ESTIMATION OF PROCESS CAPABILITY IN BAYESIAN PARADIGM

 

Chetan Malagavi ¹ , Sharada V. Bhat ²

 

¹ Department of Mathematics, GITAM Deemed to be University, Bengaluru-561203. Karnataka, India and Research Scholar, Department of Statistics, Karnatak University, Dharwad-580003. Karnataka, India

² Department of Statistics. Karnatak University, Dharwad-580003. 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

 

                                                                               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 𝑪_𝒑

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 … X_n is a random sample of size n from ), then the density function of X_i 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 𝑪̂_𝒑𝒄

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

                                                                          

Figure 1  for  and various values of

 

Figure 2

                                                                            

Figure 2  for with  and various values of

Figure 3

                                                                         

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 𝑝_𝑐. 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 𝑝_𝑐. 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

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

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

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

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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