# Max-EWMA Chart Using Beta and Simplex Distributions for Time and Magnitude Monitoring

Authors: Sajid Ali; Muhammad Farhan Akram; Ismail Shah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Control charts are used to detect assignable causes in different manufacturing and nonmanufacturing processes. This study presents a new maximum exponentially weighted moving average (Max-EWMA) chart for joint unit interval time and magnitude monitoring. To this end, beta distribution is considered for time whereas simplex distribution is used for magnitude. Average run length (ARL), standard deviation of run length (SDRL), and different quantiles are used to evaluate the performance of the Max-EWMA chart. A real data example is also included in the study to show the application of the proposed chart. The results provide evidences that the Max-EWMA chart is efficient in detecting small- to medium-sized shifts.


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## Body

## 1. Introduction

Statistical methods play an important role to monitor and control the quality of manufactured and nonmanufactured products. A control chart is a graphical device, based on the history of the process, to access the performance of a process. There are two types of control charts in statistical process control (SPC), namely, attribute charts and variable charts. The main aim of a control chart is to distinguish between assignable and natural causes. Currently, charts are used in disease surveillance systems, monitoring the quality of service(s) or monitoring stock-price stability. The fundamental technique used in SPC for monitoring a process is Shewhart approach in which we monitor a process based on the value of a statistic obtained from current sample. In this technique, the historic information about the process is not taken into account. Generally, this technique works well with the processes having large shifts. The memory-less nature of this scheme is its main shortcoming when dealing with small shifts. To overcome this shortcoming, the memory-type control charts are adopted. These include exponentially weighted moving average (EWMA) chart and cumulative sum (CUSUM) chart.Processes with extremely small defect rate, also called high-quality processes, cannot be monitored with typical attribute control charts. For high-quality process such as monitoring the critical ingredients in a lifesaving drug, a chart based on the inter-arrival time or time-between-event (TBE) is suggested in the literature. In many processes, event occurrence time and magnitude are recorded and monitoring them jointly is a challenging task.Attribute charts likep and np based on Bernoulli distribution are commonly used to monitor the fraction of the number of nonconforming items [1]. Lee Ho and Quinino [2] discussed an attribute control chart to monitor the variability of a process. There are distributions in the literature which can be utilized to deal with proportional data. One of such distributions is the beta distribution. Sant’Anna and Ten Caten [3] used this distribution to construct Shewhart chart for monitoring fractional data type. Lima-Filho et al. [4] introduced an inflated beta control chart. Simplex distribution is somehow similar to exponential dispersion family as well as to normal distribution. That is, if the dispersion parameter of simplex distribution is small, this distribution is similar to the normal distribution and becomes flexible to cover many features of data if the dispersion parameter is large [5]. This distribution also has applications in medical sciences such as mapping the proportion of remaining gas volume relative to its initial volume in the eyes of patients [6–9]. Zhang and Qiu [10] presented a regression analysis for proportion data using this distribution. Zhang et al. [11] presented an R package for regression analysis of proportional data using this distribution.The TBE charts have recently been of interest for many researchers because these charts are very effective in monitoring high-quality processes such as high-yield production lines with extremely small defect rates [12]. Several TBE control charts are available in the literature [13]. These charts also overcome the drawbacks of traditional Shewhart charting schemes, such as high false alarm rate, negative lower limit, and the limitation of normality. Ali [14] proposed TBE control chart based on renewal process assuming continuous exponentiated family of distributions. The author concluded that the proposed chart is more efficient and flexible when compared to the existing TBE charts based on exponential distribution. Some recent work on TBE can be seen in Xie et al. [15], Zhang et al. [16], Mahmood and Xie [17], Mahmood [18], Talib et al. [19, 20], Akram et al. [21], and Shafae et al. [22].Wu et al. [23] developed a rate chart for monitoring time and magnitude simultaneously by using the ratio of magnitude and time as the plotting statistic. Practically, the damage or disturbance of an event causes is determined by its frequency or by its magnitude. Thus, this arises the need to monitor both of these characteristics (frequency and magnitude) simultaneously. Taking this into account, Wu et al. [24] adopted Shewhart control charting technique for monitoring the frequency and magnitude of an event simultaneously. In earlier studies, simultaneous monitoring was done using two separate charting statistics. This technique was later found to be less effective when compared to the charts based on single statistic except for some exceptional situations. Most of works on simultaneous monitoring are based on Shewhart mechanism [25, 26]. However, Shewhart charts are of memory-less nature; the historical information of the process is ignored as the new points are plotted [27]. As a result, these charts cannot detect small shifts in process parameters satisfactorily. Consequently, memory-type control charts such as exponentially weighted moving average (EWMA) chart and cumulative sum (CUSUM) charts are required.Comparing to Shewhart chart, the EWMA chart is more sensitive as it takes into account the history of the process. In fact, EWMA chart is a Markov process and simpler than the cumulative sum (CUSUM) chart. It is not so sensitive to non-normal distributed data with small smoothing parameter values [28]. Recently, Sanusi et al. [29] introduced maximum EWMA (Max-EWMA) control chart to simultaneously monitor the frequency and magnitude of events where the frequency was assumed to follow exponential distribution and the magnitude was assumed to follow gamma distribution.In many real-life situations, we encounter data of rates and proportions which is bounded by the interval (0,1). Although Bernoulli distribution is generally assumed for binary (or dichotomous) data, the charts constructed using Bernoulli or binomial distribution do not react timely. Also, the Bernoulli distribution has the largest dispersion (1/n) in the binomial family, while the Poisson distribution has the smallest dispersion when n⟶∞. Thus, alternative distributions are needed to construct control charts for unit interval data. The importance of unit interval may be seen in proportional nature such as TV rating, unemployment rate, and area designated to agriculture (in fraction). There are a number of practical situations where one might be interested in the monitoring of proportions or rates of constituents in a product; however, the available data set may not follow the Bernoulli distribution [30]. Such data can be found extensively in many fields and often analyzed using a normal distribution due to simplicity associated with the normal distribution, yet misleading conclusions [8].The rest of the article is organized as follows. A review on beta and simplex probability distributions is given in Section2. Section 3 presents the control chart construction for time and magnitude monitoring. The performance of the chart is presented in Section 4, while a real-life data example is discussed in Section 5. Finally, some concluding remarks are given in Section 6.


## 2. Beta and Simplex Distributions

To deal with the unit interval data for monitoring jointly time and magnitude, we use beta distribution for time and simplex distribution for magnitude.


### 2.1. Beta Distribution

A random variableT is said to have a beta distribution with first shape parameter (shape1) δ>0 and second shape parameter (shape2) γ>0, i.e., X∼Bδ,γ if it has the following probability density function:(1)ft|δ,γ=tδ−11−tγ−1Bδ,γ,0<t<1,where Bδ,γ=ΓδΓγ/Γδ+γ denotes the beta function and Γδ=∫0∞ωδ−1e−ωdt. The mean and variance of T are(2)ET=μ=δδ+γ,VarT=δγδ+γ2δ+γ+1,respectively. Suppose, the parameters, instead of random variable, in (1) are transformed as(3)δ=μϕ,γ=1−μϕ.then the density function of beta distribution takes the following form:(4)ft|δ,γ=tμϕ−11−t1−μϕ−1Bμϕ,1−μϕ,0<t<1,This gives mean and variance asET=μ and VarT=μ1−μ/ϕ+1.


### 2.2. Simplex Distribution

A random variableX follows a (standard) simplex distribution with location parameter 0<μ<1 and dispersion parameter σ2>0, i.e., X∼Sμ,σ2 if it exhibits the following density function:(5)fx|μ,σ2=2πσ2x1−x3−1/2exp−12σ2dx;μ,0<x<1,where dx;μ=x−μ2/x1−x1−μ2μ2 is called the deviance function of this distribution. The mean and variance of X are EX=μ and VarX=1/2σ2exp1/2σ21−μ2μ2Γ1/2,1/2σ21−μ2μ2, respectively. Here, Γa,b represents the incomplete gamma function which may be defined by Γa,b=∫b∞sa−1e−sds. It is noted that the general simplex distribution has the large dispersion asymptotic distribution than the beta distribution.


## 2.1. Beta Distribution

A random variableT is said to have a beta distribution with first shape parameter (shape1) δ>0 and second shape parameter (shape2) γ>0, i.e., X∼Bδ,γ if it has the following probability density function:(1)ft|δ,γ=tδ−11−tγ−1Bδ,γ,0<t<1,where Bδ,γ=ΓδΓγ/Γδ+γ denotes the beta function and Γδ=∫0∞ωδ−1e−ωdt. The mean and variance of T are(2)ET=μ=δδ+γ,VarT=δγδ+γ2δ+γ+1,respectively. Suppose, the parameters, instead of random variable, in (1) are transformed as(3)δ=μϕ,γ=1−μϕ.then the density function of beta distribution takes the following form:(4)ft|δ,γ=tμϕ−11−t1−μϕ−1Bμϕ,1−μϕ,0<t<1,This gives mean and variance asET=μ and VarT=μ1−μ/ϕ+1.


## 2.2. Simplex Distribution

A random variableX follows a (standard) simplex distribution with location parameter 0<μ<1 and dispersion parameter σ2>0, i.e., X∼Sμ,σ2 if it exhibits the following density function:(5)fx|μ,σ2=2πσ2x1−x3−1/2exp−12σ2dx;μ,0<x<1,where dx;μ=x−μ2/x1−x1−μ2μ2 is called the deviance function of this distribution. The mean and variance of X are EX=μ and VarX=1/2σ2exp1/2σ21−μ2μ2Γ1/2,1/2σ21−μ2μ2, respectively. Here, Γa,b represents the incomplete gamma function which may be defined by Γa,b=∫b∞sa−1e−sds. It is noted that the general simplex distribution has the large dispersion asymptotic distribution than the beta distribution.


## 3. Max-EWMA Chart

To jointly monitor time and magnitude on unit interval, we use the Max-EWMA control chart. The maximum absolute values of two EWMA statistics constitute the plotting statistic of the proposed Max-EWMA control chart; therefore, out-of-control signal diagnostic is much easier with this chart. To construct the chart, the TBE is assumed to follow the beta distribution,T∼Bδ,γ, and the magnitude is assumed to follow the simplex distribution, X∼Sμ,σ2. We also suppose that time and magnitude are independent for the sake of simplicity. Let the process start at t=0, and IC process parameters of time and magnitude are denoted by δ0,γ0 and μ0,σ02, respectively. To denote the shifted parameters of T and X, we use δ=Δδδ0, γ=Δγγ0, μ=Δμμ0, and σ2=Δσ2σ02 where Δδ,Δγ,Δμ, and Δσ2 represent the shifts in the shape parameters of the beta distribution, and mean and dispersion parameters of the simplex distribution, respectively. To construct the chart, first we standardized time and magnitude values as(6)Uj=Xj−μ1/2σ2exp1/2σ21−μ2μ2Γ1/2,1/2σ21−μμ2,Vj=Tj−μμ1−μ/ϕ+1.Then, the two independent EWMA statistics are defined as(7)Yj=1−λeYj−1+λeUj,0<λe≤1,and(8)Zj=1−λeZj−1+λeVj,0<λe≤1,where j=1,2,⋯ and λe∈0,1 is known as the smoothing constant. Typically, a large value of smoothing constant is considered suitable to detect large shift while a small value λe∈0.05,0.25 is suggested to detect small shift [31]. Another parameter of the chart is called the constant multiplier L, which controls the width of the chart and is fixed to have the desired average run length (ARL) values. The initial values of both statistics are Y0=EUj=0 and Z0=EVj=0. The plotting statistic of the Max-EWMA is given by(9)Mj=maxYj,Zj.When a pure shift occurs in eitherX (location parameter of the simplex distribution) or T (first shape parameter of the beta distribution) or simultaneous shift in both of these parameters, the value of Mj statistic would be greater than the in-control (IC) parameters’ case. The proposed chart needs only the upper control limit (UCL) because Mj takes only real and positive values. In addition, this makes the Max-EWMA chart suitable for detecting downward (DW) as well as upward (UW) shifts in either distribution parameters. The UCL can be computed as(10)UCL=EM+LVarM,where E(M) and Var(M) represent the IC mean and variance of the charting statistic M. As the Max-EWMA statistic involves maximum of two EWMA statistics, an explicit distribution of M is difficult to obtain and therefore we calculate mean and variance using the IC Phase I data.


### 3.1. Case 1(a)

We assume shifts in the mean of simplex distribution along with the shift in the first shape parameter of beta distribution. It is worth mentioning that ifΔμ=Δδ=1, the process is said to be IC.


### 3.2. Case 1(b)

The second case of the analysis is considering shifts in the second shape parameter of the beta distribution and in the dispersion parameter of the simplex distribution. The process in this case will be termed as IC ifΔγ=Δσ2=1.


### 3.3. Charting Procedure of the Max-EWMA Control Chart

The following steps are used to construct the Max-EWMA chart for jointly monitoring time and magnitude:(1)
InitializeY0=Z0=0 and specify the desired IC ARL, i.e., ARL0. Further, specify the appropriate values of smoothing parameter λe and constant multiplier L to obtain ARL0. For different values of δ and γ in equation (3) corresponding to different combinations of μ, ϕ, and σ2, different values of constant multiplier are shown in Table1.(2)
Calculate the UCL value corresponding to equation (6) using IC data mean and variance.(3)
Obtain thejth test observation Xj,Tj,j=1,2,⋯ where Xj is the jth observation of magnitude and Tj represents the TBE between the jth and j−1th occurrences of an event.(4)
Forjth test observation, calculate Mj based on Yj and Zj.(5)
PlotMj against the counter-value j. If Mj≤UCL, the process is said to be IC and we proceed with the monitoring; otherwise, process is considered as OOC.(6)
In the case of out-of-control (OOC), investigate the OOC signal to identify whether the process actually has an assignable cause or not. If an assignable exists, make necessary adjustments to remove it, then continue with the monitoring, and go to Step 3.Table 1
The value ofL to achieve IC ARL with T∼Bδ,γ, X∼Sμ,σ2, μ  = 0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15,0.20,0.25,0.30,0.35,0.40,0.45,0.50.
λeKARL0SDRL0q10q25q50q75q90KARL0SDRL0q10q25q50q75q90ARL0=370ARL0=500Case 1ϕ=290,σ2=0.370.052.720370.152357.29154.00118.00259.00503.00827.102.924500.328485.43465.00153.00348.50686.001142.000.103.017370.480363.93046.00111.00257.00507.25837.003.196500.776490.83561.00151.00345.00690.001144.200.153.201371.871366.66443.00112.00259.00519.00848.103.368500.900498.51056.00144.00346.00701.251149.100.203.288373.905370.44343.00114.00259.00513.25851.003.458499.743489.23356.00147.00345.00696.001161.000.253.356367.145359.24842.00109.00256.00509.00829.003.533497.660490.87858.00151.00350.00690.001128.000.303.451373.730383.40642.90107.00255.00514.00871.003.622498.463490.96355.00147.00351.00690.001142.100.353.489371.382369.28042.00108.00262.00516.00854.103.673503.375498.19155.00146.00354.00696.251145.000.403.500370.896369.57340.00105.00256.00518.00869.003.687500.532491.57457.00152.00355.00690.251135.300.453.515367.001366.02140.00105.00254.00507.00842.003.698502.474493.69760.00154.00354.00690.001154.100.503.538367.103362.55440.00107.00256.00507.00839.003.719500.483499.77753.90147.00350.00685.001155.20Case 2ϕ=148,σ2=0.500.052.707370.731356.27154.00118.00263.00507.25831.002.898499.123487.76867.00154.00349.00688.001120.000.103.045369.013351.86645.00113.00263.00508.00829.003.220501.674492.19462.00148.00354.50695.001124.100.153.197369.567365.66543.00109.00253.00506.25840.003.374500.237493.92057.00147.00346.00690.001143.100.203.334373.194363.13445.00114.00266.00518.00844.103.493500.013489.58358.00152.00351.00694.241146.000.253.345373.967365.54445.00113.00262.00524.00852.103.630500.142496.73356.00145.00346.00695.001143.200.303.461367.567367.37241.00106.00255.00505.25840.103.657503.463492.53656.00149.00356.00699.001158.100.353.552372.473364.49842.00109.00256.00520.00862.003.750500.005492.16456.00151.00347.00694.001156.000.403.545368.830370.07639.00106.00252.00516.25854.003.734502.620510.33056.00146.00345.00689.001153.100.453.581367.860366.65039.00106.00254.00505.00839.003.769500.174491.47358.00150.00350.00688.001160.000.503.630370.312361.71640.00111.00262.00516.00841.103.759497.050490.47354.00144.00343.00695.251139.00Case 3ϕ=80,σ2=0.710.052.728370.823356.20655.00121.00261.00508.25816.102.909501.458485.84967.00159.00354.50691.001135.000.103.050370.960367.28247.00111.00255.00509.00850.003.230501.485486.20561.00154.00358.00698.001137.100.153.201370.680366.81443.00112.00259.00510.00852.103.386500.238483.67559.00150.00357.00697.251130.000.203.395373.536369.47642.00108.00261.00517.00845.003.582503.270495.41755.00147.75354.00696.001158.000.253.395372.622372.67241.00111.00261.00517.00852.103.574503.578504.67555.00149.00348.00694.001158.100.303.472371.473374.57340.00109.00256.00517.00856.003.671501.321499.94355.00144.00346.00695.001173.000.353.598372.650373.99441.00105.00258.00520.20875.203.779500.190499.67054.00145.00345.00688.001170.000.403.613370.201370.87941.00106.00254.00513.00857.003.807498.473496.09853.00140.00348.00690.001127.100.453.676370.377369.58639.00108.00259.00513.00844.103.869500.434498.47253.00142.00346.00703.001158.100.503.714372.503375.38740.00108.00255.00515.00855.003.915502.548499.85755.00147.00350.00689.001162.10Case 4ϕ=31,σ2=1.200.052.738370.176353.74653.00118.00263.00506.00829.102.945500.859492.33364.00152.00348.00685.001150.000.103.060369.169358.09847.00115.00258.00508.00836.003.260500.731485.02162.00151.00357.00691.001140.300.153.263371.737369.03044.00110.00255.00512.00852.103.454500.098491.69557.00147.00354.00699.001135.000.203.360370.463371.37344.00112.00256.00501.00845.003.560503.593496.59958.00149.00354.00695.001162.100.253.471370.500366.84741.00107.00258.00518.00846.103.684500.876500.08354.00148.00352.00687.001147.000.303.618373.858367.37742.00110.00262.00515.00866.003.826500.762500.36256.00147.00342.00696.001152.100.353.683370.463372.47340.00106.00256.00505.00850.203.874497.311488.68954.00146.00345.00692.001160.000.403.725371.140370.00842.00107.00256.00513.00850.003.929498.483503.33452.00143.00344.50684.001155.000.453.787369.293370.47242.00106.00250.00504.24845.004.000500.048499.76054.00143.00345.00687.001154.100.503.875373.357371.36842.00108.00261.00518.25856.104.070497.088499.88055.00146.00341.00697.251143.00Once an OOC signal is noted down, it is important to identify the parameters responsible for an OOC signal. Thus, to label the OOC signal, we suggest to label the points exceeding the UCL as given in Table2. That is, if Uj and Vj exceed the UCL, use the label ′X′ and ′T′ to indicate shift in the magnitude and TBE, respectively. If both Uj and Vj exceed UCL, use the label X′&T′ to indicate simultaneous shift in both TBE and magnitude of the event.Table 2
Labeling out-of-control signal.
TimeMagnitudeU<UCLU>UCLV<UCLICXV>UCLTX&T


## 3.1. Case 1(a)

We assume shifts in the mean of simplex distribution along with the shift in the first shape parameter of beta distribution. It is worth mentioning that ifΔμ=Δδ=1, the process is said to be IC.


## 3.2. Case 1(b)

The second case of the analysis is considering shifts in the second shape parameter of the beta distribution and in the dispersion parameter of the simplex distribution. The process in this case will be termed as IC ifΔγ=Δσ2=1.


## 3.3. Charting Procedure of the Max-EWMA Control Chart

The following steps are used to construct the Max-EWMA chart for jointly monitoring time and magnitude:(1)
InitializeY0=Z0=0 and specify the desired IC ARL, i.e., ARL0. Further, specify the appropriate values of smoothing parameter λe and constant multiplier L to obtain ARL0. For different values of δ and γ in equation (3) corresponding to different combinations of μ, ϕ, and σ2, different values of constant multiplier are shown in Table1.(2)
Calculate the UCL value corresponding to equation (6) using IC data mean and variance.(3)
Obtain thejth test observation Xj,Tj,j=1,2,⋯ where Xj is the jth observation of magnitude and Tj represents the TBE between the jth and j−1th occurrences of an event.(4)
Forjth test observation, calculate Mj based on Yj and Zj.(5)
PlotMj against the counter-value j. If Mj≤UCL, the process is said to be IC and we proceed with the monitoring; otherwise, process is considered as OOC.(6)
In the case of out-of-control (OOC), investigate the OOC signal to identify whether the process actually has an assignable cause or not. If an assignable exists, make necessary adjustments to remove it, then continue with the monitoring, and go to Step 3.Table 1
The value ofL to achieve IC ARL with T∼Bδ,γ, X∼Sμ,σ2, μ  = 0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15,0.20,0.25,0.30,0.35,0.40,0.45,0.50.
λeKARL0SDRL0q10q25q50q75q90KARL0SDRL0q10q25q50q75q90ARL0=370ARL0=500Case 1ϕ=290,σ2=0.370.052.720370.152357.29154.00118.00259.00503.00827.102.924500.328485.43465.00153.00348.50686.001142.000.103.017370.480363.93046.00111.00257.00507.25837.003.196500.776490.83561.00151.00345.00690.001144.200.153.201371.871366.66443.00112.00259.00519.00848.103.368500.900498.51056.00144.00346.00701.251149.100.203.288373.905370.44343.00114.00259.00513.25851.003.458499.743489.23356.00147.00345.00696.001161.000.253.356367.145359.24842.00109.00256.00509.00829.003.533497.660490.87858.00151.00350.00690.001128.000.303.451373.730383.40642.90107.00255.00514.00871.003.622498.463490.96355.00147.00351.00690.001142.100.353.489371.382369.28042.00108.00262.00516.00854.103.673503.375498.19155.00146.00354.00696.251145.000.403.500370.896369.57340.00105.00256.00518.00869.003.687500.532491.57457.00152.00355.00690.251135.300.453.515367.001366.02140.00105.00254.00507.00842.003.698502.474493.69760.00154.00354.00690.001154.100.503.538367.103362.55440.00107.00256.00507.00839.003.719500.483499.77753.90147.00350.00685.001155.20Case 2ϕ=148,σ2=0.500.052.707370.731356.27154.00118.00263.00507.25831.002.898499.123487.76867.00154.00349.00688.001120.000.103.045369.013351.86645.00113.00263.00508.00829.003.220501.674492.19462.00148.00354.50695.001124.100.153.197369.567365.66543.00109.00253.00506.25840.003.374500.237493.92057.00147.00346.00690.001143.100.203.334373.194363.13445.00114.00266.00518.00844.103.493500.013489.58358.00152.00351.00694.241146.000.253.345373.967365.54445.00113.00262.00524.00852.103.630500.142496.73356.00145.00346.00695.001143.200.303.461367.567367.37241.00106.00255.00505.25840.103.657503.463492.53656.00149.00356.00699.001158.100.353.552372.473364.49842.00109.00256.00520.00862.003.750500.005492.16456.00151.00347.00694.001156.000.403.545368.830370.07639.00106.00252.00516.25854.003.734502.620510.33056.00146.00345.00689.001153.100.453.581367.860366.65039.00106.00254.00505.00839.003.769500.174491.47358.00150.00350.00688.001160.000.503.630370.312361.71640.00111.00262.00516.00841.103.759497.050490.47354.00144.00343.00695.251139.00Case 3ϕ=80,σ2=0.710.052.728370.823356.20655.00121.00261.00508.25816.102.909501.458485.84967.00159.00354.50691.001135.000.103.050370.960367.28247.00111.00255.00509.00850.003.230501.485486.20561.00154.00358.00698.001137.100.153.201370.680366.81443.00112.00259.00510.00852.103.386500.238483.67559.00150.00357.00697.251130.000.203.395373.536369.47642.00108.00261.00517.00845.003.582503.270495.41755.00147.75354.00696.001158.000.253.395372.622372.67241.00111.00261.00517.00852.103.574503.578504.67555.00149.00348.00694.001158.100.303.472371.473374.57340.00109.00256.00517.00856.003.671501.321499.94355.00144.00346.00695.001173.000.353.598372.650373.99441.00105.00258.00520.20875.203.779500.190499.67054.00145.00345.00688.001170.000.403.613370.201370.87941.00106.00254.00513.00857.003.807498.473496.09853.00140.00348.00690.001127.100.453.676370.377369.58639.00108.00259.00513.00844.103.869500.434498.47253.00142.00346.00703.001158.100.503.714372.503375.38740.00108.00255.00515.00855.003.915502.548499.85755.00147.00350.00689.001162.10Case 4ϕ=31,σ2=1.200.052.738370.176353.74653.00118.00263.00506.00829.102.945500.859492.33364.00152.00348.00685.001150.000.103.060369.169358.09847.00115.00258.00508.00836.003.260500.731485.02162.00151.00357.00691.001140.300.153.263371.737369.03044.00110.00255.00512.00852.103.454500.098491.69557.00147.00354.00699.001135.000.203.360370.463371.37344.00112.00256.00501.00845.003.560503.593496.59958.00149.00354.00695.001162.100.253.471370.500366.84741.00107.00258.00518.00846.103.684500.876500.08354.00148.00352.00687.001147.000.303.618373.858367.37742.00110.00262.00515.00866.003.826500.762500.36256.00147.00342.00696.001152.100.353.683370.463372.47340.00106.00256.00505.00850.203.874497.311488.68954.00146.00345.00692.001160.000.403.725371.140370.00842.00107.00256.00513.00850.003.929498.483503.33452.00143.00344.50684.001155.000.453.787369.293370.47242.00106.00250.00504.24845.004.000500.048499.76054.00143.00345.00687.001154.100.503.875373.357371.36842.00108.00261.00518.25856.104.070497.088499.88055.00146.00341.00697.251143.00Once an OOC signal is noted down, it is important to identify the parameters responsible for an OOC signal. Thus, to label the OOC signal, we suggest to label the points exceeding the UCL as given in Table2. That is, if Uj and Vj exceed the UCL, use the label ′X′ and ′T′ to indicate shift in the magnitude and TBE, respectively. If both Uj and Vj exceed UCL, use the label X′&T′ to indicate simultaneous shift in both TBE and magnitude of the event.Table 2
Labeling out-of-control signal.
TimeMagnitudeU<UCLU>UCLV<UCLICXV>UCLTX&T


## 4. Performance Comparison

We assess the performance of the Max-EWMA control chart using the ARL criterion. The ARL is defined as the mean value of the samples/subgroups plotted until there is an OOC signal occurs. There are two types of ARL in the literature; the in-control ARLARL0 is computed assuming no assignable cause in the process, whereas the out-of-control ARL ARL1 is computed by assuming an assignable cause in the process. Under ideal settings, a control chart should have large ARL0 so that there are no unnecessary pauses in the production process. Contrary to ARL0, ARL1 should be small to detect an assignable cause quickly. Besides ARL, the standard deviation of run length (SDRL) along with several percentiles (q10, q25, q50, q75, and q90) is also discussed in this study. Table 3 and Tables S1 and S2 are computed to give a detailed analysis of ARL, SDRL, and percentiles of run length distribution corresponding to Case 1(a), and Tables S3–S5 are listed corresponding to Case 1(b). Although our interest lies in the simultaneous monitoring of TBE and magnitude, shifts in a single parameter of one of the two distributions keeping IC the other parameters are also considered in this study. To this end, for Case 1(a), shift in the shape1 parameter of beta distribution keeping the location parameter of simplex distribution IC is considered in Table 3, shift in the location parameter of simplex distribution keeping IC the shape1 parameter of beta distribution is considered in Table S1, and simultaneous shifts in the two parameter are considered in Table S2. On the other hand, for Case 1(b), shifts in the shape2 parameter are considered in Table S3, shifts in the dispersion parameter of simplex distribution are considered in Table S4, and simultaneous shifts are considered in Table S5.Table 3
ARL results by considering pure shift in the shape1 parameter of beta distribution (Δδ∈1.0,1.1,1.2,1.3,1.4,0.9,0.8,0.7,0.6,0.5, Δμ  = 1.0) when T∼Bδ,γ, X∼Sμ,σ2, μ  = 0.20, ARL0∈370,500, and λe∈0.05,0.10,0.15.
λeKEMVMΔδΔμARLSDRLq10q25q50q75q90KARLSDRLq10q25q50q75q90ARL0  = 370ARL0  = 500Case 1ϕ=290,σ2=0.370.052.7200.18022600.00922691.001.00370.152357.29154.00118.00259.00503.00827.102.924500.328485.43465.00153.00348.50686.001142.000.052.7200.18022600.00922691.101.0020.5039.95810.0013.0018.0025.0033.002.92422.03810.82311.0014.0020.0027.0036.000.052.7200.18022600.00922691.201.008.8132.7606.007.008.0010.0012.002.9249.2072.9106.007.009.0011.0013.000.052.7200.18022600.00922691.301.005.7421.4624.005.006.007.008.002.9245.9941.5214.005.006.007.008.000.052.7200.18022600.00922691.401.004.3360.9443.004.004.005.006.002.9244.5270.9843.004.004.005.006.000.052.7200.18022600.00922690.901.0019.4958.96910.0013.0018.0024.0031.002.92420.7279.56611.0014.0019.0025.0033.000.052.7200.18022600.00922690.801.007.9622.2555.006.008.009.0011.002.9248.3982.3516.007.008.0010.0012.000.052.7200.18022600.00922690.701.004.9981.0554.004.005.006.006.002.9245.2271.0814.004.005.006.007.000.052.7200.18022600.00922690.601.003.6870.6493.003.004.004.004.002.9243.8530.6753.003.004.004.005.000.052.7200.18022600.00922690.501.002.9760.4262.003.003.003.003.002.9243.0790.4053.003.003.003.004.000.103.0170.25926750.01924661.001.00370.480363.93046.00111.00257.00507.25837.003.196500.776490.83561.00151.00345.00690.001144.200.103.0170.25926750.01924661.101.0020.75412.7758.0012.0018.0026.0038.003.19622.62214.4439.0012.0019.0029.0041.000.103.0170.25926750.01924661.201.007.6232.9844.006.007.009.0011.003.1968.0663.1505.006.007.0010.0012.000.103.0170.25926750.01924661.301.004.7941.4373.004.005.006.007.003.1964.9661.4693.004.005.006.007.000.103.0170.25926750.01924661.401.003.5650.9263.003.003.004.005.003.1963.7160.9553.003.004.004.005.000.103.0170.25926750.01924660.901.0019.57111.6168.0012.0017.0024.0034.003.19620.88412.4619.0012.0018.0026.0037.000.103.0170.25926750.01924660.801.006.8522.3324.005.006.008.0010.003.1967.1942.4245.005.007.008.0010.000.103.0170.25926750.01924660.701.004.1381.0363.003.004.005.005.003.1964.3051.0393.004.004.005.006.000.103.0170.25926750.01924660.601.003.0330.6322.003.003.003.004.003.1963.1290.6312.003.003.003.004.000.103.0170.25926750.01924660.501.002.3490.4892.002.002.003.003.003.1962.4360.5172.002.002.003.003.000.153.2010.32104690.02936701.001.00371.871366.66443.00112.00259.00519.00848.103.368500.900498.51056.00144.00346.00701.251149.100.153.2010.32104690.02936701.101.0022.62516.3837.0011.0018.0029.0044.003.36825.15718.4478.0012.0020.0032.0049.000.153.2010.32104690.02936701.201.007.3513.3594.005.007.009.0012.003.3687.7043.4654.005.007.009.0012.000.153.2010.32104690.02936701.301.004.3821.4863.003.004.005.006.003.3684.5751.5413.003.004.005.007.000.153.2010.32104690.02936701.401.003.2260.9192.003.003.004.004.003.3683.3460.9462.003.003.004.005.000.153.2010.32104690.02936700.901.0021.83315.1148.0011.0018.0028.0042.003.36823.85316.8068.0012.0019.0030.0045.100.153.2010.32104690.02936700.801.006.4872.5674.005.006.008.0010.003.3686.7742.6534.005.006.008.0010.000.153.2010.32104690.02936700.701.003.7781.0703.003.004.004.005.003.3683.9001.0783.003.004.004.005.000.153.2010.32104690.02936700.601.002.6920.6372.002.003.003.003.003.3682.7670.6572.002.003.003.004.000.153.2010.32104690.02936700.501.002.1280.3452.002.002.002.003.003.3682.1770.3902.002.002.002.003.00Case 2ϕ=148,σ2=0.500.052.7070.18094570.00935671.001.00370.731356.27154.00118.00263.00507.25831.002.898499.123487.76867.00154.00349.00688.001120.000.052.7070.18094570.00935671.101.0033.09219.77514.0019.0028.0042.0059.002.89836.42522.11415.0021.0031.0046.0065.000.052.7070.18094570.00935671.201.0013.2965.3058.0010.0012.0016.0020.002.89813.9555.4918.0010.0013.0017.0021.000.052.7070.18094570.00935671.301.008.3112.5895.006.008.0010.0012.002.8988.7272.6656.007.008.0010.0012.000.052.7070.18094570.00935671.401.006.1821.6354.005.006.007.008.002.8986.4651.7105.005.006.007.009.000.052.7070.18094570.00935670.901.0031.27117.67114.0019.0027.0039.0054.002.89834.31819.77515.0020.0029.0043.0060.000.052.7070.18094570.00935670.801.0011.9074.1387.009.0011.0014.0017.002.89812.4904.2768.009.0012.0015.0018.000.052.7070.18094570.00935670.701.007.2161.8495.006.007.008.0010.002.8987.4641.8525.006.007.009.0010.000.052.7070.18094570.00935670.601.005.1761.0674.004.005.006.007.002.8985.3701.1004.005.005.006.007.000.052.7070.18094570.00935670.501.004.0340.7053.004.004.004.005.002.8984.1950.7253.004.004.005.005.000.103.0450.25877860.01912201.001.00369.013351.86645.00113.00263.00508.00829.003.220501.674492.19462.00148.00354.50695.001124.100.103.0450.25877860.01912201.101.0037.10027.49911.0018.0029.0049.0073.003.22041.83732.28512.0019.0032.0054.0084.000.103.0450.25877860.01912201.201.0012.2585.9866.008.0011.0015.0020.003.22013.0026.5066.008.0012.0016.0021.000.103.0450.25877860.01912201.301.007.2292.7444.005.007.009.0011.003.2207.5492.8494.006.007.009.0011.000.103.0450.25877860.01912201.401.005.2041.6533.004.005.006.007.003.2205.3911.6963.004.005.006.008.000.103.0450.25877860.01912200.901.0036.23925.93412.0018.0029.0047.0071.003.22039.87228.73513.0019.7531.0052.0077.000.103.0450.25877860.01912200.801.0010.8634.7376.007.0010.0013.0017.003.22011.4905.0206.008.0010.0014.0018.000.103.0450.25877860.01912200.701.006.1221.8934.005.006.007.009.003.2206.4111.9924.005.006.007.009.000.103.0450.25877860.01912200.601.004.2721.0453.004.004.005.006.003.2204.4471.0743.004.004.005.006.000.103.0450.25877860.01912200.501.003.3170.6633.003.003.004.004.003.2203.4290.6773.003.003.004.004.000.153.1970.32138860.02951381.001.00369.567365.66543.00109.00253.00506.25840.003.374500.237493.92057.00147.00346.00690.001143.100.153.1970.32138860.02951381.101.0042.80936.16910.0017.0032.0058.0089.003.37449.43542.32812.0019.0037.0065.00106.000.153.1970.32138860.02951381.201.0012.4397.3545.007.0011.0016.0022.003.37413.2557.8106.008.0011.0017.0023.000.153.1970.32138860.02951381.301.006.8333.0024.005.006.008.0011.003.3747.2143.2284.005.007.009.0011.000.153.1970.32138860.02951381.401.004.7701.7003.004.004.006.007.003.3744.9851.7653.004.005.006.007.000.153.1970.32138860.02951380.901.0045.18237.23911.0019.0034.0060.0094.003.37451.90643.26812.0021.0039.0069.00108.000.153.1970.32138860.02951380.801.0010.8465.5825.007.0010.0013.0018.003.37411.5936.1206.007.0010.0014.0020.000.153.1970.32138860.02951380.701.005.6992.0234.004.005.007.008.003.3745.9662.1174.004.006.007.009.000.153.1970.32138860.02951380.601.003.8761.0543.003.004.004.005.003.3744.0301.0913.003.004.005.005.000.153.1970.32138860.02951380.501.002.9330.6842.003.003.003.004.003.3743.0500.6972.003.003.003.004.00Case 3ϕ=148,σ2=0.500.052.7280.18055800.00928361.001.00370.823356.20655.00121.00261.00508.25816.102.909501.458485.84967.00159.00354.50691.001135.000.052.7280.18055800.00928361.101.0052.85136.63218.0027.0043.0069.00100.002.90958.29841.71019.0029.0047.0075.00113.000.052.7280.18055800.00928361.201.0019.9139.69610.0013.0018.0024.0032.002.90921.09710.29910.0014.0019.0026.0034.000.052.7280.18055800.00928361.301.0011.9384.5447.009.0011.0014.0018.002.90912.5444.6647.009.0012.0015.0019.000.052.7280.18055800.00928361.401.008.6472.7556.007.008.0010.0012.002.9099.1062.8656.007.009.0011.0013.000.052.7280.18055800.00928360.901.0051.48035.38819.0027.0042.0066.0097.002.90957.39539.22721.0030.0046.0073.25109.000.052.7280.18055800.00928360.801.0017.4797.48010.0012.0016.0021.0027.002.90918.6657.83610.0013.0017.0022.0029.000.052.7280.18055800.00928360.701.0010.2203.1337.008.0010.0012.0014.002.90910.7633.2947.008.0010.0013.0015.000.052.7280.18055800.00928360.601.007.1631.7645.006.007.008.009.002.9097.4601.8175.006.007.008.0010.000.052.7280.18055800.00928360.501.005.4951.1104.005.005.006.007.002.9095.7141.1354.005.006.006.007.000.103.0500.25841050.01913351.001.00370.960367.28247.00111.00255.00509.00850.003.230501.485486.20561.00154.00358.00698.001137.100.103.0500.25841050.01913351.101.0062.60952.26315.0026.0047.0084.00131.003.23072.47762.65916.0029.0054.0096.00154.000.103.0500.25841050.01913351.201.0019.55612.1218.0011.0016.0025.0035.003.23021.33913.1898.0012.0018.0027.0038.000.103.0500.25841050.01913351.301.0010.9235.1616.007.0010.0013.0018.003.23011.4225.4146.008.0010.0014.0018.000.103.0500.25841050.01913351.401.007.6032.9664.005.007.009.0011.003.2307.9243.0755.006.007.0010.0012.000.103.0500.25841050.01913350.901.0066.64555.09116.0028.0050.0089.00141.003.23078.58466.43619.0032.0059.00104.00164.000.103.0500.25841050.01913350.801.0017.4389.3818.0011.0015.0022.0030.003.23018.68710.3939.0011.0016.0023.0032.000.103.0500.25841050.01913350.701.009.0953.4925.007.008.0011.0014.003.2309.5913.7216.007.009.0011.0014.000.103.0500.25841050.01913350.601.006.0671.7864.005.006.007.008.003.2306.3741.8874.005.006.007.009.000.103.0500.25841050.01913350.501.004.5621.0903.004.004.005.006.003.2304.7421.1374.004.005.005.006.000.153.2010.32158640.02966291.001.00370.680366.81443.00112.00259.00510.00852.103.386500.238483.67559.00150.00357.00697.251130.000.153.2010.32158640.02966291.101.0073.52065.81914.0027.0054.00100.00160.003.38686.99580.05615.0030.0062.00119.00191.000.153.2010.32158640.02966291.201.0021.39915.2547.0011.0017.0028.0041.003.38623.57917.1758.0012.0019.0030.0046.000.153.2010.32158640.02966291.301.0010.7635.8705.007.009.0013.2518.003.38611.6436.5565.007.0010.0014.0020.000.153.2010.32158640.02966291.401.007.2053.2984.005.007.009.0012.003.3867.6633.5154.005.007.009.0012.000.153.2010.32158640.02966290.901.0090.16181.55017.0032.0065.00122.00195.003.386114.106104.39220.0039.0082.00154.25250.000.153.2010.32158640.02966290.801.0019.40312.4348.0011.0016.0025.0036.003.38621.51014.3358.0011.0018.0027.0040.000.153.2010.32158640.02966290.701.009.0054.0925.006.008.0011.0014.003.3869.4784.3395.006.008.0011.0015.000.153.2010.32158640.02966290.601.005.6501.9214.004.005.007.008.003.3865.9412.0084.005.006.007.009.000.153.2010.32158640.02966290.501.004.1491.1413.003.004.005.006.003.3864.3241.1633.004.004.005.006.00Case 4ϕ=31,σ2=1.200.052.7380.17997700.00924401.001.00370.176353.74653.00118.00263.00506.00829.102.945500.859492.33364.00152.00348.00685.001150.000.052.7380.17997700.00924401.101.00104.43188.05124.0042.0077.00140.00220.002.945119.964102.40228.0048.0089.00159.00253.100.052.7380.17997700.00924401.201.0038.81725.00014.0021.0032.0049.0072.002.94542.61027.46016.0023.0035.0054.0078.100.052.7380.17997700.00924401.301.0022.05411.48510.0014.0019.0028.0037.002.94523.74812.05611.0015.0021.0030.0040.000.052.7380.17997700.00924401.401.0015.3576.6178.0011.0014.0019.0024.002.94516.1796.9219.0011.0015.0020.0025.000.052.7380.17997700.00924400.901.00109.37389.88928.0046.0082.00145.00225.002.945133.828113.56932.0055.0099.00178.00279.000.052.7380.17997700.00924400.801.0034.39919.92315.0020.0030.0043.0060.002.94537.72121.69216.9022.0032.0047.0066.000.052.7380.17997700.00924400.701.0018.5357.70210.0013.0017.0022.0029.002.94519.7558.26311.0014.0018.0024.0031.000.052.7380.17997700.00924400.601.0012.3253.9968.009.0012.0014.0018.002.94512.9984.1988.0010.0012.0015.0018.100.052.7380.17997700.00924400.501.009.1642.4406.007.009.0010.0012.002.9459.6112.4857.008.009.0011.0013.000.103.0600.25848640.01906791.001.00369.169358.09847.00115.00258.00508.00836.003.260500.731485.02162.00151.00357.00691.001140.300.103.0600.25848640.01906791.101.00120.023110.39221.0042.0087.00162.00262.003.260144.840134.91825.0049.00104.00198.00319.000.103.0600.25848640.01906791.201.0042.78833.36312.0019.0033.0057.0087.003.26048.58038.45613.0021.0037.0064.0098.000.103.0600.25848640.01906791.301.0022.16114.4528.0012.0018.0028.0041.003.26024.49016.0939.0013.0020.0031.0045.000.103.0600.25848640.01906791.401.0014.4837.9096.009.0013.0018.0025.003.26015.7398.6327.0010.0014.0020.0027.000.103.0600.25848640.01906790.901.00165.885153.48929.0057.00119.00224.00368.003.260222.840209.10935.0074.00160.00305.00493.000.103.0600.25848640.01906790.801.0042.23731.55414.0020.0033.0054.0082.103.26049.56138.40415.0023.0038.0064.0098.000.103.0600.25848640.01906790.701.0018.79510.1669.0012.0016.0023.0032.003.26020.49511.4159.0013.0017.0025.0035.000.103.0600.25848640.01906790.601.0011.2924.5576.008.0010.0013.0017.003.26012.2245.0667.009.0011.0015.0019.000.103.0600.25848640.01906790.501.008.0282.6185.006.008.009.0012.003.2608.3942.7865.006.008.0010.0012.000.153.2630.32089410.02949891.001.00371.737369.03044.00110.00255.00512.00852.103.454500.098491.69557.00147.00354.00699.001135.000.153.2630.32089410.02949891.101.00137.524132.63321.0044.0097.00187.00307.003.454167.461159.73024.0055.00119.00229.00373.000.153.2630.32089410.02949891.201.0050.08943.29811.0020.0036.0067.00107.003.45459.24751.82213.0022.0044.0079.00127.000.153.2630.32089410.02949891.301.0024.82218.7068.0012.0020.0032.0049.003.45427.79721.4068.0013.0022.0036.0055.000.153.2630.32089410.02949891.401.0015.1489.6456.008.0013.0019.0028.003.45416.56210.9026.009.0014.0021.0031.000.153.2630.32089410.02949890.901.00249.479242.93332.0077.00175.00343.00567.003.454353.239347.96145.00106.00245.00491.00807.000.153.2630.32089410.02949890.801.0060.53651.57014.0024.0045.0080.00127.003.45476.18367.41417.0029.0055.00100.00165.000.153.2630.32089410.02949890.701.0022.12514.6178.0012.0018.0028.0041.003.45425.14217.1039.0013.0020.0032.0047.000.153.2630.32089410.02949890.601.0011.8755.8336.008.0010.0014.0020.003.45412.8526.5366.008.0011.0016.0021.000.153.2630.32089410.02949890.501.007.8412.9755.006.007.009.0012.003.4548.3783.2215.006.008.0010.0013.00For pure or simultaneous shift, we consideredΔ∈1,1.10,1.20,1.30,1.40,0.90,0.80,0.70,0.60,0.50 as the shift sizes. The values Δ= 1, 0.90 and 1.10 mean there is no shift, 10% DW shift, and 10% UW shift in the process. The IC mean (EM) and variance (VM) of the charting statistic M are also given in the tables. In particular, the analysis is conducted under the following conditions:Case 1:ϕ=290⇒δ0=μ×ϕ=58, γ0=1−μϕ=232 and σ02=0.37Case 2:ϕ=148⇒δ0=μ×ϕ=29.6, γ0=1−μϕ=118.4 and σ02=0.50.Case 3:ϕ=80⇒δ0=μ×ϕ=16, γ0=1−μϕ=64 and σ02=0.71.Case 4:ϕ=31⇒δ0=μ×ϕ=6.2, γ0=1−μϕ=24.8 and σ02=1.20.μ0=0.20 and ARL0=370,500 are considered for all the above cases. An algorithm using R programming software, version 4.0.0 (2020-04-24), is written to compute the values listed in Table 3 and Tables S1 and S2. More specifically, shifts in the shape1 parameter of T are considered in Table 3, shifts in the location parameter of simplex distribution are considered in Table S1, and simultaneous shifts in both parameters are considered in Table S2.Table3 presents ARL results assuming λe=0.05, ARL0=370, and the set of parameters in Case 1. It is noticed that the ARL reduces by 94.46%, 97.62%, 98.45%, and 98.83% as compared to the nominal value of the ARL for UW shift of size 10%, 20%, 30%, and 40% in the shape1 parameter of beta distribution, respectively. For DW shift of 10%, 20%, 30%, 40%, and 50% sizes, the ARL reduces by 94.73%, 97.85%, 98.65%, 99.00%, and 99.20%, respectively. For the same amount of UW and DW shifts, when the smoothing parameter of the EWMA chart increases from 0.05 to 0.10, the ARL reduces by 94.39%, 97.94%, 98.70%, and 99.04%, respectively. Similar degenerating behavior of the ARL is observed for λe=0.15. In the case of ARL0=500 and λe∈0.05,0.10,0.15, the same ARL behavior is observed. For the parameter combination of Case 2, assuming ARL0=370 and λe=0.05, the ARL reduces by 91.07%, 96.41%, 97.76%, and 98.33% for the abovementioned UW shifts and by 91.57%, 96.79%, 98.05%, 98.60%, and 98.91% for the same amount of DW shift as mentioned above. For λe=0.10, the reduction in the ARL with respect to UW shift is by 89.95%, 96.68%, 98.04%, and 98.59%, and for the DW shift, it is 90.18%, 97.06%, 98.34%, 98.84%, and 99.10%, respectively. For λe=0.15, a very similar reduction is observed. The performance of the proposed chart for Case 3 and Case 4 can be interpreted in a similar fashion.For Case 1 of TableS1, when λe=0.05, ARL0=370, Δμ∈1.10,1.20,1.30,1.40 and Δμ∈0.90,0.80,0.70,0.60,0.50, the ARL reduces by 92.21%, 96.87%, 98.01%, and 98.54% and by 91.93%, 96.91%, 98.09%, 98.60%, and 98.88%, respectively. For λe=0.10, the ARL reduces by 92.00%, 97.20%, 98.31%, and 98.79% and by 90.27%, 97.17%, 98.37%, 98.84%, and 99.10%, respectively, for similar UW and DW shifts in the location parameter of X. The rest of the cases can be interpreted in the same way.Simultaneous shifts in the shape1 parameter of beta distribution and location parameter of the simplex distribution are discussed in TableS2. A degenerating pattern of ARL is observed, as in the previous tables, for simultaneous increasing and decreasing (UW and DW) shifts. More precisely, for Case 1, when λe=0.05, ARL0=370, Δδ∈1.10,1.20,1.30,1.40, and Δμ=1.10; i.e., when there are simultaneous UW shifts of 10% in the location parameter of X and UW shifts of 10%, 20%, 30%, and 40% in the shape1 parameter of T, the ARL reduces by 95.42%, 97.66%, 98.45%, and 98.82%, respectively. In the case of 10%, 20%, 30%, 40%, and 50% DW shifts in the shape1 parameter of T, the ARL reduced by 95.55%, 97.86%, 98.65%, 99.00%, and 99.20%, respectively, for similar UW shifts in the location parameter of X. In the case of a UW shift of size 20% in the location parameter of X, the ARL reduced by 97.09%, 97.89%, 98.48%, and 98.83% and by 97.13%, 98.02%, 98.66%, 99.00%, and 99.20% for the same UW and DW shifts in the first shape parameter of T. Such decreasing patterns of ARL are observed for all the cases. It is noted that the performance of the proposed control chart is exceptionally good for Case 1 and Case 2 of the study. However, it deteriorates slightly for pure as well as for simultaneous shifts in Case 3 and Case 4; that is, the performance is more effected for smaller shifts. For example, in Case 4 of Table 3 when λe=0.15, ARL0=370, and Δδ=1.10, the ARL reduces by 63.01%, whereas in Case 1, the ARL reduces by 93.92% with similar parameter values. Moreover, in the case of simultaneous shift of size Δδ,Δμ=0.90,0.90 in Case 3 of Table S2 with λe=0.15 and ARL0=500, the ARL reduces by 83.64%, whereas for a similar parameter setting in Case 1, the ARL reduces by 95.9%. Similar to the ARL, the standard deviation of run length (SDRL) also shows a decreasing pattern as a shift size increases for all of the cases. Therefore, it can be said that the ARL and the SDRL decrease when the UW and DW shifts increased, whether simultaneous or pure.The analysis of different percentiles shows that, in most cases, the run length distribution is rightly skewed as the ARL is greater than q50 (median) and less than q75 (75th percentile). For example, in Case 2 of Table 3 when λe=0.05, ARL0=370, and Δδ=0.90, the ARL1 is 31.271, median is 27.00, and q75 is 39. Also, Case 1 of Table S1 using λe=0.10, ARL0=500, and Δμ=1.10, the ARL1 is 32.826, median is 26, and q75 is 42. In some cases, the ARL is slightly less than the median meaning that the run length distribution is slightly negatively skewed. For example, in Case 2 of Table 3 with λe=0.15, ARL0=370, and Δδ=0.60, the ARL1 is 3.876 and the median is 4.The proposed chart has a higher sensitivity when the shift size and smoothing parameter values are small. However, sensitivity for moderate size shifts increases whenλe increases. For example, in Case 2 of Table S1 with λe=0.05, ARL0=370, and Δμ=1.10,0.90, the ARL1=35.306,38.327, whereas for λe=0.15 the values ARL1 are 40.762 and 75.506, respectively. Contrary to this case, for moderate size shifts, e.g., Δμ=0.50, the ARL1=4.798 for λe=0.05 and ARL1=3.527 for λe=0.15. In the case of small simultaneous shifts, e.g., Δδ,Δμ=0.90,0.90, when ARL0=500 in Case 3 of Table S2, ARL1=38.612 in the case of λe=0.05, and ARL1=81.863 in the case of λe=0.15. Thus, it can be concluded that smaller values of smoothing parameter are appropriate for the detection of smaller shifts and larger values for larger shifts [31].The performance of the chart under the setup given in Case 1(b) is evaluated in TablesS3–S5. More specifically, pure shifts in the second shape parameter are assessed in Table S3, results for pure shifts in the dispersion parameter of the simplex distribution are listed in Table S4, and simultaneous shift in the dispersion parameter of simplex distribution and shape2 parameter of the beta distribution is considered in Table S5. The combined analysis of these tables shows a similar degenerating trend with respect to ARL and SDRL as the shift size increases. However, under a pure shift in the dispersion parameter of X, the proposed chart performs poor. In most cases, the ARL values are either close to or greater than ARL0 value leading to biased ARL ARL1>ARL0. Furthermore, in the case of shifts in the TBE, it can be seen that the proposed chart is more sensitive to a pure UW shift in shape1 parameter and a pure DW shift in the shape2 parameter of beta distribution. For example, when λe=0.05, ARL0=500, and Δδ∈1.10,1.20 in Case 4 of Table 3, the ARL1 is 144.840 and 48.580, respectively, whereas the corresponding ARL1 values when Δγ∈1.10,1.20 in Table S1 are 171.201 and 55.543, respectively. Additionally, under the similar setting in Table 3 when Δδ=0.80,0.70, the ARL1 is 37.721 and 19.755, respectively, whereas ARL1 in Table S1 for Δγ∈0.80,0.70 is 29.234 and 14.920, respectively. On the other hand, the proposed chart has a much higher efficiency when dealing with pure UW or DW shifts in the location parameter of the simplex distribution as compared to the pure shifts in the dispersion parameter. In the case of simultaneous shifts, the chart performs better under Case 1(a) as compared to Case 1(b).


## 5. Real Data Application

To implement the proposed methodology on a real-life data set, a data set is collected from an electronic company reported in Wu et al. [23]. The company makes electronic chips and is interested in sales decline monitoring. The data set consists of 50 observations including two variables; one is the sales of chips (number of lots ordered by customer), and the second is the time interval between two successive orders (in hours). The sale and time variables are transformed into proportion where X represents the proportion of sales of circuit chips and T represents the proportion of time interval between successive orders. The objective is to detect an increase in the proportion of time between successive orders and/or a decrease in the proportion of number of lots sold. The variable X is fitted to simplex distribution, and T is fitted to beta distribution. The maximum likelihood criterion is used to estimate the parameters of the data set using R. We have used 50 IC observations as the Phase I data and further simulated 25 OOC observations for Phase II data. In particular, the observations are stimulated by introducing 5% UW shift in both shape1 parameter of T and location parameter of X, i.e., Δδ=Δμ=1.05. To implement the proposed charting methodology, we considered λe=0.05,L=3.80,UCL=0.185, and ARL0=50. The parameter estimates and their standard errors (in parentheses), along with AIC and BIC of 50 IC observations, are listed in Table 4.Table 4
Parameter estimates with standard errors (in parentheses), AIC, and BIC for the proportion of sales and time between successive orders.
DistributionML estimatesAICBICBetaμ^0.02000.0023-290.4247-286.6006ϕ^36.62561.2221Simplexμ^0.019980.0017-318.3522-314.5281σ^4.51720.4517Figure1 presents the Max-EWMA control chart for the combined Phase I and Phase II data. Since the Phase I data show no OOC signal, the Phase II data analysis triggered the first OOC signal at 53 rd sample. The cause of this signal is the simultaneous shift in the shape1 parameter of T and location parameter of X. As there is a decline in the sales, the company can devise a better advertisement strategy to increase the demand of its product. Additionally, timely decisions can be taken by the management team to introduce innovations to have an edge over their competitors.Figure 1
Max-EWMA chart for monitoring the circuit chips.


## 6. Conclusion

This work presents a Max-EWMA chart for monitoring rates and proportions assuming beta and simplex distributions. The performance of the chart is assessed by considering different shift sizes and values of smoothing parameters. The performance of the proposed chart in terms of the ARL criterion suggests that the proposed chart has a capability for detecting small- to medium-size shifts. The current study assumed that time and magnitude are independent which could be dependent in practice. Therefore, charts with dependent time and magnitude should be proposed in the future. Furthermore, the effect of parameter estimation on Phase I and Phase II may be investigated.



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*Source: 7306775-2022-07-20.xml*