lecture9

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MIT OpenCourseWare ____________ http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms .

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1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #9 Advanced and Multivariate SPC March 6, 2008
2 M anufacturing Agenda • Conventional Control Charts – Xbar and S • Alternative Control Charts – Moving average –EWMA – CUSUM • Multivariate SPC

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3 M anufacturing Xbar Chart Process Model: x ~ N(5,1), n = 9 • Is process in control? 4.0 4.5 5.0 5.5 6.0 Mean of x 1 2 3 4 5 6 7 8 9 10 11 12 13 Sample μ0=5.000 LCL=4.000 UCL=6.000
4 M anufacturing Run Data (n=9 sample size) 0 1 3 4 5 7 8 9 11 Run Chart of x 0 9 18 27 36 45 54 63 72 81 90 99 Run Avg=4.91 • Is process in control?

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5 M anufacturing S Chart 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Standard Deviation of x 1 2 3 4 5 6 7 8 9 10 11 12 13 Sample μ0=0.969 LCL=0.232 UCL=1.707 • Is process in control?
6 M anufacturing n measurements at sample j x Rj = 1 n x i i = j j + n S 2 = 1 n 1 ( x i i = j j + n x Rj ) 2 Running Average Running Variance • More averages/Data • Can use run data alone and average for S only • Can use to improve resolution of mean shift Alternative Charts: Running Averages

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7 M anufacturing Simplest Case: Moving Average • Pick window size (e.g., w = 9) 1 2 3 4 5 6 7 8 9 Moving Avg of x 8 16 24 32 40 48 56 64 72 80 88 96 Sample μ0=5.00 LCL UCL
8 M anufacturing y j = a 1 x j 1 + a 2 x j 2 + a 3 x j 3 + ... General Case: Weighted Averages • How should we weight measurements? – All equally? (as with Moving Average) – Based on how recent? • e.g. Most recent are more relevant than less recent?

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9 M anufacturing 0 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Consider an Exponential Weighted Average Define a weighting function W t i = r (1 r ) i Exponential Weights
10 M anufacturing Exponentially Weighted Moving Average: (EWMA) A i = rx i + (1 r ) A i 1 Recursive EWMA UCL , LCL = x ± 3 σ A A = x 2 n r 2 r 1 1 r () 2 t [] time A = x 2 n r 2 r for large t

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11 M anufacturing Effect of r on σ multiplier 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 plot of (r/(2-r)) vs. r wider control limits r
12 M anufacturing SO WHAT? • The variance will be less than with xbar, • n=1 case is valid • If r=1 we have “unfiltered” data – Run data stays run data – Sequential averages remain • If r<<1 we get long weighting and long delays – “Stronger” filter; longer response time σ A = x n r 2 r = σ x r 2 r

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13 M anufacturing EWMA vs. Xbar 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 300 xbar EWMA UCL EWMA LCL EWMA grand mean UCL LCL r=0.3 Δμ = 0.5 σ
14 M anufacturing Mean Shift Sensitivity EWMA and Xbar comparison 0 0.2 0.4 0.6 0.8 1 1.2 1 5 9 1 31 72 12 52 93 33 74 14 54 9 xbar EWMA UCL EWMA LCL EWMA Grand Mean UCL LCL 3/6/03 Mean shift = .5 σ r=0.1 n=5

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15 M anufacturing Effect of r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 xbar EWMA UCL EWMA LCL EWMA Grand Mean UCL LCL r=0.3
16 M anufacturing

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## This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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lecture9 - http:/ocw.mit.edu _ MIT OpenCourseWare Spring...

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