# Efficient Auxiliary Information–Based Control Charting Schemes for the Process Dispersion with Application of Glass Manufacturing Industry

Authors: Muhammad Aslam; Syed Masroor Anwar; Majid Khan; Nafiu Lukman Abiodun; Zahid Rasheed
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The hybrid exponentially weighted moving average (HEWMA) control chart is an enhanced version of the EWMA control chart that monitors the process parameters effectively. Similarly, the auxiliary information-based (AIB) EWMA control charts are very efficient for monitoring process parameters. The purpose of this paper is to propose two new control charts for the improved monitoring of process dispersion referred to asHEWMAAIB1 and HEWMAAIB2 control charts. A simulation study is carried out to assess the performance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts. Average run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the performance of the proposed control charts against the existing counterparts. The comparisons reveal the superiority of the proposed control charts against other competing control charts, particularly for small to moderate shifts in the process dispersion. Finally, a real-life data set from the glass industry is used to demonstrate the practical implementation of the proposed control charts.


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## Body

## 1. Introduction

There are two types of variations in manufacturing and service processes; common (random) cause variations and special (assignable) cause variations. The common cause variations are an inherent part of every process and cannot be removed entirely. However, the special cause variations are harmful and may distract the processes from their target which results a shifts in the process parameter(s) (location and/or scale). As a result, the practitioner needs to identify and eliminate the assignable cause variations in the process. The statistical control chart is a primary tool in the statistical process control (SPC) toolkit that identifies and rectifies the special cause variations in the process. Memory-less control charts introduced by Shewhart [1] are used to monitor large shifts in the process. On the other hand, the classical memory control charts like cumulative sum (CUSUM) control chart designed by Page [2] and exponentially weighted moving average (EWMA) control chart offered by Roberts [3] are used to monitor small to moderate shifts in the process.Generally, the classical EWMA control chart has been used to detect small shifts in the process mean. However, in many practical situations, the shifts may also occur in the process variance (or standard deviation); when the process variance increases, the productivity and capability of the process may be damaged. If the process variance decreases, more units will be closer to their target value, resulting in improved process functionality. If these changes in the process dispersion are not rectified quickly, unnecessary losses may occur. Numerous authors have constructed different EWMA control charts for the process variance. For example, Crowder and Hamilton [4] used a suitable log transformation to S2lnS2/σY02, and designed the EWMA control chart for monitoring the process standard deviation, where σY02 is the in-control (IC) process variance. Following the same lines, Shu and Jiang [5] suggested the new EWMA control chart in which lnS2/σY02 truncated to its IC mean on every occasion whenever it is less than zero. Similarly, Chang and Gan [6] constructed a one-sided optimal EWMA for monitoring the process variance. Likewise, Khoo [7] discussed the double sampling variance control chart features for monitoring process variability. Similarly, Castagliola [8] used three parameters logarithmic transformation to S2 to improve normality and hence the proposed bilateral EWMA control chart for monitoring the process dispersion. Later, Castagliola, et al. [9] constructed a new S2-EWMA control chart for flexible sampling intervals. Likewise, Eyvazian [10] proposed the EWMSV control chart for process dispersion when the sample size equals 1. In the same direction, Huwang, et al. [11] suggested the EWMA control charts named HHW1 and HHW2 for the process dispersion. They demonstrated that their control charts perform uniformly better against the control charts by Crowder and Hamilton [4] and Shu and Jiang [5]. Razmy and Peiris [12] designed an standardized EWMA control chart for monitoring of process dispersion. Later, Yang and Arnold [13] constructed an unbiased EWMA-p control chart for monitoring the process dispersion. Currently, Castagliola, et al. [14] proposed the double sampling S2-chart for the process variance and elaborated various features of the proposed control chart. Further related work on dispersion control charts, see Abbasi and Miller [15], Ali and Haq [16] and Saghir, et al. [17], etc.Combining the features of the different memory control charts enhances the performance of the ultimate control charts. For example, Haq [18] has constructed a new hybrid exponentially weighted moving average (HEWMA) control chart using one EWMA statistic as an input for another EWMA statistic. Subsequently, several other authors discussed some of the innovations in the HEWMA control chart. For instance, Azam, et al. [19] presented a HEWMA control chart for the process mean under repetitive sampling. Similarly, Aslam, et al. [20] proposed a HEWMA control chart for the COM-Poisson distribution. Likewise, Ali and Haq [16] suggested the generally weighted moving average and the HEWMA control charts for process dispersion. Recently, Chan, et al. [21] suggested improved double EWMA and homogeneously weighted moving average lepage schemes with real life applications. For a more detailed study, see Aslam, et al. [22], Noor-ul-Amin, et al. [23], Mukherjee, et al. [24], Song, et al. [25], and references therein.Several auxiliary information-based (AIB) control charts have been suggested for efficient monitoring of process parameters. For example, Abbas, et al. [26] introduced AIB EWMA for the process mean, named as EWMAAIB control chart. Likewise, Haq [27] recommended two new AIB EWMA control charts named as EWMA-IAIB and EWMA-IIAIB control charts that efficiently monitor the process dispersion. Similarly, Haq [28] provided AIB maximum EWMA control chart for process location and dispersion. Hussain, et al. [29] suggested EWMA control chart based on dual auxiliary information-based estimator for the monitoring of process location. Similarly, Abbasi and Riaz [30] provides the control chart using dual auxiliary information under different ranked set sampling schemes. On the same lines, Riaz, et al. [31] suggested variability control chart using dual auxiliary information-based estimators under different ranked set sampling techniques and different runs rules. Besides, Noor‐ul‐Amin, et al. [32] suggested the MaxEWMAAIB control chart for the simultaneous monitoring of the process mean and coefficient of variation. Recently, Anwar, et al. [33] introduced an AIB combined mixed EWMA-CUSUM control chart for joint monitoring of process paramters.As mentioned before, Ali and Haq [16] proposed the HEWMA control chart for the process dispersion which is more sensitive than the classical EWMA control chart. Sometimes, researchers, engineers, and practitioners are interested in utilizing the features of HEWMA control charts when the original variable carries other information, such as an auxiliary variable, to improve the process’s effectiveness. In this case, the HEWMA control chart will remain inefficient. So, to address this deficiency, this study introduces two auxiliary information-based HEWMA, symbolized as HEWMAAIB1 and HEWMAAIB2 control charts to monitor the small shifts in the process dispersion. To evaluate the performance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts against other control charts, specific performance evaluation measures such as average run length (ARL), extra quadratic loss (EQL), performance comparison index (PCI), and relative ARL (RARL) measures are considered. Besides, an algorithm is designed in R using the Monte Carlo simulations method to calculate the performance evaluation measures. Existing control charts such as HEWMA, adaptive EWMA (AEWMA), HHW1, HHW2, EWMA-IAIB, and EWMA-IIAIB control charts are considered for comparison. Moreover, the proposed control charts are also implemented with real-life applications to show the significance for practical importance.The article’s remainder is organized as follows: variable of interest, auxiliary information, transformation based on auxiliary information, and the existing HEWMA control chart are highlighted in Section2. Section 3 presents the design structure of the proposed HEWMAAIB1 and HEWMAAIB2 control charts. Besides, Section 4 highlights the performance evaluation measures. Furthermore, Section 5 consists of the performance comparison of the proposed HEWMAAIB1 and HEWMAAIB2 control charts against the existing control charts. Similarly, the real-life application of the proposed control charts is provided in Section 6. The last section presents an overall summary and conclusions.


## 2. Existing Method

This section provides insight into the variable of interest and transformation in Subsection2.1. Likewise, the methodology of the HEWMA control charts are presented in Subsection 2.2.


### 2.1. Variable of Interest and Transformation

SupposeY be normally distributed process variable, that is, Y∼NμY,σY2. It is assumed that over a certain period, the underlying process remains IC with variance σY2, but afterwards, it becomes out-of-control (OOC) with variance σY,12. Let τ=σY,1/σY be the amount of shift in process standard deviation σY. In the case of the IC process, τ=1 and OOC process, τ≠1. Also, Y¯ represents the the sample mean and SY2 denotes sample variance of the process variable Y.


### 2.2. Transformation

Suppose,X be an auxiliary information variable of Y variable, then X and Y follow a bivariate normal distribution. Suppose Yi,Xi, for i=1,2,…,n be a random sample of size n. Let X¯ and SX2 be the sample mean and the sample variance of X, respectively. According to Garcia and Cebrian [34], the unbiased regression estimator of σY2 say SY2∗, is given by(1)SY2∗=SY2+ρ2σY2σX2σX2−SX2.Where ρ is the correlation coefficient. The mean and variance of SY2∗ are given as ESY2∗=σY2, VSY2∗=2σY4/n−11−ρ4.Similarly, Haq [27] suggested the difference estimator for process dispersion given as(2)Dt=MY,t−ρ∗MX,t,where MY=Φ−1Gn−1SY2/σY2, MX=Φ−1Gn−1SX2/σX2 and t represents the sample number. Also, G⋅;n−1 is the cumulative distribution function (CDF) of chi-square distribution at n−1 degrees of freedom, and Φ−1⋅ denotes the inverse CDF of the standard normal distribution. The ρ∗ is the correlation between MY and MX. The mean and variance of Dt given by EDt=0 and VDt=1−ρ∗.


### 2.3. HEWMA Control Chart for Process Dispersion

Ali and Haq [16] proposed the HEWMA control chart for the monitoring of process dispersion. Let HEt for t≥1 be the sequence of independentely and identically distributed (IID) observations based on the other sequence MY,t, then the plotting statistic HEt, for the HEWMA control chart is defined as:(3)Et=1−λ1Et−1+λ1MY,t,0<λ1≤1,HEt=1−λ2HEt−1+λ2Et,0<λ2≤1,where HE0=E0=0, and λ1 and λ2 are smoothing constants. The mean and variance of HEt are, respectively, given as EHEt=0, and VHEt=λ1λ2/λ1−λ22∑k=121−λk21−1−λk2t/1−1−λk2−21−λ11−λ21−1−λ1t1−λ2t/1−1−λ11−λ2. The lower and upper control limits of the HEWMA control chart at the time t, are presented as(4)UCLHEWMAt,LCLHEWMAt=±LHEWMAVHEt,where LHEWMA control chart coefficient is used to adjust the IC ARL of the HEWMA control chart at a pre-specified desired level. The HEWMA control chart triggers an OOC signal whenever HEt fall outside of the control limits UCLHEWMAt,LCLHEWMAt.


## 2.1. Variable of Interest and Transformation

SupposeY be normally distributed process variable, that is, Y∼NμY,σY2. It is assumed that over a certain period, the underlying process remains IC with variance σY2, but afterwards, it becomes out-of-control (OOC) with variance σY,12. Let τ=σY,1/σY be the amount of shift in process standard deviation σY. In the case of the IC process, τ=1 and OOC process, τ≠1. Also, Y¯ represents the the sample mean and SY2 denotes sample variance of the process variable Y.


## 2.2. Transformation

Suppose,X be an auxiliary information variable of Y variable, then X and Y follow a bivariate normal distribution. Suppose Yi,Xi, for i=1,2,…,n be a random sample of size n. Let X¯ and SX2 be the sample mean and the sample variance of X, respectively. According to Garcia and Cebrian [34], the unbiased regression estimator of σY2 say SY2∗, is given by(1)SY2∗=SY2+ρ2σY2σX2σX2−SX2.Where ρ is the correlation coefficient. The mean and variance of SY2∗ are given as ESY2∗=σY2, VSY2∗=2σY4/n−11−ρ4.Similarly, Haq [27] suggested the difference estimator for process dispersion given as(2)Dt=MY,t−ρ∗MX,t,where MY=Φ−1Gn−1SY2/σY2, MX=Φ−1Gn−1SX2/σX2 and t represents the sample number. Also, G⋅;n−1 is the cumulative distribution function (CDF) of chi-square distribution at n−1 degrees of freedom, and Φ−1⋅ denotes the inverse CDF of the standard normal distribution. The ρ∗ is the correlation between MY and MX. The mean and variance of Dt given by EDt=0 and VDt=1−ρ∗.


## 2.3. HEWMA Control Chart for Process Dispersion

Ali and Haq [16] proposed the HEWMA control chart for the monitoring of process dispersion. Let HEt for t≥1 be the sequence of independentely and identically distributed (IID) observations based on the other sequence MY,t, then the plotting statistic HEt, for the HEWMA control chart is defined as:(3)Et=1−λ1Et−1+λ1MY,t,0<λ1≤1,HEt=1−λ2HEt−1+λ2Et,0<λ2≤1,where HE0=E0=0, and λ1 and λ2 are smoothing constants. The mean and variance of HEt are, respectively, given as EHEt=0, and VHEt=λ1λ2/λ1−λ22∑k=121−λk21−1−λk2t/1−1−λk2−21−λ11−λ21−1−λ1t1−λ2t/1−1−λ11−λ2. The lower and upper control limits of the HEWMA control chart at the time t, are presented as(4)UCLHEWMAt,LCLHEWMAt=±LHEWMAVHEt,where LHEWMA control chart coefficient is used to adjust the IC ARL of the HEWMA control chart at a pre-specified desired level. The HEWMA control chart triggers an OOC signal whenever HEt fall outside of the control limits UCLHEWMAt,LCLHEWMAt.


## 3. Proposed Methods

This section contains the methodology of the proposedHEWMAAIB1 and HEWMAAIB2 control charts for monitoring the process dispersion. Subsection 3.1 covers the design structure of the proposed HEWMAAIB1 control chart, whereas, the HEWMAAIB2 control chart are given in Subsection 3.2.


### 3.1.HEWMAAIB1 Control Chart

LetSY,t2∗ for t≥1 be the sequence of IID random variables, then the plotting statistic H1,t, for HEWMAAIB1 control chart using the recurrence formula, given by(5)E1,t=λ1SY,t2∗+1−λ1E1,t−1,E1,0=σY2,H1,t=1−λ2H1,t−1+λ2E1,t,H1,0=σY2.Hereλ1 and λ2∈0,1 are smoothing constants. The mean and variance of the H1,t can be given by the expression as EH1,t=σY2 and VH1,t=21−ρ4/n−1λ1λ2/λ1−λ22∑k=121−λk21−1−λk2t/1−1−λk2−21−λ11−λ21−1−λ1t1−λ2t/1−1−λ11−λ2σY4, respectively. The control limits of the proposed HEWMAAIB1 control chart are given by(6)UCLHEWMAAIB1t,LCLHEWMAAIB1t=σY2±LHEWMAAIB1VH1,t,where LHEWMAAIB1 is the coefficient for the HEWMAAIB1 control chart at a pre-specified false alarm rate. The H1,t statistic is plotted against the UCLHEWMAAIB1t and LCLHEWMAAIB1t. The process is considered to be OOC when H1,t>UCLHEWMAAIB1t or H1,t<LCLHEWMAAIB1t; otherwise, it is IC.


### 3.2.HEWMAAIB2 Control Chart

LetDt for t≥1 be the sequence of IID random variables, based on Dt, we defined a new sequence H2,t, using the recurrence formula, given by(7)E2,t=λ1Dt+1−λ1E2,t−1,E2,0=0,H2,t=1−λ2H2,t−1+λ2E2,t,H2,0=0.Hereλ1 and λ2∈0,1 are smoothing constants. The mean and variance of H2,t can be given by the expression as EH2,t=0 and VH2,t=1−ρ∗λ1λ2/λ1−λ22∑k=121−λk21−1−λk2t/1−1−λk2−21−λ11−λ21−1−λ1t1−λ2t/1−1−λ11−λ2. The control limits for the HEWMAAIB2 control chart are given by (8)UCLHEWMAAIB2t,LCLHEWMAAIB2t=±LHEWMAAIB2VH2,t,where LHEWMAAIB2 is the control chart coefficient for HEWMAAIB2 control chart at a pre-specified false alarm rate. The HEWMAAIB2 statistic H2,t is plotted against the LCLHEWMAAIB2t and LCLHEWMAAIB2t, the process is considered to be OOC when H2,t<LCLHEWMAAIB2t or H2,t>UCLHEWMAAIB2t; otherwise, IC.


## 3.1.HEWMAAIB1 Control Chart

LetSY,t2∗ for t≥1 be the sequence of IID random variables, then the plotting statistic H1,t, for HEWMAAIB1 control chart using the recurrence formula, given by(5)E1,t=λ1SY,t2∗+1−λ1E1,t−1,E1,0=σY2,H1,t=1−λ2H1,t−1+λ2E1,t,H1,0=σY2.Hereλ1 and λ2∈0,1 are smoothing constants. The mean and variance of the H1,t can be given by the expression as EH1,t=σY2 and VH1,t=21−ρ4/n−1λ1λ2/λ1−λ22∑k=121−λk21−1−λk2t/1−1−λk2−21−λ11−λ21−1−λ1t1−λ2t/1−1−λ11−λ2σY4, respectively. The control limits of the proposed HEWMAAIB1 control chart are given by(6)UCLHEWMAAIB1t,LCLHEWMAAIB1t=σY2±LHEWMAAIB1VH1,t,where LHEWMAAIB1 is the coefficient for the HEWMAAIB1 control chart at a pre-specified false alarm rate. The H1,t statistic is plotted against the UCLHEWMAAIB1t and LCLHEWMAAIB1t. The process is considered to be OOC when H1,t>UCLHEWMAAIB1t or H1,t<LCLHEWMAAIB1t; otherwise, it is IC.


## 3.2.HEWMAAIB2 Control Chart

LetDt for t≥1 be the sequence of IID random variables, based on Dt, we defined a new sequence H2,t, using the recurrence formula, given by(7)E2,t=λ1Dt+1−λ1E2,t−1,E2,0=0,H2,t=1−λ2H2,t−1+λ2E2,t,H2,0=0.Hereλ1 and λ2∈0,1 are smoothing constants. The mean and variance of H2,t can be given by the expression as EH2,t=0 and VH2,t=1−ρ∗λ1λ2/λ1−λ22∑k=121−λk21−1−λk2t/1−1−λk2−21−λ11−λ21−1−λ1t1−λ2t/1−1−λ11−λ2. The control limits for the HEWMAAIB2 control chart are given by (8)UCLHEWMAAIB2t,LCLHEWMAAIB2t=±LHEWMAAIB2VH2,t,where LHEWMAAIB2 is the control chart coefficient for HEWMAAIB2 control chart at a pre-specified false alarm rate. The HEWMAAIB2 statistic H2,t is plotted against the LCLHEWMAAIB2t and LCLHEWMAAIB2t, the process is considered to be OOC when H2,t<LCLHEWMAAIB2t or H2,t>UCLHEWMAAIB2t; otherwise, IC.


## 4. Performance Evaluation Measures

Quality experts use different performance evaluation measures to evaluate the control charts’ performance.ARL is mostly used for a single shift, while EQL, RARL, and PCI are used to assess the overall performance of control charts. An algorithm is developed in R software, and the Monte Carlo simulation technique is used to compute the numerical results. Monte Carlo simulation with 20000 iterations is performed at each shift τ, where τ= 0.5, 0.6, 0.7, 0.8 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2.0. The details of these performance measures are given in the following subsections.


### 4.1.ARL Measure

TheARL is defined as the average number of sample points plotted until the OOC signal is detected. The ARL is categorized as IC ARLARL0 and OOC ARL (ARL1). If the process is IC state, the ARL0 needed to be large enough to avoid frequent false alarms. However, the ARL1 should be small enough that it quickly detects the shift(s) in the process parameters. It is necessary for the better performance of the control chart that it should have a smaller ARL1 with fixed ARL0 at the desired level.


### 4.2. Overall Performance Measures

TheEQL, RARL, and PCI performance evaluation measures evaluate a control chart’s overall effectiveness by comparison method. The EQL evaluates the overall performance of control charts over a specific range of shifts (Raza, et al. [35]). It is based on the loss function and is defined as:(9)EQL=τmax−τmin−1∫τminτmaxτ2ARLτdτ,where ARLτ is the ARL of a particular control chart at shift τ. The EQL is a weighted average ARL over the entire shift domain τmin<τ<τmax using the square of shift τ2 as weight. A control chart with a minimum EQL value is preferred over other control charts (Anwar, et al. [36]).TheRARL is the average of the ratios among the ARL of a particular control chart with the ARL of a benchmark control chart for all desired shifts.(10)RARL=τmax−τmin−1∫τminτmaxARLτARLBMτdτ,where ARLτ and ARLBMτ symbolize the ARL of a particular control chart and a benchmark control chart for the desired shift, respectively. The benchmark control chart is the control chart with the least EQL. The RARL value for the benchmark control chart is one, and for the other control charts, it is greater than 1.ThePCI evaluates the performance of the best control chart. It is defined as the ratio between the EQL of a control chart and the EQL of the benchmark control chart.(11)PCI=EQLEQLBM,where EQLBM is the EQL of the best-performing control chart. The PCI for the benchmark control chart is 1, while the other control chart’s PCI is greater than 1.


### 4.3. Choices of Parameters

The parameters of the proposedHEWMAAIB1 and HEWMAAIB2 control charts are λ1,λ2, and ρ. Several settings of these parameters are used, and hence corresponding ARL, and standard deviation of RL (SDRL) are computed. The values of λ1,λ2 are set as (0.05,0.05), (0.05,0.1), (0.05,0.2), (0.1,0.05), (0.1,0.1), (0.1,0.2), (0.2,0.05), (0.2,0.1), (0.2,0.2), (0.3,0.05), (0.3,0.1), and (0.3,0.2) and the values of LHEWMAAIB1 and LHEWMAAIB2 are determined to obtain ARL0=200. The different settings of ρ and ρ∗ are taken from the Haq [28] given as ρ,ρ∗= (0.25, 0.0563898), (0.50, 0.2293317), (0.75, 0.5313626), (0.90, 0.7870992), (0.95, 0.8880799). The numerical results of the proposed HEWMAAIB1 and HEWMAAIB2 control charts are presented in Tables 1–8.Table 1
Run-length profile of two-sidedHEWMAAIB1 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB11.6881.691.6951.70261.705τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.53.411.023.281.142.711.291.931.081.540.820.64.691.904.481.933.581.872.381.431.791.050.77.283.806.913.695.413.143.322.152.311.510.813.678.8813.028.5810.126.825.904.163.732.690.938.4832.4936.6030.9628.8324.0217.1013.8010.548.471.0200.63237.56200.24238.26200.61239.81200.47243.01200.86244.701.133.9835.7932.4634.0825.8926.9715.7015.749.859.631.212.3912.6811.6611.949.029.245.625.513.623.311.36.736.866.476.575.025.013.182.902.191.771.44.394.394.194.153.353.162.221.851.671.151.53.203.053.102.932.542.261.771.291.410.821.62.542.292.452.202.061.701.520.991.290.641.72.101.752.031.681.781.351.370.781.190.501.81.851.431.781.351.571.071.280.641.150.431.91.651.191.611.131.440.901.210.541.120.372.01.521.011.490.951.360.791.170.471.090.31λ1=0.05,λ2=0.10LHEWMAAIB11.8201.82491.83441.84161.84180.53.8851.0323.7171.1833.0401.3852.1021.1681.6350.8960.65.2651.9175.0461.9764.0251.9642.6041.5241.9141.1290.78.0833.8197.6793.7426.0183.2393.6492.2572.5111.6030.815.0058.96414.3898.63311.0736.8946.4904.2644.0822.8010.942.48234.76840.29432.98531.96225.47818.56514.20211.3238.6251.0200.27226.37200.23227.38200.24227.47200.27226.37200.96231.611.135.30236.25733.73734.72026.89427.01235.30236.25710.4379.7201.212.82212.65112.19812.0049.4849.20812.82212.6513.8223.3621.36.9916.8606.6796.5635.2965.0676.9916.8602.3051.8511.44.5844.4084.3714.2083.5223.2254.5844.4081.7191.1851.53.3213.0993.2202.9732.6612.3393.3213.0991.4700.8751.62.6142.2962.5302.2032.1431.7502.6142.2961.3090.6661.72.1971.8202.1211.7371.8281.3782.1971.8201.2190.5421.81.8961.4671.8441.3901.6171.1071.8961.4671.1670.4561.91.7001.2361.6561.1691.4850.9631.7001.2361.1250.3862.01.5581.0491.5251.0131.3850.7861.5581.0491.0950.327λ1=0.05,λ2=0.20LHEWMAAIB11.9521.9571.9671.9741.9760.54.281.044.101.203.321.452.251.241.720.950.65.761.915.511.984.372.012.791.602.021.190.78.713.788.323.736.483.263.912.302.651.670.816.079.2115.348.8411.867.006.864.324.342.860.946.5037.5044.0935.3134.4326.7319.6614.5911.958.671.0200.14219.48200.21221.63200.88222.76200.44223.97200.44224.891.135.4535.9534.0234.5227.4827.1816.8715.8810.729.691.212.8912.5112.4111.879.609.026.095.463.963.361.37.086.736.746.405.384.983.432.992.381.871.44.664.344.484.143.653.262.401.911.771.211.53.403.073.272.932.712.311.891.391.500.901.62.672.282.612.222.201.771.611.041.340.691.72.251.832.171.731.861.381.430.841.240.571.81.941.481.891.421.671.161.330.711.180.481.91.731.241.691.181.510.951.240.581.140.402.01.591.071.551.021.400.811.200.511.100.34λ1=0.10,λ2=0.05LHEWMAAIB11.82111.82551.8351.84061.8405τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.53.891.033.721.183.041.382.101.171.630.890.65.271.925.051.984.031.962.601.521.911.130.78.093.827.683.756.023.243.652.262.511.600.815.038.9814.408.6311.086.906.484.264.082.800.942.6234.8440.3432.9931.9725.4818.5414.1811.328.621.0200.88227.13200.82227.99200.73228.04200.16228.14200.20230.481.135.3436.3333.6834.6926.9227.1716.5215.8810.449.701.212.8212.6512.2812.089.469.195.895.523.843.391.36.996.866.636.505.275.063.363.012.291.831.44.584.414.404.213.533.252.341.901.721.191.53.333.123.212.962.652.311.831.341.460.861.62.602.272.532.212.151.761.571.041.320.681.72.191.822.121.721.821.371.410.811.210.531.81.911.481.851.391.621.121.300.681.170.461.91.701.231.651.171.490.951.230.581.130.392.01.561.061.531.011.390.801.180.491.100.33λ1=0.10,λ2=0.10LHEWMAAIB11.96641.9701.9831.9911.9930.54.391.074.211.223.401.492.291.271.740.970.65.921.955.662.034.502.072.851.642.051.220.78.993.908.543.816.683.354.002.372.711.710.816.619.4615.829.1212.307.207.054.424.462.940.949.7741.4047.2038.7536.2429.0120.4215.2912.328.911.0200.09216.15200.79217.67200.16219.18200.52222.13200.92222.161.136.6737.2235.0935.4128.4828.0817.5316.3411.099.931.213.3912.7912.7712.159.969.286.295.654.053.441.37.266.896.946.585.545.123.523.062.411.921.44.774.454.604.253.743.372.411.951.801.261.53.443.113.333.022.762.381.911.411.510.911.62.742.382.622.252.221.801.621.071.340.711.72.271.852.211.791.891.421.440.861.250.571.81.971.541.901.451.681.161.330.701.180.481.91.751.271.701.211.520.971.250.601.140.402.01.601.081.571.051.420.841.210.521.100.35λ1=0.10,λ2=0.20LHEWMAAIB12.11092.11412.13142.14342.14480.54.851.074.631.243.731.542.471.351.841.030.66.471.956.182.044.892.123.091.722.181.280.79.774.019.303.917.223.444.322.442.901.780.818.5210.7017.5610.1613.447.797.614.584.773.020.959.9851.3956.5448.3842.4234.9922.7117.2213.319.491.0200.38211.05200.91213.68200.81214.91200.98216.49200.08215.611.136.9237.4535.6535.5729.3528.8118.2316.7611.5910.121.213.5912.8213.0212.2510.259.366.515.654.233.461.37.426.847.056.525.695.063.633.042.501.941.44.854.384.684.183.823.342.511.981.861.281.53.533.073.422.992.842.391.971.431.550.941.62.812.372.702.252.291.821.661.101.370.741.72.331.872.261.791.951.451.470.861.270.601.82.031.561.951.461.721.181.370.741.200.501.91.801.301.761.251.571.011.280.621.140.412.01.641.101.601.061.430.841.210.531.120.37Table 2
Run-length profile of two-sidedHEWMAAIB1 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB11.9521.9571.96721.97511.9755τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.54.281.044.101.203.321.452.261.241.720.950.65.761.915.511.984.372.012.791.602.021.190.78.713.788.323.736.483.263.912.312.651.670.816.079.2115.348.8411.867.006.864.334.342.850.946.5037.5044.0935.3134.4326.7419.6814.6011.948.661.0200.14219.48200.23221.63200.88222.78201.02224.93200.15224.441.135.4535.9534.0134.5027.4827.1916.8315.8210.719.661.212.8912.5112.4211.889.619.026.115.483.963.381.37.086.736.746.405.394.983.453.002.381.871.44.664.344.474.133.643.262.381.881.771.221.53.403.073.272.932.712.311.901.391.500.901.62.672.282.612.222.201.771.621.051.340.691.72.251.832.161.721.861.381.440.851.240.571.81.951.481.891.421.671.161.330.701.180.471.91.731.241.691.191.510.951.250.581.140.412.01.591.071.551.021.400.821.200.501.100.34λ1=0.20,λ2=0.10LHEWMAAIB12.1122.1142.13122.1432.1450.54.851.074.631.243.731.542.471.351.841.030.66.481.966.182.044.892.123.091.722.181.280.79.784.009.303.917.213.444.322.442.901.780.818.5310.7017.5610.1613.447.797.614.574.773.020.960.1051.4756.5148.3442.4235.0022.7017.2113.319.491.0200.90211.43200.89213.66200.76214.83200.78216.18200.09215.641.136.9537.4835.6535.5729.3728.8218.2616.7911.5910.111.213.5912.8513.0212.2410.259.366.515.654.223.461.37.446.847.056.525.695.063.643.042.501.941.44.864.384.674.183.823.352.501.981.861.281.53.553.073.412.992.842.391.971.431.550.951.62.812.382.702.252.281.821.661.101.370.741.72.331.882.261.801.951.451.470.871.270.601.82.021.551.951.461.721.181.360.731.200.501.91.801.301.761.251.561.011.280.621.140.412.01.631.101.591.051.440.851.210.531.120.37λ1=0.20,λ2=0.20LHEWMAAIB12.2642.2692.2892.30292.3080.55.331.095.111.274.081.612.661.421.961.090.67.112.076.792.135.362.203.341.802.341.360.710.984.6610.444.518.013.764.662.563.111.860.823.2715.4921.7914.2515.739.928.355.065.133.150.991.3284.1384.0477.2058.1551.7927.5422.5515.1111.321.0200.51205.99200.00205.80200.79207.33200.97209.96200.88209.261.137.9238.5136.7236.9630.9530.5919.4218.0412.2810.791.213.9513.2413.4212.5510.669.616.755.744.403.521.37.586.887.276.615.925.153.753.052.621.991.44.944.334.814.203.933.342.601.991.931.321.53.653.093.502.972.932.392.021.461.590.971.62.872.352.792.272.361.841.711.121.410.771.72.401.922.321.792.011.471.520.911.290.621.82.081.562.021.521.751.201.400.761.210.511.91.821.291.791.261.591.011.300.651.160.442.01.671.121.641.081.460.861.230.551.130.38λ1=0.30,λ2=0.05LHEWMAAIB12.0292.0302.0422.0512.053τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.54.471.034.281.193.451.462.331.261.770.970.65.991.895.721.964.542.012.881.612.071.210.79.033.838.573.736.683.274.022.322.731.680.816.689.3515.848.9812.267.137.014.344.442.860.948.9039.2346.2036.7335.5827.5420.2214.8912.218.781.0200.92219.58200.11219.30200.28219.35200.96224.86200.48222.501.135.2435.7733.8434.3127.5327.0316.8715.7310.809.571.212.8312.3612.2811.729.628.946.135.423.993.321.37.036.616.736.315.364.893.472.952.401.851.44.674.284.454.043.663.182.401.871.791.221.53.413.013.302.892.732.281.911.381.520.911.62.682.232.612.172.211.741.631.051.340.701.72.261.812.181.711.891.381.450.851.250.581.81.961.471.901.391.681.151.340.701.190.491.91.761.261.701.181.530.951.260.601.140.412.01.591.051.571.021.410.821.200.511.110.35λ1=0.30,λ2=0.10LHEWMAAIB12.19392.1982.2172.22912.23480.55.061.074.851.243.881.562.561.371.901.060.66.751.996.442.055.092.133.201.742.251.310.710.254.199.744.107.523.504.452.472.991.800.819.9111.7918.8811.1414.218.377.914.714.933.060.968.0458.7263.5755.1946.7038.9924.1118.4513.9310.001.0200.89209.95200.74211.15200.66209.33200.98213.29200.09215.641.136.7637.1535.5035.4329.5828.9618.4216.9711.5910.111.213.5812.7913.0012.2010.309.316.575.634.223.461.37.396.797.086.465.715.023.663.002.501.941.44.874.324.704.143.833.302.541.981.861.281.53.563.023.422.942.862.351.981.411.550.951.62.832.352.732.222.311.801.691.111.370.741.72.341.852.271.781.961.451.490.881.270.601.82.041.561.981.461.741.191.380.741.200.501.91.811.291.781.241.571.001.290.631.140.412.01.651.101.611.051.450.851.230.551.120.37λ1=0.30,λ2=0.20LHEWMAAIB12.36162.3692.38652.4012.4040.55.641.135.401.324.301.662.771.452.021.120.67.622.287.272.335.662.313.481.842.411.390.712.345.7811.665.508.634.264.892.663.211.880.829.2021.2126.9419.6118.1212.298.955.585.353.300.9131.49124.30118.63112.0873.5968.0431.7327.2516.4512.731.0200.27204.34200.85204.42200.11204.41200.88206.78200.36204.541.138.1138.6337.0637.3231.5131.3220.0918.8212.7211.201.214.1313.4413.6012.8210.859.806.875.814.473.551.37.656.907.386.635.975.183.783.032.661.991.44.984.324.854.183.973.332.631.991.951.321.53.683.063.552.932.972.382.061.471.620.981.62.922.352.812.252.401.841.741.131.420.781.72.441.922.351.812.021.471.550.941.300.631.82.101.562.041.501.781.201.400.761.220.521.91.851.291.831.271.611.011.310.661.170.452.01.701.131.641.071.480.861.240.551.140.39Table 3
Run-length profile of one-sidedHEWMAAIB1 control chart at ARL0  = 200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB11.21551.21611.21311.21531.213τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.38248.99200.08248.42200.16248.03200.74250.04200.19250.511.122.7726.3921.8725.3417.4720.0510.7512.146.897.461.28.8110.058.489.676.737.544.094.252.722.541.34.905.364.665.073.844.052.462.281.781.361.43.403.543.203.292.612.511.831.451.440.891.52.542.422.472.372.081.791.551.051.280.641.62.101.832.031.751.721.351.360.781.180.491.71.821.481.761.361.541.071.270.641.130.401.81.611.201.581.131.400.861.200.531.100.341.91.491.001.460.961.310.721.150.441.070.282.01.370.831.360.801.250.631.110.381.060.24λ1=0.05,λ2=0.10LHEWMAAIB11.4091.4061.4031.3971.3941.0200.84237.00200.61236.04200.25233.53200.31236.04200.61236.361.124.8027.2423.6925.9819.0020.5811.8012.417.577.741.29.5110.179.059.757.397.844.464.432.992.731.35.395.645.095.294.134.162.662.441.911.461.43.643.623.513.502.842.681.971.591.510.961.52.792.642.652.422.181.891.611.101.320.701.62.241.992.191.921.831.441.430.871.220.551.71.901.551.851.481.621.161.300.691.150.441.81.701.271.661.231.460.931.220.561.110.361.91.541.061.490.991.360.801.170.481.090.322.01.440.901.400.861.270.641.130.411.070.28λ1=0.10,λ2=0.05LHEWMAAIB11.40761.4051.40421.3981.394τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.10236.14200.28235.48200.84234.51200.61236.31200.61236.361.124.8327.2323.6525.8919.0320.6211.8012.437.577.741.29.4810.169.109.807.417.864.494.462.992.731.35.405.655.065.274.104.132.662.441.911.461.43.633.613.523.522.832.661.971.581.510.961.52.782.612.632.422.191.901.611.121.320.701.62.241.982.191.911.841.441.430.861.220.551.71.931.591.861.501.631.161.310.691.150.441.81.691.251.661.221.460.941.220.561.110.361.91.541.051.490.981.350.781.170.471.090.322.01.430.911.410.871.280.661.130.411.070.28λ1=0.10,λ2=0.10LHEWMAAIB11.6361.6301.6181.6051.5991.0200.79225.52200.76225.46200.00222.81200.03223.34200.70225.341.127.4528.9526.2727.4421.2121.7913.1013.088.508.151.210.4910.569.9410.058.118.094.954.673.262.881.35.965.905.695.654.514.352.942.632.061.601.44.063.943.893.703.072.822.111.681.581.041.53.012.792.902.662.392.071.721.221.380.781.62.412.102.322.011.971.561.480.911.250.591.72.061.691.971.591.701.221.350.741.180.481.81.791.361.751.321.530.991.260.611.140.411.91.631.151.591.081.420.861.200.521.100.342.01.490.971.460.921.330.731.160.451.080.29Table 4
Run-length profile of one-sidedHEWMAAIB1 control chart at ARL0  = 200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB11.58751.58251.5721.56591.56τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.88230.95200.05230.21200.10228.49200.96228.86200.63229.121.125.7927.5124.8726.2819.8720.9312.3812.578.067.821.29.9010.139.499.687.747.844.744.483.162.781.35.715.715.415.394.294.142.842.532.011.531.43.863.673.663.472.992.732.061.621.571.001.52.922.702.822.532.281.941.691.181.360.751.62.342.022.261.931.931.511.470.901.250.581.71.991.601.951.551.671.181.330.711.170.471.81.761.331.711.241.510.981.250.591.130.401.91.611.101.551.031.390.791.190.501.100.342.01.470.921.450.921.320.701.160.441.080.29λ1=0.20,λ2=0.10LHEWMAAIB11.84751.84491.8211.7991.7891.0200.96217.27200.91217.63200.48215.52200.27216.11200.78217.141.129.4030.2628.1928.8422.6522.7514.0213.569.098.311.211.2411.0010.6910.368.578.135.304.793.512.991.36.396.066.095.704.794.433.112.662.181.661.44.303.974.123.813.312.952.241.771.671.121.53.202.853.072.722.512.111.781.261.430.811.62.552.172.472.092.061.581.550.971.290.631.72.151.742.081.651.791.301.380.761.210.521.81.871.401.831.361.601.081.300.661.150.421.91.681.171.641.121.470.891.220.551.120.372.01.541.001.510.961.370.771.170.471.100.33λ1=0.30,λ2=0.05LHEWMAAIB11.68751.67941.66721.6551.648τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.70228.03200.19226.08200.00225.63200.08225.87200.60227.321.126.2327.7425.0926.3820.1620.9412.5912.568.217.821.210.0510.179.609.667.827.794.834.453.222.781.35.775.645.485.344.354.092.872.492.051.531.43.963.713.723.443.032.702.091.611.591.021.52.952.672.872.542.341.951.731.191.380.771.62.382.022.301.911.961.511.490.911.250.581.72.021.601.951.521.701.191.350.731.190.481.81.801.331.741.261.520.971.270.611.140.411.91.621.101.581.061.410.811.200.501.100.342.01.500.961.470.901.330.731.160.461.080.31λ1=0.30,λ2=0.10LHEWMAAIB11.97081.96451.9391.9091.8941.0200.10213.90200.52214.69200.16213.50200.37213.25200.49214.331.130.3030.9628.9129.3023.2923.3114.4513.849.308.411.211.5411.1511.0010.578.768.185.434.833.603.001.36.546.096.265.784.894.433.192.672.231.671.44.413.984.193.793.372.962.281.781.701.131.53.262.863.132.722.562.101.831.281.450.831.62.612.182.522.102.131.621.570.981.310.651.72.191.732.121.661.831.331.410.801.220.541.81.911.431.871.381.631.071.310.661.160.431.91.711.171.681.131.490.911.240.551.130.382.01.571.011.540.971.380.771.180.481.100.34Table 5
Run-length profile of two-sidedHEWMAAIB2 control chart at ARL0  = 200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB21.7221.7181.7151.7101.710τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.141.212.051.151.670.871.190.461.050.240.63.402.263.262.172.571.651.600.901.200.500.76.014.485.724.274.533.312.651.841.701.050.812.5210.1512.059.699.637.665.534.373.352.580.938.6536.6336.8534.7629.9427.3818.1716.0211.249.651.0200.27246.69200.19245.68199.48245.59199.45242.05200.74242.871.137.3439.8835.5638.0228.6130.3017.9318.1011.4211.211.213.8514.0313.2013.3810.2210.306.366.243.923.671.37.627.667.227.245.715.633.373.172.241.871.44.984.884.754.673.813.672.331.971.631.151.53.633.413.483.302.812.551.821.391.360.771.62.882.622.762.462.261.931.531.031.210.571.72.392.082.281.961.901.511.380.821.140.441.82.061.701.981.591.691.231.260.631.100.351.91.821.401.781.351.521.021.200.541.070.292.01.671.201.611.141.420.871.140.441.050.23λ1=0.05,λ2=0.10LHEWMAAIB21.8511.84931.84451.8451.8440.52.351.302.251.231.820.951.240.511.060.260.63.712.353.562.252.811.741.720.971.250.550.76.494.546.224.364.923.402.901.941.821.120.813.4810.3112.889.7910.317.746.004.473.652.680.941.4638.2839.9536.6732.0528.5419.5816.3212.119.781.0200.21232.77201.25232.48199.96230.16200.13225.69200.18231.851.139.5440.3337.7838.6430.6930.7520.1718.4512.2711.251.214.7014.0414.0713.4411.0210.407.146.194.253.781.38.087.657.737.306.175.733.863.262.401.951.45.324.915.084.704.063.732.622.101.721.211.53.883.523.753.382.992.652.011.491.400.831.63.072.732.952.582.391.991.661.121.250.621.72.532.142.432.062.031.591.470.891.170.481.82.181.772.091.671.751.271.330.701.110.381.91.921.481.861.401.581.071.240.591.080.302.01.721.221.691.201.460.901.180.491.060.25λ1=0.05,λ2=0.20LHEWMAAIB21.98291.9811.97691.9781.9750.52.551.352.441.281.970.991.310.561.080.300.64.002.383.822.283.041.791.841.021.300.590.76.924.536.654.355.223.413.101.971.951.170.814.2110.5313.589.9210.817.806.364.473.902.710.943.8739.7642.3738.0833.9129.8820.6016.7212.669.781.0200.52222.38200.47224.70201.08225.63200.13225.69200.84224.971.141.1840.7039.3139.2331.9131.0220.1718.4512.8311.241.215.3314.0514.5713.3211.5210.427.146.194.483.751.38.377.478.087.216.435.663.863.262.541.991.45.564.865.334.664.263.722.622.101.781.231.54.103.563.903.343.152.652.011.491.470.871.63.192.703.082.622.502.021.661.121.280.651.72.652.162.532.062.091.591.470.891.180.501.82.251.742.191.701.821.301.330.701.120.391.92.001.481.931.431.631.081.240.591.080.322.01.801.261.741.201.510.951.180.491.060.27λ1=0.10,λ2=0.05LHEWMAAIB21.8511.8491.8461.8461.845τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.351.302.251.231.820.951.240.511.060.260.63.712.353.562.252.811.741.720.971.250.550.76.494.546.224.364.933.402.901.941.821.120.813.4810.3112.889.7910.337.756.004.473.662.680.941.4638.2839.9636.6632.0828.5619.6016.3412.129.781.0200.16232.78200.90231.87200.92231.12200.90234.03200.49232.231.139.5440.3637.7738.5730.6630.7419.2618.4112.2911.271.214.7114.0414.0913.4911.0110.426.826.284.263.821.38.087.647.757.336.225.733.713.292.401.941.45.324.925.074.684.063.762.502.071.731.211.53.883.523.733.373.002.641.911.461.400.831.63.072.732.952.572.402.001.601.091.250.611.72.532.142.442.072.021.601.410.841.160.481.82.181.772.091.671.741.261.310.691.120.391.91.921.471.861.401.591.081.210.551.080.302.01.721.231.691.201.460.901.160.481.050.25λ1=0.10,λ2=0.10LHEWMAAIB21.9971.9971.9921.9951.9900.52.591.382.491.322.001.021.320.581.080.300.64.112.453.922.343.111.841.871.041.310.600.77.124.666.864.465.383.513.162.031.981.200.814.7010.7914.0810.2811.207.996.574.604.012.790.946.6842.9444.8440.9035.7731.7821.3517.3713.0910.041.0200.14218.75200.50221.26199.78220.54200.48222.59200.44221.851.143.0243.0241.3541.2533.3932.6321.0119.2113.3411.561.215.9614.5615.2313.8211.9610.757.366.344.613.871.38.687.628.307.336.645.864.003.382.602.061.45.725.045.484.804.373.812.672.161.801.271.54.173.654.023.523.242.742.041.541.480.901.63.292.793.152.662.552.071.701.171.290.651.72.672.192.602.132.131.651.470.891.190.501.82.311.822.221.731.831.321.340.751.130.401.92.011.511.961.451.661.121.250.581.090.332.01.821.311.761.241.510.961.190.511.070.28λ1=0.10,λ2=0.20LHEWMAAIB22.14062.14042.14022.13992.13970.52.831.412.711.372.181.071.400.631.110.340.64.442.484.252.373.381.892.021.091.390.650.77.634.727.304.525.773.553.422.072.141.250.815.8711.5715.1711.0411.928.297.014.644.302.800.951.9547.7849.7145.7139.7036.0123.1418.9114.0510.571.0200.52212.02201.39215.11200.50213.54199.04212.09200.08212.691.146.1145.5444.0943.4236.0434.5322.5320.1214.1711.921.216.6214.8515.9414.1012.7510.987.806.344.913.881.39.077.678.657.336.965.854.233.372.752.081.45.975.035.764.824.633.792.812.181.911.321.54.393.614.213.483.402.742.161.571.540.941.63.442.793.292.652.682.081.771.201.330.691.72.822.232.722.122.221.661.530.941.210.531.82.391.852.301.751.941.391.380.761.140.421.92.101.552.041.511.731.171.280.631.100.362.01.901.341.831.271.581.001.210.551.070.29Table 6
Run-length profile of two-sidedHEWMAAIB2 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB21.9811.9791.9761.9761.975τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.541.352.431.281.960.991.310.561.080.300.64.002.383.822.283.041.791.841.021.300.590.76.904.526.644.355.223.413.091.971.951.170.814.1910.5213.569.9210.807.796.354.463.902.710.943.7539.6742.2738.0033.8829.8720.5816.6912.669.781.0199.35221.60199.09223.38200.28224.53199.48224.77200.84224.971.141.0440.6539.1639.0131.9430.9820.1118.3912.9111.331.215.2713.9814.6013.4211.4710.357.116.204.483.701.38.417.548.047.206.485.683.883.272.531.981.45.514.825.314.644.223.692.622.101.791.261.54.103.533.903.353.152.662.001.481.460.871.63.172.703.082.582.522.031.661.111.270.611.72.652.162.542.072.081.581.460.881.180.491.82.261.772.171.691.831.311.340.721.120.391.91.981.471.931.431.621.071.240.581.080.322.01.801.271.741.231.500.931.180.501.060.27λ1=0.20,λ2=0.10LHEWMAAIB22.13982.1392.14112.1432.1400.52.831.412.711.372.181.071.400.631.110.340.64.442.474.252.373.381.892.021.101.390.650.77.634.727.304.525.783.553.432.072.141.250.815.8611.5715.1511.0311.938.297.014.644.302.800.951.9147.7549.6245.6139.7336.0123.2118.9614.0610.571.0200.01211.54201.23214.87201.02214.16200.84213.71200.19212.791.146.0545.4344.0243.3535.9934.4722.6220.1814.1711.921.216.6514.8215.9314.1112.7711.007.806.324.913.881.39.057.668.667.326.985.854.233.352.752.091.45.985.025.764.834.613.772.832.201.921.331.54.403.644.203.473.402.752.161.581.530.931.63.432.773.302.662.682.081.771.201.330.691.72.812.222.722.122.231.671.520.911.220.531.82.401.852.301.751.931.381.400.781.140.421.92.111.562.041.501.731.171.280.641.110.362.01.891.321.831.271.571.001.220.551.070.29λ1=0.20,λ2=0.20LHEWMAAIB22.2962.2912.2982.2992.3000.53.101.462.961.402.381.121.500.691.140.380.64.812.554.622.453.681.932.211.151.480.710.78.325.117.914.806.283.713.712.122.321.300.818.0813.9317.1913.1613.379.607.614.904.642.860.962.0759.0359.0655.6847.2043.6326.7522.9115.6912.211.0201.81209.14200.42207.02200.66206.57200.99208.57200.93207.671.151.2550.3248.7747.6840.6239.2325.2322.8315.4913.011.217.9215.9317.1215.1313.7911.708.216.495.283.961.39.628.029.187.597.405.944.533.462.922.111.46.295.006.014.834.883.843.042.262.021.371.54.603.624.423.453.592.742.281.631.600.961.63.612.773.472.692.852.151.861.231.360.721.72.972.272.832.152.351.731.580.971.250.571.82.521.882.431.782.021.411.430.811.170.451.92.221.612.141.531.801.201.320.671.120.372.01.991.391.911.301.631.041.250.581.080.31λ1=0.30,λ2=0.05LHEWMAAIB22.05592.0532.04942.0542.050τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.641.352.531.292.041.011.350.591.090.310.64.132.363.942.273.141.781.911.031.340.610.77.074.516.784.315.373.393.191.962.021.190.814.5110.5913.8510.0711.017.826.514.454.002.680.945.0440.6743.1138.7834.7330.3621.0216.8712.909.831.0199.79218.86199.43221.28199.94222.28200.74222.87200.18220.751.141.5741.1639.8039.2032.6231.1720.4618.5313.0011.231.215.4013.9514.6813.2411.6010.257.266.134.543.701.38.467.368.127.136.535.603.923.202.581.961.45.604.795.374.584.303.662.672.071.821.241.54.133.483.943.283.192.602.041.481.480.881.63.232.663.122.572.542.011.711.141.300.651.72.682.132.572.032.121.581.490.891.200.511.82.281.722.211.681.851.291.350.721.130.401.92.031.481.961.421.651.081.260.601.090.322.01.821.261.771.201.530.941.190.501.070.28λ1=0.30,λ2=0.10LHEWMAAIB22.22572.22492.22592.2292.2270.52.951.422.821.372.271.091.450.651.130.360.64.612.494.412.383.521.892.121.111.430.680.77.894.827.584.595.973.583.552.072.231.270.816.5412.1715.7811.5312.398.597.264.684.452.800.954.5050.3452.2648.1941.7737.8824.3119.8614.5310.871.0201.41210.44201.39212.49200.98212.26200.85211.46200.24211.491.147.3246.5045.2444.2937.2035.3723.3020.6614.5112.111.216.9314.9216.2814.2713.0311.067.926.275.033.861.39.217.698.807.297.115.844.323.322.812.061.46.064.955.814.754.703.742.922.201.971.341.54.453.564.293.433.452.682.201.551.570.941.63.482.753.352.632.742.071.801.191.340.691.72.862.212.772.102.291.671.570.951.230.541.82.451.822.371.771.981.401.410.791.160.441.92.161.562.081.491.761.181.310.661.110.362.01.931.331.851.261.601.001.230.561.080.29λ1=0.30,λ2=0.20LHEWMAAIB22.3852.38462.3842.3872.3900.53.231.473.091.422.481.131.560.711.170.410.65.022.624.822.513.831.942.301.161.530.730.78.775.478.395.206.573.893.852.142.421.310.819.8515.8818.8114.8914.4110.687.965.184.812.900.968.7866.0265.7762.4651.4648.2229.0925.5516.8613.401.0200.80203.89201.46206.61199.45205.70200.61206.90200.52205.121.153.9953.3151.9250.8342.8841.3626.7924.4516.3813.961.218.7416.7918.0015.9814.3912.248.596.755.373.971.39.878.269.477.837.606.054.643.453.012.111.46.425.036.154.844.943.833.092.252.081.371.54.693.604.513.463.682.752.331.601.630.961.63.692.803.522.632.892.131.891.221.400.741.73.012.262.922.162.401.701.641.001.260.571.82.601.882.471.782.081.431.460.811.180.461.92.271.602.191.551.831.211.350.691.130.392.02.031.391.961.321.661.021.260.581.090.32Table 7
Run-length profile of one-sidedHEWMAAIB2 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB21.23331.23021.22961.2291.225τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.00251.02200.15251.83200.76252.87200.86252.09200.27250.701.124.0628.0623.0026.8518.7522.0111.9213.787.628.631.29.3210.748.9010.307.268.444.394.892.812.851.35.215.844.995.574.064.442.582.551.771.451.43.613.863.463.682.802.841.861.581.390.901.52.712.662.602.552.141.991.521.121.210.601.62.232.102.151.961.791.501.340.821.130.451.71.911.651.841.561.581.201.240.641.090.351.81.691.351.641.281.420.941.160.501.060.261.91.541.111.501.061.320.811.120.421.040.212.01.410.911.390.901.260.661.090.351.030.19λ1=0.05,λ2=0.10LHEWMAAIB21.40011.39991.39361.39231.3911.0200.00235.98200.63239.37200.48236.35200.94237.48200.50237.161.126.0228.5225.0527.4920.5822.4213.1413.988.528.921.210.1610.839.7510.357.968.514.895.033.122.991.35.826.125.585.854.444.542.822.701.901.551.43.953.943.793.783.022.962.001.691.450.961.52.992.922.892.792.352.131.621.201.260.671.62.402.202.332.111.921.601.400.881.160.481.72.041.741.981.671.671.281.280.691.100.371.81.791.441.731.341.501.011.190.541.070.301.91.631.191.581.161.400.871.140.461.050.242.01.491.001.460.961.300.741.100.381.030.20λ1=0.10,λ2=0.05LHEWMAAIB21.4011.39871.39411.39091.390τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.43236.48200.01238.74200.61236.51200.34236.69200.19236.951.125.9928.4525.0327.4120.6022.4613.1313.998.468.861.210.1610.839.7710.397.978.504.875.013.123.001.35.856.125.575.834.454.562.832.711.901.551.43.983.993.773.753.012.952.001.701.450.951.52.992.912.902.792.362.151.621.191.260.671.62.402.212.332.111.911.591.400.881.160.491.72.031.731.971.661.671.271.270.681.100.371.81.801.441.751.371.501.011.190.551.070.301.91.631.191.571.131.400.891.140.461.050.252.01.501.011.470.961.300.741.110.391.030.20λ1=0.10,λ2=0.10LHEWMAAIB21.59251.59791.58331.57991.57651.0200.34222.30200.15222.82200.34222.72200.11221.85200.00221.711.128.9229.8727.8828.7722.8823.2714.6814.539.519.151.211.2611.2110.7810.668.798.695.505.283.503.171.36.556.336.236.044.984.763.122.852.071.671.44.414.234.264.063.403.192.191.851.551.061.53.293.053.152.932.562.241.731.281.310.731.62.642.352.542.242.101.721.470.941.190.521.72.231.872.141.771.801.391.320.751.120.411.81.921.521.881.471.601.121.230.601.080.321.91.731.271.681.211.470.951.170.491.060.272.01.581.071.541.031.360.801.130.421.040.22Table 8
Run-length profile of one-sidedHEWMAAIB2 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB21.55411.55111.54841.54191.541τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.08226.56200.00228.05200.84229.89200.37228.52200.47228.321.127.1728.6026.2527.5821.6222.4713.8814.029.038.851.210.6710.7410.2710.298.418.455.205.043.363.031.36.186.035.935.804.724.532.982.732.011.601.44.254.064.063.833.253.032.131.761.521.011.53.172.933.052.802.482.181.691.231.300.711.62.552.232.462.152.021.631.450.921.180.511.72.161.812.071.691.761.311.310.721.120.401.81.881.451.831.411.581.091.220.591.080.321.91.701.251.651.181.440.911.160.481.060.262.01.551.051.531.011.350.781.120.411.040.22λ1=0.20,λ2=0.10LHEWMAAIB21.77071.76451.75451.74811.7491.0200.84215.56200.52214.67200.28214.27200.70214.46200.43214.081.131.0231.0829.9129.7924.6524.0815.7414.8710.269.341.212.1511.4611.6310.889.328.655.905.313.803.261.37.066.476.736.125.364.893.382.922.221.751.44.744.264.524.073.613.162.341.871.641.101.53.543.103.402.982.782.331.811.311.370.771.62.822.392.732.282.251.801.541.001.230.581.72.341.892.271.831.901.431.380.781.140.431.82.051.561.981.501.681.191.260.631.100.361.91.831.341.761.251.520.981.200.541.070.292.01.661.131.621.071.410.851.150.441.050.24λ1=0.30,λ2=0.05LHEWMAAIB21.63861.63631.62961.62291.622τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.90224.11200.84225.20200.54226.22200.31224.72200.35225.541.127.6828.7226.6427.6521.9722.3614.1113.959.178.761.210.8510.6910.3710.188.558.365.314.983.433.001.36.295.936.005.714.804.473.052.692.051.591.44.304.004.163.843.313.012.181.741.551.021.53.222.873.092.752.522.131.731.241.320.721.62.612.232.512.122.071.631.480.931.200.521.72.201.772.131.711.801.331.320.721.120.401.81.931.491.871.401.601.091.240.601.090.331.91.721.231.681.181.460.911.170.491.060.272.01.581.051.551.011.370.791.130.411.040.22λ1=0.30,λ2=0.10LHEWMAAIB21.86671.8591.84591.83981.8421.0200.81210.97200.66211.65200.45211.35200.96212.53200.56212.501.131.8331.6430.6330.3625.1324.2416.1815.1510.579.421.212.4911.5911.8610.919.568.626.065.313.913.221.37.226.486.926.165.454.813.452.882.291.761.44.844.234.624.053.723.142.391.841.681.111.53.623.073.492.942.852.341.861.331.400.791.62.892.372.772.262.291.781.571.001.240.581.72.411.882.341.821.941.441.400.801.160.451.82.101.582.021.491.721.191.280.651.110.371.91.861.331.811.251.551.001.210.541.080.302.01.701.161.641.091.440.851.160.461.060.25


## 4.1.ARL Measure

TheARL is defined as the average number of sample points plotted until the OOC signal is detected. The ARL is categorized as IC ARLARL0 and OOC ARL (ARL1). If the process is IC state, the ARL0 needed to be large enough to avoid frequent false alarms. However, the ARL1 should be small enough that it quickly detects the shift(s) in the process parameters. It is necessary for the better performance of the control chart that it should have a smaller ARL1 with fixed ARL0 at the desired level.


## 4.2. Overall Performance Measures

TheEQL, RARL, and PCI performance evaluation measures evaluate a control chart’s overall effectiveness by comparison method. The EQL evaluates the overall performance of control charts over a specific range of shifts (Raza, et al. [35]). It is based on the loss function and is defined as:(9)EQL=τmax−τmin−1∫τminτmaxτ2ARLτdτ,where ARLτ is the ARL of a particular control chart at shift τ. The EQL is a weighted average ARL over the entire shift domain τmin<τ<τmax using the square of shift τ2 as weight. A control chart with a minimum EQL value is preferred over other control charts (Anwar, et al. [36]).TheRARL is the average of the ratios among the ARL of a particular control chart with the ARL of a benchmark control chart for all desired shifts.(10)RARL=τmax−τmin−1∫τminτmaxARLτARLBMτdτ,where ARLτ and ARLBMτ symbolize the ARL of a particular control chart and a benchmark control chart for the desired shift, respectively. The benchmark control chart is the control chart with the least EQL. The RARL value for the benchmark control chart is one, and for the other control charts, it is greater than 1.ThePCI evaluates the performance of the best control chart. It is defined as the ratio between the EQL of a control chart and the EQL of the benchmark control chart.(11)PCI=EQLEQLBM,where EQLBM is the EQL of the best-performing control chart. The PCI for the benchmark control chart is 1, while the other control chart’s PCI is greater than 1.


## 4.3. Choices of Parameters

The parameters of the proposedHEWMAAIB1 and HEWMAAIB2 control charts are λ1,λ2, and ρ. Several settings of these parameters are used, and hence corresponding ARL, and standard deviation of RL (SDRL) are computed. The values of λ1,λ2 are set as (0.05,0.05), (0.05,0.1), (0.05,0.2), (0.1,0.05), (0.1,0.1), (0.1,0.2), (0.2,0.05), (0.2,0.1), (0.2,0.2), (0.3,0.05), (0.3,0.1), and (0.3,0.2) and the values of LHEWMAAIB1 and LHEWMAAIB2 are determined to obtain ARL0=200. The different settings of ρ and ρ∗ are taken from the Haq [28] given as ρ,ρ∗= (0.25, 0.0563898), (0.50, 0.2293317), (0.75, 0.5313626), (0.90, 0.7870992), (0.95, 0.8880799). The numerical results of the proposed HEWMAAIB1 and HEWMAAIB2 control charts are presented in Tables 1–8.Table 1
Run-length profile of two-sidedHEWMAAIB1 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB11.6881.691.6951.70261.705τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.53.411.023.281.142.711.291.931.081.540.820.64.691.904.481.933.581.872.381.431.791.050.77.283.806.913.695.413.143.322.152.311.510.813.678.8813.028.5810.126.825.904.163.732.690.938.4832.4936.6030.9628.8324.0217.1013.8010.548.471.0200.63237.56200.24238.26200.61239.81200.47243.01200.86244.701.133.9835.7932.4634.0825.8926.9715.7015.749.859.631.212.3912.6811.6611.949.029.245.625.513.623.311.36.736.866.476.575.025.013.182.902.191.771.44.394.394.194.153.353.162.221.851.671.151.53.203.053.102.932.542.261.771.291.410.821.62.542.292.452.202.061.701.520.991.290.641.72.101.752.031.681.781.351.370.781.190.501.81.851.431.781.351.571.071.280.641.150.431.91.651.191.611.131.440.901.210.541.120.372.01.521.011.490.951.360.791.170.471.090.31λ1=0.05,λ2=0.10LHEWMAAIB11.8201.82491.83441.84161.84180.53.8851.0323.7171.1833.0401.3852.1021.1681.6350.8960.65.2651.9175.0461.9764.0251.9642.6041.5241.9141.1290.78.0833.8197.6793.7426.0183.2393.6492.2572.5111.6030.815.0058.96414.3898.63311.0736.8946.4904.2644.0822.8010.942.48234.76840.29432.98531.96225.47818.56514.20211.3238.6251.0200.27226.37200.23227.38200.24227.47200.27226.37200.96231.611.135.30236.25733.73734.72026.89427.01235.30236.25710.4379.7201.212.82212.65112.19812.0049.4849.20812.82212.6513.8223.3621.36.9916.8606.6796.5635.2965.0676.9916.8602.3051.8511.44.5844.4084.3714.2083.5223.2254.5844.4081.7191.1851.53.3213.0993.2202.9732.6612.3393.3213.0991.4700.8751.62.6142.2962.5302.2032.1431.7502.6142.2961.3090.6661.72.1971.8202.1211.7371.8281.3782.1971.8201.2190.5421.81.8961.4671.8441.3901.6171.1071.8961.4671.1670.4561.91.7001.2361.6561.1691.4850.9631.7001.2361.1250.3862.01.5581.0491.5251.0131.3850.7861.5581.0491.0950.327λ1=0.05,λ2=0.20LHEWMAAIB11.9521.9571.9671.9741.9760.54.281.044.101.203.321.452.251.241.720.950.65.761.915.511.984.372.012.791.602.021.190.78.713.788.323.736.483.263.912.302.651.670.816.079.2115.348.8411.867.006.864.324.342.860.946.5037.5044.0935.3134.4326.7319.6614.5911.958.671.0200.14219.48200.21221.63200.88222.76200.44223.97200.44224.891.135.4535.9534.0234.5227.4827.1816.8715.8810.729.691.212.8912.5112.4111.879.609.026.095.463.963.361.37.086.736.746.405.384.983.432.992.381.871.44.664.344.484.143.653.262.401.911.771.211.53.403.073.272.932.712.311.891.391.500.901.62.672.282.612.222.201.771.611.041.340.691.72.251.832.171.731.861.381.430.841.240.571.81.941.481.891.421.671.161.330.711.180.481.91.731.241.691.181.510.951.240.581.140.402.01.591.071.551.021.400.811.200.511.100.34λ1=0.10,λ2=0.05LHEWMAAIB11.82111.82551.8351.84061.8405τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.53.891.033.721.183.041.382.101.171.630.890.65.271.925.051.984.031.962.601.521.911.130.78.093.827.683.756.023.243.652.262.511.600.815.038.9814.408.6311.086.906.484.264.082.800.942.6234.8440.3432.9931.9725.4818.5414.1811.328.621.0200.88227.13200.82227.99200.73228.04200.16228.14200.20230.481.135.3436.3333.6834.6926.9227.1716.5215.8810.449.701.212.8212.6512.2812.089.469.195.895.523.843.391.36.996.866.636.505.275.063.363.012.291.831.44.584.414.404.213.533.252.341.901.721.191.53.333.123.212.962.652.311.831.341.460.861.62.602.272.532.212.151.761.571.041.320.681.72.191.822.121.721.821.371.410.811.210.531.81.911.481.851.391.621.121.300.681.170.461.91.701.231.651.171.490.951.230.581.130.392.01.561.061.531.011.390.801.180.491.100.33λ1=0.10,λ2=0.10LHEWMAAIB11.96641.9701.9831.9911.9930.54.391.074.211.223.401.492.291.271.740.970.65.921.955.662.034.502.072.851.642.051.220.78.993.908.543.816.683.354.002.372.711.710.816.619.4615.829.1212.307.207.054.424.462.940.949.7741.4047.2038.7536.2429.0120.4215.2912.328.911.0200.09216.15200.79217.67200.16219.18200.52222.13200.92222.161.136.6737.2235.0935.4128.4828.0817.5316.3411.099.931.213.3912.7912.7712.159.969.286.295.654.053.441.37.266.896.946.585.545.123.523.062.411.921.44.774.454.604.253.743.372.411.951.801.261.53.443.113.333.022.762.381.911.411.510.911.62.742.382.622.252.221.801.621.071.340.711.72.271.852.211.791.891.421.440.861.250.571.81.971.541.901.451.681.161.330.701.180.481.91.751.271.701.211.520.971.250.601.140.402.01.601.081.571.051.420.841.210.521.100.35λ1=0.10,λ2=0.20LHEWMAAIB12.11092.11412.13142.14342.14480.54.851.074.631.243.731.542.471.351.841.030.66.471.956.182.044.892.123.091.722.181.280.79.774.019.303.917.223.444.322.442.901.780.818.5210.7017.5610.1613.447.797.614.584.773.020.959.9851.3956.5448.3842.4234.9922.7117.2213.319.491.0200.38211.05200.91213.68200.81214.91200.98216.49200.08215.611.136.9237.4535.6535.5729.3528.8118.2316.7611.5910.121.213.5912.8213.0212.2510.259.366.515.654.233.461.37.426.847.056.525.695.063.633.042.501.941.44.854.384.684.183.823.342.511.981.861.281.53.533.073.422.992.842.391.971.431.550.941.62.812.372.702.252.291.821.661.101.370.741.72.331.872.261.791.951.451.470.861.270.601.82.031.561.951.461.721.181.370.741.200.501.91.801.301.761.251.571.011.280.621.140.412.01.641.101.601.061.430.841.210.531.120.37Table 2
Run-length profile of two-sidedHEWMAAIB1 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB11.9521.9571.96721.97511.9755τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.54.281.044.101.203.321.452.261.241.720.950.65.761.915.511.984.372.012.791.602.021.190.78.713.788.323.736.483.263.912.312.651.670.816.079.2115.348.8411.867.006.864.334.342.850.946.5037.5044.0935.3134.4326.7419.6814.6011.948.661.0200.14219.48200.23221.63200.88222.78201.02224.93200.15224.441.135.4535.9534.0134.5027.4827.1916.8315.8210.719.661.212.8912.5112.4211.889.619.026.115.483.963.381.37.086.736.746.405.394.983.453.002.381.871.44.664.344.474.133.643.262.381.881.771.221.53.403.073.272.932.712.311.901.391.500.901.62.672.282.612.222.201.771.621.051.340.691.72.251.832.161.721.861.381.440.851.240.571.81.951.481.891.421.671.161.330.701.180.471.91.731.241.691.191.510.951.250.581.140.412.01.591.071.551.021.400.821.200.501.100.34λ1=0.20,λ2=0.10LHEWMAAIB12.1122.1142.13122.1432.1450.54.851.074.631.243.731.542.471.351.841.030.66.481.966.182.044.892.123.091.722.181.280.79.784.009.303.917.213.444.322.442.901.780.818.5310.7017.5610.1613.447.797.614.574.773.020.960.1051.4756.5148.3442.4235.0022.7017.2113.319.491.0200.90211.43200.89213.66200.76214.83200.78216.18200.09215.641.136.9537.4835.6535.5729.3728.8218.2616.7911.5910.111.213.5912.8513.0212.2410.259.366.515.654.223.461.37.446.847.056.525.695.063.643.042.501.941.44.864.384.674.183.823.352.501.981.861.281.53.553.073.412.992.842.391.971.431.550.951.62.812.382.702.252.281.821.661.101.370.741.72.331.882.261.801.951.451.470.871.270.601.82.021.551.951.461.721.181.360.731.200.501.91.801.301.761.251.561.011.280.621.140.412.01.631.101.591.051.440.851.210.531.120.37λ1=0.20,λ2=0.20LHEWMAAIB12.2642.2692.2892.30292.3080.55.331.095.111.274.081.612.661.421.961.090.67.112.076.792.135.362.203.341.802.341.360.710.984.6610.444.518.013.764.662.563.111.860.823.2715.4921.7914.2515.739.928.355.065.133.150.991.3284.1384.0477.2058.1551.7927.5422.5515.1111.321.0200.51205.99200.00205.80200.79207.33200.97209.96200.88209.261.137.9238.5136.7236.9630.9530.5919.4218.0412.2810.791.213.9513.2413.4212.5510.669.616.755.744.403.521.37.586.887.276.615.925.153.753.052.621.991.44.944.334.814.203.933.342.601.991.931.321.53.653.093.502.972.932.392.021.461.590.971.62.872.352.792.272.361.841.711.121.410.771.72.401.922.321.792.011.471.520.911.290.621.82.081.562.021.521.751.201.400.761.210.511.91.821.291.791.261.591.011.300.651.160.442.01.671.121.641.081.460.861.230.551.130.38λ1=0.30,λ2=0.05LHEWMAAIB12.0292.0302.0422.0512.053τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.54.471.034.281.193.451.462.331.261.770.970.65.991.895.721.964.542.012.881.612.071.210.79.033.838.573.736.683.274.022.322.731.680.816.689.3515.848.9812.267.137.014.344.442.860.948.9039.2346.2036.7335.5827.5420.2214.8912.218.781.0200.92219.58200.11219.30200.28219.35200.96224.86200.48222.501.135.2435.7733.8434.3127.5327.0316.8715.7310.809.571.212.8312.3612.2811.729.628.946.135.423.993.321.37.036.616.736.315.364.893.472.952.401.851.44.674.284.454.043.663.182.401.871.791.221.53.413.013.302.892.732.281.911.381.520.911.62.682.232.612.172.211.741.631.051.340.701.72.261.812.181.711.891.381.450.851.250.581.81.961.471.901.391.681.151.340.701.190.491.91.761.261.701.181.530.951.260.601.140.412.01.591.051.571.021.410.821.200.511.110.35λ1=0.30,λ2=0.10LHEWMAAIB12.19392.1982.2172.22912.23480.55.061.074.851.243.881.562.561.371.901.060.66.751.996.442.055.092.133.201.742.251.310.710.254.199.744.107.523.504.452.472.991.800.819.9111.7918.8811.1414.218.377.914.714.933.060.968.0458.7263.5755.1946.7038.9924.1118.4513.9310.001.0200.89209.95200.74211.15200.66209.33200.98213.29200.09215.641.136.7637.1535.5035.4329.5828.9618.4216.9711.5910.111.213.5812.7913.0012.2010.309.316.575.634.223.461.37.396.797.086.465.715.023.663.002.501.941.44.874.324.704.143.833.302.541.981.861.281.53.563.023.422.942.862.351.981.411.550.951.62.832.352.732.222.311.801.691.111.370.741.72.341.852.271.781.961.451.490.881.270.601.82.041.561.981.461.741.191.380.741.200.501.91.811.291.781.241.571.001.290.631.140.412.01.651.101.611.051.450.851.230.551.120.37λ1=0.30,λ2=0.20LHEWMAAIB12.36162.3692.38652.4012.4040.55.641.135.401.324.301.662.771.452.021.120.67.622.287.272.335.662.313.481.842.411.390.712.345.7811.665.508.634.264.892.663.211.880.829.2021.2126.9419.6118.1212.298.955.585.353.300.9131.49124.30118.63112.0873.5968.0431.7327.2516.4512.731.0200.27204.34200.85204.42200.11204.41200.88206.78200.36204.541.138.1138.6337.0637.3231.5131.3220.0918.8212.7211.201.214.1313.4413.6012.8210.859.806.875.814.473.551.37.656.907.386.635.975.183.783.032.661.991.44.984.324.854.183.973.332.631.991.951.321.53.683.063.552.932.972.382.061.471.620.981.62.922.352.812.252.401.841.741.131.420.781.72.441.922.351.812.021.471.550.941.300.631.82.101.562.041.501.781.201.400.761.220.521.91.851.291.831.271.611.011.310.661.170.452.01.701.131.641.071.480.861.240.551.140.39Table 3
Run-length profile of one-sidedHEWMAAIB1 control chart at ARL0  = 200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB11.21551.21611.21311.21531.213τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.38248.99200.08248.42200.16248.03200.74250.04200.19250.511.122.7726.3921.8725.3417.4720.0510.7512.146.897.461.28.8110.058.489.676.737.544.094.252.722.541.34.905.364.665.073.844.052.462.281.781.361.43.403.543.203.292.612.511.831.451.440.891.52.542.422.472.372.081.791.551.051.280.641.62.101.832.031.751.721.351.360.781.180.491.71.821.481.761.361.541.071.270.641.130.401.81.611.201.581.131.400.861.200.531.100.341.91.491.001.460.961.310.721.150.441.070.282.01.370.831.360.801.250.631.110.381.060.24λ1=0.05,λ2=0.10LHEWMAAIB11.4091.4061.4031.3971.3941.0200.84237.00200.61236.04200.25233.53200.31236.04200.61236.361.124.8027.2423.6925.9819.0020.5811.8012.417.577.741.29.5110.179.059.757.397.844.464.432.992.731.35.395.645.095.294.134.162.662.441.911.461.43.643.623.513.502.842.681.971.591.510.961.52.792.642.652.422.181.891.611.101.320.701.62.241.992.191.921.831.441.430.871.220.551.71.901.551.851.481.621.161.300.691.150.441.81.701.271.661.231.460.931.220.561.110.361.91.541.061.490.991.360.801.170.481.090.322.01.440.901.400.861.270.641.130.411.070.28λ1=0.10,λ2=0.05LHEWMAAIB11.40761.4051.40421.3981.394τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.10236.14200.28235.48200.84234.51200.61236.31200.61236.361.124.8327.2323.6525.8919.0320.6211.8012.437.577.741.29.4810.169.109.807.417.864.494.462.992.731.35.405.655.065.274.104.132.662.441.911.461.43.633.613.523.522.832.661.971.581.510.961.52.782.612.632.422.191.901.611.121.320.701.62.241.982.191.911.841.441.430.861.220.551.71.931.591.861.501.631.161.310.691.150.441.81.691.251.661.221.460.941.220.561.110.361.91.541.051.490.981.350.781.170.471.090.322.01.430.911.410.871.280.661.130.411.070.28λ1=0.10,λ2=0.10LHEWMAAIB11.6361.6301.6181.6051.5991.0200.79225.52200.76225.46200.00222.81200.03223.34200.70225.341.127.4528.9526.2727.4421.2121.7913.1013.088.508.151.210.4910.569.9410.058.118.094.954.673.262.881.35.965.905.695.654.514.352.942.632.061.601.44.063.943.893.703.072.822.111.681.581.041.53.012.792.902.662.392.071.721.221.380.781.62.412.102.322.011.971.561.480.911.250.591.72.061.691.971.591.701.221.350.741.180.481.81.791.361.751.321.530.991.260.611.140.411.91.631.151.591.081.420.861.200.521.100.342.01.490.971.460.921.330.731.160.451.080.29Table 4
Run-length profile of one-sidedHEWMAAIB1 control chart at ARL0  = 200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB11.58751.58251.5721.56591.56τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.88230.95200.05230.21200.10228.49200.96228.86200.63229.121.125.7927.5124.8726.2819.8720.9312.3812.578.067.821.29.9010.139.499.687.747.844.744.483.162.781.35.715.715.415.394.294.142.842.532.011.531.43.863.673.663.472.992.732.061.621.571.001.52.922.702.822.532.281.941.691.181.360.751.62.342.022.261.931.931.511.470.901.250.581.71.991.601.951.551.671.181.330.711.170.471.81.761.331.711.241.510.981.250.591.130.401.91.611.101.551.031.390.791.190.501.100.342.01.470.921.450.921.320.701.160.441.080.29λ1=0.20,λ2=0.10LHEWMAAIB11.84751.84491.8211.7991.7891.0200.96217.27200.91217.63200.48215.52200.27216.11200.78217.141.129.4030.2628.1928.8422.6522.7514.0213.569.098.311.211.2411.0010.6910.368.578.135.304.793.512.991.36.396.066.095.704.794.433.112.662.181.661.44.303.974.123.813.312.952.241.771.671.121.53.202.853.072.722.512.111.781.261.430.811.62.552.172.472.092.061.581.550.971.290.631.72.151.742.081.651.791.301.380.761.210.521.81.871.401.831.361.601.081.300.661.150.421.91.681.171.641.121.470.891.220.551.120.372.01.541.001.510.961.370.771.170.471.100.33λ1=0.30,λ2=0.05LHEWMAAIB11.68751.67941.66721.6551.648τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.70228.03200.19226.08200.00225.63200.08225.87200.60227.321.126.2327.7425.0926.3820.1620.9412.5912.568.217.821.210.0510.179.609.667.827.794.834.453.222.781.35.775.645.485.344.354.092.872.492.051.531.43.963.713.723.443.032.702.091.611.591.021.52.952.672.872.542.341.951.731.191.380.771.62.382.022.301.911.961.511.490.911.250.581.72.021.601.951.521.701.191.350.731.190.481.81.801.331.741.261.520.971.270.611.140.411.91.621.101.581.061.410.811.200.501.100.342.01.500.961.470.901.330.731.160.461.080.31λ1=0.30,λ2=0.10LHEWMAAIB11.97081.96451.9391.9091.8941.0200.10213.90200.52214.69200.16213.50200.37213.25200.49214.331.130.3030.9628.9129.3023.2923.3114.4513.849.308.411.211.5411.1511.0010.578.768.185.434.833.603.001.36.546.096.265.784.894.433.192.672.231.671.44.413.984.193.793.372.962.281.781.701.131.53.262.863.132.722.562.101.831.281.450.831.62.612.182.522.102.131.621.570.981.310.651.72.191.732.121.661.831.331.410.801.220.541.81.911.431.871.381.631.071.310.661.160.431.91.711.171.681.131.490.911.240.551.130.382.01.571.011.540.971.380.771.180.481.100.34Table 5
Run-length profile of two-sidedHEWMAAIB2 control chart at ARL0  = 200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB21.7221.7181.7151.7101.710τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.141.212.051.151.670.871.190.461.050.240.63.402.263.262.172.571.651.600.901.200.500.76.014.485.724.274.533.312.651.841.701.050.812.5210.1512.059.699.637.665.534.373.352.580.938.6536.6336.8534.7629.9427.3818.1716.0211.249.651.0200.27246.69200.19245.68199.48245.59199.45242.05200.74242.871.137.3439.8835.5638.0228.6130.3017.9318.1011.4211.211.213.8514.0313.2013.3810.2210.306.366.243.923.671.37.627.667.227.245.715.633.373.172.241.871.44.984.884.754.673.813.672.331.971.631.151.53.633.413.483.302.812.551.821.391.360.771.62.882.622.762.462.261.931.531.031.210.571.72.392.082.281.961.901.511.380.821.140.441.82.061.701.981.591.691.231.260.631.100.351.91.821.401.781.351.521.021.200.541.070.292.01.671.201.611.141.420.871.140.441.050.23λ1=0.05,λ2=0.10LHEWMAAIB21.8511.84931.84451.8451.8440.52.351.302.251.231.820.951.240.511.060.260.63.712.353.562.252.811.741.720.971.250.550.76.494.546.224.364.923.402.901.941.821.120.813.4810.3112.889.7910.317.746.004.473.652.680.941.4638.2839.9536.6732.0528.5419.5816.3212.119.781.0200.21232.77201.25232.48199.96230.16200.13225.69200.18231.851.139.5440.3337.7838.6430.6930.7520.1718.4512.2711.251.214.7014.0414.0713.4411.0210.407.146.194.253.781.38.087.657.737.306.175.733.863.262.401.951.45.324.915.084.704.063.732.622.101.721.211.53.883.523.753.382.992.652.011.491.400.831.63.072.732.952.582.391.991.661.121.250.621.72.532.142.432.062.031.591.470.891.170.481.82.181.772.091.671.751.271.330.701.110.381.91.921.481.861.401.581.071.240.591.080.302.01.721.221.691.201.460.901.180.491.060.25λ1=0.05,λ2=0.20LHEWMAAIB21.98291.9811.97691.9781.9750.52.551.352.441.281.970.991.310.561.080.300.64.002.383.822.283.041.791.841.021.300.590.76.924.536.654.355.223.413.101.971.951.170.814.2110.5313.589.9210.817.806.364.473.902.710.943.8739.7642.3738.0833.9129.8820.6016.7212.669.781.0200.52222.38200.47224.70201.08225.63200.13225.69200.84224.971.141.1840.7039.3139.2331.9131.0220.1718.4512.8311.241.215.3314.0514.5713.3211.5210.427.146.194.483.751.38.377.478.087.216.435.663.863.262.541.991.45.564.865.334.664.263.722.622.101.781.231.54.103.563.903.343.152.652.011.491.470.871.63.192.703.082.622.502.021.661.121.280.651.72.652.162.532.062.091.591.470.891.180.501.82.251.742.191.701.821.301.330.701.120.391.92.001.481.931.431.631.081.240.591.080.322.01.801.261.741.201.510.951.180.491.060.27λ1=0.10,λ2=0.05LHEWMAAIB21.8511.8491.8461.8461.845τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.351.302.251.231.820.951.240.511.060.260.63.712.353.562.252.811.741.720.971.250.550.76.494.546.224.364.933.402.901.941.821.120.813.4810.3112.889.7910.337.756.004.473.662.680.941.4638.2839.9636.6632.0828.5619.6016.3412.129.781.0200.16232.78200.90231.87200.92231.12200.90234.03200.49232.231.139.5440.3637.7738.5730.6630.7419.2618.4112.2911.271.214.7114.0414.0913.4911.0110.426.826.284.263.821.38.087.647.757.336.225.733.713.292.401.941.45.324.925.074.684.063.762.502.071.731.211.53.883.523.733.373.002.641.911.461.400.831.63.072.732.952.572.402.001.601.091.250.611.72.532.142.442.072.021.601.410.841.160.481.82.181.772.091.671.741.261.310.691.120.391.91.921.471.861.401.591.081.210.551.080.302.01.721.231.691.201.460.901.160.481.050.25λ1=0.10,λ2=0.10LHEWMAAIB21.9971.9971.9921.9951.9900.52.591.382.491.322.001.021.320.581.080.300.64.112.453.922.343.111.841.871.041.310.600.77.124.666.864.465.383.513.162.031.981.200.814.7010.7914.0810.2811.207.996.574.604.012.790.946.6842.9444.8440.9035.7731.7821.3517.3713.0910.041.0200.14218.75200.50221.26199.78220.54200.48222.59200.44221.851.143.0243.0241.3541.2533.3932.6321.0119.2113.3411.561.215.9614.5615.2313.8211.9610.757.366.344.613.871.38.687.628.307.336.645.864.003.382.602.061.45.725.045.484.804.373.812.672.161.801.271.54.173.654.023.523.242.742.041.541.480.901.63.292.793.152.662.552.071.701.171.290.651.72.672.192.602.132.131.651.470.891.190.501.82.311.822.221.731.831.321.340.751.130.401.92.011.511.961.451.661.121.250.581.090.332.01.821.311.761.241.510.961.190.511.070.28λ1=0.10,λ2=0.20LHEWMAAIB22.14062.14042.14022.13992.13970.52.831.412.711.372.181.071.400.631.110.340.64.442.484.252.373.381.892.021.091.390.650.77.634.727.304.525.773.553.422.072.141.250.815.8711.5715.1711.0411.928.297.014.644.302.800.951.9547.7849.7145.7139.7036.0123.1418.9114.0510.571.0200.52212.02201.39215.11200.50213.54199.04212.09200.08212.691.146.1145.5444.0943.4236.0434.5322.5320.1214.1711.921.216.6214.8515.9414.1012.7510.987.806.344.913.881.39.077.678.657.336.965.854.233.372.752.081.45.975.035.764.824.633.792.812.181.911.321.54.393.614.213.483.402.742.161.571.540.941.63.442.793.292.652.682.081.771.201.330.691.72.822.232.722.122.221.661.530.941.210.531.82.391.852.301.751.941.391.380.761.140.421.92.101.552.041.511.731.171.280.631.100.362.01.901.341.831.271.581.001.210.551.070.29Table 6
Run-length profile of two-sidedHEWMAAIB2 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB21.9811.9791.9761.9761.975τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.541.352.431.281.960.991.310.561.080.300.64.002.383.822.283.041.791.841.021.300.590.76.904.526.644.355.223.413.091.971.951.170.814.1910.5213.569.9210.807.796.354.463.902.710.943.7539.6742.2738.0033.8829.8720.5816.6912.669.781.0199.35221.60199.09223.38200.28224.53199.48224.77200.84224.971.141.0440.6539.1639.0131.9430.9820.1118.3912.9111.331.215.2713.9814.6013.4211.4710.357.116.204.483.701.38.417.548.047.206.485.683.883.272.531.981.45.514.825.314.644.223.692.622.101.791.261.54.103.533.903.353.152.662.001.481.460.871.63.172.703.082.582.522.031.661.111.270.611.72.652.162.542.072.081.581.460.881.180.491.82.261.772.171.691.831.311.340.721.120.391.91.981.471.931.431.621.071.240.581.080.322.01.801.271.741.231.500.931.180.501.060.27λ1=0.20,λ2=0.10LHEWMAAIB22.13982.1392.14112.1432.1400.52.831.412.711.372.181.071.400.631.110.340.64.442.474.252.373.381.892.021.101.390.650.77.634.727.304.525.783.553.432.072.141.250.815.8611.5715.1511.0311.938.297.014.644.302.800.951.9147.7549.6245.6139.7336.0123.2118.9614.0610.571.0200.01211.54201.23214.87201.02214.16200.84213.71200.19212.791.146.0545.4344.0243.3535.9934.4722.6220.1814.1711.921.216.6514.8215.9314.1112.7711.007.806.324.913.881.39.057.668.667.326.985.854.233.352.752.091.45.985.025.764.834.613.772.832.201.921.331.54.403.644.203.473.402.752.161.581.530.931.63.432.773.302.662.682.081.771.201.330.691.72.812.222.722.122.231.671.520.911.220.531.82.401.852.301.751.931.381.400.781.140.421.92.111.562.041.501.731.171.280.641.110.362.01.891.321.831.271.571.001.220.551.070.29λ1=0.20,λ2=0.20LHEWMAAIB22.2962.2912.2982.2992.3000.53.101.462.961.402.381.121.500.691.140.380.64.812.554.622.453.681.932.211.151.480.710.78.325.117.914.806.283.713.712.122.321.300.818.0813.9317.1913.1613.379.607.614.904.642.860.962.0759.0359.0655.6847.2043.6326.7522.9115.6912.211.0201.81209.14200.42207.02200.66206.57200.99208.57200.93207.671.151.2550.3248.7747.6840.6239.2325.2322.8315.4913.011.217.9215.9317.1215.1313.7911.708.216.495.283.961.39.628.029.187.597.405.944.533.462.922.111.46.295.006.014.834.883.843.042.262.021.371.54.603.624.423.453.592.742.281.631.600.961.63.612.773.472.692.852.151.861.231.360.721.72.972.272.832.152.351.731.580.971.250.571.82.521.882.431.782.021.411.430.811.170.451.92.221.612.141.531.801.201.320.671.120.372.01.991.391.911.301.631.041.250.581.080.31λ1=0.30,λ2=0.05LHEWMAAIB22.05592.0532.04942.0542.050τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL0.52.641.352.531.292.041.011.350.591.090.310.64.132.363.942.273.141.781.911.031.340.610.77.074.516.784.315.373.393.191.962.021.190.814.5110.5913.8510.0711.017.826.514.454.002.680.945.0440.6743.1138.7834.7330.3621.0216.8712.909.831.0199.79218.86199.43221.28199.94222.28200.74222.87200.18220.751.141.5741.1639.8039.2032.6231.1720.4618.5313.0011.231.215.4013.9514.6813.2411.6010.257.266.134.543.701.38.467.368.127.136.535.603.923.202.581.961.45.604.795.374.584.303.662.672.071.821.241.54.133.483.943.283.192.602.041.481.480.881.63.232.663.122.572.542.011.711.141.300.651.72.682.132.572.032.121.581.490.891.200.511.82.281.722.211.681.851.291.350.721.130.401.92.031.481.961.421.651.081.260.601.090.322.01.821.261.771.201.530.941.190.501.070.28λ1=0.30,λ2=0.10LHEWMAAIB22.22572.22492.22592.2292.2270.52.951.422.821.372.271.091.450.651.130.360.64.612.494.412.383.521.892.121.111.430.680.77.894.827.584.595.973.583.552.072.231.270.816.5412.1715.7811.5312.398.597.264.684.452.800.954.5050.3452.2648.1941.7737.8824.3119.8614.5310.871.0201.41210.44201.39212.49200.98212.26200.85211.46200.24211.491.147.3246.5045.2444.2937.2035.3723.3020.6614.5112.111.216.9314.9216.2814.2713.0311.067.926.275.033.861.39.217.698.807.297.115.844.323.322.812.061.46.064.955.814.754.703.742.922.201.971.341.54.453.564.293.433.452.682.201.551.570.941.63.482.753.352.632.742.071.801.191.340.691.72.862.212.772.102.291.671.570.951.230.541.82.451.822.371.771.981.401.410.791.160.441.92.161.562.081.491.761.181.310.661.110.362.01.931.331.851.261.601.001.230.561.080.29λ1=0.30,λ2=0.20LHEWMAAIB22.3852.38462.3842.3872.3900.53.231.473.091.422.481.131.560.711.170.410.65.022.624.822.513.831.942.301.161.530.730.78.775.478.395.206.573.893.852.142.421.310.819.8515.8818.8114.8914.4110.687.965.184.812.900.968.7866.0265.7762.4651.4648.2229.0925.5516.8613.401.0200.80203.89201.46206.61199.45205.70200.61206.90200.52205.121.153.9953.3151.9250.8342.8841.3626.7924.4516.3813.961.218.7416.7918.0015.9814.3912.248.596.755.373.971.39.878.269.477.837.606.054.643.453.012.111.46.425.036.154.844.943.833.092.252.081.371.54.693.604.513.463.682.752.331.601.630.961.63.692.803.522.632.892.131.891.221.400.741.73.012.262.922.162.401.701.641.001.260.571.82.601.882.471.782.081.431.460.811.180.461.92.271.602.191.551.831.211.350.691.130.392.02.031.391.961.321.661.021.260.581.090.32Table 7
Run-length profile of one-sidedHEWMAAIB2 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.05,λ2=0.05LHEWMAAIB21.23331.23021.22961.2291.225τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.00251.02200.15251.83200.76252.87200.86252.09200.27250.701.124.0628.0623.0026.8518.7522.0111.9213.787.628.631.29.3210.748.9010.307.268.444.394.892.812.851.35.215.844.995.574.064.442.582.551.771.451.43.613.863.463.682.802.841.861.581.390.901.52.712.662.602.552.141.991.521.121.210.601.62.232.102.151.961.791.501.340.821.130.451.71.911.651.841.561.581.201.240.641.090.351.81.691.351.641.281.420.941.160.501.060.261.91.541.111.501.061.320.811.120.421.040.212.01.410.911.390.901.260.661.090.351.030.19λ1=0.05,λ2=0.10LHEWMAAIB21.40011.39991.39361.39231.3911.0200.00235.98200.63239.37200.48236.35200.94237.48200.50237.161.126.0228.5225.0527.4920.5822.4213.1413.988.528.921.210.1610.839.7510.357.968.514.895.033.122.991.35.826.125.585.854.444.542.822.701.901.551.43.953.943.793.783.022.962.001.691.450.961.52.992.922.892.792.352.131.621.201.260.671.62.402.202.332.111.921.601.400.881.160.481.72.041.741.981.671.671.281.280.691.100.371.81.791.441.731.341.501.011.190.541.070.301.91.631.191.581.161.400.871.140.461.050.242.01.491.001.460.961.300.741.100.381.030.20λ1=0.10,λ2=0.05LHEWMAAIB21.4011.39871.39411.39091.390τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.43236.48200.01238.74200.61236.51200.34236.69200.19236.951.125.9928.4525.0327.4120.6022.4613.1313.998.468.861.210.1610.839.7710.397.978.504.875.013.123.001.35.856.125.575.834.454.562.832.711.901.551.43.983.993.773.753.012.952.001.701.450.951.52.992.912.902.792.362.151.621.191.260.671.62.402.212.332.111.911.591.400.881.160.491.72.031.731.971.661.671.271.270.681.100.371.81.801.441.751.371.501.011.190.551.070.301.91.631.191.571.131.400.891.140.461.050.252.01.501.011.470.961.300.741.110.391.030.20λ1=0.10,λ2=0.10LHEWMAAIB21.59251.59791.58331.57991.57651.0200.34222.30200.15222.82200.34222.72200.11221.85200.00221.711.128.9229.8727.8828.7722.8823.2714.6814.539.519.151.211.2611.2110.7810.668.798.695.505.283.503.171.36.556.336.236.044.984.763.122.852.071.671.44.414.234.264.063.403.192.191.851.551.061.53.293.053.152.932.562.241.731.281.310.731.62.642.352.542.242.101.721.470.941.190.521.72.231.872.141.771.801.391.320.751.120.411.81.921.521.881.471.601.121.230.601.080.321.91.731.271.681.211.470.951.170.491.060.272.01.581.071.541.031.360.801.130.421.040.22Table 8
Run-length profile of one-sidedHEWMAAIB2 control chart at ARL0=200.
ρ0.250.500.750.900.95λ1=0.20,λ2=0.05LHEWMAAIB21.55411.55111.54841.54191.541τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.08226.56200.00228.05200.84229.89200.37228.52200.47228.321.127.1728.6026.2527.5821.6222.4713.8814.029.038.851.210.6710.7410.2710.298.418.455.205.043.363.031.36.186.035.935.804.724.532.982.732.011.601.44.254.064.063.833.253.032.131.761.521.011.53.172.933.052.802.482.181.691.231.300.711.62.552.232.462.152.021.631.450.921.180.511.72.161.812.071.691.761.311.310.721.120.401.81.881.451.831.411.581.091.220.591.080.321.91.701.251.651.181.440.911.160.481.060.262.01.551.051.531.011.350.781.120.411.040.22λ1=0.20,λ2=0.10LHEWMAAIB21.77071.76451.75451.74811.7491.0200.84215.56200.52214.67200.28214.27200.70214.46200.43214.081.131.0231.0829.9129.7924.6524.0815.7414.8710.269.341.212.1511.4611.6310.889.328.655.905.313.803.261.37.066.476.736.125.364.893.382.922.221.751.44.744.264.524.073.613.162.341.871.641.101.53.543.103.402.982.782.331.811.311.370.771.62.822.392.732.282.251.801.541.001.230.581.72.341.892.271.831.901.431.380.781.140.431.82.051.561.981.501.681.191.260.631.100.361.91.831.341.761.251.520.981.200.541.070.292.01.661.131.621.071.410.851.150.441.050.24λ1=0.30,λ2=0.05LHEWMAAIB21.63861.63631.62961.62291.622τARLSDRLARLSDRLARLSDRLARLSDRLARLSDRL1.0200.90224.11200.84225.20200.54226.22200.31224.72200.35225.541.127.6828.7226.6427.6521.9722.3614.1113.959.178.761.210.8510.6910.3710.188.558.365.314.983.433.001.36.295.936.005.714.804.473.052.692.051.591.44.304.004.163.843.313.012.181.741.551.021.53.222.873.092.752.522.131.731.241.320.721.62.612.232.512.122.071.631.480.931.200.521.72.201.772.131.711.801.331.320.721.120.401.81.931.491.871.401.601.091.240.601.090.331.91.721.231.681.181.460.911.170.491.060.272.01.581.051.551.011.370.791.130.411.040.22λ1=0.30,λ2=0.10LHEWMAAIB21.86671.8591.84591.83981.8421.0200.81210.97200.66211.65200.45211.35200.96212.53200.56212.501.131.8331.6430.6330.3625.1324.2416.1815.1510.579.421.212.4911.5911.8610.919.568.626.065.313.913.221.37.226.486.926.165.454.813.452.882.291.761.44.844.234.624.053.723.142.391.841.681.111.53.623.073.492.942.852.341.861.331.400.791.62.892.372.772.262.291.781.571.001.240.581.72.411.882.341.821.941.441.400.801.160.451.82.101.582.021.491.721.191.280.651.110.371.91.861.331.811.251.551.001.210.541.080.302.01.701.161.641.091.440.851.160.461.060.25


## 5. Evaluation and Performance Comparison

This section provides extensive comparisons of the proposedHEWMAAIB1 and HEWMAAIB2 control charts with the HEWMA (Ali and Haq [16]), AEWMA (Haq [37]), HHW1 and HHW2 (Huwang, et al. [11]), and EWMA-IAIB and EWMA-IIAIB (Haq [27]) control charts.


### 5.1. Proposed versus HEWMA Control Chart

The proposedHEWMAAIB1 and HEWMAAIB2 control charts are compared with the HEWMA control chart. The proposed control charts outperform the HEWMA control chart. For instance, at ARL0=200, λ2=0.1, λ1=0.5, and τ=1.1,1.2, the proposed HEWMAAIB1 and HEWMAAIB2 control charts ρ=0.5 provides the ARL1 values are (23.69, 9.05) and (25.05, 9.75), respectively, whereas the HEWMA control chart has the ARL1 values equal to (25.79, 10.15) (see Tables 3, 7 versus 9). Similarly, the proposed control charts’ superiority over the HEWMA control chart can also be visualized in Figure 1. Besides, the overall performance measures confirm the better performance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts than the HEWMA control chart. For example, the proposed HEWMAAIB1 and HEWMAAIB2 control charts have EQL, PCI, and RARL values are 13.905,1,1 and 13.918, 1.001,0.989, respectively, whereas the HEWMA control chart has EQL, PCI, and RARL values are 19.752, 1.421,2.273 (see Table 10).Table 9
Run-length profile of existing charts atARL0=200.
HEWMAAEWMAHHW1HHW2EWMA-IAIBEWMA-IIAIBρ=0.95ρ=0.95τλ1=0.05, λ2=0.10λ=0.10λ=0.10λ=0.10λ=0.10λ=0.101.0201.71200.76199.69200.12200.42200.31.125.7926.0434.5232.119.7310.961.210.1510.3514.0912.723.724.031.35.795.718.27.222.342.371.43.993.785.644.91.791.761.52.962.834.283.671.511.451.62.392.293.462.961.361.281.72.031.972.912.481.261.191.81.781.752.532.161.21.131.91.621.582.251.931.151.092.01.491.472.051.771.121.07Figure 1
ARL comparison of the proposed HEWMAAIB1andHEWMAAIB2 versus HEWMA, and AEWMA control charts at λ,λ2=0.10,λ1=0.05 and ARL0=200.Table 10
Overall performance measures of proposed charts vs existing charts.
HEWMAAEWMAHHW1HHW2EWMA-IAIBEWMA-IIAIBHEWMAAIB1HEWMAAIB2ρ=0.95ρ=0.95ρ=0.95ρ=0.95λ1=0.10, λ2=0.05λ=0.10λ=0.10λ=0.10λ=0.10λ=0.10λ1=0.10, λ2=0.05λ1=0.10, λ2=0.05EQL19.75219.60923.41321.95414.56014.63813.90513.918PCI1.4211.4101.6841.5791.0471.0531.0001.001RARL2.2732.2343.1712.7931.1451.1401.0000.989


### 5.2. Proposed versus AEWMA Control Chart

The proposedHEWMAAIB1 and HEWMAAIB2 control charts are more efficient than the AEWMA control chart. For example, with λ,λ1,λ2=0.1, ρ=0.75, δ=1.1, and ARL0=200, the proposed HEWMAAIB1 and HEWMAAIB2 control charts hold the ARL1 values 21.21 and 22.88, respectively, while the ARL1 value with the AEWMA control chart is 26.04 (see Tables 4, 7 versus 9, and Figure 1). Likewise, the HEWMAAIB1 and HEWMAAIB2 control charts have smaller EQL, PCI, and RARL values than the AEWMA control chart. For instance, the proposed HEWMAAIB1 and HEWMAAIB2 control charts provide EQL, PCI, and RARL values are 13.905, 1.000, 1.000 and 13.918, 1.001, 0.989, respectively, while the AEWMA control chart produces EQL, PCI, and RARL values are 19.609, 1.41, 2.234 (see Table 10).


### 5.3. Proposed versus HHW1 and HHW2 Control Charts

The proposedHEWMAAIB1 and HEWMAAIB2 control charts provide better performance against the HHW1 and HHW2 control charts. For example, at λ,λ2=0.1, λ1=0.05, ρ=0.5, and δ=1.1,1.2, the proposed HEWMAAIB1 and HEWMAAIB2 control charts attained the ARL1 values as (23.69, 9.05) and (25.05, 9.75), respectively, while the HHW1 and HHW2 control charts have the ARL1 values as (34.52, 14.09) and (32.11, 12.72), respectively (see Tables 3, 7 versus 9). Furthermore, Figure 2 also shows the superiority of the proposed control charts over the HHW1 and HHW2 control charts. In terms of overall effectiveness (see Table 10), the HEWMAAIB1 and HEWMAAIB2 control charts are superior to the HHW1 and HHW2 charts. For instance, the proposed HEWMAAIB1 and HEWMAAIB2 control charts have EQL 13.905 and 13.918, whereas the HHW1 and HHW2 control charts have EQL values 23.413 and 21.954, respectively (see Table 10).Figure 2
ARL comparison of the proposed HEWMAAIB1andHEWMAAIB2 versus HHW1 and HHW2 control charts at λ,λ2=0.10,λ1=0.05 and ARL0=200.


### 5.4. Proposed versusEWMA−IAIB- and EWMA−IIAIB- Control Charts

The proposedHEWMAAIB1 and HEWMAAIB2 control charts attain outstanding performance against the EWMA-IAIB and EWMA-IIAIB control charts. For example, when λ,λ2=0.1, λ2=0.05,ρ=0.95, and τ=1.1, the ARL1 values of proposed HEWMAAIB1 and HEWMAAIB2 control charts are 7.57 and 8.52, whereas, the ARL1 values of EWMA-IAIB and EWMA-IIAIB control charts are 9.73 and 10.96, respectively (see Tables 4, 7 versus 9). Additionally, the dominance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts over the EWMA-IAIB and EWMA-IIAIB control charts can be seen in Figure 3. For overall performance, the proposed HEWMAAIB1 and HEWMAAIB2 control charts have smaller EQL, PCI, and RARL values than the EQL, PCI, and RARL values of the EWMA-IAIB and EWMA-IIAIB control charts, respectively (see Table 10).Figure 3
ARL comparison of the proposed HEWMAAIB1andHEWMAAIB2 versus EWMA−IAIB and EWMA−IIAIB control charts at λ,λ2=0.10,λ1=0.05,ρ=0.95 and ARL0=200.


### 5.5. Main Outcomes of the Study

Some interesting outcomes of the proposedHEWMAAIB1 and HEWMAAIB2 control charts are listed as follows:(i)
The use of Hybrid EWMA statistic certainly boosts the detection ability of the proposedHEWMAAIB1 and HEWMAAIB2 control charts.(ii)
The performance of the proposedHEWMAAIB1 and HEWMAAIB2 control charts improve with the induction of suitable auxiliary information in the model.(iii)
TheARL1 values of the proposed HEWMAAIB1 and HEWMAAIB2 control charts are smaller than HEWMA, AEWMA, HHW1, HHW2, EWMA-IAIB, and EWMA-IIAIB control charts.(iv)
The overall performance evaluation measures show the dominance of theHEWMAAIB1 and HEWMAAIB2 control charts against other control charts (see Subsections 5.1–5.4).(v)
The proposedHEWMAAIB1 and HEWMAAIB2 control charts provide the best performance for larger values of ρ (see Figures 4 and 5).(vi)
TheARL1 performance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts are increased for smaller values of λ1 and λ2 (see Tables 1–8).Figure 4
ARL behavior of the proposed HEWMAAIB1 control chart under different ρ for λ1,λ2=0.05 at ARL0=200.Figure 5
ARL behavior of the proposed HEWMAAIB2 control chart under different ρ for λ1,λ2=0.05 at ARL0=200.


## 5.1. Proposed versus HEWMA Control Chart

The proposedHEWMAAIB1 and HEWMAAIB2 control charts are compared with the HEWMA control chart. The proposed control charts outperform the HEWMA control chart. For instance, at ARL0=200, λ2=0.1, λ1=0.5, and τ=1.1,1.2, the proposed HEWMAAIB1 and HEWMAAIB2 control charts ρ=0.5 provides the ARL1 values are (23.69, 9.05) and (25.05, 9.75), respectively, whereas the HEWMA control chart has the ARL1 values equal to (25.79, 10.15) (see Tables 3, 7 versus 9). Similarly, the proposed control charts’ superiority over the HEWMA control chart can also be visualized in Figure 1. Besides, the overall performance measures confirm the better performance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts than the HEWMA control chart. For example, the proposed HEWMAAIB1 and HEWMAAIB2 control charts have EQL, PCI, and RARL values are 13.905,1,1 and 13.918, 1.001,0.989, respectively, whereas the HEWMA control chart has EQL, PCI, and RARL values are 19.752, 1.421,2.273 (see Table 10).Table 9
Run-length profile of existing charts atARL0=200.
HEWMAAEWMAHHW1HHW2EWMA-IAIBEWMA-IIAIBρ=0.95ρ=0.95τλ1=0.05, λ2=0.10λ=0.10λ=0.10λ=0.10λ=0.10λ=0.101.0201.71200.76199.69200.12200.42200.31.125.7926.0434.5232.119.7310.961.210.1510.3514.0912.723.724.031.35.795.718.27.222.342.371.43.993.785.644.91.791.761.52.962.834.283.671.511.451.62.392.293.462.961.361.281.72.031.972.912.481.261.191.81.781.752.532.161.21.131.91.621.582.251.931.151.092.01.491.472.051.771.121.07Figure 1
ARL comparison of the proposed HEWMAAIB1andHEWMAAIB2 versus HEWMA, and AEWMA control charts at λ,λ2=0.10,λ1=0.05 and ARL0=200.Table 10
Overall performance measures of proposed charts vs existing charts.
HEWMAAEWMAHHW1HHW2EWMA-IAIBEWMA-IIAIBHEWMAAIB1HEWMAAIB2ρ=0.95ρ=0.95ρ=0.95ρ=0.95λ1=0.10, λ2=0.05λ=0.10λ=0.10λ=0.10λ=0.10λ=0.10λ1=0.10, λ2=0.05λ1=0.10, λ2=0.05EQL19.75219.60923.41321.95414.56014.63813.90513.918PCI1.4211.4101.6841.5791.0471.0531.0001.001RARL2.2732.2343.1712.7931.1451.1401.0000.989


## 5.2. Proposed versus AEWMA Control Chart

The proposedHEWMAAIB1 and HEWMAAIB2 control charts are more efficient than the AEWMA control chart. For example, with λ,λ1,λ2=0.1, ρ=0.75, δ=1.1, and ARL0=200, the proposed HEWMAAIB1 and HEWMAAIB2 control charts hold the ARL1 values 21.21 and 22.88, respectively, while the ARL1 value with the AEWMA control chart is 26.04 (see Tables 4, 7 versus 9, and Figure 1). Likewise, the HEWMAAIB1 and HEWMAAIB2 control charts have smaller EQL, PCI, and RARL values than the AEWMA control chart. For instance, the proposed HEWMAAIB1 and HEWMAAIB2 control charts provide EQL, PCI, and RARL values are 13.905, 1.000, 1.000 and 13.918, 1.001, 0.989, respectively, while the AEWMA control chart produces EQL, PCI, and RARL values are 19.609, 1.41, 2.234 (see Table 10).


## 5.3. Proposed versus HHW1 and HHW2 Control Charts

The proposedHEWMAAIB1 and HEWMAAIB2 control charts provide better performance against the HHW1 and HHW2 control charts. For example, at λ,λ2=0.1, λ1=0.05, ρ=0.5, and δ=1.1,1.2, the proposed HEWMAAIB1 and HEWMAAIB2 control charts attained the ARL1 values as (23.69, 9.05) and (25.05, 9.75), respectively, while the HHW1 and HHW2 control charts have the ARL1 values as (34.52, 14.09) and (32.11, 12.72), respectively (see Tables 3, 7 versus 9). Furthermore, Figure 2 also shows the superiority of the proposed control charts over the HHW1 and HHW2 control charts. In terms of overall effectiveness (see Table 10), the HEWMAAIB1 and HEWMAAIB2 control charts are superior to the HHW1 and HHW2 charts. For instance, the proposed HEWMAAIB1 and HEWMAAIB2 control charts have EQL 13.905 and 13.918, whereas the HHW1 and HHW2 control charts have EQL values 23.413 and 21.954, respectively (see Table 10).Figure 2
ARL comparison of the proposed HEWMAAIB1andHEWMAAIB2 versus HHW1 and HHW2 control charts at λ,λ2=0.10,λ1=0.05 and ARL0=200.


## 5.4. Proposed versusEWMA−IAIB- and EWMA−IIAIB- Control Charts

The proposedHEWMAAIB1 and HEWMAAIB2 control charts attain outstanding performance against the EWMA-IAIB and EWMA-IIAIB control charts. For example, when λ,λ2=0.1, λ2=0.05,ρ=0.95, and τ=1.1, the ARL1 values of proposed HEWMAAIB1 and HEWMAAIB2 control charts are 7.57 and 8.52, whereas, the ARL1 values of EWMA-IAIB and EWMA-IIAIB control charts are 9.73 and 10.96, respectively (see Tables 4, 7 versus 9). Additionally, the dominance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts over the EWMA-IAIB and EWMA-IIAIB control charts can be seen in Figure 3. For overall performance, the proposed HEWMAAIB1 and HEWMAAIB2 control charts have smaller EQL, PCI, and RARL values than the EQL, PCI, and RARL values of the EWMA-IAIB and EWMA-IIAIB control charts, respectively (see Table 10).Figure 3
ARL comparison of the proposed HEWMAAIB1andHEWMAAIB2 versus EWMA−IAIB and EWMA−IIAIB control charts at λ,λ2=0.10,λ1=0.05,ρ=0.95 and ARL0=200.


## 5.5. Main Outcomes of the Study

Some interesting outcomes of the proposedHEWMAAIB1 and HEWMAAIB2 control charts are listed as follows:(i)
The use of Hybrid EWMA statistic certainly boosts the detection ability of the proposedHEWMAAIB1 and HEWMAAIB2 control charts.(ii)
The performance of the proposedHEWMAAIB1 and HEWMAAIB2 control charts improve with the induction of suitable auxiliary information in the model.(iii)
TheARL1 values of the proposed HEWMAAIB1 and HEWMAAIB2 control charts are smaller than HEWMA, AEWMA, HHW1, HHW2, EWMA-IAIB, and EWMA-IIAIB control charts.(iv)
The overall performance evaluation measures show the dominance of theHEWMAAIB1 and HEWMAAIB2 control charts against other control charts (see Subsections 5.1–5.4).(v)
The proposedHEWMAAIB1 and HEWMAAIB2 control charts provide the best performance for larger values of ρ (see Figures 4 and 5).(vi)
TheARL1 performance of the proposed HEWMAAIB1 and HEWMAAIB2 control charts are increased for smaller values of λ1 and λ2 (see Tables 1–8).Figure 4
ARL behavior of the proposed HEWMAAIB1 control chart under different ρ for λ1,λ2=0.05 at ARL0=200.Figure 5
ARL behavior of the proposed HEWMAAIB2 control chart under different ρ for λ1,λ2=0.05 at ARL0=200.


## 6. Real-Life Application

To demonstrate the practical implementation of the proposedHEWMAAIB1 and HEWMAAIB2 control charts, a real-life data set of glass thickness (X), and its impact on the stress strength (Y) of glass bottles is considered from Asadzadeh and Kiadaliry [38]. This data set contains 40 samples, each of size 5, of stress strength (kg/cm2), thickness (cm). The proposed control charts are constructed under the assumption of known parameters. However, in real-life data application of the proposed control chart, the population parameters are not available. Therefore, for the practical implementation of the control charts, the estimated parameters are used for the empirical quantification of the quantities required to show the proposed control charts’ implementation. The estimates of process parameters are given as: μ^Y=6.36, σ^Y=8.92, μ^X=1.38, σ^X=0.62, and ρ^=0.905. In the data set, the first 20 samples are treated as IC, while the rest of the 20 samples are considered OOC. Following Anwar, et al. [39], the Y is multiplied by 1.3 for the OOC scenario.The parameters of proposedHEWMAAIB1 and HEWMAAIB2 control charts are set on λ1=0.2, λ2=0.2, LHEWMAAIB1=2.3029, LHEWMAAIB2=2.299 and ρ=0.905 with ARL0=200. Similarly, the control chart parameters of the existing EWMA-IAIB and EWMA-IIAIB control charts are λ,LEWMA−IAIB,LEWMA−IIAIB=0.2,2.785,0.2,2.677 at ρ=0.905 and ARL0=200.TheEWMA-IAIB, EWMA-IIAIB, proposed HEWMAAIB1, and proposed HEWMAAIB2 control charts detect the first OOC signal at sample number 29 (see Figures 6–9). Overall, the existing EWMA-IAIB control chart detects 2 OOC signals while the HEWMAAIB1 control chart detects 8 OOC signals. In the same manner, the EWMA-IIAIB control chart detects a total of 2 OOC signals, and the HEWMAAIB2 control chart detects 5 OOC signals. This indicates that the proposed HEWMAAIB1 and HEWMAAIB2 control charts are more efficient than the existing EWMA-IAIB and EWMA-IIAIB control charts.Figure 6
The existingEWMA-IAIB control chart for real-life data using ρ=0.905, λ=0.2,LEWMA−IAIB=2.785 and ARL0=200.Figure 7
The proposedHEWMAAIB1 control chart for real-life data using ρ=0.905, λ1=0.2,λ2=0.2,LHEWMAAIB1=2.3029 and ARL0=200.Figure 8
The existingEWMA-IIAIB control chart for real-life data using ρ=0.905, λ=0.2LEWMA−IIAIB=2.677 and ARL0=200.Figure 9
The proposedHEWMAAIB2 control chart for real-life data using ρ=0.905, λ1=0.2,λ2=0.2,LHEWMAAIB2=2.299 and.ARL0=200.


## 7. Concluding remarks

This study presented the two new auxiliary information-based hybrid EWMA control charts, namedHEWMAAIB1 and HEWMAAIB2 control charts for process dispersion. The HEWMAAIB1 control chart used the auxiliary information through the regression estimator for the population variance, whereas the HEWMAAIB2 control chart used the auxiliary information through the difference estimator. The HEWMAAIB1 and HEWMAAIB2 control charts are constructed by combining the features of the AIB dispersion estimators with the HEWMA control chart. The proposed control charts’ performance based on average run length, extra quadratic loss, relative average run length, and performance comparison index measures reveal the superiority over the competitive control charts. It is worth mentioning that the proposed HEWMAAIB1 and HEWMAAIB2 control charts performed very well to monitor small to moderate shifts in process dispersion, especially for large correlation coefficient values. Besides, a real-life application is also provided for users and practitioners to demonstrate the implementation of the proposed study from a practical perspective. This work can be extended to a non-normal process(s) and a multivariate case.



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*Source: 1265204-2022-05-30.xml*