ESTIMATION OF PROCESS CAPABILITY IN BAYESIAN PARADIGM

The process capability index (PCI), CCCC examines the capability of control charts. Bayesian techniques to estimate CCCC are desirable when prior information about a process characteristic is available. In this paper, an estimator of CCCC under normality with process variance having conjugate prior in Bayesian scenario is proposed. Its performance is studied and compared with Bayesian estimator developed by Cheng and Spiring (1989) An illustrative example is provided. Received 20 May 2022 Accepted 29 June 2022 Published 19 July 2022 Corresponding Author Sharada V. Bhat, bhat_sharada@yahoo.com DOI 10.29121/ijetmr.v9.i7.2022.1193 Funding: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Copyright: © 2022 The Author(s). This work is licensed under a Creative Commons Attribution 4.0 International License. With the license CC-BY, authors retain the copyright, allowing anyone to download, reuse, re-print, modify, distribute, and/or copy their contribution. The work must be properly attributed to its author.


INTRODUCTION
Quality is a prudent characteristic in manufacturing industries. plays pivotal role in deciding about capability of a process preset to meet the quality requirements. It is given by where is process standard deviation (sd), UCL, LCL are upper, lower control limits and USL, LSL are upper, lower specification limits of a control chart. is a unitless measure and a process is considered as capable if > where k is a positive constant ≥ 1. When is estimated by sample sd 's', an estimator for is given by ̂= − 6 .

Equation 2
Kane (1986) compared various PCIs and Montgomery (1996) carried out a detailed discussion on PCIs along with their illustrations. Bayesian procedures for PCI use prior information about the process parameters involved. Cheng and Spiring (1989) using Bayesian approach propose an estimator ̂, under normal model when process sd has noninformative prior. The posterior distribution of ̂ derived by them is where ̂ is realized by y. They investigate the performance of their measure in terms of minimum value of ̂ required to ensure the probability that process achieves the desired specifications along with an illustration. Chan et al. (1988) proposed a measure to process capability which accounts both target value and process variation simultaneously. They examined sampling distribution of the proposed measure with its practical applications to industrial data. Spiring (1995) outlined assessment of process capability as a tool of management. Shiau et al. (1999) studied Bayesian procedure for process capability by assuming noninformative and gamma priors for 2 . Kotz and Johnson (2002) reviewed some articles on PCIs studied during 1992 -2000 from widely scattered sources and record their interpretations along with comments. Pearn and Wu (2005) studied estimation of by Bayesian approach using multiple samples.
In this paper, we establish an estimator of in Bayesian paradigm under normality when process variance has conjugate prior. In section 2, we propose ̂ and derive its posterior distribution. We study about its performance in section 3, illustrate its performance in section 4 and record our conclusions in section 5. The computed values supporting performance of ̂ are given in tables provided in appendix.

PROPOSED BAYESIAN ESTIMATOR OF
In this section, we propose a Bayesian estimator ̂ for when process variance has a conjugate prior distribution. That is, ̂ is given by Equation 2 with an assumption that, 2 has a conjugate prior.
Suppose, X 1 , X 2 … X n is a random sample of size n from ( , 2 ), then the density function of X i is given by The likelihood function of the sample = (X 1 , X 2 … X n ) is given by

Equation 5
Also, = We assume that ( 2 ) ∼ ( , ), > 0, > 0 where is shape parameter and is scale parameter. IG stands for inverse gamma which is a conjugate prior. ( Using Equation 1 and Equation 2, z can be written as

Equation 7
Thus, realizing ̂ by y, we have the posterior distribution of 2 | given by

Equation 8
Using appropriate transformation, ( | ) is given by When = =0, Equation 9 reduces to Equation 3 indicating that the posterior distribution of ̂ due to Cheng and Spiring (1989) is a particular case of posterior distribution of ̂ given in Equation 9. From Bhat and Gokhale (2014) Bhat and Gokhale (2016) and Gokhale (2017) we observe that, where in 2 in left hand side is replaced by 2 in right hand side.

PERFORMANCE OF ̂
In this section, we evaluate the performance of ̂ by obtaining minimum value of ̂ needed to assure �( > )| �. That is, = minimum ̂| .

Equation 13
By using, Wilson-Hilferty (1931) transformation, (13) can be written as Estimation of Process Capability in Bayesian Paradigm

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Where Φ(•) is cumulative distribution function of standard normal variate.
On simplifying Equation 14 we get

Equation 16
Where = − . In order to evaluate , one need to specify , n, , k, and . To compute minimum obtained in Equation 16, the denominator has to be greater than zero.
That is, �

Equation 17
By taking = Table 1, we furnish w as ⌈ ⌉ least upper integer greater than . In Table 2, we calculate for higher values of w given in Table 1 Table 1 we observe that, increases as , increase and decreases as increases. For fixed , increases as increases, for fixed , it decreases as increases and for = , it increases for increasing values of and . From Figure 1 and Table 2 we observe that, for = , is higher for higher values of . It is increasing for increasing value of k when n is small and remains nearly same when n is large. Also, is smaller for higher values of and . Figure 2 and Figure 3 depict that decreases respectively as increases for < and as increases for > . Also, from all the three figures it is observed that, increases as k increases along with increase in . Table 2 shows that, for fixed values of = 0, there is no considerable change in values of for various values of n, k and for increasing .
. Then for = , using (16), for different values of , and n, � is given by ̂ n=300 n=50 n=300 n=50 In Example 1, it is seen that for different values of , and n=300, ̂ is lesser than ̂, whereas for n=50, ̂ is near to ̂ when = 0 and is lesser than ̂ for other values of and . In Example 2, ̂ is near to ̂ for n=79 when = 0, whereas for n=5, ̂ is lesser than ̂ for various values of and . It is also observed that, sample sd is smaller in Example 1 when compared to sample sd in Example 2

CONCLUSIONS
In this section, we furnish our conclusions based on our observations.
• Under Bayesian approach, the proposed estimator ̂ includes ̂ due to Cheng and Spiring (1989) as its particular case in the sense that, the posterior distribution of ̂ reduces to that of ̂ when hyper parameters and are zero.
• For all the values of and under consideration, the minimum value of ̂ needed to assure , the probability that process is capable given the sample, increases along with increasing values of k and for smaller values of n.
• For = , is decreasing as and are increasing.
• For < , decreases as increases and for > , it decreases as increases.
• ̂<̂ when sample sd is small, n is large and also when sample sd is large, n is small. Table 1 Table 2