Normal approximation to the distribution of the estimated yield index S-pk

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Normal Approximation to the Distribution of the

Estimated Yield Index S

pk

W. L. PEARN1_{, G. H. LIN}2_{and K. H. WANG}3

1_{Department of Industrial Engineering & Management, National Chiao Tung University;} 2_{Department of Communication Engineering, National Penghu Institute of Technology;} 3_{Department of Applied Mathematics, National Chung Hsing University, Taiwan, R.O. China}

Abstract. Process yield is the most common criterion used in the manufacturing industry for

measur-ing process performance. A measurement index, called Spk, has been proposed to calculate the yield

for normal processes. The measurement index Spkestablishes the relationship between the

manufac-turing specifications and the actual process performance, which provides an exact measure on process

yield. Unfortunately, the sampling distribution of the estimated Spkis mathematically intractable.

Therefore, process performance testing cannot be performed. In this paper; we consider a normal

approximation to the distribution of the estimated Spk, and investigate its accuracy computationally.

We compare the critical values calculated from the approximate distribution with those obtained using the standard simulation technique, for various commonly used quality requirements. Extensive computational results are provided and analyzed. The investigation is useful to the practitioners for making decisions in testing process performance based on the yield.

Key words: critical value, process yield

1. Introduction

Process yield has longtime been a standard criterion used in the manufacturing industry as a common measure on process performance. Process yield is currently defined as the percentage of processed product unit passing the inspection. That is, the product characteristic must fall within the manufacturing tolerance. For product units rejected (nonconformities), additional costs would be incurred to the factory for scrapping or repairing the product. All passed product units are equally accep-ted by the producer, which requires the factory no additional cost. For processes with two-sided manufacturing specifications, the process yield can be calculated as %Yield= F (USL) − F (LSL), where USL and LSL are the upper and the lower specification limits, respectively, and F (·) is the cumulative distribution function of the process characteristic. If the process characteristic follows the normal distribu-tion, then the process yield can be alternatively expressed as Yield%= (USL − µ)/σ ]− [(LSL − µ)/σ ], where µ is the process mean, σ is the process standard deviation, and (·) is the cumulative distribution function of the standard normal distribution N (0, 1).

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Table I. Some S_pkvalues and the cor-responding nonconformities. Spk Yield PPM 1.00 0.9973002039 2699.796 1.10 0.9990331517 966.848 1.20 0.9996817828 318.217 1.30 0.9999038073 96.193 1.33 0.9999339267 66.073 1.40 0.9999733085 26.691 1.50 0.9999932047 6.795 1.60 0.9999984133 1.587 1.67 0.9999994557 0.544 1.70 0.9999996603 0.340 1.80 0.9999999334 0.067 1.90 0.9999999880 0.012 2.00 0.9999999980 0.002

Based on the expression of process yield, Boyles (1994) considered the yield measurement index Spkfor normal processes (defined in the following). The index

Spk establishes the relationship between the manufacturing specifications and the

actual process performance, which provides an exact measure on the process yield. If Spk = c, then the process yield can be expressed as %Yield = 2(3c) − 1.

Obviously, there is a one-to-one correspondence between Spk and the process

yield. Thus, Spk provides an exact (rather than approximate) measure of the

pro-cess yield. Table I summarizes the propro-cess yield, nonconformities (in PPM) as a function of the measurement index Spk, for Spk = 1.00(0.1)2.00, including the

most commonly-used performance requirements, 1.00, 1.33, 1.50, 1.67, and 2.00. For example, if a process has yield measure Spk = 1.33, then the corresponding

nonconformities is roughly 66 PPM (parts per million).

Spk = 1 3 −11 2 USL− µ σ +1 2 µ− LSL σ .

2. Approximate Distribution of the Estimated Spk

To estimate the yield measurement index Spk, we consider the following natural

estimator ˆSpk, where ¯X = (

n

i=1Xi)/n, and S = [(n−1)−1

n

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P K

the sample mean, and the sample standard deviation, the conventional estimators of µ and σ , respectively, which may be obtained from a stable process.

ˆSpk = 1 3 −11 2 USL− ¯X S + 1 2 ¯X− LSL S .

The exact distribution of Spkis mathematically intractable. Therefore, testing of

the process performance cannot be performed. On the other hand, Lee et al. (2002) obtained a normal approximation to the distribution of Spkusing Taylor expansion

technique. By taking the first order of the Taylor expansion, it is shown that the estimator can be expressed approximately as:

ˆSpk ∼= Spk+ 1 6√n W φ(3Spk) , where W = √ n(S2_{− σ}2₎ σ2 USL− µ 2σ φ USL− µ σ + µ− LSL 2σ φ µ− LSL σ − √ n( ¯X− µ) σ φ USL− µ σ − φ µ− LSL σ = √ n(S2_{− σ}2₎ σ2 d− (µ − m) 2σ φ d− (µ − m) σ +d+ (µ − m) 2σ φ d+ (µ − m) σ − √ n( ¯X− µ) σ φ d− (µ − m) σ − φ d+ (µ − m) σ .

We note that W = (√n/2)[a(S2− σ2)/σ2] −√n[b( ¯X − µ)/σ ] for µ < m, and W = (√n/2)[a(S2 _{− σ}2_)/σ2_{] +}√_n_{[b( ¯X − µ)/σ ] for µ > m, where a}

and b are functions of µ and σ , as defined in the following, and φ is the prob-ability density function of the standard normal distribution N (0, 1). Thus, the statistic W is normally distributed as N (0, a2 + b2), and the natural estimator ˆSpk is approximately distributed as N (µS, σS2), where E( ˆSpk) = µS = Spk and

Var( ˆSpk)= σS2= (a2+b2){36n[φ(3Spk)]2}−1, which can be expressed as functions

of the widely used precision index Cp= (USL−LSL)/6σ , and the accuracy index

Ca= 1 − |µ − m|/d.

The parameter Cp is defined as Cp = (USL − LSL)/6σ , a function of the

process standard deviation, which measures the overall process variation relat-ive to the specification tolerance therefore only reflecting process potential. The parameter Ca is defined as Ca = 1 − |µ − m|/d (see Pearn et al. (1998)), a

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where m = (USL + LSL)/2 is the mid- point between the upper and the lower specification limits, and d = (USL − LSL)/2 is half of the length of the specific-ation interval. The parameter Caalerts the user if the process mean deviates from

its target value. In fact, a mathematical relationship among the three measurements can be established as (3Spk)= {(3CpCa)+ [3Cp(2− Ca)]}/2. a = √1 2 USL− µ σ φ USL− µ σ +µ− LSL σ φ µ− LSL σ = √1 2 d− (µ − m) σ φ d− (µ − m) σ +d+ (µ − m) σ φ d+ (µ − m) σ = √1 2{3Cp(2− Ca)φ(3Cp(2− Ca))+ 3CpCaφ(3CpCa)}. b = φ USL− µ σ − φ µ− LSL σ = φ d− (µ − m) σ − φ d+ (µ − m) σ = φ{3Cp(2− Ca)} − φ(3CpCa).

The probability density function of the approximate distribution, N (µS, σS2),

can be expressed as:

f (x) = 18n π φ(3Spk) √ a2+ b2exp −18n(φ(3Spk))2 a2+ b2 ×(x − Spk)2 , −∞ < x < ∞.

3. Calculation of the Critical Values

The formula of the normal approximation obtained for the distribution of ˆSpk is

rather complicate, and the calculation is cumbersome to deal with. For the ap-proximation to be useful to the practitioners, we calculate the critical values co

computationally using the Maple-V programming software (see Appendix). Since the critical values cois a function of the parameters Cpand Ca, we have considered

the factor of the two parameters in the calculations to ensure that the critical values obtained are reliable.

3.1. NORMAL APPROXIMATION

Tables II(a)–II(e) display the parameters of the process characteristics used in the critical value calculations, covering the most commonly used performance require-ments Spk = 1.00 (capable), 1.33 (satisfactory), 1.50 (good), 1.67 (excellent),

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P K

Table 2a. S_pk= 1.00 with Cp= 1.0(0.1)2.0, the corresponding µ, σ , Ca, and the calculated a, b,

and nVar( ˆSpk). Note: φ (3Spk)= 0.004431848.

Cp Ca µ σ a b nVar( ˆSpk) 1.0 1.000000000 15.00000000 1.666666667 0.018802740 0.000000000 0.500000072 1.1 0.845650984 15.77174508 1.515151515 0.016792051 −0.007843357 0.485784252 1.2 0.772993431 16.13503285 1.388888889 0.016414728 −0.008282188 0.478071906 1.3 0.713386252 16.43306874 1.282051282 0.016369929 −0.008317128 0.476815000 1.4 0.662422888 16.68788556 1.190476190 0.016266525 −0.008319133 0.476704578 1.5 0.618261111 16.90869445 1.111111111 0.016366355 −0.008319213 0.476698592 1.6 0.579619785 17.10190108 1.041666667 0.016366350 −0.008319216 0.476698431 1.7 0.545524504 17.27237748 0.980392157 0.016366349 −0.008319216 0.476698383 1.8 0.515217587 17.42391207 0.925925926 0.016366349 −0.008319216 0.476698383 1.9 0.488100872 17.55949564 0.877192982 0.016366349 −0.008319216 0.476698383 2.0 0.463695828 17.68152086 0.833333333 0.016366349 −0.008319216 0.476698383

Table 2b. Spk= 1.33 with Cp = 1.33, 1.4(0.1)2.3, the corresponding µ, σ , Ca, and the calculated

a, b, and nVar( ˆSpk). Note: φ (3Spk)= 0.000139285.

1.0(01)2.0, the corresponding Ca, and (µ, σ ) for Spk= 1.00. Table II(b)

summar-izes the process characteristics with Cp = 1.33, 1.4(0.1)2.3, the corresponding Ca,

and (µ, σ ) for Spk= 1.33. Table II(c) summarizes the process characteristics with

Cp = 1.5(0.1)2.5, the corresponding Ca, and (µ, σ ) for Spk = 1.50. Table II(d)

summarizes the process characteristics Cp = 1.67, 1.7(0.1)2.6, the corresponding

Ca, and (µ, σ ) for Spk = 1.67. Table II(e) summarizes the process characteristics

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Table 2c. S_pk= 1.50 with Cp= 1.5(0.1)2.5, the corresponding µ, σ , Ca, and the calculated a, b,

Table 2d. Spk= 1.67 with Cp= 1.7, 1.7(0.1)2.6, the corresponding µ, σ , Ca, and the calculated a,

b, and nVar( ˆSpk). Note: φ (3Spk)= 0.000001414.

Table III displays the critical values co computed from the normal

approxim-ation. We note that for given fixed performance requirement Spk, the differences

among those calculated critical values corresponding to various parameter values of Cp and Ca, are sufficiently small and can be neglected. To justify this result,

we also calculated the main factor in the variance of the normal approximation, (a2 + b2){36[φ(3Spk)]2}−1 which determines the magnitude of the critical value

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P K

Table 2e. S_pk= 2.00 with Cp= 2.0(0.1)3.0, the corresponding µ, σ , Ca, and the calculated a, b,

Tables II(a)–II(e), which indicate that the factor (a2+ b2){36[φ(3Spk)]}−1 is

in-sensitive to the value changes of Cp and Ca in all cases, except for Spk = Cp

(Ca = 1.00, the process is perfectly centered). Consequently, the critical values co

may be considered as a constant, which is independent of the process characterist-ics Cpand Cafor fixed performance requirement Spk. Such behavior of the normal

approximation can be expected, as it is a linear approximation obtained from taking the first term in the Taylor expansion.

3.2. SIMULATION TECHNIQUE

To assess the accuracy of the normal approximation, we also obtain the critical values using the standard simulation technique to compare with the critical values obtained from the normal approximation, under the same performance require-ments. We note that the natural estimator Spk can be rewritten and expressed as

a function of the two parameters Cp and Ca. In fact, a mathematical relationship

among the three measurements can be established as (3Spk) = {(3CpCa)+

[3Cp(2− Ca)]}/2. Spk = 1 3 −11 2 USL− µ σ +1 2 µ− LSL σ = 1 3 −11 2 d− |µ − m| σ +1 2 d+ |µ − m| σ = 1 3 −11 2 1− |µ − m|/d σ/d +1 2 1+ |µ − m|/d σ/d

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T a b le III. A ppr oxi mat e co fo r v ar ious Spk , n = 5 (5 )200, and α = 0 .05, 0. 025, 0. 01. n 1. 00 1. 33 1. 50 1. 67 2. 00 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 5 1 .5 2 1 .6 2 1 .7 4 2 .0 2 2 .1 5 2 .3 1 2 .2 8 2 .4 3 2 .6 0 2 .5 4 2 .7 1 2 .9 0 3 .0 4 3 .2 4 3 .4 7 10 1. 37 1. 44 1. 52 1. 82 1. 91 2. 02 2. 05 2. 16 2. 28 2. 29 2. 40 2. 54 2. 74 2. 88 3. 04 15 1. 30 1. 36 1. 43 1. 73 1. 81 1. 90 1. 95 2. 04 2. 14 2. 17 2. 27 2. 38 2. 60 2. 72 2. 85 20 1. 26 1. 31 1. 37 1. 68 1. 74 1. 82 1. 89 1. 97 2. 05 2. 11 2. 19 2. 29 2. 52 2. 62 2. 74 25 1. 23 1. 28 1. 33 1. 64 1. 70 1. 77 1. 85 1. 92 1. 99 2. 06 2. 13 2. 22 2. 47 2. 55 2. 66 30 1. 21 1. 25 1. 30 1. 61 1. 67 1. 73 1. 82 1. 88 1. 95 2. 03 2. 09 2. 17 2. 43 2. 51 2. 60 35 1. 20 1. 23 1. 28 1. 59 1. 64 1. 70 1. 80 1. 85 1. 92 2. 00 2. 06 2. 14 2. 39 2. 47 2. 56 40 1. 18 1. 22 1. 26 1. 58 1. 62 1. 68 1. 78 1. 83 1. 89 1. 98 2. 04 2. 11 2. 37 2. 44 2. 52 45 1. 17 1. 21 1. 25 1. 56 1. 61 1. 66 1. 76 1. 81 1. 87 1. 96 2. 02 2. 08 2. 35 2. 41 2. 49 50 1. 16 1. 20 1. 23 1. 55 1. 59 1. 64 1. 75 1. 79 1. 85 1. 95 2. 00 2. 06 2. 33 2. 39 2. 47 55 1. 16 1. 19 1. 22 1. 54 1. 58 1. 63 1. 74 1. 78 1. 83 1. 93 1. 98 2. 04 2. 31 2. 37 2. 44 60 1. 15 1. 18 1. 21 1. 53 1. 57 1. 61 1. 73 1. 77 1. 82 1. 92 1. 97 2. 03 2. 30 2. 36 2. 43 65 1. 14 1. 17 1. 20 1. 52 1. 56 1. 60 1. 72 1. 76 1. 81 1. 91 1. 96 2. 01 2. 29 2. 34 2. 41 70 1. 14 1. 17 1. 20 1. 52 1. 55 1. 59 1. 71 1. 75 1. 80 1. 90 1. 95 2. 00 2. 28 2. 33 2. 39 75 1. 13 1. 16 1. 19 1. 51 1. 54 1. 58 1. 70 1. 74 1. 79 1. 89 1. 94 1. 99 2. 27 2. 32 2. 38 80 1. 13 1. 16 1. 18 1. 50 1. 54 1. 58 1. 70 1. 73 1. 78 1. 89 1. 93 1. 98 2. 26 2. 31 2. 37 85 1. 13 1. 15 1. 18 1. 50 1. 53 1. 57 1. 69 1. 73 1. 77 1. 88 1. 92 1. 97 2. 25 2. 30 2. 36 90 1. 12 1. 15 1. 17 1. 49 1. 52 1. 56 1. 68 1. 72 1. 76 1. 88 1. 91 1. 96 2. 25 2. 29 2. 35 95 1. 12 1. 14 1. 17 1. 49 1. 52 1. 55 1. 68 1. 71 1. 75 1. 87 1. 91 1. 95 2. 24 2. 28 2. 34 100 1. 12 1. 14 1. 16 1. 49 1. 51 1. 55 1. 67 1. 71 1. 75 1. 86 1. 90 1. 95 2. 23 2. 28 2. 33 105 1. 11 1. 14 1. 16 1. 48 1. 51 1. 54 1. 67 1. 70 1. 74 1. 86 1. 90 1. 94 2. 23 2. 27 2. 32 110 1. 11 1. 13 1. 16 1. 48 1. 51 1. 54 1. 67 1. 70 1. 74 1. 86 1. 89 1. 93 2. 22 2. 26 2. 31 115 1. 11 1. 13 1. 15 1. 47 1. 50 1. 53 1. 66 1. 69 1. 73 1. 85 1. 89 1. 93 2. 22 2. 26 2. 31 120 1. 11 1. 13 1. 15 1. 47 1. 50 1. 53 1. 66 1. 69 1. 73 1. 85 1. 88 1. 92 2. 21 2. 25 2. 30 125 1. 10 1. 12 1. 15 1. 47 1. 50 1. 53 1. 66 1. 69 1. 72 1. 84 1. 88 1. 92 2. 21 2. 25 2. 29 130 1. 10 1. 12 1. 14 1. 47 1. 49 1. 52 1. 65 1. 68 1. 72 1. 84 1. 87 1. 91 2. 20 2. 24 2. 29 135 1. 10 1. 12 1. 14 1. 46 1. 49 1. 52 1. 65 1. 68 1. 71 1. 84 1. 87 1. 91 2. 20 2. 24 2. 28

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P K T a b le III. C ont inued. n 1. 00 1. 33 1. 50 1. 67 2. 00 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 140 1. 10 1. 12 1. 14 1. 46 1. 49 1. 52 1. 65 1. 68 1. 71 1. 83 1. 87 1. 90 2. 20 2. 23 2. 28 145 1. 10 1. 12 1. 14 1. 46 1. 48 1. 51 1. 65 1. 67 1. 71 1. 83 1. 86 1. 90 2. 19 2. 23 2. 27 150 1. 10 1. 11 1. 13 1. 46 1. 48 1. 51 1. 64 1. 67 1. 70 1. 83 1. 86 1. 89 2. 19 2. 23 2. 27 155 1. 09 1. 11 1. 13 1. 45 1. 48 1. 51 1. 64 1. 67 1. 70 1. 83 1. 86 1. 89 2. 19 2. 22 2. 26 160 1. 09 1. 11 1. 13 1. 45 1. 48 1. 50 1. 64 1. 66 1. 70 1. 82 1. 85 1. 89 2. 18 2. 22 2. 26 165 1. 09 1. 11 1. 13 1. 45 1. 47 1. 50 1. 64 1. 66 1. 69 1. 82 1. 85 1. 88 2. 18 2. 22 2. 26 170 1. 09 1. 11 1. 13 1. 45 1. 47 1. 50 1. 63 1. 66 1. 69 1. 82 1. 85 1. 88 2. 18 2. 21 2. 25 175 1. 09 1. 11 1. 12 1. 45 1. 47 1. 50 1. 63 1. 66 1. 69 1. 82 1. 85 1. 88 2. 18 2. 21 2. 25 180 1. 09 1. 10 1. 12 1. 45 1. 47 1. 49 1. 63 1. 66 1. 68 1. 82 1. 84 1. 88 2. 17 2. 21 2. 25 185 1. 09 1. 10 1. 12 1. 44 1. 47 1. 49 1. 63 1. 65 1. 68 1. 81 1. 84 1. 87 2. 17 2. 20 2. 24 190 1. 08 1. 10 1. 12 1. 44 1. 46 1. 49 1. 63 1. 65 1. 68 1. 81 1. 84 1. 87 2. 17 2. 20 2. 24 195 1. 08 1. 10 1. 12 1. 44 1. 46 1. 49 1. 63 1. 65 1. 68 1. 81 1. 84 1. 87 2. 17 2. 20 2. 24 200 1. 08 1. 10 1. 12 1. 44 1. 46 1. 49 1. 62 1. 65 1. 67 1. 81 1. 83 1. 86 2. 16 2. 20 2. 23

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= 1 3 −11 2[3CpCa] + 1 2[3Cp(2− Ca)] .

The simulation was carried out using the SAS programming software, with N = 10000 replications for each sample size of n, on the same set of performance requirements Spk= 1.00, 1.33, 1.50, 1.67, and 2.00, displayed in Tables II(a)–II(e).

The simulation results indicate that the critical values are more sensitive to the two parameters Cp and Ca than those from the normal approximation. For example,

given fixed Spk = 1.67 with n = 20, co = 2.24 for Cp = 1.67, and co = 2.30

for Cp = 1.9. For practical purpose, we may take the maximal values of coamong

those parameters of Cp and Ca we investigated, to obtain conservative bounds on

the critical values for reliability purpose. This approach ensures that the decisions made based on the critical values have the risk of wrongly concluding an incapable process as capable, no greater than the preset type I error α. Table IV summarizes the critical values co (the maximal ones among those with different Cp and Ca)

obtained from the simulation.

4. Accuracy of the Normal Approximation

Table V displays a comparison of the critical values co generated by the normal

approximation and the simulation technique for various selected sample sizes n = 5(5)50, 60, 65, 75, 90, 110, 130, and 150, with risk α = 0.05. It is noted that the normal approximation significantly under- approximates the critical values, particularly, for small sample sizes n < 40, as the magnitude of the underes-timation exceeds 0.10. Therefore, for short run applications (such as accepting a supplier providing short production runs in QS-9000 certification), one should avoid using the normal approximation. It is also noted that the underestimation can be as large as 0.07 for n= 50, 0.03 for n = 110, and 0.02 for n = 150. Therefore, in real applications a sample of size greater than 150 is recommended.

4.1. CONVERGENCE OF THE APPROXIMATION

Table VI displays the sample sizes required for the normal approximation to con-verge to Spk within a sampling error less than 0.10, 0.09, 0.08, 0.07, 0.06, 0.05,

0.04, 0.03, 0.02, 0.01, 0.00 respectively (with accuracy up to the second decimal point, or 5× 10−3). For example, for Spk= 1.33 with risk α = 0.05, a sample size

of n≥ 95909 ensures that the sampling error is no greater than 5 × 10−3which is negligible. Thus, if ˆSpk >1.33, then we may conclude that the actual performance

Spk > 1.33. The investigation is not for practical purpose. But, the computations

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P K T abl e IV . Simu la te d co fo r v ar ious Spk , n, n = 5 (5 )200, and α = 0 .05, 0. 025, 0. 01. n 1. 00 1. 33 1. 50 1. 67 2. 00 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 5 2 .0 6 2 .3 2 2 .5 5 3 .1 0 3 .7 7 4 .8 6 3 .5 1 4 .2 6 5 .7 2 3 .9 3 4 .8 8 6 .2 4 4 .8 1 5 .9 1 7 .4 4 10 1. 62 1. 78 2. 05 2. 18 2. 45 2. 80 2. 48 2. 74 3. 12 2. 74 3. 03 3. 46 3. 26 3. 66 4. 19 15 1. 46 1. 57 1. 75 1. 94 2. 10 2. 33 2. 19 2. 37 2. 59 2. 44 2. 64 2. 88 2. 91 3. 14 3. 47 20 1. 37 1. 45 1. 57 1. 82 1. 94 2. 11 2. 05 2. 19 2. 37 2. 30 2. 47 2. 68 2. 74 2. 92 3. 15 25 1. 31 1. 38 1. 48 1. 75 1. 84 1. 98 1. 98 2. 09 2. 24 2. 20 2. 33 2. 49 2. 63 2. 77 2. 97 30 1. 28 1. 34 1. 43 1. 70 1. 79 1. 90 1. 93 2. 02 2. 15 2. 14 2. 25 2. 40 2. 57 2. 71 2. 86 35 1. 25 1. 31 1. 38 1. 67 1. 74 1. 84 1. 89 1. 98 2. 08 2. 10 2. 19 2. 32 2. 51 2. 62 2. 76 40 1. 23 1. 28 1. 35 1. 64 1. 72 1. 80 1. 85 1. 93 2. 04 2. 06 2. 15 2. 25 2. 47 2. 57 2. 71 45 1. 22 1. 26 1. 32 1. 61 1. 68 1. 76 1. 82 1. 90 1. 98 2. 02 2. 12 2. 22 2. 43 2. 53 2. 66 50 1. 20 1. 24 1. 30 1. 60 1. 66 1. 73 1. 80 1. 87 1. 96 2. 01 2. 08 2. 18 2. 40 2. 40 2. 61 55 1. 19 1. 23 1. 28 1. 58 1. 65 1. 71 1. 79 1. 86 1. 93 1. 99 2. 06 2. 14 2. 38 2. 47 2. 57 60 1. 18 1. 22 1. 27 1. 57 1. 63 1. 69 1. 77 1. 83 1. 91 1. 98 2. 04 2. 13 2. 36 2. 44 2. 55 65 1. 17 1. 20 1. 26 1. 56 1. 61 1. 67 1. 76 1. 82 1. 89 1. 96 2. 02 2. 10 2. 34 2. 42 2. 51 70 1. 16 1. 20 1. 24 1. 55 1. 60 1. 66 1. 77 1. 80 1. 87 1. 95 2. 01 2. 08 2. 33 2. 40 2. 50 75 1. 15 1. 19 1. 23 1. 54 1. 59 1. 65 1. 74 1. 79 1. 85 1. 94 2. 00 2. 07 2. 31 2. 39 2. 48 80 1. 15 1. 18 1. 22 1. 53 1. 58 1. 63 1. 73 1. 78 1. 93 1. 93 1. 98 2. 05 2. 31 2. 37 2. 46 85 1. 14 1. 17 1. 21 1. 53 1. 57 1. 63 1. 72 1. 77 1. 83 1. 92 1. 97 2. 04 2. 30 2. 36 2. 43 90 1. 14 1. 17 1. 21 1. 52 1. 56 1. 61 1. 71 1. 76 1. 81 1. 91 1. 96 2. 02 2. 28 2. 34 2. 42 95 1. 14 1. 17 1. 20 1. 51 1. 55 1. 61 1. 71 1. 75 1. 80 1. 90 1. 95 2. 01 2. 27 2. 34 2. 42 100 1. 13 1. 16 1. 20 1. 50 1. 54 1. 59 1. 70 1. 74 1. 80 1. 89 1. 94 2. 00 2. 27 2. 33 2. 41 105 1. 13 1. 16 1. 19 1. 50 1. 54 1. 58 1. 70 1. 74 1. 79 1. 89 1. 93 1. 99 2. 26 2. 32 2. 38 110 1. 13 1. 15 1. 19 1. 50 1. 53 1. 58 1. 69 1. 73 1. 78 1. 89 1. 93 1. 98 2. 25 2. 31 2. 37 115 1. 12 1. 15 1. 18 1. 49 1. 53 1. 57 1. 69 1. 73 1. 77 1. 88 1. 99 1. 97 2. 25 2. 30 2. 36 120 1. 12 1. 14 1. 18 1. 49 1. 52 1. 57 1. 68 1. 72 1. 76 1. 87 1. 91 1. 96 2. 24 2. 29 2. 36 125 1. 12 1. 14 1. 18 1. 49 1. 52 1. 56 1. 68 1. 71 1. 76 1. 86 1. 90 1. 95 2. 24 2. 29 2. 34 130 1. 12 1. 14 1. 17 1. 48 1. 52 1. 56 1. 68 1. 71 1. 75 1. 86 1. 90 1. 95 2. 23 2. 28 2. 34 135 1. 11 1. 14 1. 17 1. 48 1. 51 1. 55 1. 67 1. 70 1. 75 1. 86 1. 90 1. 94 2. 23 2. 27 2. 33

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T abl e IV . C ont inued. n 1. 00 1. 33 1. 50 1. 67 2. 00 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 0. 05 0. 025 0. 01 140 1. 11 1. 13 1. 16 1. 48 1. 51 1. 55 1. 67 1. 70 1. 74 1. 86 1. 89 1. 94 2. 22 2. 27 2. 33 145 1. 11 1. 13 1. 16 1. 48 1. 51 1. 54 1. 66 1. 70 1. 74 1. 85 1. 89 1. 93 2. 22 2. 26 2. 32 150 1. 11 1. 13 1. 15 1. 47 1. 50 1. 54 1. 66 1. 69 1. 73 1. 85 1. 89 1. 93 2. 21 2. 25 2. 31 155 1. 10 1. 12 1. 15 1. 47 1. 50 1. 53 1. 66 1. 69 1. 73 1. 84 1. 88 1. 92 2. 21 2. 25 2. 30 160 1. 10 1. 12 1. 15 1. 47 1. 50 1. 53 1. 65 1. 69 1. 72 1. 84 1. 88 1. 92 2. 21 2. 25 2. 30 165 1. 10 1. 12 1. 14 1. 46 1. 49 1. 52 1. 65 1. 68 1. 72 1. 84 1. 87 1. 91 2. 20 2. 24 2. 29 170 1. 10 1. 12 1. 14 1. 46 1. 49 1. 52 1. 65 1. 68 1. 71 1. 84 1. 87 1. 91 2. 20 2. 24 2. 29 175 1. 10 1. 11 1. 14 1. 46 1. 49 1. 52 1. 65 1. 68 1. 71 1. 83 1. 87 1. 91 2. 20 2. 24 2. 28 180 1. 10 1. 11 1. 14 1. 46 1. 48 1. 52 1. 65 1. 67 1. 71 1. 83 1. 86 1. 90 2. 19 2. 23 2. 28 185 1. 09 1. 11 1. 13 1. 46 1. 48 1. 52 1. 64 1. 67 1. 71 1. 83 1. 86 1. 90 2. 19 2. 23 2. 28 190 1. 09 1. 11 1. 13 1. 45 1. 48 1. 51 1. 64 1. 67 1. 70 1. 83 1. 86 1. 89 2. 19 2. 22 2. 27 195 1. 09 1. 11 1. 13 1. 45 1. 47 1. 51 1. 64 1. 67 1. 70 1. 82 1. 85 1. 89 2. 18 2. 22 2. 27 200 1. 09 1. 11 1. 13 1. 45 1. 47 1. 51 1. 64 1. 67 1. 70 1. 82 1. 85 1. 89 2. 18 2. 22 2. 26

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P K

Table V. A comparison of cobetween the normal approximation and the simulation

with α= 0.05. n 1.00 1.33 1.50 1.67 2.00 5 1.52 2.06 2.02 3.10 2.28 3.51 2.54 3.93 3.04 4.81 10 1.37 1.62 1.82 2.18 2.05 2.48 2.29 2.74 2.74 3.26 15 1.30 1.46 1.73 1.94 1.95 2.19 2.17 2.44 2.60 2.91 20 1.26 1.37 1.68 1.82 1.89 2.05 2.11 2.30 2.52 2.74 25 1.23 1.31 1.64 1.75 1.85 1.98 2.06 2.20 2.47 2.63 30 1.21 1.28 1.61 1.70 1.82 1.93 2.03 2.14 2.43 2.57 35 1.20 1.25 1.59 1.67 1.80 1.89 2.00 2.10 2.39 2.51 40 1.18 1.23 1.58 1.64 1.78 1.85 1.98 2.06 2.37 2.47 45 1.17 1.22 1.56 1.61 1.76 1.82 1.96 2.03 2.35 2.43 50 1.16 1.20 1.55 1.60 1.75 1.80 1.95 2.01 2.33 2.40 60 1.15 1.18 1.53 1.57 1.73 1.77 1.92 1.98 2.30 2.36 65 1.14 1.17 1.52 1.56 1.72 1.76 1.91 1.96 2.29 2.34 75 1.13 1.15 1.51 1.54 1.70 1.74 1.89 1.94 2.27 2.31 90 1.12 1.14 1.49 1.52 1.68 1.71 1.88 1.91 2.25 2.28 110 1.11 1.13 1.48 1.50 1.67 1.69 1.86 1.89 2.22 2.25 130 1.10 1.12 1.47 1.48 1.65 1.68 1.84 1.86 2.20 2.23 150 1.10 1.11 1.46 1.47 1.64 1.66 1.83 1.85 2.19 2.21

Table VI. Sample sizes required for the normal approximation to converge.

α 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 1.00 0.05 124 151 188 241 321 449 670 1108 2174 6053 54220 0.025 176 214 267 342 456 637 951 1573 3086 8594 76984 0.01 248 301 375 482 642 897 1340 2216 4347 12108 108455 1.33 0.05 220 266 332 426 568 793 1185 1959 3845 10707 95909 0.025 312 377 471 605 806 1126 1682 2782 5458 15202 136176 0.01 439 531 664 853 1135 1586 2369 3919 7690 21417 191846 1.50 0.05 279 338 422 542 722 1009 1507 2492 4890 13619 121994 0.025 396 480 599 770 1025 1432 2139 3538 6943 19336 173212 0.01 558 676 844 1084 1444 2017 3014 4985 9781 27241 244024 1.67 0.05 346 419 523 672 895 1250 1868 3089 6061 16881 151213 0.025 491 595 743 954 1270 1775 2652 4386 8606 23968 214699 0.01 692 838 1046 1344 1789 2500 3736 6179 12124 33766 302470 2.00 0.05 496 601 750 964 1283 1793 2679 4430 8692 24211 216878 0.025 704 853 1065 1368 1822 2545 3803 6290 12342 34376 307933 0.01 992 1201 1500 1927 2566 3585 5357 8862 17388 48429 433819

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5. An Application Example

In the following, we present an example of the LCM (liquid crystal module) man-ufacturing process. The example we investigated was taken from a manman-ufacturing factory making the Liquid Crystal Display Module (LCD Module). The LCD Mod-ule is one of the key components used in many high-tech electronic commercial devices for the display function, such as the cellular phone, the PDA (personal digital assistant), the pocket calculator, digital watch, automobile accessory visual displays, and many others. Three key components make the LCD Module func-tions properly Those include the liquid crystal display, the back lighting, and the peripheral (interface) system.

The mounting technology for the chip-on-glass makes the exposed particle overturned, with the side of circuits facing downward. Then, the electricity con-duction is joined between the IC and the panel of the liquid-crystal display through the mounting material. Currently, the mounting technology of the chip-on-glass is the best manufacturing technology for the LCD Module in terms of the mounting density. It is important to note that different mounting material requires different mounting technology of the chip-on-glass.

ACF (Anisotropic Conductive Film) is one of the several developed mounting materials, which is now the most widely used material for the chip-on-glass. For the main bonding process, the bonding precision is an essential process parameter we focused on in our study. We investigated a particular model of the LCD Module product with the upper and the lower specification limits set to USL = 15 µm, LSL = −15 µm, and the target value is set to T = 0. If the characteristic data does not fall within the tolerance (LCL, UCL), the lifetime or reliability of the LCD Module will be discounted. To ensure the production quality, the yield index for a particular model we investigated was set to Spk ≥ 1.50. If the capability

requirement fails to be met, the LCD Module product would be seriously affected on its reliability or lifetime.

To test the process performance, we consider the hypothesis testing H0: Spk ≤

1.33 (process is incapable), versus the alternative H1: Spk > 1.33 (process is

capable) with type I error α = 0.05. According to the process control plan of the COG the random sample data in consecutive four day are collected, which are displayed in Table VII. The 160 sample observations are obtained through the inspection, using the microscope by visual, which were collected eight pieces per every two hours. These 160 observations were justified taken from a stable process (in control), and the characteristic distribution is shown to be approximate normal. The calculated mean and the standard deviation of the 160 sample observations are summarized in the following. Checking Table III we obtain the critical value 1.45 for risk α= 0.05. ¯X = 0.1754, S = 3.1570, USL− ¯X S = 0.9999,

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P K T abl e V II . T h e col lect ed sampl e dat a w it h 160 obser v at ions. − 1. 40 1. 70 − 0. 17 − 3. 70 0. 40 − 1. 30 0. 48 3. 10 0. 38 3. 00 2. 10 0. 8 2 .2 0 5 .6 0 − 4. 20 1. 90 5. 90 − 2. 60 − 3. 40 2. 10 2. 50 − 0. 25 − 2. 20 1. 90 1. 20 − 0. 90 − 0. 82 − 5. 10 − 2. 20 3. 40 6. 70 3. 90 − 1. 30 7. 50 1. 90 − 2. 30 2. 40 − 4. 40 0. 09 1. 30 − 4. 90 − 3. 80 2. 30 0. 03 − 2. 80 2. 60 − 4. 60 − 5. 00 − 4. 70 0. 66 − 0. 17 1. 50 − 1. 20 0. 20 3. 00 1. 30 − 3. 80 1. 50 − 0. 43 − 0. 96 − 1. 90 − 8. 40 0. 05 − 0. 69 − 3. 70 − 5. 40 1. 40 2. 00 5. 00 0. 25 − 0. 84 4. 80 2. 70 − 1. 60 4. 40 5. 60 − 3. 30 − 4. 70 0. 81 8. 00 3. 80 2. 30 3. 00 − 1. 20 0. 70 − 2. 20 − 1. 30 1. 40 − 2. 60 3. 40 1. 00 − 0. 07 0. 79 3. 10 4. 10 − 0. 27 3. 40 3. 00 2. 20 − 2. 20 − 2. 30 − 3. 70 3. 00 − 6. 30 0. 72 2. 40 1. 50 − 0. 42 − 5. 20 − 1. 60 − 3. 90 − 0. 40 − 3. 90 − 1. 70 0. 79 7. 70 3. 30 − 1. 40 0. 59 1. 50 0. 44 3. 10 1. 30 1. 60 − 2. 20 3. 20 − 3. 80 − 1. 30 3. 80 0. 06 1. 60 − 3. 80 − 5. 40 − 2. 20 0. 33 − 2. 70 − 1. 70 0. 82 3. 20 3. 50 3. 00 6. 50 2. 50 − 0. 20 1. 80 0. 33 − 0. 31 0. 54 0. 84 − 1. 70 0. 37 − 1 1 .0 3 .5 0 1 .6 0 − 3. 90 − 3. 20 0. 11 3. 90 1. 70

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¯X− LSL S = 0.9999, ˆSpk = 1 3 −11 2 USL− ¯X S +1 2 ¯X− LSL S = 1.5814. Since the calculated ˆSpk from the sample data, 1.58, is greater than the

crit-ical value 1.45, then we may conclude, with 95% confidence the process meets the performance requirement Spk > 1.33. The probability of wrongly judging an

incapable process as a capable one is no greater than 5%. 6. Conclusions

Process yield is the most common criterion used in the manufacturing industry for measuring process performance. A measurement index, called Spk, has been

proposed to calculate the yield for normal processes. The index Spk establishes

the relationship between the manufacturing specification and the actual process performance, which provides an exact measure on process yield. Unfortunately, the distributional properties of the estimated Spk are mathematically intractable.

In this paper, we considered a normal approximation to the distribution of the estimated Spk, and investigated its accuracy computationally. We compared the

critical values calculated from the approximate distribution with those obtained using the standard simulation technique, for some commonly used quality require-ments. Computational results are provided and analyzed. The results indicated that a sample size of n > 150 is required for the approximation to be accurate. The investigation is useful to the practitioners for making reliable decisions in testing process performance based on the yield.

Appendix with(stats): n := : alpha := : cp := : ca := : x1 := 3∗cp∗(2− ca): x2 := 3∗cp∗ca: p1 := statevalf[cdf, normald[0, 1]J(xl): p2 := statevalf[cdf, normald[0, 1]](x2): p3 := (p1 + p2)/2: spk := (1/3)∗statevalf[icdf; normald[0, 1]](p3): f1 := statevalf[pdf, normald[0, 1]](x1): f2 := statevalf[pdf; normald[0, 1]](x2): f3 := statevalf[pdf, normald[0, 1]](3∗spk): a := 2∧(−0.5)∗(x1∗f1+ x2∗f2): b := f1 − f2: var := ((a∧2+ b∧2)/n)∗(6∗f3)∧(−2):

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P K

c := statevalf[icdf, normald[0, 1]](1 − alpha)∗var∧0.5: c0 := spk + c;

References

Boyles, R. A. (1994). Process capability with asymmetric tolerances. Communication in Statistics:

Simulation and Computation 23(3): 615–643.

Lee, J. C., Hung, H. N., Pearn, W. L. & Kueng, T. L. (2002). On the distribution of the estimated

process yield index Spk. Quality & Reliability Engineering International. To appear.

Pearn, W. L., Lin, G. H. & Chen, K. S. (1998). Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory & Methods 27(4): 985–1000.

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