STATISTICAL PROCESS STATISTICAL PROCESS CONTROL AND ITS CONTROL AND ITS

APPLICATION TO STEADY-APPLICATION TO STEADY-STATE SIMULATION DATASTATE SIMULATION DATA

SPCSPC

Born in the ’20’s Born in the ’20’s Walter A. ShewhartWalter A. ShewhartApplied to Manufacturing Processes with Applied to Manufacturing Processes with

Product Characteristics Measured at Product Characteristics Measured at IntervalsIntervals

Brought to Japan in the ’50’s by DemmingBrought to Japan in the ’50’s by Demming

SET UPSET UP A Process takes in perfect pieces of workA Process takes in perfect pieces of work Output is finished product with a characteristic Output is finished product with a characteristic

measured: X1, X2, X3,... iidmeasured: X1, X2, X3,... iid The Process starts off working correctly, but has The Process starts off working correctly, but has

a tendancy to go out-of-whack after a while, a tendancy to go out-of-whack after a while, producing different X’s.producing different X’s. Most common transition is to a different E[X] or a Most common transition is to a different E[X] or a

different VAR[X]different VAR[X] The goalsThe goals

Detect this transition as soon as it takes placeDetect this transition as soon as it takes place Don’t generate false alarmsDon’t generate false alarms

VENACULARVENACULAR

The Process is initially “in control”The Process is initially “in control”After the change, the process is “out of After the change, the process is “out of

control”control”Assumptions:Assumptions:

Data is iid NormalData is iid Normal

SIMPLEST CHARTSIMPLEST CHARTTHE X CHARTTHE X CHART

Estimate the E[X] and Estimate the E[X] and VAR[X] using the VAR[X] using the beginning of the data beginning of the data stream.stream.

Set: Set:

X

X

kXLCL

XCenter

kXUCL

ˆ

ˆ

PROCEDUREPROCEDURE

Baseline X’s to estimate E[X] and VAR[X]Baseline X’s to estimate E[X] and VAR[X]Begin sampling Xi’s Begin sampling Xi’s When Xi departs the control limits, declare When Xi departs the control limits, declare

OUT OF CONTROL OUT OF CONTROL Stop the process and investigateStop the process and investigate

Sequential Hypothesis Testing!Sequential Hypothesis Testing!

FALSE ALARMFALSE ALARM

is the P[type 1 error] = P[reject|H0]is the P[type 1 error] = P[reject|H0]= P[False alarm]= P[False alarm]

Using UCL = Using UCL = + 3 + 3Let p = Let p =

P[X> P[X> + 3 + 3 |E[X] and VAR[X] are true] |E[X] and VAR[X] are true]

=P[Z>3] where Z is a standard Normal=P[Z>3] where Z is a standard Normal

=0.001323=0.001323

...more FALSE ALARM...more FALSE ALARM

Expected number of samples before a Expected number of samples before a false alarm occurs is called...false alarm occurs is called...

ARL (Average Run Length)ARL (Average Run Length)ARL = 1/(2p) for symetric CL’sARL = 1/(2p) for symetric CL’sARL = 378 in the previous exampleARL = 378 in the previous example

ADDING RULESADDING RULES

Any set of rules can be used for detection of Any set of rules can be used for detection of OUT OF CONTROLOUT OF CONTROL

Balance sensitivity with P[false alarm]Balance sensitivity with P[false alarm] Western Electric Company RulesWestern Electric Company Rules

Any point outside 3sAny point outside 3s 2 out of the last 3 outside 2s2 out of the last 3 outside 2s 4 of the last 5 outside s4 of the last 5 outside s 8 on the same side of the center8 on the same side of the center

Increases sensitivity but reduces ARL to 92Increases sensitivity but reduces ARL to 92

OTHER CONTROL LIMIT OTHER CONTROL LIMIT SCHEMESSCHEMES

More sensitive than Shewhart with higher ARLMore sensitive than Shewhart with higher ARL Apply a “V-mask” on the trail of CUSUM Apply a “V-mask” on the trail of CUSUM

pointspoints V-mask dictates control limits and probability V-mask dictates control limits and probability

of false alarmof false alarm

n

iin XCUSUM

1

)(

CUSUMCUSUM

TYPICAL CUSUM CHARTTYPICAL CUSUM CHART

EXPONENTIALLY-WEIGHTED EXPONENTIALLY-WEIGHTED MOVING AVERAGEMOVING AVERAGE

)(2

)1(

22

1

k

XZZ

XZ

ttt

(k) is the autocorrelation of lag k

AR(1) PROCESSAR(1) PROCESS

Autoregressive Process, lag = 1Autoregressive Process, lag = 1Used to mimic all sorts of data without Used to mimic all sorts of data without

having the real process’s particularshaving the real process’s particularsControlled by Controlled by

)( 1tt XX

ZHANG’S PAPERZHANG’S PAPER

2500 runs of AR(1)2500 runs of AR(1)Compare ARL and sensitivity for...Compare ARL and sensitivity for...

basic X chartbasic X chartCUSUM chartCUSUM chartEWMA technique: EWMA technique: (k) all assumed 0 all assumed 0EWMAST technique: estimate first few EWMAST technique: estimate first few (k)

Comments on other methodsComments on other methods

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

ADDITIONAL COMMENTSADDITIONAL COMMENTS

EWMAST is the BEST!EWMAST is the BEST!EWMAST requires at least a 50-sample EWMAST requires at least a 50-sample

baseline for estimating baseline for estimating (k), 100 if possible, 100 if possibleRecommend Recommend =0.2 and a 3=0.2 and a 3 control chart control chart

Previous Zhang work appeared in Previous Zhang work appeared in Journal Journal of Applied Statsof Applied Stats and and Technometrics, Technometrics, both both solossolos