MONITORING THE COEFFICIENT OF VARIATION USING EWMA ... - PUC · MONITORING THE COEFFICIENT OF...

MONITORING THE COEFFICIENT OFVARIATION USING EWMA CHARTS

Philippe CASTAGLIOLA 1, Giovanni CELANO 2, Stelios PSARAKIS 3

1Universite de Nantes & IRCCyN UMR CNRS 6597, France2Universita di Catania, Catania, Italy

3Athens University of Economics and Business, Athens, Greece

ISSPC 2011, July 13–14, Rio de Janeiro, Brazil

Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHARTS

Coefficient of variation



Definition



Definition

If X > 0 is a random variable with mean µ and standard-deviation σ, bydefinition the coefficient of variation γ is defined as

γ =σ

µ



Definition


γ =σ

µ

Used to compare data sets having different units or widely different means(ex : finance, chemical and biological assays, materials engineering).



Definition


γ =σ

µ


Sample coefficient of variation



Definition


γ =σ

µ



If {X1, . . . ,Xn} is a sample of n normal i.i.d. (µ, σ) random variablesthen a “natural” estimator of γ is

γ =S

X



Definition


γ =σ

µ




γ =S

X

where



Definition


γ =σ

µ




γ =S

X

where X =1

n

n∑

i=1

Xi



Definition


γ =σ

µ




γ =S

X

where X =1

n

n∑

i=1

Xi and S =

√

√

√

√

1

n − 1

n∑

i=1

(Xi − X )2


Basic properties of the sample coefficient of variation



c.d.f and inverse c.d.f. of γ




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)

F−1γ (α|n, γ) =

√n

F−1t

(

1 − α∣

∣

∣n − 1,√

n

γ

)




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)

F−1γ (α|n, γ) =

√n

F−1t

(

1 − α∣

∣

∣n − 1,√

n

γ

)

where Ft(.) and F−1t (.) are the c.d.f. and the inverse c.d.f. of the

noncentral t distribution.




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)

F−1γ (α|n, γ) =

√n

F−1t

(

1 − α∣

∣

∣n − 1,√

n

γ

)



c.d.f and inverse c.d.f. of γ2




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)

F−1γ (α|n, γ) =

√n

F−1t

(

1 − α∣

∣

∣n − 1,√

n

γ

)




Fγ2(x |n, γ) = 1 − FF

(

n

x

∣

∣

∣ 1, n − 1,n

γ2

)




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)

F−1γ (α|n, γ) =

√n

F−1t

(

1 − α∣

∣

∣n − 1,√

n

γ

)




Fγ2(x |n, γ) = 1 − FF

(

n

x

∣

∣

∣ 1, n − 1,n

γ2

)

F−1γ2 (α|n, γ) =

n

F−1F

(

1 − α∣

∣

∣1, n − 1, nγ2

)




Fγ(x |n, γ) = 1 − Ft

(√n

x

∣

∣

∣

∣

n − 1,

√n

γ

)

F−1γ (α|n, γ) =

√n

F−1t

(

1 − α∣

∣

∣n − 1,√

n

γ

)




Fγ2(x |n, γ) = 1 − FF

(

n

x

∣

∣

∣ 1, n − 1,n

γ2

)

F−1γ2 (α|n, γ) =

n

F−1F

(

1 − α∣

∣

∣1, n − 1, nγ2

)

where FF (.) and F−1F (.) are the c.d.f. and the inverse c.d.f. of the

noncentral F distribution.


SH-γ chart (Kang et al., JQT 2007)



General assumptions



General assumptions

subgroups {Xk,1,Xk,2, . . . ,Xk,n} of size n are observed at timek = 1, 2, . . ..



General assumptions


Xk,j ∼ N(µk , σk).



General assumptions



from one subgroup to another, µk and σk may change, but they areconstrained by the relation γk = σk

µk= γ0 when the process is

in-control.



General assumptions





in-control.

Control limits



General assumptions





in-control.

Control limits

LCLSH = F−1γ

(

α0

2 |n, γ0

)

UCLSH = F−1γ

(

1 − α0

2 |n, γ0

)



General assumptions





in-control.

Control limits

LCLSH = F−1γ

(

α0

2 |n, γ0

)

UCLSH = F−1γ

(

1 − α0

2 |n, γ0

)

where α0 is the type I error rate (ex : α0 = 0.0027).



General assumptions





in-control.

Control limits

LCLSH = F−1γ

(

α0

2 |n, γ0

)

UCLSH = F−1γ

(

1 − α0

2 |n, γ0

)


Comments



General assumptions





in-control.

Control limits

LCLSH = F−1γ

(

α0

2 |n, γ0

)

UCLSH = F−1γ

(

1 − α0

2 |n, γ0

)


Comments

Simple two-sided “Shewhart-type” control chart.



General assumptions





in-control.

Control limits

LCLSH = F−1γ

(

α0

2 |n, γ0

)

UCLSH = F−1γ

(

1 − α0

2 |n, γ0

)


Comments

Simple two-sided “Shewhart-type” control chart.

Unefficient for detecting small change in γ.


EWMA-γ chart (Hong et al., JSKISE 2008)


EWMA-γ chart (Hong et al., JSKISE 2008)Monitored statistic



Zk = (1 − λ)Zk−1 + λγk



Zk = (1 − λ)Zk−1 + λγk

Control limits



Zk = (1 − λ)Zk−1 + λγk

Control limits

LCLEWMA−γ = µ0(γ) − K

√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)

UCLEWMA−γ = µ0(γ) + K

√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)



Zk = (1 − λ)Zk−1 + λγk

Control limits


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)

Approximations for µ0(γ) and σ0(γ)

µ0(γ) ≃ γ0

(

1 +1

n

(

γ20 −

1

4

)

+1

n2

(

3γ40 −

γ20

4−

7

32

)

+1

n3

(

15γ60 −

3γ40

4−

7γ20

32−

19

128

))

σ0(γ) ≃ γ0

√

1

n

(

γ20 +

1

2

)

+1

n2

(

8γ40 + γ2

0 +3

8

)

+1

n3

(

69γ60 +

7γ40

2+

3γ20

4+

3

16

)



Zk = (1 − λ)Zk−1 + λγk

Control limits


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


µ0(γ) ≃ γ0

(

1 +1

n

(

γ20 −

1

4

)

+1

n2

(

3γ40 −

γ20

4−

7

32

)

+1

n3

(

15γ60 −

3γ40

4−

7γ20

32−

19

128

))

σ0(γ) ≃ γ0

√

1

n

(

γ20 +

1

2

)

+1

n2

(

8γ40 + γ2

0 +3

8

)

+1

n3

(

69γ60 +

7γ40

2+

3γ20

4+

3

16

)

Comments



Zk = (1 − λ)Zk−1 + λγk

Control limits


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


µ0(γ) ≃ γ0

(

1 +1

n

(

γ20 −

1

4

)

+1

n2

(

3γ40 −

γ20

4−

7

32

)

+1

n3

(

15γ60 −

3γ40

4−

7γ20

32−

19

128

))

σ0(γ) ≃ γ0

√

1

n

(

γ20 +

1

2

)

+1

n2

(

8γ40 + γ2

0 +3

8

)

+1

n3

(

69γ60 +

7γ40

2+

3γ20

4+

3

16

)

Comments

More efficient than the SH-γ chart.



Zk = (1 − λ)Zk−1 + λγk

Control limits


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


√

λ(1 − (1 − λ)2k )

2 − λσ0(γ)


µ0(γ) ≃ γ0

(

1 +1

n

(

γ20 −

1

4

)

+1

n2

(

3γ40 −

γ20

4−

7

32

)

+1

n3

(

15γ60 −

3γ40

4−

7γ20

32−

19

128

))

σ0(γ) ≃ γ0

√

1

n

(

γ20 +

1

2

)

+1

n2

(

8γ40 + γ2

0 +3

8

)

+1

n3

(

69γ60 +

7γ40

2+

3γ20

4+

3

16

)

Comments

More efficient than the SH-γ chart.

The paper itself does not provide any thorough investigations(results obtained through simulation only).


New one-sided EWMA-γ2 charts


New one-sided EWMA-γ2 chartsWe suggest to ...



1 monitor γ2 instead of γ



1 monitor γ2 instead of γ (more efficient to monitor S2 than S).




2 define 2 EWMA one-sided charts




2 define 2 EWMA one-sided charts (detect shifts more efficiently).





EWMA-γ2 chart





EWMA-γ2 chart

Upward EWMA-γ2 chart

Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k )





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)

UCLEWMA−γ2 = µ0(γ

2) + K

+

√

λ+

2 − λ+σ0(γ

2)





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)


2) + K

+

√

λ+

2 − λ+σ0(γ

2)

Downward EWMA-γ2 chart

Z−

k = min(µ0(γ2), (1 − λ

−

)Z−

k−1 + λ−

γ2k )





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)


2) + K

+

√

λ+

2 − λ+σ0(γ

2)


Z−

k = min(µ0(γ2), (1 − λ

−

)Z−

k−1 + λ−

γ2k ), Z

−

0 = µ0(γ2)





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)


2) + K

+

√

λ+

2 − λ+σ0(γ

2)


Z−

k = min(µ0(γ2), (1 − λ

−

)Z−

k−1 + λ−

γ2k ), Z

−

0 = µ0(γ2)

LCLEWMA−γ2 = µ0(γ

2) − K

−

√

λ−

2 − λ−

σ0(γ2)





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)


2) + K

+

√

λ+

2 − λ+σ0(γ

2)


Z−

k = min(µ0(γ2), (1 − λ

−

)Z−

k−1 + λ−

γ2k ), Z

−

0 = µ0(γ2)


2) − K

−

√

λ−

2 − λ−

σ0(γ2)

Approximations for µ0(γ2) and σ0(γ

2)





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)


2) + K

+

√

λ+

2 − λ+σ0(γ

2)


Z−

k = min(µ0(γ2), (1 − λ

−

)Z−

k−1 + λ−

γ2k ), Z

−

0 = µ0(γ2)


2) − K

−

√

λ−

2 − λ−

σ0(γ2)


2)

µ0(γ2) ≃ γ2

0

(

1 −3γ

20

n

)





EWMA-γ2 chart


Z+k = max(µ0(γ

2), (1 − λ

+)Z

+k−1 + λ

+γ

2k ), Z

+0 = µ0(γ

2)


2) + K

+

√

λ+

2 − λ+σ0(γ

2)


Z−

k = min(µ0(γ2), (1 − λ

−

)Z−

k−1 + λ−

γ2k ), Z

−

0 = µ0(γ2)


2) − K

−

√

λ−

2 − λ−

σ0(γ2)


2)

µ0(γ2) ≃ γ2

0

(

1 −3γ

20

n

)

, σ0(γ2) ≃

√

γ40

(

2n−1 + γ2

0

(

4n

+ 20n(n−1)

+75γ

20

n2

))

− (µ0(γ2) − γ20 )2


ARL “local” optimization


ARL “local” optimizationShift τ



γ0 = in-control/nominal coefficient of variation.




γ1 = out-of-control coefficient of variation.





τ = γ1

γ0denotes the shift size.





τ = γ1


τ ∈ (0, 1) → decrease of the nominal coefficient of variation.





τ = γ1



τ > 1 → increase of the nominal coefficient of variation.





τ = γ1




Optimization





τ = γ1




Optimization

ARL = average number of samples before a control chart signals an“out-of-control” condition or issues a false alarm.





τ = γ1




Optimization


Find out the optimal couples (λ∗,K∗) such that :

(λ∗,K∗) = argmin(λ,K)

ARL(γ0, τγ0, λ,K , n),





τ = γ1




Optimization





subject to the constraint :

ARL(γ0, γ0, λ∗,K∗, n) = ARL0 = 370.4.





τ = γ1




Optimization





subject to the constraint :

ARL(γ0, γ0, λ∗,K∗, n) = ARL0 = 370.4.

ARL is evaluated using a Brook & Evans’s type Markov chainapproach.Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHARTS

ARL “local” optimization (Markov chain)



Divide the interval between LCL = µ0(γ2) and UCL into p

subintervals of width 2δ, where δ = (UCL − µ0(γ2))/(2p).

Hi

Hi−1

Hi+1

H1

Hp

µ0(γ2)

UCL

2δ





Hi

Hi−1

Hi+1

H1

Hp

µ0(γ2)

UCL

2δ

Hj , j = 1, . . . , p, represents the midpoint of the jth subinterval.





Hi

Hi−1

Hi+1

H1

Hp

µ0(γ2)

UCL

2δ

Hj , j = 1, . . . , p, represents the midpoint of the jth subinterval.

H0 = µ0(γ2) corresponds to the “restart state” feature.



The transition probability matrix




P =

Q r

0T 1

=

Q0,0 Q0,1 · · · Q0,p r0Q1,0 Q1,1 · · · Q1,p r1

......

......

Qp,0 Qp,1 · · · Qp,p rp0 0 · · · 0 1




P =

Q r

0T 1

=

Q0,0 Q0,1 · · · Q0,p r0Q1,0 Q1,1 · · · Q1,p r1

......

......

Qp,0 Qp,1 · · · Qp,p rp0 0 · · · 0 1

Transient probabilities




P =

Q r

0T 1

=

Q0,0 Q0,1 · · · Q0,p r0Q1,0 Q1,1 · · · Q1,p r1

......

......

Qp,0 Qp,1 · · · Qp,p rp0 0 · · · 0 1


Q+i,0 = Fγ2

(

µ0(γ2) − (1 − λ+)Hi

λ+

∣

∣

∣

∣

n, γ1

)




P =

Q r

0T 1

=

Q0,0 Q0,1 · · · Q0,p r0Q1,0 Q1,1 · · · Q1,p r1

......

......

Qp,0 Qp,1 · · · Qp,p rp0 0 · · · 0 1


Q+i,0 = Fγ2

(

µ0(γ2) − (1 − λ+)Hi

λ+

∣

∣

∣

∣

n, γ1

)

Q−

i,0 = 1 − Fγ2

(

µ0(γ2) − (1 − λ−)Hi

λ−

∣

∣

∣

∣

n, γ1

)




P =

Q r

0T 1

=

Q0,0 Q0,1 · · · Q0,p r0Q1,0 Q1,1 · · · Q1,p r1

......

......

Qp,0 Qp,1 · · · Qp,p rp0 0 · · · 0 1


Q+i,0 = Fγ2

(

µ0(γ2) − (1 − λ+)Hi

λ+

∣

∣

∣

∣

n, γ1

)

Q−

i,0 = 1 − Fγ2

(

µ0(γ2) − (1 − λ−)Hi

λ−

∣

∣

∣

∣

n, γ1

)

Qi,j = Fγ2

(

Hj + δ − (1 − λ)Hi

λ

∣

∣

∣

∣

n, γ1

)

− Fγ2

(

Hj − δ − (1 − λ)Hi

λ

∣

∣

∣

∣

n, γ1

)




P =

Q r

0T 1

=

Q0,0 Q0,1 · · · Q0,p r0Q1,0 Q1,1 · · · Q1,p r1

......

......

Qp,0 Qp,1 · · · Qp,p rp0 0 · · · 0 1


Q+i,0 = Fγ2

(

µ0(γ2) − (1 − λ+)Hi

λ+

∣

∣

∣

∣

n, γ1

)

Q−

i,0 = 1 − Fγ2

(

µ0(γ2) − (1 − λ−)Hi

λ−

∣

∣

∣

∣

n, γ1

)

Qi,j = Fγ2

(

Hj + δ − (1 − λ)Hi

λ

∣

∣

∣

∣

n, γ1

)

− Fγ2

(

Hj − δ − (1 − λ)Hi

λ

∣

∣

∣

∣

n, γ1

)

Vector of initial probabilities q = (1, 0, . . . , 0)T .



Definition

The number of steps L until the process reaches the absorbing state is aDiscrete PHase-type (or DPH) random variable of parameters (Q,q).



Definition


ARL, SRDL



Definition


ARL, SRDL

ν1(L) = qT (I − Q)−11



Definition


ARL, SRDL

ν1(L) = qT (I − Q)−11

ν2(L) = 2qT (I − Q)−2Q1



Definition


ARL, SRDL

ν1(L) = qT (I − Q)−11

ν2(L) = 2qT (I − Q)−2Q1

and

ARL = ν1(L)



Definition


ARL, SRDL

ν1(L) = qT (I − Q)−11

ν2(L) = 2qT (I − Q)−2Q1

and

ARL = ν1(L)

SDRL =√

ν2(L) − ν21(L) + ν1(L)


Optimal (λ∗, K ∗) and ARL for EWMA-γ2 and SH-γ charts

n = 7, ARL0 = 370.4τ γ0 = 0.05 γ0 = 0.1 γ0 = 0.15 γ0 = 0.2

0.50 (0.5671, 1.8734) (0.5637, 1.8480) (0.5608, 1.8043) (0.5539, 1.7474)(3.4, 18.4) (3.4, 18.6) (3.5, 18.9) (3.5, 19.3)

0.65 (0.2951, 2.1229) (0.2902, 2.0932) (0.2854, 2.0416) (0.2792, 1.9709)(6.4, 69.3) (6.4, 69.9) (6.4, 70.8) (6.5, 72.1)

0.80 (0.1104, 2.2582) (0.1088, 2.2142) (0.1032, 2.1413) (0.0976, 2.0414)(15.3, 212.1) (15.4, 213.2) (15.5, 215.0) (15.6, 217.5)

1.25 (0.1092, 3.0381) (0.1101, 3.0831) (0.1097, 3.1504) (0.1087, 3.2443)(11.3, 32.4) (11.4, 32.9) (11.7, 33.8) (12.0, 35.1)

1.50 (0.2646, 3.5219) (0.2603, 3.5538) (0.2531, 3.6078) (0.2443, 3.6873)(4.3, 7.2) (4.3, 7.4) (4.4, 7.6) (4.6, 8.0)

2.00 (0.5852, 3.9768) (0.5725, 4.0146) (0.5520, 4.0781) (0.5212, 4.1644)(1.8, 2.1) (1.8, 2.1) (1.9, 2.2) (2.0, 2.3)


(λ∗, K ∗) nomograms

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

λ

τ

γ0 = 0.05

λ−∗ λ+∗

1.5

2

2.5

3

3.5

4

4.5

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

K

τ

γ0 = 0.05

K−∗ K+∗

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

λ

τ

λ−∗ λ+∗

γ0 = 0.1

1.5

2

2.5

3

3.5

4

4.5

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

K

τ

K−∗ K+∗

γ0 = 0.1


(λ∗, K ∗) nomograms

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

λ

τ

γ0 = 0.15

λ−∗ λ+∗

1.5

2

2.5

3

3.5

4

4.5

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

K

τ

γ0 = 0.15

K−∗ K+∗

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

λ

τ

λ−∗ λ+∗

γ0 = 0.2

1.5

2

2.5

3

3.5

4

4.5

0.6 0.8 1 1.2 1.4 1.6 1.8 2

n=5n=7

n=10n=15

K

τ

K−∗ K+∗

γ0 = 0.2


EWMA-γ2 chart v.s. EWMA-γ (Hong et al., 2008) chart

n = 5τ γ0 = 0.05 γ0 = 0.1 γ0 = 0.15 γ0 = 0.2

0.50 (4.8, 4.7) (4.8, 4.7) (4.8, 4.8) (4.8, 4.8)0.65 (8.7, 8.8) (8.8, 8.9) (8.8, 8.9) (8.8, 9.0)0.80 (20.6, 21.1) (20.6, 21.2) (20.7, 21.3) (20.9, 21.5)0.90 (53.2, 56.2) (53.7, 56.4) (54.5, 56.8) (55.8, 57.3)1.10 (51.0, 51.5) (51.2, 51.8) (51.7, 52.3) (52.4, 52.9)1.25 (15.0, 15.5) (15.2, 15.6) (15.4, 15.8) (15.9, 16.0)1.50 (5.7, 5.9) (5.8, 5.9) (5.9, 6.0) (6.1, 6.2)2.00 (2.4, 2.4) (2.4, 2.4) (2.5, 2.5) (2.6, 2.6)

n = 7τ γ0 = 0.05 γ0 = 0.1 γ0 = 0.15 γ0 = 0.2

0.50 (3.4, 3.4) (3.4, 3.4) (3.5, 3.4) (3.5, 3.5)0.65 (6.4, 6.4) (6.4, 6.4) (6.4, 6.5) (6.5, 6.5)0.80 (15.3, 15.6) (15.4, 15.6) (15.5, 15.8) (15.6, 16.0)0.90 (40.4, 41.8) (40.7, 42.0) (41.2, 42.4) (42.0, 42.9)1.10 (39.2, 39.7) (39.5, 40.0) (40.1, 40.4) (40.9, 41.1)1.25 (11.3, 11.5) (11.4, 11.6) (11.7, 11.8) (12.0, 12.1)1.50 (4.3, 4.3) (4.3, 4.4) (4.4, 4.5) (4.6, 4.6)2.00 (1.8, 1.8) (1.8, 1.8) (1.9, 1.9) (2.0, 2.0)


ARL “global” optimization



Drawback of “local” optimization




Usually the quality practitioner does not know in advance the entityof the next shift size because of the lack of related historical data.





The shift size is not deterministic and varies accordingly to someunknown stochastic model.






New objective function and constraint









EARL(γ0, τγ0, λ,K , n)










withEARL =

∫

fτ (τ)ARL(γ0, τγ0, λ,K , n)dτ.










withEARL =

∫


subject to the constraint

EARL(γ0, γ0, λ,K , n) = ARL(γ0, γ0, λ,K , n) = ARL0,










withEARL =

∫




fτ (τ) is the p.d.f. of the shift τ










withEARL =

∫




fτ (τ) is the p.d.f. of the shift τ → uniform distribution over [0.5, 1)(decreasing case)










withEARL =

∫




fτ (τ) is the p.d.f. of the shift τ → uniform distribution over [0.5, 1)(decreasing case) and over (1, 2] (increasing case).



n = 7, ARL0 = 370.4γ0 = 0.05 γ0 = 0.1 γ0 = 0.15 γ0 = 0.2

DECREASING(0.0502, 2.1940) (0.0500, 2.1392) (0.0500, 2.0530) (0.0500, 1.9401)

(23.2) (23.3) (23.5) (23.7)0.50 6.7 (3.4) 6.6 (3.4) 6.5 (3.5) 6.4 (3.5)0.65 9.1 (6.4) 9.0 (6.4) 8.9 (6.4) 8.8 (6.5)0.80 16.5 (15.3) 16.4 (15.4) 16.4 (15.5) 16.3 (15.6)

INCREASING(0.0500, 2.6456) (0.0513, 2.7059) (0.0529, 2.7999) (0.0556, 2.9342)

(12.5) (12.7) (12.8) (13.1)1.25 11.9 (11.3) 12.0 (11.4) 12.3 (11.7) 12.6 (12.0)1.50 5.2 (4.3) 5.3 (4.3) 5.4 (4.4) 5.5 (4.6)2.00 2.5 (1.8) 2.5 (1.8) 2.5 (1.9) 2.6 (2.0)



n = 7, ARL0 = 370.4γ0 = 0.05 γ0 = 0.1 γ0 = 0.15 γ0 = 0.2

DECREASING(0.0502, 2.1940) (0.0500, 2.1392) (0.0500, 2.0530) (0.0500, 1.9401)

(23.2) (23.3) (23.5) (23.7)0.50 6.7 (3.4) 6.6 (3.4) 6.5 (3.5) 6.4 (3.5)0.65 9.1 (6.4) 9.0 (6.4) 8.9 (6.4) 8.8 (6.5)0.80 16.5 (15.3) 16.4 (15.4) 16.4 (15.5) 16.3 (15.6)

INCREASING(0.0500, 2.6456) (0.0513, 2.7059) (0.0529, 2.7999) (0.0556, 2.9342)

(12.5) (12.7) (12.8) (13.1)1.25 11.9 (11.3) 12.0 (11.4) 12.3 (11.7) 12.6 (12.0)1.50 5.2 (4.3) 5.3 (4.3) 5.4 (4.4) 5.5 (4.6)2.00 2.5 (1.8) 2.5 (1.8) 2.5 (1.9) 2.6 (2.0)

Conclusion : EARL based parameters seem robust alternatives.


An illustrative example



A sintering (frittage) process manufacturing mechanical parts




Produced parts are required to guarantee a pressure test drop timeTpd from 2 bar to 1.5 bar larger than 30 sec.





A Regression study demonstrated the presence of a constantproportionality σpd = γpd × µpd between the standard deviation ofthe pressure drop time and its mean.





A Regression study demonstrated the presence of a constantproportionality σpd = γpd × µpd between the standard deviation ofthe pressure drop time and its mean.⇒ the coefficient of variation γpd will be monitored.






Phase I dataset






Phase I dataset

m = 20 sample data, each having sample size n = 5.






Phase I dataset


Estimation of the nominal coefficient of variation γ0 = 0.417.






Phase I dataset



Control limits of the SH-γ chart






Phase I dataset



Control limits of the SH-γ chart

LCLSH = F−1γ

(

0.00272 |5, 0.417

)

= 0.064725,

UCLSH = F−1γ

(

1 − 0.00272 |5, 0.417

)

= 1.216527.


An illustrative example (SH-γ chart, Phase I)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20

Sample Number

LCL=0.0647

UCL=1.2165

γ0 = 0.417

γk

SH-γ chart


An illustrative example (SH-γ chart, Phase I)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20

Sample Number

LCL=0.0647

UCL=1.2165

γ0 = 0.417

γk

SH-γ chart, The sintering process seems in-control.


An illustrative example (EWMA-γ2 chart)



Optimal parameters



Optimal parameters

Accordingly to the process engineer experience, an increase of 25%in the coefficient of variation should be interpreted as a signal thatsomething is going wrong.



Optimal parameters

Accordingly to the process engineer experience, an increase of 25%in the coefficient of variation should be interpreted as a signal thatsomething is going wrong.⇒ τ = 1.25.



Optimal parameters


Optimizing algorithm yields λ+∗ = 0.0793 and K+∗ = 4.3699.



Optimal parameters



Upper Control Limit of the EWMA-γ2 chart



Optimal parameters




Approximations yield µ0(γ2) = 0.1557 and σ0(γ

2) = 0.1643.



Optimal parameters




Approximations yield µ0(γ2) = 0.1557 and σ0(γ

2) = 0.1643.

UCLEWMA−γ2 = 0.1557 + 4.3699 ×√

0.0793

2 − 0.0793× 0.1643 = 0.3016.


An illustrative example (EWMA-γ chart, Phase I)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20

Sample Number

UCL=0.3016

γ2 k

EWMA-γ2 chart


An illustrative example (EWMA-γ chart, Phase I)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20

Sample Number

UCL=0.3016

γ2 k

EWMA-γ2 chart, The sintering process seems in-control too.


An illustrative example (SH-γ chart, Phase II)



Phase II : 20 new samples of size n = 5 taken from the process after theoccurrence of a special cause increasing process variability.




0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20

Sample Number

LCL=0.0647

UCL=1.2165

γ0 = 0.417

γk

SH-γ chart




0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20

Sample Number

LCL=0.0647

UCL=1.2165

γ0 = 0.417

γk

SH-γ chart, The sintering process seems in-control...


An illustrative example (EWMA-γ2 chart, Phase II)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20

Sample Number

UCL=0.3016

γ2 k

EWMA-γ2 chart


An illustrative example (EWMA-γ2 chart, Phase II)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 5 10 15 20

Sample Number

UCL=0.3016

γ2 k

EWMA-γ2 chart, ... but in fact it is not !


An illustrative example (Xk , Phase II)

200

400

600

800

1000

1200

1400

1600

0 5 10 15 20

Sample Number

823.55

X

Xk


An illustrative example (Sk , Phase II)

0

200

400

600

800

1000

1200

1400

1600

1800

0 5 10 15 20

Sample Number

331.5

Abnormal pattern

S

Sk


Conclusions


Conclusions

Many situations in which the sample mean and standard deviationvary naturally in a proportional manner when the process isin-control


Conclusions

Many situations in which the sample mean and standard deviationvary naturally in a proportional manner when the process isin-control → X and S control charts cannot be implemented !


Conclusions


Alternative : monitor the coefficient of variation.


Conclusions



Proposition of the EWMA-γ2 chart (two one-sided EWMA charts).


Conclusions




Outperforms both the SH-γ and EWMA − γ charts.


Conclusions





We provide tables and nomograms in order to select the optimalchart parameters.


Conclusions





We provide tables and nomograms in order to select the optimalchart parameters.

Application on real industrial data.

To be published in Journal of Quality Technology.


MONITORING THE COEFFICIENT OF VARIATION USING EWMA ... - PUC · MONITORING THE COEFFICIENT OF...

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