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1 力 C 理 路 mhshu@npic.edu.tw 力 C 數 量 C 良率 不 料 數 力 C C 來 量 良率 C 率 度 數 (PDF) 累 數 (CDF) 率 來 C 力 良率 C Matlab 了 力 例 立 理論 力 力 C p, C, C pm, C pmk, ((Kane (1986), Chan et al. (1988)), 數 量 : C p = USL LSL, 6σ 91

2 USL µ µ LSL C = min,, 3σ 3σ USL LSL C pm =, 6 σ + ( µ T ) C pmk = min 3 USL µ σ + ( µ T ) µ LSL,, 3 σ + ( µ T ) USL LSL µ 數 σ T USL LSL 不 µ σ 料 數 X = n [ i= 1(X i n i X i / n S= = 1 X ) /(n-1)] 1/ 來 C X S Ĉ 數 力 C 理 Ĉ 來 量 良率 力 良率 了 Ĉ 率 度 數 (PDF) 累 數 (CDF) Ĉ t (bivariate non-central t - distributed) 數聯 率 Pearn et al. (199)foldednormal) chi-square) 兩 數聯 率 PDF CDF 不易 Ĉ 不 利 Ĉ [1] Zhang et al. (1990) 100γ%Ĉ L(Zh) C = n 1 ( n 1)( Γ(( n ) / )) C 1 Z( γ ), n 3 ( Γ(( n 1) / )) 9

3 Z(γ) γ 數 lower percentile of the standard normal distribution). [] Bissell (1990) 100γ%Ĉ : 1 C L( Bi) C = C ( ) + Z γ (1) 9n n [3) Nagata and Nagahata (1994) (1) 100γ% Ĉ L( NN ) C = ˆ ˆ 1 C 1 C Z( γ ) +. 5( n 1) 9n ( n 1) [4] Chou et al. (1990) and Levinson (1997) 不 t (bivariate non-central t -distributed) 數聯 率 k 1 = k 100γ% Pr(C L(Ch) C Ĉ )=Pr(C pl C L(Ch), C pu C L(Ch) X k1s µ µ L( Ch) X ks L( Ch) Pr C, C = γ, 3σ 3σ Pr[ T n 1 ( δ1) t1 T n 1 ( δ ) t ] = γ. )=γ T n 1 ( δ )~ t non-central t distribution) 度 n 1 參數 noncentrality parameter) δ = k n t = k n δ = t L( Ch) nc δ k 1 = k = 3C pl = 3C pu 了 Cˆ pu = ( USL X ) 3S = 3 L( Ch) nc Cˆ pl = ( X LSL) 3S 不 Franklin and Wasserman (199)Kushler and Hurley (199)Rodridguez (199) Nagata and Nagahata (1994)Pr( Cˆ = Cˆ ) 率 0 C 良率 Φ(3C ) 1 yield Φ(3C ) 力 C < 1.00 pu pl 93

4 論 不 降 異 不良率 700 (ppm) 不 力 1.00 C < 1.33 論 R S 降 異 不良率 (ppm) 不 力 1.33 C < 1.67 論 不良率 (ppm) 不 力 1.67 C <.00 論 不良率 (ppm) 不 C.00 論 不良率 0.00 (ppm) 不 1. 論 力 不 數 1. 力 不 數 Condition C values ppm Inadequate C < 1.00 NC > 700 Capable 1.00 C < 1.33 NC < 700 Satisfactory 1.33 C < 1.67 NC < 66 Excellent 1.67 C <.00 NC < 0.54 Super.00 C NC < 0.00 (Montgomery (001)) 力. 力 良率 類 C 良率 Existing Processes % New Processes % Existing Processes on Safety, Strength, or Critical Parameters (such as manufacturing soft drinks or chemical solution bottled with glass containers) New Processes on Safety, Strength, or Critical Parameters % % 94

5 C PDF CDF 率 Matlab 來 C 例 來 Ĉ 率 了 Ĉ 率 (1) D = (n 1) 1/ d/σ () a = [(n 1)/n] 1/ (3) K = (n 1)S /σ K~ χ n 1, (4) Z = n 1/ ( X T)/σ Z~N(δ, 1) δ = n 1/ (µ T)/σ, (5) Y = Z Y 率 度 數 : 1 f Y (y) = ( fz ( y ) + fz ( y ) Ĉ y D a Y Cˆ = 3 K : y > 0 () F (x) = Pr( Ĉ Ĉ x) = P( D a Y 3 K x) D a Y = 1 P( K Y = y) fy ( y) dy 0 3x K ~ χ n 1 Pr( K D a y ) = 0 y > (D/a) x > 0. 3x F ( D / a (x) =1 Ĉ ) Pr( D a K x 0 3 y ) f Y ( y ) dy ( D / a) ( D a y) = 1 F K 0 9x f ( y) dy Y () 不 率 累 度 數 95

6 ( ) D a ˆ ( x) = C D a y 1 0 K 9x F F 1 ( f Z ( y ) + fz ( y ) y dy (3) ( D a y ) D a ( D a y ) ˆ ( x) = C f K ( fz ( y ) + f Z ( y ) dy (4) 0 3 9x 9x y f (3) 數 y = t Ĉ 累 度 數 Chi-square distribution) normal distribution): (x) F ( n 1)( b n t) 9nx ξ 0 b n = 1 - G [ φ( t + ξ n) + φ( t n) ] dt, Ĉ x > 0 b = d/σ F K (.)ordinary central Chi-square distribution) χ n 1 累 度 數, φ(.)standard normal distribution) N(0, 1) 率 度 數. 1(a)-1(d) Ĉ PDF CDF ξ= 0.0 1b = 3d = 不 數 n = (a). Ĉ PDF ξ = 0.0, b =3, d = n=10, 0, 50 (). 1(b). Ĉ PDFξ = 1.0, b=3, d =, n=10, 0, 50 (). 1(c). Ĉ CDFξ = 0.5, b = 3, d =, n=10, 0, 50 (). 1(d). Ĉ CDFξ = 1.0, b = 3, d =, n=10, 0, 50 (). 96

7 C C = C b = d/σ b = 3C + ξ. C ξ 數. C = d µ m 3σ = d/σ ξ, where ξ = (µ T)/σ. 3 數 n γ Ĉ 參數 ξ 列 (5) 數 參數 ξ=(µ T)/σ ξˆ =( X T)/S(5)ξ 數 ξˆ =ξˆ 0 ξˆ =ξˆ 0 都 b 0 n G ( n 1)( b 9nC n t) φ ( t + ξ n) + φ( t ξ n dt =1 γ. (5) ) (Lower Confidence Bounds; LCB) 了 C 率 LCB 數 [i] χ F n 1 K ( )[ii] N(0,1)φ( )[iii] recursive adaptive Simpson quardrature quad 數 數 t 0 t 1 (1 > t 0 > t 1 > 0). LCBMatlab (available upon request 力 Step 1. 讀 料 (X 1, X,, X n ) LSLUSLT and γ Step. 料 X Sξˆ Ĉ Step 3. 利 Ĉ recursive adaptive Simpson quardrature quad ( t 0 ) Step 4. t 1 Ĉ C Simpson quardrature quad Tol 10-4 recursive adaptive Step 5. 論 The true value of the manufacturing capability C is no less than the C with 100γ % level of confidence 97

8 C 參數 ξ 參數 µ σ ξ=(µ T)/σ 若 數 µ σ 更 不 行 量數 ξ=0(0.05)3.00n=10(5)00ĉ γ=0.95 =0.7(0.1)3.0 C 參數 ξ [i] C ξ n [ii] C ξ = 1.00 ξ 1.00(5)Ĉ nγ ξˆ =ξ=1.00 來 C 論 數 數 例 -5 C v.s. ξ Ĉ γ 0.95 n=30, 50, 70, 100, 150, Ĉ = 0.7(0.1)3.0n=10(5)00γ=0.95 了 力 例 Ĉ =1.5 n= 力 C 不 1.315; C > C v.s. ξ Ĉ =0.9, γ =0.95 n=30, 50, 70, 100, 150, 00 (). 3. C v.s. ξ Ĉ =1., γ =0.95 n=30, 50, 70, 100, 150, 00 ( ). 98

9 4. C v.s. ξ Ĉ =.0, γ =0.95 n=30, 50, 70, 100, 150, 00 (). 5. C v.s. ξ Ĉ =.5, γ =0.95 n=30, 50, 70, 100, 150, 00 ( ). PVCHigh-Impact PVC PVC PVC 良 度 易 流 力 不 廉 便 料 PVC 1 1 '' 例 6 USL 48.4mm LSL47.6mm T48mm 48 ± 0. 4 mm 6. PVC 1 1 '' (mm)

10 PVC 料 料 5 料 料 Histogram) normal probability plot) 料 更 Shapiro-Wilk 量 W=0.989 p-value= 料 論 料 了 行 Matlab 讀 料 (X 1, X,, X n )LSL= 47.6 mmusl=48.4mmt=48mm γ =0.95 ξ=(µ T)/σ=1.0 數 X = S=0.0608Ĉ =1.68 C =1.48 行 論 95% 力 C 不 1.48 ; C >1.48 不良率 不 19(ppm) 不 Matlab Execution Input & Output Output: The Sample Mean is The Sample Standard Deviation is The ξ is 1.0. The Estimate of C is The true value of the manufacturing capability C is no less than 1.48 with 0.95 level of confidence 料來 六 100

11 PVC X S 數 狀 X S 來 量 力 來 料 數 X S 來 了 異 between-sample variability 料來 m s 數 n i = ( x i, xi,..., xin i m 1 ) 數 N s 1ni 數 X i S i = i= X i = x n ni j =1 i ij S i = ni j= 1 ( xij X i ) n 1 i, 數 X S P X = X m ms i =1 s i S P = ms i= 1( ni 1) Si ms ( ni 1) i= 1 Pearn and Shu (003) (6) 率 累 度 數 臨 數 率 累 度 數類 M Ĉ = pm min( USL X, LSL X ) (6) 3 S P 101

12 更 PVC 不 PVC Singhal (1991)Pearn and Chen (1997) Pearn et al. (00) 不 力 multi-process performance analysis chart (MPPAC)) 數 落 良率 離 度 (Accuracy) 異 度 (Precision) 參 力 來 MPCAC 不 論 利 來 力 (MPCAC) 來了 力 良率 更 力 論 C 率 度 數 (PDF) 累 數 (CDF) 率 來 Ĉ 力 參數 ξ=(µ T)/σ ξ=1.0 不 Type I (1γ) 理 良率 例 料來 來 參 參 1 V. E. Kane, 1986, Process capability indices, Journal of Quality Technology, vol. 18, no. 1, pp L. K. Chan, S. W. Cheng and F. A. Spiring, 1988, A new measure of process capability: C pm, Journal of Quality Technology, vol. 0, no. 3, pp W. L. Pearn, S, Kotz and N. L. Johnson, 199, Distributional and inferential properties of process capability indices, Journal of Quality Technology, vol. 4, no. 4, pp A. F. Bissel, 1990, How reliable is your capability index?, Applied Statistics, vol. 39, no. 3, pp

13 5 N. F. Zhang, G. A. Stenback and D. M. Wardrop, 1990, Interval estimation of process capability index C, Communications in Statistics: Theory and Methods, vol. 19, pp Nagata, Y. and Nagahata, H. (1994). Approximation formulas for the lower confidence limits of process capability indices. Okayama Economic Review, 5, Y. M. Chou, D. B. Owen and A. S. Borrego, 1990, Lower confidence limits on process capability indices, Journal of Quality Technology, vol., pp W. Levinson, 1997, Exact confidence limits for process capabilities, Quality Engineering, vol. 9, no. 3, pp L. A. Franklin and G. S. Wasserman, 199, Bootstrap lower confidence limits for capability indices, Journal of Quality Technology, vol. 4, no. 4, pp R. Kushler and P. Hurley, 199, Confidence bounds for capability indices, Journal of Quality Technology, vol. 4, no. 4, pp R. N. Rodriguze, 199, Recent developments in process capability analysis., Journal of Quality Technology, vol. 4, no. 4, pp W. L. Pearn and K. S. Chen, 1997, A practical implementation of the process capability index C, Quality Engineering, vol. 9, no. 4, pp W. L. Pearn, C. H. Ko and K. H. Wang, 00, A multiprocess performance analysis chart based on the incapability index C pp : an application to the chip resistors, Microelectronics Reliability, vol. 4, no. 7, pp W. L. Pearn and M. H. Shu, 004, Testing process performance using C based on multiple samples, Working Paper. 15 S. C. Singhal, 1991, Multiprocess performance analysis chart (MPPAC) with Capability Zones. Quality Engineering, vol. 4, no. 1, pp

14 3. Ĉ γ = 0.95 C Ĉ = 0.7(0.1)1.8 n = 10(5)00,. n

15 4. Ĉ γ = 0.95 C Ĉ = 1.9(0.1)3.0 n = 10(5)00. n

16 Considering the Sampling Variant of Process Capability Index C for Quality Assurance with Application M. H. Shu Department of Commerce Automation & Management, National Pingtung Institute of Commerce, 51 Min-Sheng E. Road, Pingtung, Taiwan 900, R. O. C. Abstract Process capability index C has been the most widespread one used in the manufacturing industry providing numerical measures on process performance. The index C offers a lower bound on the process yield, a widely used criterion for measuring process capability. From a practical perspective the true index C unobtainable, sample data have to collect and a great degree of uncertainty is introduced into capability assessments due to the sampling variant. By simply looking at the C estimated index and then making a conclusion on whether the given process condition, is highly unreliable since the sampling variant has been neglected. In this paper we use the exact distribution of the estimation of C, and develop an efficient algorithm to compute the lower confidence bounds on C. The lower confidence bound estimates the minimum capability of the process based on the sample data, which is essential to quality and reliability assurance. We also provide tables for the engineers/practitioners to use in measuring their processes. The applications of the control chart combining with lower confidence bound are discussed. Our implementation of the existing statistical theory for capability assessment bridges the gap between the theoretical development and the in-plant applications. Keywords: process capability index, lower confidence bound, sample size determination, control chart. 106