An algorithm for calculating the lower confidence bounds of C-PU and C-PL with application to low-drop-out linear regulators

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An algorithm for calculating the lower conﬁdence bounds

of C

PU

and C

PL

with application to low-drop-out

linear regulators

W.L. Pearn

a,*

, Ming-Hung Shu

b

a

Department of Industrial Engineering & Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsin Chu 30050 Taiwan, ROC

b

Department of Management Science, Chinese Military Academy, Taiwan, ROC Received 3 July 2002; received in revised form 13 August 2002

Abstract

In assessing the performance of normal stable manufacturing processes with one-sided speciﬁcation limits, process capability indices CPUand CPLhave been widely used to measure the process capability. The purpose of this paper is to

develop an algorithm to compute the lower conﬁdence bounds on CPUand CPLusing the UMVUEs of CPUand CPL. The

lower conﬁdence bound presents a measure on the minimum capability of the process based on the sample data. We also provide tables for the engineers/practitioners to use in measuring their processes. A real-world example taken from a microelectronics device manufacturing process is investigated to illustrate the applicability of the algorithm. Imple-mentation of the existing statistical theory for capability assessment ﬁlls the gap between the theoretical development and the in-plant applications.

1. Introduction

Process capability indices are used to measure the capability of a process to reproduce items within the speciﬁed tolerance preset by the product designers or customers. Several capability indices, Cp, CPU, CPL, Cpk,

and Cpm, have been developed for the manufacturing

industry (Kane [1], Chan et al. [2], Pearn et al. [3]). Those indices essentially compare the speciﬁcation tol-erance range with the actual production toltol-erance range, which have been deﬁned as

Cp¼ USL LSL 6r ; CPU¼ USL l 3r ; CPL¼ l LSL 3r ; Cpk¼ min USL l 3r ; l LSL 3r ; Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2_{þ ðl T Þ}2 q ;

where USL is the upper speciﬁcation limit, LSL is the lower speciﬁcation limit, l is the process mean, r is the process standard deviation (overall process variation), and T is the preset target value.

The indices Cp, Cpk, and Cpm are appropriate for

processes with two-sided speciﬁcation limits. Some procedures have been developed based on these indices (Cheng [4], Pearn and Chen [5], Pearn and Chen [6]) for the practitioners to use in making decision on whether their processes meet the preset capability requirement. On the other hand, the indices CPUand CPLare designed

speciﬁcally for processes with one-sided speciﬁcation limit. Pearn and Chen [7] considered the indices CPUand

CPL, and obtained their uniformly minimum-variance

www.elsevier.com/locate/microrel

*

Corresponding author. Tel.: 714261; fax: +886-35-722392.

E-mail address:roller@cc.nctu.edu.tw(W.L. Pearn).

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unbiased estimators (UMVUEs). Lin and Pearn [8] de-veloped some eﬃcient SAS/Maple computer programs for calculating the critical values and the p-values using those UMVUEs in capability testing, particularly for normal processes.

Critical values are used for making decisions in ca-pability testing with designated type-I error a, the risk of misjudging an incapable process (H0: CPU6C) as a

ca-pable one (H1: CPU> C). The p-values are used for

making decisions in capability testing, which presents the actual risk of misjudging an incapable process (H0: CPU6C) as a capable one (H1: CPU> C). Thus, if

p <athen we reject the null hypothesis, and conclude that the process is capable with actual type-I error p (rather than a). Both approaches, the critical values and the p-values, do not convey any information regarding the minimal value (lower confidence bound) of the ac-tual process capability. The development of the lower confidence bound on the actual process capability is essential. The lower confidence bound not only gives us a clue on the minimal level of the actual performance of the process which is tightly related to the nonconform-ing units (product units fallout the specification limit USL or LSL), but is also useful in making decisions for capability testing. Table 1 displays some capability val-ues of CPU (or CPL) and the corresponding

noncon-forming unit (in ppm).

Montgomery [9] recommended some minimum quality requirements on CPUand CPL, shown in Table 2,

for speciﬁc process types which must run under some designated capability conditions. Therefore, it would be desirable to determine a bound which practitioners

would be expected to ﬁnd the true value of the process capability no less than the bound value with certain level of conﬁdence.

2. Estimations of CPUand CPL

A random sample is taken from a stable process. The conventional estimates of l and r are:

X ¼X n i¼1 Xi=n; S¼ ðn " 1Þ1X n i¼1 ðXi X Þ 2 #1=2 :

Thus, we may consider the natural estimators to esti-mate the indices CPUand CPLwhich can be deﬁned as the

following: b C CPU¼ USL X 3S b C CPL¼ X LSL 3S :

Chou and Owen [10] showed that under normality assumption the estimator bCCPUand bCCPLare distributed as

ð3pffiffiffinÞ1Tn1ðdÞ, where Tn1ðdÞ is distributed as the

noncentral t distribution with n 1 degrees of freedom and noncentrality parameter d¼ 3p bCCffiffiffin PU and d¼

3p bCCffiffiffin PL, respectively. Both estimators are biased (Pearn

and Chen [7]). But, Pearn and Chen [7] showed that by adding the well-known correction factor

bn1¼ 2 n 1 1=2 C n 1 2 C n 2 2 1

to bCCPU and bCCPL, the unbiased estimators bn1CCbPU and

bn1CCbPL can be obtained which have been denoted

as eCCPU and eCCPL. Thus, we have Eð eCCPUÞ ¼ CPU and

Eð eCCPLÞ ¼ CPL. S ince bn1<1 for n > 2, then Varð eCCPUÞ <

Varð bCCPUÞ and Varð eCCPLÞ < Varð bCCPLÞ. Pearn and Chen

[7] further showed that eCCPU and eCCPL are the UMVUEs

of CPU and CPL, respectively. The probability density

function of eCCPU and eCCPL can be expressed as

fðxÞ ¼3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n=ðn 1Þ p 2n=2 bn1 ffiffiffip p C½ðn 1Þ=2 Z 1 0 yðn2Þ=2exp ( 1 2 y " þ 3x ffiffiffiffiffiny p bn1 ffiffiffiffiffiffiffiffiffiffiffi n 1 p d 2#) dy:

3. Lower conﬁdence bound on CPU(or CPL)

Chou et al. [11] established the lower conﬁdence bounds on CPUand CPLbased on the natural estimator

b C

CPU and bCCPL. We note that those two estimators are

Table 1

Various CPUvalues and the corresponding nonconformities

CPU ppm 0.5 66 807 0.7 17 864 0.9 3467 1.1 484 1.3 48.10 1.5 3.40 1.7 0.1698 1.9 0.0060 2.1 0.0001488 2.3 0.0000026 Table 2

Some minimum capability requirements on CPUand CPL

CPU(or CPL) Process types

1.25 Existing processes

1.45 New processes, or existing processes on safety, strength, or critical parameters 1.6 New processes on safety, strength, or critical

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Table 3

Lower conﬁdence bounds CU of CPU for eCCPU¼ 0:7ð0:1Þ1:8, n ¼ 5ð5Þ200, c ¼ 0:95 (Panel A) and eCCPU¼ 1:9ð0:1Þ3:0, n ¼ 5ð5Þ200,

c¼ 0:95 (Panel B) n eCCPU 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Panel A 5 0.304 0.364 0.423 0.481 0.538 0.594 0.650 0.705 0.760 0.815 0.870 0.924 10 0.414 0.486 0.557 0.627 0.697 0.766 0.835 0.943 0.972 1.040 1.108 1.175 15 0.465 0.542 0.618 0.693 0.768 0.843 0.917 0.991 1.065 1.139 1.213 1.286 20 0.495 0.575 0.654 0.733 0.811 0.889 0.966 1.044 1.121 1.198 1.275 1.352 25 0.516 0.598 0.679 0.760 0.840 0.920 1.000 1.080 1.159 1.239 1.318 1.397 30 0.531 0.614 0.697 0.780 0.862 0.944 1.025 1.107 1.188 1.269 1.350 1.431 35 0.543 0.628 0.712 0.795 0.879 0.962 1.045 1.127 1.210 1.293 1.375 1.457 40 0.553 0.638 0.723 0.808 0.892 0.977 1.061 1.144 1.228 1.312 1.395 1.478 45 0.561 0.647 0.733 0.819 0.904 0.989 1.074 1.158 1.243 1.327 1.412 1.496 50 0.568 0.655 0.741 0.828 0.914 0.999 1.085 1.170 1.256 1.341 1.426 1.511 55 0.574 0.661 0.748 0.835 0.922 1.008 1.094 1.181 1.267 1.352 1.438 1.524 60 0.579 0.667 0.755 0.842 0.929 1.016 1.103 1.190 1.276 1.363 1.449 1.535 65 0.583 0.672 0.760 0.848 0.936 1.023 1.110 1.197 1.285 1.371 1.458 1.545 70 0.588 0.677 0.765 0.853 0.941 1.029 1.117 1.205 1.292 1.379 1.467 1.554 75 0.591 0.681 0.770 0.858 0.947 1.035 1.123 1.211 1.299 1.387 1.474 1.562 80 0.595 0.684 0.774 0.862 0.951 1.040 1.128 1.217 1.305 1.393 1.481 1.569 85 0.598 0.688 0.777 0.866 0.956 1.044 1.133 1.223 1.311 1.399 1.488 1.577 90 0.600 0.691 0.781 0.870 0.959 1.049 1.138 1.228 1.316 1.405 1.494 1.583 95 0.603 0.693 0.784 0.873 0.963 1.053 1.142 1.233 1.321 1.411 1.500 1.589 100 0.605 0.696 0.786 0.877 0.966 1.056 1.146 1.238 1.326 1.416 1.505 1.595 105 0.608 0.699 0.789 0.879 0.970 1.060 1.150 1.242 1.330 1.420 1.510 1.601 110 0.610 0.701 0.792 0.882 0.972 1.063 1.153 1.246 1.334 1.424 1.515 1.606 115 0.612 0.703 0.794 0.855 0.975 1.066 1.156 1.249 1.338 1.428 1.519 1.610 120 0.613 0.705 0.796 0.887 0.978 1.068 1.159 1.253 1.341 1.432 1.523 1.614 125 0.615 0.707 0.798 0.889 0.980 1.071 1.162 1.256 1.344 1.435 1.527 1.618 130 0.617 0.709 0.800 0.891 0.982 1.073 1.164 1.259 1.347 1.438 1.530 1.622 135 0.618 0.710 0.802 0.893 0.984 1.075 1.166 1.261 1.350 1.441 1.533 1.625 140 0.620 0.712 0.804 0.895 0.986 1.077 1.169 1.264 1.352 1.444 1.536 1.628 145 0.621 0.713 0.805 0.897 0.988 1.079 1.171 1.266 1.355 1.446 1.539 1.631 150 0.622 0.715 0.807 0.899 0.990 1.081 1.173 1.268 1.357 1.449 1.541 1.634 155 0.624 0.716 0.808 0.900 0.992 1.083 1.175 1.271 1.359 1.451 1.543 1.636 160 0.625 0.717 0.810 0.902 0.994 1.085 1.177 1.273 1.361 1.453 1.546 1.639 165 0.626 0.719 0.811 0.903 0.995 1.087 1.178 1.275 1.363 1.456 1.548 1.641 170 0.627 0.720 0.812 0.905 0.997 1.088 1.180 1.277 1.365 1.458 1.550 1.643 175 0.628 0.721 0.814 0.906 0.998 1.090 1.182 1.279 1.367 1.460 1.552 1.645 180 0.629 0.722 0.815 0.908 1.000 1.092 1.184 1.280 1.369 1.462 1.554 1.648 185 0.630 0.723 0.816 0.909 1.001 1.093 1.185 1.282 1.371 1.464 1.556 1.650 190 0.631 0.724 0.817 0.910 1.002 1.094 1.187 1.284 1.373 1.465 1.558 1.652 195 0.632 0.725 0.818 0.911 1.004 1.096 1.188 1.286 1.374 1.467 1.560 1.653 200 0.632 0.726 0.819 0.912 1.005 1.097 1.189 1.287 1.376 1.469 1.562 1.655 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 Panel B 5 0.978 1.032 1.086 1.140 1.194 1.247 1.301 1.355 1.408 1.462 1.515 1.568 10 1.243 1.310 1.378 1.445 1.512 1.580 1.647 1.714 1.781 1.848 1.915 1.982 15 1.359 1.433 1.510 1.579 1.652 1.725 1.798 1.871 1.944 2.017 2.090 2.163 20 1.429 1.506 1.582 1.659 1.736 1.812 1.889 1.965 2.041 2.118 2.194 2.271 25 1.477 1.556 1.635 1.714 1.793 1.872 1.950 2.029 2.108 2.187 2.266 2.344 30 1.512 1.593 1.673 1.754 1.835 1.916 1.996 2.077 2.157 2.238 2.318 2.399 35 1.539 1.622 1.704 1.786 1.868 1.950 2.032 2.114 2.196 2.278 2.359 2.441 40 1.562 1.645 1.728 1.811 1.895 1.978 2.061 2.144 2.227 2.310 2.393 2.476 45 1.580 1.664 1.749 1.833 1.917 1.980 2.085 2.169 2.252 2.336 2.420 2.504 50 1.596 1.681 1.766 1.851 1.935 1.999 2.105 2.190 2.274 2.359 2.443 2.528 (continued on next page)

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biased. In the following, we use eCCPU and eCCPL, the

UMVUEs of CPU and CPL, to obtain the lower

conﬁ-dence bound on CPU and CPL.

Let USL¼ X þ k1S and LSL¼ X k2S, so k1¼

3 eCCPU=bn1¼ 3 bCCPU and k2¼ 3 eCCPL=bn1¼ 3 bCCPL. A

100c% lower conﬁdence bound CU for CPU satisﬁes

PrðCPUP CUÞ ¼ c. It can be written as:

Pr USL l 3r P CU ¼ Pr Xþ k1S l 3r P CU ¼ Pr Z 3 ffiffiffi n p CU S=r P k1 ffiffiffin p ¼ Pr Z 3 ffiffiffi n p CU S=r P 3 eCCPU bn1 ffiffiffi n p ! ¼ PrðTn1ðd1Þ P t1Þ ¼ c;

and PrðTn1ðd1Þ 6 t1Þ ¼ 1 c. Similarly, a 100c% lower

conﬁdence bound CLfor CPLsatisﬁes PrðCPLP CLÞ ¼ c.

It can be shown as PrðTn1ðd2Þ 6 t2Þ ¼ c, where Z is

distributed as Nð0; 1Þ, Tn1ðdÞ is the noncentral t

distri-bution with n 1 degrees of freedom and noncentrality parameter d, t1¼ k1 ffiffiffin p , t2¼ k2 ffiffiffin p , d1¼ 3 ffiffiffin p CU, and d2¼ 3 ffiffiffin p

CL. Thus, to obtain the lower conﬁdence

bound (LCB), we may proceed as follows: Procedure of obtaining the LCB

(a) Determine the c level of conﬁdence (normally set to 0.95) for the lower conﬁdence bound.

(b) Calculate the value of estimator, eCCPU(or eCCPL), from

the sample data.

(c) Check the appropriate table listed in Table 3(a) and (b) and ﬁnd the corresponding CU(or CL) based on

e C

CPU(or eCCPL) and n.

(d) Conclude that the true value of the process capabil-ity is no less than the CU(or CL) with 100c% level of

conﬁdence.

4. An algorithm for calculating the lower conﬁdence bound Based on the procedure described above, a Matlab algorithm, called LCB, is developed for computing the

Table 3 (continued) n CCePU 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 55 1.610 1.695 1.781 1.866 1.952 2.017 2.123 2.208 2.293 2.379 2.464 2.549 60 1.621 1.707 1.794 1.880 1.966 2.031 2.137 2.223 2.309 2.395 2.481 2.567 65 1.632 1.718 1.805 1.891 1.978 2.044 2.150 2.237 2.323 2.410 2.496 2.582 70 1.641 1.728 1.815 1.902 1.988 2.055 2.162 2.249 2.335 2.422 2.509 2.596 75 1.649 1.737 1.824 1.911 1.998 2.065 2.173 2.260 2.347 2.434 2.521 2.608 80 1.657 1.745 1.833 1.921 2.008 2.075 2.183 2.271 2.358 2.446 2.533 2.621 85 1.665 1.753 1.841 1.929 2.017 2.085 2.193 2.281 2.369 2.457 2.545 2.633 90 1.672 1.761 1.849 1.938 2.026 2.094 2.203 2.291 2.379 2.468 2.556 2.644 95 1.679 1.768 1.857 1.946 2.034 2.103 2.212 2.301 2.389 2.478 2.567 2.656 100 1.685 1.774 1.864 1.953 2.042 2.111 2.221 2.310 2.399 2.488 2.577 2.666 105 1.691 1.780 1.870 1.960 2.050 2.119 2.229 2.319 2.408 2.498 2.587 2.677 110 1.696 1.786 1.877 1.967 2.057 2.127 2.237 2.327 2.417 2.507 2.597 2.687 115 1.701 1.792 1.882 1.973 2.064 2.134 2.245 2.335 2.425 2.516 2.606 2.696 120 1.706 1.797 1.888 1.979 2.070 2.140 2.252 2.343 2.433 2.524 2.615 2.706 125 1.710 1.801 1.893 1.984 2.076 2.147 2.258 2.350 2.441 2.532 2.623 2.714 130 1.714 1.805 1.897 1.989 2.081 2.152 2.265 2.357 2.448 2.540 2.631 2.723 135 1.717 1.809 1.902 1.994 2.086 2.158 2.270 2.363 2.455 2.547 2.639 2.731 140 1.721 1.813 1.906 1.998 2.091 2.163 2.276 2.369 2.461 2.554 2.646 2.738 145 1.724 1.816 1.909 2.002 2.095 2.167 2.281 2.374 2.467 2.560 2.653 2.746 150 1.727 1.819 1.912 2.006 2.099 2.172 2.285 2.379 2.472 2.566 2.659 2.752 155 1.729 1.822 1.916 2.009 2.102 2.175 2.290 2.384 2.477 2.571 2.665 2.758 160 1.732 1.825 1.918 2.012 2.106 2.179 2.294 2.388 2.482 2.576 2.670 2.764 165 1.734 1.827 1.921 2.015 2.109 2.182 2.297 2.392 2.486 2.581 2.675 2.770 170 1.737 1.830 1.924 2.018 2.112 2.185 2.300 2.395 2.490 2.585 2.680 2.775 175 1.739 1.832 1.926 2.020 2.114 2.188 2.304 2.399 2.493 2.589 2.684 2.779 180 1.741 1.834 1.928 2.023 2.117 2.191 2.306 2.402 2.497 2.592 2.688 2.783 185 1.743 1.837 1.931 2.025 2.119 2.194 2.309 2.405 2.500 2.596 2.691 2.787 190 1.745 1.839 1.933 2.027 2.122 2.196 2.312 2.407 2.503 2.599 2.695 2.791 195 1.747 1.841 1.935 2.029 2.124 2.198 2.314 2.410 2.505 2.602 2.698 2.794 200 1.749 1.843 1.937 2.031 2.126 2.200 2.316 2.412 2.508 2.604 2.701 2.797

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lower conﬁdence bounds, CU(or CL), on CPU (or CPL).

Four auxiliary functions for evaluating CU(or CL), the

normal distribution function, the Gamma function, polynomial roots function, and the noncentral t distri-bution function, are required. In order to accelerate the computation, Levinson [12] provided a good initial guess of d (below the real d) for numerical iteration. This equation is used to solve for d if t1 is given.

t1 dRn1þ Zc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 n1þ ð1 R2n1Þðd 2 Z2 cÞ q R2 n1 Z2cð1 R 2 n1Þ ; where Rn1¼ ½2=ðn 1Þ 1=2Cðn=2Þ=C½ðn 1Þ=2 and Zc¼

100c% of the standard normal distribution. Algorithm for the LCB

Step 1. Input the sample data (X1; X2; . . . ; Xn), USL (or

LSL), and c.

Step 2. Calculate X , S, bn1, and eCCPU (or eCCPL).

Step 3. Compute a good initial guess for d.

Step 4. Find the lower conﬁdence bound CU(or CL) on

CPU (or CPL), through numerical iterations.

Step 5. Output the conclusive message, ‘‘The true value of the process capability CPU (or CPL) is no less

than the CU (or CL) with 100c% level of

conﬁ-dence’’.

We implement the algorithm, and develop a Matlab program to compute the lower conﬁdence bounds (see Appendix A). Table 3(a) and (b) tabulate the lower conﬁdence bounds CU (or CL) for eCCPU (or eCCPLÞ ¼

0:7ð0:1Þ3:0, for n ¼ 5ð5Þ200, and c ¼ 0:95. Table 4 compares the lower conﬁdence bounds obtained by Chou et al. [11] with one obtained by our approach. It is noted that with the same conﬁdence level a, our ap-proach provides better bound (particularly for small n), which is closer to the real eCCPU (or eCCPL), and therefore

should be recommended for real-world applications.

5. An application example of low-drop-out linear regula-tors

The positive linear series pass voltage regulators are tailored for low-drop-out applications where low qui-escent power is considered important. These regulators include the reverse voltage sensing that prevents current in the reverse direction. The regulators are fabricated with the BiCMOStechnology, which is ideally suited for the low input-to-output diﬀerential applications. These products are speciﬁcally designed to provide well-regu-lated supply for low IC applications such as high-speed bus termination, low current logic supply, and VGA cards.

The products investigated here are low-drop-out 3A linear regulators with a low dropout voltage and short-circuit protection. These regulators are in three-lead packages and ﬁve-lead packages, as depicted in Fig. 1. The three-lead packages have preset outputs at 3.3 V or 5.0 V. The output voltage is regulated to 1.5% at room temperature. The ﬁve-lead packages regulate the output

Table 4

Comparisons of the LCB CUbetween the ChouÕs and our approach (New), for eCCPU¼ 0:7ð0:1Þ1:5, n ¼ 10ð10Þ50, c ¼ 0:95

n CCePU 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 10 New 0.41 0.49 0.56 0.63 0.70 0.77 0.84 0.93 0.97 Chou 0.37 0.44 0.50 0.57 0.63 0.70 0.76 0.82 0.88 20 New 0.50 0.58 0.65 0.73 0.81 0.89 0.97 1.04 1.12 Chou 0.47 0.55 0.63 0.70 0.78 0.85 0.93 1.00 1.08 30 New 0.53 0.61 0.70 0.78 0.86 0.94 1.03 1.11 1.19 Chou 0.52 0.60 0.68 0.76 0.84 0.92 1.00 1.08 1.16 40 New 0.55 0.64 0.72 0.81 0.89 0.98 1.06 1.14 1.23 Chou 0.54 0.63 0.71 0.79 0.87 0.96 1.04 1.12 1.20 50 New 0.57 0.66 0.74 0.83 0.91 1.00 1.09 1.17 1.26 Chou 0.56 0.64 0.73 0.81 0.90 0.98 1.07 1.15 1.24

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voltage programmed by an external resistor ratio. Short-circuit current is internally limited. The device responds a sustained overcurrent condition by turning oﬀ after a tON time delay. The device then stay oﬀ for a period,

tOFF, that is 32 times the tONdelay. The device then

be-gins pulsing on and oﬀ at the tON=ðtONþ tOFFÞ duty cycle

of 3%. This drastically reduces the power dissipation during the short-circuit, which means that the heat sinks need only accommodating the normal operation.

The quiescent current is an essential product char-acteristic, which has significant impact to product quality. For the quiescent current of a particular model of low-drop-out 3A linear regulators, the upper specifi-cation limit, USL, is set to 650 lA. Sample data of 80 observations are collected, which are displayed in Table 5. Fig. 2 displays the histogram of the 80 observations. Fig. 3 displays the normal probability plot of the sample data, and from this figure the sample data appears to be normal. Further, we perform Shapiro–Wilk test to check whether the sample data is normal. The statistic W , is found to be 0.9926, we can conclude that the sample data can be regarded as taken from normal process.

In order to obtain the lower conﬁdence bound on CPU, we execute the Matlab program attached in

Ap-pendix A. The program reads the sample data ﬁle, and input of n¼ 80, USL ¼ 650, and c ¼ 0:95, then output

with the sample mean X ¼ 398:85, the sample standard deviation S¼ 61:65, the correction factor bn1¼ 0:9905,

the estimator eCCPU¼ 1:371, and the lower conﬁdence

bound CU¼ 1:191. The corresponding execution output

is included in Appendix B. We therefore conclude that the true value of the process capability CPU is no less

than 1.191 with 95% level of conﬁdence.

6. Conclusions

In assessing the performance of normal stable man-ufacturing processes with one-sided speciﬁcation limits, process capability indices CPUand CPLhave been widely

used to measure the process capability. In this paper, we developed an algorithm to compute the lower conﬁdence bounds on CPUand CPLusing the UMVUEs of CPUand

CPL. The lower conﬁdence bound presents a measure on

the minimum capability of the process based on the sample data. We also provided tables for the engineers/ practitioners to use in measuring their processes. The lower bounds we provided are closer to the true CPUand

CPL index values than the existing ones. A real-world

example taken from a microelectronics device manu-facturing process is investigated to illustrate the appli-cability of the algorithm. Implementation of the existing statistical theory for capability assessment ﬁlls the gap between the theoretical development and the in-plant applications.

Table 5

The 80 sample observations

428.63 408.09 417.62 317.20 519.40 438.75 380.42 457.60 474.87 395.22 344.88 497.17 373.81 337.62 430.78 363.31 475.43 270.65 357.53 454.24 402.19 462.12 403.40 463.31 513.74 433.62 361.49 360.67 400.46 540.87 361.72 303.60 377.56 376.22 379.27 332.91 258.16 357.04 352.24 432.00 432.66 395.20 371.64 315.74 330.47 405.25 324.87 337.89 435.80 399.28 433.73 358.42 422.63 419.25 387.12 277.72 464.39 388.89 384.68 455.70 387.57 337.50 444.43 494.75 472.60 369.92 393.09 492.78 429.29 403.41 274.69 324.70 528.84 443.18 331.91 428.18 423.13 397.66 440.76 332.47

Fig. 2. Histogram of the 80 observations.

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Appendix A

% -% Input the sample dataðX1; X2; . . . ; XnÞ, USL, and c

% -function lcb(n,usl,r) t¼ 0; b¼ 0; z¼ 0; e¼ 0; f¼ 0; g¼ 0; rt¼ zerosð2; 1Þ; x¼ 0; x1¼ 0; x2¼ 0; y¼ 0; y1¼ 0; delta ¼ 0; delta1 ¼ 0; x3 ¼ 0; data¼ zeros(n,1); ½datað1 : n; 1Þ ¼ textread(Ôlingen.datÕ,Ô%fÕ,n); % -% Compute X , S, b_n1, and eCCPU. % -mdata¼ mean(data); stddata¼ std(data); b¼ sqrtð2=ðn 1ÞÞ gammaððn 1Þ=2Þ= gammaððn 2Þ=2ÞÞ;

ecpu¼ ðusl mdataÞ=ð3 b stddata); fprintf(ÔThe sample Mean is %g.nnÕ,mdata)

fprintf(ÔThe sample standard Deviation is %g.nnÕ,std-data)

fprintf(ÔThe Correction Factor is %g.nnÕ,b) fprintf(ÔThe UMVUE of Cpu is %g.nnÕ,ecpu)%% % -% Compute a good initial value of d.

% -t¼ 3 ðecoy=bÞ sqrtðnÞ; R¼ sqrtð2=ðn 1ÞÞ gammaðn=2Þ= gammaððn 1Þ=2Þ; z¼ norminv(r,0,1); e¼ R^₂_z^₂ _{ð1 R}^_2Þ; f¼ 2 R ðt R^2 z^₂ _{ð1 R}^_{2Þ tÞ;} g¼ ðt R^₂_z^₂ _{ð1 R}^_{2Þ tÞ}^ 2 z^₂ _R^₂_{þ z}^₄ _{ð1 R}^_2Þ; pol¼ ½e f g ; rt¼ roots(pol); x¼ nctinvðr; n 1; rtð2ÞÞ; x1¼ nctinvðr; n 1; rtð2Þ þ 0:01Þ; x2¼ ðx1 xÞ=9:5; y¼ absðx sqrtðnÞ 3 ecpu=bÞ; y1¼ y=x2; delta1¼ rtð2Þ þ y1 0:001; % -% Evaluate the lower conﬁdence bound cU(or cL) on

CPU(or CPL), by numerical iterations.

delta¼ delta1 þ 0:001 : 0:001 : delta1 þ 0:5; for i¼ 1 : 1 : 500

x3¼ nctinvðr; n 1; deltaðiÞÞ;

if(absðx3 sqrtðnÞ 3 ecpu=bÞÞ <¼ 0:01 cu¼ deltaðiÞ=ð3 sqrtðnÞÞ;

fprintf(ÔThe Lower Conﬁdence Bound is %g.nnÕ,cu)

break end end

% -% Output the conclusive message, ‘‘The true value of % the process capability CPU(or CPL) is no less

% than cU (or cL) with 100c% level of conﬁdence’’

% -fprintf(ÔThe true value of the process capability Cpu is no less than %gÕ, cu)

fprintf(Ôwith %gÕ,r) fprintf(Ôlevel of conﬁdenceÕ) % -% The End % -Appendix B Input: >>lcb(80,650,0.95) Output:

The sample Mean is 398.85.

The Sample Standard Deviation is 61.6503. The Correction Factor is 0.990471. The UMVUE of Cpu is 1.37099. The Lower Conﬁdence Bound is 1.19104.

The true value of the process capability Cpu is no less than 1.19104 with 0.95 level of conﬁdence

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