Supplier Selection Critical Decision Values for
Processes with Multiple Independent Lines
W. L. Pearn and C. H. Wu*
†The process yield is the most common criterion considered for decision making in supplier selection problem. For normally distributed processes with multiple independent lines, theSMpk index provides an exact measurement for the overall yield. Therefore, the SMpk index can be implemented to deal with the supplier selection problem with processes having multiple independent lines. In this article, a test statistic obtained by a division method is employed to establish a hypothesis testing procedure, with two phases, which is developed to determine whether two suppliers are equally capable or not. The sampling distribution and the probability density function of the test statistic are derived. For various minimum requirements of process capability, number of lines, sample sizes, magnitudes of the difference between the two suppliers and the type I error, the critical values for decision making are presented. The required sample sizes for various designated powers at given type I error are tabulated. A thin-film transistor type liquid-crystal display application example is provided to demonstrate the testing procedure. Copyright © 2012 John Wiley & Sons, Ltd.
Keywords: critical value; multiple independent lines; supplier selection problem
1. Introduction
S
upplier selection is a problem of comparing two or even more suppliers and selecting the one that has a significantly higher process capability. Process capability indices (PCI) have been widely used to be a criterion for dealing with supplier selection problem in manufacturing industries. Process yield is an important factor that needs to be considered in supplier selection problem. Boyles1 proposed the Spkindex to provide an exact measurement for the process yield. However, processes with multiple characteristics ormultiple independent lines often occur in practice. For processes with multiple independent characteristics, Chen et al.2firstly introduced the ST
pkindex to evaluate the process performance. Pearn and Cheng
3
investigated the relationship between process parameters and the sampling distribution of natural estimator of ST
pk. Pearn et al.
4
derived the asymptotic distribution for the natural estimator of ST pkindex
under multiple samples. Recently, more investigations for processes with multiple characteristics include Pearn et al.5–7 For normally distributed processes with multiple independent lines, Tai et al.8proposed the overall yield index SM
pkto establish the
relationship between the actual overall process yield and the manufacturing specifications. Thus, the SM
pk index can be used as a
benchmark for evaluating process performance. Tai et al.8developed an effective method to measure the manufacturing yield for photolithography processes with multiple independent manufacturing lines by SMpkindex.
For the supplier selection problem, Tai et al.9investigated the glass substrate processes selection problem in thin-film transistor
type liquid-crystal display (TFT-LCD) manufacturing industries. Lin and Pearn10developed an analytical approach based on the yield index Spkto compare two processes. Lin and Pearn11extended the results of Lin and Pearn10to cases with multiple independent
manufacturing lines. Yum and Kim12and Wu et al.13provided some reviews and overviews for PCI.
2. The
S
Mpkindex for multiple independent lines
For a multiple independent lines process with k identical lines (flows), an overall capability index was proposed by Tai et al.8designed as follows: SMpk¼1 3Φ 1 1 k Xk j¼1 2Φ 3Spkj 1 þ 1 " # =2 ( ) ; (1)
where Spkj, j = 1,. . ., k is the Spkindex value of the jth line.Φ() means the cumulative distribution function of the standard normal
distribution. For normally distributed processes, the yield of the jth line Pj, j = 1, 2,. . ., k can be obtained by
Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, Taiwan
*Correspondence to: Chia-Huang Wu, Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, 300 Taiwan.
†E-mail: [email protected]
Research Article
(wileyonlinelibrary.com) DOI: 10.1002/qre.1449 Published online 13 September 2012 in Wiley Online Library
Yield¼ Pj¼ 2Φ 3Spkj
1: (2)
Consequently, from Equation (1), a one-to-one relationship between the index SM
pkand the overall process yield P can be presented
as follows: P¼1 k Xk j¼1 Pj¼ 1 k Xk j¼1 2Φ 3Spkj 1 ¼ 2Φ 3SM pk 1: (3) Hence, the SM
pk index provides an exact measurement of the yield for normally distributed processes with multiple independent
lines. That is, the SM
pkcan be used to deal with the supplier selection problem on the basis of the overall process yield. Because the
process parameters such as means and variances are unknown, in general, SM
pksample data should be collected to estimate the SMpk
index. The natural estimator of SM
pk, ^SpkM, can be expressed as ^SpkM¼1 3Φ 1 1 k Xk j¼1 2Φ 3^Spkj 1 þ 1 " # =2 ( ) ; (4) where ^Spkj¼ 1 3Φ 1 1 2 Φ USLj Xj Sj þ Φ Xj LSLj Sj ; j ¼ 1; 2; . . . ; k; (5) is the natural estimator of Spkindex value of the jth line1. The exact sampling distribution of ^SpkMis mathematically intractable. Tai et al.
8
used thefirst-order Taylor expansion for multiple variables to derive the asymptotic distribution of ^SpkMas ^SpkM N SM pk; D2f 3Dð Þ 2k2nf2 3SM pk 0 @ 1 A; (6) where D¼1 3Φ 1 k 2Φ 3SM pk 1 k 2ð Þ h i =2 n o : (7)
The results mentioned earlier can be implemented to compare two suppliers with multiple independent lines and normally distributed processes.
3.
Supplier selection for processes with multiple independent lines
Consider two suppliers, suppliers I and II, supplier II claims that it has a significantly higher capability than supplier I. Our main object is to compare two suppliers and make a reliable decision at a given significance level a. Based on given data from the two suppliers with multiple independent lines, ^Spk1M and ^Spk2M would befirst calculated. The quotient R ¼ ^Spk2M=^Spk1M would then be considered. If the quotient R¼ ^Spk2M =^Spk1M is sufficiently large, then it is clear that supplier II is better than supplier I, and supplier II in this case would be selected. The critical decision values, however, must be determined by statistical hypothesis testing. When the suppliers have SM
pkindex values SMpk1and SMpk2, the testing of the hypothesis
H0: SMpk1⩾SMpk2;
H1: SMpk1< SMpk2:
(8) is considered to handle the supplier selection problem. Next, the probability density function of the test statistic R is derived explicitly.
4.
Test statistic quotient
R
In this section, we implement a test statistic R to investigate the hypothesis testing mentioned earlier. Firstly, Equation (8) can be represented as H0: SM pk2 SM pk1 ⩽1; H1: SM pk2 SM pk1 >1: (9)
900
The ratio of the two estimators, R¼ ^Spk2M =^Spk1M, is applied to deal with the hypothesis testing of Equation (9). From Equation (6), the natural estimator ^SpkMis an asymptotic, normally distributed random variable. Consequently, the sampling distribution of the test sta-tistic R is as follows: R¼^S M pk2 ^SM pk1 N SM pk1; D12f 3Dð 1Þ 2k12n1f2 3SMpk1 N SM pk2; D22f 3Dð 2Þ 2k22n2f2 3SMpk2 : (10)
Thus, the distribution of the test statistic R is the quotient of two independent, normally distributed random variables and is related to the Cauchy distribution. Let XeN m1; s21
and YeN m2; s22
be two independent random variables with normal distribution. Using the Jocobian transformation technique, the probability density function of R = Y/X can be represented as
fRð Þ ¼r 1 2ps1s2 2s23exp m 2 3 2s2 3 þ m3s3 ffiffiffiffiffiffi 2p p 1 2Φ m3 s3 exp 12 m21 s2 1 þm22 s2 2 m23 s2 3 ; (11) where m3¼ m1=s21þ rm2=s22 1=s2 1þ r2=s22 ¼rm2s21þ m1s22 r2s2 1þ s22 ; and s2 3¼ 1 s2 1 þsr22 2 1 ¼ s21s22 r2s2 1þ s22 : (12)
Finally, the probability distribution of R¼ ^Spk2M =^Spk1M, can be established by substituting the parameters as m1¼ SMpk1; m2¼ SMpk2; s 2 1¼ D12f 3Dð 1Þ 2k12n1f2 3SMpk1 ; and s2 2¼ D22f 3Dð 2Þ 2k22n2f2 3SMpk2 : (13)
In the next section, on the basis of the sampling distribution of R developed in Equation (11), a procedure having two phases is proposed to deal with the supplier selection problem.
5. Supplier selection procedure
Let C denote the minimum requirement of SM
pkvalues for all suppliers. When the existing supplier, supplier I, has achieved the process
requirement (i.e., SM
pk1⩾C), a new supplier, supplier II claims that its capability is better than suppler I. Our object is to compare two
suppliers and make a reliable decision at a given significance level a risk.
5.1. Phase I: selecting supplier with higher capability
In thefirst phase, the hypothesis testing: H0: SMpk2⩽SMpk1versus H1: SMpk2> SMpk1 is considered to test whether supplier II has a better
process capability than supplier I or not. On the basis of the testing statistic R¼ ^Spk2M =^Spk1M, and a given significance level a, the decision rule is to reject H0if R⩾ c0. The critical value c0satisfies the following equation:
Type I Error¼ P R⩾cð 0jH0: SMpk2⩽SMpk1; n1; n2; k1; k2and SMpk1⩾CÞ⩽a: (14)
That is, the probability that falsely rejects H0is no more thana. Because the smaller the value of SMpk2=SMpk1, the larger the type I error
is, then, we calculate the critical value c0under the conditions SMpk1¼ SMpk2¼ C. Therefore, the critical value c0can be obtained by
solving the following equation
P Rð ⩾c0jSpk1M ¼ Spk2M ¼ C; n1; n2; k1; k2Þ ¼ a: (15)
Table I shows the critical values to test H1: SMpk2> SMpk1for various values of k1= k2= k and n1= n2= n = 30(10)200 ata = 0.05. It is to
be noted that the critical value is the same as the result in Lin and Pearn.10
5.2. Phase II: magnitude outperformed measurement In Phase I, the decision is based solely on the two SM
pkvalues without further regard to the magnitude of the difference between the
two suppliers. In practice, owing to the high cost of the process replacement, the supplier change is considered only if the new supplier significantly outperforms the existing supplier’s capability by a given magnitude h > 0. In this case, the proposed approach can be modified to test the corresponding hypothesis:
H0: Spk2M ⩽ Spk1M þ h;
H1: Spk2M > SMpk1þ h:
(16)
The decision rule is similar to Phase I. We will reject H0and accept that SMpk2> SMpk1þ h when the test statistic R is greater than or
equal to the critical value c0, where c0satisfies the following:
Type I Error¼ P R⩾cð 0jH0: Spk2M ⩽ Spk1M þ h; n1; n2; k1; k2 and SMpk1⩾CÞ⩽a: (17)
Similarly, the critical value c0is obtained by keeping the type I error less thana under the conditions SMpk1¼ C and SMpk2¼ C þ h. That
is, c0is obtained by solving the following equations
P Rð ⩾c0jSpk1M ¼ C; SMpk2¼ C þ h; n1; n2; k1; k2Þ⩽a: (18)
If the decision is rejecting the null hypothesis (16), then we have sufficient evidence to conclude that supplier II is significantly better than supplier I by a magnitude of h. Table II shows the critical values for given numbers of lines k1= k2= k = 2(1)5, sample sizes
n1= n2= n = 30(10)200, the magnitude of the difference between the two suppliers h = 0.1(0.1)0.5, and minimum requirements of
suppliers C = 1.00, 1.33, 1.50.
5.3. Required sample size
The decision making in Phases I and II solely depends on the given type I errora risk, but does not consider the type II error b risk (or power), which is the probability of falsely accepting H0. It is an unfavorable risk for the competing supplier. To decrease theb risk,
then increasing the decision power at a givena risk, sample sizes need to be increased. Obviously, the larger the sample size, the smaller theb risk (the larger the power of test) is. By calculating the power for a specific combination of SM
pk1; SMpk2
, the minimal sample size required for various given power (orb risk) and a risk can be established. The required sample size can be obtained by solving the following two constraints
Type I Error¼ P R ⩾ cð 0jH0: SMpk2⩽ SMpk1; n1; n2; k1; k2and SMpk1⩾ CÞ ⩽a; (19)
Power¼ P R ⩽ cð 0jH1: SMpk2> SMpk1; n1; n2; k1; k2 and SMpk1⩾ CÞ⩾1 b: (20)
For application, Table III tabulates the required sample sizes for various minimal capability requirements C = 1.00, 1.30, 1.50, 1.67 and magnitude of difference h =0.15(0.05)1.00 with given power = 0.90, 0.95, 0.975, 0.99 when the sample size and the number of lines of two suppliers are the same (i.e., n1= n2, k1= k2). For example, if two suppliers both have k = 3 lines, the minimal capability
requirement C = 1.30, the designateda risk is 0.05, and the expected magnitude of difference h ¼ SM
pk2 SMpk1¼ 0:3, then it requires
183 samples from both suppliers to reach a testing power of 0.95 (i.e.,b risk = 0.05). Table I. Critical values for rejecting SM
pk1⩽Spk2M with n1= n2= n = 30(10)200 ata = 0.05 k n 1 2 3 4 5 6 7 8 9 10 30 1.3581 1.3037 1.2734 1.2526 1.2368 1.2242 1.2137 1.2046 1.1968 1.1899 40 1.3019 1.2571 1.2321 1.2148 1.2016 1.1910 1.1822 1.1747 1.1680 1.1622 50 1.2653 1.2266 1.2048 1.1897 1.1782 1.1690 1.1613 1.1547 1.1489 1.1438 60 1.2391 1.2046 1.1852 1.1717 1.1613 1.1531 1.1461 1.1402 1.1350 1.1304 70 1.2192 1.1879 1.1701 1.1578 1.1484 1.1409 1.1345 1.1291 1.1243 1.1201 80 1.2034 1.1746 1.1582 1.1468 1.1381 1.1311 1.1252 1.1202 1.1158 1.1118 90 1.1906 1.1637 1.1485 1.1378 1.1297 1.1231 1.1176 1.1129 1.1088 1.1051 100 1.1799 1.1546 1.1403 1.1303 1.1226 1.1164 1.1112 1.1068 1.1029 1.0994 110 1.1707 1.1469 1.1333 1.1238 1.1166 1.1107 1.1058 1.1016 1.0979 1.0946 120 1.1628 1.1402 1.1272 1.1182 1.1113 1.1057 1.1011 1.0970 1.0935 1.0904 130 1.1559 1.1343 1.1219 1.1133 1.1067 1.1014 1.0969 1.0930 1.0897 1.0867 140 1.1498 1.1290 1.1172 1.1090 1.1026 1.0975 1.0932 1.0895 1.0862 1.0833 150 1.1443 1.1244 1.1130 1.1050 1.0990 1.0940 1.0899 1.0863 1.0832 1.0804 160 1.1394 1.1202 1.1092 1.1015 1.0956 1.0909 1.0869 1.0835 1.0804 1.0778 170 1.1349 1.1164 1.1057 1.0983 1.0926 1.0881 1.0842 1.0809 1.0779 1.0753 180 1.1309 1.1129 1.1026 1.0954 1.0899 1.0854 1.0817 1.0785 1.0756 1.0732 190 1.1271 1.1097 1.0998 1.0928 1.0874 1.0831 1.0794 1.0763 1.0736 1.0711 200 1.1237 1.1068 1.0971 1.0903 1.0851 1.0809 1.0774 1.0743 1.0717 1.0693
902
Table II. Critical values for rejecting SM pk1⩽Spk2M þ h with a = 0.05 and h = 0.1(0.1)0.5 h 0.10 0.20 0.30 0.40 0.50 0.10 0.20 0.30 0.40 0.50 C = 1.00 n k = 2 k = 3 30 1.4374 1.5709 1.7044 1.8377 1.9710 1.4060 1.5384 1.6707 1.8028 1.9348 40 1.3857 1.5142 1.6426 1.7709 1.8990 1.3598 1.4873 1.6148 1.7421 1.8694 50 1.3518 1.4769 1.6020 1.7269 1.8518 1.3293 1.4537 1.5780 1.7022 1.8263 60 1.3274 1.4501 1.5728 1.6953 1.8178 1.3073 1.4294 1.5514 1.6733 1.7951 70 1.3089 1.4297 1.5505 1.6712 1.7919 1.2906 1.4109 1.5311 1.6513 1.7713 80 1.2941 1.4135 1.5328 1.6521 1.7713 1.2772 1.3962 1.5150 1.6337 1.7524 90 1.2820 1.4002 1.5184 1.6365 1.7545 1.2663 1.3841 1.5018 1.6194 1.7369 100 1.2719 1.3891 1.5063 1.6234 1.7404 1.2571 1.3740 1.4907 1.6073 1.7238 110 1.2633 1.3796 1.4959 1.6122 1.7283 1.2493 1.3653 1.4812 1.5970 1.7127 120 1.2558 1.3714 1.4870 1.6025 1.7179 1.2426 1.3578 1.4730 1.5881 1.7031 130 1.2493 1.3642 1.4791 1.5940 1.7088 1.2366 1.3512 1.4658 1.5802 1.6946 140 1.2435 1.3578 1.4722 1.5864 1.7007 1.2313 1.3454 1.4594 1.5733 1.6871 150 1.2383 1.3521 1.4660 1.5797 1.6935 1.2266 1.3402 1.4537 1.5671 1.6804 160 1.2336 1.3470 1.4604 1.5737 1.6869 1.2224 1.3355 1.4485 1.5615 1.6744 170 1.2294 1.3424 1.4553 1.5682 1.6810 1.2185 1.3312 1.4438 1.5564 1.6689 180 1.2255 1.3381 1.4507 1.5632 1.6756 1.2150 1.3273 1.4396 1.5518 1.6640 190 1.2220 1.3342 1.4464 1.5586 1.6706 1.2118 1.3238 1.4357 1.5475 1.6594 200 1.2187 1.3306 1.4425 1.5543 1.6661 1.2088 1.3205 1.4321 1.5436 1.6551 C = 1.00 n k = 4 k = 5 30 1.3844 1.5161 1.6476 1.7790 1.9101 1.3681 1.4993 1.6303 1.7611 1.8917 40 1.3419 1.4689 1.5958 1.7225 1.8491 1.3283 1.4550 1.5814 1.7077 1.8339 50 1.3138 1.4377 1.5615 1.6852 1.8087 1.3019 1.4256 1.5490 1.6723 1.7955 60 1.2935 1.4152 1.5367 1.6582 1.7795 1.2829 1.4043 1.5255 1.6467 1.7677 70 1.2779 1.3979 1.5177 1.6375 1.7572 1.2682 1.3880 1.5076 1.6271 1.7464 80 1.2655 1.3842 1.5026 1.6211 1.7393 1.2566 1.3750 1.4932 1.6114 1.7294 90 1.2554 1.3728 1.4902 1.6075 1.7247 1.2470 1.3643 1.4815 1.5985 1.7155 100 1.2469 1.3634 1.4798 1.5962 1.7124 1.2390 1.3554 1.4716 1.5878 1.7038 110 1.2396 1.3553 1.4710 1.5865 1.7019 1.2322 1.3477 1.4632 1.5785 1.6938 120 1.2333 1.3483 1.4633 1.5781 1.6929 1.2262 1.3411 1.4558 1.5705 1.6851 130 1.2278 1.3422 1.4565 1.5707 1.6849 1.2210 1.3352 1.4494 1.5635 1.6774 140 1.2229 1.3367 1.4505 1.5642 1.6778 1.2164 1.3301 1.4437 1.5572 1.6707 150 1.2185 1.3318 1.4451 1.5583 1.6714 1.2122 1.3254 1.4386 1.5517 1.6647 160 1.2145 1.3274 1.4403 1.5531 1.6658 1.2085 1.3213 1.4340 1.5466 1.6592 170 1.2109 1.3234 1.4359 1.5483 1.6606 1.2051 1.3175 1.4298 1.5420 1.6542 180 1.2076 1.3198 1.4319 1.5439 1.6559 1.2020 1.3140 1.4260 1.5379 1.6497 190 1.2046 1.3165 1.4282 1.5399 1.6515 1.1991 1.3109 1.4225 1.5340 1.6455 200 1.2019 1.3134 1.4248 1.5362 1.6475 1.1965 1.3079 1.4193 1.5305 1.6417 C = 1.33 n k = 2 k = 3 30 1.4261 1.5272 1.6283 1.7294 1.8304 1.4067 1.5073 1.6079 1.7083 1.8087 40 1.3718 1.4690 1.5661 1.6631 1.7602 1.3559 1.4526 1.5493 1.6459 1.7425 50 1.3363 1.4309 1.5253 1.6198 1.7142 1.3226 1.4167 1.5109 1.6050 1.6990 60 1.3108 1.4035 1.4961 1.5887 1.6812 1.2986 1.3910 1.4833 1.5755 1.6678 70 1.2915 1.3827 1.4739 1.5650 1.6562 1.2803 1.3713 1.4623 1.5531 1.6440 80 1.2761 1.3662 1.4562 1.5463 1.6363 1.2658 1.3557 1.4455 1.5353 1.6250 90 1.2635 1.3527 1.4418 1.5309 1.6200 1.2539 1.3429 1.4318 1.5207 1.6096 100 1.2530 1.3414 1.4297 1.5181 1.6064 1.2440 1.3322 1.4204 1.5085 1.5966 110 1.2440 1.3317 1.4194 1.5071 1.5948 1.2355 1.3231 1.4106 1.4981 1.5855 120 1.2363 1.3234 1.4106 1.4976 1.5847 1.2282 1.3152 1.4022 1.4891 1.5760 130 1.2295 1.3161 1.4027 1.4893 1.5759 1.2218 1.3083 1.3947 1.4811 1.5676 140 1.2234 1.3097 1.3958 1.4820 1.5681 1.2161 1.3022 1.3882 1.4742 1.5601 150 1.2181 1.3039 1.3897 1.4754 1.5611 1.2110 1.2967 1.3823 1.4679 1.5535 (Continues)
903
Table II. Continued. h 0.10 0.20 0.30 0.40 0.50 0.10 0.20 0.30 0.40 0.50 160 1.2132 1.2987 1.3841 1.4695 1.5549 1.2064 1.2917 1.3770 1.4623 1.5475 170 1.2088 1.2940 1.3791 1.4641 1.5492 1.2022 1.2872 1.3722 1.4571 1.5421 180 1.2048 1.2897 1.3745 1.4592 1.5440 1.1984 1.2831 1.3678 1.4525 1.5371 190 1.2011 1.2857 1.3703 1.4548 1.5392 1.1950 1.2794 1.3638 1.4482 1.5326 200 1.1978 1.2821 1.3664 1.4506 1.5349 1.1918 1.2760 1.3601 1.4442 1.5284 C = 1.33 n k = 4 k = 5 30 1.3933 1.4935 1.5937 1.6937 1.7937 1.3829 1.4829 1.5828 1.6826 1.7823 40 1.3448 1.4413 1.5377 1.6340 1.7302 1.3363 1.4325 1.5287 1.6248 1.7208 50 1.3129 1.4069 1.5008 1.5947 1.6885 1.3055 1.3993 1.4931 1.5868 1.6803 60 1.2900 1.3822 1.4743 1.5663 1.6584 1.2833 1.3754 1.4674 1.5593 1.6512 70 1.2725 1.3633 1.4541 1.5448 1.6354 1.2665 1.3572 1.4478 1.5384 1.6289 80 1.2586 1.3483 1.4380 1.5276 1.6172 1.2530 1.3427 1.4322 1.5217 1.6112 90 1.2472 1.3361 1.4248 1.5136 1.6023 1.2420 1.3308 1.4195 1.5081 1.5967 100 1.2377 1.3258 1.4138 1.5018 1.5897 1.2328 1.3209 1.4088 1.4967 1.5845 110 1.2296 1.3170 1.4044 1.4918 1.5791 1.2250 1.3124 1.3997 1.4869 1.5741 120 1.2225 1.3094 1.3962 1.4831 1.5698 1.2182 1.3050 1.3917 1.4784 1.5651 130 1.2164 1.3028 1.3891 1.4754 1.5617 1.2122 1.2985 1.3848 1.4710 1.5572 140 1.2109 1.2969 1.3828 1.4687 1.5545 1.2068 1.2928 1.3787 1.4644 1.5502 150 1.2060 1.2916 1.3771 1.4626 1.5481 1.2021 1.2877 1.3731 1.4586 1.5439 160 1.2016 1.2868 1.3720 1.4572 1.5423 1.1979 1.2830 1.3682 1.4533 1.5383 170 1.1976 1.2825 1.3674 1.4522 1.5370 1.1940 1.2789 1.3637 1.4485 1.5332 180 1.1939 1.2786 1.3632 1.4478 1.5323 1.1904 1.2750 1.3596 1.4441 1.5285 190 1.1906 1.2750 1.3593 1.4436 1.5279 1.1872 1.2715 1.3558 1.4400 1.5243 200 1.1875 1.2716 1.3558 1.4398 1.5238 1.1842 1.2683 1.3523 1.4363 1.5203 C = 1.50 n k = 2 k = 3 30 1.4213 1.5112 1.6010 1.6908 1.7805 1.4057 1.4951 1.5846 1.6740 1.7633 40 1.3663 1.4525 1.5388 1.6250 1.7112 1.3534 1.4394 1.5254 1.6113 1.6972 50 1.3303 1.4142 1.4982 1.5820 1.6659 1.3192 1.4029 1.4866 1.5702 1.6538 60 1.3045 1.3868 1.4690 1.5512 1.6334 1.2947 1.3767 1.4587 1.5407 1.6227 70 1.2849 1.3659 1.4468 1.5278 1.6087 1.2760 1.3568 1.4375 1.5183 1.5990 80 1.2694 1.3493 1.4293 1.5092 1.5891 1.2611 1.3409 1.4207 1.5004 1.5802 90 1.2567 1.3358 1.4149 1.4940 1.5731 1.2490 1.3280 1.4069 1.4859 1.5648 100 1.2460 1.3245 1.4029 1.4813 1.5597 1.2388 1.3171 1.3954 1.4737 1.5519 110 1.2370 1.3148 1.3926 1.4705 1.5483 1.2301 1.3079 1.3856 1.4632 1.5409 120 1.2291 1.3064 1.3838 1.4611 1.5384 1.2226 1.2999 1.3771 1.4542 1.5314 130 1.2223 1.2991 1.3760 1.4528 1.5297 1.2161 1.2928 1.3696 1.4463 1.5231 140 1.2162 1.2927 1.3691 1.4456 1.5220 1.2103 1.2867 1.3630 1.4393 1.5157 150 1.2108 1.2869 1.3630 1.4391 1.5152 1.2051 1.2811 1.3571 1.4331 1.5091 160 1.2059 1.2817 1.3575 1.4332 1.5090 1.2004 1.2761 1.3518 1.4275 1.5031 170 1.2015 1.2770 1.3525 1.4279 1.5034 1.1961 1.2716 1.3470 1.4224 1.4978 180 1.1974 1.2727 1.3479 1.4231 1.4983 1.1922 1.2675 1.3426 1.4177 1.4928 190 1.1937 1.2687 1.3437 1.4187 1.4936 1.1887 1.2636 1.3386 1.4135 1.4884 200 1.1903 1.2650 1.3398 1.4146 1.4894 1.1854 1.2601 1.3349 1.4095 1.4842 C = 1.50 n k = 4 k = 5 30 1.3947 1.4840 1.5731 1.6622 1.7512 1.3863 1.4754 1.5643 1.6532 1.7420 40 1.3444 1.4302 1.5160 1.6017 1.6873 1.3375 1.4232 1.5088 1.5943 1.6798 50 1.3114 1.3950 1.4785 1.5619 1.6453 1.3054 1.3889 1.4723 1.5556 1.6388 60 1.2877 1.3696 1.4515 1.5334 1.6152 1.2824 1.3642 1.4460 1.5277 1.6094 70 1.2697 1.3503 1.4310 1.5116 1.5922 1.2648 1.3454 1.4259 1.5065 1.5869 80 1.2553 1.3350 1.4147 1.4943 1.5739 1.2508 1.3304 1.4100 1.4896 1.5691 90 1.2435 1.3224 1.4013 1.4802 1.5590 1.2394 1.3182 1.3970 1.4758 1.5545 100 1.2337 1.3119 1.3901 1.4683 1.5464 1.2298 1.3079 1.3861 1.4642 1.5422 (Continues)
904
Table II. Continued. h 0.10 0.20 0.30 0.40 0.50 0.10 0.20 0.30 0.40 0.50 110 1.2253 1.3030 1.3806 1.4582 1.5357 1.2216 1.2992 1.3768 1.4543 1.5318 120 1.2181 1.2952 1.3723 1.4494 1.5265 1.2146 1.2917 1.3687 1.4457 1.5227 130 1.2117 1.2884 1.3651 1.4418 1.5184 1.2084 1.2850 1.3616 1.4382 1.5148 140 1.2061 1.2824 1.3587 1.4350 1.5112 1.2029 1.2791 1.3554 1.4316 1.5078 150 1.2011 1.2770 1.3530 1.4289 1.5048 1.1979 1.2739 1.3498 1.4256 1.5015 160 1.1965 1.2722 1.3478 1.4234 1.4990 1.1935 1.2692 1.3447 1.4203 1.4958 170 1.1924 1.2678 1.3431 1.4185 1.4937 1.1895 1.2648 1.3401 1.4154 1.4907 180 1.1886 1.2637 1.3389 1.4139 1.4890 1.1858 1.2609 1.3360 1.4110 1.4860 190 1.1852 1.2601 1.3349 1.4098 1.4846 1.1825 1.2573 1.3322 1.4069 1.4817 200 1.1820 1.2567 1.3313 1.4060 1.4806 1.1794 1.2540 1.3286 1.4032 1.4778
Table III. Sample size required for testing H0:Spk2M ⩽Spk1M versus H1: Spk2M ⩽Spk1M ata = 0.05
Power Power SM pk1 SMpk2 0.90 0.95 0.975 0.99 SMpk1 SMpk2 0.90 0.95 0.975 0.99 k = 2 k = 2 1.00 1.15 338 429 515 626 1.30 1.45 612 774 930 1129 1.20 200 254 305 371 1.50 357 452 544 660 1.25 135 171 205 250 1.55 238 301 361 439 1.30 98 124 150 182 1.60 171 217 260 316 1.35 76 96 115 140 1.65 130 165 198 241 1.40 61 77 92 113 1.70 103 131 157 191 1.45 50 63 76 93 1.75 85 107 129 157 1.50 42 54 65 79 1.80 71 90 108 131 1.55 36 46 56 68 1.85 61 77 92 112 1.60 32 41 49 60 1.90 53 67 80 97 1.65 28 36 43 53 1.95 46 59 71 86 1.70 25 32 39 48 2.00 41 52 63 76 1.75 23 29 35 43 2.05 37 47 56 69 1.80 21 27 32 39 2.10 34 42 51 62 1.85 19 25 30 36 2.15 31 39 47 57 1.90 18 23 27 34 2.20 28 36 43 52 1.95 17 21 26 31 2.25 26 33 40 48 2.00 16 20 24 29 2.30 24 31 37 45 k= 2 k= 2 1.50 1.65 834 1054 1267 1537 1.67 1.82 1047 1324 1589 1929 1.70 485 613 736 894 1.87 607 767 921 1118 1.75 320 405 487 591 1.92 400 505 607 737 1.80 230 291 349 424 1.97 286 361 434 527 1.85 174 220 265 322 2.02 216 273 328 398 1.90 138 174 209 254 2.07 170 215 258 314 1.95 112 142 170 207 2.12 138 174 210 255 2.00 93 118 142 173 2.17 115 145 174 212 2.05 80 101 121 147 2.22 97 123 148 180 2.10 69 87 105 127 2.27 84 106 128 155 2.15 60 76 92 112 2.32 74 93 112 136 2.20 54 68 81 99 2.37 65 82 99 120 2.25 48 61 73 89 2.42 58 73 88 107 2.30 43 55 66 80 2.47 52 66 80 97 2.35 39 50 60 73 2.52 48 60 72 88 2.40 36 46 55 67 2.57 44 55 66 81 2.45 33 42 51 61 2.62 40 50 60 73 (Continues)
905
Table III. Continued. Power Power SM pk1 SMpk2 0.90 0.95 0.975 0.99 SMpk1 SMpk2 0.90 0.95 0.975 0.99 2.50 31 39 47 57 2.67 36 46 55 67 k = 3 k = 3 1.00 1.15 286 362 436 531 1.30 1.45 553 700 841 1022 1.20 170 215 259 316 1.50 323 410 493 599 1.25 114 145 175 213 1.55 215 273 328 399 1.30 84 106 128 156 1.60 155 197 237 288 1.35 65 82 99 121 1.65 118 150 181 220 1.40 52 66 80 97 1.70 94 119 144 175 1.45 43 55 66 81 1.75 77 98 118 143 1.50 36 46 56 69 1.80 65 82 99 120 1.55 31 40 49 59 1.85 55 70 85 103 1.60 28 35 43 52 1.90 48 61 74 90 1.65 25 31 38 46 1.95 42 54 65 79 1.70 22 28 34 42 2.00 38 48 58 70 1.75 20 26 31 38 2.05 34 43 52 63 1.80 18 23 28 35 2.10 31 39 47 58 1.85 17 22 26 32 2.15 28 36 43 53 1.90 16 20 24 30 2.20 26 33 40 48 1.95 15 19 23 28 2.25 24 30 37 45 2.00 14 17 21 26 2.30 22 28 34 42 k = 3 k = 3 1.50 1.65 772 977 1174 1425 1.67 1.82 984 1244 1495 1815 1.70 450 569 683 831 1.87 571 721 867 1053 1.75 297 376 453 550 1.92 376 476 572 695 1.80 213 270 325 395 1.97 269 340 409 497 1.85 162 205 247 300 2.02 203 257 309 376 1.90 128 162 195 237 2.07 160 203 244 296 1.95 104 132 159 193 2.12 130 165 198 241 2.00 87 110 133 162 2.17 108 137 165 201 2.05 74 94 113 138 2.22 92 117 140 170 2.10 64 81 98 119 2.27 79 101 121 147 2.15 56 71 86 105 2.32 70 88 106 129 2.20 50 63 76 93 2.37 62 78 94 114 2.25 45 57 68 83 2.42 55 70 84 102 2.30 41 51 62 75 2.47 50 63 76 92 2.35 37 47 56 69 2.52 45 57 69 84 2.40 34 43 52 63 2.57 41 53 63 77 2.45 31 39 48 58 2.62 38 48 58 70 2.50 29 37 44 54 2.67 36 45 55 67 k = 4 k = 4 1.00 1.15 251 319 384 468 1.30 1.45 513 650 781 949 1.20 149 190 229 279 1.50 301 381 458 557 1.25 101 129 155 190 1.55 200 254 306 372 1.30 74 94 114 139 1.60 145 183 221 269 1.35 57 73 88 108 1.65 110 140 169 206 1.40 46 59 71 87 1.70 88 111 134 164 1.45 38 49 59 73 1.75 72 91 110 134 1.50 33 42 50 62 1.80 60 77 93 113 1.55 28 36 44 54 1.85 52 66 79 97 1.60 25 32 39 47 1.90 45 57 69 84 1.65 22 28 34 42 1.95 40 50 61 74 1.70 20 25 31 38 2.00 35 45 54 66 1.75 18 23 28 35 2.05 32 41 49 60 1.80 17 21 26 32 2.10 29 37 44 54 1.85 15 20 24 29 2.15 26 34 41 50 (Continues)
906
Table III. Continued. Power Power SM pk1 SMpk2 0.90 0.95 0.975 0.99 SMpk1 SMpk2 0.90 0.95 0.975 0.99 1.90 14 18 22 27 2.20 24 31 37 46 1.95 13 17 21 26 2.25 23 29 35 42 2.00 12 16 19 24 2.30 21 27 32 39 k = 4 k = 4 1.50 1.65 730 924 1111 1349 1.67 1.82 941 1190 1429 1735 1.70 425 538 647 787 1.87 546 690 830 1008 1.75 282 356 429 521 1.92 360 456 548 665 1.80 202 256 308 375 1.97 258 326 392 476 1.85 154 195 234 285 2.02 195 247 297 361 1.90 121 154 185 225 2.07 154 194 234 284 1.95 99 126 151 184 2.12 125 158 190 231 2.00 83 105 126 154 2.17 104 132 158 193 2.05 71 89 108 131 2.22 88 112 135 164 2.10 61 77 93 114 2.27 76 97 116 142 2.15 54 68 82 100 2.32 67 85 102 124 2.20 48 60 73 89 2.37 59 75 90 110 2.25 43 54 65 80 2.42 53 67 81 98 2.30 39 49 59 72 2.47 48 60 73 89 2.35 35 45 54 66 2.52 43 55 66 81 2.40 32 41 49 60 2.57 40 50 61 74 2.45 30 38 45 56 2.62 36 46 55 67 2.50 28 35 42 51 2.67 33 42 50 61 k = 5 k = 5 1.00 1.15 226 287 346 422 1.30 1.45 483 612 736 895 1.20 135 172 207 253 1.50 283 359 432 526 1.25 91 117 141 172 1.55 189 240 289 351 1.30 67 86 104 127 1.60 137 173 209 254 1.35 52 67 81 99 1.65 104 133 160 195 1.40 42 54 65 80 1.70 83 105 127 155 1.45 35 45 54 67 1.75 68 87 104 127 1.50 30 38 46 57 1.80 57 73 88 107 1.55 26 33 40 49 1.85 49 62 75 92 1.60 23 29 35 44 1.90 43 54 66 80 1.65 20 26 32 39 1.95 38 48 58 71 1.70 18 24 29 35 2.00 34 43 52 63 1.75 17 21 26 32 2.05 30 39 47 57 1.80 15 20 24 30 2.10 28 35 42 52 1.85 14 18 22 27 2.15 25 32 39 48 1.90 13 17 21 25 2.20 23 30 36 44 1.95 12 16 19 24 2.25 21 27 33 41 2.00 11 15 18 22 2.30 20 25 31 38 k = 5 k = 5 1.50 1.65 698 884 1063 1291 1.67 1.82 908 1148 1380 1675 1.70 407 515 620 754 1.87 527 666 802 974 1.75 270 342 411 500 1.92 348 440 529 643 1.80 194 246 295 360 1.97 249 315 379 461 1.85 147 187 225 273 2.02 188 239 287 349 1.90 116 148 178 217 2.07 148 188 226 275 1.95 95 121 145 177 2.12 121 153 184 224 2.00 79 101 121 148 2.17 101 128 154 187 2.05 68 86 104 126 2.22 86 108 131 159 2.10 59 75 90 109 2.27 74 94 113 137 2.15 52 65 79 96 2.32 65 82 99 120 2.20 46 58 70 86 2.37 57 73 87 107 2.25 41 52 63 77 2.42 51 65 78 95 (Continues)
907
6.
Supplier selection for thin-
film transistor type liquid-crystal display
Manufacturing yield has been the most basic common criterion used in the manufacturing industry for measuring process performance. Because offiercer competition in the global TFT-LCD industry, the supplier must have very low fraction of defectives, normally measured by parts per million (ppm) or parts per billion (ppb). Therefore, the multiple independent lines yield index SM
pkcan be used as a criterion to
select the suppliers. For the investigated model of TFT-LCD10, the target thickness value of a glass substrate is set to T = 0.70 mm with upper specification limit USL = 0.77 mm and lower specification limit LSL = 0.63 mm. When the minimum requirement of the supplier is SM
pk¼ 1:00, and two suppliers both have k = 4 lines, n1= n2= 150 data are collected for suppliers I and II. The calculated sample means,
sample standard deviations and the estimated Spkiindex values for each line are summarized in Table IV.
6.1. Phase I: select a supplier with higher capability
To determine whether supplier II has a better process capability than supplier I or not, the hypothesis testing: H0: SMpk2⩽SMpk1versus
H1: SMpk2> SMpk1is considered. From Table IV, we have SMpk1¼ 1:055755, SMpk2¼ 1:407204 and thus R = 1.332889. At a = 0.05, k = 4 and
n1= n2= 150, from Table I, the critical value is c0= 1.1050. Because the test statistic R = 1.332889> 1.1050, we conclude that supplier
II is better than supplier I with a 95% confidence level. Next, the second-phase testing would investigate the magnitude of the capability difference between the two suppliers.
6.2. Phase II: magnitude outperformed
The hypothesis testing H0: SMpk2⩽SMpk1þ h versus H1: SMpk2> SMpk1þ h is performed. For various values of the magnitude h, the
decisions of the hypotheses are shown in Table V (a = 0.05). The decision maker would replace the existing supplier when supplier II (competition) significantly outperforms supplier I by a magnitude of 0.20. On the basis of the testing result in Table V, we conclude that supplier II (competition) has a manufacturing capability that is significantly better than Supplier I by a magnitude of 0.20, that is, SM
pk2> SMpk1þ 0:2. Consequently, the supplier replacement would be suggested.
Table IV. Estimated values of capability indices for suppliers I and II
Suppliers Lines x s ^Spki
I 1 0.7098303 0.0192028 1.108760 2 0.7104621 0.0215073 0.992385 3 0.7104065 0.0192131 1.099091 4 0.7140126 0.0187125 1.065612 II 1 0.7001798 0.0142802 1.633835 2 0.6969854 0.0166799 1.378086 3 0.6976766 0.0172959 1.337498 4 0.7001785 0.0137853 1.692482
Table III. Continued.
Power Power SM pk1 SMpk2 0.90 0.95 0.975 0.99 SMpk1 SMpk2 0.90 0.95 0.975 0.99 2.30 37 47 57 69 2.47 46 59 71 86 2.35 34 43 52 63 2.52 42 53 64 79 2.40 31 39 48 58 2.57 38 49 59 72 2.45 29 36 44 54 2.62 35 44 53 64 2.50 27 34 41 50 2.67 31 40 48 58
Table V. Critical values and decisions of testing the two suppliers (a = 0.05, k = 4, n = 150)
Test cases I II III IV V
SMpk1 1.00 1.00 1.00 1.00 1.00
SM
pk2 1.10 1.20 1.21 1.22 1.23
h 0.10 0.20 0.21 0.22 0.23
c0 1.21847 1.33183 1.343152 1.354492 1.365816
Decision Reject H0 Reject H0 Non-Reject H0 Non-Reject H0 Non-Reject H0
6.3. Sample size required for designated power
For the cases in which the minimal requirement C = 1.00 and number of lines k = 4, the decision maker would replace the existing supplier with designated power 0.95 when the new supplier has an SMpk index value significantly higher than the existing process by a scale of 0.20. The required sample size is 190 as shown in Table III. In the application example mentioned earlier, because the sample sizes of two suppliers are smaller than the required sample size (150< 190), the power would be less than 0.95. In fact, the power of test for SM
pk2¼ 1:20 is 90.18%. That is, the b risk is up to 9.82%. To reduce the b risk and increase the decision power, we
would suggest the decision maker to collect more samples as recommended in Table III.
7. Conclusions
In this article, the supplier selection problem for normal processes with multiple independent lines was investigated; the overall yield index SM
pkprovided a one-to-one relationship between the specification limits and the overall process yield. A two-phase procedure on
the basis of the quotient test statistic was proposed to deal with the supplier selection problem. The probability density function of the test statistic was also established. For applications, some tables of the critical values for decision making were presented under various minimal capability requirements, magnitudes of difference of two suppliers, number of lines, and sample sizes. The required sample sizes to make a reliable decision were also provided for various given power. A TFT-LCD application was presented.
References
1. Boyles RA. Process capability with asymmetric tolerances. Communications in Statistics - Simulation and Computation 1994; 23(3):615–635. DOI:
10.1080/03610919408813190
2. Chen KS, Pearn WL, Lin PC. Capability measures for processes with multiple characteristics. Quality and Reliability Engineering International 2003;
19:101–110. DOI: 10.1002/qre.513
3. 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
4. Pearn WL, Yen CH, Yang DY. Production yield measure for multiple characteristics processes based on ST
pkunder multiple samples. Central European
Journal of Operational Research 2012; 20:65–85. DOI: 10.1007/s10100-010-0152-9
5. 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
6. Pearn WL, Wu CH, Tsai MC. A note on“Capability Assessment for Process with Multiple Characteristics: A Generalization of the Popular Index Cpk”.
Quality and Reliability Engineering International 2012, Feb 8 Online. DOI: 10.1002/qre.1295
7. Pearn WL, Wu CH, Hung HN, Kao CM. An extension of the production acceptance determination for one-sided process with multiple characteristics. Quality and Reliability Engineering International 2012, Mar 16 Online. DOI: 10.1002/qre1378
8. Tai YT, Pearn WL, Kao CM. Measuring the manufacturing yield for processes with multiple manufacturing lines. IEEE Transactions on Semiconductor
Manufacturing 2012; 25(2):284–290. DOI: 10.1109/TSM.2011.2179568
9. Tai YT, Pearn WL, You SK. An effective test for supplier selection evaluation with multiple characteristics. Journal of Testing and Evaluation 2011; 39(6):1–9. DOI: 10.1520/JTE103937
10. Lin CJ, Pearn WL. Process selection for higher production yield based on capability index Spk. Quality and Reliability Engineering International 2010;
26(3):247–258. DOI: 10.1002/qre.1051
11. Lin CJ, Pearn WL. Group selection for production yield among k manufacturing lines. Journal of Statistical Planning and Inference 2011; 141(4):1510–1518. DOI: 10.1016/j.bbr.2011.03.031
12. 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
13. 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. DOI: 10.1016/j.ijpe.2008.11.008
Authors' biographies
Wen-Lea Pearn received his PhD 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 Labora-tories, 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 Sta-tistical Society, Series C, Journal of Quality Technology, European Journal of Operational Research, Journal of the Operational Research Society, Operations Research Letters, OmegaI 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.
