ISyE 512 hapter 9 USUM and EWMA ontrol harts Instructor: Prof. Kabo Lu Department of Industral and Systems Engneerng UW-Madson Emal: klu8@wsc.edu Offce: Room 317 (Mechancal Engneerng Buldng) ISyE 512 Instructor: Kabo Lu 1
Lst of topcs n hapter 9 USUM Motvaton What s t? How to determne the control lmts EWMA control chart MA hart ISyE 512 Instructor: Kabo Lu 2
Motvaton for USUM & EWMA ontrol harts To expedte detecton of a small mean shft n the process. Shewhart chart takes a long tme to detect a small mean shft (shft<1.5) only uses the nformaton about process contaned n the last plotted pont and gnores any nformaton gven by the entre sequence of ponts s not sutable for the sample wth a sngle observaton Shewhart chart wth other supplemental senstzng rules can ncrease detecton senstvty but reduce smplcty and ease of nterpretaton of the Shewhart control chart and ncrease type I error (sometmes dramatcally) ISyE 512 Instructor: Kabo Lu 3
An Motvaton Example onsder the followng data. Ths process has a mean ncrease snce sample 21, do you see t? ISyE 512 Instructor: Kabo Lu 4
What f ( X j j1 1) ISyE 512 Instructor: Kabo Lu 5
What s USUM hart? The USUM chart was frst proposed by Page (1954). USUM chart: drectly ncorporates all the nformaton n the sequence of sample values by plottng the cumulatve sums (USUM) of devatons of the sample values from a target value x j (x j ) j1 : the average of the jth sample : the target for the process mean : the cumulatve sum up to and ncludng the th sample n1: cusum could be constructed for ndvdual observatons ISyE 512 Instructor: Kabo Lu 6
Interpretaton of the USUM chart 1 (x j ) (x j ) (x ) 1 (x ) j1 j1 =, s a random walk wth mean zero >, s an upward drft trend <, s a downward drft trend Remark: a trend of s an ndcaton of the process mean shft. ISyE 512 Instructor: Kabo Lu 7
How to onstruct a USUM ontrol hart? Montor the mean of a process : Tabular (algorthmc) cusum (preferable way) V-mask form of cusum usum can be constructed for both ndvdual observatons and for the averages of ratonal subgroups. For ndvdual observaton: j1 x x (x j ) 1 (x ) ISyE 512 Instructor: Kabo Lu 8
onstruct a USUM ontrol hart Tabular USUM max[, max[, x ( ( K) x K) 1 1 ] ] Statstc / : one sde upper/lower cusum / : accumulate devatons from o that are greater than K, wth both quanttes reset to zero upon becomng negatve K: reference value (allowance or slack value) Often chosen about halfway between the target o and the out-of-control value of the mean 1 that we are nterested n detectng quckly Decson Rules: If ether or exceeds the decson nterval H (A common choce H=5), the process s consdered to be out of control 1 1 1 K 2 2 ISyE 512 Instructor: Kabo Lu 9
ISyE 512 Instructor: Kabo Lu 1
ISyE 512 Instructor: Kabo Lu 11
usum Status hart (Fgure 9.3a) ISyE 512 Instructor: Kabo Lu 12
Procedures for onstructon of USUM Select K and H onstruct one sde upper and lower cusum and then represent n the two separate columns of the table alculate x -( +K) and -K- x alculate the accumulatve devatons + and - ount the number of consecutve perods that the cusum + or - have been nonzero, whch are ndcated by N + and N - respectvely ISyE 512 Instructor: Kabo Lu 13
Interpretaton of USUM Fnd the data pont out at whch or exceeds the decson nterval H (H=5) If the out-of-control data corresponds to an assgnable cause, then to determne the locaton of the last n-control data n = out - N + out or n = out - N - out where N + out and N - out correspond to N + and N - at data pont out Estmate the new process mean ˆ K N K N out out H H Plot a USUM status chart for vsualzaton f f however, the other senstzng rules cannot be safely used for the USUM chart because and are not ndependent ISyE 512 Instructor: Kabo Lu 14
Desgn of USUM Based on ARL The reference value of K and the decson nterval H have an effect on ARL and ARL 1 k=.5 (K=k): to mnmze the ARL 1 value for fxed ARL choose h (H=h): to obtan the desred n-control ARL performance ARL ARL 1 Not too senstve Shewhart chart ARL 1 =43.96 ISyE 512 Instructor: Kabo Lu 15
ARL of the USUM hart The most wdely used ARL s the Segmund's approxmaton. ISyE 512 Instructor: Kabo Lu 16
A numercal example (1) Try h = 3.25 ARL s not large enough so we wll ncrease h a lttle and try agan. ISyE 512 Instructor: Kabo Lu 17
ISyE 512 Instructor: Kabo Lu Standardzed USUM Advantage of a standardzed cusum: does not depend on. So, many cusum charts can now have the same values of k and h naturally represent the process varablty ; ] y k max[, ] k y max[, ; x y 1 1 18
Improvement of USUM Fast Intal Response (FIR): set If a shft occurred at the begnnng, FIR can detect the shft more quckly to decrease ARL 1 If n control at the begnnng, cusum wll quckly drop to zero, lttle effect on the performance Example: =1, K=3, H=12, 5% headstart value In-control data =1 1 =15 H / 2 6 ISyE 512 Instructor: Kabo Lu 19
USUM for Montorng Process Varablty reate a new standardzed quantty (Hawkns, 1981,1993), whch s senstve to varance changes. In-control dstrbuton of s approxmately N(,1) S S y.822 ; ~.349 max[, k S 1] max[, k S 1 N(,1); ] y x S S ; Selecton of h and k and the nterpretaton of cusum are smlar to the cusum for controllng the process mean ISyE 512 Instructor: Kabo Lu 2
EXPONENTIALLY WEIGHTED MOVING AVERAGE (EWMA) To exponentally forget the past data, we want to attach more weght to the most recent data It s a weghted average: a geometrc seres of weghts Z X (1 )Z 1 ; Z X <1-1 and (1-) j gettng smaller when j ncrease Z 1 j (1 ) j X j (1 ) <1, Z = Z j= j=1 j=1 X 1 ) 1 ( X 1 1 (1 ) X More weght on most recent data 1 Assume x t are ndependent random varables, wth E(x t )=, Var(x t )= 2 ISyE 512 Instructor: Kabo Lu 21
ISyE 512 Instructor: Kabo Lu 22
How to onstruct a EWMA hart? Assume x t are ndependent random varables, wth E(x Var (Z t ) = 2 n 2 ( 1 (1 ) 2t) t )=, Var(x t )= 2 t As t becomes large: Var (Z t ) = 2 n 2 Note: for =1 we have Shewhart hart. In general, 2t UL Zt = + L [1 (1 ) ] (2 )n L= 2t LL Zt = L [1 (1 ) ] (2 )n Steady-state control lmts UL L LL L (2 ) n (2 ) n Note : Ths s the general equatons for ratonal subgroups (sample) of observatons (n>1). For ndvdual samples, n=1. ISyE 512 Instructor: Kabo Lu 23
ISyE 512 Instructor: Kabo Lu 24
ISyE 512 Instructor: Kabo Lu 25
ISyE 512 Instructor: Kabo Lu 26
Desgn of the EWMA ISyE 512 Instructor: Kabo Lu 27
Performance of EWMA ontrol hart ompared to Shewhart chart and USUM chart, EWMA chart s effectve on detecton of small mean shfts lke USUM, less effectve on larger shft detecton than the Shewhart chart, but generally superor to the USUM chart (partcularly f >.1) EWMA s very nsenstve to the normalty assumpton. So, t s an deal control chart for ndvdual observatons. ISyE 512 Instructor: Kabo Lu 28
EWMA for Montorng Process Varablty The exponentally weghted mean square error (EWMS) 2 2 2 2 S (x ) (1 ) S 1 x ~ N(, ) For larger and ndependent observaton, t follows the chsquare dstrbuton wth (2- )/ degrees of freedom and onstruct exponentally weghted root mean square (EWRMS) 2 control chart to check data pont n-control or not. E(S ) UL LL, / 2 To be nsenstve to process mean change, t s suggested to replace wth z (a predcton of x +1 ) xˆ 1,1 / 2 z 2 S 2 ISyE 512 Instructor: Kabo Lu 29
Example: An EWMA control chart uses =.4. How wde wll the lmts be on the x-bar control chart (n sgma-unts), expressed as a multple of the wdth of the steady-state EWMA lmts (n sgma-unts)? ISyE 512 Instructor: Kabo Lu 3
USUM vs. EWMA Smlarty between USUM & EWMA To expedte detecton of a small mean shft n the process Drectly ncorporates ALL the nformaton n the sequence of sample values (past data) Dfference between USUM & EWMA USUM: all the past data are EQUALLY mportant EWMA: exponentally forget the past data, we want to attach more weght to the most recent data (more emphasze on the sequence) ISyE 512 Instructor: Kabo Lu
Movng Average ontrol harts Sample Statstc x Dfferent from EWMA, use an unweghted movng average M t = ( X t + X t-1 + X t-2 +...+ X t-w+1 ) / w Ths wndow of sze w, ncorporates some of the memory of the past data nformaton by droppng the oldest data and addng the newest data 1.7 1.6 1.5 1.4 1.3 w=4 UL= L= LL= k k n n 1 Tme (hour) 2 ISyE 512 Instructor: Kabo Lu 32
onstruct Movng Average ontrol harts The movng averatge can be wrtten recursvely as, M t = M t-1 + ( X t X t-w )/w w = wndow sze, n = sample sze 1 t Var (M t ) = w 2 Var( X ) = 2 nw =t-w+1 UL = + 3 Mt nw At the begnnng, f <w LL Mt = - 3 nw 3 n ISyE 512 Instructor: Kabo Lu 33
ISyE 512 Instructor: Kabo Lu 34
ISyE 512 Instructor: Kabo Lu 35