users and practitioners to demonstrate the implementation of the proposed study from a practical perspective. This work can be extended to a non-normal process(s) and a multivariate case.
The data is available in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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