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STATISTICAL PROCESS CONTROL AND ITS APPLICATION TO STEADY-STATE SIMULATION DATA

STATISTICAL PROCESS CONTROL AND ITS APPLICATION TO STEADY-STATE SIMULATION DATA. SPC. Born in the ’20’s Walter A. Shewhart Applied to Manufacturing Processes with Product Characteristics Measured at Intervals Brought to Japan in the ’50’s by Demming. SET UP.

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STATISTICAL PROCESS CONTROL AND ITS APPLICATION TO STEADY-STATE SIMULATION DATA

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  1. STATISTICAL PROCESS CONTROL AND ITS APPLICATION TO STEADY-STATE SIMULATION DATA

  2. SPC • Born in the ’20’s • Walter A. Shewhart • Applied to Manufacturing Processes with Product Characteristics Measured at Intervals • Brought to Japan in the ’50’s by Demming

  3. SET UP • A Process takes in perfect pieces of work • Output is finished product with a characteristic measured: X1, X2, X3,... iid • The Process starts off working correctly, but has a tendancy to go out-of-whack after a while, producing different X’s. • Most common transition is to a different E[X] or a different VAR[X] • The goals • Detect this transition as soon as it takes place • Don’t generate false alarms

  4. VENACULAR • The Process is initially “in control” • After the change, the process is “out of control” • Assumptions: • Data is iid Normal

  5. SIMPLEST CHARTTHE X CHART • Estimate the E[X] and VAR[X] using the beginning of the data stream. • Set:

  6. PROCEDURE • Baseline X’s to estimate E[X] and VAR[X] • Begin sampling Xi’s • When Xi departs the control limits, declare OUT OF CONTROL • Stop the process and investigate • Sequential Hypothesis Testing!

  7. FALSE ALARM • a is the P[type 1 error] = P[reject|H0] • = P[False alarm] • Using UCL = m + 3s... Let p = P[X> m + 3s |E[X] and VAR[X] are true] =P[Z>3] where Z is a standard Normal =0.001323

  8. ...more FALSE ALARM • Expected number of samples before a false alarm occurs is called... • ARL (Average Run Length) • ARL = 1/(2p) for symetric CL’s • ARL = 378 in the previous example

  9. ADDING RULES • Any set of rules can be used for detection of OUT OF CONTROL • Balance sensitivity with P[false alarm] • Western Electric Company Rules • Any point outside 3s • 2 out of the last 3 outside 2s • 4 of the last 5 outside s • 8 on the same side of the center • Increases sensitivity but reduces ARL to 92

  10. OTHER CONTROL LIMIT SCHEMES • More sensitive than Shewhart with higher ARL • Apply a “V-mask” on the trail of CUSUM points • V-mask dictates control limits and probability of false alarm

  11. CUSUM

  12. TYPICAL CUSUM CHART

  13. EXPONENTIALLY-WEIGHTED MOVING AVERAGE r(k) is the autocorrelation of lag k

  14. AR(1) PROCESS • Autoregressive Process, lag = 1 • Used to mimic all sorts of data without having the real process’s particulars • Controlled by f

  15. ZHANG’S PAPER • 2500 runs of AR(1) • Compare ARL and sensitivity for... • basic X chart • CUSUM chart • EWMA technique: r(k) all assumed 0 • EWMAST technique: estimate first few r(k) • Comments on other methods

  16. what is the desired behavior of a superior method of detection?

  17. ADDITIONAL COMMENTS • EWMAST is the BEST! • EWMAST requires at least a 50-sample baseline for estimating r(k), 100 if possible • Recommend l=0.2 and a 3s control chart • Previous Zhang work appeared in Journal of Applied Stats and Technometrics, both solos

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