# Homogeneously Mixed Memory Charts with Application in the Substrate Production Process

Authors: Zahid Rasheed; Hongying Zhang; Syed Masroor Anwar; Babar Zaman
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. 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. The 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. The 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 MECAIB, MCEAIB, and combined MECAIB (CMECAIB) charts, respectively. The MECAIB, MCEAIB, and CMECAIB 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.The 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. The HWMA chart assigns specific weights to the current observation and remaining weights to the previous observations homogeneously. The 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. This 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. These 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 proposedMHWHC and MHCHW charts are executed in a real-life application to ensure their practicability.The remainder of the paper is organized as follows: Section2 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.


## 2. Existing Methods

Here, Section2.1 explains the variable of interest. The classical CUSUM chart is explained in Section 2.2. In Section 2.3, the classical EWMA chart is demonstrated. The 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.


### 2.1. Variable of Interest

Suppose thatX is the process variable that follows a normal distribution, that is, X∼NμX+δσX,σX. If the process is in control (IC), then δ=0; otherwise, if the process is out of control (OOC), δ≠0. LetX¯t=∑i=1nXit/n and SXt2=∑i=1nXit−X¯t2/n−1 be the mean and variance of X, respectively, of ith observation of tth sample of size n. For the IC scenario, X¯t and SXt2 are mutually independent identically distributed, that is, X¯t∼NμX,σX2/n and SXt2∼σX2/n−1χn−12.


### 2.2. Classical CUSUM Chart

Page [2] introduced the classical CUSUM chart and it is used to monitor small and moderate shifts in the process location. The classical CUSUM chart’s plotting statistics are defined as follows:(1)Ct+=max0,X¯t−μ0−K+Ct−1+,Ct−=max0,−X¯t−μ0−K+Ct−1−.TheC0+ and C0− statistics are equal to zero. Also, K=kσ/n is a slack value and is usually taken as k=δ/2. Here δ denotes shift, and it is defined as δ=μ1−μ0/σ/n, where μ1 is deviated value of μ0. The Ct− and Ct+ statistics are plotted against H=hσ/n and h is a control limit coefficient. If Ct+>H or Ct−>H, the process is considered OOC; otherwise, it is considered IC.


### 2.3. Classical EWMA Chart

The 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(2)Zt=λX¯t+1−λZt−1,where Z0=0andλ 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:(3)LCLEWMAt=μ0−LEWMAσX¯λ2−λ1−1−λ2t,UCLEWMAt=μ0+LEWMAσX¯λ2−λ1−1−λ2t.HereLEWMA is the control limit coefficient. The Zt statistic is plotted against LCLEWMAt andUCLEWMAt. The process is considered OOC whenever Zt<LCLEWMAt or Zt>UCLEWMAt; otherwise, it is considered to be IC.


### 2.4. MEC Chart

Abbas, Riaz, and Does [14] proposed the MEC chart by combining the classical EWMA and CUCUM charts' properties. The classical EWMA statistic is used as an element in the classical CUSUM statistic to construct the MEC chart. The plotting statistics of the MEC chart are as follows:(4)MECt+=max0,Zt−μ0−KZt+MECt−1+,MECt−=max0,−Zt−μ0−KZt+MECt−1−,where KZt is defined as(5)KZt=k×VarZt=kσλn2−λ1−1−λ2t.TheMECt+ and MECt− statistics for the MEC chart are plotted against HZt, which is defined as(6)HZt=h×VarZt=hσλn2−λ1−1−λ2t.Here,h is the control limit coefficient. The MEC chart will provide an OOC signal if MECt+>HZt or MECt−>HZt.


### 2.5. 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(7)MCEt+=λCt++1−λMCEt−1+,MCEt−=λCt−+1−λMCEt−1−,where Ct+ and Ct− are the classical CUSUM statistics, λ0<λ≤1 is the smoothing parameter of the proposed chart, and MCE0+=MCE0−=μ. The mean and variance of the plotting statistics are given as MeanCt+=MeanCt−=μt and VarCt+=VarCt−=σt2, respectively. The control limit for the MCE chart is as follows:(8)UCLMCEt=μt+LMCEσλn2−λ1−1−λ2t,where LMCE represents the control limit coefficient. If MCEt+>UCLMCEt or MCEt−>UCLMCEt, 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. The plotting statistic of the HWMA chart is given as(9)Et=λX¯t+1−λX¯¯t−1,where X¯t is the sample average of tth group and λ0<λ≤1 is the smoothing constant and X¯¯t−1=∑i=1t−1X¯t/t−1 is the mean of sample average of previous t−1 samples and X¯¯0=μ0. The UCL and LCL of the HWMA chart are given as(10)LCLHWMAt=μ0−LHWMAσλ2n,ift=1,μ0−LHWMAσ1nλ2+1−λ2t−1,ift>1,UCLHWMAt=μ0+LHWMAσλ2n,ift=1,μ0+LHWMAσ1nλ2+1−λ2t−1,ift>1,where LHWMA is the chart's coefficient, and its value is based on the IC ARL. The process is to be OOC if Et<LCLHWMAt or Et>UCLHWMAt.


## 2.1. Variable of Interest

Suppose thatX is the process variable that follows a normal distribution, that is, X∼NμX+δσX,σX. If the process is in control (IC), then δ=0; otherwise, if the process is out of control (OOC), δ≠0. LetX¯t=∑i=1nXit/n and SXt2=∑i=1nXit−X¯t2/n−1 be the mean and variance of X, respectively, of ith observation of tth sample of size n. For the IC scenario, X¯t and SXt2 are mutually independent identically distributed, that is, X¯t∼NμX,σX2/n and SXt2∼σX2/n−1χn−12.


## 2.2. Classical CUSUM Chart

Page [2] introduced the classical CUSUM chart and it is used to monitor small and moderate shifts in the process location. The classical CUSUM chart’s plotting statistics are defined as follows:(1)Ct+=max0,X¯t−μ0−K+Ct−1+,Ct−=max0,−X¯t−μ0−K+Ct−1−.TheC0+ and C0− statistics are equal to zero. Also, K=kσ/n is a slack value and is usually taken as k=δ/2. Here δ denotes shift, and it is defined as δ=μ1−μ0/σ/n, where μ1 is deviated value of μ0. The Ct− and Ct+ statistics are plotted against H=hσ/n and h is a control limit coefficient. If Ct+>H or Ct−>H, the process is considered OOC; otherwise, it is considered IC.


## 2.3. Classical EWMA Chart

The 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(2)Zt=λX¯t+1−λZt−1,where Z0=0andλ 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:(3)LCLEWMAt=μ0−LEWMAσX¯λ2−λ1−1−λ2t,UCLEWMAt=μ0+LEWMAσX¯λ2−λ1−1−λ2t.HereLEWMA is the control limit coefficient. The Zt statistic is plotted against LCLEWMAt andUCLEWMAt. The process is considered OOC whenever Zt<LCLEWMAt or Zt>UCLEWMAt; otherwise, it is considered to be IC.


## 2.4. MEC Chart

Abbas, Riaz, and Does [14] proposed the MEC chart by combining the classical EWMA and CUCUM charts' properties. The classical EWMA statistic is used as an element in the classical CUSUM statistic to construct the MEC chart. The plotting statistics of the MEC chart are as follows:(4)MECt+=max0,Zt−μ0−KZt+MECt−1+,MECt−=max0,−Zt−μ0−KZt+MECt−1−,where KZt is defined as(5)KZt=k×VarZt=kσλn2−λ1−1−λ2t.TheMECt+ and MECt− statistics for the MEC chart are plotted against HZt, which is defined as(6)HZt=h×VarZt=hσλn2−λ1−1−λ2t.Here,h is the control limit coefficient. The MEC chart will provide an OOC signal if MECt+>HZt or MECt−>HZt.


## 2.5. 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(7)MCEt+=λCt++1−λMCEt−1+,MCEt−=λCt−+1−λMCEt−1−,where Ct+ and Ct− are the classical CUSUM statistics, λ0<λ≤1 is the smoothing parameter of the proposed chart, and MCE0+=MCE0−=μ. The mean and variance of the plotting statistics are given as MeanCt+=MeanCt−=μt and VarCt+=VarCt−=σt2, respectively. The control limit for the MCE chart is as follows:(8)UCLMCEt=μt+LMCEσλn2−λ1−1−λ2t,where LMCE represents the control limit coefficient. If MCEt+>UCLMCEt or MCEt−>UCLMCEt, 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. The plotting statistic of the HWMA chart is given as(9)Et=λX¯t+1−λX¯¯t−1,where X¯t is the sample average of tth group and λ0<λ≤1 is the smoothing constant and X¯¯t−1=∑i=1t−1X¯t/t−1 is the mean of sample average of previous t−1 samples and X¯¯0=μ0. The UCL and LCL of the HWMA chart are given as(10)LCLHWMAt=μ0−LHWMAσλ2n,ift=1,μ0−LHWMAσ1nλ2+1−λ2t−1,ift>1,UCLHWMAt=μ0+LHWMAσλ2n,ift=1,μ0+LHWMAσ1nλ2+1−λ2t−1,ift>1,where LHWMA is the chart's coefficient, and its value is based on the IC ARL. The process is to be OOC if Et<LCLHWMAt or Et>UCLHWMAt.


## 3. Design Structure of the Proposed Charts

Section3.1 sets out the structure of the HCUSUM chart. The 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(11)HCt+=max0,X¯t−μ0−K+X¯t−1,(12)HCt−=max0,−X¯t−μ0−K+X¯t−1,where X¯¯0=μ0 and K=kσ/n is a slack value as mentioned in CUSUM chart. The HCUSUM statistics HCt− and HCt+ are plotted against HHCUSUMt=hσ/n, and the process is considered to be OOC if HCt+>HHCUSUMt or HCt−>HHCUSUMt; otherwise, it is considered to be IC.


### 3.2. ProposedMHWHC Chart

The proposed MHWHC chart is constructed by mixing HWMA and HCUSUM charts. The 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).(13)MHWHCt+=max0,Et−μ0−KMHWHCt+X¯¯t−1,(14)MHWHCt−=max0,Et−μ0−KMHWHCt+X¯¯t−1,where the reference value KMHWHCt is defined as follows:(15)KMHWHCt=kσλ2n,ift=1,kσ1nλ2+1−λ2t−1,ift>1.MHWHCt+ and MHWHCt− are the statistics of MHWHC chart. These statistics are plotted against the control limit HMHWHCt, which is defined as(16)HMHWHCt=hσλ2n,ift=1,hσ1nλ2+1−λ2t−1,ift>1,where h is the predicted false alarm risk coefficient. If MHWHCt+<HMHWHCt or MHWHCt−<HMHWHCt, the process is IC; otherwise, it is OOC.


### 3.3. ProposedMHCHW Chart

The proposedMHCHW chart is constructed by merging the HCUSUM and HWMA charts. The 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(17)MHCHWt+=λHCt++1−λMHCHWt−1+,(18)MHCHWt−=λHCt−+1−λMHCHWt−1−,where the MHCHWt+ and MHCHWt− statistics are the lower and upper statistics of the proposed MHCHW chart and λ0<λ≤1 is the smoothing parameter of the proposed MHCHW chart. The mean and variance of the MHCHW statistics are given as(19)MeanMHCHWt+=MeanMHCHWt−=μMHCHWt,VarMHCHWt+=VarMHCHWt−=σMHCHWt2.The UCL for the proposedMHCHW chart is given as(20)UCLMHCHWt=μMHCHWt+LMHCHWσMHCHWtλ2−λ1−1−λ2t,where LMHCHW is control limit coefficient, which decides the predefined false alarm rate. The MHCHW chart will detect OOC signal if MHCHWt+>UCLMHCHWt or MHCHWt+>UCLMHCHWt.


## 3.1. Proposed HCUSUM Chart

Following Abbas [22], the plotting statistics of the HCUSUM are defined as(11)HCt+=max0,X¯t−μ0−K+X¯t−1,(12)HCt−=max0,−X¯t−μ0−K+X¯t−1,where X¯¯0=μ0 and K=kσ/n is a slack value as mentioned in CUSUM chart. The HCUSUM statistics HCt− and HCt+ are plotted against HHCUSUMt=hσ/n, and the process is considered to be OOC if HCt+>HHCUSUMt or HCt−>HHCUSUMt; otherwise, it is considered to be IC.


## 3.2. ProposedMHWHC Chart

The proposed MHWHC chart is constructed by mixing HWMA and HCUSUM charts. The 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).(13)MHWHCt+=max0,Et−μ0−KMHWHCt+X¯¯t−1,(14)MHWHCt−=max0,Et−μ0−KMHWHCt+X¯¯t−1,where the reference value KMHWHCt is defined as follows:(15)KMHWHCt=kσλ2n,ift=1,kσ1nλ2+1−λ2t−1,ift>1.MHWHCt+ and MHWHCt− are the statistics of MHWHC chart. These statistics are plotted against the control limit HMHWHCt, which is defined as(16)HMHWHCt=hσλ2n,ift=1,hσ1nλ2+1−λ2t−1,ift>1,where h is the predicted false alarm risk coefficient. If MHWHCt+<HMHWHCt or MHWHCt−<HMHWHCt, the process is IC; otherwise, it is OOC.


## 3.3. ProposedMHCHW Chart

The proposedMHCHW chart is constructed by merging the HCUSUM and HWMA charts. The 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(17)MHCHWt+=λHCt++1−λMHCHWt−1+,(18)MHCHWt−=λHCt−+1−λMHCHWt−1−,where the MHCHWt+ and MHCHWt− statistics are the lower and upper statistics of the proposed MHCHW chart and λ0<λ≤1 is the smoothing parameter of the proposed MHCHW chart. The mean and variance of the MHCHW statistics are given as(19)MeanMHCHWt+=MeanMHCHWt−=μMHCHWt,VarMHCHWt+=VarMHCHWt−=σMHCHWt2.The UCL for the proposedMHCHW chart is given as(20)UCLMHCHWt=μMHCHWt+LMHCHWσMHCHWtλ2−λ1−1−λ2t,where LMHCHW is control limit coefficient, which decides the predefined false alarm rate. The MHCHW chart will detect OOC signal if MHCHWt+>UCLMHCHWt or MHCHWt+>UCLMHCHWt.


## 4. Performance Evaluation Measures

This 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.


### 4.1. Average Run Length

Usually, theARL measures are used to assess the performance of charts. The ARL can be classified into two types, that is, ARL0 andARL1. ARL0 is an IC ARL and ARL1 is an OOCARL. If a process is working IC scenario, the desired ARL0 should be large enough to avoid frequent false alarms. However, ARL1should be small enough; it quickly detects the shift. A chart is preferred when it has smaller ARL1 than that of its competitors.


### 4.2. Overall Performance Assessment Measures

The 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. The specifics of these approaches are outlined in the subsequent subsections.


#### 4.2.1. Extra Quadratic Loss

The 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:(21)EQL=δmax−δmin−1∫δminδmaxδ2ARLδdδ,whereδmin and δmax are the minimum and maximum shift values, respectively. ARLδ is the ARL value at specific δ. The chart with higher EQL values is assumed to be less effective than the other charts.


#### 4.2.2. Relative Average Run Length

Mathematically,RARL can be defined as(22)RARL=δmax−δmin−1∫δminδmaxARLδARLδ∗dδ,where ARLδ is the ARL of competing chart and ARLδ∗ is the ARL of the benchmark chart at shift δ. The benchmark chart's RARL measure is equal to one but it is higher than one for other contending charts [16].


#### 4.2.3. 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:(23)PCI=EQLEQLbenchmark.ThePCI of the benchmark chart should be 1. If the PCI > 1, the benchmark chart is superior to the competing charts.


### 4.3. Monte Carlo Simulation

ObservationsXit (where i=1,2,…,n and t>1) are normally distributed of different parameters with increasing shifts in the process location. The nature of the shift is as follows; μ toμ+δσ, whereδ= 0.25, 0.50, 0.75, 1.00, 1.50, and 2.00. The Monte Carlo simulation methods are used for the computation of the numerical results. The simulation work is done in the R software that consists of 50,000 replications at shift δ. The numerical results are presented in Tables 1–6. The 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 ofλ(iii)
Assume the value ofh(iv)
Calculate the plotting statistics of theMHWHC chart as given in equations (13) and (14)(v)
Calculate the control limitHMHWHCt as mentioned in equation (16)(vi)
Plot theMHWHCt± statistics against the control limit HMHWHCt(vii)
If the plotting statisticMHWHCt+>HMHWHCt or MHWHCt−>HMHWHCt, then note this sample number as an RL(viii)
Repeat this process from step (i) to step (vii)105 times to obtain ARL0(ix)
To compute theARL1 measures, again draw a sample from a normal distribution using shifted location parameter and repeat this process from step (ii) to step (viii)Table 1
ARL measures of the proposed MHWHC and MHCHW charts fork = 0.50 at ARL0≅370.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00371.46369.44370.27369.49370.07370.07369.93370.410.05167.51205.47260.52290.37131.23214.42267.72294.620.1083.44115.9173.07223.2263.05133.79194.03232.000.2031.7044.580.48126.5523.2860.82102.22139.220.2522.5630.8157.2393.7917.0043.5676.46106.200.508.209.6414.9826.176.2314.0322.5933.550.754.805.216.7910.523.847.169.6213.431.003.503.584.115.642.934.595.517.041.502.472.462.492.812.252.782.913.162.002.052.031.992.061.972.132.102.152.501.751.761.741.751.761.791.761.773.001.491.531.521.521.561.541.531.525.001.021.031.031.031.031.031.021.02h2.4352.5542.5492.53LMHCHW1.5624.20755.20715.508Table 2
ARL measures of the proposed MHWHC and MHCHW charts fork=1.00 at ARL0≅370.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00370.35371.67371.67369.74369.23372.65370.74372.240.05155.94202.11258.60290.77133.95213.15256.82291.530.1074.15108.90174.51222.1964.04131.33182.08226.320.2027.0441.2179.49124.2623.7360.6497.64135.690.2519.7428.8655.1592.3717.0943.6672.64104.530.507.208.8514.5526.646.214.1221.8332.950.754.274.746.5410.663.847.179.7813.251.003.223.374.045.602.954.595.56.851.502.372.372.452.792.262.782.873.132.002.012.001.992.051.972.132.12.162.501.761.771.741.741.771.791.761.773.001.521.541.531.521.561.531.521.525.001.021.031.021.021.031.021.021.02h2.0192.08322.05612.031LMHCHW1.5544.17085.155.475Table 3
ARL measures of the proposed MHWHC and MHCHW charts fork = 1.50 at ARL0≅370.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00371.57370.54369.34370.54369.41369.53370.52368.880.05144.82196.86256.09291.18130.34215.29268.06283.910.1065.49103.49172.99222.2061.98133.26190.46220.170.2023.4438.6578.57124.1022.9859.86101.38132.170.2516.7626.8953.9692.3617.0143.9774.98104.250.506.047.9014.3426.546.1614.1622.3332.440.753.834.416.3410.603.777.159.7613.051.002.953.183.915.542.904.625.576.781.502.282.292.392.772.252.792.883.142.002.001.971.972.051.972.132.112.152.501.771.771.741.741.761.791.761.763.001.551.541.531.521.561.531.531.525.001.031.031.031.021.031.021.021.02h1.6061.61081.5621.5324LMHCHW1.5184.11545.22055.442Table 4
ARL measures of the proposed MHWHC and MHCHW charts fork = 0.50 at ARL0≅500.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00500.19501.94500.70499.35500.38501.62500.37499.480.05204.54265.67345.65391.52167.23283.44359.71432.540.1092.05141.27227.74292.3076.58169.05254.55337.570.2035.1150.6598.93159.4727.3672.67130.21196.020.2524.6934.5366.55118.7119.3752.3495.89148.720.508.6610.2516.6931.386.8115.7025.9943.210.754.955.477.2111.884.057.7710.6616.091.003.623.704.296.103.034.935.947.831.502.552.512.532.902.312.862.973.332.002.102.072.032.102.002.192.152.252.501.801.811.781.791.801.841.791.813.001.561.581.561.561.591.581.561.575.001.031.031.031.031.041.031.031.03h2.5632.6552.6392.6195LMHCHW1.7834.4435.4285.7288Table 5
ARL measures of the proposed MHWHC and MHCHW charts fork = 1.00 at ARL0≅500.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00499.87501.18500.67501.18500.37501.06500.88503.330.05191.60262.10344.30392.73193.17282.39360.03412.470.1083.05136.10226.78292.8596.16168.26254.38322.710.2030.3848.0197.65159.9837.7671.95129.39187.910.2521.2032.4065.23118.7726.7451.7395.08141.020.507.539.5815.9931.129.3815.4525.5841.430.754.455.017.0311.765.257.6710.3815.691.003.303.464.176.093.704.865.857.761.502.422.402.492.892.562.872.933.322.002.052.022.022.092.082.172.142.212.501.801.801.771.771.791.841.801.823.001.581.581.561.561.551.571.561.565.001.031.041.031.031.031.031.031.03h2.136332.182.14462.1215LMHCHW4.7636.86857.5565.75Table 6
ARL measures of the proposed MHWHC and MHCHW charts fork = 1.50 at ARL0≅500.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00499.76501.00499.90499.43502.83499.48500.09498.390.05177.17255.11342.56391.34194.25278.94358.87382.760.1074.05130.68224.51292.3596.77164.08253.16299.960.2026.3245.6396.00159.2237.4568.46127.62173.760.2518.4729.5564.30118.2126.4948.9294.05135.590.506.528.5215.5031.019.1514.4924.8539.680.753.994.636.6711.715.167.0710.2015.021.003.063.254.036.053.714.515.587.611.502.332.332.452.862.572.722.873.262.002.022.002.002.102.102.122.122.202.501.811.801.771.771.791.811.791.803.001.591.591.571.561.551.581.571.565.001.041.031.031.031.031.031.031.03h1.71431.70611.65021.6213LMHCHW8.65510.3210.72565.7257Similarly, the procedure to construct the proposedMHCHW chart is given in the following steps:(i)
Draw a sample from a normal distribution using IC parameters(ii)
Select specific values of design parametersλandLMHCHW(iii)
Determine plotting statistics of theMHCHW chart from equations (17) and (18)(iv)
Find out the upper control limitUCLMHCHWt from equation (20)(v)
Plot the statisticsMHCHWt± against the upper control limit UCLMHCHWt(vi)
Record the sample number as anRL whenever MHCHWt+>UCLMHCHWt or MHCHWt−>UCLMHCHWt(vii)
Continue this process105 times and record RLs(viii)
Find the average of105 times noted as RLs, called ARL0(ix)
ForARL1values, take a random sample from a normal distribution with shifted process location and repeat the steps from (ii) to (viii)


### 4.4. Effect of Parameter Choices

Parametersλ,h,andk of the MHWHC and λ,k,andLMHCHW of the MHCHW charts have a certain impact on the detection ability. In this regard, various settings of the parameters are used and hence corresponding ARLs are calculated. Parameter λ is taken as 0.10,0.25,0.50, and 0.75, and k is chosen as 0.50,1.00,and1.50. Similarly, h and LMHCHW are chosen in such a way to obtain prespecified ARL0≅370 and 500. Numerical results of the proposed MHWHC andMHCHW charts are given in Tables 1–6.


### 4.5. Special Cases of the ProposedMHWHC and MHCHW Charts

The proposedMHWHC and MHCHW charts are reduced to the HCUSUM chart for the specific parameters' values. Exceptional cases are presented here with their testimonies.


#### 4.5.1. A Special Case of the Proposed MHWHC Chart

The proposed MHWHC chart is reduced to theHCUSUM chart for λ=1.Proof:.
The plotting statistics of the proposedMHWHC chart are(24)MHWHCt+=max0,Et−μ0−KMHWHCt+X¯¯t−1,MHWHCt−=max0,Et−μ0−KMHWHCt+X¯¯t−1,and Et, defined in equation (9), is given as(25)Et=λX¯t+1−λX¯¯t−1.
Whenλ=1, Etcan be written as(26)Et∗=X¯t.
Replacing the resulting value ofEt∗ in statistics of the proposed MHWHC chart, it will become(27)MHWHCt+1=max0,Et∗−μ0−KMHWHCt+X¯¯t−1,(28)MHWHCt−1=max0,Et∗−μ0−KMHWHCt+X¯¯t−1,
and equations (27) and (28) show that the proposed MHWHC chart approaches the HCUSUM chart when λ=1.


#### 4.5.2. A Special Case of the Proposed MHCHW Chart

The proposed MHCHW chart reduces to theHCUSUM chart for λ=1.Proof.
The plotting statistics of the proposedMHCHW chart are(29)MHCHWt+=λHCt++1−λMHCHWt−1+,MHCHWt−=λHCt−+1−λMHCHWt−1−.
Whenλ=1, the above statistics are reduced to(30)MHCHWt+2=HCt+,(31)MHCHWt−2=HCt−.
The 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, theMHWHC and MHCHW charts reduce to the HCUSUM chart whenλ=1. Thus, if someone is interested in obtaining ARL values of the HCUSUM chart, he will directly use the MHWHC or MHCHW chart with λ=1.


### 4.6. Sensitivity Analysis of the Parameters

A sensitivity analysis is used to examine the impact of different parameter values on the performance of the proposed MHWHC and MHCHW charts. The following are the important findings of the proposed charts:(i)
The proposedMHWHC and MHCHW charts are very sensitive to λ values. A small change in the λ values significantly affects the ARL performance. For instance, at δ=0.05,k=0.50, and ARL0=370, the ARL values of the proposed MHWHC chart are 205.47 and 167.51 for λ=0.25 and λ=0.10, respectively (see Table 1). Similarly, at δ=0.05,k=0.50, and ARL0=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).(ii)
For allλ, the ARL values of the proposed MHWHC and MHCHW charts are smaller than the ARL values of the classical CUSUM and EWMA, MEC, MCE, and HWMA charts (see Sections 5.1–5.5).(iii)
The performance of the proposedMHWHC and MHCHW charts increases as k increases. For example, at δ=0.10,λ=0.10, and ARL0=370, the ARL value of the proposed MHWHC chart is 74.15 and 65.49 for k=1.00 and k=1.50, respectively (see Tables 2 and 3). Likewise, for δ=0.05,λ=0.10, and ARL0=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)
The overall performance measures show that the proposed MHWHC and MHCHW charts are superior to existing (used in this study) charts for allλ (see Table 7).Table 7
EQL, PCI, and RARL measures of the proposed versus existing charts.
Measuresλ=0.10MHWHCMHCHWHWMAMECMCEEWMACUSUMk=0.50k=0.50k=0.50k=0.50k=0.50EQL3.983.528.7920.3412.067.8810.27PCI1.131.002.505.783.432.242.92RARL1.171.002.885.333.902.803.41λ=0.25EQL4.125.079.2815.2010.6710.2710.27PCI1.001.232.263.692.592.502.50RARL1.001.292.433.362.742.882.59λ=0.5EQL4.796.1713.2512.1010.2015.7010.27PCI1.001.292.772.532.133.282.14RARL1.001.312.852.362.093.402.08λ=0.75EQL6.578.1123.2511.139.4124.7410.27PCI1.001.243.541.701.433.771.56RARL1.001.223.441.661.423.661.54


## 4.1. Average Run Length

Usually, theARL measures are used to assess the performance of charts. The ARL can be classified into two types, that is, ARL0 andARL1. ARL0 is an IC ARL and ARL1 is an OOCARL. If a process is working IC scenario, the desired ARL0 should be large enough to avoid frequent false alarms. However, ARL1should be small enough; it quickly detects the shift. A chart is preferred when it has smaller ARL1 than that of its competitors.


## 4.2. Overall Performance Assessment Measures

The 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. The specifics of these approaches are outlined in the subsequent subsections.


### 4.2.1. Extra Quadratic Loss

The 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:(21)EQL=δmax−δmin−1∫δminδmaxδ2ARLδdδ,whereδmin and δmax are the minimum and maximum shift values, respectively. ARLδ is the ARL value at specific δ. The chart with higher EQL values is assumed to be less effective than the other charts.


### 4.2.2. Relative Average Run Length

Mathematically,RARL can be defined as(22)RARL=δmax−δmin−1∫δminδmaxARLδARLδ∗dδ,where ARLδ is the ARL of competing chart and ARLδ∗ is the ARL of the benchmark chart at shift δ. The benchmark chart's RARL measure is equal to one but it is higher than one for other contending charts [16].


### 4.2.3. 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:(23)PCI=EQLEQLbenchmark.ThePCI of the benchmark chart should be 1. If the PCI > 1, the benchmark chart is superior to the competing charts.


## 4.2.1. Extra Quadratic Loss

The 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:(21)EQL=δmax−δmin−1∫δminδmaxδ2ARLδdδ,whereδmin and δmax are the minimum and maximum shift values, respectively. ARLδ is the ARL value at specific δ. The chart with higher EQL values is assumed to be less effective than the other charts.


## 4.2.2. Relative Average Run Length

Mathematically,RARL can be defined as(22)RARL=δmax−δmin−1∫δminδmaxARLδARLδ∗dδ,where ARLδ is the ARL of competing chart and ARLδ∗ is the ARL of the benchmark chart at shift δ. The benchmark chart's RARL measure is equal to one but it is higher than one for other contending charts [16].


## 4.2.3. 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:(23)PCI=EQLEQLbenchmark.ThePCI of the benchmark chart should be 1. If the PCI > 1, the benchmark chart is superior to the competing charts.


## 4.3. Monte Carlo Simulation

ObservationsXit (where i=1,2,…,n and t>1) are normally distributed of different parameters with increasing shifts in the process location. The nature of the shift is as follows; μ toμ+δσ, whereδ= 0.25, 0.50, 0.75, 1.00, 1.50, and 2.00. The Monte Carlo simulation methods are used for the computation of the numerical results. The simulation work is done in the R software that consists of 50,000 replications at shift δ. The numerical results are presented in Tables 1–6. The 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 ofλ(iii)
Assume the value ofh(iv)
Calculate the plotting statistics of theMHWHC chart as given in equations (13) and (14)(v)
Calculate the control limitHMHWHCt as mentioned in equation (16)(vi)
Plot theMHWHCt± statistics against the control limit HMHWHCt(vii)
If the plotting statisticMHWHCt+>HMHWHCt or MHWHCt−>HMHWHCt, then note this sample number as an RL(viii)
Repeat this process from step (i) to step (vii)105 times to obtain ARL0(ix)
To compute theARL1 measures, again draw a sample from a normal distribution using shifted location parameter and repeat this process from step (ii) to step (viii)Table 1
ARL measures of the proposed MHWHC and MHCHW charts fork = 0.50 at ARL0≅370.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00371.46369.44370.27369.49370.07370.07369.93370.410.05167.51205.47260.52290.37131.23214.42267.72294.620.1083.44115.9173.07223.2263.05133.79194.03232.000.2031.7044.580.48126.5523.2860.82102.22139.220.2522.5630.8157.2393.7917.0043.5676.46106.200.508.209.6414.9826.176.2314.0322.5933.550.754.805.216.7910.523.847.169.6213.431.003.503.584.115.642.934.595.517.041.502.472.462.492.812.252.782.913.162.002.052.031.992.061.972.132.102.152.501.751.761.741.751.761.791.761.773.001.491.531.521.521.561.541.531.525.001.021.031.031.031.031.031.021.02h2.4352.5542.5492.53LMHCHW1.5624.20755.20715.508Table 2
ARL measures of the proposed MHWHC and MHCHW charts fork=1.00 at ARL0≅370.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00370.35371.67371.67369.74369.23372.65370.74372.240.05155.94202.11258.60290.77133.95213.15256.82291.530.1074.15108.90174.51222.1964.04131.33182.08226.320.2027.0441.2179.49124.2623.7360.6497.64135.690.2519.7428.8655.1592.3717.0943.6672.64104.530.507.208.8514.5526.646.214.1221.8332.950.754.274.746.5410.663.847.179.7813.251.003.223.374.045.602.954.595.56.851.502.372.372.452.792.262.782.873.132.002.012.001.992.051.972.132.12.162.501.761.771.741.741.771.791.761.773.001.521.541.531.521.561.531.521.525.001.021.031.021.021.031.021.021.02h2.0192.08322.05612.031LMHCHW1.5544.17085.155.475Table 3
ARL measures of the proposed MHWHC and MHCHW charts fork = 1.50 at ARL0≅370.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00371.57370.54369.34370.54369.41369.53370.52368.880.05144.82196.86256.09291.18130.34215.29268.06283.910.1065.49103.49172.99222.2061.98133.26190.46220.170.2023.4438.6578.57124.1022.9859.86101.38132.170.2516.7626.8953.9692.3617.0143.9774.98104.250.506.047.9014.3426.546.1614.1622.3332.440.753.834.416.3410.603.777.159.7613.051.002.953.183.915.542.904.625.576.781.502.282.292.392.772.252.792.883.142.002.001.971.972.051.972.132.112.152.501.771.771.741.741.761.791.761.763.001.551.541.531.521.561.531.531.525.001.031.031.031.021.031.021.021.02h1.6061.61081.5621.5324LMHCHW1.5184.11545.22055.442Table 4
ARL measures of the proposed MHWHC and MHCHW charts fork = 0.50 at ARL0≅500.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00500.19501.94500.70499.35500.38501.62500.37499.480.05204.54265.67345.65391.52167.23283.44359.71432.540.1092.05141.27227.74292.3076.58169.05254.55337.570.2035.1150.6598.93159.4727.3672.67130.21196.020.2524.6934.5366.55118.7119.3752.3495.89148.720.508.6610.2516.6931.386.8115.7025.9943.210.754.955.477.2111.884.057.7710.6616.091.003.623.704.296.103.034.935.947.831.502.552.512.532.902.312.862.973.332.002.102.072.032.102.002.192.152.252.501.801.811.781.791.801.841.791.813.001.561.581.561.561.591.581.561.575.001.031.031.031.031.041.031.031.03h2.5632.6552.6392.6195LMHCHW1.7834.4435.4285.7288Table 5
ARL measures of the proposed MHWHC and MHCHW charts fork = 1.00 at ARL0≅500.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00499.87501.18500.67501.18500.37501.06500.88503.330.05191.60262.10344.30392.73193.17282.39360.03412.470.1083.05136.10226.78292.8596.16168.26254.38322.710.2030.3848.0197.65159.9837.7671.95129.39187.910.2521.2032.4065.23118.7726.7451.7395.08141.020.507.539.5815.9931.129.3815.4525.5841.430.754.455.017.0311.765.257.6710.3815.691.003.303.464.176.093.704.865.857.761.502.422.402.492.892.562.872.933.322.002.052.022.022.092.082.172.142.212.501.801.801.771.771.791.841.801.823.001.581.581.561.561.551.571.561.565.001.031.041.031.031.031.031.031.03h2.136332.182.14462.1215LMHCHW4.7636.86857.5565.75Table 6
ARL measures of the proposed MHWHC and MHCHW charts fork = 1.50 at ARL0≅500.
MHWHCMHCHWλλShift0.100.250.500.750.100.250.500.750.00499.76501.00499.90499.43502.83499.48500.09498.390.05177.17255.11342.56391.34194.25278.94358.87382.760.1074.05130.68224.51292.3596.77164.08253.16299.960.2026.3245.6396.00159.2237.4568.46127.62173.760.2518.4729.5564.30118.2126.4948.9294.05135.590.506.528.5215.5031.019.1514.4924.8539.680.753.994.636.6711.715.167.0710.2015.021.003.063.254.036.053.714.515.587.611.502.332.332.452.862.572.722.873.262.002.022.002.002.102.102.122.122.202.501.811.801.771.771.791.811.791.803.001.591.591.571.561.551.581.571.565.001.041.031.031.031.031.031.031.03h1.71431.70611.65021.6213LMHCHW8.65510.3210.72565.7257Similarly, the procedure to construct the proposedMHCHW chart is given in the following steps:(i)
Draw a sample from a normal distribution using IC parameters(ii)
Select specific values of design parametersλandLMHCHW(iii)
Determine plotting statistics of theMHCHW chart from equations (17) and (18)(iv)
Find out the upper control limitUCLMHCHWt from equation (20)(v)
Plot the statisticsMHCHWt± against the upper control limit UCLMHCHWt(vi)
Record the sample number as anRL whenever MHCHWt+>UCLMHCHWt or MHCHWt−>UCLMHCHWt(vii)
Continue this process105 times and record RLs(viii)
Find the average of105 times noted as RLs, called ARL0(ix)
ForARL1values, take a random sample from a normal distribution with shifted process location and repeat the steps from (ii) to (viii)


## 4.4. Effect of Parameter Choices

Parametersλ,h,andk of the MHWHC and λ,k,andLMHCHW of the MHCHW charts have a certain impact on the detection ability. In this regard, various settings of the parameters are used and hence corresponding ARLs are calculated. Parameter λ is taken as 0.10,0.25,0.50, and 0.75, and k is chosen as 0.50,1.00,and1.50. Similarly, h and LMHCHW are chosen in such a way to obtain prespecified ARL0≅370 and 500. Numerical results of the proposed MHWHC andMHCHW charts are given in Tables 1–6.


## 4.5. Special Cases of the ProposedMHWHC and MHCHW Charts

The proposedMHWHC and MHCHW charts are reduced to the HCUSUM chart for the specific parameters' values. Exceptional cases are presented here with their testimonies.


### 4.5.1. A Special Case of the Proposed MHWHC Chart

The proposed MHWHC chart is reduced to theHCUSUM chart for λ=1.Proof:.
The plotting statistics of the proposedMHWHC chart are(24)MHWHCt+=max0,Et−μ0−KMHWHCt+X¯¯t−1,MHWHCt−=max0,Et−μ0−KMHWHCt+X¯¯t−1,and Et, defined in equation (9), is given as(25)Et=λX¯t+1−λX¯¯t−1.
Whenλ=1, Etcan be written as(26)Et∗=X¯t.
Replacing the resulting value ofEt∗ in statistics of the proposed MHWHC chart, it will become(27)MHWHCt+1=max0,Et∗−μ0−KMHWHCt+X¯¯t−1,(28)MHWHCt−1=max0,Et∗−μ0−KMHWHCt+X¯¯t−1,
and equations (27) and (28) show that the proposed MHWHC chart approaches the HCUSUM chart when λ=1.


### 4.5.2. A Special Case of the Proposed MHCHW Chart

The proposed MHCHW chart reduces to theHCUSUM chart for λ=1.Proof.
The plotting statistics of the proposedMHCHW chart are(29)MHCHWt+=λHCt++1−λMHCHWt−1+,MHCHWt−=λHCt−+1−λMHCHWt−1−.
Whenλ=1, the above statistics are reduced to(30)MHCHWt+2=HCt+,(31)MHCHWt−2=HCt−.
The 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, theMHWHC and MHCHW charts reduce to the HCUSUM chart whenλ=1. Thus, if someone is interested in obtaining ARL values of the HCUSUM chart, he will directly use the MHWHC or MHCHW chart with λ=1.


## 4.5.1. A Special Case of the Proposed MHWHC Chart

The proposed MHWHC chart is reduced to theHCUSUM chart for λ=1.Proof:.
The plotting statistics of the proposedMHWHC chart are(24)MHWHCt+=max0,Et−μ0−KMHWHCt+X¯¯t−1,MHWHCt−=max0,Et−μ0−KMHWHCt+X¯¯t−1,and Et, defined in equation (9), is given as(25)Et=λX¯t+1−λX¯¯t−1.
Whenλ=1, Etcan be written as(26)Et∗=X¯t.
Replacing the resulting value ofEt∗ in statistics of the proposed MHWHC chart, it will become(27)MHWHCt+1=max0,Et∗−μ0−KMHWHCt+X¯¯t−1,(28)MHWHCt−1=max0,Et∗−μ0−KMHWHCt+X¯¯t−1,
and equations (27) and (28) show that the proposed MHWHC chart approaches the HCUSUM chart when λ=1.


## 4.5.2. A Special Case of the Proposed MHCHW Chart

The proposed MHCHW chart reduces to theHCUSUM chart for λ=1.Proof.
The plotting statistics of the proposedMHCHW chart are(29)MHCHWt+=λHCt++1−λMHCHWt−1+,MHCHWt−=λHCt−+1−λMHCHWt−1−.
Whenλ=1, the above statistics are reduced to(30)MHCHWt+2=HCt+,(31)MHCHWt−2=HCt−.
The 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, theMHWHC and MHCHW charts reduce to the HCUSUM chart whenλ=1. Thus, if someone is interested in obtaining ARL values of the HCUSUM chart, he will directly use the MHWHC or MHCHW chart with λ=1.


## 4.6. Sensitivity Analysis of the Parameters

A sensitivity analysis is used to examine the impact of different parameter values on the performance of the proposed MHWHC and MHCHW charts. The following are the important findings of the proposed charts:(i)
The proposedMHWHC and MHCHW charts are very sensitive to λ values. A small change in the λ values significantly affects the ARL performance. For instance, at δ=0.05,k=0.50, and ARL0=370, the ARL values of the proposed MHWHC chart are 205.47 and 167.51 for λ=0.25 and λ=0.10, respectively (see Table 1). Similarly, at δ=0.05,k=0.50, and ARL0=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).(ii)
For allλ, the ARL values of the proposed MHWHC and MHCHW charts are smaller than the ARL values of the classical CUSUM and EWMA, MEC, MCE, and HWMA charts (see Sections 5.1–5.5).(iii)
The performance of the proposedMHWHC and MHCHW charts increases as k increases. For example, at δ=0.10,λ=0.10, and ARL0=370, the ARL value of the proposed MHWHC chart is 74.15 and 65.49 for k=1.00 and k=1.50, respectively (see Tables 2 and 3). Likewise, for δ=0.05,λ=0.10, and ARL0=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)
The overall performance measures show that the proposed MHWHC and MHCHW charts are superior to existing (used in this study) charts for allλ (see Table 7).Table 7
EQL, PCI, and RARL measures of the proposed versus existing charts.
Measuresλ=0.10MHWHCMHCHWHWMAMECMCEEWMACUSUMk=0.50k=0.50k=0.50k=0.50k=0.50EQL3.983.528.7920.3412.067.8810.27PCI1.131.002.505.783.432.242.92RARL1.171.002.885.333.902.803.41λ=0.25EQL4.125.079.2815.2010.6710.2710.27PCI1.001.232.263.692.592.502.50RARL1.001.292.433.362.742.882.59λ=0.5EQL4.796.1713.2512.1010.2015.7010.27PCI1.001.292.772.532.133.282.14RARL1.001.312.852.362.093.402.08λ=0.75EQL6.578.1123.2511.139.4124.7410.27PCI1.001.243.541.701.433.771.56RARL1.001.223.441.661.423.661.54


## 5. Evaluation and Performance Comparison

This section contains a detailed performance comparison of the proposedMHWHC and MHCHW charts with the existing charts. These include the classical CUSUM [2], the classical EWMA [3], MEC [14], MCE [9], and HWMA [22] charts.


### 5.1. Proposed versus Classical CUSUM Charts

It is observed that the proposedMHWHC and MHCHW charts work well, unlike the classical CUSUM chart. To illustrate atλ=0.50,k=0.50,andδ=0.50, the ARL1 values of the proposed MHWHC and MHCHW charts are 16.69 and25.99, respectively, while the ARL1 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).Table 8
ARL measures of the existing charts at ARL0≅500.
λ0.000.250.500.751.001.502.00HWMA0.10499.7881.4828.6114.859.354.983.320.25498.06113.3433.7916.199.714.943.200.50498.56218.0669.0427.8714.085.673.200.75499.89311.99132.5258.1528.339.154.25MEC (k=0.50)0.10501.0079.0035.0024.0019.0014.0011.000.25502.0083.0031.0019.0014.0010.008.000.50503.00100.0030.0017.0012.007.006.000.75499.00120.0034.0017.0011.006.005.00MCE (k=0.50)0.10499.00126.0036.0018.0012.007.005.000.25500.00137.0037.0017.0010.006.004.000.50503.00144.0038.0017.0010.005.004.000.75499.00144.0037.0016.009.005.003.00CUSUM (k=0.50)0.50500.09108.9734.0916.4010.275.794.04EWMA0.10500.00103.3228.8113.618.214.172.660.25500.00169.3447.3619.3110.414.782.940.50500.00254.5388.4335.5717.186.273.390.75500.00320.92140.1762.4030.509.814.46Figure 1
ARL comparison of HWMA,MEC,MCE,EWMA,and CUSUM charts with the proposed MHWHC chart at λ=0.10,k=0.50, and ARL≅500.Figure 2
ARL comparison of HWMA,MEC,MCE,EWMA,and CUSUM charts with the proposed MHCHW chart at λ=0.10,k=0.50, and ARL≅500.


### 5.2. Proposed versus Classical EWMA Chart

The proposedMHWHC and MHCHW charts are more responsive than the EWMA chart to detect a shift in the process location. For all combinations of shift (δandλ, the ARL1 values for the proposed charts are lower than the ARL1 values of classical EWMA chart. For example, for λ=0.10,δ=0.25, and k=0.50, ARL1 value of the classical EWMA chart is 103.32, whereas ARL1 of the proposed MHWHC and MHCHW charts are 24.69and19.37, 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, atλ=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).


### 5.3. Proposed versus MEC Chart

The comparison of the proposedMHWHC and MHCHW charts to the MEC chart shows that the proposed MHWHC and MHCHW charts are superior. For example, at λ=0.25,δ=0.25,0.50,0.75,1.00,1.50,2.00, the ARL1 values for the MEC chart are 83, 31, 19, 14, 10, and 8, whereas, for MHWHC and MHCHW charts, the ARL1 values are 34.53, 10.25, 5.47, 3.7, 2.51, and 2.07 and 52.34, 15.7, 7.77, 4.93, 2.86, and 2.19, respectively (see Table 4 versus Table 8). Likewise, the proposed MHWHC and MHCHW charts’ efficiency against the MEC chart is demonstrated in Figures 1 and 2. Similarly, in overall performance measures, the proposed MHWHC and MHCHW charts also show better performance compared to the MEC chart. As an illustration, at λ=0.10,andk=0.50,the EQL, PCI, and RARL values of MHWHC,MHCHW, and MEC charts are 3.98,1.13,1.17, (3.52,1.00,1.00, and 20.34,5.78,5.33, respectively (see Table 7).


### 5.4. Proposed versus MCE Chart

The proposedMHWHC and MHCHW charts have smaller ARL1 values for all λ in comparison with the MCE chart. As illustration at λ=0.10 and δ=0.25,0.50,0.75, ARL1 for the proposed charts are 24.69, 8.66, and 4.95 and 19.37, 6.81, and 4.05, respectively, whereas ARL1 for the MCE chart are 127.63, 36.03, and 17.61 (see Tables 4 versus 8). Similarly, Figures 2 and 3 also highlight the MHWHC and MHCHW charts’ superiority over the MCE chart. The proposed MHWHC and MHCHW charts also show better performance compared to the MCE chart based on the EQL,PCI,and RARL measures. To illustrate, at λ=0.50 and k=0.50, the EQL, PCI, and RARL values of the MHWHC and MHCHW charts are 4.79,1.00,1.00, and 6.17,1.29,1.31, respectively, whereas for the MCE chart they are 10.11,2.13,and2.09 (see Table 7).Figure 3
ARL measures of the proposed MHWHC chart at different values of λ.


### 5.5. Proposed versus HWMA Chart

The proposedMHWHC and MHCHW charts have smaller ARL1 values than the HWMA chart for λ. For example, if δ=0.25andλ=0.10,ARL1 of the proposed MHWHC and MHCHW charts are 24.69 and 19.37, respectively, while the HWMA chart has ARL1 value of 81.48 (see 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. The RARL values of MHWHC, MHCHW, and HWMA charts at λ=0.10 are 1.17,1.00and2.90, respectively (see Table 7). They indicate the inferiority of the HWMA chart.Figure 4
ARL measures of the proposed MHCHW chart at different values of λ.


## 5.1. Proposed versus Classical CUSUM Charts

It is observed that the proposedMHWHC and MHCHW charts work well, unlike the classical CUSUM chart. To illustrate atλ=0.50,k=0.50,andδ=0.50, the ARL1 values of the proposed MHWHC and MHCHW charts are 16.69 and25.99, respectively, while the ARL1 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).Table 8
ARL measures of the existing charts at ARL0≅500.
λ0.000.250.500.751.001.502.00HWMA0.10499.7881.4828.6114.859.354.983.320.25498.06113.3433.7916.199.714.943.200.50498.56218.0669.0427.8714.085.673.200.75499.89311.99132.5258.1528.339.154.25MEC (k=0.50)0.10501.0079.0035.0024.0019.0014.0011.000.25502.0083.0031.0019.0014.0010.008.000.50503.00100.0030.0017.0012.007.006.000.75499.00120.0034.0017.0011.006.005.00MCE (k=0.50)0.10499.00126.0036.0018.0012.007.005.000.25500.00137.0037.0017.0010.006.004.000.50503.00144.0038.0017.0010.005.004.000.75499.00144.0037.0016.009.005.003.00CUSUM (k=0.50)0.50500.09108.9734.0916.4010.275.794.04EWMA0.10500.00103.3228.8113.618.214.172.660.25500.00169.3447.3619.3110.414.782.940.50500.00254.5388.4335.5717.186.273.390.75500.00320.92140.1762.4030.509.814.46Figure 1
ARL comparison of HWMA,MEC,MCE,EWMA,and CUSUM charts with the proposed MHWHC chart at λ=0.10,k=0.50, and ARL≅500.Figure 2
ARL comparison of HWMA,MEC,MCE,EWMA,and CUSUM charts with the proposed MHCHW chart at λ=0.10,k=0.50, and ARL≅500.


## 5.2. Proposed versus Classical EWMA Chart

The proposedMHWHC and MHCHW charts are more responsive than the EWMA chart to detect a shift in the process location. For all combinations of shift (δandλ, the ARL1 values for the proposed charts are lower than the ARL1 values of classical EWMA chart. For example, for λ=0.10,δ=0.25, and k=0.50, ARL1 value of the classical EWMA chart is 103.32, whereas ARL1 of the proposed MHWHC and MHCHW charts are 24.69and19.37, 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, atλ=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).


## 5.3. Proposed versus MEC Chart

The comparison of the proposedMHWHC and MHCHW charts to the MEC chart shows that the proposed MHWHC and MHCHW charts are superior. For example, at λ=0.25,δ=0.25,0.50,0.75,1.00,1.50,2.00, the ARL1 values for the MEC chart are 83, 31, 19, 14, 10, and 8, whereas, for MHWHC and MHCHW charts, the ARL1 values are 34.53, 10.25, 5.47, 3.7, 2.51, and 2.07 and 52.34, 15.7, 7.77, 4.93, 2.86, and 2.19, respectively (see Table 4 versus Table 8). Likewise, the proposed MHWHC and MHCHW charts’ efficiency against the MEC chart is demonstrated in Figures 1 and 2. Similarly, in overall performance measures, the proposed MHWHC and MHCHW charts also show better performance compared to the MEC chart. As an illustration, at λ=0.10,andk=0.50,the EQL, PCI, and RARL values of MHWHC,MHCHW, and MEC charts are 3.98,1.13,1.17, (3.52,1.00,1.00, and 20.34,5.78,5.33, respectively (see Table 7).


## 5.4. Proposed versus MCE Chart

The proposedMHWHC and MHCHW charts have smaller ARL1 values for all λ in comparison with the MCE chart. As illustration at λ=0.10 and δ=0.25,0.50,0.75, ARL1 for the proposed charts are 24.69, 8.66, and 4.95 and 19.37, 6.81, and 4.05, respectively, whereas ARL1 for the MCE chart are 127.63, 36.03, and 17.61 (see Tables 4 versus 8). Similarly, Figures 2 and 3 also highlight the MHWHC and MHCHW charts’ superiority over the MCE chart. The proposed MHWHC and MHCHW charts also show better performance compared to the MCE chart based on the EQL,PCI,and RARL measures. To illustrate, at λ=0.50 and k=0.50, the EQL, PCI, and RARL values of the MHWHC and MHCHW charts are 4.79,1.00,1.00, and 6.17,1.29,1.31, respectively, whereas for the MCE chart they are 10.11,2.13,and2.09 (see Table 7).Figure 3
ARL measures of the proposed MHWHC chart at different values of λ.


## 5.5. Proposed versus HWMA Chart

The proposedMHWHC and MHCHW charts have smaller ARL1 values than the HWMA chart for λ. For example, if δ=0.25andλ=0.10,ARL1 of the proposed MHWHC and MHCHW charts are 24.69 and 19.37, respectively, while the HWMA chart has ARL1 value of 81.48 (see 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. The RARL values of MHWHC, MHCHW, and HWMA charts at λ=0.10 are 1.17,1.00and2.90, respectively (see Table 7). They indicate the inferiority of the HWMA chart.Figure 4
ARL measures of the proposed MHCHW chart at different values of λ.


## 6. Real-Life Application

The 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. This section offers the application of the proposedMHWHC 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. These subgroups are used for real-life applications to compare existing HWMA and proposed MHWHC and MHCHW charts. For monitoring phase-II, the population mean and standard deviation values are taken as μ=1.5056 and sp=0.13943, respectively [22].The parameters of the proposedMHWHC and MHCHW charts are taken as λ=025,h=2.655,k=0.50andλ=0.50,LMHCHW=3.191,h=6,k=0.50, respectively, to obtain ARL0=500, while the parameters of the existing HWMA chart are taken as λ=0.25,L=3.075,at ARL0=500. Tables 9 and 10 and Figures 5–7 show that the HWMA chart detects shifts at the 20th sample, whereas the proposed MHWHC and MHCHW charts detect shifts at the 18th sample. Moreover, the proposed MHWHC and MHCHW charts detect three OOC points, whereas the HWMA chart detects only one OOC point. Hence, the performances of the proposed MHWHC and MHCHW charts are superior to that of the existing HWMA chart. Similarly, the proposed charts can also be compared with other charts.Table 9
Application of the proposed MHWHC and MHCHW charts versus HWMA chart.
X1X2X3X4X511.4481.5461.4541.4301.62121.5441.6901.5831.3361.41931.5181.3451.4721.6661.66641.5451.0931.4071.5041.52651.4421.5061.5121.4621.62661.4301.2731.5951.5401.52571.4981.4511.6171.5841.49681.3011.5061.6231.5831.64591.4131.4601.5811.7111.731101.3821.3141.4951.4891.460111.5771.7011.4031.2771.454121.4941.4371.5141.4811.529131.5731.6741.5051.5651.747141.8091.5511.8251.4391.656151.6241.5391.6741.8701.504161.4121.7931.7351.6391.779171.7371.5661.4911.7811.550181.5971.7391.6831.6681.797191.4301.6541.9131.7271.437201.6221.8221.7921.6741.940Table 10
Application of the proposed MHWHC and MHCHW charts versus HWMA chart.
MHWHCMHCHWHWMAMHWHC1MHWHC2HMHWHCtMHCHW1MHCHW2UCLMHCHWtHWMAUCLHWMAtLCLHWMAt10.000.000.111.511.511.561.501.461.5520.000.000.341.491.491.661.501.351.6630.000.000.241.511.491.621.511.391.6240.000.000.201.491.531.611.491.411.6050.000.000.171.481.481.601.501.421.5960.000.000.151.481.491.591.491.431.5970.000.000.141.491.481.581.501.431.5880.000.000.131.491.481.581.511.431.5890.000.000.121.511.471.581.521.441.58100.000.000.121.481.521.581.491.441.57110.000.000.111.491.501.571.501.441.57120.000.000.111.491.501.571.501.441.57130.000.000.101.521.461.571.531.441.57140.010.000.101.541.461.571.541.441.57150.020.000.101.541.481.571.551.441.57160.050.000.101.561.481.571.561.451.57170.070.000.091.561.501.571.561.451.57180.110.000.091.581.491.571.561.451.56190.140.000.091.571.511.571.561.451.56200.210.000.091.611.481.571.611.451.56Figure 5
The proposedMHWHC chart limits for real-life data.Figure 6
The proposedMHCHW chart limits for real-life data.Figure 7
The existing HWMA chart limits for real-life data.


## 7. 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. This 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. The 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. The 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. This 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.



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*Source: 2582210-2021-10-14.xml*