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
Equation 1
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
where k is a positive constant ≥ 1. When
𝜎 is estimated by sample sd ‘s’, an estimator for 𝐶𝑝 is
given by
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
Equation 3
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
. 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.
2. PROPOSED BAYESIAN ESTIMATOR OF 𝑪𝒑
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, 𝜎2 has a conjugate prior.
Suppose, X1, X2 … Xn is a
random sample of size n from
),
then the density function of Xi is given by
Equation 4
The likelihood function of the sample
is given by
Equation 5
Also,
where
We assume that
where 𝜂 is shape parameter and 𝛿 is
scale parameter. IG stands for inverse gamma which is a conjugate prior.
Equation 6
Using Equation 1 and Equation 2,
z can be written as
Equation 7
Thus, realizing
by y, we have
the posterior distribution of 𝑦2|𝑿 given by
Equation 8
Using appropriate transformation,
is given by
Equation 9
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,
And
Equation 10
where in
in left hand
side is replaced by 𝑠2 in right hand side.
3. PERFORMANCE OF 𝑪̂𝒑𝒄
In this section, we evaluate the performance of
by obtaining
minimum value of
needed to
assure
. That is,
𝜏 = minimum
.
Here, 𝑝𝑐 =
Equation 11
which is equivalent
to finding
Equation 12
Where
.
By taking
,
and
and
proceeding on the lines of Chan et al. (1988), we express (12) as
Equation 13
By using, Wilson-Hilferty (1931) transformation, (13) can
be written as
Equation 14
Where
is cumulative
distribution function of standard normal variate.
On simplifying Equation 14 we get
Equation 15
Therefore,
Equation 16
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,
Equation 17
By taking
in Table 1, we furnish w as
least
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
for
and
various values of
|
Figure
2
for with
and various values of
|
Figure
3
for
and various values of
|
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
, it increases for increasing values of 𝜂
and 𝛿. From Figure 1 and Table 2 we observe that, for 𝜂
= 𝛿, 𝜏 is higher for higher values of 𝑝𝑐.
It is increasing for increasing value of k when n is small and remains nearly
same when n is large. Also, 𝜏 is smaller for higher values of 𝜂
and 𝛿. Figure 2 and Figure 3 depict that 𝜏
decreases respectively as 𝛿 increases for 𝜂 < 𝛿 and
as 𝜂 increases for 𝜂 > 𝛿. Also, from all the three
figures it is observed that, 𝜏 increases as k increases along with
increase in 𝑝𝑐. Table 2 shows that, for fixed
values of 𝜂 = 0, there is no considerable change in values of 𝜏
for various values of n, k and 𝑝𝑐 for increasing 𝛿.
4. ILLUSTRATION
In this section, we consider example given in Kane (1986) and discussed in Cheng
and Spiring (1989)
Example
1 For n=300, s=4.3,
and
. Then for
, using (16),
for different values of
and n,
is given by
|
𝑪̂𝒑𝒄
|
𝜼
|
𝜹
|
n=300
|
n=50
|
𝜼
|
𝜹
|
n=300
|
n=50
|
0
|
5
|
1.174
|
1.5479
|
10
|
0
|
1.167
|
1.4214
|
0
|
10
|
1.1746
|
1.5565
|
5
|
5
|
1.1707
|
1.4782
|
5
|
0
|
1.1701
|
1.4709
|
10
|
10
|
1.1682
|
1.4347
|
Example
2 For n=79, s=7.8,
and
For
,
is given by
|
𝑪̂𝒑𝒄
|
𝜼
|
𝜹
|
n=79
|
n=5
|
𝜼
|
𝜹
|
n=79
|
n=5
|
0
|
5
|
0.8453
|
0.5091
|
10
|
0
|
0.8584
|
0.6783
|
0
|
10
|
0.8462
|
0.5129
|
5
|
5
|
0.8528
|
0.6386
|
5
|
0
|
0.8519
|
0.6314
|
10
|
10
|
0.8602
|
0.6966
|
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
and is lesser
than
for other
values of
and
In Example 2,
is near to
for n=79 when
, 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
5. 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 𝜂 and 𝛿 are zero.
·
For all the values of 𝜂 and 𝛿
under consideration, 𝜏 the minimum value of
needed to assure
the
probability that process is capable given the sample, increases along with
increasing values of k and 𝑝𝑐 for smaller values of n.
·
For
is decreasing
as 𝜂 and 𝛿 are increasing.
·
For
,
decreases as 𝛿
increases and for 𝜂 > 𝛿, it decreases as 𝜂
increases.
·
when sample
sd is small, n is large and also when sample sd is large, n is small.
6. APPENDIX
Table 1
for various values of n,
k,
and
|
|
k
|
n
|
5
|
15
|
25
|
50
|
75
|
100
|
|
k
|
n
|
5
|
15
|
25
|
50
|
75
|
100
|
|
|
pc
|
|
|
|
|
|
|
|
|
pc
|
|
|
|
|
|
|
0,5
|
1
|
0.9
|
17
|
7
|
5
|
4
|
3
|
3
|
0,10
|
1
|
0.9
|
24
|
10
|
7
|
5
|
4
|
3
|
|
|
0.95
|
21
|
8
|
5
|
4
|
3
|
3
|
|
|
0.95
|
29
|
13
|
8
|
5
|
4
|
4
|
|
|
0.99
|
35
|
9
|
6
|
4
|
3
|
3
|
|
|
0.99
|
49
|
13
|
8
|
5
|
4
|
4
|
|
1.33
|
0.9
|
22
|
9
|
7
|
5
|
4
|
3
|
|
1.33
|
0.9
|
31
|
13
|
9
|
6
|
5
|
4
|
|
|
0.95
|
28
|
10
|
7
|
5
|
4
|
3
|
|
|
0.95
|
39
|
14
|
10
|
7
|
5
|
5
|
|
|
0.99
|
46
|
12
|
8
|
5
|
4
|
4
|
|
|
0.99
|
65
|
17
|
11
|
7
|
6
|
5
|
|
1.66
|
0.9
|
28
|
11
|
8
|
6
|
5
|
4
|
|
1.66
|
0.9
|
39
|
16
|
12
|
8
|
6
|
5
|
|
|
0.95
|
34
|
12
|
9
|
6
|
5
|
4
|
|
|
0.95
|
48
|
17
|
12
|
8
|
6
|
6
|
|
|
0.99
|
57
|
15
|
10
|
6
|
5
|
4
|
|
|
0.99
|
80
|
21
|
14
|
9
|
7
|
6
|
5,5
|
1
|
0.9
|
12
|
7
|
5
|
4
|
3
|
3
|
5, 10
|
1
|
0.9
|
17
|
9
|
7
|
5
|
4
|
3
|
|
|
0.95
|
18
|
8
|
5
|
4
|
3
|
3
|
|
|
0.95
|
25
|
10
|
8
|
5
|
4
|
4
|
|
|
0.99
|
27
|
9
|
6
|
4
|
3
|
3
|
|
|
0.99
|
38
|
12
|
8
|
5
|
4
|
4
|
|
1.33
|
0.9
|
16
|
9
|
7
|
5
|
4
|
3
|
|
1.33
|
0.9
|
23
|
12
|
9
|
6
|
5
|
4
|
|
|
0.95
|
24
|
10
|
7
|
5
|
4
|
3
|
|
|
0.95
|
34
|
15
|
10
|
7
|
5
|
5
|
|
|
0.99
|
35
|
12
|
8
|
5
|
4
|
4
|
|
|
0.99
|
50
|
16
|
11
|
7
|
6
|
5
|
|
1.66
|
0.9
|
20
|
11
|
8
|
6
|
5
|
4
|
|
1.66
|
0.9
|
29
|
15
|
11
|
8
|
6
|
5
|
|
|
0.95
|
30
|
12
|
9
|
6
|
5
|
4
|
|
|
0.95
|
42
|
17
|
12
|
8
|
6
|
6
|
|
|
0.99
|
44
|
14
|
10
|
6
|
5
|
4
|
|
|
0.99
|
62
|
20
|
14
|
9
|
7
|
6
|
10,5
|
1
|
0.9
|
12
|
6
|
5
|
4
|
3
|
3
|
10,10
|
1
|
0.9
|
16
|
9
|
7
|
5
|
4
|
4
|
|
|
0.95
|
17
|
7
|
5
|
4
|
3
|
3
|
|
|
0.95
|
23
|
10
|
7
|
5
|
4
|
4
|
|
|
0.99
|
23
|
9
|
6
|
4
|
3
|
3
|
|
|
0.99
|
32
|
12
|
8
|
5
|
4
|
4
|
|
1.33
|
0.9
|
15
|
8
|
6
|
5
|
4
|
3
|
|
1.33
|
0.9
|
21
|
12
|
9
|
6
|
5
|
4
|
|
|
0.95
|
22
|
10
|
7
|
5
|
4
|
3
|
|
|
0.95
|
31
|
14
|
10
|
7
|
5
|
5
|
|
|
0.99
|
30
|
11
|
8
|
5
|
4
|
4
|
|
|
0.99
|
42
|
16
|
11
|
7
|
6
|
5
|
|
1.66
|
0.9
|
19
|
10
|
8
|
6
|
5
|
4
|
|
1.66
|
0.9
|
26
|
14
|
11
|
8
|
6
|
5
|
|
|
0.95
|
27
|
12
|
9
|
6
|
5
|
4
|
|
|
0.95
|
38
|
17
|
12
|
8
|
6
|
6
|
|
|
0.99
|
38
|
14
|
10
|
6
|
5
|
4
|
|
|
0.99
|
53
|
20
|
13
|
9
|
7
|
6
|
Table 2
for various values of
and n
|
|
n
|
k=1
|
k=1.33
|
k=1.66
|
|
|
pc=0.90
|
pc=0.95
|
pc=0.99
|
pc=0.90
|
pc=0.95
|
pc=0.99
|
pc=0.90
|
pc=0.95
|
pc=0.99
|
0,0
|
5
|
1.7372
|
2.1531
|
3.5876
|
2.3105
|
2.8636
|
4.7715
|
2.8838
|
3.5741
|
5.9555
|
|
15
|
1.2947
|
1.4108
|
1.6818
|
1.722
|
1.8764
|
2.2368
|
2.1493
|
2.342
|
2.7918
|
|
25
|
1.2128
|
1.29
|
1.4589
|
1.613
|
1.7157
|
1.9404
|
2.0132
|
2.1414
|
2.4218
|
|
50
|
1.142
|
1.1897
|
1.2886
|
1.5188
|
1.5822
|
1.7139
|
1.8957
|
1.9748
|
2.1391
|
|
75
|
1.1133
|
1.1502
|
1.2251
|
1.4807
|
1.5298
|
1.6294
|
1.8481
|
1.9093
|
2.0337
|
|
100
|
1.0969
|
1.1279
|
1.19
|
1.4589
|
1.5001
|
1.5827
|
1.8209
|
1.8723
|
1.9754
|
0,5
|
5
|
1.7382
|
2.1549
|
3.596
|
2.3127
|
2.8678
|
4.7912
|
2.8881
|
3.5823
|
5.9938
|
|
15
|
1.2949
|
1.411
|
1.682
|
1.7223
|
1.8767
|
2.2373
|
2.1498
|
2.3426
|
2.7929
|
|
25
|
1.2129
|
1.2901
|
1.459
|
1.6131
|
1.7158
|
1.9406
|
2.0135
|
2.1417
|
2.4223
|
|
50
|
1.142
|
1.1897
|
1.2887
|
1.5189
|
1.5823
|
1.7139
|
1.8958
|
1.9749
|
2.1393
|
|
75
|
1.1134
|
1.1502
|
1.2251
|
1.4808
|
1.5298
|
1.6294
|
1.8482
|
1.9094
|
2.0337
|
|
100
|
1.0969
|
1.1279
|
1.19
|
1.4589
|
1.5001
|
1.5827
|
1.821
|
1.8723
|
1.9754
|
0,10
|
5
|
1.7382
|
2.1549
|
3.5961
|
2.3128
|
2.8679
|
4.7916
|
2.8882
|
3.5825
|
5.9946
|
|
15
|
1.2949
|
1.411
|
1.682
|
1.7223
|
1.8768
|
2.2374
|
2.1498
|
2.3427
|
2.7929
|
|
25
|
1.2129
|
1.2901
|
1.459
|
1.6132
|
1.7158
|
1.9406
|
2.0135
|
2.1417
|
2.4223
|
|
50
|
1.142
|
1.1897
|
1.2887
|
1.5189
|
1.5823
|
1.7139
|
1.8958
|
1.975
|
2.1393
|
|
75
|
1.1134
|
1.1502
|
1.2251
|
1.4808
|
1.5298
|
1.6294
|
1.8482
|
1.9094
|
2.0337
|
|
100
|
1.0969
|
1.1279
|
1.19
|
1.4589
|
1.5001
|
1.5827
|
1.821
|
1.8723
|
1.9754
|
5,0
|
5
|
1.2617
|
1.8611
|
2.7631
|
1.678
|
2.4753
|
3.6749
|
2.0944
|
3.0895
|
4.5868
|
|
15
|
1.2072
|
1.3868
|
1.6384
|
1.6056
|
1.8445
|
2.1791
|
2.0039
|
2.3022
|
2.7197
|
|
25
|
1.1718
|
1.2809
|
1.4436
|
1.5585
|
1.7035
|
1.92
|
1.9452
|
2.1262
|
2.3964
|
|
50
|
1.1275
|
1.187
|
1.2844
|
1.4996
|
1.5787
|
1.7082
|
1.8717
|
1.9704
|
2.1321
|
|
75
|
1.1055
|
1.1488
|
1.223
|
1.4704
|
1.528
|
1.6266
|
1.8352
|
1.9071
|
2.0302
|
|
100
|
1.0919
|
1.127
|
1.1887
|
1.4522
|
1.4989
|
1.581
|
1.8126
|
1.8709
|
1.9732
|
5,5
|
5
|
1.2805
|
1.9233
|
2.9799
|
1.7049
|
2.5641
|
3.9862
|
2.1237
|
3.1859
|
4.9218
|
|
15
|
1.2118
|
1.3939
|
1.65
|
1.6122
|
1.8545
|
2.1957
|
2.0112
|
2.3131
|
2.7379
|
|
25
|
1.1743
|
1.2841
|
1.4482
|
1.562
|
1.7081
|
1.9266
|
1.949
|
2.1313
|
2.4036
|
|
50
|
1.1286
|
1.1882
|
1.286
|
1.5011
|
1.5805
|
1.7105
|
1.8733
|
1.9723
|
2.1345
|
|
75
|
1.1062
|
1.1496
|
1.2239
|
1.4713
|
1.529
|
1.6279
|
1.8362
|
1.9083
|
2.0316
|
|
100
|
1.0924
|
1.1276
|
1.1893
|
1.4529
|
1.4997
|
1.5818
|
1.8133
|
1.8717
|
1.9742
|
5,10
|
5
|
1.2801
|
1.922
|
2.9751
|
1.7001
|
2.548
|
3.9265
|
2.1296
|
3.2062
|
4.9975
|
|
15
|
1.2117
|
1.3938
|
1.6498
|
1.611
|
1.8528
|
2.1928
|
2.0126
|
2.3153
|
2.7415
|
|
25
|
1.1742
|
1.284
|
1.4481
|
1.5614
|
1.7073
|
1.9254
|
1.9498
|
2.1323
|
2.405
|
|
50
|
1.1286
|
1.1882
|
1.2859
|
1.5009
|
1.5801
|
1.7101
|
1.8737
|
1.9727
|
2.135
|
|
75
|
1.1062
|
1.1496
|
1.2239
|
1.4712
|
1.5289
|
1.6277
|
1.8364
|
1.9085
|
2.0319
|
|
100
|
1.0924
|
1.1275
|
1.1893
|
1.4528
|
1.4996
|
1.5817
|
1.8135
|
1.8719
|
1.9744
|
10,0
|
5
|
1.1598
|
1.6933
|
2.3537
|
1.5425
|
2.2521
|
3.1304
|
1.9252
|
2.8109
|
3.9071
|
|
15
|
1.1649
|
1.366
|
1.6012
|
1.5494
|
1.8168
|
2.1296
|
1.9338
|
2.2676
|
2.658
|
|
25
|
1.1467
|
1.2724
|
1.4295
|
1.5251
|
1.6923
|
1.9013
|
1.9035
|
2.1122
|
2.373
|
|
50
|
1.1165
|
1.1844
|
1.2803
|
1.4849
|
1.5752
|
1.7028
|
1.8534
|
1.9661
|
2.1253
|
|
75
|
1.0991
|
1.1475
|
1.221
|
1.4618
|
1.5262
|
1.6239
|
1.8245
|
1.9049
|
2.0268
|
|
100
|
1.0875
|
1.1262
|
1.1874
|
1.4464
|
1.4978
|
1.5793
|
1.8053
|
1.8695
|
1.9711
|
10,5
|
5
|
1.1961
|
1.8128
|
2.7109
|
1.6002
|
2.4441
|
3.7191
|
1.9943
|
3.0402
|
4.6058
|
|
15
|
1.1751
|
1.3825
|
1.6279
|
1.5654
|
1.8429
|
2.1719
|
1.9531
|
2.2989
|
2.7087
|
|
25
|
1.1523
|
1.2801
|
1.4405
|
1.534
|
1.7045
|
1.9186
|
1.9142
|
2.1268
|
2.3938
|
|
50
|
1.119
|
1.1874
|
1.2841
|
1.4889
|
1.58
|
1.7088
|
1.8582
|
1.9718
|
2.1325
|
|
75
|
1.1007
|
1.1493
|
1.2232
|
1.4643
|
1.5291
|
1.6273
|
1.8275
|
1.9083
|
2.031
|
|
100
|
1.0887
|
1.1275
|
1.1889
|
1.4483
|
1.4999
|
1.5817
|
1.8075
|
1.8719
|
1.974
|
10,10
|
5
|
1.2009
|
1.8295
|
2.768
|
1.5969
|
2.4323
|
3.678
|
1.9974
|
3.0509
|
4.6435
|
|
15
|
1.1764
|
1.3846
|
1.6314
|
1.5645
|
1.8414
|
2.1695
|
1.9539
|
2.3002
|
2.7109
|
|
25
|
1.153
|
1.2811
|
1.4419
|
1.5335
|
1.7038
|
1.9176
|
1.9146
|
2.1274
|
2.3946
|
|
50
|
1.1193
|
1.1878
|
1.2846
|
1.4887
|
1.5797
|
1.7085
|
1.8584
|
1.972
|
2.1328
|
|
75
|
1.1009
|
1.1496
|
1.2234
|
1.4641
|
1.5289
|
1.6272
|
1.8276
|
1.9085
|
2.0311
|
|
100
|
1.0888
|
1.1276
|
1.1891
|
1.4482
|
1.4998
|
1.5815
|
1.8076
|
1.872
|
1.9741
|
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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