An improved Bayesian Modified-EWMA location chart and its applications in mechanical and sport industry

Control charts are popular tools in the statistical process control toolkit and the exponentially weighted moving average (EWMA) chart is one of its essential component for efficient process monitoring. In the present study, a new Bayesian Modified-EWMA chart is proposed for the monitoring of the location parameter in a process. Four various loss functions and a conjugate prior distribution are used in this study. The average run length is used as a performance evaluation tool for the proposed chart and its counterparts. The results advocate that the proposed chart performs very well for the monitoring of small to moderate shifts in the process and beats the existing counterparts. The significance of the proposed scheme has proved through two real-life examples: (1) For the monitoring of the reaming process which is used in the mechanical industry. (2) For the monitoring of golf ball performance in the sports industry.


1: Introduction
Statistical process control (SPC) is an important and powerful technique for the enhancement of the product and process quality. The control charts are the familiar tool used within SPC to detect the out of control situations and hence monitor variation in a process [1]. Shewhart [2] introduced the basic control chart named Shewhart control chart for the monitoring of the process. The Shewhart charts are widely used to detect changes in the quality of the process due to it is simple to use without any substantial statistical pieces of training [3]. Practically, these control charts are not very effective for the monitoring of the process when a small shift occurs in the process [4,5].
This limitation of the Shewhart control chart is that it considers information from the current data and ignores the previous information contained in the data sequence [6].
The exponentially weighted moving average (EWMA) charts are widely used for the monitoring of the small shift in the process as compared to the Shewhart charts [7]. The EWMA charts accumulate current information as well as previous information and detect a small change in the process. The EWMA charts are extensively used and have many applications in the field of process monitoring. Various contributions are made to the literature by many authors, for example [8,9,10,11,12,13,14,15,16,17,18].
The normal distribution is a very common and important distribution in Statistics. It is commonly used in the social and natural phenomena. Due to special nice properties, it has attracted a lot the attention of the researcher belongs to different fields. For example, Ajadi a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 EWMA chart using various loss functions. The performance comparisons of the proposed chart are discussed in section 4. The last section is made for conclusions and recommendations.

2: Bayesian approach
A Bayesian approach is an alternative approach to the frequentist (classical) approach. Here, the parameter is considered as a "random variable" and is assumed to follow a prior distribution having certain parameters, known as hyperparameters. Two types of prior distributions are used to construct the posterior distribution, known as non-informative and informative priors. The Jeffreys and uniforms priors are commonly used non-informative priors and conjugate prior is commonly used as a family of informative prior [48].
The posterior distribution is constructed by combing the sampling distribution (likelihood function) and prior distribution. Hence the posterior distribution is proportional to the product of the prior distribution and likelihood function. The posterior distribution for unknown parameter μ given y is defined by where f(μ) is the prior distribution, L(�)is the likelihood function and g(μ|y) is the posterior distribution for the parameter. The premium advantage of using the Bayesian approach is that it is naturally suitable for prediction of the future value. The posterior predictive distribution for the future variable X given Y is defined by

2.1: Loss functions
Loss Functions have a significant role in Bayesian inference. The loss function measures the losses produced by an erroneous estimation of a parameter. To obtain the best estimates in decision theory, a loss function must be specified. The loss functions may be symmetric or asymmetric. We have used the following loss functions in our study. The Square Error loss function (SELF) is a symmetric loss function and extensively used loss function in Bayesian inference. If X is predictive variable and μ SELF is its estimate than SELF is defined as L(X,μ SELF ) = (X,μ SELF ) 2 . The estimator μ SELF which minimizes L(X,μ SELF ) is given by Varian [49] proposed asymmetric loss function known as LINEX loss function (LLF), which is used in the case when overestimation is more significant. It is defined as L(X,μ LLF ) = exp{c(μ LLF −μ)}−c(μ LLF −μ)−1, where c 6 ¼ 0. When c is greater than zero, it is the case of overestimation. The predictive Bayes estimator of X under LLF is Rodrigues [50] introduced Weighted Balance Loss function (WBLF) is widely used in the Bayesian Inference can be defined as L X; The predictive Bayes estimator of X which minimizes L(X,μ WBLF ) is given as The Precautionary Loss Function (PLF) was introduced by Norstrøm [51]. When we are interested in study the low failure rate, then PLF is used to prevent underestimation. This is asymmetric loss function and easy to handle. PLF can be written as L X; m PLF ð Þ ¼ ðXÀ m PLF Þ 2 m PLF . The predictive Bayes estimator of X under PLF case is

3: The proposed monitoring scheme
In the following subsections, the proposed scheme is explained, followed by some existing control charts related to this era.

3.1: Classical Modified-EWMA chart
Khan et al. [45] introduced a Modified-EWMA Chart for the process mean. Suppose that the quality characteristic Y belongs to the normal distribution with mean μ and variance σ 2 . At the time (or subgroup) t, a random sample of size n is taken and measure its quality characteristic. Estimating the following Modified-EWMA statistic at subgroup t, expressed by X t .
here λ 2 (0,1) is the smoothing constant and ψ is an additional constant concerned with the Modified-EWMA statistic. By continuous substitution X t−i , the simplified form of Modified-EWMA statistic is given by here, X 0 is assumed to be equal to μ 0 (i.e. in-control process mean). The mean and variance of the Modified-EWMA statistic given in Eq (7) can be written as E(X t ) = μ 0 and where L is the coefficient of control chart, and UCL and LCL are upper and lower control limits. The value ψ may be independent of λ, but according to Khan et al. [45], ψ is derived by minimizing the Var(X t ), i.e. c ¼ À l 2 . Also, for ψ = 0, the Modified-EWMA chart is reduced to the classical EWMA chart.

3.2: Proposed bayesian Modified-EWMA chart under various loss functions
Suppose that quality characteristic Y belongs to the normal distribution with mean μ and variance σ 2 .i.e. g y; m; s 2 ð Þ ¼ 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2ps 2 p exp À 1 2s 2 ðy À mÞ The likelihood function for a random sample of size n is given by We assumed the informative prior for the parameter μ, which is also a normal distribution with mean a and variance b 2 , i.e g m; a; b 2 ð Þ ¼ 1 ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 2pb 2 The prior distribution and the likelihood function are combined by using Eq (1), to construct the posterior distribution given as g mjy ð Þ ¼ 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2ps 2 where m n ¼ as 2 þnb 2 � Y s 2 þnb 2 and s 2 n ¼ b 2 s 2 s 2 þnb 2 . Now, the posterior predictive distribution is obtained, by using Eq (2) given as Note that, the distribution of the predictive mean � Xjy is also normal with mean μ n and variance s 2 X ¼ s 2 n þ s 2 n . Based on the existing Modified-EWMA statistic given above, now we introduce the Bayesian Modified-EWMA chart by inserting the predictive mean � X t jy instead of the sample mean � Y t in the Modified-EWMA statistic given in (7). Hence, the Bayesian Modified-EWMA statistic is defined by here λ 2 (0,1) is the smoothing constant and ψ is an additional constant concerned with the proposed Bayesian Modified-EWMA statistic. The proposed Bayesian Modified-EWMA chart is reduced to the Bayesian EWMA chart by Riaz et al. [18] for ψ = 0. Here, Z t is the plotting statistic, which is updated with the predictive estimator � X t jy. For the Bayesian Modified-EWMA chart, we discuss it under different loss functions given as. (14), the posterior predictive estimator under SELF, using (3) is written as m SELF ¼ as 2 þnb 2 � Y s 2 þnb 2 . Under SELF, the posterior predictive time-varying control limits using Bayesian Modified-EWMA statistic in (15) are defined as

3.2.2: Bayesian Modified-EWMA under LLF.
For LLF, the posterior predictive estimator The posterior predictive time-varying control limits under LLF using Bayesian Modified-EWMA statistic in (15) are written as . Under WBLF, the posterior predictive time-varying control limits using Bayesian Modified-EWMA statistic in (15) are written as

3.2.4: Bayesian Modified-EWMA under PLF.
For PLF, the posterior predictive estimator The posterior predictive time-varying control limits using Bayesian Modified-EWMA statistic under PLF are presented as

4: Performance evaluation and comparisons
In this section, we used the Monte Carlo Simulation with 10,000 iterations to compute the average run length (ARL) of the Bayesian Modified-EWMA chart. In this regard, we considered various values of smoothing constant (0.05, 0.15, 0.30 and 0.70) and different sample sizes (5, 10 and 20). The ARL 0 and ARL 1 are the in-control (IC) and out-of-control (OC) ARL, respectively. A control chart with smaller ARL 1 value is more efficient as compared to the competitor control charts [23]. Without any loss of generality, we choose μ = 0 and σ = 1. Suppose, because of some assignable cause, the IC process to be shifted as μ � = μ + δσ. For comparison purpose, we assumed the same hyper-parameters i.e. (a = 10,b = 4), and the same current and future sample size as taken by Riaz et al. [18]. Following [52,18], we assumed c = 4 in LLF. The ARL values of the proposed chart are presented in the Tables 1-3.
• The control chart coefficient, L of the Bayesian Modified-EWMA chart increases by increasing the smoothing constant for all loss functions to be used in this study. • For the fixed value of λ, the performance of the proposed chart increases by increasing the sample size under SELF, WBLF, PLF, and LLF. This implies that an OC signal is noticed early for a large sample size as compared to a small sample size (cf. Fig 3).   • The proposed chart under SELF and WBLF performs better than the proposed chart under LLF and PLF for different choices of smoothing constant and sample size (cf. Fig 2).

4.1: Comparison of proposed with existing chart
Here, we present a comparative study of the proposed Bayesian Modified-EWMA chart with some existing control charts present in the literature.

4.1.1: Proposed chart vs. existing Bayesian EWMA chart
Here, we compared the proposed Bayesian Modified-EWMA chart with the existing Bayesian EWMA chart suggested by [18], which can be obtained by setting ψ = 0 in the proposed Bayesian Modified-EWMA chart. The results of the existing Bayesian EWMA chart for ARL 0 = 370 are presented in  Table 1 and Table 4 can be compared, (cf. Fig 4).

4.1.2: Proposed chart vs. existing classical EWMA chart
Here, we compared the proposed Bayesian Modified-EWMA chart with the existing classical EWMA chart proposed by Roberts [7]. The ARL 1 of classical EWMA chart are presented in Table 5 for ARL 0 =370, λ = 0.05 and n = 5. The comparison reveals that the proposed Bayesian Modified-EWMA chart performs better than the existing classical EWMA chart for the monitoring of the process location.  Table 5). In addition, the dominance of the proposed chart is presented in Fig 5.

4.1.3: Proposed chart vs. existing Modified-EWMA chart
The Modified-EWMA chart is proposed by Khan et al. [45] and some results of this control chart are reported in  Table 5). In addition, the better performance of the proposed chart from the Modified-EWMA chart can be visualized in Fig 6.

4.2: Real-life application 1
Reaming is the machining process which is used a multi-edged fluted cutting to smooth or accurate size adjusting specified instrument in the existing hole in the machine. It is used to cut the hole within +0.0005 inch of tool size and provide finishing to 32 micro inches [53]. According to [53], the screw threads are used for various purposes and have many applications in the machine tool industry. To hold and fasten parts together, screws, bolts, and nuts are used in the mechanical industry especially in engines of different vehicles and airplanes. According to Decheffre et al. [54], the quality of the reaming process is much affected by the alignment of the machine parts, reamer geometry, cutting condition, and lubrication. Any variation in alignment may detract from the accuracy of the hole. All these reamers provide smooth and accurate holes in metals to maintain high quality. The quality of the reaming process is evaluated through surface roughness, diameter, and roundness, etc. [54]. Zhang [55] and Riaz and Ali [41] used a data set in the form of the summary statistics (mean and variance) for 20 samples of n = 5, which is about the surface roughness of reamed holes in a particular metal part. These summary statistics are presented in Table 6. The data set is well fitted by a normal distribution with mean and variance are 32.1 and 21.833, respectively [41]. Following Riaz and Ali [41], we assume prior distribution as μ~N (30,20) and posterior distribution as μ|y~Normal (31.72367,3.584086). Using these information and real-life data given in Table 6, the graphical display of the proposed and existing charts is presented in     7).

4.2: Real-life Application 2
Manufacturing companies of golf balls spend a lot of money on the research to improve the performance of golf balls for maximum flight distance and used the latest techniques to improve the performance of these balls. The performance of the golf balls depends upon the conditions, material (silicone, titanium and several kinds of urethane), engineer expertise and the equipment to be used for the manufacturing of balls (http://www.all-science-fair-projects. com/print_project_1221_57). For more detailed information on golf ball see [56]. Savran et al. [57] designed the technical report entitled "The Golf Ball Detector" at the Department of Mechanical, Industrial and Manufacturing Engineering, Northeastern University. They stated that the United States Golf Association (USGA) performs different tests to check the golf ball and club design and its performance. The USGA uses the robotic arm called "Iron Byron" and the golf club to check the performance of a golf ball in terms of swing and movement of the golf ball. The USGA also detects the strength of thermoplastic, covering, inertia and the compressing ability (named coefficient of restitution) of the ball. A simple bonce of a golf ball test is performed for the coefficient of restitution. A bounce test is also performed to check the effect of tag on the behavior of the prototype.
In this subsection, we presented the implementation and performance of the proposed Bayesian Modified-EWMA control chart under various loss functions. For this purpose, we considered a real-life data set from the book entitled, "Introduction to Bayesian Statistics (second edition)", by William M. Bolstad page: 236, [58]. The data explain the golf ball first bonce when the 10 new golf balls dropped down from the height of 1 meter. The bounces of these 10 balls are given in Table 7.
Assuming the height (in cm) a golf ball bounces when dropped from the height of one meter is Normal(μ,σ 2 ), where σ = 12. Using a prior distribution for μ is Normal(75,10 2 ). We set the ARL 0 as 370 for a valid comparison of existing Bayesian EWMA with the proposed Bayesian Modified-EWMA chart. For this purpose, we considered the real-life data and the prior     https://doi.org/10.1371/journal.pone.0229422.g008 Table 7. Real-life data set-2 (Golf ball first bounces of 10 balls from the distance of 1-meter measurement in cm).

5: Summary and conclusions
This article presents a new Bayesian Modified-EWMA chart based on posterior predictive distribution under various loss functions for efficient process monitoring of the location parameter. The ARL is used as a tool to measure the performance of the proposed chart. A simulation study is conducted to judge the behavior of ARL. We used conjugate prior and hyper-parameters from the previous study. The performance of the proposed chart under SELF and WBLF is better than the proposed chart under LLF and PLF. Similarly, the proposed chart performed better for smaller smoothing constant and for larger sample sizes under all these loss functions to be used here. It is observed that the proposed Bayesian Modified-EWMA chart under different loss functions detects earlier shifts that occur in the process location. A detailed comparison of the proposed chart with the existing charts is also provided. The study is also supported by two real-life applications. This study may be extended to the non-normal distributions and authors are already working on these sides. The multivariate structure of the proposed chart is worthwhile for practical purpose and need to be explored.