Alternatives to Shewhart Charts

Transcription

1 Alternatves to Shewhart Charts CUSUM & EWMA S Wongsa

2 Overvew Revstng Shewhart Control Charts Cumulatve Sum (CUSUM) Control Chart Eponentally Weghted Movng Average (EWMA) Control Chart 2

3 Revstng Shewhart Control Charts Eample of Process Varatons Chance/Random causes (natural varablty) Systematc/Assgnable causes (there s a cause) Source: MVE45 - Statstcal qualty control 3

4 Revstng Shewhart Control Charts Hypothess Test & Errors H H 0 Type II error Type I error µ = µ µ 0 = µ µ 0 Type I error: H 0 s rejected when t s true,.e. rejectng the qualfed product. H 0 Type II error: s not rejected when t s false,.e. acceptng nonconformng product. 4

5 Revstng Shewhart Control Charts Hypothess Test & Errors H H 0 Type II error Type I error µ = µ µ 0 = µ µ 0 α = P(Type I error) = P(reject H0 H0 s true) false alarm rate β = P(Type II error) = P(fal to reject H0 H0 s false) mssed alarm rate 5

6 Revstng Shewhart Control Charts A Typcal Control Chart Sample number or tme 6

7 Revstng Shewhart Control Charts Control Lmts µ and σ are known: ~ 2 N ( µ, σ ) 2 ~ N ( µ, σ ) σ = σ n UCL =µ+ Lσ LCL =µ Lσ The popular lmt: L = 3 (.e. the three-sgma lmts), 99.7% of the observatons of a stable process are n-control. 7

8 Revstng Shewhart Control Charts But µ and σ are not known and need to be estmated m = ˆµ = = m n j= = n, j ˆ σ = ˆ σ = R d 2 ˆ σ n or ˆ σ = s c 4 m R = R= m R = ma(, j ) mn(, j ), j=,... n UCL LCL = = + 3σˆ 3σˆ s m s = = m See Append VI of D.C. Montgomery, Introducton to Statstcal Qualty Control for constants d 2,c 4. s = n n j= (, j ) 2 8

9 Revstng Shewhart Control Charts 9

10 Revstng Shewhart Control Charts Average Run Length ARL s a performance measure that s frequently used to evaluate the performance of the control charts. ARL s the average number of ponts that must be plotted before a pont ndcates an out-of-control condton. ARL can be calculated from ARL= p where p s the probablty that any pont eceeds the control lmts. In control ARL: the average run when the process s n control ARL 0 = α Out of control ARL: to detect shft when t actually occurs ARL = β 0

11 Revstng Shewhart Control Charts Average Run Length For the X chart wth 3-sgma lmts, p= s the probablty that a sngle pont falls outsde the lmts when the process s n control. In-control ARL = /0.0027=370.37

12 Revstng Shewhart Control Charts Average Run Length Assume that a σ ncrease n the mean occurs when an X chart s used For the UCL z= For the LCL z= [ µ + 3σ ( µ + σ )] σ [ µ 3σ ( µ + σ )] σ = 2 =4 ARL = P( z > 2) + P( z <4) =

13 Revstng Shewhart Control Charts Shewhart Control Charts Detect large shfts. Not senstve for small shfts, say on the order of about.5σ or less. Sample number or tme 3

14 Alternatve Charts How to ncrease senstvty to small, persstent mean shft? Runnng ntegraton of the mean shft CUSUM (The Cumulatve Sum Control Chart) Weghted average of multple observatons EWMA (The Eponentally Weghted Movng Average Control Chart) 4

15 The CUSUM Control Chart for Montorng the Process Mean CUSUM (The Cumulatve Sum Control Chart) Add up devatons from mean j C j = ( µ 0 ) = Snce E{-µ}=0, ths sum should stay near zero when n control Any bas,.e. mean shft, n wll show as a trend. 5

16 The CUSUM Control Chart for Montorng the Process Mean Eample I: A small shft Shewhart Control Chart ~ N (0,) ~ N (,) Source: D.C. Montgomery, Statstcal Qualty Control: A Modern Introducton, 6 th Edton. 6

17 The CUSUM Control Chart for Montorng the Process Mean Eample I: A small shft CUSUM Cumulatve Sum ~ N (0,) ncorporates all the nformaton n the sequence of sample values by plottng the cumulatve sums of the devatons of the sample values from a target value. C = = j= j= ( j ( = j 0) 0) + ( C + ( 0) 0) ~ N (,) Source: D.C. Montgomery, Statstcal Qualty Control: A Modern Introducton, 6 th Edton. 7

18 The CUSUM Control Chart for Montorng the Process Mean The Standardsed CUSUM The varable s standardsed before performng the calculatons. z = µ 0 σ µ 0 s the target value for the qualty characterstc. Detecton of mean ncreases: C + + = ma( 0, C + ( z k)); C0 + = Detecton of mean decreases: 0 k s usually selected to be onehalf of the mean shft (n z unts) that one wshes to detect quckly. h s typcally chosen to be 4 or 5. C = ma( 0, C ( z+ k)); C0 = 0 + Alarm f C > h or C > h k slack varable h decson threshold 8

19 The CUSUM Control Chart for Montorng the Process Mean Eample II: CUSUM Versus Chart X Snce σ =, σ = / 4 = 0.5, so the shft s a -sgma shft. Source: T.M. Ryan, Statstcal Methods for Qualty Improvement. 9

20 The CUSUM Control Chart for Montorng the Process Mean Eample II: CUSUM Versus Chart X Source: Adapted from T.M. Ryan, Statstcal Methods for Qualty Improvement. 20

21 The CUSUM Control Chart for Montorng the Process Mean h, k and ARL for CUSUM X Chart Source: T.M. Ryan, Statstcal Methods for Qualty Improvement. 2

22 The Eponentally Weghted Movng Average Control Chart EWMA Control Chart for Montorng the Process Mean EWMA s defned as w = + ( ) w where 0< s the weght gven to the most recent observaton and the startng value ( 0) s the process target ( µ ). w 0 It can be shown that w j = ) j j= 0 ( + ( ) w 0 22

23 The Eponentally Weghted Movng Average Control Chart EWMA Control Chart for Montorng the Process Mean w j = ) j j= 0 ( + ( ) w j The weghts, whch sum to one ( ( ) + ( ) = ), decrease geometrcally wth the age of the sample j= 0 0 Weghts ( Eponental weghts j ) : weght of sample j =0.2 hghest weght s assgned to the most recent observatons Weghts decreasng eponentally for less recent observatons j 23

24 The Eponentally Weghted Movng Average Control Chart EWMA Control Lmts 2 If the observatons are ndependent random varables wth varance σ, then the varance of s w σ = σ 2 [ ( ) ] w 2 If 0.02, ( ) wll be close to zero f 5, so could then be appromated by. Constant control lmts can be obtaned usng σ σ w UCL CL LCL = µ + = 0 µ 0 Lσ w = µ 0 Lσ w 24

25 The Eponentally Weghted Movng Average Control Chart Effect of - Control lmts σ 2 w σ Wder control lmts

26 The Eponentally Weghted Movng Average Control Chart Effect of - Flterng w = + ( ) w If = we have unfltered data. If <<, we get long weghtng and long delays Stronger flter & longer response tme 26

27 The Eponentally Weghted Movng Average Control Chart Effect of - Flterng.5 Mean shft = 0.5σ = σ n=4, = EWMA Xbar Subgroup ~ N (0,) ~ N (0.5,) 27

28 The Eponentally Weghted Movng Average Control Chart Effect of - Flterng.5 Mean shft = 0.5σ = σ n=4, = EWMA Xbar Subgroup ~ N (0,) ~ N (0.5,) 28

29 The Eponentally Weghted Movng Average Control Chart Eample III: EWMA versus CUSUM Lucas & Saccucc (990) suggested that the one reasonable combnaton of L and would be L=3.00 and =0.25. Ths combnaton would provde an n-control ARL of and ARL of 0.95 for detectng the -sgma shft. These are very close to the correspondng values for a CUSUM procedure wth h=5 and k=0.5. Source: T.M. Ryan, Statstcal Methods for Qualty Improvement. 29

30 The Eponentally Weghted Movng Average Control Chart Eample III: EWMA versus CUSUM [ ] n 2 ) ( 2 ˆ 3 σ ± Eact Control Lmts L=3 Source: T.M. Ryan, Statstcal Methods for Qualty Improvement. Appro. Control Lmts σ ± 2 ˆ 3 n 30

31 The Eponentally Weghted Movng Average Control Chart Eample III: EWMA versus CUSUM L= UCL 0.4 EWMA Subgroup LCL Source: T.M. Ryan, Statstcal Methods for Qualty Improvement. 3

32 The Eponentally Weghted Movng Average Control Chart Desgn of an EWMA Control Chart Choose L andfor wanted ARL NB: - Specfyng the desred n-control and out-of-control ARL and the magntude of the process shft that s antcpated, and then select the combnaton of and L that provde the desred ARL performance. - A good rule of thumb s to use smaller values of to detect smaller shfts. Source: D.C. Montgomery, Statstcal Qualty Control: A Modern Introducton, 6 th Edton. 32

33 Conclusons Remarks The amount of requred computaton s about the same for CUSUM and EWMA. CUSUM and EWMA charts do not match the ablty of Shewhart control charts to detect larger shft. The combned Shewhart-CUSUM scheme could be consdered to make the CUSUM and EWMA procedures to have good propertes for detectng both small and large shfts. To acheve a small ARL for mean shfts, fast ntal response (FIR) CUSUM s recommended. 33

34 Conclusons Recommended Readng 34

35 Supplementary MATLAB command to fnd tal area -normcdf(z) 35

36 Supplementary Weghts of EWMA [ ] ) ( ) ( ) ( ) ( ) ( + + = + + = + = w w w w Contnung to substtute recursvely for,..., = 2,3 j j w ) ( ) ( w w j j j + = =