ch 9 CUSUM and EWMA

# ch 9 CUSUM and EWMA - 321:QualityControl Chapter9...

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1 321:  Quality Control Chapter 9 Cumulative Sum and Exponentially Weighted  Moving Average Control Charts Instructor :  Linda Boyle Industrial and Systems Engineering  University of Washington

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2 Introduction   To   detect large abrupt shifts (1.5 σ  to 2 σ  shifts  or larger) Shewhart Control Charts. To detect small shifts CUSUM: CUSUM:  cumulative sum  control chart  EWMA: EWMA:  exponentially weighted moving  average
3 Example Process mean =10 std dev=1 Process mean =11 std dev=1 Subgroup x Subgroup x 1 9.5   9 11.2 2 7.0 10 10.3 3 9.3 11 11.1 4 11.7 12 10.4 5 12.2 13 11.6 6 10.2 14 11.9 7 8.0 15 10.9 8 11.5

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4 Example 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 UCL(new) UCL(orig) CL(new) CL(orig) LCL(new) LCL(orig)
5 CUSUM chart CUSUM chart Incorporates all information in the sequence  of sample values  by plotting the  cumulative sums  of the deviations  of the sample values from a target value. Where C i  = cumulative sum up to and including  the i th  sample and n> 1 μ - = = i 1 j 0 j i ) x ( C

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6 CUSUM Using previous  data Where  μ o =10 μ - = = i 1 j 0 j i ) x ( C Subgroup x x 1 9.5 9 11.2 2 7.0 10 10.3 3 9.3 11 11.1 4 11.7 12 10.4 5 12.2 13 11.6 6 10.2 14 11.9 7 8.0 15 10.9 8 11.5 Board discussion
7 CUSUM Subgroup x i C i Subgroup x i C i 1 9.5 -0.5 9 11.2 0.6 2 7.0 -3.5 10 10.3 0.9 3 9.3 -4.2 11 11.1 2.0 4 11.7 -2.5 12 10.4 2.4 5 12.2 -0.3 13 11.6 4.0 6 10.2 -0.1 14 11.9 5.9 7 8.0 -2.1 15 10.9 6.8 8 11.5 -0.6 Plot C i

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8 CUSUM -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Sample number C i μ = 10 μ = 11
9 CUSUM Plot Is not a control chart Lacks statistical control limits Two types of CUSUM representation Tabular cusum (preferred method) Can add Fast Initial Response (FIR) or Headstart V-mask Error probability consideration,  α β The book strongly advises against using it (see pg  416)

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10 Tabular Cusum Let x i  be the i th  observation on the process If the process is in control then x i  has a  normal distribution Assume  σ  is known or can be estimated Upper cusum: Lower cusum: C i + = max 0, x i - ( m 0 + K ) + C i - 1 + [ ] C i - = max 0,( m 0 - K ) - x i + C i - 1 - [ ]
11 Tabular Cusum Upper Cusum The accumulated change from the target  μ 0   above  the target with one statistic, C + Lower Cusum The accumulated change from the target  μ 0   below  the target with another statistic, C

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12 Tabular Cusum starting values are If either statistic exceed a decision interval  H , the process is  considered to be out of control. Often taken as  H  = 5 σ C i + = max 0, x i - ( m 0 + K ) + C i - 1 + [ ] C i - = max 0,( m 0 - K ) - x i + C i - 1 - [ ] 0 0 0 = = - + C C Where K = reference value (or allowance or slack value )
13 Tabular Cusum   Selecting  K K  is often chosen as halfway between target  μ 0  and,  out-of-control value we are interested in detecting (mean  μ 1 Shift is expressed in standard deviation units as  μ 1 = μ 0 +δσ , then  K  is: 2 2 K 0 1 μ - μ = σ δ =

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14 Tabular Cusum Example   Subgroup
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ch 9 CUSUM and EWMA - 321:QualityControl Chapter9...

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