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A NEW BAYESIAN CONTROL CHART FOR PROCESS MEAN USING EMPIRICAL BAYES ESTIMATES
Souradeep Das1 and Sudhansu S. Maiti2
•
department of Statistics, Charuchandra College
Kolkata-700029, West Bengal, India 2Department of Statistics, Visva-Bharati University
Santiniketan-731 235, West Bengal, India dassouradeep6@gmail.com, dssml@rediffmail.com,
Abstract
This article develops a new control chart for the mean using empirical Bayes estimates. We assume that the quality characteristic of the proposed control chart follows a normal distribution with unknown mean and variance. Both the parameters have known prior probability distributions. In practice, the parameters of priors are unknown and are estimated using the empirical Bayes approach. For the performance assessment of the new control chart, the Average Run Length (ARL) procedure is used while the process is in control and out of control. A real-life example is also considered to evaluate the performance of the proposed control chart.
Keywords: Average Run Length, Empirical Bayes, Mean Chart, Posterior, Statistical Process Control.
l. Introduction
Statistical Process Control (SPC) is a popular methodology for monitoring and assessing the quality of a manufacturing process. The main objective of SPC is to minimize the process variability. A control chart is the main technique SPC uses to measure whether a manufacturing process is in control. Dr. Walter Shewhart first proposed the control chart technique in the 1920s. If the quality characteristic under study is quantifiable, we use variable control charts like XX, R, and S charts, etc. For these control charts, it is assumed that the quality characteristic follows a normal distribution. Over the years, researchers have developed control charts for means by considering different aspects. [5] proposed a XX chart when the quality characteristic follows a skewed distribution. [9] introduced a new XX chart by considering variable sample size and sampling intervals, which can detect the shift in the process mean in less time than a traditional XX chart. [8] gave an idea of the Max chart by combining the XX chart and S chart. [18] proposed a new control chart for mean based on variable and attribute inspections.
The Bayesian approach has recently become very popular among researchers for constructing control charts. Using empirical Bayes, [ll] developed a multivariate process control chart. [19] compared the effectiveness of different mean charts under the Bayesian approach. [13] have constructed a new control chart for the coefficient of variation using prior information when the mean is variable, and the variance is the function of the mean. [17] designed a two-sided XX control chart for mean. [4] developed a new control chart for mean using posterior distribution. [10] measured the performance of a Bayesian Control Chart using empirical Bayes based on
Weibull data. [2] proposed a mean control chart using a uniform prior. [12] used Empirical Bayes methods based on loss functions for a sequential sampling plan. [6] used the Bayesian model for constructing predictive control charts. [1] designed a Bayesian Shewhart-type control chart for the Maxwell distributed process.
The entire article is arranged in the following way. Section 2 discusses the Shewhart X chart. In section 3, a discussion is made on the posterior mean control chart. Section 4 briefly describes the empirical Bayes method. In section 5, we explain the construction of the new control chart for mean using empirical Bayes estimates. In section 6, the performance of the proposed control chart is evaluated concerning the Average Run Length values. In section 7, a real-life dataset is taken to analyze the performance of the proposed control chart for mean. In the last section (section 8), concluding remarks are given.
2. X-bar Control Chart
Let X1,X2,...,X n be n observations of a quality characteristic X following a normal distribution with mean, p and variance, a2 of a manufacturing process. Then, according to W. Shewhart, the 3-sigma control limits of XX chart are
UCL = p + 3 -—= Vn
LCL = u — 3 —-= Vn
3. Posterior Control Chart for Mean
[4] proposed a new posterior XX control chart for process mean. Suppose X1, X2,..., Xn be n observations of a quality characteristic X. It is assumed that X/s are independently and identically distributed normal variables with mean p and variance a2(known). Here, the process average p has normal prior with known parameters.Then Xj's ~ N(p, a2) and p ~ N(9, A2), where 9 and A are known. So, the posterior mean ao = xZo + 9(1 — Zo) and the posterior variance is po = On^
where Zo = n"2+a2. Hence, the three-sigma control limits of the posterior control chart for the
mean are
UCL = xZo + 9(1 — Zo) + 3 —n VTo CL = xZo + 9(1 — Zo) LCL = xZo + 9(1 — Zo) — 3 —n VTo
4. Empirical Bayes method
In the Bayesian method, the probability distribution function's unknown parameters are considered the random variables. Suppose X 1, X 2, . . . , Xn are n observations from f(9). Here, the parameter 9 has some prior information. , 9 has the prior distribution n(9\w), where &> is the hyperparameter. The Bayes' theorem states that the posterior distribution of 9 can be expressed as proportional multiplication of the likelihood L(9) and the prior distribution n(9|w). Symbolically,
h(9\x) = «L(9)n(9M
The Bayesian method is different from the frequentist method. In the parametric empirical Bayes method, the prior distribution n(9\w) takes parametric form, where the prior distribution parameters are unknown. [7] estimates the prior parameters using the observed data. These parameters could be estimated using the empirical Bayes procedure (see [14] and [3]). Given the observations, the joint likelihood distributions have been compared with the joint prior distributions. The joint likelihood distributions are just the multiplication of the likelihood distribution of X, and
Souradeep Das and Sudhansu S. Maiti RT&A, No 2 (78) A NEW BAYESIAN CONTROL CHART FOR PROCESS MEAN_Volume 19, June, 2024
joint prior distributions are the multiplication of the prior distributions of the parameters. We can estimate the prior parameters by comparing them individually with their corresponding likelihood functions.
5. Control Chart for Mean using Empirical Bayes
Using the empirical Bayes method, we propose a new control chart for the mean. Suppose X is a quality characteristic of a manufacturing process and is assumed to follow a normal distribution with mean, f and variance, a2. The location parameter, f, has a normal prior with unknown parameters, and a follows an inverse gamma distribution with unknown parameters. Let X has n observations X1, X2,..., Xn, such that Xi ~ N(f, a2). here f ~ N(f 0, a2) and a ~ InverseGamma(a, ft).
g(x\u,a2) =-e_202 E(x_f)2 _ ^ < f < a2 > 0
(a^2n )n
g(f\a2, x) = —e_ 2a2 (f-f0) _ to < f0 < to av 2n
P>a ft
g(a2\x) = ra(a2)_a_1ea2 a > 0,ft > 0
Hence, the posterior distribution of (f, a2) is given by,
g(x\f, a2)g(f\a2)
g(f, a2\x)
/0°° fZo g(x\u a2)g(f\a2)dfda2
So, posterior mean E(f\x) = E n+if0. The empirical Bayes estimate of f0 is XX. So, E(t\x) = ^ = x.
Now,
E(a2\x) = J ag(f, a2\x)da2
!• TO
= ag(a2 \x)g(f\°2, ^^ J0
!• TO
= ag(a2\x)g(f\a2, x)da2 0
n+1 + a
_ r(n + a) w12 +a
= r(n+1 + a) r(n + a) ^
r( n+1 + a)
2n+2«_1r(« + a)
/nT(n + 2a)
VWi
here wi = E x,2 + 2ft + f0 _ n+T.
The empirical Bayes procedure will be used to estimate the parameters. So the estimated values of the parameters of the likelihood function of a2 are a = (n _ 3)/2 and ft = E'i=1 (X _ X)2/2 (see [15]). Therefore
r(n + n__3) 1 n
TV n+1 1 n_3\
1 + ~ ) V i=1
E(a\x) = ;fn++ ./2 E(xi _ x)2
r( ^)
r( ^ ) V =1
2 E (xi _ x)
2
r( ) 1 n f^?1^ )v i=1
^./2 E(xi _ x)2
Hence, the control limits of the proposed control chart for the mean are
r( 2n-3 ) I n
UCL = x + , 2 V (xi - x)2
r(2n-2 +i )V ¿i )
CL = x
r( 2n-3 ) I n
LCL = x - Lr( 2-2 +)i ^2 V(Xi - x)2
Here, L is the control chart coefficient.
6. Evaluation of Performance and Comparisons
This section uses Monte Carlo Simulation to compute the proposed control chart's Average Run Length (ARL) for mean using empirical Bayes. We consider different sample values of n and compare the computed in-control-ARL (ARLo) and out-of-control ARL (ARL1) with the existing posterior mean control chart and Shewhart Control Chart. The decision is based on the value of ARLi. The control chart with a smaller ARL1 value is more efficient in detecting a shift in the process mean than other control charts. We consider the shift in process mean as p* = p + ca. Algorithm for construction of UCL and LCL is as follows.
• Step 1: Select a random sample of size n, say, Xi, x2,..., xn from N(p, a2) distribution. Here we assume that p has a normal prior and a has an Inverse Gamma prior with unknown parameters.
• Step 2: Estimate the posterior distribution parameters using the empirical Bayes procedure.
• Step 3: For given values of n and fixed in-control Average Run Length(ARL), say ro, find the control chart coefficient L.
• Step 4: Find UCL and LCL for each i, i = 1,2,..., n. The process is in control if all the
values of Xj fall within the UCL and LCL of the proposed mean chart.
• Step 5: Next, we find the ARLo value for a particular choice of the process mean p.
• Step 6: We shift the process mean p to a certain amount, say c and compute ARL1 by repeating steps 1 to steps 5.
Here, we fixed the ARLo at 37o. The ARL values of the proposed control chart are given in Table 1 - Table 3.
We can see the proposed control chart for mean using empirical Bayes estimators has the least ARL among all the control charts under consideration. The ARL1 of the proposed chart decreases quickly for a small shift in the process mean. As we increase the sample size, the ARL1 values of the control chart decrease. Therefore, we can conclude that the new control chart for using empirical Bayes estimators is more efficient than the posterior control chart and Shewhart XX control chart.
7. Illustrative Example
In recent trends, SPC researchers use both simulated and real-life data to evaluate the performance of a control chart. In this study, we have considered a real dataset from [16] to evaluate the performance of the new control chart for mean using empirical Bayes estimates. The data set is given in the appendix section. Here, we have filled out height data in ten subgroups of size 1o. It is assumed that the control chart statistic, fill height, follows normal distribution where the parameters p and a have unknown prior distribution. Using the empirical Bayes procedure, the
Table 1: Comparison of average run lengths of Empirical Bayes control chart for Mean with Posterior Mean Control Chart and Shewhart X Control Chart for n = 10 and ARL = 370
Shift Empirical Bayes Mean Chart Posterior Mean Chart Shewhart XX Chart
L = 3 L = 3 L = 3
0.0 370.398 370.398 370.398
0.05 281.397 312.467 328.011
0.1 183.248 221.991 249.167
0.15 128.813 144.631 181.701
0.2 86.003 94.297 123.981
0.25 58.238 68.997 87.457
0.3 34.184 39.832 59.301
0.4 9.327 18.115 28.034
0.7 2.265 3.168 4.387
0.9 1.183 1.719 2.814
Table 2: Comparison of average run lengths of Empirical Bayes control chart for Mean with Posterior Mean Control Chart and Shewhart XX Control Chart for n = 20 and ARL = 370
Shift Empirical Bayes Mean Chart Posterior Mean Chart Shewhart XX Chart
L = 3 L = 3 L = 3
0.0 370.398 370.398 370.398
0.05 236.234 289.754 300.373
0.1 131.197 168.103 185.559
0.15 67.469 91.476 106.358
0.2 34.482 50.893 61.539
0.25 21.107 29.555 36.807
0.3 12.893 17.985 22.885
0.4 5.221 7.663 9.959
0.7 1.934 2.988 3.824
0.9 1.021 1.504 2.357
Table 3: Comparison of average run lengths of Empirical Bayes control chart for Mean with Posterior Mean Control Chart and Shewhart XX Control Chart for n = 30 and ARL = 370
Shift Empirical Bayes Mean Chart Posterior Mean Chart Shewhart XX Chart
L = 3 L = 3 L = 3
0.0 370.398 370.398 370.398
0.05 183.609 262.736 271.659
0.1 96.064 130.865 142.164
0.15 34.627 63.37 71.433
0.2 24.922 32.409 37.614
0.25 9.088 17.731 20.860
0.3 3.167 10.384 12.343
0.4 2.081 4.338 5.163
0.7 1.355 2.805 3.532
0.9 1.008 1.244 1.841
prior parameters are estimated. The UCL and LCL of the new control chart for mean based on empirical Bayes for the data set are o.7288446 and -o.6888446, respectively. From figure 2, we can see that the proposed control chart based on empirical Bayes can detect an out-of-control observation more precisely than the posterior control chart for mean and Shewhart XX chart. In figure 4, the Average Run Lengths of the proposed chart and other control charts are drawn.
Figure 1: Comparison of ARLs of Empirical Bayes Mean Control Chart for different sample values
Table 4: Comparison of average run lengths of Empirical Bayes control chart for Mean with Posterior Mean Control Chart and Shewhart XX Control Chart
Shift Empirical Bayes Mean Chart Posterior Mean Chart Shewhart XX Chart
L = 3 L = 3 L = 3
0.0 370.398 370.398 370.398
0.05 297.351 322.097 334.916
0.1 180.398 227.721 257.719
0.15 101.842 147.533 182.071
0.2 58.247 94.044 124.894
0.25 34.533 60.687 85.584
0.3 21.331 40.032 59.301
0.4 9.217 18.786 29.912
0.7 1.867 3.538 5.911
0.9 1.198 1.829 2.822
From table 4 and figure 2, we can conclude that the proposed mean chart using the empirical Bayes estimator has smaller ARL values than the posterior mean control chart as well as Shewhart XX control chart when there occurs a shift. In figure 1, we can see that the width of control limits for the proposed control chart is narrower than the posterior mean control chart and Shewhart XX chart. This implies that the new mean chart based on empirical Bayes estimate can detect an 'out of control state' of a process mean earlier.
8. Conclusion
This article proposes a new control chart for mean using the empirical Bayes approach. For the mean, we compare the performance of the new control chart with that of the existing control charts. We used ARL values to measure the performance of the control charts for the mean. It is observed that the proposed control chart can detect the smaller shift in the process mean quickly than the posterior mean control chart and Shewhart XX control chart. It was also noted that the
Bayesian Control Chart
UCL Posterior Chart
UCL Empirical Bayes
0.5 --______
zj (D
-0.5
LCL Empirical Bayes
LCL Posterior Chart
2.5 5.0 7.5 10.0
Sample Number
Figure 2: Empirical Bayes control chart with that of Posterior Control Chart
Bayesian Control Chart
1.0 UCL X-bar Chart
UCL Empirical Bayes
LCL X-bar Chart
2.5 5.0 7.5 10.0
Sample Number
Figure 3: Empirical Bayes Control Chart and Shewhart XX control chart
proposed control chart performed better for larger sample sizes. Declarations
Disclosure of Conflicts of Interest/Competing Interests: The authors declare no conflict of interest. Authors contributions: Each author has an equal contribution. All authors jointly write, review, and edit the manuscript.
Funding: The authors received no specific funding for this study.
Data Availability Statements: All cited data analyzed in the article are included in References. Data sets are also provided in the article.
Ethical Approval: This article does not contain any studies with human participants performed by authors.
Code availability: Codes are available on request.
Comparison of Average Run Lengths of different Control Charts
colour
X-bar Chart
Mean Chart using empirical Bayes estimates Posterior Mean Chart
0.00
0.25
0.50
0.75
Shift In Process Mean
Figure 4: Comparison of ARLs of Empirical Bayes Mean Control Chart with other Control Charts for the example
[1] Alshahrani, F., Almanjahie, I.M., Khan, M., Anwar, S.M., Rasheed, Z., Cheema, A.N. (2023): On Designing of Bayesian Shewhart-Type Control Charts for Maxwell Distributed Processes with Application of Boring Machine. Mathematics, 11(5),1126.
[2] Aunali, A. S., Venkatesan, D. (2019): Bayesian Control Charts Using Uniform Prior. Journal of Information and Computational Science, 9(11), 295-301.
[3] Awad, A. M. and Gharraf, M. K. (1986): Estimation of P(Y < X) in the Burr case: a comparative study. Communications in Statistics-Simulation and Computation, 15(2), 389-403.
[4] Bhat, S. V., Gokhale, K. D. (2014): Posterior control chart for process average under conjugate prior distribution. Economic Quality Control, 29(1),19"27.
[5] Bai, D. S., Choi, T. S. (1995): X and R Control Charts for Skewed Populations. Journal of Quality Technology, 22(2),120-131.
[6] Bourazas, K., Kiagias, D., Tsiamyrtzis, P. (2022): Predictive Control Charts (PCC): A Bayesian approach in online monitoring of short runs. Journal of Quality Technology, 54(4), 367-391.
[7] Carlin, B. P., Louis T. A. (2000): Bayes and Empirical Bayes Methods for Data Analysis. CRC Press.
[8] Chen, G., Cheng, S. W. (1998): Max chart: combining X-bar chart and S chart. Statistica Sinica, 8(1),263-272.
[9] Costa, A. F. B. (1997): X chart with variable sample size and sampling intervals. Journal of Quality Technology. 29(2),197-204.
[10] Erto, P., Pallotta, G., Palumbo, B., Mastrangelo, C. M. (2018): The performance of semi-empirical Bayesian control charts for monitoring Weibull data. Quality Technology & Quantitative Management, 15(1), 69-86.
[11] Feltz, C. J., Shiau, J. J. H. (2001): Statistical process monitoring using an empirical Bayes multivariate process control chart. Quality and Reliability Engineering International, 17(2), 119-124.
dataset
References
[12] Jampachaisri, K., Tinochai, K., Sukparungsee, S., Areepong, Y. (2020): Empirical Bayes based on squared error loss and precautionary loss functions in sequential sampling plan. IEEE Access, 8, 51460-51469.
[13] Kang, C. W., Lee, M. S., Seong, Y. J., Hawkins, D. M. (2007): A control chart for the coefficient of variation. Journal of Quality Technology, 39(2),151-158.
[14] Lindley, D. V. (1969): Introduction to Probability and Statistics from a Bayesian Viewpoint. Cambridge University Press, Cambridge, UK.
[15] Maiti, S. S., Saha, M. (2012): Bayesian estimation of generalized process capability indices. Journal of Probability and Statistics.
[16] Montgomery, D. C. (2018): Introduction to statistical quality control. John Wiley & Sons.
[17] Nenes, G., Tagaras, G. (2007): The economically designed two-sided Bayesian XX control chart. European Journal ofOperational Research, 183(1), 263-277.
[18] Quinino, R. C., Cruz, F. R., Quinino, V. B. (2021). Control chart for process mean monitoring combining variable and attribute inspections. Computers & Industrial Engineering, 152, 106996.
[19] Tagaras, G., Nikolaidis, Y. (2002): Comparing the effectiveness of various Bayesian XX control charts. Operations Research, 50(5), 878-888.
