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An Extension of the Product Acceptance

Determination for One-Sided Process with

Multiple Characteristics

W. L. Pearn,

a

C. H. Wu,

a

* H. N. Hung

b

and C. M. Kao

b

In statistical quality control, product acceptance determination is an important problem for producer and consumer. In practical, particularly more than one quality characteristic must be simultaneously considered to improve the product quality because of the product design. In this article, we investigate the lot sentencing problem for normally distributed process with one-sided specification and multiple characteristics. We not only provide a simple procedure to help practi-tioners make reliable decision easily but also tabulate the required sample size and the corresponding critical acceptance value for various producer’s and consumer’s risks with the capability requirements AQL (acceptable quality level) and LTPD (lot tolerance percent defective). Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: critical values; multiple characteristics; one-sided specification; product acceptance determination

1. Introduction

P

roduct acceptance determination can be used for quality assurance applications involving quality contract on product orders between producers and consumers. Product acceptance determination basically consists of a sample size for inspection and an acceptance criterion. The operating characteristic (OC) curve plots the probability of accepting the lot versus the lot fraction defective. The producer is primarily interested in insuring that good lots would be accepted, and the consumer wants to be reason-ably sure that bad products would be rejected. Therefore, a product acceptance determination usually focus on two designated points, (AQL,1 a) and (LTPD,b), on the OC curve. The symbols a and b denote the risk of producers and consumers, respectively. AQL presents the poorest level of quality for the vendor’s process average, and LTPD is the poorest quality level that the consumer is willing to accept.

Pearn and Wu1developed a clear product acceptance determination procedure for a one-sided process with single characteristic. The results attended are very practical for industrial application but are restricted to process with only one quality characteristic. Hence, we extend the results to cases with multiple independent quality characteristics. The generalized indices CT

puand CTpkfor processes with

one-sided and two-sided specifications were proposed by Wu and Pearn2and Pearn et al.3

2. Multiple characteristics

Hsu et al.4discussed the sample size determination problem for production yield estimation with multiple quality characteristics. Then,

Pearn et al.5implemented the process capability index with multiple characteristics to deal with the photolithography control in wafer fabrication. Later, Pearn and Cheng6proposed a process capability index for measuring production yield with multiple characteristics.

Recently, Awad and Kovach7investigated a multiresponse optimization problem using multivariate process capability index. Then, Goethals and Cho8developed a target-focused index for process with multiple characteristics. Kotz and Johnson,9Wu et al.,10and

Yum and Kim11provided some reviews and overviews about process capability index. Recent studies on process capability include White and Borror,12Spiring,13and Negrin et al.14

a

Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, R.O.C.

bInstitute of Statistics, National Chiao Tung University, Taiwan, R.O.C.

*Correspondence to: C. H. Wu, Department of Industrial Engineering and Management, National Chiao Tung University, 1001 University Road, Hsinchu, Taiwan 300, R.O.C.

E-mail: hexjacal.iem96g@nctu.edu.tw

(wileyonlinelibrary.com) DOI: 10.1002/qre.1378 Published online 16 March 2012 in Wiley Online Library

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Wu and Pearn2proposed the overall capability index for one-sided processes, designed as CT pu¼ 1 3 Φ-1 Ym j¼1 Φ 3Cpuj   ( ) ; (1)

where Cpujdenotes the Cpuvalue of the jth characteristic for j = 1, 2,. . ., m, and m is the number of characteristics. f() is the

cumu-lative distribution function of the standard normal distribution. A one-to-one correspondence relationship between the index CT puand

the overall process yield P can be established as P¼ Πm j¼1Pj¼ Π m j¼1Φ 3Cpuj   ¼Φð3CT puÞ¼Φð3cÞ (2)

Hence, the new index CT

puprovides an exact measure on the overall process yield.

3.

Statistical properties of ^

C

puT

To handle the issue for cases with multiple quality characteristics, we applied the similar technique used by Pearn et al.3to derive the asymptotic distribution for the natural estimator of CT

pu, which is defined as ^CpuT¼1 3Φ 1 Πm j¼1Φ 3^Cpuj     ¼ 1 3Φ 1 Ym j¼1 Φ USL Xj Sj   ( ) (3)

where Xjand Sjdenote the sample mean and the sample variance of jth characteristic. Using the Taylor expansion technique for

m-variate and taking thefirst order, the asymptotic distribution of ^CpuT

is ^CpuT  N CT pu; 1 9n f 3 CT pu   2 Xm j¼1 aj2þ bj2   0 B @ 1 C A (4) where aj¼ Y i¼1 i 6¼ j m Φ 3Cpui   f 3Cpuj   and bj¼ 3Cpuj ffiffiffi 2 p aj; j ¼ 1; . . . ; m

f() represents the probability density function of the standard normal distribution. Later, Pearn et al.3

also showed that the con-servative lower confidence bound can be obtained by setting Cpu1¼ CpuT and others Cpuj=1 (j = 2, 3, . . ., m). The approximate 100

(1 a)% lower confidence bound for CT

pucan be expressed as CT LB pu ¼ 2^CTpu ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4z2 a 9nþ 2z2 a n ^CTpu2 2z4 a 9n2 q 2 z2 a=n (5) Therefore, we would derive the reliable critical value c0with such parameter setting. At this time, the approximate sampling

dis-tribution of ^CpuT can be rewritten as a simpler form, that is,

^CpuT N CT pu; 1 9nþ CT pu  2 2n 0 B @ 1 C A: (6)

4.

Product acceptance determination

The CT

puindex can be used as a quality benchmark for acceptance of a product lot. Let (AQL,1 a) and (LTPD,b) be the two points on

the OC curve of interest. Two conditions are considered: Pr{Reject the lot| P≥ AQL} = Pr{Reject the lot| CT

pu≥CAQL}≤ a

Pr{Accepting the lot| P≤ LTPD} = Pr{Accepting the lot| CT

pu≤CLTPD}≤ b

That is, for the protection of producers, the probability of rejecting acceptable lots is less thana. At the same time, for the protec-tion of consumers, the probability of accepting unqualified (unacceptable) lots is less than b. Therefore, our object is solving the

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following two nonlinear simultaneous equations and then obtaining the required inspection sample size n and critical acceptance value c0of ^Cpu T a≥Z c0 0 ffiffiffi n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 9þ CAQL2   q exp n xð  CAQLÞ 2 2 9þ CAQL2   " # dx (7) b≥Z 1 c0 ffiffiffi n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 9þ CLTPD2   q exp n xð  CLTPDÞ 2 2 9þ CLTPD2   " # dx (8)

where CAQLand CLTPDrepresent the capability requirements corresponding to the AQL and the LTPD on the basis of the CTpuindex,

respectively. Although the two previously mentioned equations are satisfied, the product acceptance determination procedure would judge the lots under controllable risks. To solve Equations (7) and (8), we let

S1ðn; c0Þ ¼ Z 1 c0 ffiffiffi n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 9þ CAQL2   q exp n xð  CAQLÞ 2 2 9þ CAQL2   " # dx 1  að Þ (9) S2ðn; c0Þ ¼ Z 1 c0 ffiffiffi n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 9þ CLTPD2   q exp n xð  CLTPDÞ 2 2 9þ CLTPD2   " # dx b (10)

For CAQL= 1.33 and CLTPD= 1.00, Figures 1a and 1b and Figures 2a and 2b display the surface and the contour plots of Equations (9)

and (10), respectively, witha = b = 0.05. Then, Figure 3a presents the two surfaces simultaneously, and Figure 3b shows the two contour plots simultaneously.

From Figure 3b, we can see that the interaction of S1(n, c0) and S2(n, c0) contour curves at level 0 is (n, c0)= (79, 1.14502), which is the

solution to nonlinear simultaneous Equations (9) and (10).That is, in this case, the minimum required sample size n=79, and the corresponding critical acceptance value c0=1.14502 of the product acceptance determination on the basis of the capability index

CpuT . For the purpose of practical applications, we performed extensive calculations to obtain the solution of Equations (9) and (10) then tabulated the critical acceptance values (c0) and the required sample sizes (n) for the product acceptance on the basis of CTpuindex

in Table AI.

From the results presented in Table AI, we observed that the greater the risk (a and/or b) the producer or customer could suffer, the smaller the required sample size n. This phenomenon can be interpreted intuitively: if we expect that the chance of wrongly conclud-ing a bad process as good or good lots as bad ones is smaller, then more sample information is needed to judge the lots. Further, for fixed a-risk, b-risk, and CAQL, the required sample sizes become larger when the CLTPDbecomes larger. This can also be explained by

the same reasoning. The required sample size is smaller when the difference between CAQLand CLTPDis significant because it is relatively

easier to reach the correct decision. For the proposed product acceptance determination procedure to be practical and convenient to use, a step-by-step algorithm is provided as follows:

(a) (b)

Figure 1. (a) Surface plot of S1(n,c0). (b) Contour plot of S1(n,c0)

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Step 1: Decide the process capability requirements (i.e. set the values of CAQLand CLTPD) and choose thea-risk, the chance of

wrongly concluding a capable lot as incapable, and theb-risk, the chance of wrongly concluding a bad lot as good one. Step 2: Check Table AI tofind the critical value (or acceptance criterion) and the required number of product units for inspection

(n,c0) on the basis of the given values ofa-risk,b-risk,CAQL, and CLTPD.

Step 3: Calculate the value of ^CpuT (sample estimator) from these n inspected samples.

Step 4: Make decisions to accept the entire lot if the estimated ^CpuT value is greater than the critical value c0(^Cpu

T > c

0). Otherwise,

reject the entire product.

5.

Case study

The thin-film transistor–type LCD has been intensively developed for electric appliances such as cell phones, personal digital assistants, notebook computers, and monitors. In recent years, LCDs are applied to information equipment and visual equipment, including personal computers, as thin, light, low-power-consuming, wide-viewing angle, high color purity, and lower-power panel display. Response time (rising and falling) and brightness uniformity are essential product quality characteristics, which has significant impact to product quality. For a 15-inch thin-film transistor LCD module, the data collected from a panel plant have three interesting characteristics: response time (rising), response time (falling), and brightness nonuniformity, with upper specification limits USL1= 7 ms, USL2= 18 ms, and

(a) (b)

Figure 2. (a) Surface plot of S2(n,c0). (b) Contour plot of S2(n,c0)

(a) (b)

Figure 3. (a) Surface plot of S1and S2. (b) Contour plot of S1and S2

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USL3= 15%. In contract, CAQLand CLTPDare set to 1.33 (which is equivalent to no more than 33 ppm fraction of defectives for the product)

and 1.00 (which is equivalent to 1349 ppm fraction of defectives for the product), witha-risk = 0.05 and b-risk = 0.05. Therefore, we find the acceptance critical values and inspected sample sizes of product acceptance determination (n,c0) = (79, 1.1450), as shown in Table AI.

The 79 required data of product items for inspection are taken from the process randomly, using microscope visually for inspec-tion, and the observed measurements are displayed in Table I. On the basis of these data, sample mean, sample standard deviainspec-tion, and Cpuindices for the three characteristics are listed in Table II. Therefore, the consumer would“reject” the entire products because

the sample estimator from the inspection, 0.9218, is smaller than the critical acceptance value, 1.1450. We developed an effective pro-duct acceptance determination procedure on the basis of process capability index CT

puto deal with lot sentencing problem with very

low fraction of defectives and multiple characteristics. The required sample size and the corresponding critical acceptance value for various producer’s and consumer’s risks with the capability requirements AQL and LTPD are also tabulated.

References

1. Pearn WL, Wu CW. Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives. Omega, The Inter-national Journal of Management Science 2006; 34:90–101. DOI: 10.1016/j.omega.2004.08.001

2. Wu CW, Pearn WL. Measuring manufacturing capability for couplers and wavelength division multiplexers. The International Journal of Advanced Manufacturing Technology 2005; 25:533–541. DOI: 10.1007/s00170–003–1793–9

3. Pearn WL, Shiau JJH, Tai YT, Li MY. Capability assessment for processes with multiple characteristics : A generalization of the popular index Cpk.

Quality and Reliability Engineering International 2011; 27(8):1119–1129. DOI: 10.1002/qre.1200

4. Hsu YC, Pearn WL, Chuang YF. Sample size determination for production yield estimation with multiple independent process characteristics. European Journal of Operational Research 2009; 196:968–978. DOI: 10.1016/j.ejor.2008.04.029

Table I. Sample data of 79 observations Response time (rising)

5.975 5.824 5.870 6.455 5.679 5.626 6.032 5.755 5.867 5.939 6.146 5.917 6.067 6.164 6.454 7.314 6.132 6.107 6.442 5.884 5.719 6.381 6.298 6.426 6.384 6.387 6.320 6.162 6.360 5.524 5.772 6.234 6.053 5.816 6.129 6.297 6.226 5.973 5.964 5.889 6.008 6.056 6.031 6.220 6.030 6.619 6.172 6.396 5.241 6.142 5.594 5.341 6.119 6.024 6.028 6.403 6.159 5.451 5.305 6.191 5.453 6.099 6.368 6.006 6.035 6.468 6.062 5.566 5.678 5.883 5.535 5.849 6.154 5.936 6.488 5.930 5.677 6.722 5.542

Response time (falling)

15.413 14.844 14.484 14.835 14.821 14.709 15.202 15.212 14.941 15.511 15.171 14.834 15.309 15.299 15.265 15.505 15.456 15.259 15.045 15.397 14.960 14.182 14.740 14.748 14.839 15.413 15.134 14.824 15.291 14.653 14.691 14.795 15.524 15.028 14.348 15.240 14.337 15.398 14.874 14.855 14.585 15.426 15.386 15.201 15.047 14.750 15.030 14.568 15.591 14.404 15.241 15.298 14.376 15.468 14.481 15.618 16.048 14.661 15.188 14.255 15.582 14.950 15.057 15.347 14.691 14.737 15.266 15.115 15.135 15.042 14.997 15.357 15.143 14.829 15.216 14.773 15.027 14.790 15.421 Brightness nonuniformity 12.196 13.604 12.648 13.295 13.077 13.147 12.916 13.193 13.042 12.933 12.702 12.463 12.778 13.588 13.779 13.360 12.684 12.586 12.556 13.104 12.565 12.750 13.951 12.443 12.874 12.849 13.425 13.411 12.379 12.904 13.181 12.921 12.845 12.215 12.323 12.979 12.816 13.381 12.957 13.118 13.544 12.201 12.727 12.747 13.404 13.171 12.705 12.773 12.870 13.241 12.872 13.748 12.743 12.973 12.667 13.404 13.067 12.795 13.743 12.924 12.444 12.752 13.176 12.253 12.802 14.111 12.217 12.742 13.386 12.722 12.928 12.293 13.141 13.439 12.852 13.145 13.441 13.028 13.709

Table II. The calculated statistic of three characteristics

xi Si ^Cpuj

Response time (rising) 6.037239 0.348145 0.921801

Response time (falling) 15.03144 0.368657 2.68412

Brightness nonuniformity 12.9726 0.429825 1.572267

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5. Pearn WL, Kang HY, Lee AHI, Liao MY. Photolithography control in wafer fabrication based on process capability indices with multiple characteristics. IEEE Transactions on Semiconductor Manufacturing 2009; 22(3):351–356. DOI: 10.1109/TSM.2009.2024851

6. Pearn WL, Cheng YC. Measuring production yield for processes with multiple characteristics. International Journal of Production Research 2010; 48(15):4519–4536. DOI: 10.1080/00207540903036313

7. Awad MI, Kovach JV. Multiresponse optimization using multivariate process capability index. Quality and Reliability Engineering International 2011; 27:465–477. DOI: 10.1002/qre.1141

8. Goethals PL, Cho BR. The development of a target-focused process capability index with multiple characteristics. Quality and Reliability Engineering International 2011; 27:297–311. DOI: 10.1002/qre.1120

9. Kotz S, Johnson NL. Process capability indices-review, 1992–2000. Journal of Quality Technology 2002; 34(1):1–19. DOI: 10.1061/(ASCE)0733– 9429(1998)124:6(561)

10. Wu CW, Pearn WL, Kotz S. An overview of theory and practice on process capability indices for quality assurance. International Journal of Production Economics 2009; 117:338–359.

11. Yum BJ, Kim KW. A bibliography of the literature on process capability indices : 2000–2009. Quality and Reliability Engineering International 2011; 27:251–268. DOI: 10.1002/qre.1115

12. White TK, Borror CM. Two-dimensional guidelines for measurement system indices. Quality and Reliability Engineering International 2011; 27(4):479–487. DOI: 10.1002/qre.1144

13. Spiring F. Exploring process capability with Mathematics. Quality and Reliability Engineering International 2011; 27(3):369–387. DOI: 10.1002/qre.1112 14. Negrin I, Parmet Y, Schechtman E. Developing a sampling plan based on Cpk– Unknown variance. Quality and Reliability Engineering International

2011; 27(1):3–14. DOI: 10.1002/qre.1094

Appendix A

Table A1. Required sample sizes (n) and critical acceptance values (c0) for variousa-risk and b-risk with selected CAQLand CLTPD

a b

CAQL= 1.33 CAQL= 1.50 CAQL= 1.50 CAQL= 1.67 CAQL= 1.67 CAQL= 2.00

CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.33 CLTPD= 1.33 CLTPD= 1.50 CLTPD= 1.67 n C0 n C0 n C0 n C0 n C0 n C0 0.01 0.01 159 1.1455 79 1.2070 839 1.4102 233 1.4824 1034 1.5811 360 1.8213 0.02 142 1.1348 71 1.1914 749 1.4053 211 1.4727 920 1.5757 322 1.8105 0.03 134 1.1289 67 1.1797 692 1.4014 195 1.4648 849 1.5718 298 1.8037 0.04 127 1.1230 64 1.1758 652 1.3984 184 1.4590 800 1.5688 284 1.7988 0.05 121 1.1172 61 1.1680 621 1.3960 178 1.4551 763 1.5664 270 1.7930 0.06 116 1.1133 59 1.1621 596 1.3936 170 1.4492 735 1.5645 258 1.7891 0.07 112 1.1094 57 1.1563 572 1.3916 164 1.4453 702 1.5620 249 1.7852 0.08 109 1.1055 55 1.1504 553 1.3896 158 1.4424 677 1.5601 241 1.7813 0.09 106 1.1035 55 1.1504 542 1.3887 154 1.4395 660 1.5586 235 1.7773 0.10 102 1.0996 53 1.1406 519 1.3862 149 1.4355 637 1.5566 228 1.7754 0.02 0.01 138 1.1553 68 1.2227 737 1.4158 204 1.4932 906 1.5859 317 1.8320 0.02 124 1.1455 61 1.2070 654 1.4102 181 1.4824 805 1.5811 280 1.8213 0.03 115 1.1387 57 1.1953 601 1.4067 169 1.4756 739 1.5771 258 1.8140 0.04 109 1.1328 54 1.1875 565 1.4038 159 1.4697 694 1.5742 244 1.8086 0.05 103 1.1270 52 1.1797 538 1.4014 152 1.4648 659 1.5713 232 1.8037 0.06 99 1.1230 50 1.1758 512 1.3989 146 1.4609 629 1.5693 221 1.7988 0.07 95 1.1191 48 1.1699 491 1.3965 139 1.4551 605 1.5674 213 1.7949 0.08 92 1.1152 46 1.1621 472 1.3945 134 1.4512 579 1.5649 205 1.7910 0.09 89 1.1123 45 1.1592 458 1.3931 131 1.4492 562 1.5635 198 1.7871 0.10 86 1.1084 44 1.1553 441 1.3911 127 1.4453 541 1.5615 192 1.7842 0.05 0.01 111 1.1738 54 1.2500 597 1.4250 164 1.5122 735 1.5955 253 1.8496 0.02 97 1.1631 48 1.2358 523 1.4199 144 1.5015 643 1.5903 222 1.8394 0.03 90 1.1567 44 1.2236 476 1.4160 132 1.4941 586 1.5864 203 1.8320 0.04 84 1.1504 41 1.2139 444 1.4131 123 1.4878 546 1.5835 190 1.8262 0.05 79 1.1450 41 1.2139 418 1.4104 116 1.4824 513 1.5808 180 1.8213 0.06 76 1.1416 39 1.2031 396 1.4080 112 1.4790 487 1.5784 170 1.8164 0.07 73 1.1377 37 1.1953 378 1.4058 106 1.4736 464 1.5762 163 1.8125 0.08 70 1.1328 35 1.1904 362 1.4038 102 1.4697 445 1.5742 157 1.8091 0.09 67 1.1289 34 1.1855 348 1.4019 98 1.4658 428 1.5723 150 1.8047 0.10 65 1.1260 33 1.1797 335 1.3999 96 1.4629 411 1.5703 145 1.8013 0.06 0.01 105 1.1785 52 1.2598 570 1.4275 155 1.5166 702 1.5979 242 1.8545 0.02 92 1.1680 46 1.2441 495 1.4222 136 1.5059 609 1.5925 210 1.8440 0.03 85 1.1616 42 1.2324 451 1.4185 124 1.4985 554 1.5889 193 1.8369 0.04 79 1.1553 39 1.2227 420 1.4155 116 1.4927 516 1.5859 180 1.8311 (Continues)

282

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Authors' biographies

W. L. Pearn received the PhD degree in Operations Research from the University of Maryland, College Park. He is a Professor of Opera-tions Research and Quality Assurance at National Chiao-Tung University (NCTU), Hsinchu, Taiwan, R.O.C. He was with AT&T Bell Laboratories as a member of the quality research staff before joining NCTU. His research interests include process capability, network optimization, and production management.

Table A1. Continued.

a b

CAQL= 1.33 CAQL= 1.50 CAQL= 1.50 CAQL= 1.67 CAQL= 1.67 CAQL= 2.00

CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.33 CLTPD= 1.33 CLTPD= 1.50 CLTPD= 1.67 n C0 n C0 n C0 n C0 n C0 n C0 0.05 75 1.1504 37 1.2148 394 1.4128 110 1.4878 485 1.5833 170 1.8262 0.06 71 1.1455 35 1.2075 373 1.4104 104 1.4824 459 1.5808 160 1.8213 0.07 68 1.1416 34 1.2031 356 1.4082 99 1.4780 437 1.5786 153 1.8164 0.08 65 1.1367 33 1.1953 341 1.4063 95 1.4741 419 1.5767 147 1.8135 0.09 63 1.1343 32 1.1934 327 1.4043 92 1.4707 402 1.5747 141 1.8096 0.10 60 1.1294 30 1.1836 314 1.4023 89 1.4668 386 1.5728 136 1.8062 0.07 0.01 100 1.1826 48 1.2627 546 1.4297 148 1.5210 670 1.6000 230 1.8584 0.02 88 1.1729 43 1.2495 473 1.4243 130 1.5107 582 1.5947 201 1.8486 0.03 80 1.1650 39 1.2363 429 1.4207 118 1.5032 528 1.5911 183 1.8413 0.04 75 1.1597 37 1.2300 399 1.4177 110 1.4971 490 1.5881 170 1.8350 0.05 70 1.1538 35 1.2188 374 1.4150 104 1.4922 460 1.5854 160 1.8301 0.06 67 1.1499 33 1.2139 353 1.4126 98 1.4868 434 1.5830 153 1.8262 0.07 64 1.1455 32 1.2090 336 1.4104 94 1.4829 413 1.5808 145 1.8218 0.08 61 1.1411 31 1.2051 322 1.4084 90 1.4790 395 1.5786 139 1.8179 0.09 59 1.1377 30 1.2002 308 1.4063 86 1.4746 379 1.5769 133 1.8135 0.10 57 1.1348 28 1.1895 296 1.4045 83 1.4707 364 1.5750 127 1.8096 0.08 0.01 96 1.1868 46 1.2695 523 1.4316 143 1.5254 645 1.6021 221 1.8625 0.02 84 1.1768 41 1.2559 453 1.4265 124 1.5146 558 1.5969 192 1.8525 0.03 76 1.1689 37 1.2422 411 1.4229 114 1.5083 506 1.5933 176 1.8457 0.04 71 1.1631 35 1.2354 380 1.4198 105 1.5015 467 1.5901 162 1.8394 0.05 67 1.1582 33 1.2275 356 1.4170 99 1.4966 438 1.5874 152 1.8340 0.06 64 1.1543 31 1.2192 337 1.4148 95 1.4922 414 1.5852 144 1.8296 0.07 61 1.1504 30 1.2109 319 1.4125 89 1.4863 394 1.5830 137 1.8254 0.08 58 1.1455 29 1.2090 305 1.4104 85 1.4824 375 1.5808 131 1.8213 0.09 56 1.1406 28 1.2046 292 1.4084 82 1.4795 359 1.5789 126 1.8179 0.10 54 1.1387 27 1.1953 280 1.4065 79 1.4756 344 1.5769 122 1.8149 0.09 0.01 92 1.1904 45 1.2773 505 1.4336 137 1.5293 620 1.6039 213 1.8662 0.02 80 1.1802 39 1.2607 435 1.4285 119 1.519 536 1.5989 185 1.8567 0.03 73 1.1729 36 1.2510 394 1.4248 109 1.5122 485 1.5952 167 1.8491 0.04 68 1.1675 33 1.2402 365 1.4219 100 1.5054 449 1.5923 155 1.8433 0.05 64 1.1626 31 1.2319 341 1.4192 94 1.5000 418 1.5895 145 1.8379 0.06 61 1.1582 30 1.2275 321 1.4167 89 1.4951 395 1.5872 137 1.8330 0.07 58 1.1523 29 1.2227 305 1.4146 85 1.4915 375 1.5850 131 1.8296 0.08 55 1.1494 27 1.2129 290 1.4124 81 1.4863 357 1.5828 124 1.8250 0.09 53 1.1460 26 1.207 279 1.4106 78 1.4834 341 1.5808 119 1.8213 0.10 51 1.1406 25 1.2017 266 1.4084 75 1.4795 327 1.5789 114 1.8174 0.10 0.01 89 1.1943 43 1.2822 486 1.4353 133 1.5332 599 1.6057 205 1.8699 0.02 77 1.1841 37 1.2656 419 1.4302 114 1.5225 516 1.6007 177 1.8599 0.03 70 1.1768 34 1.2554 378 1.4266 104 1.5156 467 1.5972 160 1.8525 0.04 65 1.1709 32 1.2461 349 1.4236 96 1.5093 429 1.5940 148 1.8467 0.05 61 1.1660 30 1.2393 326 1.4210 90 1.5042 401 1.5914 139 1.8418 0.06 58 1.1602 28 1.2305 308 1.4187 85 1.4990 379 1.5891 131 1.8372 0.07 55 1.1575 27 1.2256 292 1.4165 81 1.4951 359 1.5869 125 1.8330 0.08 52 1.1523 26 1.2188 277 1.4143 77 1.4902 341 1.5847 119 1.8291 0.09 50 1.1489 25 1.2148 265 1.4124 74 1.4871 326 1.5828 114 1.8252 0.10 49 1.1472 24 1.2090 255 1.4106 71 1.4824 312 1.5808 109 1.8215

283

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C. H. Wu received his MS degree in Applied Mathematics from National Chung-Hsing University. Currently, he is a PhD candidate at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, ROC.

H. N. Hung received his BS in mathematics from National Taiwan University, MS in mathematics from National Tsing-Hua University and PhD. in statistics from the University of Chicago. He is an associate professor in the Institute of Statistics at National Chiao Tung University. His interests include statistical inference, statistical computing and industrial statistics.

C. M. Kao received her MS degree in Institute of Statistics, National Chiao Tung University, Taiwan, ROC.

數據

Table I. Sample data of 79 observations Response time (rising)
Table A1. Required sample sizes (n) and critical acceptance values (c 0 ) for various a-risk and b-risk with selected C AQL and C LTPD
Table A1. Continued.

參考文獻

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