Homogeneously Mixed Memory Charts with Application in the Substrate Production Process

The cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) charts are renowned classical memory charts used to monitor small and moderate shifts in the process(s). Mixed memory charts like mixed EWMA-CUSUM (MEC) and mixed CUSUM-EWMA (MCE) are the advanced forms of classical memory charts used to identify shifts quickly in process parameters (location and/or dispersion). Similarly, the homogeneously weighted moving average (HWMA) chart is used for improved process monitoring. It will be worthwhile to combine the HWMA chart features with the existing mixed memory (MCE and MEC) charts to enhance the effectiveness of the mixed memory charts. Therefore, we proposed new charts: mixed HWMA-homogeneously CUSUM (MHWHC) and mixed homogeneously CUSUM-HWMA (MHCHW) charts. The Monte Carlo simulations are used to evaluate the proposed charts’ effectiveness. The average run length (ARL) is utilized to compare the proposed MHWHC and MHCHW charts’ performance with existing charts such as classical CUSUM and EWMA, MEC, MCE, and HWMA charts. The comparison revealed that the proposed mixed charts are superior to the existing counterparts, specifically monitoring small and moderate shifts. Finally, a real-life application using the manufacturing process’s data set is also provided from a practical point of view.


Introduction
Natural and unnatural variations are part of every manufacturing/nonmanufacturing process. Natural variations regularly occur and are essential part of any stable process, whereas unnatural variations damage the quality process caused by process deficiencies. Statistical process control (SPC) is a valuable toolkit that allows one to monitor unnatural variations. e charts are popular SPC tools to monitor shifts in the process parameters. Generally, charts are divided into two groups: memoryless and memory-type charts. Shewhart [1] proposed the memoryless charts used to monitor large shifts in the process parameters. On the other hand, the classical memory charts like the cumulative sum (CUSUM) chart designed by Page [2] and the exponentially weighted moving average (EWMA) chart offered by Roberts [3] are used for the efficient handling of small and moderate shifts.
Different modifications and improvements in SPC literature are continuously used in the classical memory charts to enhance process monitoring. One modification is mixed memory charts, which improve conventional memory-type charts' performance. Abbas [4] and Abbas et al. [5] proposed mixed EWMA-CUSUM (MEC) charts for the process location and dispersion shift monitoring, respectively. e MEC chart is effective in monitoring shifts in the process location. Similarly, Aslam et al. [6] designed a mixed chart based on a variable sampling scheme that outperforms the traditional nonparametric chart in terms of reduced sample size. Likewise, Aslam [7] offered the mixed EWMA-CUSUM chart for Weibull distribution. After that, Aslam et al. [8] proposed a mixed hybrid EWMA-CUSUM control chart for the monitoring of the normal process. Subsequently, Zaman et al. [9] suggested a reverse version of the MEC chart, named the mixed CUSUM-EWMA (MCE) chart. Furthermore, Zaman et al. [10] suggested the MEC chart for monitoring process location and dispersion. In addition, Ajadi and Riaz [11] presented the mixed multivariate EWMA-CUSUM chart for efficient process monitoring. Osei-Aning et al. [12] constructed the MEC and MCE charts to monitor the first-order autoregressive processes. Later on, Ali and Haq [13] extended the work of Abbas et al. [14] and introduced a mixed generally weighted moving average-CUSUM chart to monitor process mean. Also, Rao et al. [15] constructed a new mixed EWMA-CUSUM chart for monitoring small shifts relying on the COM-Poisson distribution. Anwar et al. [16] and Anwar et al. [17] recently introduced mixed charts through single auxiliary information labeled as MEC AIB , MCE AIB , and combined MEC AIB (CMEC AIB ) charts, respectively. e MEC AIB , MCE AIB , and CMEC AIB charts perform better than the existing counterparts. Zaman et al. [18] proposed a multivariate mixed memory chart for improved process monitoring. Likewise, Aslam and Anwar [19] and Anwar et al. [20] proposed Bayesian modified-EWMA and auxiliary information-based modified-EWMA charts to boost process location monitoring, respectively. e limitation of the classical EWMA chart is that the EWMA plotting statistic assigned more weights to current observations than the previous observations [21]. Abbas [22] suggests the homogeneous weighted moving average (HWMA) chart to overcome this limitation. e HWMA chart assigns specific weights to the current observation and remaining weights to the previous observations homogeneously. e distribution of weights in such a way improves the HWMA chart's performance ability compared to its competitor's charts. Later on, Adegoke et al. [23] generalized the idea of Abbas [22] and introduced an auxiliary-based HWMA (AHWMA) chart. Similarly, Abid et al. [24] extended the work of Abbas [22] and advocated a double HWMA (DHWMA) chart. Adeoti and Koleoso [25] offered the hybrid HWMA (HHWMA) chart for efficient monitoring shift in the process location. After that, Adeoti et al. [26] introduced the HWMA chart for count data. is newly introduced structure is based on a COM-Poisson distribution and named as CMP-HWMA chart. Moreover, Abid et al. [27] proposed an MHC chart for monitoring shifts in the process location. Also, Riaz et al. [28] presented a triple HWMA (THWMA) chart for monitoring the process location parameter. Recently, Anwar et al. [29] introduced auxiliary information-based DHWMA chart for enhanced process location monitoring.
As mentioned earlier, the mixed memory charts (MEC and MCE) are practiced to monitor small and moderate shifts. Similarly, the homogeneous memory charts are the enhanced versions of classical memory charts for efficient monitoring of process parameters. To the best of our knowledge, no single chart combining the features of mixed memory chart along with homogeneously memory charts has been investigated previously. It is a clear research gap and needs to be explored. So, to address this research gap, this study introduces mixed memory charts based on homogeneous features for monitoring process location. ese charts are named the mixed HWMA-homogenously CUSUM and mixed homogenously CUSUM-HWMA charts, symbolized as MHWHC and MHCHW charts. It is expected that combining features of mixed memory charts along with homogeneously charts will further enhance the detection ability of the ultimate charts. Average run length (ARL), extra quadratic Loss (EQL), performance comparison index (PCI), and relative ARL (RARL) measurements are used to test the effectiveness of the suggested charts. Besides, an algorithm is designed in R software using the Monte Carlo simulations technique to obtain numerical results. A variety of existing charts, including the classical EWMA, CUSUM, MEC, MCE, and HWMA charts, are compared with the proposed charts. Moreover, the proposed MHWHC and MHCHW charts are executed in a real-life application to ensure their practicability. e remainder of the paper is organized as follows: Section 2 provides the variable of interest and methodologies of the existing charts. Similarly, the design structures for HCUSUM, MHWHC, and MHCHW charts are presented in Section 3. Likewise, Section 4 defines the various performance evaluation measures. Also, the performance comparisons of the proposed charts to the existing CUSUM, EWMA, MEC, MCE, and HWMA are given in Section 5. Furthermore, Section 6 contains real-life data analysis for practical implementation of the proposed charts. Lastly, the summary, conclusion, and recommendation are set out in Section 7.

Existing Methods
Here, Section 2.1 explains the variable of interest. e classical CUSUM chart is explained in Section 2.2. In Section 2.3, the classical EWMA chart is demonstrated. e MEC and MCE charts' design structures are presented in Sections 2.4 and 2.5, respectively, whereas Section 2.6 contains the HWMA chart's methodology.

Variable of Interest.
Suppose that X is the process variable that follows a normal distribution, that is, X ∼ N(μ X + δσ X , σ X ). If the process is in control (IC), then δ � 0; otherwise, if the process is out of control (OOC), δ ≠ 0. Let X t � n i�1 X it /n and S 2 Xt � n i�1 (X it − X t ) 2 /(n − 1) be the mean and variance of X, respectively, of i th observation of t th sample of size n. For the IC scenario, X t and S 2 Xt are mutually independent identically distributed, that is, X t ∼ N(μ X , σ 2 X /n) and S 2 Xt ∼ σ 2 X /(n − 1)(χ 2 n− 1 ).

Classical CUSUM Chart.
Page [2] introduced the classical CUSUM chart and it is used to monitor small and moderate shifts in the process location. e classical CUSUM chart's plotting statistics are defined as follows: e C + 0 and C − 0 statistics are equal to zero. Also, ) is a slack value and is usually taken as k � δ/2. Here δ denotes shift, and it is defined as δ � |μ 1 ) and h is a control limit coefficient. If C + t > H or C − t > H, the process is considered OOC; otherwise, it is considered IC.

Classical EWMA Chart.
e classical EWMA chart was introduced by Roberts [3]. It is also used to monitor small and moderate shifts in the process location. For the classical EWMA chart, the plotting statistic is given by where Z 0 � 0 and λ is a smoothing constant such that 0 < λ ≤ 1. For the conventional EWMA chart, the upper control limit (UCL) and lower control limit (LCL) are as follows: Here L EWMA is the control limit coefficient. e Z t statistic is plotted against LCL (EWMA)t and UCL (EWMA)t . e process is considered OOC whenever Z t < LCL (EWMA)t or Z t > UCL (EWMA)t ; otherwise, it is considered to be IC.

MEC Chart.
Abbas, Riaz, and Does [14] proposed the MEC chart by combining the classical EWMA and CUCUM charts' properties. e classical EWMA statistic is used as an element in the classical CUSUM statistic to construct the MEC chart. e plotting statistics of the MEC chart are as follows: where K Z t is defined as e MEC + t and MEC − t statistics for the MEC chart are plotted against H Z t , which is defined as Here, h is the control limit coefficient. e MEC chart will provide an OOC signal if MEC + t > H Z t or MEC − t > H Z t .

MCE Chart.
Zaman et al. [30] proposed a reverse of the MEC chart. Unlike the MEC chart, the design structure of the MCE chart offers the classical CUSUM statistics as input for the EWMA chart given as where C + t and C − t are the classical CUSUM statistics, λ(0 < λ ≤ 1) is the smoothing parameter of the proposed chart, and MCE + 0 � MCE − 0 � μ. e mean and variance of the plotting statistics are given as Mean ( respectively. e control limit for the MCE chart is as follows: where L MCE represents the control limit coefficient. If the process is not IC; otherwise, it remains as IC.
2.6. HWMA Chart. Abbas [22] introduced the HWMA chart to enhance process location monitoring. Unlike the classical EWMA chart, the HWMA chart equally allocates weights to all the previous observations. e plotting statistic of the HWMA chart is given as where X t is the sample average of t th group and λ(0 < λ ≤ 1) is the smoothing constant and X t− 1 � t− 1 i�1 X t /t − 1 is the mean of sample average of previous t − 1 samples and X 0 � μ 0 . e UCL and LCL of the HWMA chart are given as where L HWMA is the chart's coefficient, and its value is based on the IC ARL. e process is to be OOC if E t < LCL (HWMA)t or E t > UCL (HWMA)t .

Design Structure of the Proposed Charts
Section 3.1 sets out the structure of the HCUSUM chart. e MHWHC chart's methodology is defined in Section 3.2, while Section 3.3 defines the MHCHW chart's methodology.
3.1. Proposed HCUSUM Chart. Following Abbas [22], the plotting statistics of the HCUSUM are defined as Mathematical Problems in Engineering 3 where X 0 � μ 0 and ) is a slack value as mentioned in CUSUM chart. e HCUSUM statistics HC − t and , and the process is considered to be OOC if HC + t > H (HCUSUM)t or HC − t > H (HCUSUM)t ; otherwise, it is considered to be IC.

Proposed MHWHC Chart.
e proposed MHWHC chart is constructed by mixing HWMA and HCUSUM charts.
e plotting statistics of the proposed MHWHC chart are obtained by using the HWMA statistic in equation (9) as input for the HCUSUM statistics in equations (11) and (12).
where the reference value K (MHWHC) t is defined as follows: MHWHC + t and MHWHC − t are the statistics of MHWHC chart. ese statistics are plotted against the control limit H (MHWHC) t , which is defined as where h is the predicted false alarm risk coefficient. If

Proposed MHCHW Chart.
e proposed MHCHW chart is constructed by merging the HCUSUM and HWMA charts. e plotting statistics are obtained for the MHCHW chart by using the plotting statistics of equations (11) and (12) as input statistics in equation (9). Consequently, the plotting statistics of the proposed MHCHW chart are given as where the MHCHW + t and MHCHW − t statistics are the lower and upper statistics of the proposed MHCHW chart and λ(0 < λ ≤ 1) is the smoothing parameter of the proposed MHCHW chart. e mean and variance of the MHCHW statistics are given as e UCL for the proposed MHCHW chart is given as where L MHCHW is control limit coefficient, which decides the predefined false alarm rate. e MHCHW chart will detect

Performance Evaluation Measures
is section consists of the performance evaluation measures of the proposed charts. ARL is explained in Section 4.1, and the overall performance evaluation measures, that is, EQL, RARL, and PCI, are outlined in Section 4.2. Likewise, the Monte Carlo simulation is defined in Section 4.3. Parameters' roles and choices and special cases of the proposed MHWHC and MHCHW charts are given in Sections 4.4 and 4.5, respectively. Finally, a sensitivity analysis of the proposed MHWHC and MHCHW charts is given in Section 4.6.

Average Run Length.
Usually, the ARL measures are used to assess the performance of charts. e ARL can be classified into two types, that is, ARL 0 and ARL 1 . ARL 0 is an IC ARL and ARL 1 is an OOC ARL. If a process is working IC scenario, the desired ARL 0 should be large enough to avoid frequent false alarms. However, ARL 1 should be small enough; it quickly detects the shift. A chart is preferred when it has smaller ARL 1 than that of its competitors. e ARL is used for measuring the performance of a chart at a specific shift, but some other performance measurements, like EQL, RARL, and PCI, are preferred over a specific range of shifts. e specifics of these approaches are outlined in the subsequent subsections.

Extra Quadratic Loss.
e EQL is the weighted average of ARL over its entire shift domain, and the square shift is used as a weight (Wu et al. [31]). It is written as follows: where δ min and δ max are the minimum and maximum shift values, respectively. ARL(δ) is the ARL value at specific δ. e chart with higher EQL values is assumed to be less effective than the other charts.

Relative Average Run Length.
Mathematically, RARL can be defined as where ARL(δ) is the ARL of competing chart and ARL * (δ) is the ARL of the benchmark chart at shift δ. e benchmark chart's RARL measure is equal to one but it is higher than one for other contending charts [16].

Performance Comparison Index.
Ou et al. [32] introduced the PCI measure for a chart's ultimate performance evaluation. PCI is the ratio of the EQL of the competing chart with the EQL of the benchmark chart. Symbolically, it can be defined as follows: e PCI of the benchmark chart should be 1. If the PCI > 1, the benchmark chart is superior to the competing charts.

Monte Carlo Simulation.
Observations X it (where i � 1, 2, . . . , n and (t > 1)) are normally distributed of different parameters with increasing shifts in the process location. e nature of the shift is as follows; μ to μ + δσ, where δ � 0.25, 0.50, 0.75, 1.00, 1.50, and 2.00. e Monte Carlo simulation methods are used for the computation of the numerical results. e simulation work is done in the R software that consists of 50,000 replications at shift δ.
e numerical results are presented in Tables 1-6. e simulation algorithm to develop the proposed MHWHC chart follows the following steps: (i) Generate a random sample from a normal distribution with IC parameters (ii) Choose a specific value of λ (iii) Assume the value of h (iv) Calculate the plotting statistics of the MHWHC chart as given in equations (13) and (14) (v) Calculate the control limit (H (MHWHC) t ) as mentioned in equation (16) (vi) Plot the MHWHC ± t statistics against the control then note this sample number as an RL (viii) Repeat this process from step (i) to step (vii) 10 5 times to obtain ARL 0 (ix) To compute the ARL 1 measures, again draw a sample from a normal distribution using shifted location parameter and repeat this process from step (ii) to step (viii) Similarly, the procedure to construct the proposed MHCHW chart is given in the following steps: (i) Draw a sample from a normal distribution using IC parameters (ii) Select specific values of design parameters λ and L MHCHW (iii) Determine plotting statistics of the MHCHW chart from equations (17) and (18) and E t , defined in equation (9), is given as When λ � 1, E t can be written as Replacing the resulting value of E ( * ) t in statistics of the proposed MHWHC chart, it will become and equations (27) and (28) show that the proposed MHWHC chart approaches the HCUSUM chart when λ � 1. e proposed MHCHW chart reduces to the HCUSUM chart for λ � 1.

Proof.
e plotting statistics of the proposed MHCHW chart are When λ � 1, the above statistics are reduced to e statistics in equations (30) and (31) are similar to those in equations (11) and (12), except their notations, showing that the proposed MHCHW chart is reduced to the HCUSUM chart.
As both proposed, the MHWHC and MHCHW charts reduce to the HCUSUM chart when λ � 1. us, if someone is interested in obtaining ARL values of the HCUSUM chart, he will directly use the MHWHC or MHCHW chart with λ � 1.   Table 1). Similarly, at δ � 0.05, k � 0.50, and ARL 0 � 370, the ARL values of the proposed MHCHW chart are 214.42 and 131.23 for λ � 0.25 and λ � 0.10, respectively (see Table 1).  Tables 2 and 3). Likewise, for δ � 0.05, λ � 0.10, and ARL 0 � 370, the ARL value of the proposed MHCHW chart is 64.04 and 61.98 for k � 1.00 and k � 1.50, respectively (see Tables 2 and 3). (iv) e overall performance measures show that the proposed MHWHC and MHCHW charts are superior to existing (used in this study) charts for all λ (see Table 7).

Evaluation and Performance Comparison
is section contains a detailed performance comparison of the proposed MHWHC and MHCHW charts with the existing charts. ese include the classical CUSUM [2], the classical EWMA [3], MEC [14], MCE [9], and HWMA [22] charts.

Proposed versus Classical CUSUM Charts.
It is observed that the proposed MHWHC and MHCHW charts work well, unlike the classical CUSUM chart. To illustrate at λ � 0.50, k � 0.50, and δ � 0.50 , the ARL 1 values of the proposed MHWHC and MHCHW charts are 16.69 and 25.99, respectively, while the ARL 1 value of the classical CUSUM chart is 34.09 (see Table 4 versus Table 8). Figures 1 and 2 also show that the MHWHC and MHCHW charts are superior to the classical CUSUM chart. Similarly, the proposed MHWHC and MHCHW charts also have a better overall performance than the classical CUSUM chart. As an illustration, at λ � 0.05, the RARL values of the proposed MHWHC and MHCHW charts are 1.00 and 1.31, whereas the RARL measure of the classical CUSUM chart is 2.08 (see Table 7).

Proposed versus Classical EWMA Chart.
e proposed MHWHC and MHCHW charts are more responsive than the EWMA chart to detect a shift in the process location. For all combinations of shift (δ) and λ, the ARL 1 values for the proposed charts are lower than the ARL 1 values of classical EWMA chart. For example, for λ � 0.10, δ � 0.25, and k � 0.50, ARL 1 value of the classical EWMA chart is 103.32, whereas ARL 1 of the proposed MHWHC and MHCHW charts are 24.69 and 19.37, respectively (see Tables 4  versus 8). Visual presentation in Figures 1 and 2 also confirms that the proposed MHWHC and MHCHW charts are superior to the classical EWMA chart. Similarly, the EQL, RARL, and PCI values provided in Table 7 suggested the better performance of the proposed MHWHC and MHCHW charts compared to the classical EWMA chart. For example, at λ � 0.75, the EQL values of MHWHC, MHCHW, and EWMA charts are 6.57, 8.11, and 24.74, respectively, showing the inferiority of the classical EWMA charts (see Table 7).  Table 4 versus Table 8). Likewise, the proposed MHWHC and MHCHW charts' efficiency against the MEC chart is demonstrated in Figures 1 and 2 Table 7).

Proposed versus MCE Chart.
e proposed MHWHC and MHCHW charts have smaller ARL 1 values for all λ in comparison with the MCE chart. As illustration at λ � 0.10 and δ � 0. 25 Tables 4  versus 8). Additionally, Figure 4 shows the proposed charts' dominance over the HWMA chart. In terms of overall performance, the MHWHC and MHCHW charts perform better than the HWMA chart. e RARL values of       Table 7). ey indicate the inferiority of the HWMA chart.

Real-Life Application
e application of the proposed charts is primarily associated with manufacturing processes and finished goods, but they can also be applied to a variety of other fields like health, planning, accounts, neutrosophic statistics, banking, and so forth.
is section offers the application of the proposed MHWHC and MHCHW charts versus the existing HWMA chart using a semiconductor manufacturing data set in the production process [33]. In the phase-I study, the variable of interest is taken to measure the "flow width of the resist" from 25 subgroups of size five each. In the phase-II study, there are 20 more subgroups. ese subgroups are used for real-life applications to compare existing HWMA and

Summary, Conclusions, and Recommendations
Memory charts are used to monitor small and moderate shifts in the process. Likewise, the homogeneous memory charts represent the updated version of the conventional memory-type chart for efficiently monitoring process parameters. is study presents the design structures of new charts, that is, the mixed homogeneously weighted moving average-homogeneously cumulative sum (MHWHC) and the mixed homogeneously cumulative sum-homogeneously weighted moving average (MHCHW) charts for the monitoring of the process location. e MHWHC chart is formed using the homogeneously weighted moving average (HWMA) chart statistic to input the homogeneously cumulative sum (HCUSUM) statistic. In contrast, the MHCHW chart is developed using the HCUSUM statistics as input to the HWMA statistic. Different performance measures like the average run length, extra quadratic loss, relative average run length, and performance comparison index are used to assess the proposed MHWHC and MHCHW charts' performance. e study shows that, for small and moderate shifts, the proposed MHWHC and MHCHW charts provide better performance compared to various existing charts; these include the classical CUSUM and EWMA, mixed EWMA-CUSUM, mixed CUSUM-EWMA, and HWMA charts. A real-life data analysis is also given to demonstrate the proposed chart's practical implementation. is study is carried out where the process variable follows the univariate normal distribution. So the proposed MHWHC and MHCHW charting schemes can be used to enhance the monitoring of high-quality processes, time between events [34], nonnormal processes, and multivariate processes and scenarios.

Data Availability
e real-life data used to support the findings of this study are included within the article.

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