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 HEWMA ( 1 ) AIB and HEWMA ( 2 ) AIB control charts. A simulation study is carried out to assess the performance of the proposed HEWMA ( 1 ) AIB and HEWMA ( 2 ) AIB 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.


Introduction
ere are two types of variations in manufacturing and service processes; common (random) cause variations and special (assignable) cause variations. e 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. e statistical control chart is a primary tool in the statistical process control (SPC) toolkit that identi es and recti es the special cause variations in the process. Memory-less control charts introduced by Shewhart [1] 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 [2] and exponentially weighted moving average (EWMA) control chart o ered by Roberts [3] 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 [4] used a suitable log transformation to S 2 (ln(S 2 /σ 2 Y0 )), and designed the EWMA control chart for monitoring the process standard deviation, where σ 2 Y0 is the in-control (IC) process variance. Following the same lines, Shu and Jiang [5] suggested the new EWMA control chart in which ln(S 2 /σ 2 Y0 ) truncated to its IC mean on every occasion whenever it is less than zero. Similarly, Chang and Gan [6] constructed a one-sided optimal EWMA for monitoring the process variance. Likewise, Khoo [7] discussed the double sampling variance control chart features for monitoring process variability. Similarly, Castagliola [8] used three parameters logarithmic transformation to S 2 to improve normality and hence the proposed bilateral EWMA control chart for monitoring the process dispersion. Later, Castagliola, et al. [9] constructed a new S 2 -EWMA control chart for flexible sampling intervals. Likewise, Eyvazian [10] proposed the EWMSV control chart for process dispersion when the sample size equals 1. In the same direction, Huwang, et al. [11] suggested the EWMA control charts named HHW1 and HHW2 for the process dispersion.
ey demonstrated that their control charts perform uniformly better against the control charts by Crowder and Hamilton [4] and Shu and Jiang [5]. Razmy and Peiris [12] designed an standardized EWMA control chart for monitoring of process dispersion. Later, Yang and Arnold [13] constructed an unbiased EWMA-p control chart for monitoring the process dispersion. Currently, Castagliola, et al. [14] proposed the double sampling S 2 -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 [15], Ali and Haq [16] and Saghir, et al. [17], etc.
Combining the features of the different memory control charts enhances the performance of the ultimate control charts. For example, Haq [18] 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. [19] presented a HEWMA control chart for the process mean under repetitive sampling. Similarly, Aslam, et al. [20] proposed a HEWMA control chart for the COM-Poisson distribution. Likewise, Ali and Haq [16] suggested the generally weighted moving average and the HEWMA control charts for process dispersion. Recently, Chan, et al. [21] suggested improved double EWMA and homogeneously weighted moving average lepage schemes with real life applications. For a more detailed study, see Aslam, et al. [22], Noor-ul-Amin, et al. [23], Mukherjee, et al. [24], Song, et al. [25], and references therein.
Several auxiliary information-based (AIB) control charts have been suggested for efficient monitoring of process parameters. For example, Abbas, et al. [26] introduced AIB EWMA for the process mean, named as EWMA AIB control chart. Likewise, Haq [27] recommended two new AIB EWMA control charts named as EWMA-I AIB and EWMA-II AIB control charts that efficiently monitor the process dispersion. Similarly, Haq [28] provided AIB maximum EWMA control chart for process location and dispersion. Hussain, et al. [29] suggested EWMA control chart based on dual auxiliary informationbased estimator for the monitoring of process location. Similarly, Abbasi and Riaz [30] provides the control chart using dual auxiliary information under different ranked set sampling schemes. On the same lines, Riaz, et al. [31] 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. [32] suggested the Max EWMA AIB control chart for the simultaneous monitoring of the process mean and coefficient of variation. Recently, Anwar, et al. [33] introduced an AIB combined mixed EWMA-CUSUM control chart for joint monitoring of process paramters.
As mentioned before, Ali and Haq [16] 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 informationbased HEWMA, symbolized as HEWMA (1) AIB and HEWMA (2) AIB control charts to monitor the small shifts in the process dispersion. To evaluate the performance of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts against other control charts, specific performance evaluation measures such as average run length (ARL), extra quadratic loss (EQL), performance comparison index (PCI), and relative ARL (RARL) 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, EWMA-I AIB , and EWMA-II AIB 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. e 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 HEWMA (1) AIB and HEWMA (2) AIB control charts. Besides, Section 4 highlights the performance evaluation measures. Furthermore, Section 5 consists of the performance comparison of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts against the existing control charts. Similarly, the real-life application of the proposed control charts is provided in Section 6. e last section presents an overall summary and conclusions.

Existing Method
is section provides insight into the variable of interest and transformation in Subsection 2.1. Likewise, the methodology of the HEWMA control charts are presented in Subsection 2.2.

Variable of Interest and Transformation.
Suppose Y be normally distributed process variable, that is, . It is assumed that over a certain period, the underlying process remains IC with variance σ 2 Y , but afterwards, it becomes out-of-control (OOC) with variance σ 2 Y,1 . Let τ � σ Y,1 /σ Y be the amount of shift in process standard deviation σ Y . In the case of the IC process, τ � 1 and OOC process, τ ≠ 1. Also, Y represents the the sample mean and S 2 Y denotes sample variance of the process variable Y.

Transformation.
Suppose, X be an auxiliary information variable of Y variable, then X and Y follow a bivariate normal distribution. Suppose (Y i , X i ), for i � 1, 2, . . . , n be a random sample of size n. Let X and S 2 X be the sample mean and the sample variance of X, respectively. According to Garcia and Cebrian [34], the unbiased regression estimator of σ 2 Y say S 2 * Y , is given by Where ρ is the correlation coefficient. e mean and variance of S 2 * Y are given as Similarly, Haq [27] suggested the difference estimator for process dispersion given as where and t represents the sample number. Also, G(·; n − 1) is the cumulative distribution function (CDF) of chi-square distribution at n − 1 degrees of freedom, and Φ − 1 (·) denotes the inverse CDF of the standard normal distribution. e ρ * is the correlation between M Y and M X . e mean and variance of D t given by E(D t ) � 0 and V(D t ) � 1 − ρ * .

HEWMA Control Chart for Process Dispersion. Ali and
Haq [16] proposed the HEWMA control chart for the monitoring of process dispersion. Let HE t for t ≥ 1 be the sequence of independentely and identically distributed (IID) observations based on the other sequence M Y,t , then the plotting statistic HE t , for the HEWMA control chart is defined as: where HE 0 � E 0 � 0, and λ 1 and λ 2 are smoothing constants. e mean and variance of HE t are, respectively, given ]. e lower and upper control limits of the HEWMA control chart at the time t, are presented as (4) where L HEWMA control chart coefficient is used to adjust the IC ARL of the HEWMA control chart at a pre-specified desired level. e HEWMA control chart triggers an OOC signal whenever HE t fall outside of the control limits (UCL (HEWMA)t , LCL (HEWMA)t ).

Proposed Methods
is section contains the methodology of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts for monitoring the process dispersion. Subsection 3.1 covers the design structure of the proposed HEWMA (1) AIB control chart, whereas, the HEWMA (2) AIB control chart are given in Subsection 3.2. (1) AIB Control Chart. Let S 2 * Y,t for t ≥ 1 be the sequence of IID random variables, then the plotting statistic H 1,t , for HEWMA (1) AIB control chart using the recurrence formula, given by

HEWMA
Here λ 1 and λ 2 ∈∈(0, 1] are smoothing constants. e mean and variance of the H 1,t can be given by the expression as Y , respectively. e control limits of the proposed HEWMA (1) AIB control chart are given by where L HEWMA (1) AIB is the coefficient for the HEWMA (1) AIB control chart at a pre-specified false alarm rate. e H 1,t statistic is plotted against the UCL (HEWMA (1) AIB )t and LCL (HEWMA (1) AIB )t . e process is considered to be OOC when H 1,t > UCL (HEWMA (1) AIB )t or H 1,t < LCL (HEWMA (1) AIB )t ; otherwise, it is IC.

HEWMA (2)
AIB Control Chart. Let D t for t ≥ 1 be the sequence of IID random variables, based on D t , we defined a new sequence H 2,t , using the recurrence formula, given by Mathematical Problems in Engineering 3 Here λ 1 and λ 2 ∈∈(0, 1] are smoothing constants. e mean and variance of H 2,t can be given by the expression as e control limits for the HEWMA (2) AIB control chart are given by where L HEWMA (2) AIB is the control chart coefficient for HEWMA (2) AIB control chart at a pre-specified false alarm rate. e HEWMA (2) AIB statistic H 2,t is plotted against the LCL (HEWMA (2) AIB )t and LCL (HEWMA (2) AIB )t , the process is considered to be OOC when H 2,t < LCL (HEWMA (2) AIB )t or H 2,t > UCL (HEWMA (2) AIB )t ; otherwise, IC. e ARL is categorized as IC ARL (ARL 0 ) and OOC ARL (ARL 1 ). If the process is IC state, the ARL 0 needed to be large enough to avoid frequent false alarms. However, the ARL 1 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 ARL 1 with fixed ARL 0 at the desired level.

Overall Performance Measures.
e EQL, RARL, and PCI performance evaluation measures evaluate a control chart's overall effectiveness by comparison method. e EQL evaluates the overall performance of control charts over a specific range of shifts (Raza,et al. [35]). It is based on the loss function and is defined as: where ARL(τ) is the ARL of a particular control chart at shift τ. e EQL is a weighted average ARL over the entire shift domain (τ min < τ < τ max ) using the square of shift (τ 2 ) as weight. A control chart with a minimum EQL value is preferred over other control charts (Anwar, et al. [36]). e RARL is the average of the ratios among the ARL of a particular control chart with the ARL of a benchmark control chart for all desired shifts.
where ARL(τ) and ARL BM (τ) symbolize the ARL of a particular control chart and a benchmark control chart for the desired shift, respectively. e benchmark control chart is the control chart with the least EQL. e RARL value for the benchmark control chart is one, and for the other control charts, it is greater than 1. e PCI evaluates the performance of the best control chart. It is defined as the ratio between the EQL of a control chart and the EQL of the benchmark control chart.
where EQL BM is the EQL of the best-performing control chart. e PCI for the benchmark control chart is 1, while the other control chart's PCI is greater than 1.

Proposed versus HHW1 and HHW2 Control Charts.
e proposed HEWMA (1) AIB and HEWMA (2) AIB control charts provide better performance against the HHW1 and HHW2 control charts. For example, at (λ, λ 2 ) � 0.1,  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 HEWMA (1) AIB and HEWMA (2) AIB control charts are superior to the HHW1 and HHW2 charts. For instance, the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts have EQL 13.905 and 13.918, whereas the HHW1 and HHW2 control charts have EQL values 23.413 and 21.954, respectively (see Table 10).

Main
Outcomes of the Study. Some interesting outcomes of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts are listed as follows: (i) e use of Hybrid EWMA statistic certainly boosts the detection ability of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts. (ii) e performance of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts improve with the induction of suitable auxiliary information in the model.

Real-Life Application
To demonstrate the practical implementation of the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts, a real-life data set of glass thickness (X), and its impact on the stress strength (Y) of glass bottles is considered from Asadzadeh and Kiadaliry [38]. is data set contains 40 samples, each of size 5, of stress strength (kg/cm 2 ), thickness (cm). e 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. erefore, 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. e estimates of process parameters are given as: μ Y � 6.36, σ Y � 8.92, μ X � 1.38, σ X � 0.62, and ρ � 0.905. 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. [39], the Y is multiplied by 1.3 for the OOC scenario. e parameters of proposed HEWMA (1) AIB and HEWMA (2) AIB control charts are set on λ 1 � 0.2, λ 2 � 0. e EWMA-I AIB , EWMA-II AIB , proposed HEWMA (1) AIB , and proposed HEWMA (2) AIB control charts detect the first OOC signal at sample number 29 (see Figures 6-9). Overall, the existing EWMA-I AIB control chart detects 2 OOC signals while the HEWMA (1) AIB control chart detects 8 OOC signals. In the same manner, the EWMA-II AIB control chart detects a total of 2 OOC signals, and the HEWMA (2) AIB control chart detects 5 OOC signals. is indicates that the proposed HEWMA (1) AIB and HEWMA (2) AIB control charts are more efficient than the existing EWMA-I AIB and EWMA-II AIB control charts.

Concluding remarks
is study presented the two new auxiliary informationbased hybrid EWMA control charts, named HEWMA (1) AIB and HEWMA (2) AIB control charts for process dispersion. e HEWMA (1) AIB control chart used the auxiliary information through the regression estimator for the population variance, whereas the HEWMA (2) AIB control chart used the auxiliary information through the difference estimator. e HEWMA (1) AIB and HEWMA (2) AIB control charts are constructed by combining the features of the AIB dispersion estimators with the HEWMA control chart. e 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 HEWMA (1) AIB and HEWMA (2) AIB 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. is work can be extended to a non-normal process(s) and a multivariate case.
Data Availability e data is available in the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

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Mathematical Problems in Engineering