SC 93246H327032
(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
. (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
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) 3. 3.. (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
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
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 2.5 4.0 ( 5
200) 25 λ = 2 p = 0.2 20 λ = 3 p = 0.25 : : λ p E(X) V(X) 2 0.20 2.5 3.75 25 3 0.25 4.0 6.67 20 ARL ARL 2 2.5 4.0 EWMA K = 3.3 w = 0. 2 2: EWMA K w h L h U ARL ARL 2 3.3 0..3030 3.6970 345.39 3.55 EWMA ( 25 (y) (x) 20 ) EWMA 5.5 4.5 3.5 h U = 4.4633 2.5 u 0 = 2.5.5 0.5 5 9 3 7 2 25 29 33 37 4 45 h L =.7867 EWMA 29 ( Z 29 = 4.205) 2.5 4.0 4 ( 29 25 = 4 ) 3.55( ARL2 = 3.55) K = 3.3 w = 0. ARL 782 h U =4.4633h L =.7867 ARL =782 (I 6
728 ) K ( w K=2.84w = 0.5 ARL 383.6 K=2.84w=0.5 ARL 2.5 5.36 ( 728 2.5 ) I 5 EWMA EWMA (w K) w K w Z t K EWMA w 0.05 0.25 w = 0.05 w = 0.0 w = 0.5w = 0.20 K K = 3 EWMA w w 0. K 3 2.6 2.8 (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 4 4 2.5 4.0 ARL K w h L h U ARL ARL 2 3.30 0.000.3030 3.6970 2.3940 345.39 3.55 3.25 0.0880.3976 3.6024 2.2048 345.0 3.65 3.20 0.0800.4672 3.5328 2.0656 343.34 3.74 3.5 0.072.543 3.4569.938 345.26 3.95 3.0 0.0620.6233 3.3767.7534 346.58 4.7 3.00 0.0500.7404 3.2596.592 333.39 4.46 7
(ARL, ARL 2 )(345.39, 3.55) ARL ARL 2 w K 0. 3.3 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 2 3 7-2 2 I EWMA Brook Evans RL 3 EWMA 8
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