The recently introduced triple exponentially weighted moving average (TEWMA) chart is the extended form of the classical EWMA and double EWMA (DEWMA) charts. On the other hand, the auxiliary information-based (AIB) homogeneously weighted moving average () chart is used for the monitoring of process location shifts efficiently as compared to the HWMA chart. Combining TEWMA with HWMA features is a new idea and is not seen in the literature until now. So, the objective of this study is to combine the idea of and TEWMA charts and propose an AIB triple HWMA, symbolized as chart to further improve the process location shift monitoring. The proposed chart is developed by combining the AIB double HWMA plotting statistic into the other HWMA chart. The numerical results are computed using the Monte Carlo simulation method. Average run length (), relative , performance comparison index, and extra quadratic loss are used as comparison tools for the proposed chart with the classical EWMA, TEWMA, mixed HWMA-CUSUM (MHC), AIB EWMA (), HWMA, double HWMA (DHWMA), , AIB mixed EWMA-CUSUM (), and AIB mixed CUSUM-EWMA () charts. Finally, a practical application is also provided for users to demonstrate the proposed study’s vitality.

1. Introduction

Variations are an integral part of all kinds of production and non-production processes. These variations are categorized as common and special variations. Generally, these special variations can result in process parameters (location and/or dispersion) shifts. These shifts are classified into three sizes as small, moderate, and large. Control charts are widely used in the statistical process control (SPC) toolkit to identify these special variations. For tracking large shifts, Shewhart charts [1] are widely used; however, for small to moderate shifts, cumulative sum (CUSUM) [2] and exponentially weighted moving average (EWMA) [3] are used.

In the SPC literature, many enhancements and modifications on control charts continuously fulfil the practitioner’s requirement of quickly detecting shifts. In this regard, Shamma and Shamma [4] extended the classical EWMA chart and suggested an improved double EWMA (DEWMA) chart. The DEWMA chart is more responsive than the classical EWMA chart. Alevizakos, et al. [5] proposed a triple EWMA (TEWMA) chart, the extended version of the EWMA and DEWMA charts for process location. The TEWMA chart is handy for detecting smaller process location shifts. After that, Alevizakos, et al. [6] recommended one-sided and two-sided TEWMA charts for the time between events, regarded as charts, and showed that the charts are more sensitive than the and charts. Similarly, Sukparungsee, et al. [7], Taboran, et al. [8], and Talordphop, et al. [9] suggested mixed EWMA-moving average (MA), Tukey MA-DEWMA, modified EWMA-MA control charts for process location, respectively. Also, Taboran, et al. [10] introduced non-parametric Tukey MA-EWMA control chart for detecting process location shifts. To monitor process dispersion, Chatterjee, et al. [11] developed a TEWMA chart using three parameters of logarithmic transformation to and named as -TEWMA chart. Subsequently, Alevizakos, et al. [12] formulated a non-parametric TEWMA sign chart for process median. Recently, Alevizakos, et al. [13] suggested non-parametric TEWMA sign ranked charts. Similarly, Rasheed, et al. [14] introduced Nonparametric Triple EWMA Wilcoxon Signed-Rank Control Chart for process location.

Similarly, using auxiliary information with the original study variable enhances the control chart’s detection ability [15]. Many researchers identified various features of the auxiliary information-based (AIB) memory charts. For illustration, Abbas, et al. [16] initiated an AIB EWMA chart to monitor process location. Likewise, Adegoke, et al. [17] introduced an AIB EWMA chart for monitoring process location shifts using various sampling schemes. Similarly, Noor-ul-Amin, et al. [18] presented the AIB HEWMA () chart for the phase-II process monitoring of the location parameter. Abbasi and Haq [15] recommended AIB optimal and adaptive CUSUM chart for process location. Similarly, Haq and Khoo [19] designed the AIB multivariate chart for the mean vector. Subsequently, Anwar, et al. [20] and Aslam and Anwar [21] provided AIB modified-EWMA and Bayesian modified-EWMA charts to improve process location monitoring. Also, to monitor process location, Anwar, et al. [22] formulated two new charts using auxiliary information named mixed EWMA-CUSUM () and mixed CUSUM-EWMA charts. Recently, a combined chart is proposed for the simultaneous monitoring of process location and dispersion parameters [23]. Haq, et al. [24] suggested adaptive CUSUM and EWMA under variable sampling intervals using auxiliary information. More details of AIB memory charts are available in Lee, et al. [25], Haq [26], Haq [27], Adegoke, et al. [28] and the references therein.

Hunter [29] points out the drawback of classical EWMA statistic is that the freshest observations are given more weight than previous observations. To resolve this drawback, Abbas [30] proposed a homogeneous weighted moving average (HWMA) chart that assigns particular weights to recent process values and homogeneously allocates the remaining weights to the old process values. This approach enhances the effectiveness of the HWMA chart as compared to its competitor’s charts. Later Adegoke, et al. [28] extended the Abbas [30] work to propose the AIB HWMA () chart is more effective for smaller shift monitoring. Subsequently, Adeoti and Koleoso [31] and Abid, et al. [32] enhanced the existing work by introducing a hybrid-HWMA (HHWMA) and double HWMA (DHWMA) charts, respectively. Also, Thanwane, et al. [33] presented an HWMA chart for the autocorrelated process under the assumption of estimated parameters. Unlike process location monitoring, Riaz, et al. [34] utilized the HWMA chart concept for efficient process dispersion monitoring. Recently, Abid, et al. [35] introduced a mixed HWMA-CUCUM (MHC) chart for process location for efficient process monitoring. Also, Rasheed, et al. [36] introduced homogeneously mixed memory control chart for process location.

Motivated by the extraordinary performances of TEWMA and charts, we are aimed to combine the features of TEWMA and charts and proposed a more efficient AIB triple HWMA (symbolized as ) chart for process location. The merging of TEWMA with HWMA features in the presence of auxiliary information is a new idea and is not seen in the literature till now and it is expected that the combining of TEWMA and charts features will further boost the performance of proposed chart. Average run length (), relative (), extra quadratic loss (), and performance comparison index () are used for the performance comparison with existing counterparts. Besides, the Monte Carlo simulation method is used for obtaining numerical measures. The classical EWMA, TEWMA, MHC, , HWMA, DHWMA, , , and charts are considered for the comparison. Additionally, the proposed chart is implemented in a real-world scenario to demonstrate its utility in practice.

The remainder of the research article is structured as follows: existing charts are described in Section 2. Additionally, Section 3 enlisted the design structure and special cases of proposed chart. Furthermore, the next section provides the performance evaluation and comparison against classical EWMA, TEWMA, MHC, , HWMA, DHWMA,, , and charts. Also, the application of chart is included in Section 5. The last section describes the overall summary, conclusions, and recommendations.

2. Existing Methods

This section describes the detailed methodology of the existing and TEWMA charts for process location monitoring.

2.1. Variable of Interest and Auxiliary Information

Assume that the process variable is normally distributed with a mean and variance . Let and represents the sample mean and variance of of size . If , the process is in-control (IC); otherwise, it is out-of-control (OOC). So, for the IC situation, and are mutually independent identically distributed. Let be an auxiliary variable of . The and follow a bivariate normal distribution (BND) (i.e., , where represents the mean and represents the standard deviation of . Also, the is the correlation coefficient corresponding to and . The AIB regression estimator for process location is given as follows:where, , and .

2.2. Chart

Adegoke, et al. [28] introduced chart to track changes in process location. The chart is the extended form of HWMA chart, used when interest variable is observed along with auxiliary information variable. The plotting statistic of the HWMA chart given as:where is the sample average of sample and is the smoothing constant, and is the mean of preceding observations The control limits of the chart are:

The represents the coefficient of control limits. If or , the process is OOC; otherwise, IC.

2.3. TEWMA Chart

Alevizakos et al. [5] designed a TEWMA chart for efficient process location monitoring. The plotting statistics of the TEWMA chart are defined aswhere is a TEWMA constant. The initial values of are all equal to . The mean and variance of the are. andrespectively, where . The time-varying control limits of the TEWMA chart are defined as:where is the control limits coefficient of the TEWMA chart. The process will be OOC if any .

3. Proposed Method

The methodology for the proposed chart is described in this section. Subsection 3.1 covers the design structure of the proposed chart. Besides, the special cases of the chart are given in Subsection 3.2.

3.1. Proposed Chart

To construct the proposed chart, the plotting statistic of the chart is given as:

We assume by following Shamma and Shamma [4] and Abid, et al. [32], then after simplification, the statistic can be written as:

The statistic in Equation (9) can be expressed as:

Also, for , , , . The proposed chart’s control limits are written as:where is the control limits co-efficient. The statistic is plotted along with and . The process is declared IC if or ; otherwise, OOC.

3.2. Special Cases of Chart

The proposed chart reduced to some existing charts including HWMA, DHWMA, and charts by considering special values of parameters. The special cases, along with their proves, are provided here.

Case 1. When and , the proposed chart tends to the DHWMA chart.

Proof. When , then the difference estimator reduces toNow substitute the resulted in Equation (6) to obtainThen the Equation (7) can be written asNow substitute Equation (14) in Equation (8) which provides the following:Also, put in Equation (15) then the reduced as follows:This shows that statistic of proposed chart becomes the statistic of DHWMA by Abid, et al. [32], when and .

Case 2. The proposed chart converts to THWMA chart at .

Proof. Consider estimator which is given in Equation (12) when . Based on , the of Equation (13) will be as follows:Substituting the in the plotting statistic of the , (presented as in this case) we get: The in Equation (20) is the chart with no correlation. Hence, the proposed chart reduced to the statistic of the chart for .

Case 3. When and , the proposed chart tends to the chart.

Proof. When then the plotting statistic of the proposed chart, symbolized as can be written asThe in Equation (19) is similar to the statistic proposed by Adegoke, et al. [28] except for their notations. Hence, the statistic of proposed chart becomes the statistic of when and .

Case 4. The proposed chart converts to the HWMA chart at , , and

Proof. Consider estimator which is given in Equation (12) when . Based on , the of Equation (13) will be as follows:Substituting the in the plotting statistic of Equation (7) and (8), we getNow put then the plotting statistic of the proposed chart, symbolized as can be written asThe in Equation (22) is identical to the HWMA statistic. This shows that the chart is identical to the HWMA chart by Abbas (2018) for , , and.

Case 5. When , the proposed chart tends to the double chart.

Proof. When then the plotting statistic of the proposed chart, symbolized as can be written asThe in Equation (23) is a statistic. Hence, the statistic of proposed chart becomes the statistic of for .

4. Performance Evaluation Measures

This section introduces the performance evaluation measures to analyze the charts’ performance. The Monte Carlo simulation detail is given in Subsection 4.1. Likewise, the description of the is enlisted in Subsection 4.2. Similarly, the overall performance evaluation measures are defined in Subsection 4.3. The choices of parameters of the proposed chart is given in Subsection 4.4.

4.1. Monte Carlo Simulation

The Monte Carlo simulation procedure is regarded as a computational technique for obtaining numerical results for evaluating the performance of the proposed chart. Monte Carlo simulation with 105 iterations is conducted for each displacement of using R software to obtain the of the proposed chart. The shift is reflected in the process mean as: to , where  = 0.00, 0.05, 0.10, 0.20, 0.25, 0.50, 1.00, 1.50, 2.00, 2.50, 3.00, and 5.00. The proposed chart is constructed by following given below guidelines:(i)Generate random observations from .(ii)Calculate the estimator from Equation (1).(iii)Calculate the statistics of the chart from Equation (6) using estimator.(iv)Use the statistic as an input in Equation (8) to obtain the statistic.(v)Chose along with other desired parameters for desired IC denoted as .(vi)Compute and based on and (vii)Plot the statistic against the and .(viii)If > or < , record sequence order which is called run length (ix)Repeat steps (i)-(viii) 105 times.(x)Calculate the average of 105 which is . If it is desired ; otherwise, adjust the accordingly and repeat from steps (i)–(ix) until will not get desired .(xi)For OOC values, considered , ) and repeat from steps (ii)-(x).

4.2. Individual Performance Measure

The average run length is commonly used to evaluate a chart’s performance at a single shift. The is listed as IC and OOC . For IC state, the is chosen to be sufficiently large to eliminate the effect of the false alarm rate. On the other hand, the should be small enough to detect a shift quickly. A chart is preferred than the other competing charts if it should have a smaller value at predefined .

4.3. Overall Performance Evaluation Measures

The , , and performance evaluation measures are used to evaluate a chart’s overall effectiveness. More information can be found here.

The is the weighted average over the domain of shifts with as a weight [37]. It is defined as:where is the at specific ; and are the smallest and largest shift values of the domain, respectively. The lower the value signifies, the better the chart’s performance.

The , like is also used to assess the efficiency of a chart. It can be defined as follows:

The is the of the competing chart. The is the of benchmark chart at a . A chart is decided as a benchmark chart for a smaller at specific [22]. The value of the benchmark chart is assumed to be one. If the competing chart has >1, the benchmark chart is more efficient than competing.

The corresponds to the ratio of the specific chart to the of the benchmark chart. Here represents the of the benchmark chart, whereas the represents the of the competing chart. According to Ou, et al. [38], the is presented as:

The of the benchmark chart should be one. If the >1, the benchmark chart is superior against the competing chart [20].

4.4. Effect of Parameters Choices

The design parameters () of the proposed chart has its effect on the detection ability. Various combinations of these parameters are chosen, and hence corresponding , the standard deviation of RL (), and median of RL () are computed. The parameter is set as 0.10, 0.25, 0.50, and 0.75 to find the values of , to obtain . Various like 0.00, 0.25, 0.50, 0.75, and 0.95 are assumed for this study. Numerical results of the proposed chart are presented in Tables 14.

5. Evaluation and Performance Comparison

This section presents comprehensive comparisons of the proposed chart with classical EWMA [3], [16], HWMA [30], [28], DHWMA [32], TEWMA [5], and [22], and MHC [35] charts. More detail is provided here.

5.1. Proposed versus Classical EWMA Chart

The proposed chart provides superior detection ability for various values of against the classical EWMA chart. For instance, at the proposed chart has an value of 7.74, while the classical EWMA chart has an value of 47.36 (see Tables 2 and 5). The supremacy of the proposed chart compared to the classical EWMA chart can be seen in Figure 1. Evaluating the overall efficiency with , the chart has smaller (i.e., 9.38, 1.000, 1.000) values against the classical EWMA chart (i.e., 13.38, 1.43, 2.23) values (see Table 6).

5.2. Proposed versus TEWMA Chart

The proposed chart is superior to the TEWMA chart. For example, at , the proposed chart () yields , whereas the TEWMA chart gives equal to 74.21 (see Table 1 and 5). The superiority of the chart than the TEWMA chart can also be found in Figure 1. Additionally, for a specific range of shifts, the values for proposed chart are smaller than the TEWMA chart. For example, at for and TEWMA charts, the values are (13.03, 1.000, 1.000) and (15.29, 1.26, 1.17), respectively (see Table 6).

5.3. Proposed versus MHC Chart

The proposed chart is better for tracking changes in process location than the MHC chart. For example, with and , the values of the MHC chart are (25.00, 13.00, 9.00), while the values of proposed chart ) are (13.34, 6.01, 3.83) (see Table 1 and 5). Figure 1 also highlighted the dominant position of the chart on the MHC chart. Also, the values are significantly higher for the MHC chart as compared to chart. For illustration, at , the chart has  = 10.35,  = 1.00, and  = 1.00, whereas the MHC chart has measures (see Table 6).

5.4. Proposed versus Chart

The proposed chart offers good performance against the chart. As an illustration, at and (0.25, 0.50), the (80.66, 22.04) values of the chart are larger than the (13.34, 6.01) values of the chart (see Tables 1 and 5). Visual representation confirms the supremacy of the proposed chart over chart (see Figure 2). The dominance of the chart against the chart is seen in the values for a certain range of shifts. As an illustration, at , the proposed chart has , while the chart has (see Table 6). This reveals the superiority of the proposed chart.

5.5. Proposed versus HWMA Chart

The proposed chart has lower values than the HWMA chart. Suppose that, for , the value of the proposed chart with is 68.01, while the value of the HWMA chart is 218.06 (see Tables 3 and 5). In other words, the proposed chart can track a quick shift in the process parameter than the HWMA chart (see Figure 3). The overall performance metrics suggest that the effectiveness of proposed than the HWMA chart. For instance, at , the values for HWMA and are 16.25 and 13.03, respectively (see Table 6).

5.6. Proposed versus DHWMA Chart

The proposed chart is better as compared to the DHWMA chart. As an illustration, if 5, the values of for DHWMA and are 34.76 and 17.40, respectively (see Table 1 versus Table 5 and Figure 3). The highlight the superiority of the chart. For instance, at , the EQL, PCI, and RARL values of the DHWMA and charts are presented as (10.76, 1.15, and 1.38), and (9.38, 1.000, and 1.000), respectively (see Table 6).

5.7. Proposed versus Chart

The proposed chart detects earlier shift than the chart. If , , and the values of the charts are (65.89, 22.63, 11.79) and (13.34, 6.01, 3.83) respectively, for and charts (see Tables 1 and 5). Figure 3 also reveals the better performance of the proposed against the chart. Likewise, the chart EQL, RARL, and PCI (i.e., 14.55, 1.12, and 1.23) values are higher than the EQL, RARL, and PCI (13.03, 1.000, and 1.000) values of the chart at (see Table 6).

5.8. Proposed versus Chart

By comparing the measures, the proposed chart with the chart shows that the chart is more sensitive for all and values. For instance, with , , the for chart, while the for () chart (see Tables 2 and 5). The proposed chart has smaller measures as compared to the chart (see Table 6). Figure 2 illustrates that the proposed is functioning well and detects shifts earlier in the process. These findings demonstrate that the proposed chart performs better than the chart.

5.9. Proposed versus Chart

The study indicates that the proposed chart works much better than the chart for each of the choices formed by (see Tables 14 versus 5). As an illustration, Table 3 and 5 shows that at , the values of the proposed chart are (68.01, 23.02, 11.93), and the corresponding values of for chart are (111.47, 28.62, 13.55). Figure 2 indicates that the proposed chart is better than the chart. In the overall performance scenario, the proposed chart also exhibits overperformance than chart (see Table 6).

5.10. Main Findings of the Study

Important findings of the proposed chart are listed below:(i)The statistic improves the detection ability of the proposed chart.(ii)The efficiency of the proposed chart is strengthened with the appropriate inclusion of auxiliary information in the structure (see. Tables 15).(iii)Unlike the classic CUSUM, MHC, EWMA, HWMA, DHWMA, , , and charts, the values of the proposed chart are smaller at various parameter values.(iv)Overall performance evaluations demonstrate that the chart is dominant over the other competing charts included in this research (see Subsections 5.15.9).(v)The proposed chart provides optimal results for larger values .(vi)The efficiency of the measures is increased by using the proposed chart.(vii)As increases, the control limit coefficient () of the proposed chart also increases.(viii)The proposed chart is more effective for larger and smaller in term of performance (see Figures 4 and 5). For example, the minimal value of is 4.11 at , , and .

6. Case Study

This section contains the case study of the versus DHWMA, and HWMA charts. In this regard, a data set is considered from the study of Constable, et al. [39]. The data represents the measurements for adjacent parts of the braking system of vehicles, which contains the study variable and auxiliary variable . Forty-five data values are taken from the IC process, and this data set is used to estimate unknown parameters. These estimates are given as; , ,  = 1.23,  = 1.17, and  = 0.54. In view of these estimates as the known parameters, two datasets were generated from a BND. Data set-I consists of fifteen samples with , , , and and data set-II consists of fifteen samples with , , , and (see Table 7). This fashion of perturbing the parameters in such a way can be seen from the study of Anwar, et al. [40].

For the comparison, the proposed chart is considered along with the existing DHWMA, , and HWMA charts. Furthermore, the parameters of the proposed chart are ,  = 0.25, with and the parameters of the DHWMA chart are ,  = 0.25, with . Likewise, the parameters of chart are  = 0.25, with and the parameters of the HWMA chart are ,  = 0.25, with . From Table 7 and Figures 69 the proposed chart tracksthe first OOC signal at the 5th sample, whereas the DHWMA, , and HWMA charts track the first OOC signal at the 18th, 8th, and 9th samples, respectively. Overall, the proposed chart highlights 21 OOC points, while the DHWMA, , and HWMA charts highlight 13, 9, and 9 OOC points, respectively. This show that the proposed chart is more sensitive than the existing DHWMA, , and HWMA charts.

7. Summary, Conclusions, And Recommendations

The objective of this study is to enhance homogeneously weighted moving average (HWMA) and double HWMA (DHWMA) charts and propose an auxiliary-information-based triple HWMA, symbolized as chart to further improve the process location shift monitoring. To evaluate the performance of the proposed chart against other charts, an algorithm is developed in R software using the Monte Carlo simulation technique to obtain numerical results. The analysis based on average run length, extra quadratic loss, performance comparison index, and relative average run length reveals the proposed chart over-performed against EWMA, , HWMA, , DHWMA, TEWMA, , and MHC charts. Finally, to demonstrate the proposed chart’s utility from a practical perspective, a real-world application is included. This study can be extended for multivariate and more than one auxiliary characteristic in the model.

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.

Authors’ Contributions

Conceptualization (Syed Masroor Anwar, Zahid Rasheed). Formal Analysis (Syed Masroor Anwar, Nafiu Lukman Abiodun). Funding Acquisition (Somayya Komal, Ammara Nawaz Cheema). Investigation (Somayya Komal, Ammara Nawaz Cheema). Methodology (Syed Masroor Anwar, Majid Khan). Project Administration (Nafiu Lukman Abiodun, Majid Khan). Resources (Somayya Komal, Zahid Rasheed). Supervision (Nafiu Lukman Abiodun, Majid Khan). Validation (Somayya Komal, Ammara Nawaz Cheema). Visualization (Syed Masroor Anwar, Zahid Rasheed). Writing–Original Draft Preparation (Syed Masroor Anwar, Nafiu Lukman Abiodun). Writing–Review and Editing (Somayya Komal, Ammara Nawaz Cheema, Zahid Rasheed).