An application example - 多品質特性製程的製程能力指標 CpkT 的分布和推論

Table 13 gives the most conservative sample sizes required for the estimator of C^T_pkto be within a sample error less than 0.005, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1 for various C^T_pk and significance level α. The most conservative case is used in calculation.

(i.e., C^T_pk =C_pk₁ o C^T_pkr=C_pk₂).

For example, for C^T_pk =1.33 with α= 0.05, a sample size≥193 ensures that the sampling error of C^^Tpk is no greater than 0.09.

6. An application example

For illustration, we consider a real example presented in Pearn and Wu (2005b), which is taken from an optical communication manufacturing factory located in Science-based Industrial Park in Taiwan. The example involves a process manufacturing dual-fiber tips, a component used in making fiber optic cables.

Figure 12 depicts a sample of the dual-fiber tips. Sixty dual-fiber tips were taken from a stable (i.e., in statistical control) process in the factory, and two product quality characteristics were measured, (i) Capillary length and (ii) Wedge. For a particular model of dual-fiber tips, the specifications of characteristics are listed in Table 14. According to Pearn and Wu (2005b), it is reasonable to assume that these 60 data were from a normal distribution with two independent quality characteristics. The sample mean, standard deviation, and specifications along with the individual C^pk of each characteristic are summarized in Table 14.

If the quality requirement was predefined as C^T_pk ≥1.33, then we can make some statistical inferences on C^T_pk by using hypothesis testing and interval estimation. For testing the null hypothesis Ho as given in (4.5) with c =1.33, the testing statistic T given in (4.6) is 2.321008 > Z_0.05 = 1.645. Thus, H_o is rejected at α = 0.05. We conclude that the process meets the capability requirement of C^T_pk >1.33 with 95% confidence.

Moreover, C = 1.702917^Tpk and C^{T LB}_pk =1.438560by (4.3) and (4.8), respectively. Thus, we have 95% confidence to say C^T_pk is no less than 1.43856, or equivalently, there are no more than 16 PPM of non-conformities as given in (3.5).

7. Conclusions

Process yield is the most common criterion used in the manufacturing industry for measuring process performance. The widely used capability index C_pk is a yield-related index, in the sense that it can provide a lower bound for the yield of a process with single characteristic. But in many real applications, the process has multiple characteristics.

In this paper, we extend Cpk to an index C^T_pk to assess the yield of processes with multiple characteristics. It is shown that 2 (3Φ C^T_pk) 1 % (3− ≤ Yield≤ Φ C^T_pk), a property holds for the univariate Cpk. Based on the new index C^T_pk, the practitioners can make reliable decisions for capability testing and monitoring the overall performance of all process characteristics.

Unfortunately, the distributional properties of the natural estimator C^^Tpk are mathematically intractable. We derive a normal approximation to the distribution of the C^^Tpk

by the first-order Taylor expansion and investigate the accuracy and precision of C^^Tpk by simulation.

Applying the asymptotic distribution of C^^Tpk, hypothesis testing, confidence interval, and a confidence lower bound C^{T LB}_pk are constructed. We investigate the behavior of C^{T LB}_pk versus C_pk1and C_pk2 for given C^T_pk^′s and find that the most conservative lower bound can be obtained by setting one of C_pk′s at the given C^T_pk and the other at infinity. We also provide tables for engineers or practitioners to use in assessing their processes. On the other hand, it is also found that C^{T LB}_pk reaches its absolute maximum when Cpk1 = Cpk2.

As an illustrative example, an application example on dual-fiber tips taken from Pearn and Wu (2005b) is employed. The practical implementation of the statistical theory for manufacturing capability assessment bridges the gap between the theoretical development and the in-plant applications.

For the future research, we could consider the following topics:

 Use the second-order expansion of Taylor series to approximate the distribution of

^T_pk

C to get a more accurate approximation.

 Generalize C^T_pk for processes with asymmetric tolerances.

 Explore the similar research to C^T_pk for Cp, CPU, CPL, Cpk, Cpm, Cpmk.

 Develop appropriate process capability measurement based on C^T_pk when gauge measurement errors exist.

Followings are some other potential research topics:

 develop a powerful test for on-sided or two-sided supplier selection problem.

 develop a decision making method for product acceptance.

 develop tool replacement strategies for production with a low fraction of defectives.

References

1. Bickel, P. J. and Doksum, K. A. (2007). Mathematical Statistics Basic Ideas and Selected Topics, Volume I, 322－324. Pearson Prentice Hall.

2. Bothe, D. R. (1992). A capability study for an entire product. ASQC Quality Congress Transactions, 172-178.

3. Boyles, R. A. (1991). The Taguchi capability index. Journal of Quality Technology, 23, 17-26.

4. Boyles, R. A. (1994). Process capability with asymmetric tolerances. Communications in Statistics: Simulations and Computation, 23 (3), 615-643.

5. Boyles, R.A. (1996). Multivariate process analysis with lattice data. Technometrics, 38 (1), 37-49.

6. Chan, L. K., Cheng, S.W., and Spiring, F. A. (1988). A new measure of process capability: Cpm. Journal of Quality Technology, 20(3), 162-175.

7. Chen, H. (1994). A multivariate process capability index over a rectangular solid tolerance zone. Statistica Sinica, 4, 749-758.

8. Chen, K. S., Pearn, W. L., and Lin, P. C. (2003). Capability measures for processes with multiple characteristics. Quality and Reliability Engineering International, 19, 101-110.

9. Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18, 41-52.

10. Kotz, S. and Johnson, N. L. (1993). Process Capability Indices. Chapman and Hall, London, U.K.

11. Kotz, S. and Johnson, N. L. (2002). Process capability indices－review, 1992－2000.

Journal of Quality Technology, 34 (1), 1-19.

12. Kotz, S. and Lovelace, C. (1998). Process Capability Indices in Theory and Practice.

Arnold, London, U.K.

13. Pearn, W. L. and Kotz, S. (2006). Encyclopedia and Handbook of Process Capability Indices. World Scientific.

14. Pearn, W. L., Kotz, S., and Johnson, N. L. (1992). Distributional and inferential properties of process capability indices. Journal of Quality Technology, 24(4), 216-231.

15. Pearn, W. L. and Shu, M. H. (2003). Lower confidence bounds with sample size information for Cpm applied to production yield assurance. International Journal of Production Research, 41(15), 3581-3599.

16. Pearn, W. L. and Wu, C. W. (2005a). Measuring manufacturing capability for couplers and wavelength division multiplexers. International Journal of Advanced Manufacturing Technology, 25, 533-541.

17. Pearn, W. L. and Wu, C. W. (2005b). Production quality and yield assurance for processes with multiple independent characteristics. European Journal of Operational Research, 173, 637-647.

18. Wang, F. K. and Chen, J. (1998-1999). Capability index using principal component analysis. Quality Engineering, 11, 21-27.

19. Wang, F. K. and Du, T. C. T. (2000). Using principal component analysis in process performance for multivariate data. Omega－The international Journal of Management Science, 28, 185-194.

20. Wang, F. K., Hubele, N. F., Lawrence, P., Miskulin, J. D., and Shahriari, H. (2000).

Comparison of three multivariate process capability indices. Journal of Quality Technology, 32(3), 263-275.

where the first inequality holds by (A4) and the second inequality holds by (A3).

This completes the induction.

Then it suffices to show that

Proof. By definition, we have

| | for i=1,…,m. By definition,

( )

Employing the first-order expansion of m-variates Taylor series, we can obtain

  

Appendix C

Another proof of Theorem 1.

Since C^T_pk = f(µ₁,...,µ σ_m; ₁²,...,σ_m²) and C^^Tpk = f(^µ1,...,µ σ^{ }m; 1²,...,σ^²m), where µ^i =Xi

By Theorem 5.3.5. in Bickel, and Doksum (2007), we can obtain the desired result:

 2

(

² ²

)

Cpk

Figure 1. Upper bounds on NCPPM values of C^T_pk.

Figure 2. Comparison of the probability density function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =60, 200, 500, 1000 (C^T_pk =1.0 and µ₁≠m₁ and µ₂ ≠m₂).

Figure 3. Comparison of the distribution function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =50, 200, 500, 1000 (C^T_pk =1.0 and

1 m1 and 2 m2

µ ≠ µ ≠ ).

Figure 4. Comparison of the probability density function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =60, 200, 500, 1000 (C^T_pk =1.0 and

1 = 1 and 2= 2

µ m µ m ).

Figure 5. Comparison of the distribution function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =50, 200, 500, 1000 (C^T_pk =1.0 and

1= 1 and 2= 2

µ m µ m ).

Figure 6. Comparison of the probability density function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =60, 200, 500, 1000 (C^T_pk = 1.33 and µ₁≠m₁ and µ₂ ≠m ). ₂

Figure 7. Comparison of the distribution function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =50, 200, 500, 1000 (C^T_pk =1.33 and

1 1 and 2 2

µ ≠m µ ≠m ).

Figure 8. Comparison of the probability density function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =60, 200, 500, 1000 (C^T_pk =1.33 and µ₁=m₁ and µ₂ =m ). ₂

Figure 9. Comparison of the distribution function (dash line) of C^^Tpk obtained by simulation and its normal approximation (solid line) for n =50, 200, 500, 1000 (C^T_pk = 1.33 and

1 1 and 2 2

µ =m µ =m ).

(a)1.0≤C_pk₁≤2.0,1.0≤C_pk₂≤2.0and C^T_pk =1 (b)1.33≤C_pk₁≤2.0,1.33≤C_pk₂≤2.0andC^T_pk =1.33

Figure 10. Curves of C^{T LB}_pk versus ( C_pk1, C_pk2) with α = 0.05 and n =10(20)90 (from bottom to top in plot).

(a) (b)

T LB

Cpk C^{T LB}_pk

T LB

Cpk T LB

Cpk 2

Cpk Cpk2

Cpk 2

Cpk

(a) (b)

(a) 1.0≤C_pk₁≤2.0,1.0≤C_pk₂≤2.0and

T 1 Cpk =

(b)1.33≤C_pk₁≤2.0,1.33≤C_pk₂≤2.0and

=1.33

Cpk

T 1.5 Cpk =

(d)1.67≤C_pk₁≤2.0,1.67≤C_pk₂ ≤2.0and

=1.67

Cpk .

Figure 11. Curves of C^{T LB}_pk versus Cpk1 for various C^T_pk, α = 0.05, and n=10(20)90 (from bottom to top in plot).

T LB

Cpk

T LB

Cpk T LB

Cpk

T LB

Cpk

Figure 12. A sample of the dual-fiber tips. (from Pearn and Wu (2005b)).

Table 1. Corresponding upper bounds of NCPPM for some specific values of C^T_pk.

Cpk Upper bound of NCPPM

1.00 2699.796

1.25 176.835

1.33 66.073

1.45 13.614

1.50 6.795

1.60 1.587

1.67 0.544

2.00 0.002

Table 2. Coverage rate and 90% confidence interval length (in parentheses) for various cases of C^T_pk=1.0 with n =30, 50, 100, 500, 1000.

sample size n coverage rate (confidence interval length)

30 0.9436

Table 3. Coverage rate and 90% confidence interval length (in parentheses) for various cases of C^T_pk=1.33 with n =30, 50, 100, 500, 1000.

sample size n coverage rate (confidence interval length)

30 0.9295

Table 4. Coverage rate and 90% confidence interval length (in parentheses) for various cases of C^T_pk=1.5 with n =30, 50, 100, 500, 1000.

sample size n coverage rate (confidence interval length)

30 ^0.9340

Table 5. Coverage rate and 90% confidence interval length (in parentheses) for various cases of C^T_pk=2.0 with n =30, 50, 100, 500, 1000.

sample size n coverage rate (confidence interval length)

100 ^0.9227

Table 6. The most conservative 95% lower confidence bounds C^{T LB}_pk of C^T_pk for C^^Tpk=1(0.1)2, n =10(10)200.

n C^T_pk

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

10 0.5934 0.6598 0.7258 0.7914 0.8567 0.9217 0.9865 1.0511 1.1156 1.1800 1.2442 20 0.7125 0.7888 0.8647 0.9404 1.0158 1.0911 1.1662 1.2412 1.3161 1.3909 1.4656 30 0.7652 0.8459 0.9262 1.0064 1.0863 1.1661 1.2458 1.3254 1.4049 1.4843 1.5637 40 0.7967 0.8799 0.9629 1.0457 1.1283 1.2108 1.2933 1.3756 1.4578 1.5400 1.6221 50 0.8182 0.9032 0.9879 1.0725 1.1570 1.2414 1.3256 1.4098 1.4939 1.5780 1.6620 60 0.8340 0.9203 1.0064 1.0924 1.1782 1.2639 1.3495 1.4351 1.5206 1.6061 1.6915 70 0.8463 0.9336 1.0208 1.1078 1.1946 1.2814 1.3681 1.4548 1.5413 1.6279 1.7144 80 0.8562 0.9444 1.0323 1.1202 1.2079 1.2955 1.3831 1.4706 1.5580 1.6454 1.7328 90 0.8645 0.9533 1.0419 1.1305 1.2189 1.3072 1.3955 1.4837 1.5719 1.6600 1.7481 100 0.8714 0.9608 1.0500 1.1392 1.2282 1.3171 1.4060 1.4948 1.5836 1.6723 1.7610 110 0.8774 0.9673 1.0570 1.1466 1.2362 1.3256 1.4150 1.5044 1.5937 1.6829 1.7721 120 0.8826 0.9729 1.0631 1.1532 1.2432 1.3331 1.4229 1.5127 1.6024 1.6922 1.7818 130 0.8872 0.9779 1.0685 1.1589 1.2493 1.3396 1.4298 1.5200 1.6102 1.7003 1.7904 140 0.8913 0.9824 1.0733 1.1641 1.2548 1.3454 1.4360 1.5266 1.6171 1.7076 1.7980 150 0.8950 0.9863 1.0776 1.1687 1.2597 1.3507 1.4416 1.5325 1.6233 1.7141 1.8049 160 0.8983 0.9900 1.0815 1.1728 1.2642 1.3554 1.4466 1.5378 1.6289 1.7200 1.8111 170 0.9014 0.9932 1.0850 1.1766 1.2682 1.3597 1.4512 1.5426 1.6340 1.7254 1.8167 180 0.9042 0.9963 1.0882 1.1801 1.2719 1.3637 1.4554 1.5471 1.6387 1.7303 1.8219 190 0.9067 0.9990 1.0912 1.1833 1.2754 1.3673 1.4593 1.5511 1.6430 1.7348 1.8266 200 0.9091 1.0016 1.0940 1.1863 1.2785 1.3707 1.4628 1.5549 1.6470 1.7390 1.8310

Table 7. The largest possible 95% lower confidence bounds C^{T LB}_pk of C^T_pk for C^^Tpk= 1(0.1)2, n =10(10)200.

^T_pk C

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

10 0.6788 0.7573 0.8353 0.9127 0.9898 1.0666 1.1430 1.2192 1.2952 1.3709 1.4466 20 0.7729 0.8577 0.9421 1.0262 1.1100 1.1935 1.2768 1.3600 1.4430 1.5259 1.6087 30 0.8146 0.9021 0.9894 1.0764 1.1632 1.250 1.3361 1.4224 1.5085 1.5945 1.6805 40 0.8394 0.9287 1.0176 1.1064 1.1949 1.2833 1.3715 1.4596 1.5476 1.6355 1.7233 50 0.8564 0.9467 1.0369 1.1268 1.2166 1.3062 1.3956 1.4850 1.5742 1.6634 1.7525 60 0.8689 0.9601 1.0511 1.1419 1.2325 1.3230 1.4134 1.5037 1.5939 1.6840 1.7741 70 0.8786 0.9705 1.0621 1.1536 1.2450 1.3362 1.4273 1.5183 1.6092 1.7000 1.7908 80 0.8865 0.9788 1.0710 1.1631 1.2550 1.3468 1.4384 1.5300 1.6215 1.7129 1.8043 90 0.8929 0.9858 1.0784 1.1709 1.2633 1.3555 1.4477 1.5398 1.6317 1.7236 1.8155 100 0.8984 0.9916 1.0847 1.1775 1.2703 1.3629 1.4555 1.5480 1.6404 1.7327 1.8250 110 0.9032 0.9967 1.0900 1.1832 1.2763 1.3693 1.4622 1.5550 1.6478 1.7405 1.8331 120 0.9073 1.0011 1.0947 1.1882 1.2816 1.3749 1.4681 1.5612 1.6543 1.7473 1.8402 130 0.9109 1.0050 1.0988 1.1926 1.2862 1.3798 1.4732 1.5666 1.6600 1.7533 1.8465 140 0.9142 1.0084 1.1025 1.1965 1.2904 1.3842 1.4779 1.5715 1.6651 1.7586 1.8521 150 0.9171 1.0115 1.1058 1.2000 1.2941 1.3881 1.4820 1.5759 1.6696 1.7634 1.8571 160 0.9197 1.0143 1.1088 1.2032 1.2975 1.3916 1.4857 1.5798 1.6738 1.7677 1.8616 170 0.9221 1.0169 1.1115 1.2061 1.3005 1.3949 1.4892 1.5834 1.6776 1.7717 1.8658 180 0.9243 1.0192 1.1140 1.2087 1.3033 1.3978 1.4923 1.5867 1.6810 1.7753 1.8696 190 0.9263 1.0214 1.1163 1.2112 1.3059 1.4006 1.4951 1.5897 1.6842 1.7786 1.8730 200 0.9282 1.0234 1.1184 1.2134 1.3083 1.4031 1.4978 1.5925 1.6871 1.7817 1.8762

Table 8. The precision R for the most conservative 95% lower confidence bounds C^{T LB}_pk of

Cpk for C^^Tpk=1(0.1)2, n =10(10)200.

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

10 0.5934 0.5998 0.6048 0.6088 0.6119 0.6145 0.6166 0.6183 0.6198 0.6211 0.6221 20 0.7125 0.7171 0.7206 0.7234 0.7256 0.7274 0.7289 0.7301 0.7312 0.7321 0.7328 30 0.7652 0.7690 0.7718 0.7742 0.7759 0.7774 0.7786 0.7796 0.7805 0.7812 0.7819 40 0.7967 0.7999 0.8024 0.8044 0.8059 0.8072 0.8083 0.8092 0.8099 0.8105 0.8111 50 0.8182 0.8211 0.8233 0.8250 0.8264 0.8276 0.8285 0.8293 0.8299 0.8305 0.8310 60 0.8340 0.8366 0.8387 0.8403 0.8416 0.8426 0.8434 0.8442 0.8448 0.8453 0.8458 70 0.8463 0.8487 0.8507 0.8522 0.8533 0.8543 0.8551 0.8558 0.8563 0.8568 0.8572 80 0.8562 0.8585 0.8603 0.8617 0.8628 0.8637 0.8644 0.8651 0.8656 0.8660 0.8664 90 0.8645 0.8666 0.8683 0.8696 0.8706 0.8715 0.8722 0.8728 0.8733 0.8737 0.8741 100 0.8714 0.8735 0.8750 0.8763 0.8773 0.8781 0.8788 0.8793 0.8798 0.8802 0.8805 110 0.8774 0.8794 0.8808 0.8820 0.8830 0.8837 0.8844 0.8849 0.8854 0.8857 0.8861 120 0.8826 0.8845 0.8859 0.8871 0.8880 0.8887 0.8893 0.8898 0.8902 0.8906 0.8909 130 0.8872 0.8890 0.8904 0.8915 0.8924 0.8931 0.8936 0.8941 0.8946 0.8949 0.8952 140 0.8913 0.8931 0.8944 0.8955 0.8963 0.8969 0.8975 0.8980 0.8984 0.8987 0.8990 150 0.8950 0.8966 0.8980 0.8990 0.8998 0.9005 0.9010 0.9015 0.9018 0.9022 0.9025 160 0.8983 0.9000 0.9013 0.9022 0.9030 0.9036 0.9041 0.9046 0.9049 0.9053 0.9056 170 0.9014 0.9029 0.9042 0.9051 0.9059 0.9065 0.9070 0.9074 0.9078 0.9081 0.9084 180 0.9042 0.9057 0.9068 0.9078 0.9085 0.9091 0.9096 0.9101 0.9104 0.9107 0.9110 190 0.9067 0.9082 0.9093 0.9102 0.9110 0.9115 0.9121 0.9124 0.9128 0.9131 0.9133 200 0.9091 0.9105 0.9117 0.9125 0.9132 0.9138 0.9143 0.9146 0.9150 0.9153 0.9155

σ 3.2050 4.0062 3.5611 5.3416 3.5611 3.5611 3.2050 3.2050 5.3416 3.2050

2 2

σ 3.2050 4.0062 3.5611 5.3416 5.3416 4.0062 3.5611 5.3416 4.0062 4.0062

1 1

− 0 0.8012 0.3561 2.1366 2.1366 0.8012 0.3561 2.1366 0.8012 0.8012

n Estimate of E(C^^Tpk)and its standard error (in parentheses)

Table 11. Estimates of E(C^^Tpk)and their standard errors (in parentheses) for some cases of

T 1.5

Table 12. Estimates of E(C^^Tpk)and their standard errors (in parentheses) for some cases of

T 2.0

Table 13. Sample sizes required for a specified margin of sampling error.

Cpk _α_{\ error}

0.05 0.06 0.07 0.08 0.09 0.1

1.00

0.05 413 287 211 162 128 104

0.025 587 408 300 230 181 147

0.01 826 574 422 323 255 207

1.33

0.05 624 434 319 244 193 156

0.025 886 616 452 346 274 222

0.01 1248 867 637 488 386 312

1.50

0.05 750 521 383 293 232 188

0.025 1065 740 544 416 329 267

0.01 1501 1042 766 587 464 376

1.67

0.05 903 627 461 353 279 226

0.025 1282 891 654 501 396 321

0.01 1806 1254 922 706 558 452

2.00

0.05 1224 850 625 478 378 306

0.025 1738 1207 887 679 537 435

0.01 2448 1700 1249 957 756 612

Table 14. Sample mean, sample standard deviation, specifications of individual characteristics for the dual-fiber tips, and the estimated capability indices.

Characteristic X S LSL USL C_pk

Capillary length (mm) 6.255 0.04035 6.00 6.50 2.024

Wedge (^o) 7.99 0.0959 7.5 8.5 1.703

在文檔中多品質特性製程的製程能力指標 CpkT 的分布和推論 (頁 21-54)