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

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

6. SIMPLEST CHART THE X CHART
Estimate the E[X] and VAR[X] using the beginning of the data stream.
Set:

7. 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!

8. 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 =

9. ...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

10. 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

11. 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

12. CUSUM

13. TYPICAL CUSUM CHART

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

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

17. 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

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

21. 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