W6INSE6220 - 1 INSE 6220 Week 6 Advanced Statistical...

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Unformatted text preview: 1 INSE 6220 -- Week 6 Advanced Statistical Approaches to Quality Cumulative Sum (CUSUM) Control Charts Exponentially Weighted Moving Average (EWMA) Control Charts Midterm Review Dr. A. Ben Hamza Concordia University 2 Although Shewhart charts with 3 limits can quickly detect large process changes, they are ineffective for small, sustained process changes (for example, changes smaller than 1.5 ) Two alternative control charts have been developed to detect small changes: the CUSUM and EWMA control charts. They also can detect large process changes (for example, 3 shifts), but detection is usually somewhat slower than for Shewhart charts. CUSUM Control Chart The cumulative sum ( CUSUM ) is defined to be a running summation of the deviations of the plotted variable from its target. If the sample mean is plotted, the cumulative sum is where is the target for i j j i x C 1 ) ( x The Cumulative Sum (CUSUM) Control Chart 3 Interpretation of the CUSUM chart = , C i is a random walk with mean zero > , C i is an upward drift trend < , C i is a downward drift trend Remark: a trend of C i is an indication of the process mean shift. ) x ( C ) x ( ) x ( ) x ( C i 1 i i 1 i 1 j j i 1 j j i 4 1 ( 10) i i j j C x Using CUSUM 5 K is usually called the reference value A deviation from the target that is larger than K increases either C + or C- A control limit violation occurs when either C + or C- exceeds a specified control limit (or threshold ) H . After a limit violation occurs, that sum is reset to zero or to a specified value. and are accumulation of deviations above and below the target mean is called: one-sided upper cusum is called: one-sided lower cusum cusum control chart plots the values of and for each sample The Tabular CUSUM for Monitoring the Process Mean i C i C i C i C i C i C 1 1 1 max 0, ( ) max 0,( ) where the starting values ar : | | 2 2 i i i i i i C x K C C K x C e C C K 6 The Standardized CUSUM / and / H h K k Recommendations for CUSUM Design: where is the standard deviation of the sample variable used in forming the CUSUM. 7 Exponentially Weighted Moving Average (EWMA) Control Chart 1 1 (1 ) i i i z x z 2 2 2 1 1 (1 ) 2 (1 ) (1 ) i i z i j i i i j j z z x z 8 Steady-state control limits 5 10 15 20 25 30 9.9 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 3 26 27 UCL LCL CL Exponentially Weighted Moving Average (EWMA) Chart Sample Number EWMA L = width of control limits (given)...
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W6INSE6220 - 1 INSE 6220 Week 6 Advanced Statistical...

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