A Bayesian Control Chart for Monitoring Process Variance

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The results in this article are applicable to the signal detection in a service process and the monitoring of an intelligent automated process.

Abstract

Automation in the service industry is emerging as a new wave of industrial revolution. Standardization and consistency of service quality is an important part of the automation process. The quality control methods widely used in the manufacturing industry can provide service quality measurement and service process monitoring. In particular, the control chart as an online monitoring technique can be used to quickly detect whether a service process is out of control. However, the control of the service process is more difficult than that of the manufacturing process because the variability of the service process comes from widespread and complex factors. First of all, the distribution of the service process is usually non-normal or unknown. Moreover, the skewness of the process distribution can be time-varying, even if the process is in control. In this study, a Bayesian procedure is applied to construct a Phase II exponential weighted moving average (EWMA) control chart for monitoring the variance of a distribution-free process. We explore the sampling properties of the new monitoring statistic, which is suitable for monitoring the time-varying process distribution. The average run lengths (ARLs) of the proposed Bayesian EWMA variance chart are calculated, and they show that the chart performs well. The simulation studies for a normal process, exponential process, and the mixed process of normal and exponential distribution prove that our chart can quickly detect any shift of a process variance. Finally, a numerical example of bank service time is used to illustrate the application of the proposed Bayesian EWMA variance chart and confirm the performance of the process control.

1. Introduction

Due to the rapid development of artificial intelligence, service automation is emerging as a new wave of industrial revolution. Standardization and consistency of service quality are important parts of the automation process. The quality control methods widely used in the manufacturing industry can provide service quality measurement and service process monitoring. In particular, online process monitoring of the control chart can be used to quickly detect signals that indicates when the service process is out of control. However, the control of a service process is more difficult than that of a manufacturing process because the variability of a service process comes from widespread and complex factors [1,2,3]. First of all, the distribution of the service process is usually non-normal or unknown. Moreover, the skewness of process distribution is time-varying, even if the process is in control.

Because Shewhart variables control charts depend on the normality assumption, several studies have searched for effective methods to measure the performance of control charts with non-normal or unknown distribution data. Among them, most studies deal with process location monitoring, while fewer studies deal with process variability monitoring. Readers interested in process location monitoring can refer to Ferrell [4], Bakir and Reynolds [5], Amin et al. [6], Altukife [7,8], Bakir [9,10], Chakraborti and Eryilmaz [11], Chakraborti and Graham [12], Chakraborti et al. [13], Li et al. [14], Zou and Tsung [15], and Graham et al. [16,17]. As for the process variability monitoring, Zou and Tsung [15] developed the nonparametric likelihood-ratio (NLE) exponential weighted moving average (EWMA) control chart and the combination of EWMA mean chart and EWMA variance chart (CEW) to monitor the changes in process variance. Jones-Farmer and Champ [18] proposed a distribution-free Phase I scale chart, which can be used to define the in-control state of the process variability. Zombade and Ghute [19] developed two Shewhart-type nonparametric control charts based on run rules for monitoring the changes in the process variability. The above non-parametric approaches can deal with non-normal or unknown distributed data, but they are not easy to apply for practitioners. The practitioners, usually not statisticians, do not quite understand the proper way to implement the control scheme.

To achieve ease of use, Yang et al. [20] proposed a new Sign Chart for variables data to monitor the deviation of process measurements from the target without the assumption of a normal process distribution or a distribution of known form. Similarly, a new Mean Chart and an improved Mean Chart for variables data were proposed by Yang et al. [21] and Yang [22], respectively, to monitor the process mean also without the assumption of a normal process distribution or a distribution of known form. As for the process variability monitoring, Yang and Arnold [23] proposed a new distribution-free variance chart for variables data to monitor the process variance. Yang and Jiang [24] further developed the interquartile range control chart for detecting the out-of-control variance of a critical quality characteristic that exhibits a non-normal or unknown distribution. The above easy-to-use approaches can deal with non-normal or unknown distributed data under the assumption that the skewness of the distribution is fixed. However, the assumption of a fixed skewness is not true for a process with the mixture distribution. Hence, the question arises: ‘How can one measure the performance of control charts for monitoring a process in which skewness of the distribution may vary randomly by time?’

The Bayesian approach is usually applied to address the uncertainty about the parameter of interest. Girshick and Rubin [25] used the Bayesian approach to deal with a class of statistical quality control procedures and continuous inspection procedures. Menzefricke [26] proposed a Bayesian approach to obtaining control charts, where a predictive distribution based on a Bayesian approach was used to derive the control limits. Furthermore, based on the predictive distributions, the control charts for generalized variance, variance, and coefficient of variation have been done [27,28]. It is worth mentioning that Menzefricke [29] developed exponentially weighted moving average (EWMA) control charts for mean and variance of a normal distribution with the Bayesian approach, where the posterior predictive distribution has incorporated the parameter uncertainty. Saghir [30] used informative and non-informative priors for updating the process mean and obtained the X-bar chart based on posterior control limits. Khan et al. [31] proposed a new design for a control chart using a modified EWMA statistic under the assumption that the quality characteristic of interest follows the normal distribution. Based on the framework of Khan et al. [31], Saghir et al. [32] proposed a modified EWMA control chart for monitoring the process variance. Recently, Aslam and Anwar [33] proposed a new Bayesian Modified-EWMA chart for the monitoring of the location parameter in a process, and applied the proposed scheme to the mechanical and sport industry. From the reviews of the above literatures, we found that most of the Bayesian control charts were discussed under normality, and almost no literature addresses Bayesian control charts under distribution-free processes.

This study will focus on the Phase II monitoring. Based on the framework of Yang and Arnold [23], a Bayesian procedure is applied to obtain the control limits for monitoring the process variance without the assumption of a normal process distribution, a distribution of known form, or a distribution with fixed skewness. We explore the sampling properties of the new monitoring statistic and find that it is suitable for monitoring the time-varying process. The average run lengths (ARLs) of the proposed Bayesian control chart are calculated and show that the chart performs well. The simulation studies for a normal process, exponential process, and the mixed process of normal and exponential distribution prove that our chart can quickly detect any shift of process variance. Finally, a numerical example of service time is used to illustrate the application of the proposed Bayesian EWMA variance chart and confirm the performance of process control.

The article is organized as follows. In Section 2, we describe the preliminary settings of our control scheme and the distribution of a new statistic derived from the Bayesian approach. Then, the construction of a newly proposed Bayesian variance chart and its performance are discussed in Section 3. In Section 4, we propose a Bayesian exponentially weighted moving average (EWMA) variance chart. In Section 5, a simulation study for a normal process, exponential process, and the mixed process of normal and exponential distributions is performed to test the detection performance of the proposed Bayesian EWMA variance chart. In Section 6, a numerical example of a service system in a bank branch is used to illustrate how to construct the proposed variance chart. Finally, we summarize the findings and provide recommendations in Section 7.

2. Preliminary Settings

Assuming that a service process is of concern and a critical quality characteristic to be monitored, X, has a mean μ and variance σ². In order to monitor the process variance, a random sample of size 2n, X₁, X₂, …, X_2n is taken from the service process. For convenience in the calculation of Equation (1), the sample size is even; if not, one observation is deleted. Thus, we define that:

$$Y_j=\frac{{\left(X_{2j}-X_{2j-1}\right)}^2}{2},\hspace{1em}\hspace{1em}j=1, 2, \dots , n.$$

(1)

It is easy to show that $E(Y_j)={\sigma }^2 \mathrm{for} j=1, 2, \dots ,n.$ When M is the total number of instances in which Y_j > σ², then M follows from a binomial distribution with parameters n and p, which is defined as:

$$p=P\left(Y_j>{\sigma }^2\right)$$

(2)

The value of p depends on the population of X and is always between 0.2 and 0.5 [23]. If the population of X is normal distribution, then p = 0.3173, which is independent of variance parameter σ²; if the population of X is exponential distribution, then $p=e^{-\sqrt{2}}=0.2431$, which is also independent of rate parameter and hence σ². When p is fixed and known, the binomial distribution can be used to determine the control limits in a process control scheme. However, p is usually unknown in practice because the process distribution may be non-normal and unknown, especially in service processes. Moreover, in some special situations, p may not be fixed even under in-control. The reason for this is that the value of p is not only associated with the shift of variance but also with the skewness of the distribution. Randomized skewness of the distribution will disturb or mislead the judgment about whether the process variance shifts. Thus, we will consider the random behavior of p from the Bayesian perspective. The Bayesian approach views parameter p as a random variable that describes the previous information about parameter p stated in a prior distribution from which we can obtain the posterior distribution or posterior predictive distribution.

Because the beta distribution is the conjugate prior probability distribution for the binomial distributions, it is a suitable model for the random behavior of parameter p. Assuming that the prior distribution of p is a beta distribution:

$$p~Beta\left(\alpha , \beta \right),$$

(3)

where the shape parameters α and β are chosen to reflect any existing belief or information. The statistic M defined above follows a binomial distribution with parameters n and p:

$$M~Binomial\left(n, p\right)$$

(4)

Therefore, the marginal distribution of M is the beta-binomial distribution with parameters n, α, and β:

$$M~Beta-Binomial\left(n, \alpha , \beta \right).$$

(5)

Given the prior distribution of p, the probability function of M is then written as:

$$f_{M|n, \alpha , \beta }\left(m\right)=\left(\begin{array}{c}n\\ m\end{array}\right)\frac{\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }(\alpha )\mathsf{\Gamma }(\beta )}\frac{\mathsf{\Gamma }\left(m+\alpha \right)\mathsf{\Gamma }\left(n-m+\beta \right)}{\mathsf{\Gamma }\left(n+\alpha +\beta \right)},m=0, 1, \dots , n,$$

(6)

and the mean as well as variance are:

$$E\left(M\right)=n\times \frac{\alpha }{\alpha +\beta }$$

(7)

and

$$Var\left(M\right)=\frac{n\alpha \beta \left(\alpha +\beta +n\right)}{{\left(\alpha +\beta \right)}^2\left(\alpha +\beta +1\right)},$$

(8)

respectively. Notice that the variability of the beta-binomial distribution is larger than that of the binomial distribution with p = α/(α + β), and the difference will get smaller as (α + β) increases or n decreases. Figure 1 depicts several probability functions of the beta-binomial distributions for comparison. Panel A of Figure 1 shows that, compared with the binomial distribution, the beta-binomial distributions are more dispersive. Panel B of Figure 1 shows that the skewness of the beta-binomial distributions increases as α/(α + β) decreases from 0.5.

On the other hand, the prior distribution of p will be updated to a posterior distribution:

$$p~Beta\left(\alpha +M, \beta +n-M\right),$$

(9)

because statistic M has been observed. Now, we then propose to collect a new sample and obtain a new statistic, M_new, which also follows a binomial distribution:

$$M_{new}~Binomial\left(n, p\right).$$

(10)

Then, the posterior predictive distribution of M_new is a beta-binomial distribution:

$$M_{new}~Beta-Binomial\left(n, \alpha +M, \beta +n-M\right)$$

(11)

3. The Construction of the Shewhart-Type Variance Chart with the Bayesian Approach

Assuming that the prior distribution of p is the beta distribution with parameters α and β, the expected value of p is expressed as E(p) = α/(α + β). Once process variance σ² shifts upwards/downwards, the expected process proportion E(p) will change upwards/downwards. Thus, monitoring process variance shifts is equivalent to monitoring the changes in the expected process proportion.

3.1. The Parameters of the Prior Distribution of p and Process Variance Are Known

In this subsection, we first develop the variance chart under known process variance σ₀² and parameters of the prior distribution of p, α₀, and β₀. For the in-control process, the monitoring statistic, M_t, defined as the number of Y_j’s > σ₀² at time t, follows from the beta-binomial distribution with parameters n, α₀, and β₀, written as:

$$M_t~Beta-Binomial\left(n, {\alpha }_0, {\beta }_0\right).$$

(12)

The mean and standard deviation of M_t are

$$E_0\left(M_t\right)=n\times \frac{{\alpha }_0}{{\alpha }_0+{\beta }_0}$$

(13)

and

$$S{D}_0\left(M_t\right)=\sqrt{\frac{n{\alpha }_0{\beta }_0\left({\alpha }_0+{\beta }_0+n\right)}{{\left({\alpha }_0+{\beta }_0\right)}^2\left({\alpha }_0+{\beta }_0+1\right)}},$$

(14)

respectively.

Thus, we propose a Shewhart-type Bayesian variance chart with the following limits:

$$C{L}_M=\frac{n{\alpha }_0}{{\alpha }_0+{\beta }_0},$$

$$UC{L}_M=\frac{n{\alpha }_0}{{\alpha }_0+{\beta }_0}+3\sqrt{\frac{n{\alpha }_0{\beta }_0\left({\alpha }_0+{\beta }_0+n\right)}{{\left({\alpha }_0+{\beta }_0\right)}^2\left({\alpha }_0+{\beta }_0+1\right)}},$$

(15)

$$LC{L}_M=\frac{n{\alpha }_0}{{\alpha }_0+{\beta }_0}-3\sqrt{\frac{n{\alpha }_0{\beta }_0\left({\alpha }_0+{\beta }_0+n\right)}{{\left({\alpha }_0+{\beta }_0\right)}^2\left({\alpha }_0+{\beta }_0+1\right)}},$$

(16)

and plot M_t, t = 1, 2, …. If either M_t ≥ UCL_M or M_t ≤ LCL_M, the process is deemed to show some out-of-variance-control signals.

3.1.1. The In-Control ARL of the Shewhart-Type Bayesian Variance Chart

The average run length (ARL) is used to measure the performance of a control chart. The in-control ARL, denoted by ARL₀, of the Bayesian variance chart is associated with n, α₀, and β₀. Under the in-control process, the chance that the proposed chart alarms a false signal is calculated by:

$$ALAR{M}_0=P\left(M_t\le LC{L}_M \mathrm{or} M_t\ge UC{L}_M|M_t~Beta-Binomial\left(n, {\alpha }_0, {\beta }_0\right)\right).$$

(17)

So, ARL₀ = 1/ALARM₀.

Table 1. The in-control average run lengths (ARL₀s) of the proposed Bayesian variance chart.

(α0, β0)	E0(p)	n = 2	n = 3	n = 5	n = 10	n = 15	n = 20	n = 25
(1, 1)	0.5000	∞	∞	∞	∞	∞	∞	∞
(1, 2)	0.3333	∞	∞	∞	∞	∞	∞	∞
(1, 3)	0.2500	∞	∞	∞	286.00	204.00	177.10	163.80
(1, 4)	0.2000	∞	∞	126.00	200.20	110.74	84.33	113.10
(1, 5)	0.1667	∞	56.00	42.00	143.00	123.05	67.08	71.18
(2, 2)	0.5000	∞	∞	∞	∞	∞	∞	∞
(2, 3)	0.4000	∞	∞	∞	∞	∞	∞	∞
(2, 4)	0.3333	∞	∞	∞	∞	969.00	526.04	1131.00
(2, 5)	0.2875	∞	∞	∞	728.00	180.28	222.23	257.80
(2, 6)	0.2500	∞	∞	132.00	273.92	151.19	240.66	174.65
(3, 3)	0.5000	∞	∞	∞	∞	∞	∞	∞
(3, 4)	0.4286	∞	∞	∞	∞	∞	∞	∞
(3, 5)	0.3750	∞	∞	∞	∞	1254.00	693.23	1703.37
(3, 6)	0.3333	∞	∞	∞	663.00	572.80	567.09	577.14
(3, 7)	0.3000	∞	∞	∞	204.83	333.89	468.32	270.28
(10, 10)	0.5000	∞	∞	∞	∞	∞	3441.00	1607.81
(8, 10)	0.4444	∞	∞	∞	∞	3317.17	2130.37	1798.07
(6, 10)	0.3750	∞	∞	∞	1088.50	1177.06	399.88	562.37
(4, 10)	0.2857	∞	∞	153.00	460.20	270.31	214.86	404.65
(2, 10)	0.1666	26.00	91.00	78.00	149.08	91.49	140.96	106.46
(100, 100)	0.5000	∞	∞	∞	413.00	709.04	1426.63	589.89
(80, 100)	0.4444	∞	∞	∞	2473.90	609.75	424.17	573.01
(60, 100)	0.3750	∞	∞	∞	757.85	642.55	655.36	725.14
(40, 100)	0.2857	∞	∞	442.91	618.83	258.22	620.79	418.48
(20, 100)	0.1666	34.57	191.71	242.62	270.86	406.05	174.27	297.27

presents the ARL₀s of the proposed Bayesian variance chart for various combinations of n, α₀, and β₀.

There are many infinite ARL₀s in Table 1, indicating that the proposed Bayesian variance chart does not release false alarm under an in-control process. However, there also appear many ARL₀s smaller than 370.4, indicating that the proposed chart frequently releases false alarm under the in-control process. The inconsistent results may be attributed to the discontinuity of the monitoring statistic.

3.1.2. The Out-of-Control ARL of the Shewhart-Type Bayesian Variance Chart

When the process is out-of-variance-control and parameters α₀ and β₀ are changed to α₁ and β₁, the chance that the proposed chart alarms a true signal is calculated by:

$$ALAR{M}_1=P\left(M_t\le LC{L}_M \mathrm{or} M_t\ge UC{L}_M|M_t~Beta-Binomial\left(n, {\alpha }_1, {\beta }_1\right)\right).$$

(18)

So, ARL₁ = 1/ALARM₁.

Table 2. The ARL₁s of the proposed Shewhart-type Bayesian variance chart.

Panel A:	When	(α0, β0) =	(5, 10)
(α1, β1)	E1(p)	n = 2	n = 3	n = 5	n = 10	n = 15	n = 20	n = 25
(6, 11)	0.3529	∞	∞	∞	212.26	423.95	711.36	401.99
(7, 10)	0.4118	∞	∞	∞	91.49	155.39	231.17	130.77
(6, 10)	0.3750	∞	∞	∞	141.98	257.23	399.88	226.12
(5, 10)a	0.3333	∞	∞	∞	240.62	468.98	765.58	433.45
(5, 9)	0.3571	∞	∞	∞	153.86	270.31	407.36	231.39
(5, 8)	0.3846	∞	∞	∞	96.21	152.44	212.41	121.84
Panel B:	When	(α0, β0) =	(3, 9)
(α1, β1)	E1(p)	n = 2	n = 3	n = 5	n = 10	n = 15	n = 20	n = 25
(4, 10)	0.2857	∞	∞	153.00	98.96	270.31	214.86	189.35
(5, 9)	0.3571	∞	∞	68.00	38.51	85.33	65.18	55.79
(4, 9)	0.3077	∞	∞	110.50	66.73	162.15	125.90	109.03
(3, 9) a	0.2500	∞	∞	208.00	136.39	368.68	292.32	256.96
(3, 8)	0.2727	∞	∞	143.00	86.90	206.94	160.26	138.47
(3, 7)	0.3000	∞	∞	95.33	53.99	113.11	86.08	73.45
Panel C:	When	(α0, β0) =	(46, 100)
(α1, β1)	E1(p)	n = 2	n = 3	n = 5	n = 10	n = 15	n = 20	n = 25
(70, 100)	0.4118	∞	∞	77.86	57.21	65.44	29.00	39.88
(60, 100)	0.3750	∞	∞	121.93	104.50	137.11	61.39	94.91
(50, 100)	0.3333	∞	∞	213.75	225.07	354.80	164.14	289.74
(46, 100) a	0.3151	∞	∞	279.22	325.57	562.03	265.77	477.10
(46, 90)	0.3382	∞	∞	196.80	199.24	301.99	138.61	237.21
(46, 80)	0.3651	∞	∞	135.10	118.39	157.31	70.67	109.53
(46, 70)	0.3966	∞	∞	89.95	68.11	79.34	35.32	48.98
Panel D:	When	(α0, β0) =	(32, 100)
(α1, β1)	E1(p)	n = 2	n = 3	n = 5	n = 10	n = 15	n = 20	n = 25
(70, 100)	0.4118	∞	13.97	77.86	14.37	8.20	6.06	8.87
(60, 100)	0.3750	∞	18.39	121.93	23.22	13.59	10.26	17.01
(50, 100)	0.3333	∞	25.96	213.75	43.25	26.64	21.05	41.21
(40, 100)	0.2857	∞	40.70	442.91	99.65	67.31	57.74	142.05
(32, 100) a	0.2424	∞	65.52	954.75	246.38	187.73	179.41	565.37
(32, 90)	0.2623	∞	51.83	647.82	154.85	110.15	98.86	269.28
(32, 80)	0.2857	∞	40.18	425.45	94.42	62.89	53.20	124.71
(32, 70)	0.3137	∞	30.43	268.81	55.67	34.92	28.00	56.30

^a denotes the in-control process.

presents the ARL₁s of the proposed Bayesian variance chart for various combinations of n, α₁, and β₁ under the in-control parameters (α₀, β₀) = (5, 10), (3, 9), (46,100), and (32, 100), respectively.

There are many infinite ARL₁s in Table 2, which indicates that the proposed Bayesian variance chart fails to correctly detect any shift of process variance under the out-of-control process. A larger ARL₁ shows that the proposed chart has poor detecting ability for a shift of process variance.

3.2. The Parameters of the Prior Distribution of p and Process Variance Are Unknown

When the in-control process variance, σ₀², is unknown, the following preliminary sample data can be used to estimate it. Assuming that

$$X_{t,1}, X_{t,2}, \dots , X_{t,2n},\hspace{1em}\hspace{1em}t=1, 2, \dots , T$$

(19)

are obtained from T sampling periods, each with 2n observations, then

$$S_t^2=\frac{{{\displaystyle \sum }}_{i=1}^{2n}{\left(X_{t,i}-{\overline{X}}_t\right)}^2}{2n-1} \mathrm{and} {\hat{\mathsf{\sigma }}}^2=\frac{{\sum }_{t=1}^T{S}_t^2}{T}.$$

(20)

On the other hand, when there is no information about the parameters of a prior distribution of p, a non-informative prior distribution, Beta(1, 1), is suggested initially. Then, the preliminary sample data can be used to obtain the posterior distribution of p:

$$p|n,T,m_1, \dots , m_T ~ Beta\left(1+{\sum }_{i=1}^T{m}_t, 1+nT{\sum }_{i=1}^T{m}_t\right),$$

where m_t represents the number of Y_j’s > ${\widehat{\sigma }}^2$ at the tth sampling period. Because a new statistic, M_T+t, follows from a binomial distribution with parameters n and p, the posterior predictive distribution of M_T+t is the beta-binomial distribution with parameters n, $\widehat{\alpha }$, and $\widehat{\beta }$, written as:

$$M_{T+t}~Beta-Binomial\left(n,\widehat{\alpha },\widehat{\beta }\right), \mathrm{where} \widehat{\alpha }=1+{\sum }_{t=1}^T{m}_t, \widehat{\beta }=1+nT-{\sum }_{t=1}^T{m}_t.$$

(21)

Thus, the in-control expected value of p is estimated with:

$$\widehat{E_0\left(p\right)}=\widehat{\alpha }/\left(\widehat{\alpha }+\widehat{\beta }\right).$$

(22)

The upper and lower control limits of M_T+t are obtained, respectively, as follows:

$$UC{L}_M=\frac{n\widehat{\alpha }}{\widehat{\alpha }+\widehat{\beta }}+3\sqrt{\frac{n\widehat{\alpha }\widehat{\beta }\left(\widehat{\alpha }+\widehat{\beta }+n\right)}{{\left(\widehat{\alpha }+\widehat{\beta }\right)}^2\left(\widehat{\alpha }+\widehat{\beta }+1\right)}} ,$$

(23)

$$LC{L}_M=\frac{n\widehat{\alpha }}{\widehat{\alpha }+\widehat{\beta }}-3\sqrt{\frac{n\widehat{\alpha }\widehat{\beta }\left(\widehat{\alpha }+\widehat{\beta }+n\right)}{{\left(\widehat{\alpha }+\widehat{\beta }\right)}^2\left(\widehat{\alpha }+\widehat{\beta }+1\right)}}.$$

(24)

The M_t for the T preliminary samples are plotted on the resulting Bayesian variance chart. If no points fall outside the control limits, then the process is regarded as the in-control state. Thus, the M_T+t for the subsequent monitoring samples are plotted in order. If either M_T+t ≥ UCL_M or M_T+t ≤ LCL_M, the process is deemed to show some out-of-variance-control signals. In the following discussions, we assume that the number of preliminary samples, T, is large enough so that ${\widehat{\sigma }}^2$ and $\frac{\widehat{\alpha }}{\widehat{\alpha }+\widehat{\beta }}$ are almost the same as the true values of σ₀² and E₀(p), respectively.

4. The Construction of the EWMA Variance Chart with the Bayesian Approach

Because observed statistic M_t is discrete and the beta-binomial distribution is asymmetric, the in-control ARLs of the proposed Shewhart-type Bayesian variance chart cannot always satisfy the required in-control ARLs, such as 370.4. The same reason results in the anomaly that the out-of-control ARLs of the proposed Shewhart-type Bayesian variance chart do not inversely change with n as they normally should.

The EWMA chart has been used to monitor small shifts of process variance quickly and effectively. Moreover, the characteristics of the moving average of observed statistics may ease the disadvantages of discreteness and asymmetry.

We define a new EWMA statistic as:

$$EWM{A}_t=\lambda M_t+\left(1-\lambda \right)EWM{A}_{t-1},\hspace{1em}\hspace{1em}0<\lambda \le 1,$$

(25)

where λ is the smoothing parameter. Because $E\left(M_t\right)=\frac{n\alpha }{\alpha +\beta }$ and M₀ is set to be the mean of M_t, the mean of EWMA_t is specified as:

$$E\left(EWM{A}_t\right)=\frac{n\alpha }{\alpha +\beta }\times \left(1-{\left(1-\lambda \right)}^t\right).$$

(26)

On the other hand, because $Var\left(M_t\right)=\frac{n\alpha \beta \left(\alpha +\beta +n\right)}{{\left(\alpha +\beta \right)}^2\left(\alpha +\beta +1\right)}$, the variance of EWMA_t is specified as:

$$Var\left(EWM{A}_t\right)=\frac{n\alpha \beta \left(\alpha +\beta +n\right)}{{\left(\alpha +\beta \right)}^2\left(\alpha +\beta +1\right)} \times \frac{\lambda }{2-\lambda } \times \left(1-{\left(1-\lambda \right)}^{2t}\right).$$

(27)

Thus, central limit theorem would ensure that the EWMA_t will asymptotically follow from a continuous symmetric distribution with [15]

$$E\left(EWM{A}_t\right)=\frac{n\alpha }{\alpha +\beta }$$

(28)

and

$$Var\left(EWM{A}_t\right)=\frac{n\alpha \beta \left(\alpha +\beta +n\right)}{{\left(\alpha +\beta \right)}^2\left(\alpha +\beta +1\right)} \times \frac{\lambda }{2-\lambda }.$$

(29)

4.1. Construction of the EWMA Variance Control Chart

Assuming that the in-control process variance is known to be σ₀² and the prior distribution of p is also known to be Beta(α₀, β₀), we can further construct a Bayesian EWMA variance chart with the following control limits:

$$C{L}_{EWMA}=\frac{n{\alpha }_0}{{\alpha }_0+{\beta }_0} ,$$

(30)

$$UC{L}_{EWMA}=\frac{n{\alpha }_0}{{\alpha }_0+{\beta }_0}+k_1\sqrt{\frac{n{\alpha }_0{\beta }_0\left({\alpha }_0+{\beta }_0+n\right)}{{\left({\alpha }_0+{\beta }_0\right)}^2\left({\alpha }_0+{\beta }_0+1\right)}\times \frac{\lambda }{2-\lambda }},$$

(31)

$$LC{L}_{EWMA}=\frac{n{\alpha }_0}{{\alpha }_0+{\beta }_0}-k_2\sqrt{\frac{n{\alpha }_0{\beta }_0\left({\alpha }_0+{\beta }_0+n\right)}{{\left({\alpha }_0+{\beta }_0\right)}^2\left({\alpha }_0+{\beta }_0+1\right)}\times \frac{\lambda }{2-\lambda }},$$

(32)

where (k₁, k₂) are coefficients of the control limits. The adjustment allowing unequal coefficients of the control limits conforms to the asymmetry of beta-binomial distribution. Next, we will discuss how to choose suitable coefficients (k₁, k₂) to satisfy a required in-control average run length.

The ARLs are again used to measure the performance of the proposed Bayesian EWMA variance chart. We evaluate the ARLs of the proposed Bayesian EWMA variance chart by a Monte Carlo simulation approach. In order to ensure that the in-control ARLs are at least 370.4, the coefficients of the control limits (k₁, k₂) are chosen with a procedure as follows.

1.

Given α₀, β₀, n, and λ, then UCL_EWMA and LCL_EWMA can be expressed as the function of k₁ and k₂ by Equations (31) and (32), respectively;

2.

Let EWMA₀ be equal to the mean of EWMA, i.e., nα₀/(α₀+β₀);

3.

Simulate random numbers M_t from Beta-Binomial(n,α₀,β₀) and compute EWMA_t by Equation (25) until EWMA_t > UCL_EWMA, then record the run length;

4.

Repeat step 3 1000 times and obtain the average run length, ARL(k₁);

5.

Determine the least k₁ to make sure the ARL(k₁) is larger than 740.8;

6.

Given k₁, simulate random numbers M_t from Beta-Binomial(n,α₀,β₀) and compute EWMA_t using Equation (25) until EWMA_t > UCL_EWMA or EWMA_t < LCL_EWMA, then record the run length;

7.

Repeat step 6 1000 times and obtain the average run length, ARL(k₂);

8.

Determine the least k₂ to make sure the ARL(k₂) is larger than 370.4.

Table 3. The coefficients of the control limits (k₁, k₂) of the proposed Bayesian exponential weighted moving average (EWMA) variance chart (λ = 0.05, ARL₀ = 370.4).

(α0, β0)	E0(p)	n = 2	n = 3	n = 5	n = 10	n = 15	n = 20	n = 25
(1, 1)	0.5000	(2.95, 3.00)	(2.94, 3.03)	(2.92, 3.01)	(2.94, 2.98)	(2.93, 2.99)	(2.94, 2.98)	(2.94, 2.99)
(1, 2)	0.3333	(3.12, 2.75)	(3.06, 2.79)	(3.08, 2.82)	(3.05, 2.83)	(3.05, 2.80)	(3.05, 2.80)	(3.06, 2.81)
(1, 3)	0.2500	(3.16, 2.66)	(3.20, 2.66)	(3.17, 2.69)	(3.12, 2.71)	(3.11, 2.73)	(3.13, 2.71)	(3.13, 2.74)
(1, 4)	0.2000	(3.25, 2.62)	(3.24, 2.57)	(3.26, 2.64)	(3.24, 2.69)	(3.21, 2.68)	(3.20, 2.71)	(3.19, 2.66)
(1, 5)	0.1667	(3.26, 2.53)	(3.25, 2.56)	(3.35, 2.58)	(3.24, 2.62)	(3.26, 2.64)	(3.24, 2.64)	(3.22, 2.66)
(2, 2)	0.5000	(2.90, 3.07)	(2.96, 3.04)	(2.95, 3.03)	(2.92, 3.04)	(2.93, 3.03)	(2.94, 3.03)	(2.94, 3.04)
(2, 3)	0.4000	(2.96, 2.84)	(3.01, 2.93)	(2.97, 2.92)	(3.00, 2.93)	(2.98, 2.91)	(3.00, 2.91)	(2.99, 2.92)
(2, 4)	0.3333	(3.19, 2.76)	(3.10, 2.81)	(3.08, 2.86)	(3.04, 2.83)	(3.03, 2.85)	(3.06, 2.85)	(3.04, 2.85)
(2, 5)	0.2857	(3.16, 2.65)	(3.09, 2.81)	(3.06, 2.84)	(3.08, 2.84)	(3.08, 2.81)	(3.05, 2.83)	(3.07, 2.81)
(2, 6)	0.2500	(3.15, 2.65)	(3.21, 2.66)	(3.09, 2.80)	(3.06, 2.76)	(3.12, 2.75)	(3.10, 2.81)	(3.11, 2.82)
(3, 3)	0.5000	(2.91, 3.05)	(2.95, 3.02)	(3.02, 3.01)	(2.90, 3.03)	(2.92, 3.06)	(2.94, 3.05)	(2.93, 3.05)
(3, 4)	0.4286	(2.93, 2.92)	(2.97, 3.00)	(2.97, 2.95)	(2.98, 2.94)	(2.96, 2.99)	(2.97, 2.97)	(2.96, 2.98)
(3, 5)	0.3750	(3.01, 2.85)	(3.08, 2.93)	(3.01, 2.86)	(2.99, 2.95)	(3.00, 2.94)	(3.00, 2.92)	(3.01, 2.93)
(3, 6)	0.3333	(3.16, 2.76)	(3.05, 2.83)	(3.08, 2.89)	(3.04, 2.87)	(3.06, 2.88)	(3.01, 2.89)	(3.05, 2.90)
(3, 9)	0.2500	(3.14, 2.65)	(3.25, 2.73)	(3.05, 2.83)	(3.08, 2.83)	(3.10, 2.79)	(3.08, 2.81)	(3.09, 2.83)
(10, 10)	0.5000	(2.87, 3.07)	(3.03, 3.01)	(3.02, 3.09)	(2.96, 3.07)	(2.94, 3.05)	(2.94, 3.08)	(2.93, 3.11)
(8, 10)	0.4444	(2.89, 2.97)	(2.97, 3.05)	(2.97, 3.01)	(2.98, 3.02)	(2.95, 3.05)	(2.96, 3.02)	(2.97, 3.03)
(6, 10)	0.3750	(3.05, 2.83)	(3.05, 2.93)	(3.01, 2.87)	(3.01, 2.95)	(3.02, 2.98)	(2.99, 2.99)	(3.01, 2.98)
(4, 10)	0.2857	(3.21, 2.71)	(3.07, 2.81)	(3.07, 2.85)	(3.07, 2.85)	(3.03, 2.87)	(3.03, 2.89)	(3.07, 2.90)
(2, 10)	0.1667	(3.21, 2.58)	(3.18, 2.63)	(3.26, 2.63)	(3.21, 2.74)	(3.11, 2.74)	(3.18, 2.76)	(3.18, 2.72)
(100, 100)	0.5000	(2.87, 3.07)	(3.04, 2.97)	(3.02, 3.06)	(2.94, 3.07)	(2.94, 3.04)	(2.91, 3.09)	(2.96, 3.07)
(80, 100)	0.4444	(2.87, 2.98)	(2.95, 3.02)	(2.94, 3.07)	(2.99, 3.09)	(2.95, 3.07)	(2.94, 3.07)	(2.93, 3.07)
(60, 100)	0.3750	(3.03, 2.90)	(2.95, 2.96)	(3.02, 2.90)	(3.01, 3.00)	(2.97, 3.00)	(3.01, 3.02)	(2.99, 3.01)
(40, 100)	0.2857	(3.27, 2.79)	(3.10, 2.81)	(3.10, 2.94)	(3.02, 2.90)	(3.00, 2.98)	(3.00, 3.02)	(2.97, 2.99)
(20, 100)	0.1667	(3.26, 2.59)	(3.15, 2.61)	(3.17, 2.73)	(3.12, 2.82)	(3.12, 2.85)	(3.03, 2.90)	(3.06, 2.93)

shows the coefficients of the control limits (k₁, k₂) of the proposed Bayesian EWMA variance chart with λ = 0.05, where the ARL₀s are slightly beyond 370.4, for various different n, α₀, and β₀.

The results in Table 3 show that k₁ and k₂ are close, while E₀(p) = 0.5; however, k₁ changes inversely with E₀(p) and k₂ changes positively with E₀(p). The in-control area gets bigger due to the increasing skewness of the beta-binomial distribution.

4.2. Evaluation of the EWMA Variance Control Chart

To realize the performance of the proposed Bayesian EWMA variance chart, the out-of-control ARL will be calculated. First of all, the coefficients of the control limits, k₁ and k₂, are determined from the procedure described in the previous subsection. When the process is out-of-variance-control and the parameters of the prior distribution of p are changed to α₁ and β₁, the ARL₁s of the proposed Bayesian EWMA variance chart are evaluated by the Monte Carlo simulation approach. The procedure of evaluation is presented as follows:

• Given α₀, β₀, n, λ, k₁, and k₂, then UCL_EWMA and LCL_EWMA can be calculated by Equations (31) and (32), respectively;
• Let EWMA₀ be equal to the mean of EWMA, i.e., nα₀/(α₀+β₀);
• Simulate random numbers M_t from Beta-Binomial(n,α₁,β₁) and compute EWMA_t by Equation (25) until EWMA_t > UCL_EWMA or EWMA_t < LCL_EWMA, then record the run length;
• Repeat step 3 1000 times and obtain the average run length, ARL₁.

Table 4. The ARL₁s of the proposed Bayesian EWMA variance chart (λ = 0.05).

Panel A:	When (α0, β0) = (5, 10) and E0(p) = 0.3333
(α1, β1)	E1(p)	n = 2 (3.21, 2.76) b	n = 3 (3.02, 2.91)	n = 5 (3.04, 2.86)	n = 10 (3.02, 2.95)	n = 15 (3.01, 2.96)	n = 20 (3.01, 2.94)	n = 25 (3.03, 2.95)
(10, 10)	0.5000	2.32	1.40	1.11	1.01	1.00	1.00	1.00
(6, 10)	0.3750	101.42	49.93	33.33	18.01	12.90	10.99	9.86
(5, 10) a	0.3333	378.79	371.75	373.13	374.53	380.23	383.14	378.79
(5, 9)	0.3571	220.75	130.72	96.90	57.67	46.69	38.52	37.84
(5, 5)	0.5000	2.29	1.41	1.13	1.01	1.00	1.00	1.00
(1, 10)	0.0909	1.01	1.00	1.00	1.00	1.00	1.00	1.00
Panel B:	When (α0, β0) = (3, 9) and E0(p) = 0.2500
(α1, β1)	E1(p)	n = 2 (3.14, 2.65) b	n = 3 (3.25, 2.73)	n = 5 (3.05, 2.83)	n = 10 (3.08, 2.83)	n = 15 (3.10, 2.79)	n = 20 (3.08, 2.81)	n = 25 (3.09, 2.83)
(9, 9)	0.5000	1.07	1.01	1.00	1.00	1.00	1.00	1.00
(4, 9)	0.3077	29.39	25.41	11.20	6.56	4.98	4.11	3.73
(3, 9) a	0.2500	377.36	383.14	370.37	378.79	373.13	375.94	375.94
(3, 8)	0.2727	139.28	174.83	83.40	57.84	49.48	41.67	39.62
(3, 3)	0.5000	1.08	1.02	1.00	1.00	1.00	1.00	1.00
(1, 9)	0.1000	1.43	1.15	1.03	1.00	1.00	1.00	1.00
Panel C:	When (α0, β0) = (46, 100) and E0(p) = 0.3151
(α1, β1)	E1(p)	n = 2 (3.16, 2.75) b	n = 3 (2.97, 2.89)	n = 5 (3.08, 2.93)	n = 10 (3.00, 2.94)	n = 15 (3.03, 2.98)	n = 20 (3.00, 2.99)	n = 25 (3.00, 3.01)
(100, 100)	0.5000	1.60	1.13	1.01	1.00	1.00	1.00	1.00
(60, 100)	0.3750	35.56	15.69	9.46	3.29	2.09	1.55	1.32
(46, 100) a	0.3151	374.53	370.37	384.62	377.36	380.23	375.94	378.79
(46, 80)	0.3651	54.05	25.03	16.01	5.52	3.34	2.31	1.85
(46, 46)	0.5000	1.60	1.13	1.01	1.00	1.00	1.00	1.00
(10, 100)	0.0909	1.01	1.00	1.00	1.00	1.00	1.00	1.00
Panel D:	When (α0, β0) = (32, 100) and E0(p) = 0.2424
(α1, β1)	E1(p)	n = 2 (3.10, 2.65) b	n = 3 (3.19, 2.74)	n = 5 (3.02, 2.93)	n = 10 (3.09, 2.92)	n = 15 (3.05, 2.94)	n = 20 (3.03, 2.98)	n = 25 (2.98, 3.00)
(100, 100)	0.5000	1.03	1.00	1.00	1.00	1.00	1.00	1.00
(50, 100)	0.3333	7.51	5.08	2.18	1.24	1.05	1.01	1.00
(32, 100) a	0.2424	377.36	380.23	373.13	375.94	371.75	387.60	377.36
(32, 80)	0.2857	46.82	38.49	16.07	7.35	4.17	2.85	2.15
(32, 32)	0.5000	1.03	1.00	1.00	1.00	1.00	1.00	1.00
(10,100)	0.0909	1.30	1.07	1.00	1.00	1.00	1.00	1.00

^a denotes the in-control process; ^b denotes the coefficients of the control limits (k₁, k₂).

shows the ARL₁s of the proposed Bayesian EWMA variance chart with the combination of k₁ and k₂ for various different n, α₁, and β₁ under the in-control parameters (α₀, β₀) = (5, 10), (3, 9), (46,100), (32, 100), and smoothing parameter λ = 0.05.

The results in Table 4 show that the ARL₁ decreases quickly as E₁(p) shifts away from E₀(p) in each set of sample size, documenting that the proposed Bayesian EWMA variance chart possesses excellent detecting power for any shift of the process variance.

4.3. Performance Comparison of the EWMA Variance Control Chart

In general, the EWMA type control chart performs better than the Shewhart-type chart for small shifts. The single sampling EWMA variance (SS EWMA-V) chart has proved better performance than the NLE, CEW, NP-M, and IRC charts from previous studies [23,34,35]. To demonstrate that the proposed Bayesian EWMA variance chart outperforms the existing charts, the SS EWMA-V chart is considered for comparison.

Table 5. The out-of-control ARL comparison of the proposed Bayesian EWMA variance chart and the single sampling EWMA variance (SS EWMA-V) chart.

p0 = E0(p) = 0.3333			p0 = E0(p) = 0.3151			p0 = E0(p) = 0.2500			p0 = E0(p) = 0.2424
p1/ E1(p)	Bayesian EWMA Chart	SS EWMA- V Chart	p1/ E1(p)	Bayesian EWMA Chart	SS EWMA- V Chart	p1/ E1(p)	Bayesian EWMA Chart	SS EWMA- V Chart	p1/ E1(p)	Bayesian EWMA Chart	SS EWMA- V Chart
0.5	1.11	13.92	0.5	1.01	12.41	0.5	1.00	8.12	0.5	1.00	7.76
0.375	33.33	85.92	0.3750	9.46	57.97	0.3077	11.20	56.92	0.3333	2.18	28.96
0.3571	96.90	158.03	0.3651	16.01	74.92	0.2727	83.40	187.40	0.2857	16.07	84.13
0.3333 a	373.13	373.08	0.3151 a	384.62	369.96	0.2500 a	370.37	388.39	0.2424 a	373.13	370.15
0.0909	1.00	9.21	0.0909	1.00	9.45	0.1000	1.03	14.56	0.0909	1.00	13.84

Notes: sample size is 10 (n = 5); smoothing parameter is 0.05 (λ = 0.05); ^a Denotes an in-control process.

presents the out-of-control average run lengths of the proposed Bayesian EWMA variance chart and the SS EWMA-V chart with a sample size of 10 (n = 5) and smoothing parameter 0.05 (λ = 0.05). The results of Table 5 show that, while E₀(p) and E₁(p) are both equal to p₀ and p₁, respectively, the ARL₁s of the Bayesian EWMA variance chart are all smaller than those of the SS EWMA-V chart, showing that our proposed chart could detect a shift of variance more quickly than the SS EWMA-V chart.

Hence, we conclude that the out-of-control detection performance of the proposed Bayesian EWMA variance chart is always better than those of the SS EWMA-V chart, no matter whether the shift from p₀/E₀(p) to p₁/E₁(p) is small, medium, or large.

5. Process Simulations

The out-of-control variance detection performance of our proposed Bayesian EWMA variance chart is further evaluated under the normal process, exponential process, and the mixed process of normal and exponential distribution. The process with mixed distribution demonstrates a situation where the process sometimes is normally distributed and sometimes is exponentially distributed, both with the same variance but with different skewness.

For any normal distribution with standard deviation σ, while σ shifts to dσ, the probability that statistic Y_j is larger than σ² can be computed as:

$$\begin{array}{cc}{p}_N\hfill & =P\left(Y_j>{\sigma }^2|X_{2j}, X_{2j-1}~Normal\left(\mu , {(d\sigma )}^2\right)\right)\hfill \\ \hfill & =P\left(\frac{{\left(X_{2j}-X_{2j-1}\right)}^2}{2}>{\sigma }^2|X_{2j}, X_{2j-1}~Normal\left(\mu , {(d\sigma )}^2\right)\right)\hfill \\ \hfill & =2\times \left(1-\mathsf{\Phi }\left(\frac{1}{d}\right)\right),\hfill \end{array}$$

(33)

where Φ(.) represents the cumulative probability function of a standard normal variable and d represents a shift factor. Panel A of

Table 6. The relationship between the shift factor d and p.

Panel A:	Normal	Process		Panel B:	Exponential	Process
da	p	d	p	d	p	d	p
1	0.3173			1	0.2431
1.1	0.3633	0.9	0.2665	1.1	0.2765	0.9	0.2078
1.2	0.4047	0.8	0.2113	1.2	0.3077	0.8	0.1707
1.3	0.4418	0.7	0.1531	1.3	0.3369	0.7	0.1326
1.4	0.4751	0.6	0.0956	1.4	0.3642	0.6	0.0947
1.5	0.5050	0.5	0.0455	1.5	0.3895	0.5	0.0591
1.6	0.5320	0.4	0.0124	1.6	0.4132	0.4	0.0291
1.7	0.5564	0.3	0.0009	1.7	0.4352	0.3	0.0090
1.8	0.5785	0.2	0.0000	1.8	0.4558	0.2	0.0008
1.9	0.5987			1.9	0.4751
2.0	0.6171			2.0	0.4931

^ad = 1 denotes an in-control process, d > 1 denotes an upward shift of variance, d < 1 denotes a downward shift of variance.

presents the relationship between d and p in the normal process to look at how p varies with the shifts of process variance.

For any exponential distribution with standard deviation σ, while σ shifts to dσ, the probability that statistic Y_j is larger than σ² can be computed as:

$$\begin{array}{cc}{p}_E\hfill & =P\left(Y_j>{\sigma }^2|X_{2j}, X_{2j-1}~Exponential\left(\frac{1}{d\sigma }\right)\right)\hfill \\ \hfill & =P\left(\frac{{\left(X_{2j}-X_{2j-1}\right)}^2}{2}>{\sigma }^2|X_{2j}, X_{2j-1}~Exponential\left(\frac{1}{d\sigma }\right)\right)\hfill \\ \hfill & =e^{-\sqrt{2}/d},\hfill \end{array}$$

(34)

where d represents a shift factor. Panel B of Table 6 presents the relationship between d and p in the exponential process to look at how p varies with the shifts of process variance. Both processes in Table 6 indicate that p is positively associated with d.

Suppose that the mixed process is an equally-weighted mixture of a normal distribution and exponential distribution with same variance σ². Thus, for the mixed distribution with standard deviation σ, while σ shifts to dσ, the probability that statistic Yj is larger than σ² is expressed as:

$$p_M=\frac{p_N+p_E}{2}.$$

(35)

Because probability p_N is the same for any normal distribution, the in-control normal process is set a standard normal distribution in the following simulations, without loss of generality. Similarly, probability p_E is the same for any exponential distribution, so the in-control exponential process is set an exponential distribution with standard deviation 1, without loss of generality.

The following simulation procedure will be applied to three processes to evaluate the performance of the proposed Bayesian EWMA variance chart:

• Set the process variance σ² equal to 1;
• α₀ and β₀ are chosen such that E₀(p) is as close to 0.3173 as possible in the normal process (0.2431 in the exponential process or 0.2802 in the mixed process);
• k₁ and k₂ are determined by the procedure described in Section 4.1;
• Given α₀, β₀, n, λ, k₁, and k₂, then UCL_EWMA and LCL_EWMA can be calculated by Equations (31) and (32), respectively;
• Let EWMA₀ be equal to the mean of EWMA, i.e., nα₀/(α₀+β₀);
• Simulate random samples of size 2n, X_t1, X_t2, …, X_t,2n from Normal(0, d²) (Exponential(1/d), or equally-weighted mixture of Noraml(0, d²) and Exponential(1/d)), compute Y_t1, Y_t2, …, Y_tn by Equation (1), obtain the statistic M_t by counting the number of instances in which Y_tj > 1, and compute EWMA_t by Equation (25) until EWMA_t > UCL_EWMA or EWMA_t < LCL_EWMA, then record the run length;
• Repeat step 6 1000 times and obtain the average run length, ARL₁.

While simulating random samples of size 2n from the equally-weighted mixed process, we assume that observations within one sample come from the same distribution; that is, the skewness of distribution varies only between samples.

Table 7. The simulation results of the ARL₁s of the proposed Bayesian EWMA variance chart under three different processes (λ = 0.05).

Panel A:	Normal process: E0(p) = 0.3173 and (α0, β0) = (317, 683)
d	E1(p)	n = 2 (3.16, 2.77) b 381.68 c	n = 10 (2.98, 2.94) 375.94	n = 25 (2.96, 3.05) 375.94
0.6	0.0956	1.02	1.00	1.00
0.7	0.1531	1.46	1.00	1.00
0.8	0.2113	5.08	1.05	1.00
0.9	0.2665	44.38	5.03	1.60
1 a	0.3173	394.01	368.32	376.08
1.1	0.3633	65.10	6.18	1.84
1.2	0.4047	12.57	1.33	1.00
1.3	0.4418	4.35	1.01	1.00
1.4	0.4751	2.28	1.00	1.00
1.5	0.5050	1.55	1.00	1.00
Panel B:	Exponential process: E0(p) = 0.2431 and (α0, β0) = (243, 757)
d	E1(p)	n = 2 (3.10, 2.65) b 374.53 c	n = 10 (3.05, 2.93) 370.37	n = 25 (2.99, 3.00) 377.36
0.6	0.0947	1.36	1.00	1.00
0.7	0.1326	2.98	1.00	1.00
0.8	0.1707	11.36	1.48	1.01
0.9	0.2078	70.27	12.60	3.18
1 a	0.2431	283.85	369.69	388.50
1.1	0.2765	75.05	13.47	3.51
1.2	0.3077	18.68	2.15	1.06
1.3	0.3369	6.87	1.14	1.00
1.4	0.3642	3.43	1.01	1.00
1.5	0.3895	2.15	1.00	1.00
Panel C:	Mixed process: E0(p) = 0.2802 and (α0, β0) = (280, 720)
d	E1(p)	n = 2 (3.26, 2.74) b 383.14 c	n = 10 (3.04, 2.95) 373.13	n = 25 (2.98, 3.04) 377.36
0.6	0.0952	1.10	1.00	1.00
0.7	0.1429	2.00	1.00	1.00
0.8	0.1910	7.73	1.17	1.00
0.9	0.2372	56.60	7.56	2.16
1 a	0.2802	413.39	310.56	200.84
1.1	0.3199	97.59	8.67	2.41
1.2	0.3562	19.16	1.65	1.03
1.3	0.3893	6.46	1.06	1.00
1.4	0.4196	3.15	1.00	1.00
1.5	0.4473	1.98	1.00	1.00

^a denotes the in-control process; ^b denotes the coefficients of the control limits (k₁, k₂); ^c denotes the preset ARL₀.

shows the simulation results of the ARL₁s of the proposed Bayesian EWMA variance chart under a normal process, exponential process, and the equally-weighted mixed process of normal and exponential distributions.

Panel A of Table 7 indicates that the simulated ARL₀s are around the preset ARL₀s in a normal process. Panel B of Table 7 indicates that the simulated ARL₀s, except for n = 2, are around the preset ARL₀s in an exponential process. However, Panel C of Table 7 indicates that the simulated ARL₀s are significantly different from the ARL₀s in the mixed process. The less robust ARL₀s in the mixed process result from the possibility that the prior distribution of p may be bimodal. Nevertheless, the ARL₁s in Table 7 reversely (positively) change with d when d > 1 (<1) and quickly reduce to 1 as d departs from 1, demonstrating that the proposed Bayesian EWMA variance chart possesses excellent detecting power for any upward or downward shift of the process variance.

6. An Example for Demonstration

A banking example from Yang et al. [21] is applied to illustrate the proposed Bayesian EWMA variance chart. The banking industry regards service time as an important quality characteristic. Under the demand for standardization and uniformity of service time, a control chart will be constructed to monitor the service time and detect whether the variance is offset.

Service time usually comes from a complex service process, whose distribution may not be normal. Sometimes, the distribution of the service time is time-varying even under the in-control condition. Because we know little about the nature of the service process, the proposed Bayesian EWMA variance is applicable to the banking case.

In order to measure the efficiency in the service system of a bank branch, the in-control sampling service time was measured from 10 counters every day for 15 days. So, 15 samples of size 2n = 10 were collected. A histogram shows that these samples are right-skewed.

Table 8. The service time from 10 counters in a bank branch.

t	X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	S2	M	EWMA
1	0.88	0.78	5.06	5.45	2.93	6.11	11.59	1.2	0.89	3.21	11.59	1	1.4688
2	3.82	13.4	5.16	3.2	32.27	3.68	3.14	1.58	2.72	7.71	86.35	2	1.4954
3	1.4	3.89	10.88	30.85	0.54	8.4	5.1	2.63	9.17	3.94	77.86	2	1.5206
4	16.8	8.77	8.36	3.55	7.76	1.81	1.11	5.91	8.26	7.19	19.77	1	1.4946
5	0.24	9.57	0.66	1.15	2.34	0.57	8.94	5.54	11.69	6.58	18.47	1	1.4699
6	4.21	8.73	11.44	2.89	19.49	1.2	8.01	6.19	7.48	0.07	31.88	2	1.4964
7	15.08	7.43	4.31	6.14	10.37	2.33	1.97	1.08	4.27	14.08	24.85	2	1.5215
8	13.89	0.3	3.21	11.32	9.9	4.39	10.5	1.7	10.74	1.46	25.00	4	1.6455
9	0.03	12.76	2.41	7.41	1.67	3.7	4.31	2.45	3.57	3.33	12.78	1	1.6132
10	12.89	17.96	2.78	3.21	1.12	12.61	4.23	6.18	2.33	6.92	31.47	1	1.5825
11	7.71	1.05	1.11	0.22	3.53	0.81	0.41	3.73	0.08	2.55	5.62	0	1.5034
12	5.81	6.29	3.46	2.66	4.02	10.95	1.59	5.58	0.55	4.1	8.51	0	1.4282
13	2.89	1.61	1.3	2.58	18.65	10.77	18.23	3.13	3.38	6.34	44.71	2	1.4568
14	1.36	1.92	0.12	11.08	8.85	3.99	4.32	1.71	1.77	1.94	12.63	1	1.4340
15	21.52	0.63	8.54	3.37	6.94	3.44	3.37	6.37	1.28	12.83	39.96	2	1.4623

shows the service time from 10 counters in a bank branch. The variance of service time is estimated with ${\widehat{\sigma }}^2$ = 30.0969 through Equation (20). The prior distribution of p is Beta(1, 1). After incorporating the information of 15 preliminary samples, we get $\hat{\alpha }=1+\sum _{t=1}^T{m}_t$ = 23 and $\widehat{\beta }=1+nT-\sum _{t=1}^T{m}_t$ = 54 by Equation (21) as well as $\widehat{E_0\left(p\right)}=23/77$ = 0.2987 by Equation (22). Past experience has confirmed that 30.0969 is very close to the true value of in-control process variance, σ₀², and 0.2987 is also very close to the true value of the in-control mean of p, E₀(p).

When the smoothing parameter λ is 0.05, referring to the procedure described in Section 4.1, we obtain k₁ = 3.07 and k₂ = 2.86. Thus, the control limits are calculated as UCL_EWMA = 2.0094 and LCL_EWMA = 1.0129 through Equations (31) and (32). From Equation (30), we preset EWMA₀ = 1.4935, then EWMA_t is computed in order by Equation (25) and presented in Table 8.

Figure 2 shows the proposed Bayesian EWMA variance chart. The observed statistics EWMA_ts are plotted in order. Because all the EWMA_ts fall into the region between control limits, the chart reveals that the process is in control.

In order to illustrate the detecting power for the variance change, ten new samples of size 10 were collected from a new automatic service system of the bank branch. Installing an automatic service system makes more consistent service time. The process variance will be reduced significantly.

Table 9. The new service time from 10 counters in a bank branch.

t	X1	X2	X3	X4	X5	X6	X7	X8	X9	X10	S2	M	EWMA
1	3.54	0.01	1.33	7.27	5.52	0.09	1.84	1.04	2.91	0.63	5.83	0	0.8602
2	0.86	1.61	1.15	0.96	0.54	3.05	4.11	0.63	2.37	0.05	1.62	0	0.8172
3	1.45	0.19	4.18	0.18	0.02	0.7	0.8	0.97	3.6	2.94	2.30	0	0.7764
4	1.37	0.14	1.54	1.58	0.45	6.01	4.59	1.74	3.92	4.82	4.15	0	0.7375
5	3	2.46	0.06	1.8	3.25	2.13	2.22	1.37	2.13	0.25	1.10	0	0.7007
6	1.59	3.88	0.39	0.54	1.58	1.7	0.68	1.25	6.83	0.31	4.11	0	0.6656
7	5.01	1.85	3.1	1	0.09	1.16	2.69	2.79	1.84	2.62	1.85	0	0.6324
8	4.96	0.55	1.43	4.12	4.06	1.42	1.43	0.86	0.67	0.13	3.02	0	0.6007
9	1.08	0.65	0.91	0.88	2.02	2.88	1.76	2.87	1.97	0.62	0.75	0	0.5707
10	4.56	0.44	5.61	2.79	1.73	2.46	0.53	1.73	7.02	2.13	4.70	0	0.5422

lists the service time of the ten new out-of-variance-control samples, M_t, and EWMA_t.

Note that the out-of-control process variance is estimated with ${\widehat{\sigma }}^2$ = 2.9437, showing a significant downward shift in the process variance. Hence, the observed M_ts are all 0 and the observed EWMA_ts decrease with t.

We then plotted the ten new observed statistics EWMA_ts in order on the proposed control chart, see Figure 3. The proposed Bayesian EWMA variance chart detected out-of-control signals from the first sample onward (samples 1–10). That is, our chart can quickly detect an out-of-control signal when the variability of the new service time is significantly reduced because of the improved new automatic service system.

7. Conclusions

In this paper, we propose a new control chart, the Bayesian EWMA variance chart, to monitor changes in the process variance when the distribution of a critical quality characteristic is non-normal, unknown, or skewness-time-varying. The average run lengths (ARLs) of the proposed chart are calculated and show that the chart performs well. When compared with the well-performing SS EWMA-V chart, the out-of-control detection performance of the proposed Bayesian EWMA variance chart is always better than that of the SS EWMA-V chart, no matter whether the shift is small, medium, or large. The simulation studies for a normal process, exponential process, and the mixed process of normal and exponential distribution prove that our chart can quickly detect a small, medium, or large shift of process variance. Moreover, a numerical example of service time is used to illustrate the application of the proposed Bayesian EWMA variance chart and confirm the performance of process control.

However, when monitoring the mixed process, the in-control ARLs seem to be biased due to the occurrence of the bimodal random behavior of p, which will limit the use of the proposed chart, especially in situations that have a large dispersion of skewness. A future study could consider multimodal prior distribution to catch the random behavior of p resulting from the skewness-time-varying process. Moreover, this study does not test or compare the effects of different weights under a mixed distribution. This may be a good issue for further studies.