# Efficient Control Charting Scheme for the Process Location with Application in Automobile Industry

Authors: Syed Masroor Anwar; Somayya Komal; Ammara Nawaz Cheema; Nafiu Lukman Abiodun; Zahid Rasheed; Majid Khan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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 (HWMAAIB) 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 HWMAAIB and TEWMA charts and propose an AIB triple HWMA, symbolized as THWMAAIB chart to further improve the process location shift monitoring. The proposed THWMAAIB 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 (ARL), relative ARL, performance comparison index, and extra quadratic loss are used as comparison tools for the proposed THWMAAIB chart with the classical EWMA, TEWMA, mixed HWMA-CUSUM (MHC), AIB EWMA (EWMAAIB), HWMA, double HWMA (DHWMA), HWMAAIB, AIB mixed EWMA-CUSUM (MECAIB), and AIB mixed CUSUM-EWMA (MCEAIB) charts. Finally, a practical application is also provided for users to demonstrate the proposed study’s vitality.


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## Body

## 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 TEWMATBE charts, and showed that the TEWMATBE charts are more sensitive than the DEWMATBE and EWMATBE 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 S2 and named as S2-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 EWMAAIB 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 (HEWMAAIB) 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 (MCEAIB) and mixed CUSUM-EWMA MECAIB charts. Recently, a combined MECAIB 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 (HWMAAIB) 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 andHWMAAIB charts, we are aimed to combine the features of TEWMA and HWMAAIB charts and proposed a more efficient AIB triple HWMA (symbolized as THWMAAIB) 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 HWMAAIB charts features will further boost the performance of proposed THWMAAIB chart. Average run length (ARL), relative ARL (RARL), extra quadratic loss (EQL), and performance comparison index (PCI) 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, EWMAAIB, HWMA, DHWMA, HWMAAIB, MECAIB, and MCEAIB charts are considered for the comparison. Additionally, the proposed THWMAAIB 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 Section2. Additionally, Section 3 enlisted the design structure and special cases of proposed THWMAAIB chart. Furthermore, the next section provides the performance evaluation and comparison against classical EWMA, TEWMA, MHC, EWMAAIB, HWMA, HWMAAIB DHWMA,, MECAIB, and MCEAIB charts. Also, the application of THWMAAIB 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 existingHWMAAIB and TEWMA charts for process location monitoring.


### 2.1. Variable of Interest and Auxiliary Information

Assume that the process variableY is normally distributed with a mean μY+δσY and variance σY2. Let Y¯t=∑i=1nYit/n and SYt2=∑i=1nYit−Y¯t2/n−1 represents the sample mean and variance of Y of size n. If δ=0, the process is in-control (IC); otherwise, it is out-of-control (OOC). So, for the IC situation, Y¯t and SYt2 are mutually independent identically distributed. Let X be an auxiliary variable of Y. The X and Y follow a bivariate normal distribution (BND) (i.e., Y,X∼NμY,μX,σY,σX,ρ, where μX represents the mean and σX represents the standard deviation of X. Also, the ρ is the correlation coefficient corresponding to X and Y. The AIB regression estimator for process location is given as follows:(1)Rt=Y¯t+bYXμX−X¯t,where, bYX=ρσY/σX, ERt=μY and VarRt=σY2−bYX2σX2/n.


### 2.2.HWMAAIB Chart

Adegoke, et al. [28] introduced HWMAAIB chart to track changes in process location. The HWMAAIB 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:(2)Ht=λ1Rt+1−λ1R¯t−1,where Y¯t is the sample average of tth sample and λ1∈0,1 is the smoothing constant, and R¯t−1=∑i=1t−1Ri/t−1 is the mean of preceding t−1 observations The control limits of the HWMAAIB chart are:(3)LCLHWMAAIBt=μ0−LHWMAAIBσYλ2/n1−ρ2,ift=1μ0−LHWMAAIBσY1nλ2+1−λ2t−11−ρ2,ift>1UCLHWMAAIBt=μ0+LHWMAAIBσYλ2/n1−ρ2,ift=1μ0+LHWMAAIBσY1nλ2+1−λ2t−11−ρ2,ift>1.TheLHWMAAIB represents the coefficient of control limits. If Ht>UCLHWMAAIBt or Ht<LCLHWMAAIBt, 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 as(4)Et=λYt+λEt−1,DEt=λEt+λDEt−1,TEt=λDEt+λTEt−1,where λ∈0,1 is a TEWMA constant. The initial values of E0,DE0,TE0 are all equal to μ0. The mean and variance of the TEt are.ETEt=μ0 and(5)varTEt/σ2=θ3λ6/4−tt2−1t−2θt−3/1−θ−4tt2−1θt−2/1−θ2−12tt+1θt−1/1−θ3−24t+1θt/1−θ4+241−θt+1/1−θ5+2θ2λ6−tt2−1θt−2/1−θ−3tt+1θt−1/1−θ2−6t+1θt/1−θ3+61−θt+1/1−θ4+7θλ6/2−tt+1θt−1/1−θ−2t+1θt/1−θ2+21−θt+1/1−θ3+λ61−θt+1/1−θ2−t+1θt/1−θ,respectively, where θ=1−λ2. The time-varying control limits of the TEWMA chart are defined as:(6)LCLTEWMAt=μ0−LTEWMAvarTEt,UCLTEWMAt=μ0+LTEWMAvarTEt,where LTEWMALTEWMA>0 is the control limits coefficient of the TEWMA chart. The process will be OOC if any TEt>UCLTEWMAtorTEt<LCLTEWMAt.


## 2.1. Variable of Interest and Auxiliary Information

Assume that the process variableY is normally distributed with a mean μY+δσY and variance σY2. Let Y¯t=∑i=1nYit/n and SYt2=∑i=1nYit−Y¯t2/n−1 represents the sample mean and variance of Y of size n. If δ=0, the process is in-control (IC); otherwise, it is out-of-control (OOC). So, for the IC situation, Y¯t and SYt2 are mutually independent identically distributed. Let X be an auxiliary variable of Y. The X and Y follow a bivariate normal distribution (BND) (i.e., Y,X∼NμY,μX,σY,σX,ρ, where μX represents the mean and σX represents the standard deviation of X. Also, the ρ is the correlation coefficient corresponding to X and Y. The AIB regression estimator for process location is given as follows:(1)Rt=Y¯t+bYXμX−X¯t,where, bYX=ρσY/σX, ERt=μY and VarRt=σY2−bYX2σX2/n.


## 2.2.HWMAAIB Chart

Adegoke, et al. [28] introduced HWMAAIB chart to track changes in process location. The HWMAAIB 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:(2)Ht=λ1Rt+1−λ1R¯t−1,where Y¯t is the sample average of tth sample and λ1∈0,1 is the smoothing constant, and R¯t−1=∑i=1t−1Ri/t−1 is the mean of preceding t−1 observations The control limits of the HWMAAIB chart are:(3)LCLHWMAAIBt=μ0−LHWMAAIBσYλ2/n1−ρ2,ift=1μ0−LHWMAAIBσY1nλ2+1−λ2t−11−ρ2,ift>1UCLHWMAAIBt=μ0+LHWMAAIBσYλ2/n1−ρ2,ift=1μ0+LHWMAAIBσY1nλ2+1−λ2t−11−ρ2,ift>1.TheLHWMAAIB represents the coefficient of control limits. If Ht>UCLHWMAAIBt or Ht<LCLHWMAAIBt, 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 as(4)Et=λYt+λEt−1,DEt=λEt+λDEt−1,TEt=λDEt+λTEt−1,where λ∈0,1 is a TEWMA constant. The initial values of E0,DE0,TE0 are all equal to μ0. The mean and variance of the TEt are.ETEt=μ0 and(5)varTEt/σ2=θ3λ6/4−tt2−1t−2θt−3/1−θ−4tt2−1θt−2/1−θ2−12tt+1θt−1/1−θ3−24t+1θt/1−θ4+241−θt+1/1−θ5+2θ2λ6−tt2−1θt−2/1−θ−3tt+1θt−1/1−θ2−6t+1θt/1−θ3+61−θt+1/1−θ4+7θλ6/2−tt+1θt−1/1−θ−2t+1θt/1−θ2+21−θt+1/1−θ3+λ61−θt+1/1−θ2−t+1θt/1−θ,respectively, where θ=1−λ2. The time-varying control limits of the TEWMA chart are defined as:(6)LCLTEWMAt=μ0−LTEWMAvarTEt,UCLTEWMAt=μ0+LTEWMAvarTEt,where LTEWMALTEWMA>0 is the control limits coefficient of the TEWMA chart. The process will be OOC if any TEt>UCLTEWMAtorTEt<LCLTEWMAt.


## 3. Proposed Method

The methodology for the proposed chart is described in this section. Subsection3.1 covers the design structure of the proposed THWMAAIB chart. Besides, the special cases of the THWMAAIB chart are given in Subsection 3.2.


### 3.1. ProposedTHWMAAIB Chart

To construct the proposedTHWMAAIB chart, the plotting statistic of the THWMAAIB chart is given as:(7)Ht=λ1Rt+1−λ1R¯t−1,(8)DHt=λ2Ht+1−λ2R¯t−1,(9)THt=λ3DHt+1−λ3R¯t−1.We assumeλ1=λ2=λ3=λ by following Shamma and Shamma [4] and Abid, et al. [32], then after simplification, the THt statistic can be written as:(10)THt=λ3Rt+1−λ3R¯t−1.TheTHt statistic in Equation (9) can be expressed as:(11)THt=λ3Rt+1−λ3i−1Rt−1+1−λ3i−1Rt−2+⋯+1−λ3i−1R1.Also, forTHt, ETHt=μ0, varTHt=λ6/n1−ρ2σY2fort=1, varTHt=λ6/n1−ρ2σY2+1−λ32/nt−11−ρ2σY2fort>1. The proposed THWMAAIB chart’s control limits are written as:(12)LCLTHWMAAIBt=μ0−LTHWMAAIBσYλ6n1−ρ2,ift=1μ0−LTHWMAAIBσY1nλ6+1−λ32t−11−ρ2,ift>1UCLTHWMAAIBt=μ0+LTHWMAAIBσYλ6n1−ρ2,ift=1μ0+LTHWMAAIBσY1nλ6+1−λ32t−11−ρ2,ift>1,where LTHWMAAIB is the control limits co-efficient. The THt statistic is plotted along with LCLTHWMAAIBt and UCLTHWMAAIBt. The process is declared IC if THt>LCLTHWMAAIBt or THt<UCLTHWMAAIBt; otherwise, OOC.


### 3.2. Special Cases ofTHWMAAIB Chart

The proposedTHWMAAIB chart reduced to some existing charts including HWMA, DHWMA, and HWMAAIB charts by considering special values of parameters. The special cases, along with their proves, are provided here.Case 1.
Whenρ=0 and λ3=1, the proposed THWMAAIB chart tends to the DHWMA chart.Proof.
Whenρ=0, then the difference Rt estimator reduces to(13)Rt1=Y¯t.
Now substitute the resultedRt1 in Equation (6) to obtain(14)Ht1=λ1Rt1+1−λ1Rt−11,Then the Equation (7) can be written as(15)DHt1=λ2Ht1+1−λ2Rt−11.
Now substitute Equation (14) in Equation (8) which provides the following:(16)THt1=λ3DHt1+1−λ3Rt−11.
Also, putλ3=1 in Equation (15) then the THt1 reduced as follows:(17)THt1=DHt1.
This shows that statistic of proposedTHWMAAIB chart becomes the statistic of DHWMA by Abid, et al. [32], when ρ=0 and λ3=1.Case 2.
The proposedTHWMAAIB chart converts to THWMA chart at ρ=0.Proof.
Consider estimatorRt1 which is given in Equation (12) when ρ=0i.e.,Rt1=Y¯t. Based on Rt1, the Ht1 of Equation (13) will be as follows:(18)Ht1=λ1Rt1+1−λ1Rt−11.
Substituting theHt1 in the plotting statistic of the DHt, (presented as DHt2 in this case) we get: (19)DHt2=λ2Ht1+1−λ2Rt−11,(20)THt2=λ3DHt1+1−λ3Rt−11.
TheTHt2in Equation (20) is the THWMA chart with no correlation. Hence, the proposed THWMAAIB chart reduced to the statistic of the THWMA chart for ρ=0.Case 3.
Whenλ2=1 and λ3=1, the proposed THWMAAIB chart tends to the HWMAAIB chart.Proof.
Whenλ2=1andλ3=1, then the plotting statistic of the proposed THWMAAIB chart, symbolized as THt3 can be written as(21)THt3=λ1Rt+1−λ1Rt−1.
TheTHt3 in Equation (19) is similar to the HWMAAIB statistic proposed by Adegoke, et al. [28] except for their notations. Hence, the statistic of proposed THWMAAIB chart becomes the statistic of HWMAAIB when λ2=1 and λ3=1.Case 4.
The proposedTHWMAAIB chart converts to the HWMA chart at ρ=0, λ2=1, and λ3=1.Proof.
Consider estimatorRt1 which is given in Equation (12) when ρ=0i.e.,Rt1=Y¯t. Based on Rt1, the Ht1 of Equation (13) will be as follows:(22)Ht4=λ1Rt1+1−λ1Rt−11.
Substituting theHt1 in the plotting statistic of Equation (7) and (8), we get(23)DHt4=λ2Ht4+1−λ2Rt−11,THt4=λ3DHt4+1−λ3Rt−11.
Now putλ2=1andλ3=1, then the plotting statistic of the proposed THWMAAIB chart, symbolized as THt4 can be written as(24)THt4=λ1Rt1+1−λ1Rt−11.
TheTHt4 in Equation (22) is identical to the HWMA statistic. This shows that the THWMAAIB chart is identical to the HWMA chart by Abbas (2018) for ρ=0, λ2=1, and. λ3=1.Case 5.
Whenλ3=1, the proposed THWMAAIB chart tends to the double HWMAAIBDHWMAAIB chart.Proof.
Whenλ3=1, then the plotting statistic of the proposed THWMAAIB chart, symbolized as THt5 can be written as(25)THt5=λ2DHt+1−λ2Rt−1.
TheTHt5 in Equation (23) is a DHWMAAIB statistic. Hence, the statistic of proposed THWMAAIB chart becomes the statistic of DHWMAAIB for λ3=1.


## 3.1. ProposedTHWMAAIB Chart

To construct the proposedTHWMAAIB chart, the plotting statistic of the THWMAAIB chart is given as:(7)Ht=λ1Rt+1−λ1R¯t−1,(8)DHt=λ2Ht+1−λ2R¯t−1,(9)THt=λ3DHt+1−λ3R¯t−1.We assumeλ1=λ2=λ3=λ by following Shamma and Shamma [4] and Abid, et al. [32], then after simplification, the THt statistic can be written as:(10)THt=λ3Rt+1−λ3R¯t−1.TheTHt statistic in Equation (9) can be expressed as:(11)THt=λ3Rt+1−λ3i−1Rt−1+1−λ3i−1Rt−2+⋯+1−λ3i−1R1.Also, forTHt, ETHt=μ0, varTHt=λ6/n1−ρ2σY2fort=1, varTHt=λ6/n1−ρ2σY2+1−λ32/nt−11−ρ2σY2fort>1. The proposed THWMAAIB chart’s control limits are written as:(12)LCLTHWMAAIBt=μ0−LTHWMAAIBσYλ6n1−ρ2,ift=1μ0−LTHWMAAIBσY1nλ6+1−λ32t−11−ρ2,ift>1UCLTHWMAAIBt=μ0+LTHWMAAIBσYλ6n1−ρ2,ift=1μ0+LTHWMAAIBσY1nλ6+1−λ32t−11−ρ2,ift>1,where LTHWMAAIB is the control limits co-efficient. The THt statistic is plotted along with LCLTHWMAAIBt and UCLTHWMAAIBt. The process is declared IC if THt>LCLTHWMAAIBt or THt<UCLTHWMAAIBt; otherwise, OOC.


## 3.2. Special Cases ofTHWMAAIB Chart

The proposedTHWMAAIB chart reduced to some existing charts including HWMA, DHWMA, and HWMAAIB charts by considering special values of parameters. The special cases, along with their proves, are provided here.Case 1.
Whenρ=0 and λ3=1, the proposed THWMAAIB chart tends to the DHWMA chart.Proof.
Whenρ=0, then the difference Rt estimator reduces to(13)Rt1=Y¯t.
Now substitute the resultedRt1 in Equation (6) to obtain(14)Ht1=λ1Rt1+1−λ1Rt−11,Then the Equation (7) can be written as(15)DHt1=λ2Ht1+1−λ2Rt−11.
Now substitute Equation (14) in Equation (8) which provides the following:(16)THt1=λ3DHt1+1−λ3Rt−11.
Also, putλ3=1 in Equation (15) then the THt1 reduced as follows:(17)THt1=DHt1.
This shows that statistic of proposedTHWMAAIB chart becomes the statistic of DHWMA by Abid, et al. [32], when ρ=0 and λ3=1.Case 2.
The proposedTHWMAAIB chart converts to THWMA chart at ρ=0.Proof.
Consider estimatorRt1 which is given in Equation (12) when ρ=0i.e.,Rt1=Y¯t. Based on Rt1, the Ht1 of Equation (13) will be as follows:(18)Ht1=λ1Rt1+1−λ1Rt−11.
Substituting theHt1 in the plotting statistic of the DHt, (presented as DHt2 in this case) we get: (19)DHt2=λ2Ht1+1−λ2Rt−11,(20)THt2=λ3DHt1+1−λ3Rt−11.
TheTHt2in Equation (20) is the THWMA chart with no correlation. Hence, the proposed THWMAAIB chart reduced to the statistic of the THWMA chart for ρ=0.Case 3.
Whenλ2=1 and λ3=1, the proposed THWMAAIB chart tends to the HWMAAIB chart.Proof.
Whenλ2=1andλ3=1, then the plotting statistic of the proposed THWMAAIB chart, symbolized as THt3 can be written as(21)THt3=λ1Rt+1−λ1Rt−1.
TheTHt3 in Equation (19) is similar to the HWMAAIB statistic proposed by Adegoke, et al. [28] except for their notations. Hence, the statistic of proposed THWMAAIB chart becomes the statistic of HWMAAIB when λ2=1 and λ3=1.Case 4.
The proposedTHWMAAIB chart converts to the HWMA chart at ρ=0, λ2=1, and λ3=1.Proof.
Consider estimatorRt1 which is given in Equation (12) when ρ=0i.e.,Rt1=Y¯t. Based on Rt1, the Ht1 of Equation (13) will be as follows:(22)Ht4=λ1Rt1+1−λ1Rt−11.
Substituting theHt1 in the plotting statistic of Equation (7) and (8), we get(23)DHt4=λ2Ht4+1−λ2Rt−11,THt4=λ3DHt4+1−λ3Rt−11.
Now putλ2=1andλ3=1, then the plotting statistic of the proposed THWMAAIB chart, symbolized as THt4 can be written as(24)THt4=λ1Rt1+1−λ1Rt−11.
TheTHt4 in Equation (22) is identical to the HWMA statistic. This shows that the THWMAAIB chart is identical to the HWMA chart by Abbas (2018) for ρ=0, λ2=1, and. λ3=1.Case 5.
Whenλ3=1, the proposed THWMAAIB chart tends to the double HWMAAIBDHWMAAIB chart.Proof.
Whenλ3=1, then the plotting statistic of the proposed THWMAAIB chart, symbolized as THt5 can be written as(25)THt5=λ2DHt+1−λ2Rt−1.
TheTHt5 in Equation (23) is a DHWMAAIB statistic. Hence, the statistic of proposed THWMAAIB chart becomes the statistic of DHWMAAIB for λ3=1.


## 4. Performance Evaluation Measures

This section introduces the performance evaluation measures to analyze the charts’ performance. The Monte Carlo simulation detail is given in Subsection4.1. Likewise, the description of the ARL 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 THWMAAIB 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 proposedTHWMAAIB chart. Monte Carlo simulation with 105 iterations is conducted for each displacement of δ using R software to obtain the ARL of the proposed THWMAAIB chart. The shift is reflected in the process mean as: μY to μY+δσY, where δ  = 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 THWMAAIB chart is constructed by following given below guidelines:(i)
Generate random observations fromYit,Xit∼NμY,μX,σY,σX,ρt=1,2,3,….(ii)
Calculate theRt estimator from Equation (1).(iii)
Calculate theHt statistics of the THWMAAIB chart from Equation (6) using Rt estimator.(iv)
Use theDHt statistic as an input in Equation (8) to obtain the THt statistic.(v)
ChoseLTHWMAAIB along with other desired parameters λ,ρ for desired IC ARL denoted as ARL0.(vi)
ComputeLCLTHWMAAIBt and UCLTHWMAAIBt based on LTHWMAAIB and λ(vii)
Plot theTHt statistic against the LCLTHWMAAIBt and UCLTHWMAAIBt.(viii)
IfTHt> UCLTHWMAAIBt or THt < LCLTHWMAAIBt, record sequence order which is called run length RL(ix)
Repeat steps (i)-(viii) 105 times.(x)
Calculate the average of 105RL, which is ARL0. If it is desired ARL0; otherwise, adjust the LProposed accordingly and repeat from steps (i)–(ix) until will not get desired ARL0.(xi)
For OOCARL values, considered Yit,Xit∼NμY+δσY,μX, σY,σX,ρ) and repeat from steps (ii)-(x).


### 4.2. Individual Performance Measure

The average run lengthARL is commonly used to evaluate a chart’s performance at a single shift. The ARL is listed as IC ARLARL0 and OOC ARLARL1. For IC state, the ARL0 is chosen to be sufficiently large to eliminate the effect of the false alarm rate. On the other hand, the ARL1 should be small enough to detect a shift quickly. A chart is preferred than the other competing charts if it should have a smaller ARL1 value at predefined ARL0.


### 4.3. Overall Performance Evaluation Measures

TheEQL, RARL, and PCI performance evaluation measures are used to evaluate a chart’s overall effectiveness. More information can be found here.TheEQL is the weighted average ARL over the domain of shifts with δ2 as a weight [37]. It is defined as:(26)EQL=δmax−δmin−1∫δminδmaxδ2ARLδdδ,where ARLδ is the ARL at specific δ; δmin and δmax are the smallest and largest shift values of the domain, respectively. The lower the EQL value signifies, the better the chart’s performance.TheRARL, like EQL is also used to assess the efficiency of a chart. It can be defined as follows:(27)RARL=δmax−δmin−1∫δminδmaxARLδARL∗δdδ.TheARLδ is the ARL of the competing chart. The ARL∗δ is the ARL of benchmark chart at a δ. A chart is decided as a benchmark chart for a smaller ARL at specific δ [22]. The RARL value of the benchmark chart is assumed to be one. If the competing chart has RARL >1, the benchmark chart is more efficient than competing.ThePCI corresponds to the EQL ratio of the specific chart to the EQL of the benchmark chart. Here EQL∗ represents the EQL of the benchmark chart, whereas the EQL represents the EQL of the competing chart. According to Ou, et al. [38], the PCI is presented as:(28)PCI=EQLEQL∗.ThePCI of the benchmark chart should be one. If the PCI >1, the benchmark chart is superior against the competing chart [20].


### 4.4. Effect of Parameters Choices

The design parameters (λ,LTHWMAAIB) of the proposed THWMAAIB chart has its effect on the detection ability. Various combinations of these parameters are chosen, and hence corresponding ARL, the standard deviation of RL (SDRL), and median of RL (MDRL) are computed. The parameter λ is set as 0.10, 0.25, 0.50, and 0.75 to find the values of LTHWMAAIB, to obtain ARL0=500. Various ρ like 0.00, 0.25, 0.50, 0.75, and 0.95 are assumed for this study. Numerical results of the proposed THWMAAIB chart are presented in Tables 1–4.Table 1
ARL values of the proposed THWMAAIB chart for λ=0.10 and ARL0  = 500.
LTHWMAAIB  = 1.2855ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL501.48102.7139.1423.3217.037.814.663.502.151.541.231.101.00SDRL2641.60340.7188.0639.8526.728.664.292.631.520.990.660.440.00MDRL9.0009.0007.0008.0007.0005.004.003.001.001.001.001.001.000.25ARL503.1894.3543.7223.5017.407.354.753.362.171.491.231.061.00SDRL2862.44320.4790.3739.8126.077.744.082.541.520.950.670.350.00MDRL8.0008.5009.008.007.0005.004.003.001.001.001.001.001.000.50ARL501.1680.2637.6217.2513.346.013.832.801.801.321.111.031.00SDRL2702.05252.8284.7627.8619.376.073.212.091.230.780.460.240.00MDRL8.0008.0007.0006.006.0004.003.003.001.001.001.001.001.000.75ARL503.0565.8928.3612.049.6704.392.932.111.351.081.011.001.00SDRL2603.27188.8961.4616.1211.843.842.291.490.810.400.140.090.00MDRL8.0008.0006.006.005.004.003.001.001.001.001.001.001.000.90ARL503.1037.5917.978.216.092.871.831.371.041.001.001.001.00SDRL2658.1575.7930.218.816.822.141.280.830.290.060.000.000.00MDRL7.0008.0006.005.004.003.001.001.001.001.001.001.001.000.95ARL501.3126.8511.175.684.112.001.271.051.001.001.001.001.00SDRL2733.7149.4913.865.353.601.390.700.310.000.000.000.000.00MDRL8.0007.0006.004.003.001.001.001.001.001.001.001.001.00Table 2
ARL values of the proposed THWMAAIB chart for λ=0.25 and ARL0  = 500.
LTHWMAAIB  = 1.900ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL503.61319.46161.7961.6944.2414.527.795.183.052.111.581.271.00SDRL612.20400.52194.9568.2247.1113.266.123.631.901.330.980.690.06MDRL226.00153.0089.0038.0028.0010.006.004.003.001.001.001.001.000.25ARL502.50313.03155.7658.7742.0113.847.424.932.932.031.511.231.00SDRL610.66391.94186.7564.5244.5012.505.773.421.831.270.930.640.04MDRL225.00153.0086.0036.0027.0010.006.004.003.001.001.001.001.000.50ARL501.40286.40134.7550.2035.0711.656.314.262.541.751.321.111.00SDRL609.85358.25159.4553.9436.4010.154.682.831.581.100.750.450.02MDRL223.00142.0076.0032.0023.009.005.004.003.001.001.001.001.000.75ARL504.97221.9893.3732.6622.837.744.383.021.791.261.061.011.00SDRL610.64274.09106.5533.7122.436.072.931.881.120.680.320.110.00MDRL230.00115.0056.0021.0015.006.004.003.001.001.001.001.001.000.90ARL502.67135.4050.0616.7111.754.322.561.771.121.011.001.001.00SDRL611.51159.9953.9215.5810.282.871.591.110.470.110.010.000.00MDRL230.0076.0031.0012.009.004.003.001.001.001.001.001.001.000.95ARL503.7186.1929.819.877.102.811.661.191.001.001.001.001.00SDRL612.4397.8030.488.255.451.751.040.580.080.000.000.000.00MDRL229.0051.0020.008.006.003.001.001.001.001.001.001.001.00Table 3
ARL values of the proposed THWMAAIB chart for λ=0.50 and ARL0  = 500.
LTHWMAAIB  = 2.992ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL499.25401.97259.88115.6684.1429.0215.039.455.003.332.451.881.03SDRL428.13344.00216.6287.4361.1418.378.785.202.421.521.170.980.18MDRL387.00311.00202.0095.0070.0025.0013.009.005.003.003.001.001.000.25ARL496.96396.92251.97110.3380.3027.4914.368.974.763.182.351.791.02SDRL427.47339.79210.1083.1757.9517.218.314.892.291.451.140.940.15MDRL384.00306.00197.0090.5067.0024.0013.008.004.003.003.001.001.000.50ARL498.49378.78226.0194.1168.0123.0211.937.504.062.751.991.501.00SDRL428.74325.16186.0769.4047.7014.126.763.971.891.271.020.790.06MDRL384.00291.00177.0078.0058.0020.0011.007.004.003.002.001.001.000.75ARL498.97324.43166.9763.2345.0114.817.714.962.811.861.321.081.00SDRL427.80276.15133.2244.1929.988.644.132.411.300.970.650.320.00MDRL385.00251.00133.0053.0039.0013.007.005.003.001.001.001.001.000.90ARL498.09227.8895.2133.3323.297.574.112.761.521.071.001.001.00SDRL428.96188.0170.3821.3514.304.031.911.280.800.300.060.010.00MDRL387.00179.0079.0029.0021.007.004.003.001.001.001.001.001.000.95ARL498.09155.1458.1319.5413.544.552.601.701.041.001.001.001.00SDRL428.96155.1458.1319.5413.544.552.601.701.041.001.001.001.00MDRL387.00123.0140.0711.737.832.171.220.900.230.020.000.000.00Table 4
ARL values of the proposed THWMAAIB chart for λ=0.75 and ARL0  = 500.
LTHWMAAIB  = 3.086ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL499.00472.53394.17240.47181.1854.1122.2711.895.223.112.191.681.03SDRL496.67471.44391.22234.98175.6148.7818.268.843.221.641.040.740.16MDRL347.00328.00275.00169.00128.0040.0017.0010.004.003.002.002.001.000.25ARL499.12467.64390.36231.45173.2649.9120.8111.154.902.972.091.611.02SDRL494.39465.74385.16227.52166.1844.8716.858.242.961.530.980.700.13MDRL347.00323.00273.00163.00123.0037.0016.009.004.003.002.001.001.000.50ARL498.26459.87370.16202.16146.8339.6216.318.774.002.471.771.391.00SDRL495.53458.28367.02195.49140.8634.9312.776.102.271.210.800.570.06MDRL346.00319.00258.00143.00104.0030.0013.007.004.002.002.001.001.000.75ARL498.33434.97311.17137.4493.1921.969.095.142.551.671.261.081.00SDRL494.82433.55306.48131.9087.3117.946.383.141.260.740.470.270.00MDRL347.00303.00217.0098.0067.0017.007.004.002.002.001.001.001.000.90ARL499.04371.46204.3663.9240.248.904.052.501.411.071.001.001.00SDRL496.89368.42199.1658.4135.426.232.321.220.580.250.070.010.00MDRL348.00259.00144.0047.0030.007.004.002.001.001.001.001.001.000.95ARL498.05297.88124.6931.7619.364.612.331.541.041.001.001.001.00SDRL494.39293.94119.6727.2815.502.731.120.660.200.020.000.000.00MDRL347.50208.0089.0024.0015.004.002.001.001.001.001.001.001.00


## 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 proposedTHWMAAIB chart. Monte Carlo simulation with 105 iterations is conducted for each displacement of δ using R software to obtain the ARL of the proposed THWMAAIB chart. The shift is reflected in the process mean as: μY to μY+δσY, where δ  = 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 THWMAAIB chart is constructed by following given below guidelines:(i)
Generate random observations fromYit,Xit∼NμY,μX,σY,σX,ρt=1,2,3,….(ii)
Calculate theRt estimator from Equation (1).(iii)
Calculate theHt statistics of the THWMAAIB chart from Equation (6) using Rt estimator.(iv)
Use theDHt statistic as an input in Equation (8) to obtain the THt statistic.(v)
ChoseLTHWMAAIB along with other desired parameters λ,ρ for desired IC ARL denoted as ARL0.(vi)
ComputeLCLTHWMAAIBt and UCLTHWMAAIBt based on LTHWMAAIB and λ(vii)
Plot theTHt statistic against the LCLTHWMAAIBt and UCLTHWMAAIBt.(viii)
IfTHt> UCLTHWMAAIBt or THt < LCLTHWMAAIBt, record sequence order which is called run length RL(ix)
Repeat steps (i)-(viii) 105 times.(x)
Calculate the average of 105RL, which is ARL0. If it is desired ARL0; otherwise, adjust the LProposed accordingly and repeat from steps (i)–(ix) until will not get desired ARL0.(xi)
For OOCARL values, considered Yit,Xit∼NμY+δσY,μX, σY,σX,ρ) and repeat from steps (ii)-(x).


## 4.2. Individual Performance Measure

The average run lengthARL is commonly used to evaluate a chart’s performance at a single shift. The ARL is listed as IC ARLARL0 and OOC ARLARL1. For IC state, the ARL0 is chosen to be sufficiently large to eliminate the effect of the false alarm rate. On the other hand, the ARL1 should be small enough to detect a shift quickly. A chart is preferred than the other competing charts if it should have a smaller ARL1 value at predefined ARL0.


## 4.3. Overall Performance Evaluation Measures

TheEQL, RARL, and PCI performance evaluation measures are used to evaluate a chart’s overall effectiveness. More information can be found here.TheEQL is the weighted average ARL over the domain of shifts with δ2 as a weight [37]. It is defined as:(26)EQL=δmax−δmin−1∫δminδmaxδ2ARLδdδ,where ARLδ is the ARL at specific δ; δmin and δmax are the smallest and largest shift values of the domain, respectively. The lower the EQL value signifies, the better the chart’s performance.TheRARL, like EQL is also used to assess the efficiency of a chart. It can be defined as follows:(27)RARL=δmax−δmin−1∫δminδmaxARLδARL∗δdδ.TheARLδ is the ARL of the competing chart. The ARL∗δ is the ARL of benchmark chart at a δ. A chart is decided as a benchmark chart for a smaller ARL at specific δ [22]. The RARL value of the benchmark chart is assumed to be one. If the competing chart has RARL >1, the benchmark chart is more efficient than competing.ThePCI corresponds to the EQL ratio of the specific chart to the EQL of the benchmark chart. Here EQL∗ represents the EQL of the benchmark chart, whereas the EQL represents the EQL of the competing chart. According to Ou, et al. [38], the PCI is presented as:(28)PCI=EQLEQL∗.ThePCI of the benchmark chart should be one. If the PCI >1, the benchmark chart is superior against the competing chart [20].


## 4.4. Effect of Parameters Choices

The design parameters (λ,LTHWMAAIB) of the proposed THWMAAIB chart has its effect on the detection ability. Various combinations of these parameters are chosen, and hence corresponding ARL, the standard deviation of RL (SDRL), and median of RL (MDRL) are computed. The parameter λ is set as 0.10, 0.25, 0.50, and 0.75 to find the values of LTHWMAAIB, to obtain ARL0=500. Various ρ like 0.00, 0.25, 0.50, 0.75, and 0.95 are assumed for this study. Numerical results of the proposed THWMAAIB chart are presented in Tables 1–4.Table 1
ARL values of the proposed THWMAAIB chart for λ=0.10 and ARL0  = 500.
LTHWMAAIB  = 1.2855ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL501.48102.7139.1423.3217.037.814.663.502.151.541.231.101.00SDRL2641.60340.7188.0639.8526.728.664.292.631.520.990.660.440.00MDRL9.0009.0007.0008.0007.0005.004.003.001.001.001.001.001.000.25ARL503.1894.3543.7223.5017.407.354.753.362.171.491.231.061.00SDRL2862.44320.4790.3739.8126.077.744.082.541.520.950.670.350.00MDRL8.0008.5009.008.007.0005.004.003.001.001.001.001.001.000.50ARL501.1680.2637.6217.2513.346.013.832.801.801.321.111.031.00SDRL2702.05252.8284.7627.8619.376.073.212.091.230.780.460.240.00MDRL8.0008.0007.0006.006.0004.003.003.001.001.001.001.001.000.75ARL503.0565.8928.3612.049.6704.392.932.111.351.081.011.001.00SDRL2603.27188.8961.4616.1211.843.842.291.490.810.400.140.090.00MDRL8.0008.0006.006.005.004.003.001.001.001.001.001.001.000.90ARL503.1037.5917.978.216.092.871.831.371.041.001.001.001.00SDRL2658.1575.7930.218.816.822.141.280.830.290.060.000.000.00MDRL7.0008.0006.005.004.003.001.001.001.001.001.001.001.000.95ARL501.3126.8511.175.684.112.001.271.051.001.001.001.001.00SDRL2733.7149.4913.865.353.601.390.700.310.000.000.000.000.00MDRL8.0007.0006.004.003.001.001.001.001.001.001.001.001.00Table 2
ARL values of the proposed THWMAAIB chart for λ=0.25 and ARL0  = 500.
LTHWMAAIB  = 1.900ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL503.61319.46161.7961.6944.2414.527.795.183.052.111.581.271.00SDRL612.20400.52194.9568.2247.1113.266.123.631.901.330.980.690.06MDRL226.00153.0089.0038.0028.0010.006.004.003.001.001.001.001.000.25ARL502.50313.03155.7658.7742.0113.847.424.932.932.031.511.231.00SDRL610.66391.94186.7564.5244.5012.505.773.421.831.270.930.640.04MDRL225.00153.0086.0036.0027.0010.006.004.003.001.001.001.001.000.50ARL501.40286.40134.7550.2035.0711.656.314.262.541.751.321.111.00SDRL609.85358.25159.4553.9436.4010.154.682.831.581.100.750.450.02MDRL223.00142.0076.0032.0023.009.005.004.003.001.001.001.001.000.75ARL504.97221.9893.3732.6622.837.744.383.021.791.261.061.011.00SDRL610.64274.09106.5533.7122.436.072.931.881.120.680.320.110.00MDRL230.00115.0056.0021.0015.006.004.003.001.001.001.001.001.000.90ARL502.67135.4050.0616.7111.754.322.561.771.121.011.001.001.00SDRL611.51159.9953.9215.5810.282.871.591.110.470.110.010.000.00MDRL230.0076.0031.0012.009.004.003.001.001.001.001.001.001.000.95ARL503.7186.1929.819.877.102.811.661.191.001.001.001.001.00SDRL612.4397.8030.488.255.451.751.040.580.080.000.000.000.00MDRL229.0051.0020.008.006.003.001.001.001.001.001.001.001.00Table 3
ARL values of the proposed THWMAAIB chart for λ=0.50 and ARL0  = 500.
LTHWMAAIB  = 2.992ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL499.25401.97259.88115.6684.1429.0215.039.455.003.332.451.881.03SDRL428.13344.00216.6287.4361.1418.378.785.202.421.521.170.980.18MDRL387.00311.00202.0095.0070.0025.0013.009.005.003.003.001.001.000.25ARL496.96396.92251.97110.3380.3027.4914.368.974.763.182.351.791.02SDRL427.47339.79210.1083.1757.9517.218.314.892.291.451.140.940.15MDRL384.00306.00197.0090.5067.0024.0013.008.004.003.003.001.001.000.50ARL498.49378.78226.0194.1168.0123.0211.937.504.062.751.991.501.00SDRL428.74325.16186.0769.4047.7014.126.763.971.891.271.020.790.06MDRL384.00291.00177.0078.0058.0020.0011.007.004.003.002.001.001.000.75ARL498.97324.43166.9763.2345.0114.817.714.962.811.861.321.081.00SDRL427.80276.15133.2244.1929.988.644.132.411.300.970.650.320.00MDRL385.00251.00133.0053.0039.0013.007.005.003.001.001.001.001.000.90ARL498.09227.8895.2133.3323.297.574.112.761.521.071.001.001.00SDRL428.96188.0170.3821.3514.304.031.911.280.800.300.060.010.00MDRL387.00179.0079.0029.0021.007.004.003.001.001.001.001.001.000.95ARL498.09155.1458.1319.5413.544.552.601.701.041.001.001.001.00SDRL428.96155.1458.1319.5413.544.552.601.701.041.001.001.001.00MDRL387.00123.0140.0711.737.832.171.220.900.230.020.000.000.00Table 4
ARL values of the proposed THWMAAIB chart for λ=0.75 and ARL0  = 500.
LTHWMAAIB  = 3.086ρδ0.000.050.100.200.250.500.751.001.502.002.503.005.000.00ARL499.00472.53394.17240.47181.1854.1122.2711.895.223.112.191.681.03SDRL496.67471.44391.22234.98175.6148.7818.268.843.221.641.040.740.16MDRL347.00328.00275.00169.00128.0040.0017.0010.004.003.002.002.001.000.25ARL499.12467.64390.36231.45173.2649.9120.8111.154.902.972.091.611.02SDRL494.39465.74385.16227.52166.1844.8716.858.242.961.530.980.700.13MDRL347.00323.00273.00163.00123.0037.0016.009.004.003.002.001.001.000.50ARL498.26459.87370.16202.16146.8339.6216.318.774.002.471.771.391.00SDRL495.53458.28367.02195.49140.8634.9312.776.102.271.210.800.570.06MDRL346.00319.00258.00143.00104.0030.0013.007.004.002.002.001.001.000.75ARL498.33434.97311.17137.4493.1921.969.095.142.551.671.261.081.00SDRL494.82433.55306.48131.9087.3117.946.383.141.260.740.470.270.00MDRL347.00303.00217.0098.0067.0017.007.004.002.002.001.001.001.000.90ARL499.04371.46204.3663.9240.248.904.052.501.411.071.001.001.00SDRL496.89368.42199.1658.4135.426.232.321.220.580.250.070.010.00MDRL348.00259.00144.0047.0030.007.004.002.001.001.001.001.001.000.95ARL498.05297.88124.6931.7619.364.612.331.541.041.001.001.001.00SDRL494.39293.94119.6727.2815.502.731.120.660.200.020.000.000.00MDRL347.50208.0089.0024.0015.004.002.001.001.001.001.001.001.00


## 5. Evaluation and Performance Comparison

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


### 5.1. Proposed versus Classical EWMA Chart

The proposedTHWMAAIB chart provides superior detection ability for various values of λ against the classical EWMA chart. For instance, at ρ=0.75,λ=0.25,δ=0.50, the proposed THWMAAIB chart has an ARL1 value of 7.74, while the classical EWMA chart has an ARL1 value of 47.36 (see Tables 2 and 5). The supremacy of the proposed THWMAAIB chart compared to the classical EWMA chart can be seen in Figure 1. Evaluating the overall efficiency with λ=0.10, the THWMAAIB chart has smaller EQL,RARL,and PCI (i.e., 9.38, 1.000, 1.000) values against the classical EWMA chart EQL,RARL,and PCI (i.e., 13.38, 1.43, 2.23) values (see Table 6).Table 5
ARL values of some existing charts when ARL0=500.
δ0.000.250.500.751.001.502.002.503.005.00λ  = 0.10TEWMA499.3174.2124.0512.637.853.852.291.771.251.00MHC (k=0.50)501.0025.0013.009.007.005.004.004.004.003.00EWMA500.00103.3228.8113.618.214.172.661.921.511.02HWMA500.0081.4828.6114.859.354.983.322.451.871.03DHWMA499.1034.7611.976.634.562.721.931.461.191.00HWMAAIBρ=0.50501.4765.8922.6311.797.434.042.722.431.831.01EWMAAIBρ=0.50499.8180.6622.0410.546.433.322.151.581.281.00MECAIBρ=0.50,k=0.50500.6356.2919.9811.427.734.383.002.002.001.00MCEAIBρ=0.50,k=0.50499.47100.2527.9814.5110.206.775.294.433.873.12λ  = 0.25TEWMA500.31110.7430.0413.968.494.352.692.021.431.05MHC (k=0.50)499.0031.0012.008.006.004.004.003.003.002.00EWMA500.00169.3447.3619.3110.414.782.942.091.621.02HWMA500.00113.3433.7916.199.714.943.202.331.791.03DHWMA501.3776.8626.7713.888.694.663.122.321.771.02HWMAAIBρ=0.50504.8889.8625.7712.527.593.952.612.301.751.01EWMAAIBρ=0.50499.69135.7734.5814.117.803.722.351.701.351.00MECAIBρ=0.50,k=0.50498.7265.2622.1412.618.655.323.802.992.241.99MCEAIBρ=0.50,k=0.50500.69107.4928.2213.839.245.934.563.793.302.71λ  = 0.50TEWMA500.80173.2148.3119.3110.194.632.852.191.571.15MHC (k=0.50)501.0052.0016.009.006.004.003.003.003.002.00EWMA500.00254.5388.4335.5717.186.273.392.261.701.03HWMA500.00218.0669.0427.8714.085.673.202.201.681.03DHWMA502.39113.0234.2916.389.784.953.192.321.781.03HWMAAIBρ=0.50498.55180.9150.4319.8710.064.212.502.171.641.01EWMAAIBρ=0.50500.79216.1065.4024.8011.914.562.591.801.401.00MECAIBρ=0.50,k=0.50502.7777.8823.4212.538.455.163.763.012.431.01MCEAIBρ=0.50,k=0.50500.11111.4728.6213.558.655.263.933.242.802.16Figure 1
ARL comparison of the proposed THWMAAIB versus classical EWMA, TEWMA, and MHC charts when λ=0.1 and ARL0=500.Table 6
EQL,PCI,and RARL measures of the proposed versus existing charts.
λ=0.10THWMAMHCEWMAHWMADHWMAHWMAAIBEWMAAIBMECAIBMCEAIBTHWMAAIBEQL12.1830.2413.3815.0710.7614.0611.7814.2632.559.38PCI1.303.221.431.611.151.501.261.523.471.00RARL1.933.042.232.421.382.111.822.093.841.00λ=0.25THWMAMHCEWMAHWMADHWMAHWMAAIBEWMAAIBMECAIBMCEAIBTHWMAAIBEQL13.5422.0514.7514.9914.4513.8812.7221.0628.2710.35PCI1.312.131.431.451.401.341.232.042.731.00RARL1.581.981.891.791.651.571.532.022.641.00λ=0.50THWMAMHCEWMAHWMADHWMAHWMAAIBEWMAAIBMECAIBMCEAIBTHWMAAIBEQL15.2921.7917.3416.2514.9714.5514.4216.5923.6213.03PCI1.171.671.331.251.151.121.111.271.811.00RARL1.261.451.611.451.211.231.281.291.651.00


### 5.2. Proposed versus TEWMA Chart

The proposedTHWMAAIB chart is superior to the TEWMA chart. For example, at δ=0.25, the proposed THWMAAIB chart (λ=0.10,ρ=0.50) yields ARL1=13.34, whereas the TEWMA chart gives ARL1 equal to 74.21 (see Table 1 and 5). The superiority of the THWMAAIB chart than the TEWMA chart can also be found in Figure 1. Additionally, for a specific range of shifts, the EQL,RARL,and PCI values for proposed THWMAAIB chart are smaller than the TEWMA chart. For example, at λ=0.50, for THWMAAIB and TEWMA charts, the EQL,RARL,and PCI 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 proposedTHWMAAIB chart is better for tracking changes in process location than the MHC chart. For example, with λ=0.10 and δ=0.25,0.50,0.75, the ARL1 values of the MHC chart are (25.00, 13.00, 9.00), while the ARL1 values of proposed THWMAAIB chart ρ=0.50) are (13.34, 6.01, 3.83) (see Table 1 and 5). Figure 1 also highlighted the dominant position of the THWMAAIB chart on the MHC chart. Also, the EQL,PCI,and RARL values are significantly higher for the MHC chart as compared to THWMAAIB chart. For illustration, at λ=0.25, the THWMAAIB chart has EQL  = 10.35, PCI  = 1.00, and RARL  = 1.00, whereas the MHC chart has EQL=22.05,PCI=2.13,and RARL=1.98 measures (see Table 6).


### 5.4. Proposed versusEWMAAIB Chart

The proposedTHWMAAIB chart offers good performance against the EWMAAIB chart. As an illustration, at λ=0.10,ρ=0.50 and δ∈ (0.25, 0.50), the ARL1 (80.66, 22.04) values of the EWMAAIB chart are larger than the ARL1 (13.34, 6.01) values of the THWMAAIB chart (see Tables 1 and 5). Visual representation confirms the supremacy of the proposed THWMAAIB chart over EWMAAIB chart (see Figure 2). The dominance of the THWMAAIB chart against the EWMAAIB chart is seen in the EQL,RARL,and PCI values for a certain range of shifts. As an illustration, at λ=0.10, the proposed THWMAAIB chart has EQL=9.38,PCI=1.00,and  RARL=1.00, while the EWMAAIB chart has EQL=11.78,PCI=1.26,RARL=1.82 (see Table 6). This reveals the superiority of the proposed THWMAAIB chart.Figure 2
ARL comparison of the proposed THWMAAIB versus EWMA, MECAIB, and MCEAIB charts when λ=0.25 and ARL0=500.


### 5.5. Proposed versus HWMA Chart

The proposedTHWMAAIB chart has lower ARL1 values than the HWMA chart. Suppose that, for λ=0.50,δ=0.25, the ARL1 value of the proposed THWMAAIB chart with ρ=0.50 is 68.01, while the ARL1 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 THWMAAIB than the HWMA chart. For instance, at λ=0.50, the EQL values for HWMA and DHWMAAIB are 16.25 and 13.03, respectively (see Table 6).Figure 3
ARL comparison of the proposed THWMAAIB versus HWMA, DHWMA, and HWMAAIB charts when λ=0.25 and ARL0=500.


### 5.6. Proposed versus DHWMA Chart

The proposedTHWMAAIB chart is better as compared to the DHWMA chart. As an illustration, if λ=0.10,δ=0.2 5, the values of ARL1 for DHWMA and THWMAAIBρ=0.25 are 34.76 and 17.40, respectively (see Table 1 versus Table 5 and Figure 3). The overall performance measures highlight the superiority of the THWMAAIB chart. For instance, at λ=0.10, the EQL, PCI, and RARL values of the DHWMA and THWMAAIB 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 versusHWMAAIB Chart

The proposedTHWMAAIB chart detects earlier shift than the HWMAAIB chart. If λ=0.10, ρ=0.50, and δ=0.25,0.50,0.75 the ARL1 values of the charts are (65.89, 22.63, 11.79) and (13.34, 6.01, 3.83) respectively, for HWMAAIB and THWMAAIB charts (see Tables 1 and 5). Figure 3 also reveals the better performance of the proposed THWMAAIB against the HWMAAIB chart. Likewise, the HWMAAIB 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 THWMAAIB chart at λ=0.50,ρ=0.50 (see Table 6).


### 5.8. Proposed versusMECAIB Chart

By comparing theARL measures, the proposed THWMAAIB chart with the MECAIB chart shows that the THWMAAIB chart is more sensitive for all δ and λ values. For instance, with λ=0.25,ρ=0.50, δ=0.25,0.50,0.75, the ARL1=35.07,11.65,6.31 for THWMAAIB chart, while the ARL1=65.26,22.14,12.61 for MECAIB (k=0.50) chart (see Tables 2 and 5). The proposed THWMAAIB chart has smaller EQL,RARL,and PCI measures as compared to the MECAIB chart (see Table 6). Figure 2 illustrates that the proposed THWMAAIB is functioning well and detects shifts earlier in the process. These findings demonstrate that the proposed THWMAAIB chart performs better than the MECAIB chart.


### 5.9. Proposed versusMCEAIB Chart

TheARL study indicates that the proposed THWMAAIB chart works much better than the MCEAIB chart for each of the choices formed by λ,δ (see Tables 1–4 versus 5). As an illustration, Table 3 and 5 shows that at λ=0.50,δ=0.25,0.50,0.75,ρ=0.50, the ARL1 values of the proposed THWMAAIB chart are (68.01, 23.02, 11.93), and the corresponding values of ARL1 for MCEAIB chart k=0.50 are (111.47, 28.62, 13.55). Figure 2 indicates that the proposed THWMAAIB chart is better than the MCEAIB chart. In the overall performance scenario, the proposed THWMAAIB chart also exhibits overperformance than MCEAIB chart (see Table 6).


### 5.10. Main Findings of the Study

Important findings of the proposedTHWMAAIB chart are listed below:(i)
TheTHWMAAIB statistic improves the detection ability of the proposed chart.(ii)
The efficiency of the proposedTHWMAAIB chart is strengthened with the appropriate inclusion of auxiliary information in the structure (see. Tables 1–5).(iii)
Unlike the classic CUSUM, MHC, EWMA, HWMA, DHWMA,HWMAAIB, EWMAAIB, MECAIB and MCEAIB charts, the ARL1 values of the proposed THWMAAIB chart are smaller at various parameter values.(iv)
Overall performance evaluations demonstrate that theTHWMAAIB chart is dominant over the other competing charts included in this research (see Subsections 5.1–5.9).(v)
The proposedTHWMAAIB chart provides optimal results for larger values ρ.(vi)
The efficiency of theARL1 measures is increased by using the proposed THWMAAIB chart.(vii)
Asλ increases, the control limit coefficient (LTHWMAAIB) of the proposed THWMAAIB chart also increases.(viii)
The proposedTHWMAAIB chart is more effective for larger ρ and smaller λ in term of ARL1 performance (see Figures 4 and 5). For example, the minimal value of ARL1 is 4.11 at δ=0.25, λ=0.10, and ρ=0.95.Figure 4
ARL comparison of the proposed THWMAAIB chart for various ρ when λ=0.25 and ARL0=500.Figure 5
ARL comparison of the proposed THWMAAIB chart for various λ when ρ=0.50 and ARL0=500.


## 5.1. Proposed versus Classical EWMA Chart

The proposedTHWMAAIB chart provides superior detection ability for various values of λ against the classical EWMA chart. For instance, at ρ=0.75,λ=0.25,δ=0.50, the proposed THWMAAIB chart has an ARL1 value of 7.74, while the classical EWMA chart has an ARL1 value of 47.36 (see Tables 2 and 5). The supremacy of the proposed THWMAAIB chart compared to the classical EWMA chart can be seen in Figure 1. Evaluating the overall efficiency with λ=0.10, the THWMAAIB chart has smaller EQL,RARL,and PCI (i.e., 9.38, 1.000, 1.000) values against the classical EWMA chart EQL,RARL,and PCI (i.e., 13.38, 1.43, 2.23) values (see Table 6).Table 5
ARL values of some existing charts when ARL0=500.
δ0.000.250.500.751.001.502.002.503.005.00λ  = 0.10TEWMA499.3174.2124.0512.637.853.852.291.771.251.00MHC (k=0.50)501.0025.0013.009.007.005.004.004.004.003.00EWMA500.00103.3228.8113.618.214.172.661.921.511.02HWMA500.0081.4828.6114.859.354.983.322.451.871.03DHWMA499.1034.7611.976.634.562.721.931.461.191.00HWMAAIBρ=0.50501.4765.8922.6311.797.434.042.722.431.831.01EWMAAIBρ=0.50499.8180.6622.0410.546.433.322.151.581.281.00MECAIBρ=0.50,k=0.50500.6356.2919.9811.427.734.383.002.002.001.00MCEAIBρ=0.50,k=0.50499.47100.2527.9814.5110.206.775.294.433.873.12λ  = 0.25TEWMA500.31110.7430.0413.968.494.352.692.021.431.05MHC (k=0.50)499.0031.0012.008.006.004.004.003.003.002.00EWMA500.00169.3447.3619.3110.414.782.942.091.621.02HWMA500.00113.3433.7916.199.714.943.202.331.791.03DHWMA501.3776.8626.7713.888.694.663.122.321.771.02HWMAAIBρ=0.50504.8889.8625.7712.527.593.952.612.301.751.01EWMAAIBρ=0.50499.69135.7734.5814.117.803.722.351.701.351.00MECAIBρ=0.50,k=0.50498.7265.2622.1412.618.655.323.802.992.241.99MCEAIBρ=0.50,k=0.50500.69107.4928.2213.839.245.934.563.793.302.71λ  = 0.50TEWMA500.80173.2148.3119.3110.194.632.852.191.571.15MHC (k=0.50)501.0052.0016.009.006.004.003.003.003.002.00EWMA500.00254.5388.4335.5717.186.273.392.261.701.03HWMA500.00218.0669.0427.8714.085.673.202.201.681.03DHWMA502.39113.0234.2916.389.784.953.192.321.781.03HWMAAIBρ=0.50498.55180.9150.4319.8710.064.212.502.171.641.01EWMAAIBρ=0.50500.79216.1065.4024.8011.914.562.591.801.401.00MECAIBρ=0.50,k=0.50502.7777.8823.4212.538.455.163.763.012.431.01MCEAIBρ=0.50,k=0.50500.11111.4728.6213.558.655.263.933.242.802.16Figure 1
ARL comparison of the proposed THWMAAIB versus classical EWMA, TEWMA, and MHC charts when λ=0.1 and ARL0=500.Table 6
EQL,PCI,and RARL measures of the proposed versus existing charts.
λ=0.10THWMAMHCEWMAHWMADHWMAHWMAAIBEWMAAIBMECAIBMCEAIBTHWMAAIBEQL12.1830.2413.3815.0710.7614.0611.7814.2632.559.38PCI1.303.221.431.611.151.501.261.523.471.00RARL1.933.042.232.421.382.111.822.093.841.00λ=0.25THWMAMHCEWMAHWMADHWMAHWMAAIBEWMAAIBMECAIBMCEAIBTHWMAAIBEQL13.5422.0514.7514.9914.4513.8812.7221.0628.2710.35PCI1.312.131.431.451.401.341.232.042.731.00RARL1.581.981.891.791.651.571.532.022.641.00λ=0.50THWMAMHCEWMAHWMADHWMAHWMAAIBEWMAAIBMECAIBMCEAIBTHWMAAIBEQL15.2921.7917.3416.2514.9714.5514.4216.5923.6213.03PCI1.171.671.331.251.151.121.111.271.811.00RARL1.261.451.611.451.211.231.281.291.651.00


## 5.2. Proposed versus TEWMA Chart

The proposedTHWMAAIB chart is superior to the TEWMA chart. For example, at δ=0.25, the proposed THWMAAIB chart (λ=0.10,ρ=0.50) yields ARL1=13.34, whereas the TEWMA chart gives ARL1 equal to 74.21 (see Table 1 and 5). The superiority of the THWMAAIB chart than the TEWMA chart can also be found in Figure 1. Additionally, for a specific range of shifts, the EQL,RARL,and PCI values for proposed THWMAAIB chart are smaller than the TEWMA chart. For example, at λ=0.50, for THWMAAIB and TEWMA charts, the EQL,RARL,and PCI 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 proposedTHWMAAIB chart is better for tracking changes in process location than the MHC chart. For example, with λ=0.10 and δ=0.25,0.50,0.75, the ARL1 values of the MHC chart are (25.00, 13.00, 9.00), while the ARL1 values of proposed THWMAAIB chart ρ=0.50) are (13.34, 6.01, 3.83) (see Table 1 and 5). Figure 1 also highlighted the dominant position of the THWMAAIB chart on the MHC chart. Also, the EQL,PCI,and RARL values are significantly higher for the MHC chart as compared to THWMAAIB chart. For illustration, at λ=0.25, the THWMAAIB chart has EQL  = 10.35, PCI  = 1.00, and RARL  = 1.00, whereas the MHC chart has EQL=22.05,PCI=2.13,and RARL=1.98 measures (see Table 6).


## 5.4. Proposed versusEWMAAIB Chart

The proposedTHWMAAIB chart offers good performance against the EWMAAIB chart. As an illustration, at λ=0.10,ρ=0.50 and δ∈ (0.25, 0.50), the ARL1 (80.66, 22.04) values of the EWMAAIB chart are larger than the ARL1 (13.34, 6.01) values of the THWMAAIB chart (see Tables 1 and 5). Visual representation confirms the supremacy of the proposed THWMAAIB chart over EWMAAIB chart (see Figure 2). The dominance of the THWMAAIB chart against the EWMAAIB chart is seen in the EQL,RARL,and PCI values for a certain range of shifts. As an illustration, at λ=0.10, the proposed THWMAAIB chart has EQL=9.38,PCI=1.00,and  RARL=1.00, while the EWMAAIB chart has EQL=11.78,PCI=1.26,RARL=1.82 (see Table 6). This reveals the superiority of the proposed THWMAAIB chart.Figure 2
ARL comparison of the proposed THWMAAIB versus EWMA, MECAIB, and MCEAIB charts when λ=0.25 and ARL0=500.


## 5.5. Proposed versus HWMA Chart

The proposedTHWMAAIB chart has lower ARL1 values than the HWMA chart. Suppose that, for λ=0.50,δ=0.25, the ARL1 value of the proposed THWMAAIB chart with ρ=0.50 is 68.01, while the ARL1 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 THWMAAIB than the HWMA chart. For instance, at λ=0.50, the EQL values for HWMA and DHWMAAIB are 16.25 and 13.03, respectively (see Table 6).Figure 3
ARL comparison of the proposed THWMAAIB versus HWMA, DHWMA, and HWMAAIB charts when λ=0.25 and ARL0=500.


## 5.6. Proposed versus DHWMA Chart

The proposedTHWMAAIB chart is better as compared to the DHWMA chart. As an illustration, if λ=0.10,δ=0.2 5, the values of ARL1 for DHWMA and THWMAAIBρ=0.25 are 34.76 and 17.40, respectively (see Table 1 versus Table 5 and Figure 3). The overall performance measures highlight the superiority of the THWMAAIB chart. For instance, at λ=0.10, the EQL, PCI, and RARL values of the DHWMA and THWMAAIB 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 versusHWMAAIB Chart

The proposedTHWMAAIB chart detects earlier shift than the HWMAAIB chart. If λ=0.10, ρ=0.50, and δ=0.25,0.50,0.75 the ARL1 values of the charts are (65.89, 22.63, 11.79) and (13.34, 6.01, 3.83) respectively, for HWMAAIB and THWMAAIB charts (see Tables 1 and 5). Figure 3 also reveals the better performance of the proposed THWMAAIB against the HWMAAIB chart. Likewise, the HWMAAIB 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 THWMAAIB chart at λ=0.50,ρ=0.50 (see Table 6).


## 5.8. Proposed versusMECAIB Chart

By comparing theARL measures, the proposed THWMAAIB chart with the MECAIB chart shows that the THWMAAIB chart is more sensitive for all δ and λ values. For instance, with λ=0.25,ρ=0.50, δ=0.25,0.50,0.75, the ARL1=35.07,11.65,6.31 for THWMAAIB chart, while the ARL1=65.26,22.14,12.61 for MECAIB (k=0.50) chart (see Tables 2 and 5). The proposed THWMAAIB chart has smaller EQL,RARL,and PCI measures as compared to the MECAIB chart (see Table 6). Figure 2 illustrates that the proposed THWMAAIB is functioning well and detects shifts earlier in the process. These findings demonstrate that the proposed THWMAAIB chart performs better than the MECAIB chart.


## 5.9. Proposed versusMCEAIB Chart

TheARL study indicates that the proposed THWMAAIB chart works much better than the MCEAIB chart for each of the choices formed by λ,δ (see Tables 1–4 versus 5). As an illustration, Table 3 and 5 shows that at λ=0.50,δ=0.25,0.50,0.75,ρ=0.50, the ARL1 values of the proposed THWMAAIB chart are (68.01, 23.02, 11.93), and the corresponding values of ARL1 for MCEAIB chart k=0.50 are (111.47, 28.62, 13.55). Figure 2 indicates that the proposed THWMAAIB chart is better than the MCEAIB chart. In the overall performance scenario, the proposed THWMAAIB chart also exhibits overperformance than MCEAIB chart (see Table 6).


## 5.10. Main Findings of the Study

Important findings of the proposedTHWMAAIB chart are listed below:(i)
TheTHWMAAIB statistic improves the detection ability of the proposed chart.(ii)
The efficiency of the proposedTHWMAAIB chart is strengthened with the appropriate inclusion of auxiliary information in the structure (see. Tables 1–5).(iii)
Unlike the classic CUSUM, MHC, EWMA, HWMA, DHWMA,HWMAAIB, EWMAAIB, MECAIB and MCEAIB charts, the ARL1 values of the proposed THWMAAIB chart are smaller at various parameter values.(iv)
Overall performance evaluations demonstrate that theTHWMAAIB chart is dominant over the other competing charts included in this research (see Subsections 5.1–5.9).(v)
The proposedTHWMAAIB chart provides optimal results for larger values ρ.(vi)
The efficiency of theARL1 measures is increased by using the proposed THWMAAIB chart.(vii)
Asλ increases, the control limit coefficient (LTHWMAAIB) of the proposed THWMAAIB chart also increases.(viii)
The proposedTHWMAAIB chart is more effective for larger ρ and smaller λ in term of ARL1 performance (see Figures 4 and 5). For example, the minimal value of ARL1 is 4.11 at δ=0.25, λ=0.10, and ρ=0.95.Figure 4
ARL comparison of the proposed THWMAAIB chart for various ρ when λ=0.25 and ARL0=500.Figure 5
ARL comparison of the proposed THWMAAIB chart for various λ when ρ=0.50 and ARL0=500.


## 6. Case Study

This section contains the case study of theTHWMAAIB versus DHWMA, HWMAAIB 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 Y and auxiliary variable X. 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; X¯=210.24, Y¯=210.18, sX  = 1.23, sY  = 1.17, and r  = 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 μX=210.24, μY=201.18, σX=1.23, σY=1.17 and ρ=0.54 and data set-II consists of fifteen samples with μX=210.24, μY=201.88, σX=1.23, σY=1.17 and ρ=0.54 (see Table 7). This fashion of perturbing the parameters in such a way can be seen from the study of Anwar, et al. [40].Table 7
Application of proposedTHWMAAIB versus HWMA, DHWMA, and HWMAAIB control charts.
123456789101112131415YY1199.851200.078201.572201.09203.035201.641199.389203.604201.341203.875201.415201.327199.378203.229203.666Y2201.717204.336202.807201.157201.386200.303201.456202.498201.745199.945201.245203.002200.117199.989199.652Y3201.063200.958200.288199.513199.515200.546199.954199.737198.347203.028201.669200.467200.626202.224200.987Y4200.479200.14201.337199.568200.601201.503201.243202.127199.917199.884201.104200.754200.817201.418202.245Y5204.204201.173197.843210.842196.682202.145203.873208.894213.552194.211204.549192.111205.174193.41201.814XX1208.872209.19210.33209.683211.605211.848210.226212.204208.388212.249210.695208.663210.937210.735212.075X2210.02210.728211.123208.749210.736208.962210.265212.804211.209209.336210.526210.047210.042211.139208.57X3211.784211.257210.103210.499208.057209.695210.796208.956209.715211.107210.2208.407209.959211.377209.29X4209.839208.857209.364208.411209.868209.755211.28209.029209.08210.553209.253210.413211.014210.837210.234X5204.431215.078201.396217.27205.493205.348200.342212.286217.71205.579218.152212.33211.71217.177202.595161718192021222324252627282930YY1201.195201.373202.659201.39199.658201.601204.071201.609201.854199.816202.422199.059201.486204.192202.843Y2203.174203.178201.947200.477200.397201.367202.385202.564200.649201.083201.805199.116202.648202.087201.181Y3203.254200.807203.02201.453202.249202.728203.604202.916199.401202.103201.462204.669203.934201.164201.924Y4199.653201.482202.333202.191202.226202.085201.484203.841202.158203.57199.9201.098201.21203.186200.475Y5212.546205.377206.285199.63204.52211.787198.042188.259210.226210.174189.097198.165197.983201.197208.504XX1210.228211.909210.683209.256210.279210.797211.296210.406209.824210.091210.5206.871207.802210.201210.436X2211.06209.807211.567209.924209.518209.562210.242208.725209.318210.692209.075209.056211.222210.487211.24X3211.268210.188209.629211.333210.889209.11211.793209.951209.517210.299210.303210.042211.854208.353210.735X4208.717209.953209.783210.569210.19209.616208.771209.781210.098212.661209.076209.911208.757211.35209.472X5216.49216.396209.758202.26214.384215.914216.095195.214211.286217.99206.575214.198212.917208.099214.285123456789101112131415HWMAPlotting statistic201.25201.43201.24201.5201.19201.24201.23201.77201.87201.3201.64201.06201.35201.05201.38LCLHWMAt200.71199.69200.08200.24200.33200.39200.44200.47200.49200.51200.53200.55200.56200.57200.58UCLHWMAt201.65202.67202.28202.12202.03201.97201.92201.89201.87201.85201.83201.81201.8201.79201.78HWMAAIBPlotting statistic201.4201.79201.55201.69201.46201.57201.66201.96201.96201.5201.6201.17201.38200.87201.6LCLHWMAAIBt200.71199.69200.08200.24200.33200.39200.44200.47200.49200.51200.53200.55200.56200.57200.58UCLHWMAAIBt201.65202.67202.28202.12202.03201.97201.92201.89201.87201.85201.83201.81201.8201.79201.78DHWMAPlotting statistic201.2201.45201.36201.27201.42201.25201.24201.37201.6201.58201.55201.44201.38201.3201.31LCLDHWMAt201.08199.6200.06200.27200.39200.47200.53200.58200.61200.64200.67200.69200.71200.73200.75UCLDHWMAt201.28202.76202.3202.09201.97201.89201.83201.78201.75201.72201.69201.67201.65201.63201.61THWMAAIBPlotting statistic201.19202.06201.52201.56201.68201.51201.56201.64201.8201.84201.71201.65201.5201.43201.32LCLTHWMAAIBt201.16200.22200.5200.62200.7200.75200.79200.82200.84200.86200.87200.89200.9200.91200.92UCLTHWMAAIBt201.2202.14201.86201.74201.66201.61201.57201.54201.52201.5201.49201.47201.46201.45201.44161718192021222324252627282930HWMAPlotting statistic201.97201.72201.96201.48201.65202.18201.77201.25201.95202.11201.06201.35201.57201.79201.97LCLHWMAt200.58200.59200.6200.6200.61200.61200.62200.62200.62200.63200.63200.63200.64200.64200.64UCLHWMAt201.78201.77201.76201.76201.75201.75201.74201.74201.74201.73201.73201.73201.72201.72201.72HWMAAIBPlotting statistic201.86201.56201.94201.66201.57202.09201.58201.64202201.88201.19201.38201.55201.87201.86LCLHWMAAIBt200.58200.59200.6200.6200.61200.61200.62200.62200.62200.63200.63200.63200.64200.64200.64UCLHWMAAIBt201.78201.77201.76201.76201.75201.75201.74201.74201.74201.73201.73201.73201.72201.72201.72DHWMAPlotting statistic201.48201.54201.64201.59201.61201.75201.73201.61201.72201.8201.59201.58201.6201.65201.71LCLDHWMAt200.76200.77200.78200.79200.8200.81200.82200.83200.84200.84200.85200.85200.86200.86200.87UCLDHWMAt201.6201.59201.58201.57201.56201.55201.54201.53201.52201.52201.51201.51201.5201.5201.49THWMAAIBPlotting statistic201.41201.5201.54201.61201.62201.64201.69201.68201.69201.74201.72201.65201.62201.63201.66LCLTHWMAAIBt200.93200.94200.95200.95200.96200.96200.97200.97200.98200.98200.99200.99200.99201201UCLTHWMAAIBt201.43201.42201.41201.41201.4201.4201.39201.39201.38201.38201.37201.37201.37201.36201.36For the comparison, the proposedTHWMAAIB chart is considered along with the existing DHWMA, HWMAAIB, and HWMA charts. Furthermore, the parameters of the proposed THWMAAIB chart are LTHWMAAIB=1.90, λ  = 0.25, ρ=0.54 with ARL0=500, and the parameters of the DHWMA chart are LDHWMA=2.7424, λ  = 0.25, with ARL0=500. Likewise, the parameters of HWMAAIB chart are LHWMAAIB=3.075,λ  = 0.25, ρ=0.54 with ARL0=500, and the parameters of the HWMA chart are LHWMA=3.075, λ  = 0.25, with ARL0=500. From Table 7 and Figures 6–9 the proposed THWMAAIB chart tracksthe first OOC signal at the 5th sample, whereas the DHWMA, HWMAAIB, and HWMA charts track the first OOC signal at the 18th, 8th, and 9th samples, respectively. Overall, the proposed THWMAAIB chart highlights 21 OOC points, while the DHWMA, HWMAAIB, and HWMA charts highlight 13, 9, and 9 OOC points, respectively. This show that the proposed THWMAAIB chart is more sensitive than the existing DHWMA, HWMAAIB, and HWMA charts.Figure 6
Application for the existing HWMA chart whenLHWMA=3.075, λ=0.25 and ARL0=500.Figure 7
Application for the existing DHWMA chart whenLDHWMA=2.7424, λ=0.25 and ARL0=500.Figure 8
Application for the existingHWMAAIB chart when LHWMAAIB=3.075, λ=0.25,ρ=0.54, and ARL0=500.Figure 9
Application for the proposedTHWMAAIB chart when LTHWMAAIB  = 1.90, λ=0.25,ρ=0.54, and ARL0=500.


## 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 asTHWMAAIB chart to further improve the process location shift monitoring. To evaluate the performance of the proposed THWMAAIB 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 THWMAAIB chart over-performed against EWMA, EWMAAIB, HWMA, HWMAAIB, DHWMA, TEWMA, MCEAIB, MECAIB and MHC charts. Finally, to demonstrate the proposed THWMAAIB 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.



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*Source: 2938878-2022-10-15.xml*