A Close Form Solution for the Product
Acceptance Determination Based on the
Popular Index
C
pk
W. L. Pearn and C. H. Wu*
,†Product acceptance determinations are practical tools for quality control applications involving quality contract on product orders between the vendor and the buyer. It provides the vendor and the buyer rules for product acceptance to meet the preset product quality requirement. As the rapid advancement of manufacturing technology, more than one quality charac-teristic must be simultaneously considered to improve the product quality because of the product design. In this article, we introduce an efficient product acceptance procedure on the basis of the generalization CpkT index, to deal with lot sentencing problem with very low fraction of defectives. We tabulate the required sample size n and the corresponding critical acceptance valuec0for variousa-risk, b-risk, and the levels of the lot fraction of defectives that correspond to acceptance and rejecting quality levels. Practitioners can use the proposed method to make reliable decisions in product acceptance. Copyright © 2012 John Wiley & Sons, Ltd.
Keywords: critical acceptance value; multiple characteristics; product acceptance determination; required sample size
1. Introduction
P
roduct acceptance determination provides the vendor and the buyer a general criterion for lot sentencing while meeting their preset requirements on product quality. It basically consists of a sample size for inspection and an acceptance criterion and is usually based on the operating characteristic (OC) curve, which quantifies the risk for vendors and buyers. The OC curve plots the probability of accepting the lot against actual lot fraction defective, which displays the discriminatory power of the product acceptance 1 a determination rule. The vendor is primarily interested in insuring that good lots would be accepted, and the buyer wants to be reasonably sure that bad product would be rejected. Therefore, two designated points, (AQL,a) and (LTPD, b), on the OC curve are focused. Acceptable quality level (AQL) presents the poorest quality for the vendor process that consumer would consider acceptable as a process average. Lot tolerance percent defective (LTPD) is the poorest quality level that the consumer is willing to accept.a and b are called the vendor’s risk and the buyer’s risk, respectively.Pearn and Wu1developed an effective decision making method for product acceptance on the basis of the Cpkindex. The results attended are very practical but are restricted to process with only one quality characteristic. In this article, we extend the results to cases with multiple characteristics based on the generalization CT
pk index.
2. The generalization
C
pkTindex
The generalization CT
pkindex was proposed by Pearn et al.,
2 designed as CpkT ¼ 1 3Φ 1 Ym i¼1 2Φ 3Cpki 1 þ 1 " # =2 ( ) (1)
where Cpkidenotes the Cpkvalue of the ith characteristic for i = 1, 2,. . ., m, and m is the number of characteristics. Function Φ() means the cumulative distribution of standard normal distribution. For a normally distributed process with a specific CT
pk value, the lower
bound on overall process yield P can be established as
Department of Industrial Engineering and Management, 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, ROC.
†E-mail: hexjacal.iem96g@nctu.edu.tw
(wileyonlinelibrary.com) DOI: 10.1002/qre.1423 Published online 25 June 2012 in Wiley Online Library
P⩾Y m i¼1 2Φ 3Cpki 1 ¼ 2Φ 3CpkT 1 (2) The CT
pkindex provides a lower bound on the overall process yield. In practical, because the process mean and the variance for each
characteristic are unknown, the CT
pkindex is estimated by collecting the sample data. The natural estimator of the CTpkindex defined in
the following is considered:
^CpkT ¼1 3Φ 1 Ym i¼1 2Φ 3^Cpki 1 þ 1 " # =2 ( ) (3)
where ^Cpkidenotes the natural estimator of Cpkvalue of the ith characteristic. Pearn et al.2derived the asymptotic distribution of ^Cpk
T using Taylor expansion technique for multiple variables. The asymptotic distribution of ^CpkT is (see2)
^CpkT N CT pk; 1 9nþ CT 2 pk 2n ! (4)
For the processes with multiple characteristics, Hsu et al.3 applied the bootstrap method for calculating the lower confidence bounds of the capability index CT
pu and determined the sample size for the given estimation accuracy. Pearn et al.
4implemented the process capability index to deal with the photolithography production control problem with multiple quality characteristics. Pearn and Cheng5investigated the production yield measurement for processes with multiple characteristics. Awad and Kovach6purposed a simple and integrated modeling methodology for robust design on the basis of multivariate process capability vector. Wu et al.7 provided an overview for process capability indices practice of quality assurance. More recent studies on process capability index (PCI) include Goethals and Cho,8Yum and Kim,9and Pearn et al.10
3.
Product acceptance determination
The CT
pkindex can be used as a quality benchmark for product acceptance. Let (AQL,1 a) and (LTPD, b) be the two points on the OC
curve of interest. Note that AQL and LTPD are levels of the product fraction of defectives that correspond to acceptable and rejectable quality levels. To determine whether a given lot is capable, we can consider the testing hypothesis as
H0: p ¼ AQL ; H1: p ¼ LTPD (5)
where p means the process fraction of defectives. The AQL is simply a standard against which to judge the lots. It is hoped that the vendor’s process will operate at a fallout level that is considerable better than the AQL. The null hypothesis with process fraction of defectives, H0: p = AQL, is equivalent to test process the capability index with H0: CTpk⩾ CAQL, where CAQLis the level of acceptable quality for the CT
pkindex corresponding to the lot or process fraction of defectives AQL. For the production of vendors and buyers,
two conditions are considered:
Pr reject the lot p ⩾ AQL o
¼ Pr reject the lot CT pk⩾ CAQL o⩽ a n n (6)
Pr accepting the lot p ⩽ LTPDo¼ Pr accepting the lot CT pk⩽ CLTPD o⩽ b n n (7)
where CLTPDrepresents the capability requirement corresponding to the LTPD on the basis of the CpkT index.
That is, the probability of rejecting acceptable lots is no more thana. At the same time, the probability of accepting unqualified lots is no more thanb. Our object is solving the two simultaneous equations mentioned earlier and then obtaining the required inspection sample size n and the critical acceptance value c0of CpkT. By using the approximate distribution shown in Equation (4), Equations (6)
and (7) can be rewritten as
P ^CpkT< c0CpkT ⩾ CAQL ! ⩽ P Z < c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CAQL 1 9nþ C2 AQL 2n q 0 B @ 1 C A⩽ a (8) P ^CpkT⩾ c0CpkT ⩽ CLTPD ! ⩽ P Z ⩾ c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CLTPD 1 9nþ C2 LTPD 2n q 0 B @ 1 C A⩽ b (9)
720
Equations (8) and (9) imply that c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CAQL 1 9nþ C2 AQL 2n q ¼ z1a¼ za (10) c0 CLTPD ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 9nþ C2 LTPD 2n q ¼ zb (11)
From Equations (10) and (11), we have
c0 CAQL¼ za ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 9nþ C2 AQL 2n s (12) c0 CLTPD¼ zb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 9nþ C2 LTPD 2n r (13)
Subtracting Equation (13) by Equation (12) yields CAQL CLTPD¼ za ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=9 þ C2 AQL=2 q þ za ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=9 þ C2 AQL=2 q =pffiffiffin (14)
Consequently, from Equation (14), we establish the required inspection sample size n and the corresponding critical value c0as
n¼ za ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=9 þ C2 AQL=2 q þ zb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=9 þ C2LTPD=2 p CAQL CLTPD 0 @ 1 A 2 2 6 6 6 6 3 7 7 7 7 (15) c0¼ CAQL n1=2za ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=9 þ C2 AQL=2 q (16) The symboldne means the ceiling function that gains the least integer greater than or equal to n.
Remark: one-sided process
For one-sided processes with multiple characteristics, the generalization index CT
puis considered. Pearn et al.
11
have developed the asymptotic distribution of the natural estimator ^CpuT in the following11:
^CPUT N CT PU; 1 9nþ 1 2nC T PU 2 (17) We can use the same technique to establish the close form solutions of (n, c0) for one-sided processes with multiple characteristics similar to Equations (15) and (16) mentioned earlier.
4. Determination procedure
For practical application purpose, we calculate and tabulate the required sample size (n) and the critical acceptance values (c0) for various a-risk, b-risk, CAQL, and CLTPD. Table I displays (n,c0) values fora-risk = 0.01, 0.025, 0.05, 0.075, and 0.10 and b-risk = 0.01, 0.025, 0.05, 0.075, and 0.10, with various benchmarking quality levels, (CAQL,CLTPD) = (1.33, 1.00), (1.50, 1.33), (1.67, 1.33), and (2.00, 1.67).
For instance, if the requirement quality level (CAQL,CLTPD) is set to (1.50, 1.33) witha-risk = 0.01 and b-risk = 0.05, the required sample size and critical acceptance value can be obtained as (596, 1.4251). It means that the lot will be rejected if the 596 inspected product items yield measurement with ^CpkT < 1:4251. For the proposed product acceptance determination procedure to be practical and convenient to use, a step-by-step algorithm is provided as follows
Step1: decide the process capability requirements (i.e. set the values of CAQLand CLTPD) and choose thea-risk and the b-risk. Step2: check Table I tofind the critical acceptance value and the required number, (n,c0), on the basis of givena-risk, b-risk, CAQL, and CLTPD. Step3: calculate the value of ^CpkT (sample estimator) from the n inspected samples.
Step4: make decisions to accept the entire products if ^CpkT ⩾c0. Otherwise, reject the entire products.
5.
An application example
We consider a case study to demonstrate how the product acceptance determination procedure can be used in lot sentencing problem for processes with multiple characteristics. The case we investigate involves a process manufacturing the dual-fiber tips (see12), which is used in makingfiber optic cables. The quality characteristics and specifications are presented in Table II. The key quality characteristics include capillary diameter, length, wedge, and core diameter.
In the contract, the CAQLand the CLTPDare set to 1.33 and 1.00 witha-risk = 0.05 and b-risk = 0.05. First, we find the acceptance critical values and inspected sample sizes (n, c0) = (79, 1.1454) from Table I. The observations measurement and the calculated results for each characteristic are displayed in Table III. On the basis of those results, we obtain ^CpkT ¼ 0:93037. Therefore, the buyer would “reject” the entire products because the sample estimator, 0.93037, is smaller than the critical acceptance value 1.1454.
Table I. Required sample sizes (n) and critical acceptance values (c0) for variousa- and b -risks with selected CAQLand CLTPD
a b CAQL= 1.33, CLTPD= 1.00 CAQL= 1.50, CLTPD=1.33 CAQL= 1.67, CLTPD=1.33 CAQL= 2.00, CLTPD=1.67 n c0 n c0 n c0 n c0 0.01 0.01 158 1.1453 834 1.4104 232 1.4826 357 1.8211 0.025 131 1.1591 701 1.4177 194 1.4973 299 1.8353 0.05 110 1.1735 596 1.4251 163 1.5119 253 1.8497 0.075 98 1.1849 533 1.4307 145 1.5233 225 1.8606 0.10 89 1.1945 486 1.4354 132 1.5331 205 1.8699 0.025 0.01 137 1.1317 714 1.4031 201 1.4687 308 1.8074 0.025 112 1.1452 592 1.4104 165 1.4828 254 1.8213 0.05 93 1.1598 496 1.4179 137 1.4976 212 1.8359 0.075 81 1.1704 438 1.4235 120 1.5088 186 1.8466 0.10 73 1.1803 396 1.4284 108 1.5187 168 1.8563 0.05 0.01 120 1.1181 619 1.3960 175 1.4542 268 1.7935 0.025 97 1.1314 506 1.4031 142 1.4682 218 1.8071 0.05 79 1.1454 417 1.4104 116 1.4826 179 1.8214 0.075 69 1.1571 364 1.4161 101 1.4942 156 1.8326 0.10 61 1.1663 326 1.4211 90 1.5042 139 1.8421 0.075 0.01 110 1.1087 560 1.3907 160 1.4443 244 1.7836 0.025 88 1.1215 453 1.3976 128 1.4574 196 1.7966 0.05 71 1.1352 369 1.4048 104 1.4721 159 1.8105 0.075 61 1.1461 320 1.4105 89 1.4828 137 1.8213 0.10 54 1.1560 284 1.4155 79 1.4931 121 1.8307 0.10 0.01 102 1.1002 517 1.3862 148 1.4354 226 1.7752 0.025 81 1.1127 414 1.3929 118 1.4486 180 1.7877 0.05 65 1.1264 335 1.4001 95 1.4629 145 1.8015 0.075 55 1.1363 287 1.4055 81 1.4737 124 1.8122 0.10 48 1.1454 253 1.4104 71 1.4834 109 1.8216
Table II. Specifications of characteristics for the dual-fiber tips
Characteristic LSL Target USL
Capillary diameter (mm) 1.795 1.800 1.805
Capillary length (mm) 6.00 6.25 6.50
Wedge () 7.5 8 8.5
Core diameter (mm) 126 127 128
LSL: Lower Specification Limit; USL: Upper Specification Limit.
Table III. Calculated sample mean, sample standard derivation,Ĉpki, and estimated nonconformity
Characteristic x s Ĉpki Nonconformity (ppm)
Capillary diameter 1.8008 0.00106 1.320755 74.24207
Capillary length 6.2460 0.05908 1.387949 31.29299
Wedge 8.0128 0.17414 0.932583 5146.009
Core diameter 127.02 0.13482 1.594896 1.712531
6. Conclusions
In this article, we developed an effective and clear algorithm on the basis of overall yield-measure index CT
pk to deal with the lot
sentencing problem for normally distributed processes with multiple characteristics. The explicitly close form formulae of the required sample size n and the corresponding critical acceptance value c0were obtained. For various givena-risk, b-risk with capability require-ments CAQLand CLTPDvalues of (n,c0) were tabulated for practitioners to make reliable decisions.
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Authors' biographies
Wen-Lea Pearn received the Ph.D. degree in operations research from the University of Maryland, College Park. He is a Professor of Operations Research and Quality Assurance at the National Chiao-Tung University (NCTU), Hsinchu, Taiwan. He was with Bell Laboratories, Murray Hill, NJ, as a Quality Research Scientist before joining the NCTU, and others. His current research interests include process capability, network optimization, and production management.
Dr. Pearn’s publications have appeared in the Journal of the Royal Statistical Society, Series C, Journal of Quality Technology, European Journal of Operational Research, Journal of the Operational Research Society, Operations Research Letters, Omega, Networks, and the International Journal Productions Research.
Chia-Huang 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.
