Group selection for production yield among k manufacturing lines

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Group selection for production yield among k manufacturing lines

Chen-ju Lin

a,n

, Wen Lea Pearn

b

a

Department of Industrial Engineering & Management, Yuan Ze University, Chungli, Taiwan b

Department of Industrial Engineering & Management, National Chiao Tung University, Hsinchu, Taiwan

a r t i c l e

i n f o

Article history:

Received 14 September 2009 Received in revised form 18 September 2010 Accepted 2 November 2010 Available online 18 November 2010 Keywords: Group selection Production yield Quality management Bonferroni method

a b s t r a c t

Producing qualiﬁed items or products is essential to meet the requirement preset by customers. Evaluation and selection of desired manufacturing lines become challenging tasks for decision makers. Production yield is one of the important factors in measuring production performance. The goal of this paper is to screen a group of manufacturing lines and identify the best one with the highest yield. For the production lines with extremely low fraction of defectives, the yield index, Spk, is an efﬁcient indicator for quality level. This

paper considers the production selection problem by using Spkto compare k (k 42)

manufacturing lines. A subset is constructed to contain the production lines with the highest yield. A systematic approach of test order k compares selected pairs of manufacturing lines along with the Bonferroni method is proposed to solve this problem. Each pair of production yields is compared by taking ratio. The paper provides critical values and required sample sizes of the group selection procedure. An application example on evaluating four power inductor productions is presented to illustrate the practicality of the proposed approach.

1. Introduction

Production line selection is a common problem faced by engineers or managers. Decision makers need to determine the best manufacturing lines based on preferred criteria such as quality, cost, processing time, and so on. Among these attributes, production quality is usually taken as the major consideration in decision making. The manufacturing line which produces many defectives is incapable and would incur extra material costs or even delay delivery. Evaluating multiple production lines and selecting the one that has the highest yield become critical issues in managing production systems. This paper investigates the production selection problem of comparing k (k42) production lines with a focus on production yield. The k production lines may refer to different manufacturing recipes, production procedures, or potential suppliers of the critical components.

Many industries require high quality of their products due to the features of items. For example, semiconductor, automotive, and integrated circuit assembly companies may set a minimum yield level higher than 99.9%. With the progress of manufacturing technique, the fraction of defective could be well controlled in these industries nowadays. It is clear that the conventional method of counting the frequency of nonconformities becomes inappropriate in evaluating production performance. An inspection may take thousands of samples to have one defective to be observed. Thus, a new measurement is necessary for judging the manufacturing lines with property of low defective rate.

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jspi

Journal of Statistical Planning and Inference

n

Corresponding author. Tel.: + 886 3 4638800; fax: + 886 3 4638907. E-mail address: [email protected] (C.-j. Lin).

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When k equals two, classical hypothesis testing methods can be extended to evaluate the performance of two productions.

Chou (1994)presents a likelihood ratio test to solve the selection problem.Daniels et al. (2005)uses confidence intervals to investigate process equality.Pearn et al. (2004, 2009)andLin and Pearn (2010)apply two-phase test scheme to select the better process and examine the magnitude of the difference based on different performance measurements. When testing homogeneity of two or more production performance,Hubele et al. (2005)develops a Wald test for the products with one-sided specification.Chen and Chen (2006)examine process equality by Fmaxtest followed by simultaneous confidence

intervals for pairwise difference. Polansky (2006)derives a permutation method without assuming the distribution of investigated quality feature. In terms of choosing the manufacturing line with the highest capability,Tseng and Wu (1991)

applies a modiﬁed likelihood ratio selection rule to select the best production line based on process variation. However, their approach is not appropriate when production mean also shifts away from target.Huang and Lee (1995)proposes subset selection approaches to identify the best process among several manufacturing lines. The authors emphasize the cost of failing customer’s requirement. The square error loss is considered as the decision criteria.

The goal of this paper is to select a subset of production lines with the highest yield among k (k 42) manufacturing lines. The methods selecting the best subset mentioned above, however, do not directly adopt production yield as decision criterion. A yield index, Spk, proposed byBoyles (1994), instead, has a one-to-one mapping to yield. Through analyzing such

index, production yield can be identiﬁed. The following paper presents an efﬁcient method to identify the best production lines among those with low fraction of defectives by using Spkindex. Section 2 reviews the statistical inference of Spk. Section 3

proposes a systematic approach to choose a group containing the manufacturing lines associated with the highest production yield. Section 4 investigates the required sample size to eliminate the worse manufacturing lines at a pre-speciﬁed error rate and power level. Section 5 evaluates four 3D-accelerometer manufacturing lines in a factory to show practicability of the proposed method. Section 6 concludes the paper with discussions.

2. Review yield index of production evaluation

Production yield is defined as the percentage of items passing inspection. For production lines with two-sided specifications, the product must have investigated characteristic falling within the manufacturing tolerance to be considered as a qualified item. Under the definition, production yield can be calculated as Yield=F(USL)–F(LSL) where USL is the upper specification limit, LSL is the lower specification limit, and F( ) is the cumulative distribution function (cdf) of the product characteristic. When the investigated characteristic is normally distributed, the expression can be rewritten as Yield=

F

[(USL–

m

)/

s

]–

F

[(LSL–

m

)/

s

] where

m

is the production mean,

s

is the production standard deviation, and

F

( ) is the cdf of a standard normal distribution. 2.1. Yield index Spk

For the manufacturing line with extremely low fraction of defectives, it is not efﬁcient to evaluate yield by counting the number of defectives. Based on the idea of production yield mentioned above,Boyles (1994)proposed a yield index deﬁned as follows: Spk¼ 1 3

F

1 1 2

F

USL

m

s

þ1 2

F

m

LSL

s

ð1Þ The Spkindex is a convenient and effective tool for performance assurance and process improvement. The index quantiﬁes

production performance by considering production location, production variation, and manufacturing speciﬁcations all together. The Spkindex provides a one-to-one mapping to yield: Yield =2

F

(3Spk) 1. For example, Spk= 1 and 1.33 imply that

nonconformities in parts per million (ppm) is 2700 and 66, respectively. 2.2. Estimation of the yield index

Production parameters

m

and

s

are usually unknown.Lee et al. (2002)considers the general estimator of Spk

^Spk¼ 1 3

F

1 1 2

F

USLx s þ1 2

F

xLSL s ð2Þ where x and s are the sample mean and the sample standard deviation, respectively, which can be obtained from an in-controlled manufacturing lines. The exact distribution of ^Spkis mathematically intractable.Lee et al. (2002)applies the ﬁrst

order of Taylor expansion to obtain the approximate distribution of ^Spk

^SpkffiSpkþ 1 6pffiffiffin W

j

ð3SpkÞ ð3Þ where W ¼ ﬃﬃn p ðS2_s2_Þ s2 USLm 2s

j

USLm s þmLSL₂_s

j

mLSL s n o ﬃﬃn p ðXmÞ s

j

USLm s

j

mLSL s n o ð4Þ

j

( ) is the probability density function (pdf) of the standard normal distribution. By Central Limit Theorem, the W statistic is asymptotically distributed as N(0, a2+ b2) where a and b can be expressed by the precision index Cp=(USL LSL)/6

s

and the

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accuracy index Ca= 1 9

m

m9/d, d =(USL LSL)/2, m= (USL+LSL)/2,

a ¼ 1ffiffiffi 2

p 3Cpð2CaÞ

j

ð3Cpð2CaÞÞ þ3CpCa

j

ð3CpCaÞ, ð5Þ

b ¼

j

ð3Cpð2CaÞÞ

j

ð3CpCaÞ: ð6Þ

Therefore, ^Spkis asymptotically distributed as N(

m

s,

s

s2) with

m

s¼Spkand

s

2s¼

ða2_þb2_Þ

36n½

j

ð3SpkÞ2

:

3. Group selection for high production yield

Let

P

1, y,

P

kbe mutually independent manufacturing lines. The k (k42) production lines could refer to different

manufacturing recipes, production procedures, or suppliers of the investigated component. The random samples Xi,jis the

quality characteristic collected from the jth item of the ith production line, 1rirk, 1rjrni. Assume that the collected

characteristics from the same population are independent and normally distributed with mean

m

iand variance

s

i2. Under the

normality assumption, the yield index, Spk, is an effective indicator of production yield. Denote Spkias the yield level of

production line

P

iand ^Spkias the estimated yield of production line i derived from Xi,j. The goal of this paper is to select a

nonempty group of arbitrary size containing all possibly the best manufacturing lines associated with the largest Spkvalue.

3.1. Group selection procedure for the highest yield

This paper adopts the ratio statistics to compare k (k42) yields together. Let ^Spk½1r ^Spk½1r. . .^Spk½k be the ordered

estimators of Spk. Deﬁne Ri¼ ^Spk½k=^Spki, where ^Spkio ^Spk½k, as a measure of separation between the two

P

’s corresponding to

^Spk½kand ^Spki. The concept of the proposed selection procedure is to compare (k–1) production lines to the line with the largest

estimated yield. The proposed group selection procedure is as follows: Step 1: Calculate sample statistics and derive estimated yield based on Spk.

Xi¼ Xni j ¼ 1 Xi,j ni , s 2 i ¼ Xni j ¼ 1 ðXi,jXiÞ2 ni1 , ^Spki¼ 1 3

F

1 1 2

F

USLXi si ! þ1 2

F

XiLSL si ! ( ) : Step 2: Rank the estimated yields in an ascending order.

^Spk½1r ^Spk½2r. . .r ^Spk½k:

Step 3: Include

P

iin the selected subset if and only if the ratio statistic

Ri¼ ^Spk½k=^Spkioca, 1rirk:

cais a predetermined constant satisfying the overall error constraint and is greater than one. Since ^Spk½k=^Spk½kis always less

than ca, only (k–1) ratio statistics need to be tested against ca. The selected subset must contain the production line with the

highest estimated Spkby using the test procedure above.

Lin and Pearn (2010)investigates a simpler problem of testing H0: Spk2/Spk1r1 vs. H1: Spk2/Spk141. Unlike that paper

which clearly speciﬁes the test hypotheses before the experiment, here we cannot construct the hypotheses before calculating estimated yields since ^Spk½kis unavailable in advance. When testing more than two Spk’s, two fundamental issues

arise: (1) error inﬂation due to multiple tests and (2) unﬁxed population

P

iassociated with ^Spk½k. In the following sections,

Section 3.2 discusses the Bonferroni method which handles the overall error rate problem. Section 3.3 calculates the critical value, ca, under the proposed selection procedure along the Bonferroni adjustment.

3.2. The Bonferroni adjustment

The actual production yields are usually unknown in practice. Let Spk[1]rSpk[2]ryr Spk[k]be the ordered yield indices.

Neither the values of the Spk[s]nor the pairing of the

P

iwith the Spk[s](1ri, srt) is known. The best production line may not

be unique. A correct selection (CS) is the event that the production line associated with Spk[k]has been correctly chosen by the

selection rule. When evaluating more than two production lines, multiple tests are required to identify the highest yield. The Bonferroni method is an effective approach to handle multiple comparisons problem. The Bonferroni inequality states that

P [ g i ¼ 1Ei rX g i ¼ 1 PðEiÞ: ð7Þ

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Suppose that there are g tests in total. Let Eibe the event of falsely rejecting the ith test, 1rirg. If the signiﬁcance level

of individual test is controlled at

a

/g, the probability of falsely rejecting any test is less than or equal to

a

by the Bonferroni inequality P [ g i ¼ 1Ei ¼1P \ g i ¼ 1E c i ¼1 1

a

g g rg

a

_g ¼

a

ð8Þ

The Bonferroni method controls the overall error rate by adjusting p-values and has been applied to different ﬁelds. For example,Chen and Chen (2006)examines every pair of processes to evaluate multiple manufacturing lines,Ludlow et al. (2007)applies the Bonferroni correction to medical experiments. However, the goal of this paper is to select all of the best production lines instead of ordering them, not all relationship between every two productions is of interest. Instead of making all-pairwise comparisons with k(k–1)/2 tests, the proposed selection rule in Section 3.1 will only carry out (k–1) tests. As a result, the test order reduces from k2to k which is more efﬁcient.

3.3. Critical values for group selection procedure

The production line with the maximal estimated Spkis considered to have the highest yield among k lines. After ﬁxing the

value of ^Spk½k, the proposed selection procedure conducts ( k–1) tests. For the ^Spkislightly smaller than ^Spk½k, the corresponding

production line is also considered as one of the best lines. Suppose that ^Spkkhas ^Spk½k, the hypotheses of comparing the yields

of production lines k and i (1riok ) can be formed as H0: Spkk/Spkir1 vs. H1: Spkk/Spki41 where the test statistic

Ri¼ ^Spk½k=^Spkifollows a distribution of N(Spkk,

s

sk2)/N(Spki,

s

si2). By convolution, the pdf of Riis

fRiðrÞ ¼ 1 ffiffiffiffiffiffi 2

p

p ð

s

2 skþ

s

2 sir2Þ3=2 ð

s

2skSpkiþ

s

2siSpkkrÞ exp 1 2 Spki

s

2 si þSpkkr

s

2 sk !

s

2 skSpkiþ

s

2siSpkkr

s

2 skþ

s

2sir2 ! 1 2 S2 pki

s

2 si þS 2 pkk

s

2 sk ! " # 2

F

s

2 skSpkiþ

s

2siSpkkr

s

si

s

sk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s

2 skþ

s

2sir2 q 0 B @ 1 C A1 2 6 4 3 7 5þ

_p

s

si

s

sk ð

s

2 skþ

s

2 sir2Þ exp 1 2 S2 pki

s

2 si þS 2 pkk

s

2 sk ! " # , ð9Þ

where NoroN. Based on the sampling distribution of Ri¼ ^Spk½k=^Spkiand a given signiﬁcance level, we reject H0and claim

that production line i has a lower yield than line k if the test statistic RiZc_a. Otherwise, production line i may be at least as good as line k and is then selected into the subset when Rioca.

However, the test hypotheses are undetermined before data collection because the proposed selection procedure depends on the production line with ^Spk½k, which is unknown. To control the overall error rate, the type I error rate that equals to k time

the conditional type I error rate given that ^Spkk¼ ^Spk½kshould be set to

a

. In another words, the probability of falsely rejecting

any Spki,1riok, given that H0: SpkkrSpkiand ^Spkk¼ ^Spk½kshould be controlled at

a

/k. The requirement can be satisﬁed by

adjusting the type I error of individual test to

a

/[k(k–1)]

PrfRiZc_a9H₀: SpkkrSpki, ^Spkk¼ ^Spk½kgr

a

=½kðk1Þ ð10Þ

Such Bonferroni adjustment then guarantees that the probability of correct selection, P(CS), is at least 1–

a

.

The maximal value of caamong those investigated parameters of Cpand Cais chosen to obtain conservative bound on the

critical value for reliability purpose. Since ^Spk has the largest variance at Ca=1, we calculate the critical value with the

conditions of Spki= Spkk= C and Cai= Cak= 1, i.e.

PrfRiZc_a9S_pki¼Spkk¼C, Cai¼Cak¼1,ni,nkg ¼

a

=½kðk þ 1Þ: ð11Þ

For Cai=Cak= 1 and equal sample size, the probability density function of Rican be further simpliﬁed as follows:

fRiðrÞ ¼ ffiffiffiffi n

p

r 1þ r ð1 þr2_Þ3=2 2

F

ffiffiffiffiffiffiffiffiffiffiffiffi 2n 1 þr2 r ð1þ rÞ ! 1 ( ) exp nð1rÞ 2 1 þr2 ( ) þ 1

p

ð1þ r2_Þexp 2nf g ð12Þ

which is a function of n only. Therefore, the value of c0is dependent on sample size but invariant under C.

Under the proposed selection rule, we can guarantee to include all of the best production lines at a conﬁdence level of at least 1–

a

.Table 1provides the critical values for k=3, 4, y, 6, ni=n = 30, 40, y, 200, at

a

= 0.05, 0.1. The production selection

procedure can be also applied to general cases with unequal sample sizes by recalculating ca.

3.4. Correct selection under least favorable conﬁguration

When Spk1= Spk2=y = Spkk, all production lines are considered as equally the best. The selection procedure will lead to a

wrong decision if any of the k production lines is not selected into the group. For a ﬁxed sample size, such setting minimizes the probability of correct selection and thus is considered as the least favorable conﬁguration (LFC). Controlling the overall

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error rate at

a

= 0.05,Table 2analyzes the probability of correct selection under the LFC by using the proposed selection procedure with the critical values listed inTable 1.

It is clear that the probability requirement of P(CS)41

a

is attained for all settings inTable 2. When k gets larger, the Bonferroni method gets conservative where the type I error rate is less than the preset

a

level and probability of correct selection is greater than 95%. This phenomenon can be further improved by using other multiple comparisons technique in the future. Note that when applying the selection procedure to compare two production lines, probability of correct selection would be exactly 1

a

since no adjustment of signiﬁcant level is required for having one test.

4. Required sample size for speciﬁed power

The group selection procedure in section 3 guarantees that the probability requirement of P(CS)41–

a

. The probability that the selected subset contains all production lines associated with Spk[k]should be maintained. Once the sample sizes and

the

a

risk are speciﬁed, the ability of eliminating the inferior production line is determined. To increase the probability of excluding the line with lower yield and to maintain probability of correct selection at the same time, sample sizes need to be increased.

The proposed method separates k production lines into two parts. The lines in the selected subset are considered as having the highest yield while those that are not selected are considered as having lower yields. Suppose that the production line k has ^Spk½kand is selected into the subset. To exclude the line that has yield level less than Spkkfrom the selected group, the

minimal required sample size for pre-speciﬁed power q and overall error

a

can be obtained by recursively searching the

Table 1

Critical values of the group selection procedure with. ni= n= 30, 40, y, 200, k= 3, 4, 5, 6, anda= 0.05, 0.1.

n k= 3 k= 4 k= 5 k= 6 a= 0.05 30 1.577 1.660 1.722 1.771 40 1.478 1.542 1.589 1.627 50 1.415 1.469 1.508 1.539 60 1.371 1.418 1.452 1.478 70 1.338 1.380 1.410 1.434 80 1.312 1.351 1.378 1.399 90 1.292 1.327 1.352 1.371 100 1.274 1.307 1.330 1.348 110 1.260 1.290 1.312 1.329 120 1.247 1.276 1.297 1.313 130 1.236 1.264 1.283 1.298 140 1.226 1.253 1.271 1.286 150 1.218 1.243 1.261 1.275 160 1.210 1.234 1.251 1.265 170 1.203 1.226 1.243 1.256 180 1.197 1.219 1.235 1.247 190 1.191 1.213 1.228 1.240 200 1.186 1.207 1.222 1.233 a= 0.1 30 1.494 1.577 1.638 1.687 40 1.412 1.478 1.525 1.563 50 1.359 1.415 1.455 1.486 60 1.322 1.371 1.406 1.433 70 1.294 1.338 1.369 1.393 80 1.272 1.312 1.341 1.363 90 1.255 1.292 1.318 1.338 100 1.240 1.274 1.299 1.317 110 1.227 1.260 1.282 1.300 120 1.216 1.247 1.269 1.285 130 1.207 1.236 1.257 1.272 140 1.199 1.226 1.246 1.261 150 1.191 1.218 1.237 1.251 160 1.184 1.210 1.228 1.242 170 1.178 1.203 1.220 1.234 180 1.173 1.197 1.214 1.226 190 1.168 1.191 1.207 1.220 200 1.163 1.186 1.201 1.213

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following constraints

PrfRiZc_a9H₀: SpkkrSpki, ^Spkk¼ ^Spk½kgr

a

=½kðk1Þ, ð13Þ

and

PrfRiZc_a9H₁: Spkk4Spki, ^Spkk¼ ^Spk½kg Zq ð14Þ

Assume that the sample sizes are equal in every production line,Table 3shows the required sample sizes for q= 0.7, 0.8, 0.9, 0.95 and Spki= Spkk/(1 +p) where p = 0.1, 0.15, y, 0.55 is the smallest difference to detect in the ratio of Spkat

a

= 0.05. Taking

comparisons of the four production yields as an example; to reach a 70% power of rejecting the yield index that is around 2/3 of the highest yield index (p = 0.5), the required sample size is around 60 at a significant level of 0.05.Table 3shows that a smaller sample size is sufficient to reach the same power level when the gap of the yields is larger. The benefit of using the ratio statistics is that the selection procedure does not require the prior information of the value of Spk[k]. Neither the critical

points nor the sample size depends on Spklevel.

Table 3

The required sample sizes for detecting the yield index smaller than 1/(1 + p) of the largest yield index at power = 0.7, 0.8, 0.9, 0.95 anda= 0.05.

p k= 3 k=4 k= 5 k= 6 k= 3 k=4 k= 5 k= 6 power= 0.7 power= 0.8 0.1 939 1104 1226 1322 1155 1336 1469 1574 0.15 439 515 572 617 538 623 684 733 0.2 259 304 337 364 317 367 403 432 0.25 173 203 225 244 213 246 270 289 0.3 126 148 164 178 154 179 197 211 0.35 97 115 127 136 118 137 151 162 0.4 77 91 101 109 94 109 120 129 0.45 65 76 84 90 79 90 100 107 0.5 55 63 70 76 66 76 84 90 0.55 46 55 60 66 56 66 73 77 power= 0.9 power= 0.95 0.1 1489 1694 1843 1960 1798 2023 2184 2313 0.15 694 789 859 913 838 943 1018 1077 0.2 409 465 506 538 493 554 600 634 0.25 274 311 339 360 330 371 400 424 0.3 199 227 245 261 240 269 291 307 0.35 152 173 188 200 183 207 224 236 0.4 122 139 151 161 147 165 179 188 0.45 100 115 124 132 120 136 147 155 0.5 84 96 104 111 101 115 123 131 0.55 73 83 90 96 87 98 107 113 Table 2

The probability of correct selection under the least favorable conﬁguration, Spk1= Spk2= y = Spkk, at ni= n= 30, 40, y, 200, k= 3, 4, 5, 6, anda= 0.05.

n k= 3 k= 4 k= 5 k= 6 30 0.958 0.962 0.965 0.967 40 0.957 0.961 0.963 0.965 50 0.957 0.961 0.963 0.964 60 0.957 0.960 0.962 0.964 70 0.957 0.960 0.962 0.963 80 0.957 0.960 0.962 0.963 90 0.957 0.960 0.962 0.963 100 0.957 0.960 0.961 0.963 110 0.957 0.959 0.961 0.962 120 0.957 0.959 0.961 0.962 130 0.956 0.959 0.961 0.962 140 0.956 0.959 0.961 0.962 150 0.956 0.959 0.961 0.962 160 0.956 0.959 0.961 0.962 170 0.956 0.959 0.961 0.962 180 0.956 0.959 0.961 0.962 190 0.956 0.959 0.961 0.962 200 0.956 0.959 0.961 0.962

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5. An application on power inductor production 5.1. Power inductor

Power inductor is a passive electrical device composed of magnetic material surrounded by coil. The component can store energy in a magnetic ﬁeld generated by the electric current passing through it. The output is measured by inductance in the units of henries. Power inductors are used in DC–DC converters that convert voltage to desired level. The DC–DC converter parts appear in many applications such as cellular phones, light-emitting diode (LED) lighting, personal digital assistants (PDAs), and digital still cameras.

Here we investigate a speciﬁc type of digital still camera whose target inductance of the power inductor is 10 micro-henries (

m

H) with LSL =8

m

H and USL= 12

m

H. If the observed inductance falls outside of the tolerance (LSL, USL), the quality of pictures and lifetime of cameras will be discounted. Four power inductor production lines under different manufacturing recipes are evaluated in terms of production yield at

a

= 0.05.Fig. 1displays the inductance data collected from the four production lines. There are only 60 samples available from each line. Normal probability plots inFig. 2suggest that the data from all manufacturing lines could be normally distributed.

5.2. Select all production lines with the highest yield

For the inductance data displayed inFig. 1, we calculate the sample means, sample standard deviations and derive estimated yield levels of all manufacturing lines: ^Spk1¼1:316, ^Spk2¼1:035, ^Spk3¼1:888, ^Spk4¼1:545. Since production line

number three has the highest estimated yield, all the other lines are compared with line 3: H0: Spk3rSpkiversus H1: Spk34Spki,

i= 1, 2, 4. The critical value is 1.418 at

a

= 0.05 fromTable 1. By taking ratio of two estimators, only the test statistic R4is smaller

than the critical point. It implies that the performance of line 3 and 4 cannot be separated while line 3 is superior to line 1 and 2. Therefore, both line 3 and 4 are selected into the group. We conclude that the selected group contains all production lines corresponding to the highest yield with a 95% conﬁdence level. The calculation results and ﬁnal selection decisions are listed inTable 4.

Suppose that the probability of detecting the smallest difference with p= 0.5 is required to be around 0.7. In other words, the test procedure should be able to reject the yield index which is 2/3 of the highest yield index with a power about 0.7.

Table 3suggests that 60 samples from each production line could serve the purpose. By ﬁxing group number, sample size,

0 10 20 30 40 50 60 8

10 12

Raw data from process 1

Observations

µH

0 10 20 30 40 50 60 Raw data from process 2

Observations 0 10 20 30 40 50 60 Observations 0 10 20 30 40 50 60 Observations 8 10 12

µH 8 10 12 µH 8 10 12 µH

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type I error, and p, we can calculate numerical power by using eq. (9).Table 5demonstrates the powers at given k= 4, n= 60,

a

= 0.05 and p ranges from 0.5 to 0.75.

6. Conclusions

Selection of the best production lines with the highest yield is an important problem in production evaluation. The task is difﬁcult especially when the fraction of defective is extremely low. Conventional method that examines the production lines by calculating the frequency of nonconformities is inefﬁcient. This paper considers the production selection problem among multiple (k42) two-sided manufacturing lines by adopting a yield index Spk. For well-controlled normal production lines

with two-sided speciﬁcation limits, such index provides effective measures on production yield. Along with the Bonferroni method, a group selection approach is proposed to include all of the best production lines with the highest yield. Unlike

Table 4

Sample statistics and selection decisions.

Production line 1 2 3 4 x 10.415 10.985 9.691 10.369 s 0.419 0.351 0.305 0.363 ^Spk 1.316 1.035 1.888 1.545 Estimated yield 99.992119% 99.810492% 99.999999% 99.999642% Ri 1.435 1.823 1.222 Selection N N Y Y 9.5 10 10.5 11 0.01 0.05 0.25 0.50 0.75 0.95 0.99 Data Probability

Normal Probability Plot of Process 1

10 10.5 11 11.5 0.01 0.05 0.25 0.50 0.75 0.95 0.99 Data Probability

9.2 9.4 9.6 9.8 10 10.2 10.4 0.01 0.05 0.25 0.50 0.75 0.95 0.99 Data Probability

10 10.5 11 0.01 0.05 0.25 0.50 0.75 0.95 0.99 Data Probability

Fig. 2. The normal probability plots of the data collected from the four production lines.

Table 5

The powers of the selection procedure given p = 0.5, 0.55, y , 0.75 at k= 4, n= 60, and ca= 1.418.

p 0.50 0.55 0.60 0.65 0.70 0.75

(9)

all-pairwise comparisons with k(k–1)/2 tests, the selection procedure here only requires (k–1) tests. The test order k, which is lower than k2_{, further indicates efﬁciency of the proposed method. An application on selecting the best power inductor}

manufacturing lines among four candidates is presented to demonstrate the efﬁcacy of the proposed approach.

Acknowledgments

The authors are grateful to the referees for their insightful comments that greatly improve the paper. This paper is supported by the National Science Council of Taiwan, Grant NSC 98-2221-E-155-008.

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