ESTIMATION OF PROCESS CAPABILITY IN BAYESIAN PARADIGM Chetan Malagavi ^{1} ^{1}^{ }Department of Mathematics, GITAM Deemed to be University, Bengaluru561203. Karnataka, India and Research Scholar, Department of Statistics, Karnatak University, Dharwad580003. Karnataka, India ^{2} Department of Statistics. Karnatak University, Dharwad580003. Karnataka State, India
1. INTRODUCTION Quality is a prudent characteristic in manufacturing industries. 𝐶𝑝 plays pivotal role in deciding about capability of a process preset to meet the quality requirements. It is given by where 𝜎 is process
standard deviation (sd), UCL, LCL are upper, lower control limits and USL, LSL
are upper, lower specification limits of a control chart. 𝐶𝑝 is
a unitless measure and a process is considered as capable if given by 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 where 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 scattered sources and record their interpretations along
with comments. Pearn
and Wu (2005) studied estimation of In this paper, we establish an estimator of 2. PROPOSED BAYESIAN ESTIMATOR OF 𝑪_{𝒑} In this section, we propose a Bayesian estimator Suppose, X_{1}, X_{2} … X_{n} is a
random sample of size n from The likelihood function of the sample Also, We assume that Using Equation 1 and Equation 2, z can be written as Thus, realizing Using appropriate transformation, When And where in 3. PERFORMANCE OF 𝑪̂_{𝒑𝒄} In this section, we evaluate the performance of 𝜏 = minimum
Here, 𝑝𝑐 =
which is equivalent to finding Where By taking proceeding on the lines of Chan et al. (1988), we express (12) as By using, WilsonHilferty (1931) transformation, (13) can be written as Where On simplifying Equation 14 we get Therefore, Where In order to evaluate 𝜏, one need to specify 𝑤, n, 𝑝𝑐, k, 𝜂 and 𝛿. To compute minimum 𝐶̂𝑝𝑐 obtained in Equation 16, the denominator has to be greater than zero. That is,
By taking upper integer greater than 𝛾. In Table 2, we calculate 𝜏 for higher values of w given in Table 1 n= 5, 15, 25, 50, 75, 100, 𝑝𝑐 = 0.90, 0.95, 0.99, k=1, 1.33,1.66, 𝜂 = 0, 5, 10 and 𝛿 = 0, 5, 10. Using Table 2 we plot 𝜏 in Figure 1 Figure 2 and Figure 3 respectively for 𝜂 = 𝛿, 𝜂 < 𝛿 and 𝜂 > 𝛿. Figure 1
Figure 2
Figure 3
From Table 1 we observe that, 𝑤
increases as 𝑝𝑐, 𝑘 increase and decreases as 𝑛
increases. For fixed 𝜂, 𝑤 increases as 𝛿 increases, for
fixed 𝛿, it decreases as 𝜂 increases and for 4. ILLUSTRATION In this section, we consider example given in Kane (1986) and discussed in Cheng and Spiring (1989) Example 1
Example 2
In Example 1, it is seen that for
different values of 5. CONCLUSIONS In this section, we furnish our conclusions based on our observations. ·
Under Bayesian approach, the proposed estimator ·
For all the values of 𝜂 and 𝛿
under consideration, 𝜏 the minimum value of ·
For ·
For ·
6. APPENDIX Table 1
