Testing Reliability Assurance Using Capability Indices CPU and CPL Based on Multiple Samples with Variable Sample Size

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Information and Management Sciences Volume 17, Number 1, pp. 47-65, 2006

Testing Quality Assurance Using Process Capability

Indices

CP U

and

CP L

Based on Several Groups of

Samples with Unequal Sizes

M. H. Shu K. H. Lu

National Kaohsing University of National University of Kaohsiung

Applied Sciences R.O.C.

R.O.C.

B. M. Hsu K. R. Lou

Cheng Shiu University Tamkang University

R.O.C. R.O.C.

Abstract

For stably normal processes with one-sided specification limits, capability indices CP U

and CP Lhave been used to provide numerical measures for product quality assurance from

manufacturing perspective. Statistical properties of the estimators of CP U and CP L have

been investigated extensively for cases with one single sample. In this paper, we consider testing product quality assurance for cases of several groups of samples with unequal sizes. We obtain the uniform minimum variance unbiased estimators (UMVUEs) of CP U and CP L,

and develop a powerful test for that purpose. We also implement Fortran programs to compute the p-values, critical values, for testing product quality assurance. A practical procedure using the UMVUEs is provided to assist the practitioners judging whether their processes are capable of reproducing reliable products. An example of voltage limiting amplifier (VLA) is presented to illustrate the practicality of our approach to actual data collected from the real-world applications.

Keywords: Process Capability Indices, UMVUE, One-Sided Specification Limit, Several Groups of Samples, Hypothesis Testing.

1. Introduction

Process capability indices are proposed to measure the capability of a process to reproduce items satisfying the preset requirement preset by the product designers or customer’s specifications. Several capability indices, including CP, CP U, CP L, Cpk, Cpm,

Received March 2004; Revised August 2004; Accepted December 2004.

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48 Information and Management Sciences, Vol. 17, No. 1, March, 2006

and Cpmk, are developed for this purpose (Kane (1986), Chan et al. (1988), Pearn et al. (1992)). Those indices essentially compare the predefined product specifications with the actual process distribution characteristics, which have been defined as

Cp = U SL − LSL 6σ , CP U = U SL − µ 3σ , CP L= µ − LSL 3σ , Cpk = min nU SL − µ 3σ , µ − LSL 3σ o , Cpm= U SL − LSL 6p σ2_{+ (µ − T )}2, Cpmk = min n U SL − µ 3p σ2_{+ (µ − T )}2, µ − LSL 3p σ2_{+ (µ − T )}2 o ,

where U SL is the upper specification limit, LSL is the lower specification limit, µ is the process mean, σ is the process standard deviation (overall process variation), and T is the target value.

The indices Cp, Cpk, Cpm, and Cpmk are appropriate for product with two-sided specification limits. On the other hand, the indices CP Uand CP Lare designed specifically for product with one-sided specification limit. Many quality/reliability and statistics literatures have investigated the estimation of these indices based on one single sample (see Kane (1986), Chou and Owen (1989), Cheng (1992, 1994)). Pearn and Chen (2002) obtained the UMVUEs for CP U and CP L. Lin and Pearn (2002) developed SAS programs to calculate the exact critical values and p-values for testing CP U and CP L. Pearn and Shu [12] presented an algorithm for calculating the exact lower confidence bounds on CP U and CP L. In practice, however, product manufacturing information is often derived from several groups of samples rather than one single sample, particularly, when a daily-based process control plan is implemented for monitoring process stability. In this paper, we consider the estimation of the one-sided capability indices CP U and CP L for several groups of samples with unequal sizes, and develop a practical test procedure to assist the practitioners judging whether their processes are capable of reproducing reliable products.

2. Estimating CP U and CP L Based on Several Groups of Samples

To estimate the indices CP U and CP Lin the presence of one single sample, Chou and Owen (1989) considered the following natural estimators of CP U and CP L:

ˆ

CP U = U SL − X

3S , CˆP L=

X − LSL

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Testing Quality Assurance Using Process Capability Indices CP U and CP L 49

where X =Pn

i=1xi/n, S = [(n−1)−1Pni=1(xi−X)2]1/2are the conventional estimators of µ and σ, which may be obtained from a process that is demonstrably stable (in-control). Chou and Owen (1989) showed that under the normality assumption, the estimators 3√n ˆCP U and 3√n ˆCP L are distributed as t(n − 1, 3√nCP U) and t(n − 1, 3√nCP L), re-spectively, a non-central t distribution with n − 1 degrees of freedom and non-centrality parameters 3√nCP U and 3√nCP L. Pearn and Chen (2002) obtained the UMVUEs of CP U and CP L. Based on the UMVUEs, Pearn and Chen (2002) developed a practical procedure for testing CP U and CP L based on one single sample. Applying the proposed procedure, the practitioners can judge whether or not their processes are capable of reproducing reliable products.

Kirmani et al. (1991) indicated that a common practice of the process capability estimation in the manufacturing industry is to first implement a routine-basis data col-lection program for monitoring/controlling the process stability, then to analyze the past “in-control” data. For several groups of samples of ms groups, with unequal sizes ni, (xi1, xi2, . . . , xini), are chosen randomly from a stable process which follows a normal distribution N (µ, σ2) for i = 1, 2, . . . , ms. We consider the following natural estimators of CP U and CP L. Let Xi= Pni j=1xij ni and Si = s Pni j=1(xij− Xi)2 ni− 1

be the i-th sample mean and the sample standard deviation, respectively and total number of observations N = Pms

i=1ni. Then, X = Pmi=1s niXi/N and Sp2 = Pmi=1s(ni− 1)S2

i/(N − ms) are the unbiased estimators of µ and σ2, respectively, and the estimators of CP U and CP Lare: ˜ C_{P U}M = bN −ms(U SL − X) 3Sp , C˜_{P L}M = bN −ms(X − LSL) 3Sp , (1)

where bN −m is the well-known correction factor defined as:

bN −ms = s 2 N − m Γ[(N − ms)/2] Γ[(N − ms− 1)/2] . (2)

In the following, we show that the proposed estimators ˜C_{P U}M and ˜C_{P L}M are the UMVUEs (uniform minimum variance unbiased estimators) of CP U and CP L for several groups of samples with unequal sizes.

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Theorem 1. If a process follows the normal distribution N (µ, σ2_{), then the} estima-tors (3p N/h/bN −ms) ˜C M P U and(3 p N/h/bN −ms) ˜C M

P Lare distributed as the non-central t distribution with_{N −m}sdegrees of freedom and non-central parametersδU = 3pN/hCP U and δL= 3pN/h/CP L, denoted as t(N − ms, δU) and t(N − ms, δL), respectively.

Proof. If the process follows the normal distribution N (µ, σ2_{), then}

˜ C_{P U}M = bN −ms hU SL − X 3Sp i =hbN −ms 3 ihN − m_s N i1/2h(N − m_s)S_p2 σ2 i−1/2h √ N (U SL − X) σ i =hbN −ms 3 ihN − m_s N i1/2 [K]−1/2 [ZU], (3) where K = (N − ms)S 2 p σ2 is distributed as χ 2 N −ms, and ZU = h √ N (U SL − X) σ i is dis-tributed as N (3√N CP U, h) and h = NPms i=1(n1i) m2 s . Since X and S2

p are mutually independent, then ZUand K are also mutually independent. Therefore, 3√N bN −ms √ h ˜ C_{P U}M = ZU/ √ h p K/(N − ms) ∼ t(N − ms , δU). Similarly, 3 √ N bN −ms √ h ˜ C_{P L}M = ZU/ √ h p K/(N − ms) ∼ t(N − m s, δL). Q.E.D.

If the process follows the normal distribution N (µ, σ2), then, it follows from (3), the

E[K]−r/2 = Γ N −ms−r 2 ΓN −ms 2 2r/2, (4)

then the r-th moment of ˜C_{P U}M can be obtained as:

E[ ˜C_{P U}M ]r= h Γ(N −ms 2 ) ir−1 Γ(N −ms−r 2 ) [3√N ]rh_Γ(N −ms−1 2 ) ir E(ZU) r_, ₍₅₎

We note that E(ZU) = 3 √

N CP U, and E(ZU)2 = 9N CP U2 + h. Therefore, the first two moments and the variance can be obtained as follows:

E( ˜C_{P U}M ) = CP U, thus the ˜CP UM is an unbiased estimator of CP U. E( ˜C_{P U}M )2= Γ( N −ms 2 )Γ(N −ms −2 2 ) [3√N ]rh_Γ(N −ms−1 2 ) ir[9N C 2 P U + h], thus

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Testing Quality Assurance Using Process Capability Indices CP U and CP L 51 Var ( ˜C_{P U}M ) = E( ˜C_{P U}M )2_{− E}2( ˜C_{P U}M ) =nΓ( N −ms 2 )Γ(N −ms −2 2 ) Γ[N −ms−1 2 ] 2 o (C_{P U}2 + h 9N) − C 2 P U.(6)

Similarly, ˜C_{P L}M is an unbiased estimator of CP L. The first two moments, and the variance of ˜CM

P L can be similarly derived.

Theorem 2. If the process follows the normal distribution, N (µ, σ2_{), then ˜}_CM P U and ˜

C_{P L}M are the UMVUEs of CP U andCP L, respectively.

Proof. The joint probability density function of ms samples of size ni, (xi1, xi2, . . ., xini), where i = 1, 2, . . . , ms can be obtained as follows:

f (x; µ, σ2) = ms Y i=1 ni Y j=1 1 √ 2πσexp n −h(xij− µ) 2 2σ2 io = 1 (√2πσ)N exp n −h P niµ2 2σ2 − Pms i=1 Pni j=1x2ij 2σ2 + Pms i=1 Pni j=1xijµ σ2 io , (7)

which belongs to the exponential family. Then (Pms

i=1

Pni

j=1x2ij,Pmi=1s

Pni

j=1xij) is a com-plete sufficient statistic of (µ, σ2). Thus, (X, S_p2) is a complete sufficient statistic for (µ, σ2_{). Since ˜}_CM

P U and ˜CP LM are function of (X, Sp2), from the Rao-Blackwell theorem (Bain and Engehardt (1992)), it follows that ˜C_{P U}M and ˜C_{P L}M are the UMVUEs of CP U and CP L based on several groups of samples, respectively.

3. Distribution Percentage Points of ˜C_{P U}M _{and ˜}C_{P L}M Since the probability density function (PDF) of t(v, δI) is

f (t) = v v/2_e−δ2 I/2 √_{πΓ(v/2)(v + t}₂₎_(v+1)/2 ∞ X j=0 Γ[(v + j + 1)/2]δj_I j! 2t2 v + t2 j/2 , (8)

thus, from (8), the PDF and the cumulative distribution function (CDF) of the UMVUE of CP U (or CP L) can be obtained as the following, where v = N − ms and δi = δU (or δL). f (x) = 3 √ N vv/2_e−δ2 I/2 bv√πhΓ(v₂)(v+9bv−2h−1x2N )(v+1)/2 ∞ X j=0 hΓ[(v+j +1)/2]δj I j! i 2x2 b2 v(9N )−1hv+x2 j/2 , (9) F (x) = Z x −∞ 3√N vv/2_e−δ2 I/2 bv √ πhΓ(v₂)(v + 9b−2 v h−1u2N )(v+1)/2 × ∞ X j=0 Γ[(v + j + 1)/2]δj I j! 2u2 b2 v(9N )−1hv + u2 j/2 du. (10)

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Figures 1−4 plot the PDF of the UMVUE ˜CM

P U (or ˜CP LM ), for total number of obser-vations, N = 30, and number of samples, ms = 1, 5, 10, 15, 20, for CP U = 1.00, 1.33, 1.67, 2.00, respectively.

Figure 1. PDF plot of ˜C_{P U}M for N = 30, ms = 1, 5, 10, 15, 20, for CP U = 1.00.

Figure 2. PDF plot of ˜C_{P U}M for N = 30, ms= 1, 5, 10, 15, 20, for CP U = 1.33. Figure 3. PDF plot of ˜CM P U for N = 30, ms = 1, 5, 10, 15, 20, for CP U = 1.67. Figure 4. PDF plot of ˜CM P U for N = 30, ms= 1, 5, 10, 15, 20, for CP U = 2.00. It is noted that the curves all centered at E( ˜C_{P U}M ) = CP U (or E( ˜CP UM ) = CP U). But, for fixed total number of sample observations N the range of the variation increases as the number of samples increases, and increases as the value of CP U (or CP L) increases. Tables 1−4 provide the corresponding percentage points for CP U = 1.00, 1.33, 1.67, 2.00, respectively, for N = 150, with various ms and 0.10(0.05).0.95, 0.975, and 0.99th

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per-Testing Quality Assurance Using Process Capability Indices CP U and CP L 53

centile. Figures 5−8 display the surface plot of the percentage points versus the percentile and the number of samples ms. It is interesting to note that for percentile α < 0.545, the percentage point decreases as the number of samples ms increases. But, for percentile α > 0.585, the percentage point increases as the number of samples ms increases. For α in the range, 0.545 ≤ α ≤ 0.585, the percentage point increases first then decreases. These features, although can’t be proved analytically, do appear in all cases investigated including all CP U with 1.00 ≤ CP U ≤ 2.00.

Figure 5. Surface plot of the percentage point versus percentile and the number of samples ms, for N = 150, CP U = 1.00.

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54 Information and Managemen t Sciences, V ol. 17, No. 1, Marc h, 2006

Table 1. Percentage points of CP U or CP L= 1.00, for N = 150, ms= 1, 5(5)120, and 0.10(0.05).0.95, 0.975, 0.99th

percentile. 1.00 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.975 0.99 1 0.920 0.934 0.945 0.955 0.964 0.973 0.981 0.989 0.997 1.005 1.013 1.022 1.031 1.042 1.053 1.067 1.084 1.111 1.135 1.163 5 0.919 0.933 0.945 0.955 0.964 0.973 0.981 0.989 0.997 1.005 1.014 1.022 1.032 1.042 1.054 1.067 1.085 1.112 1.136 1.165 10 0.918 0.932 0.944 0.954 0.963 0.972 0.980 0.989 0.997 1.005 1.014 1.023 1.032 1.043 1.054 1.068 1.086 1.114 1.138 1.168 15 0.916 0.931 0.943 0.953 0.963 0.972 0.980 0.988 0.997 1.005 1.014 1.023 1.033 1.043 1.055 1.069 1.088 1.116 1.141 1.171 20 0.915 0.930 0.942 0.953 0.962 0.971 0.980 0.988 0.997 1.005 1.014 1.023 1.033 1.044 1.056 1.070 1.089 1.118 1.143 1.174 25 0.914 0.929 0.941 0.952 0.961 0.971 0.979 0.988 0.997 1.005 1.014 1.024 1.034 1.045 1.057 1.072 1.091 1.120 1.146 1.177 30 0.912 0.928 0.940 0.951 0.961 0.970 0.979 0.988 0.996 1.005 1.014 1.024 1.034 1.045 1.058 1.073 1.092 1.122 1.149 1.181 35 0.911 0.926 0.939 0.950 0.960 0.969 0.978 0.987 0.996 1.005 1.015 1.024 1.035 1.046 1.059 1.074 1.094 1.124 1.152 1.185 40 0.909 0.925 0.938 0.949 0.959 0.969 0.978 0.987 0.996 1.005 1.015 1.025 1.035 1.047 1.060 1.076 1.096 1.127 1.155 1.189 45 0.907 0.923 0.936 0.948 0.958 0.968 0.977 0.987 0.996 1.005 1.015 1.025 1.036 1.048 1.061 1.077 1.098 1.130 1.158 1.193 50 0.905 0.922 0.935 0.947 0.957 0.967 0.977 0.986 0.996 1.005 1.015 1.025 1.037 1.049 1.063 1.079 1.100 1.133 1.162 1.198 55 0.903 0.920 0.933 0.945 0.956 0.966 0.976 0.986 0.995 1.005 1.015 1.026 1.037 1.050 1.064 1.081 1.102 1.136 1.166 1.203 60 0.901 0.918 0.932 0.944 0.955 0.965 0.975 0.985 0.995 1.005 1.016 1.026 1.038 1.051 1.065 1.083 1.105 1.140 1.171 1.209 65 0.899 0.916 0.930 0.942 0.954 0.964 0.975 0.985 0.995 1.005 1.016 1.027 1.039 1.052 1.067 1.085 1.108 1.143 1.176 1.215 70 0.896 0.914 0.928 0.941 0.952 0.963 0.974 0.984 0.995 1.005 1.016 1.028 1.040 1.053 1.069 1.087 1.111 1.148 1.181 1.222 75 0.893 0.911 0.926 0.939 0.951 0.962 0.973 0.984 0.994 1.005 1.016 1.028 1.041 1.055 1.071 1.090 1.114 1.153 1.187 1.230 80 0.890 0.908 0.923 0.937 0.949 0.961 0.972 0.983 0.994 1.005 1.017 1.029 1.042 1.056 1.073 1.093 1.118 1.158 1.194 1.238 85 0.886 0.905 0.921 0.934 0.947 0.959 0.971 0.982 0.993 1.005 1.017 1.030 1.043 1.058 1.075 1.096 1.122 1.164 1.202 1.248 90 0.882 0.902 0.918 0.932 0.945 0.957 0.969 0.981 0.993 1.005 1.017 1.031 1.045 1.060 1.078 1.099 1.127 1.171 1.210 1.259 95 0.877 0.897 0.914 0.929 0.942 0.955 0.968 0.980 0.992 1.005 1.018 1.031 1.046 1.063 1.081 1.104 1.133 1.178 1.220 1.272 100 0.872 0.893 0.910 0.925 0.939 0.953 0.966 0.979 0.991 1.005 1.018 1.033 1.048 1.065 1.085 1.108 1.139 1.187 1.232 1.287 105 0.865 0.887 0.905 0.921 0.936 0.950 0.964 0.977 0.990 1.004 1.019 1.034 1.050 1.068 1.089 1.114 1.147 1.198 1.245 1.305 110 0.858 0.881 0.900 0.917 0.932 0.947 0.961 0.975 0.989 1.004 1.019 1.035 1.052 1.072 1.094 1.120 1.155 1.211 1.262 1.326 115 0.849 0.873 0.893 0.911 0.927 0.942 0.958 0.973 0.988 1.003 1.019 1.037 1.055 1.076 1.100 1.128 1.166 1.226 1.283 1.353 120 0.838 0.863 0.884 0.903 0.920 0.937 0.953 0.969 0.986 1.002 1.020 1.038 1.058 1.081 1.107 1.138 1.180 1.246 1.309 1.388

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T esting Qualit y Assurance Using Pro cess Capabilit y Indices C P U and C P L 55

Table 2. Percentage points of CP U or CP L = 1.33, for N = 150, ms= 1, 5(5)120, and 0.10(0.05). 0.95, 0.975, 0.99th

percentile. 1.33 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.975 0.99 1 1.228 1.246 1.398 1.273 1.284 1.295 1.306 1.316 1.326 1.336 1.347 1.358 1.370 1.383 1.398 1.415 1.437 1.471 1.502 1.539 5 1.226 1.244 1.398 1.272 1.284 1.295 1.305 1.316 1.326 1.337 1.347 1.358 1.370 1.384 1.398 1.416 1.439 1.473 1.504 1.542 10 1.225 1.243 1.400 1.271 1.283 1.294 1.305 1.315 1.326 1.337 1.347 1.359 1.371 1.384 1.400 1.417 1.440 1.476 1.507 1.545 15 1.223 1.242 1.401 1.270 1.282 1.294 1.304 1.315 1.326 1.337 1.348 1.359 1.372 1.385 1.401 1.419 1.442 1.478 1.511 1.549 20 1.221 1.240 1.402 1.269 1.281 1.293 1.304 1.315 1.326 1.337 1.348 1.360 1.372 1.386 1.402 1.420 1.444 1.481 1.514 1.554 25 1.219 1.239 1.403 1.268 1.280 1.292 1.303 1.314 1.325 1.337 1.348 1.360 1.373 1.387 1.403 1.422 1.446 1.484 1.518 1.558 30 1.217 1.237 1.404 1.267 1.279 1.291 1.303 1.314 1.325 1.337 1.348 1.361 1.374 1.388 1.404 1.424 1.449 1.487 1.521 1.563 35 1.215 1.235 1.406 1.265 1.278 1.290 1.302 1.314 1.325 1.337 1.349 1.361 1.375 1.389 1.406 1.426 1.451 1.490 1.526 1.568 40 1.213 1.233 1.407 1.264 1.277 1.290 1.301 1.313 1.325 1.337 1.349 1.362 1.375 1.390 1.407 1.428 1.454 1.494 1.530 1.574 45 1.210 1.231 1.409 1.262 1.276 1.289 1.301 1.313 1.325 1.337 1.349 1.362 1.376 1.392 1.409 1.430 1.456 1.498 1.535 1.580 50 1.208 1.229 1.411 1.261 1.275 1.287 1.300 1.312 1.324 1.337 1.349 1.363 1.377 1.393 1.411 1.432 1.459 1.502 1.540 1.586 55 1.205 1.226 1.413 1.259 1.273 1.286 1.299 1.312 1.324 1.337 1.350 1.363 1.378 1.394 1.413 1.434 1.463 1.506 1.546 1.593 60 1.202 1.224 1.415 1.257 1.272 1.285 1.298 1.311 1.324 1.337 1.350 1.364 1.379 1.396 1.415 1.437 1.466 1.511 1.552 1.601 65 1.198 1.221 1.417 1.255 1.270 1.284 1.297 1.310 1.323 1.337 1.350 1.365 1.380 1.398 1.417 1.440 1.470 1.516 1.558 1.610 70 1.195 1.218 1.419 1.253 1.268 1.282 1.296 1.309 1.323 1.337 1.351 1.366 1.382 1.399 1.419 1.443 1.474 1.522 1.566 1.619 75 1.191 1.214 1.422 1.250 1.266 1.280 1.295 1.309 1.322 1.337 1.351 1.367 1.383 1.401 1.422 1.447 1.479 1.529 1.574 1.629 80 1.186 1.210 1.425 1.248 1.264 1.279 1.293 1.308 1.322 1.337 1.352 1.368 1.385 1.404 1.425 1.451 1.484 1.536 1.583 1.641 85 1.181 1.206 1.428 1.244 1.261 1.276 1.292 1.306 1.321 1.336 1.352 1.369 1.386 1.406 1.428 1.455 1.490 1.544 1.593 1.654 90 1.176 1.201 1.432 1.241 1.258 1.274 1.290 1.305 1.321 1.336 1.353 1.370 1.388 1.409 1.432 1.460 1.496 1.553 1.605 1.669 95 1.169 1.196 1.436 1.237 1.255 1.271 1.288 1.304 1.320 1.336 1.353 1.371 1.390 1.412 1.436 1.465 1.504 1.563 1.618 1.686 100 1.162 1.190 1.441 1.232 1.251 1.268 1.285 1.302 1.319 1.336 1.354 1.372 1.393 1.415 1.441 1.472 1.512 1.576 1.634 1.706 105 1.154 1.182 1.447 1.227 1.246 1.264 1.282 1.300 1.317 1.335 1.354 1.374 1.396 1.419 1.447 1.479 1.522 1.590 1.652 1.730 110 1.144 1.174 1.453 1.220 1.240 1.260 1.279 1.297 1.316 1.335 1.355 1.376 1.399 1.424 1.453 1.488 1.534 1.607 1.675 1.759 115 1.131 1.163 1.461 1.212 1.234 1.254 1.274 1.294 1.314 1.334 1.355 1.378 1.402 1.430 1.461 1.499 1.549 1.628 1.702 1.795 120 1.117 1.150 1.471 1.202 1.225 1.247 1.268 1.289 1.311 1.333 1.356 1.380 1.407 1.437 1.471 1.512 1.567 1.655 1.737 1.842

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56 Information and Managemen t Sciences, V ol. 17, No. 1, Marc h, 2006

percentile. 1.67 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.975 0.99 1 1.544 1.566 1.584 1.600 1.614 1.627 1.640 1.653 1.665 1.678 1.691 1.705 1.719 1.735 1.753 1.775 1.802 1.844 1.882 1.927 5 1.542 1.565 1.583 1.599 1.613 1.627 1.640 1.652 1.665 1.678 1.691 1.705 1.720 1.736 1.754 1.776 1.804 1.846 1.885 1.931 10 1.540 1.563 1.581 1.597 1.612 1.626 1.639 1.652 1.665 1.678 1.691 1.706 1.721 1.737 1.756 1.778 1.806 1.850 1.889 1.936 15 1.538 1.561 1.580 1.596 1.611 1.625 1.638 1.652 1.665 1.678 1.692 1.706 1.721 1.738 1.757 1.780 1.809 1.853 1.893 1.941 20 1.536 1.559 1.578 1.595 1.610 1.624 1.638 1.651 1.665 1.678 1.692 1.707 1.722 1.739 1.759 1.782 1.811 1.856 1.897 1.946 25 1.533 1.557 1.576 1.593 1.609 1.623 1.637 1.651 1.664 1.678 1.692 1.707 1.723 1.740 1.760 1.784 1.814 1.860 1.902 1.952 30 1.531 1.555 1.575 1.592 1.607 1.622 1.636 1.650 1.664 1.678 1.693 1.708 1.724 1.742 1.762 1.786 1.817 1.864 1.907 1.958 35 1.528 1.553 1.573 1.590 1.606 1.621 1.636 1.650 1.664 1.678 1.693 1.708 1.725 1.743 1.764 1.788 1.820 1.868 1.912 1.965 40 1.525 1.550 1.571 1.588 1.605 1.620 1.635 1.649 1.664 1.678 1.693 1.709 1.726 1.745 1.766 1.791 1.823 1.873 1.917 1.972 45 1.522 1.548 1.568 1.586 1.603 1.619 1.634 1.649 1.663 1.678 1.694 1.710 1.727 1.746 1.768 1.793 1.827 1.877 1.924 1.979 50 1.519 1.545 1.566 1.584 1.601 1.617 1.633 1.648 1.663 1.678 1.694 1.711 1.728 1.748 1.770 1.796 1.830 1.883 1.930 1.988 55 1.515 1.542 1.563 1.582 1.600 1.616 1.632 1.647 1.663 1.678 1.694 1.711 1.730 1.750 1.772 1.799 1.834 1.888 1.937 1.997 60 1.511 1.538 1.560 1.580 1.598 1.614 1.630 1.646 1.662 1.678 1.695 1.712 1.731 1.752 1.775 1.803 1.839 1.895 1.945 2.006 65 1.507 1.535 1.557 1.577 1.595 1.613 1.629 1.645 1.662 1.678 1.695 1.713 1.733 1.754 1.778 1.807 1.844 1.901 1.953 2.017 70 1.502 1.531 1.554 1.574 1.593 1.611 1.628 1.644 1.661 1.678 1.696 1.714 1.734 1.756 1.781 1.811 1.849 1.909 1.963 2.029 75 1.497 1.526 1.550 1.571 1.590 1.608 1.626 1.643 1.661 1.678 1.696 1.715 1.736 1.759 1.784 1.815 1.855 1.917 1.973 2.042 80 1.491 1.521 1.546 1.568 1.587 1.606 1.624 1.642 1.660 1.678 1.697 1.717 1.738 1.761 1.788 1.820 1.862 1.926 1.985 2.057 85 1.485 1.516 1.541 1.564 1.584 1.603 1.622 1.641 1.659 1.678 1.697 1.718 1.740 1.764 1.792 1.826 1.869 1.936 1.998 2.073 90 1.478 1.510 1.536 1.559 1.580 1.600 1.620 1.639 1.658 1.678 1.698 1.719 1.742 1.768 1.797 1.832 1.877 1.948 2.012 2.092 95 1.470 1.503 1.530 1.554 1.576 1.597 1.617 1.637 1.657 1.678 1.699 1.721 1.745 1.772 1.802 1.839 1.886 1.961 2.029 2.114 100 1.461 1.495 1.523 1.548 1.571 1.593 1.614 1.635 1.656 1.677 1.699 1.723 1.748 1.776 1.808 1.847 1.897 1.976 2.049 2.139 105 1.450 1.486 1.515 1.541 1.565 1.588 1.610 1.632 1.654 1.677 1.700 1.725 1.752 1.781 1.815 1.856 1.910 1.994 2.072 2.169 110 1.437 1.475 1.506 1.533 1.558 1.582 1.606 1.629 1.652 1.676 1.701 1.727 1.756 1.787 1.824 1.867 1.925 2.016 2.100 2.206 115 1.422 1.461 1.494 1.523 1.550 1.575 1.600 1.625 1.650 1.675 1.702 1.730 1.761 1.795 1.834 1.881 1.944 2.043 2.135 2.251 120 1.403 1.445 1.479 1.510 1.539 1.566 1.593 1.619 1.646 1.674 1.702 1.733 1.766 1.803 1.846 1.898 1.967 2.076 2.179 2.310

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T esting Qualit y Assurance Using Pro cess Capabilit y Indices C P U and C P L 57

percentile. 2.00 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.975 0.99 1 1.856 1.877 1.898 1.917 1.933 1.949 1.965 1.979 1.994 2.009 2.025 2.041 2.058 2.077 2.099 2.124 2.156 2.206 2.251 2.305 5 1.849 1.875 1.897 1.916 1.933 1.949 1.964 1.979 1.994 2.009 2.025 2.041 2.059 2.078 2.100 2.126 2.159 2.209 2.255 2.309 10 1.846 1.873 1.895 1.914 1.931 1.948 1.963 1.979 1.994 2.009 2.025 2.042 2.060 2.079 2.101 2.128 2.161 2.213 2.259 2.315 15 1.844 1.871 1.893 1.912 1.930 1.947 1.963 1.978 1.994 2.009 2.026 2.043 2.061 2.081 2.103 2.130 2.164 2.217 2.264 2.321 20 1.841 1.869 1.891 1.911 1.929 1.946 1.962 1.978 1.993 2.010 2.026 2.043 2.062 2.082 2.105 2.132 2.167 2.221 2.269 2.328 25 1.838 1.866 1.889 1.909 1.927 1.944 1.961 1.977 1.993 2.010 2.026 2.044 2.063 2.084 2.107 2.135 2.171 2.225 2.275 2.335 30 1.835 1.864 1.887 1.907 1.926 1.943 1.960 1.977 1.993 2.010 2.027 2.045 2.064 2.085 2.109 2.137 2.174 2.230 2.281 2.342 35 1.832 1.861 1.884 1.905 1.924 1.942 1.959 1.976 1.993 2.010 2.027 2.046 2.065 2.087 2.111 2.140 2.178 2.235 2.287 2.350 40 1.828 1.858 1.882 1.903 1.922 1.940 1.958 1.975 1.992 2.010 2.028 2.046 2.066 2.089 2.114 2.143 2.182 2.241 2.294 2.359 45 1.824 1.855 1.879 1.901 1.920 1.939 1.957 1.974 1.992 2.010 2.028 2.047 2.068 2.090 2.116 2.147 2.186 2.247 2.301 2.368 50 1.820 1.851 1.876 1.898 1.918 1.937 1.956 1.974 1.992 2.010 2.029 2.048 2.069 2.092 2.119 2.150 2.191 2.253 2.309 2.378 55 1.816 1.847 1.873 1.896 1.916 1.936 1.954 1.973 1.991 2.010 2.029 2.049 2.071 2.095 2.122 2.154 2.196 2.260 2.318 2.388 60 1.811 1.843 1.870 1.893 1.914 1.934 1.953 1.972 1.991 2.010 2.030 2.050 2.073 2.097 2.125 2.158 2.201 2.267 2.327 2.400 65 1.806 1.839 1.866 1.889 1.911 1.932 1.951 1.971 1.990 2.010 2.030 2.051 2.074 2.100 2.128 2.162 2.207 2.275 2.337 2.413 70 1.800 1.834 1.862 1.886 1.908 1.929 1.950 1.969 1.989 2.010 2.031 2.053 2.076 2.102 2.132 2.167 2.213 2.284 2.349 2.427 75 1.794 1.829 1.857 1.882 1.905 1.927 1.948 1.968 1.989 2.010 2.031 2.054 2.079 2.105 2.136 2.173 2.220 2.294 2.361 2.443 80 1.787 1.823 1.852 1.878 1.902 1.924 1.945 1.967 1.988 2.010 2.032 2.056 2.081 2.109 2.141 2.179 2.228 2.305 2.375 2.461 85 1.780 1.816 1.847 1.873 1.898 1.921 1.943 1.965 1.987 2.009 2.033 2.057 2.083 2.113 2.146 2.185 2.237 2.317 2.391 2.481 90 1.771 1.809 1.840 1.868 1.893 1.917 1.940 1.963 1.986 2.009 2.033 2.059 2.086 2.117 2.151 2.193 2.247 2.331 2.408 2.504 95 1.761 1.801 1.833 1.862 1.888 1.913 1.937 1.961 1.985 2.009 2.034 2.061 2.090 2.121 2.158 2.201 2.258 2.347 2.429 2.530 100 1.750 1.791 1.825 1.855 1.882 1.908 1.933 1.958 1.983 2.009 2.035 2.063 2.093 2.127 2.165 2.211 2.271 2.365 2.452 2.560 105 1.737 1.780 1.815 1.846 1.875 1.902 1.929 1.955 1.981 2.008 2.036 2.066 2.098 2.133 2.173 2.222 2.286 2.387 2.480 2.596 110 1.722 1.767 1.804 1.836 1.867 1.895 1.923 1.951 1.979 2.007 2.037 2.068 2.102 2.140 2.183 2.236 2.305 2.413 2.514 2.640 115 1.704 1.751 1.790 1.824 1.856 1.887 1.916 1.946 1.976 2.006 2.038 2.072 2.108 2.149 2.195 2.252 2.327 2.445 2.556 2.695 120 1.681 1.731 1.772 1.809 1.843 1.876 1.908 1.939 1.971 2.004 2.039 2.075 2.115 2.159 2.210 2.272 2.354 2.485 2.609 2.765

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4. Statistical Testing for Product Quality Assurance

Statistical properties of the two estimator ˜C_{P U}M and ˜C_{P L}M are exactly the same. For convenience of presentation, we let CI be either CP U and CP L. In current practice, a (manufacturing) process is called “inadequate” if CI < 1.00; it indicates that the process is not adequate with respect to the production tolerances, either process variation σ2 needs to be reduced or process mean µ needs to be shifted closer to the target value T . A process is called capable if 1.00 ≤ CI < 1.33; it indicates that caution needs to be taken regarding to process distribution, some process control is required. A process is called satisfactory if 1.33 ≤ CI< 1.50; it indicates that process capability is satisfactory, material substitution may be allowed, and no stringent process control is required. A process is called excellent if 1.50 ≤ CI< 2.00; it indicates that process capability exceeds “satisfactory”. A process satisfies Motorola’s capability requirement (Harry (1988)) if Cpk≥ 1.50 and Cp ≥ 2.00. Thus, for processes with one-sided specifications, Motorola’s capability requirement is equivalent to “Excellent”. Finally, a process is called “super” if CI≥ 2.00.

In the electronics or microelectronics manufacturing industry, making product items meeting the specification limits is essential to product quality. Process capability mea-sures convey critical information regarding percentage of conforming items, meeting prod-uct design specification limits, which is a basic criterion used for judging whether the products are reliable from manufacturing perspective. Thus, if the product falls within the specification limits (LSL, U SL), then the product is considered reliable. A high value of CI implies a high quality of the product. To test whether a given process meets the preset capability requirement, we can consider the following statistical testing. A process meets the capability requirement if CI> C0 (a preset known constant), and fails to meet the capability requirement if CI ≤ C0.

H0 : CI ≤ C0 H1 : CI > C0

In the following, we calculate p-value (rejection probability), critical value, and power of the test. Suppose the observed value of the statistic ˜CM

I = C

∗

, then we can calculate those values as the following, where δI = 3pN/hC0.

p-value = p{ ˜C_IM _{≥ C}∗ |CI≤ C0} = Pn_{t(N − m}s, δI) ≥ 3√N bN −ms √ hC ∗ |CI ≤ C0 o . (11)

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The critical value, c0, is determined by the following, where δI = 3pN/hC0.

p{ ˜C_IM _{≥ c}0|CI = C0} = P n t(N − ms, δI) ≥ 3√N bN −ms √ hc0|CI = C0 o = α. Hence, we have c0= bN −ms √ h 3√N tα(N − ms, δI), (12)

where tα(N − ms, δI) is the upper α-th percentile of t(N − ms, δI) distribution. The power of the test can be computed as the following, where δI= 3pN/hCI.

p{ ˜C_IM _{≥ c}0|CI > C0} = P n t(N − ms, δi) ≥ 3√N bN −ms √ hc0|CI > C0 o = π(CI). (13) 4.1. Fortran Programs

To compute the p-values, the critical values, and the power of the test, two auxiliary functions are used in developing the Fortran program using the IMSL building functions, including (i) the CDF of non-central t distribution, (ii) the inverse of the non-central t distribution function. These Fortran programs, with real executed inputs and outputs, are included in Appendix I and Appendix II.

The Fortran programs are developed due to the limitations of the existing computing software, including SAS, MAPLE, MATLAB, MATHEMATICA, MATHCAD, in cal-culating the percentiles of non-central t distribution. Those computing software limit the degrees of freedom of the non-central t distribution to 70-100 (except for MATH-EMATICA, which handles the degrees of freedom up to 120), with non-centrality no greater than 30-35. The Fortran programs using the IMSL building functions, extend the computing capability to degrees of freedom of 150.

4.2. A Practical Procedure for Testing with non-centrality 75

To determine if the process meets the capability (reliability/quality) requirement, we first determine the capability requirement C0, the α-risk, and the total number of sample observations. Executing the Fortran program, we can find the critical value c0 based on the α-risk, C0, and total number of observations. If the estimated value ˜CIM calculated from the sample data is greater than the critical value c0, then we may conclude that the process meets the capability requirement. In this case, it can be assured that a high proportion of the product items satisfy the specification limits, and the process is

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considered reliable. For example, if CP U and CP Lis tested to be greater than 1.00, then it is assured that at least 99.865% of the product items are within the specification limits, and those product items are considered reliable. Otherwise, we do not have sufficient information to conclude that the process meets the preset capability requirement. In this case, we would tend to believe that the process is incapable. In the following, we develop a simple step-by-step procedure for the practitioners to use for their in-plant applications to obtain reliable decisions.

4.3. Test Procedure

Step 1: Determine the value of the capability requirement C0 (normally set to 1.00, 1.33, 1.67, or 2.00), the desired quality condition, and the α-risk (type-I error, normally set to 0.05, 0.025, or 0.01), the chance of incorrectly concluding a bad process (does not meet the preset capability requirement) as good one (meets the preset capability requirement).

Step 2: Calculate the value of the estimator, ˜C_IM, from the samples.

Step 3: Run the program listed in Appendix I to find the corresponding critical value, c0, based on α, C0, and total N sample observations with ms several groups of samples.

Step 4: Conclude that the process meets the capability requirement if ˜CM

I is greater than c0. Otherwise, we do not have enough information to conclude that the process meets the capability requirement. In this case, we would conclude that the process is incapable.

5. An Application Example VLA

The product investigated is a wideband, unity gain stable voltage feedback op amp that offers bipolar output voltage limiting. Two buffered limiting voltages take control of the output when it attempts to drive beyond these limits. This new output limit-ing architecture holds the limiter offset error to ±15mV. The voltage limitlimit-ing amplifier (VLA) operates linearly to within 30mV of the output limit voltages. The combination of narrow nonlinear range and low limiting offset allows the complete output limiting voltages to be set within 100mV of the desired linear output range. A fast 2.4ns recovery from limiting ensures that overdrive signals will be transparent to the signal channel. Implementing the limiting function at the output, as opposed to the input, gives the

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specified limiting accuracy for any gain, and allows the VLA to be used in all standard op amp applications. Non-linear analog signal processing will benefit from the VLA’s sharp transition from linear operation to output limiting. The quick recovery time sup-ports high-speed applications such as CCD pixel clock stripping, video sync stripping, HF mixers, and AM signal generation. The VLA is available in an industry standard pin out SO-8 package, as depicted in Figure 9. The typical performance curves of limited output response and detailed limited output voltage are shown on Figure 10.

Figure 9. The voltage limiting amplifier.

Figure 10. Typical performance curves of limited output response and limited output voltage.

The complete output limit voltage is an essential product characteristic, which has sig-nificant impact to product quality. For this particular model of VLA product, the upper

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specification limit, U SL, is 100mV. The capability requirement for this VLA product was to Satisfactory (CP U > 1.33). A total of 86 sample data with 18 groups of unequal sizess are collected from a stable process (under statistical control), which is displayed in Table 5. In order to obtain the critical values, we first calculate the overall sam-ple mean X = 88.7162, and the pooled samsam-ple variance S2

p = 5.2741. The estimator ˜

CM

P U = bN −ms(U SL − X)/(3Sp) = 1.6197. With type I error α-risk set to 0.05, we find the critical value c0 = 1.543 from the program listed in Appendix I based on N = 86, ms= 18, C0 = 1.33, and α = 0.05. Since ˜CP UM = 1.6197 is greater than the critical value c0 = 1.543 in this case, we therefore determine that the process meets the capability re-quirement “Satisfactory”. Consequently, at least 99.9967% of the VLA products satisfy the specification limit, and are considered reliable.

Table 5. A total number of 18 samples of 86 observations.

Sample Observations 1 91.08 87.74 92.10 87.29 90.35 86.25 2 87.18 88.00 89.47 92.93 — — 3 89.23 88.14 90.07 86.48 88.80 — 4 89.94 88.38 88.65 89.38 89.88 — 5 92.52 93.19 90.46 86.13 — — 6 89.76 87.81 86.81 90.46 90.02 88.59 7 88.21 87.14 85.14 85.58 — — 8 86.54 88.13 89.37 89.48 86.39 — 9 90.47 89.90 92.06 87.93 88.99 — 10 87.90 90.51 91.29 — — — 11 90.05 94.82 88.66 89.90 84.87 — 12 87.85 85.03 88.77 86.24 87.59 — 13 87.11 88.67 90.99 92.26 89.90 — 14 87.82 89.38 87.38 87.02 — — 15 86.04 85.00 88.86 90.41 89.43 — 16 88.98 89.04 87.83 92.96 88.54 — 17 86.03 88.36 87.80 88.74 — — 18 88.18 87.23 85.35 88.06 85.75 88.80 6. Conclusions

Process capability indices CP U and CP Lhave been widely used in the manufacturing industry to provide quantitative measures on process performance, particularly, for

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pro-Testing Quality Assurance Using Process Capability Indices CP U and CP L 63

cesses with one-sided specification limits. Statistical properties of the estimators of CP U and CP L have been investigated extensively for cases with one single sample. In this paper, we considered the estimation and capability testing of CP U and CP L based on several groups of samples with unequal sizes. We showed that the proposed estimators of CP U and CP Lare the UMVUEs. A simple but practical procedure based on a hypothesis testing using the proposed UMVUE is developed. The engineers/practitioners can use the proposed procedure to test whether their processes are capable of reproducing prod-ucts meeting the preset product requirement and determine the percentage of reliable product items. We also presented an example of the VLA, to illustrate how one may apply the proposed approach to the actual data collected from real-world applications for product quality testing from manufacturing perspective.

Acknowledgements

The authors would like to thank the anonymous reviewers for encouraging and helpful comments that greatly improved the paper.

Appendix I

Fortran 90 Program for p-Value

INTEGER IDF, NOUT

REAL DELTA, P, T, TNDF, b, bf EXTERNAL TNDF, UMACH

PRINT*, ’Please Enter: Total # of Sample Observations, # of Samples, Cpu, Estimate of Cpu.’

READ*, N, m, Cpu, ecpu

b=0; bf=0 ; IDF=0 ; DELTA=0; p=0; T=0 CALL UMACH (2, NOUT)

b=sqrt((N-m-1)/2.0)*(1-1.0/(4*(N-m-1))+1.0/(32*(N-m-1)**2)+& &5.0/(128*(N-m-1)**3)) bf=sqrt(2.0/(N-m))*b IDF = N-m DELTA = 3*sqrt(N/1.0)*Cpu T = 3*sqrt(N/1.0)*eCpu/bf p = 1-TNDF(T,IDF,DELTA) PRINT*, ’ The p-Value =’, p END

Fortran 90 Program for Critical Value

INTEGER IDF, NOUT

REAL DELTA, P, T, TNIN, b,bf EXTERNAL TNIN, UMACH

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Cpu, alpha-risk.’ READ*, N, m, Cpu, alph

b=0; bf=0 ; IDF=0 ; DELTA=0; P=0; T=0 CALL UMACH (2, NOUT)

b=sqrt((N-m-1)/2.0)*(1-1.0/(4*(N-m-1))+1.0/(32*(N-m-1)**2)+& &5.0/(128*(N-m-1)**3)) bf=sqrt(2.0/(N-m))*b IDF = N-m DELTA = 3*sqrt(N/1.0)*Cpu P = 1-alph T = (TNIN(P,IDF,DELTA))*bf/(3*sqrt(N/1.0)) PRINT*, ’ The Critical Value =’ , T END

Fortran 90 Program for Power of the Test

INTEGER IDF, NOUT

REAL DELTA, DELTA1, P, P1, T, TNDF, TNIN, b, bf EXTERNAL TNDF, TNIN, UMACH

PRINT*, ’Please Enter: Total # of Sample Size, # of Samples, Critical Value, Given Cpu , alpha-risk.’

READ*, N, m, Co, gCpu, alph

b=0; bf=0 ; IDF=0 ; DELTA=0; DELTA1=0; P=0; P1=0 T=0 CALL UMACH (2, NOUT)

IDF = N - m DELTA = 3*sqrt(N/1.0)*Co P = 1-alpha T = TNIN(P,IDF,DELTA) DELTA1 = 3*sqrt(N/1.0)*gCpu P1 = 1-TNDF(T,IDF,DELTA1) PRINT*, ’Power of the Test= ’, P1 END

Appendix II

[1] For p-Value:

Please Enter: Total # of Sample Observations, # of Samples, Cpu, Estimated Cpu. Input:

130, 35, 1.33, 1.67 Output:

The p-Value = 2.074778E-03 [2] For Critical Value:

Please Enter: Total # of Sample Observations, # of Samples, Cpu, alpha-risk. Input:

86, 18, 1.33, 0.05 Output:

The Critical Value = 1.542814 [3] For Power of the Test:

Please Enter: Total # of Sample Observations, # of Samples, Critical Value, Given Cpu , alpha-risk. Input:

150, 60, 1.33, 1.65, 0.025 Output:

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Authors’ Information

Ming-Hung Shu holds a Ph.D. degree from the University of Texas, Arlington, USA. He is currently an as-sociate professor of Industrial Engineering and Management in National Kaohsiung University of Applied Sciences, Kaohsiung 80778, Taiwan. Professor Shu’s research interests include statistical/engineering pro-cess control, propro-cess capability analysis, and electronic commerce. He has published several articles in International Journal of Production Research, Microelectronics Reliability, Journal of Applied Statistics, Communications in Statistics, International Journal of Advanced Manufacturing Technology, etc.. Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sci-ences, Kaohsiung 80778, Taiwan, R.O.C.

E-mail: workman@cc.kuas.edu.tw Tel: +886-7-381-4526 ext 7105

K. H. Lu, holds a Ph.D. degree from the National Chiao Tung University, Hsinchu 300 Taiwan. He is currently a professor of Department of Asia-Pacific Industrial and Business Management, National University of Kaohsiung 811, Taiwan. He has published several articles in International Journal of Pro-duction and Inventory Management, Quality Engineering, International Journal of Electronic Business

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Management, International Journal of Libraries and Information Services, Journal of Management & Systems, etc..

Department of Asia-Pacific Industrial and Business Management, National University of Kaohsiung, Kaohsiung 811, Taiwan. R.O.C.

E-mail: log@nuk.edu.tw Tel: +886-7-591-9245

Bi-Min Hsu holds a Ph.D. degree from the University of Texas, Arlington, USA. She is currently an assistant professor of Industrial Engineering & Management in Cheng Shiu University, Kaohsiung 83347, Taiwan.

Department of Industrial Engineering and Management, Cheng Shiu University, Kaohsiung 83347, Tai-wan, R.O.C.

E-mail: biminhsu2568@yahoo.com.tw Tel: + 886-7-731-0606

Kuo-Ren Lou holds a Ph.D. degree in the Department of Statistics from the University of Connectiant, U.S.A. He is an associate professor in the Department of Management Sciences & Decision Making at Tamkang University in Taiwan.

Department of Management Sciences & Decision Making, Tamkang University, Tamsui, Taipei, 251, Taiwan, R.O.C.