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恋退鄂
7 years ago
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3 (EWMA) 3 EWMA (defects) (Poisson distribution) (compound Poisson process) Brook and Evans EWMA (ARL) EWMA (geometric Poisson EWMA control schemes) The Study on Detecting Small Shifts in Quality Levels with the Geometric Poisson Process Using EWMA Control Schemes Abstract It is well known that the exponentially weighted moving average (EWMA) control schemes can better detect small shift of process mean in a short period of time than the conventional 3-sigma control charts do. However, an effective EWMA control scheme has to be with correct underling distribution in describing the characteristic of process quality. In process control, the common situation is to assume that the Poisson distribution holds for count data, such as the conventional c-chart. In contemporary modern production environment, the production process is usually more complex than others. In fact the control schemes based on compound Poisson distribution are more appropriately used in controlling the defects. Applying the Markov chain approach by Brook and Evans to calculate the average run length (ARL), we take one step further, in this study, to develop the EWMA control scheme for the geometric Poisson production process to detect a small sustained shift in the mean. Using the EWMA control schemes we can detect possible shift at the early beginning of the process change, and improve the quality level of the production process. Key words: geometric Poisson, exponentially weighted moving average (EWMA), average run length (ARL), Markov chain

4 . (number of defects) (defective item) (defects) (false alarm rate) (Albin and Friedman, 989; Gardiner, 987) Stapper (985)Albin and Friedman (989) Friedman and Albin (99) eyman s Type A Randolph and Sahinoglu (995) (geometric Poisson distribution) (software quality control) (Shewhart, 93) ( 200) (exponentially weighted moving average EWMA )( Roberts, 959) (cumulative sum control chart CUSUM )( Page, 954) (Crowder, 989; Fu et al., 2002) EWMA (average run length, ARL) (RL) ARL RL ARL ARL ARL 2 ARL ARL 2 EWMA ARL ARL 2 ARL EWMA 3 Brook Evans (972) Brook Evans (972) EWMA 2. Poisson( λ )^ geometric( p) (Randolph and Sahinoglu, 995) Sherbrooke (966), λp 2

5 λt PXt [ ( ) = 0] = e λ x PXt x p p x x y λ t ( t) e x y y [ ( ) = ] = ( ), =,2, y= y! y λ > 0, 0 < p < t 0 (t=) Chen et al. (2005) X EX ( ) = λ () p + p Var( X ) = λ ( p) 2 (2) (EWMA) EWMA EWMA Zt = wdt + ( w) Zt 0 < w, t =, 2,..., n,... w Z t t X t EWMA X t t Z t (Montgomery 996) 2 w σ 2 Z t σ ( ) (3) 2 w EWMA h h w hu = u0 + KUσ ( ) 2 w w hl = u0 KLσ ( ) (5) 2 w K U K L K K U = K L = K Z t > h U Z t < h L EWMA EWMA (ARL) EWMA ARL U L (4) 3.2. (Markov chain approach) Brook and Evans (972) EWMA ARL (number of defects) D EWMA 3

6 Z = wd + ( w) Z 0 < w, t =, 2,..., n,... t t t Z t > h U Z t < h L Borror et al. (998)Z t [ hl, hu] h L h U Subinterval jth sub-interval { m k ith sub-interval { Z t+ Z t hl L j U j (2k )( h h ) U L h + L 2 hu ( j )( h h ) U L h + L j( h h ) U L h + L [ hl, hu] i [ Li, Ui] E i mi E i hl hu Li U i i( E i ) Z E i E i (transition) t (random walk) E i E j D Z t [ hl, hu] (in control) Zt Z t+ E i E j Z t [ hl, hu] (out of control) (absorbing state) ARL Zt i (Z t= 0 E i ) (2i )( hu hl) Z0 = mi = hl + 2 E i ( Li U i ) i Li = hl + ( hu hl ) Ui = hl + i ( hu hl) Z t (Brook and Evans, 972) P ij E i E j (transition probability) P i, = Pr (E i E ) P i,j = Pr (E i E j ) P i,+ = Pr (E i E + ) P +,i = 0, P +,+ = i =, 2,, j =, 2,,. E + P 4

7 P P P P P P P P P P P P P= P P P P P P P P P P P P P P P P P P P,,2,3, j,, + 2, 2,2 2,3 2,j 2, 2, + i, i,2 i,3 i, j i, i, +,,2,3, j,, + +, +,2 +,3 +, j +, +, + P = R A 0 D s (sth factorial moment) (I R) u (s) = sru (s ) (s = 2, 3, ) (6) I u (s) D i (i =, 2,, )s R P s = (6)u = (I R) u ARL u ( + )/2 ( ) Z0 = u0 EWMA ARL u0 Borror et al. (998) E i E j P ij Pij = Pr ( Ei Ej ) (2i )( hu hl) = Pr Lj < Zt < U j Zt = hl + 2 = P L < wx + w Z < U { ( ) } r j t t j j 2i = Pr hl + ( hu hl) < wxt + ( w) [ hl + ( hu hl)] 2 < j h + L ( hu hl) hu hl = Pr hl + [ 2( j ) ( w)(2i ) ] < Xt 2w (7) hu hl < hl + [ 2 j ( w)(2i ) ] 2w w K (6)(7) EWMA ARL 4. EWMA ( 5

8 200) 25 λ = 2 p = λ = 3 p = 0.25 : : λ p E(X) V(X) ARL ARL EWMA K = 3.3 w = : EWMA K w h L h U ARL ARL EWMA ( 25 (y) (x) 20 ) EWMA h U = u 0 = h L =.7867 EWMA 29 ( Z 29 = 4.205) ( = 4 ) 3.55( ARL2 = 3.55) K = 3.3 w = 0. ARL 782 h U =4.4633h L =.7867 ARL =782 (I 6

9 728 ) K ( w K=2.84w = 0.5 ARL K=2.84w=0.5 ARL ( ) I 5 EWMA EWMA (w K) w K w Z t K EWMA w w = 0.05 w = 0.0 w = 0.5w = 0.20 K K = 3 EWMA w w 0. K (Montgomery, 996) EWMA () Z = u 0 0 (ARL ) w( 0< w < ) K (2) Brook and Evansn (972) (transition probability matrix) (4)(5)(6)(7) EWMA w K ARL (3) (h U h L ) ( 0.5σ σ) (2) EWMA ARL 2 (4) ()(3) ARL ARL 2 I II w K EWMA 2 ARL w K w K (h U h L ) ARL 2 (ARL, ARL 2 ) ARL ARL 2 w K ARL K w h L h U ARL ARL

10 (ARL, ARL 2 )(345.39, 3.55) ARL ARL 2 w K X t w w=0.5 w Z t ( h L h U ) 4 Z t Z t w 6 EWMA (geometric Poisson process) EWMA Brook and Evans EWMA ARL p [ hl, hu] Brook and Evans EWMA EWMA EWMA EWMA 7 7- (geometric Poisson EWMA control schemes) ARL I EWMA Brook Evans RL 3 EWMA 8

11 200 pp.07~3 200 Journal of the Chinese Institute of Industrial Engineers, Vol. 8, o. 6, pp. -8. Albin, S. L. and D. J. Friedman, 989, The Impact of Clustered Defect Distributions in IC Fabrication, Management Sciences, 35(9), Borror, C. M., Champ, C. W., and Rigdon, S. E., 998, Poisson EWMA control charts, Journal of Quality Technology, 30(4), Brook, D. and Evans, D. A., 972, An approach to the probability distribution of CUSUM run length, Biometrika, 59(3), Chen, C. W., Randolph, P. H., and Liou, T. S., 2005, Using CUSUM control schemes for monitoring quality levels in compound Poisson production environment: the geometric Poisson process, Quality Engineering, 7(2), Crowder, S. V., 989, Design of exponentially weighted moving average schemes, Journal of Quality Technology, 2, Friedman, D. J. and Albin, S. L., 99, Clustered defects in IC fabrication: impact on process control charts, IEEE Transactions on Semiconductor Manufacturing, 4(), Fu, J. C., Spiringa, F. A. and Xie, H., 2002, On the average run lengths of quality control schemes using a Markov chain approach, Statistics & Probability Letters, 56, Gardiner, J. S., 987, Detecting small shifts in quality levels in a near-zero defect environment for integrated circuits, Unpublished Doctoral Dissertation, University of Washington, Seattle, Washington. Montgomery, D. C., 996, Introduction to Statistical Quality Control, 3 rd ed., John Wiley & Sons, Inc., ew York, Y. Page, E. S., 954, Continuous inspection schemes, Biometrika, 4, Randolph, H. P. and Sahinoglu, M., 995, A stopping rule for a compound Poisson random variable, Applied Stochastic Models and Data Analysis,, Roberts, S. W., 959, Control charts based on geometric moving average, Technometrics,, Sherbrooke, C. C., 966, Discrete compound Poisson process and tables of the geometric Poisson distribution, Memorandum RM-483-PR, The Rand Corporation, Santa Monica, CA. Shewhart, W. A., 93, Economic Control of Quality of Manufactured Product, Van ostrand, ew York, Y. Stapper, C. H., 985, The Effects of Wafer to Wafer Defect Density Variations on Integrated Circuit Defect and Fault Distribution, IBM Journal of Research Development, 29(),

* CUSUM EWMA PCA TS79 A DOI /j. issn X Incipient Fault Detection in Papermaking Wa

* CUSUM EWMA PCA TS79 A DOI /j. issn X Incipient Fault Detection in Papermaking Wa 2 *. 20037 2. 50640 CUSUM EWMA PCA TS79 A DOI 0. 980 /j. issn. 0254-508X. 207. 08. 004 Incipient Fault Detection in Papermaking Wastewater Treatment Processes WANG Ling-song MA Pu-fan YE Feng-ying XIONG