MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHARTS Philippe - - PowerPoint PPT Presentation

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHARTS

Philippe CASTAGLIOLA 1, Giovanni CELANO 2, Stelios PSARAKIS 3

1Universit´

e de Nantes & IRCCyN UMR CNRS 6597, France

2Universit`

a 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 CHART

Coefficient of variation

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

Coefficient of variation

Definition

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

Coefficient of variation

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

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

Coefficient of variation

Definition If X > 0 is a random variable with mean µ and standard-deviation σ, by definition the coefficient of variation γ is defined as γ = σ µ Used to compare data sets having different units or widely different means (ex : finance, chemical and biological assays, materials engineering).

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

Coefficient of variation

Definition If X > 0 is a random variable with mean µ and standard-deviation σ, by definition the coefficient of variation γ is defined as γ = σ µ Used to compare data sets having different units or widely different means (ex : finance, chemical and biological assays, materials engineering). Sample coefficient of variation

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

Coefficient of variation

Definition If X > 0 is a random variable with mean µ and standard-deviation σ, by definition the coefficient of variation γ is defined as γ = σ µ Used to compare data sets having different units or widely different means (ex : finance, chemical and biological assays, materials engineering). Sample coefficient of variation If {X1, . . . , Xn} is a sample of n normal i.i.d. (µ, σ) random variables then a “natural” estimator of γ is ˆ γ = S ¯ X

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

Coefficient of variation

Definition If X > 0 is a random variable with mean µ and standard-deviation σ, by definition the coefficient of variation γ is defined as γ = σ µ Used to compare data sets having different units or widely different means (ex : finance, chemical and biological assays, materials engineering). Sample coefficient of variation If {X1, . . . , Xn} is a sample of n normal i.i.d. (µ, σ) random variables then a “natural” estimator of γ is ˆ γ = S ¯ X where

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

Coefficient of variation

Definition If X > 0 is a random variable with mean µ and standard-deviation σ, by definition the coefficient of variation γ is defined as γ = σ µ Used to compare data sets having different units or widely different means (ex : finance, chemical and biological assays, materials engineering). Sample coefficient of variation If {X1, . . . , Xn} is a sample of n normal i.i.d. (µ, σ) random variables then a “natural” estimator of γ is ˆ γ = S ¯ X where ¯ X = 1 n

n

• i=1

Xi

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

Coefficient of variation

Definition If X > 0 is a random variable with mean µ and standard-deviation σ, by definition the coefficient of variation γ is defined as γ = σ µ Used to compare data sets having different units or widely different means (ex : finance, chemical and biological assays, materials engineering). Sample coefficient of variation If {X1, . . . , Xn} is a sample of n normal i.i.d. (µ, σ) random variables then a “natural” estimator of γ is ˆ γ = S ¯ X where ¯ X = 1 n

n

• i=1

Xi and S =

• 1

n − 1

n

• i=1

(Xi − ¯ X)2

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

Basic properties of the sample coefficient of variation

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

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ

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

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• F −1

ˆ γ (α|n, γ)

= √n F −1

t

• 1 − α
• n − 1,

√n γ

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• F −1

ˆ γ (α|n, γ)

= √n F −1

t

• 1 − α
• n − 1,

√n γ

• where Ft(.) and F −1

t

(.) are the c.d.f. and the inverse c.d.f. of the noncentral t distribution.

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

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• F −1

ˆ γ (α|n, γ)

= √n F −1

t

• 1 − α
• n − 1,

√n γ

• where Ft(.) and F −1

t

(.) are the c.d.f. and the inverse c.d.f. of the noncentral t distribution. c.d.f and inverse c.d.f. of ˆ γ2

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

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• F −1

ˆ γ (α|n, γ)

= √n F −1

t

• 1 − α
• n − 1,

√n γ

• where Ft(.) and F −1

t

(.) are the c.d.f. and the inverse c.d.f. of the noncentral t distribution. c.d.f and inverse c.d.f. of ˆ γ2 Fˆ

γ2(x|n, γ)

= 1 − FF n x

• 1, n − 1, n

γ2

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• F −1

ˆ γ (α|n, γ)

= √n F −1

t

• 1 − α
• n − 1,

√n γ

• where Ft(.) and F −1

t

(.) are the c.d.f. and the inverse c.d.f. of the noncentral t distribution. c.d.f and inverse c.d.f. of ˆ γ2 Fˆ

γ2(x|n, γ)

= 1 − FF n x

• 1, n − 1, n

γ2

• F −1

ˆ γ2 (α|n, γ)

= n F −1

F

• 1 − α
• 1, n − 1, n

γ2

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

Basic properties of the sample coefficient of variation

c.d.f and inverse c.d.f. of ˆ γ Fˆ

γ(x|n, γ)

= 1 − Ft √n x

• n − 1,

√n γ

• F −1

ˆ γ (α|n, γ)

= √n F −1

t

• 1 − α
• n − 1,

√n γ

• where Ft(.) and F −1

t

(.) are the c.d.f. and the inverse c.d.f. of the noncentral t distribution. c.d.f and inverse c.d.f. of ˆ γ2 Fˆ

γ2(x|n, γ)

= 1 − FF n x

• 1, n − 1, n

γ2

• F −1

ˆ γ2 (α|n, γ)

= n F −1

F

• 1 − α
• 1, n − 1, n

γ2

• where FF(.) and F −1

F (.) are the c.d.f. and the inverse c.d.f. of the

noncentral F distribution.

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

SH-γ chart (Kang et al., JQT 2007)

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . ..

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk).

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control.

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control. Control limits

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control. Control limits LCLSH = F −1

ˆ γ

α0

2 |n, γ0

• UCLSH

= F −1

ˆ γ

• 1 − α0

2 |n, γ0

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control. Control limits LCLSH = F −1

ˆ γ

α0

2 |n, γ0

• UCLSH

= F −1

ˆ γ

• 1 − α0

2 |n, γ0

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

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control. Control limits LCLSH = F −1

ˆ γ

α0

2 |n, γ0

• UCLSH

= F −1

ˆ γ

• 1 − α0

2 |n, γ0

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

Comments

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control. Control limits LCLSH = F −1

ˆ γ

α0

2 |n, γ0

• UCLSH

= F −1

ˆ γ

• 1 − α0

2 |n, γ0

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

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

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

SH-γ chart (Kang et al., JQT 2007)

General assumptions subgroups {Xk,1, Xk,2, . . . , Xk,n} of size n are observed at time k = 1, 2, . . .. Xk,j ∼ N(µk, σk). from one subgroup to another, µk and σk may change, but they are constrained by the relation γk = σk

µk = γ0 when the process is

in-control. Control limits LCLSH = F −1

ˆ γ

α0

2 |n, γ0

• UCLSH

= F −1

ˆ γ

• 1 − α0

2 |n, γ0

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

Comments Simple two-sided “Shewhart-type” control chart. Unefficient for detecting small change in γ.

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

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

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

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

Monitored statistic

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

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk

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

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk Control limits

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

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk Control limits

LCLEWMA−γ = µ0(ˆ γ) − K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ) UCLEWMA−γ = µ0(ˆ γ) + K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ)

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

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk Control limits

LCLEWMA−γ = µ0(ˆ γ) − K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ) UCLEWMA−γ = µ0(ˆ γ) + K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ)

Approximations for µ0(ˆ γ) and σ0(ˆ γ)

µ0(ˆ γ) ≃ γ0

• 1 + 1

n

• γ2

0 − 1

4

• + 1

n2

• 3γ4

0 − γ2

4 − 7 32

• + 1

n3

• 15γ6

0 − 3γ4

4 − 7γ2 32 − 19 128

• σ0(ˆ

γ) ≃ γ0

• 1

n

• γ2

0 + 1

2

• + 1

n2

• 8γ4

0 + γ2 0 + 3

8

• + 1

n3

• 69γ6

0 + 7γ4

2 + 3γ2 4 + 3 16

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk Control limits

LCLEWMA−γ = µ0(ˆ γ) − K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ) UCLEWMA−γ = µ0(ˆ γ) + K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ)

Approximations for µ0(ˆ γ) and σ0(ˆ γ)

µ0(ˆ γ) ≃ γ0

• 1 + 1

n

• γ2

0 − 1

4

• + 1

n2

• 3γ4

0 − γ2

4 − 7 32

• + 1

n3

• 15γ6

0 − 3γ4

4 − 7γ2 32 − 19 128

• σ0(ˆ

γ) ≃ γ0

• 1

n

• γ2

0 + 1

2

• + 1

n2

• 8γ4

0 + γ2 0 + 3

8

• + 1

n3

• 69γ6

0 + 7γ4

2 + 3γ2 4 + 3 16

• Comments

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

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk Control limits

LCLEWMA−γ = µ0(ˆ γ) − K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ) UCLEWMA−γ = µ0(ˆ γ) + K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ)

Approximations for µ0(ˆ γ) and σ0(ˆ γ)

µ0(ˆ γ) ≃ γ0

• 1 + 1

n

• γ2

0 − 1

4

• + 1

n2

• 3γ4

0 − γ2

4 − 7 32

• + 1

n3

• 15γ6

0 − 3γ4

4 − 7γ2 32 − 19 128

• σ0(ˆ

γ) ≃ γ0

• 1

n

• γ2

0 + 1

2

• + 1

n2

• 8γ4

0 + γ2 0 + 3

8

• + 1

n3

• 69γ6

0 + 7γ4

2 + 3γ2 4 + 3 16

• Comments

More efficient than the SH-γ chart.

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

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

Monitored statistic Zk = (1 − λ)Zk−1 + λˆ γk Control limits

LCLEWMA−γ = µ0(ˆ γ) − K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ) UCLEWMA−γ = µ0(ˆ γ) + K

• λ(1 − (1 − λ)2k)

2 − λ σ0(ˆ γ)

Approximations for µ0(ˆ γ) and σ0(ˆ γ)

µ0(ˆ γ) ≃ γ0

• 1 + 1

n

• γ2

0 − 1

4

• + 1

n2

• 3γ4

0 − γ2

4 − 7 32

• + 1

n3

• 15γ6

0 − 3γ4

4 − 7γ2 32 − 19 128

• σ0(ˆ

γ) ≃ γ0

• 1

n

• γ2

0 + 1

2

• + 1

n2

• 8γ4

0 + γ2 0 + 3

8

• + 1

n3

• 69γ6

0 + 7γ4

2 + 3γ2 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).

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

New one-sided EWMA-γ2 charts

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

New one-sided EWMA-γ2 charts

We suggest to ...

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

monitor γ2 instead of γ

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

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

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2)

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

Downward EWMA-γ2 chart

Z −

k

= min(µ0(ˆ γ2), (1 − λ−)Z −

k−1 + λ−ˆ

γ2

k) Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

Downward EWMA-γ2 chart

Z −

k

= min(µ0(ˆ γ2), (1 − λ−)Z −

k−1 + λ−ˆ

γ2

k), Z −

= µ0(ˆ γ2)

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

Downward EWMA-γ2 chart

Z −

k

= min(µ0(ˆ γ2), (1 − λ−)Z −

k−1 + λ−ˆ

γ2

k), Z −

= µ0(ˆ γ2) LCLEWMA−γ2 = µ0(ˆ γ2) − K −

• λ−

2 − λ− σ0(ˆ γ2)

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

Downward EWMA-γ2 chart

Z −

k

= min(µ0(ˆ γ2), (1 − λ−)Z −

k−1 + λ−ˆ

γ2

k), Z −

= µ0(ˆ γ2) LCLEWMA−γ2 = µ0(ˆ γ2) − K −

• λ−

2 − λ− σ0(ˆ γ2)

Approximations for µ0(ˆ γ2) and σ0(ˆ γ2)

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

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

Downward EWMA-γ2 chart

Z −

k

= min(µ0(ˆ γ2), (1 − λ−)Z −

k−1 + λ−ˆ

γ2

k), Z −

= µ0(ˆ γ2) LCLEWMA−γ2 = µ0(ˆ γ2) − K −

• λ−

2 − λ− σ0(ˆ γ2)

Approximations for µ0(ˆ γ2) and σ0(ˆ γ2)

µ0(ˆ γ2) ≃ γ2

• 1 −

3γ2 n

• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

New one-sided EWMA-γ2 charts

We suggest to ...

1

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

2

define 2 EWMA one-sided charts (detect shifts more efficiently). EWMA-γ2 chart Upward EWMA-γ2 chart

Z +

k

= max(µ0(ˆ γ2), (1 − λ+)Z +

k−1 + λ+ˆ

γ2

k), Z + 0 = µ0(ˆ

γ2) UCLEWMA−γ2 = µ0(ˆ γ2) + K +

• λ+

2 − λ+ σ0(ˆ γ2)

Downward EWMA-γ2 chart

Z −

k

= min(µ0(ˆ γ2), (1 − λ−)Z −

k−1 + λ−ˆ

γ2

k), Z −

= µ0(ˆ γ2) LCLEWMA−γ2 = µ0(ˆ γ2) − K −

• λ−

2 − λ− σ0(ˆ γ2)

Approximations for µ0(ˆ γ2) and σ0(ˆ γ2)

µ0(ˆ γ2) ≃ γ2

• 1 −

3γ2 n

• , σ0(ˆ

γ2) ≃

• γ4
• 2

n−1 + γ2

• 4

n + 20 n(n−1) + 75γ2 n2

• − (µ0(ˆ

γ2) − γ2

0)2 Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

ARL “local” optimization

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

ARL “local” optimization

Shift τ

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation.

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation.

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

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

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

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

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

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

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

τ ∈ (0, 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “out-of-control” condition or issues a false alarm.

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

τ ∈ (0, 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “out-of-control” condition or issues a false alarm. Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

ARL(γ0, τγ0, λ, K, n),

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

τ ∈ (0, 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “out-of-control” condition or issues a false alarm. Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

ARL(γ0, τγ0, λ, K, n), subject to the constraint : ARL(γ0, γ0, λ∗, K ∗, n) = ARL0 = 370.4.

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

ARL “local” optimization

Shift τ γ0 = in-control/nominal coefficient of variation. γ1 = out-of-control coefficient of variation. τ = γ1

γ0 denotes the shift size.

τ ∈ (0, 1) → decrease of the nominal coefficient of variation. τ > 1 → increase of the nominal coefficient of variation. Optimization ARL = average number of samples before a control chart signals an “out-of-control” condition or issues a false alarm. Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

ARL(γ0, τγ0, λ, K, n), subject to the constraint : ARL(γ0, γ0, λ∗, K ∗, n) = ARL0 = 370.4. ARL is evaluated using a Brook & Evans’s type Markov chain approach.

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

ARL “local” optimization (Markov chain)

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

ARL “local” optimization (Markov chain)

Divide the interval between LCL = µ0(ˆ γ2) and UCL into p subintervals of width 2δ, where δ = (UCL − µ0(ˆ γ2))/(2p).

Hi Hi−1 Hi+1 H1 Hp µ0(ˆ γ2) UCL 2δ

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

ARL “local” optimization (Markov chain)

Divide the interval between LCL = µ0(ˆ γ2) and UCL into p subintervals of width 2δ, where δ = (UCL − µ0(ˆ γ2))/(2p).

Hi Hi−1 Hi+1 H1 Hp µ0(ˆ γ2) UCL 2δ

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

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

ARL “local” optimization (Markov chain)

Divide the interval between LCL = µ0(ˆ γ2) and UCL into p subintervals of width 2δ, where δ = (UCL − µ0(ˆ γ2))/(2p).

Hi Hi−1 Hi+1 H1 Hp µ0(ˆ γ2) UCL 2δ

Hj, j = 1, . . . , p, represents the midpoint of the jth subinterval. H0 = µ0(ˆ γ2) corresponds to the “restart state” feature.

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

ARL “local” optimization (Markov chain)

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

ARL “local” optimization (Markov chain)

The transition probability matrix

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

ARL “local” optimization (Markov chain)

The transition probability matrix P =   Q r 0T 1   =        Q0,0 Q0,1 · · · Q0,p r0 Q1,0 Q1,1 · · · Q1,p r1 . . . . . . . . . . . . Qp,0 Qp,1 · · · Qp,p rp · · · 1       

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

ARL “local” optimization (Markov chain)

The transition probability matrix P =   Q r 0T 1   =        Q0,0 Q0,1 · · · Q0,p r0 Q1,0 Q1,1 · · · Q1,p r1 . . . . . . . . . . . . Qp,0 Qp,1 · · · Qp,p rp · · · 1        Transient probabilities

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

ARL “local” optimization (Markov chain)

The transition probability matrix P =   Q r 0T 1   =        Q0,0 Q0,1 · · · Q0,p r0 Q1,0 Q1,1 · · · Q1,p r1 . . . . . . . . . . . . Qp,0 Qp,1 · · · Qp,p rp · · · 1        Transient probabilities

Q+

i,0

= Fˆ

γ2

µ0(ˆ γ2) − (1 − λ+)Hi λ+

• n, γ1
• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

ARL “local” optimization (Markov chain)

The transition probability matrix P =   Q r 0T 1   =        Q0,0 Q0,1 · · · Q0,p r0 Q1,0 Q1,1 · · · Q1,p r1 . . . . . . . . . . . . Qp,0 Qp,1 · · · Qp,p rp · · · 1        Transient probabilities

Q+

i,0

= Fˆ

γ2

µ0(ˆ γ2) − (1 − λ+)Hi λ+

• n, γ1
• Q−

i,0

= 1 − Fˆ

γ2

µ0(ˆ γ2) − (1 − λ−)Hi λ−

• n, γ1
• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

ARL “local” optimization (Markov chain)

The transition probability matrix P =   Q r 0T 1   =        Q0,0 Q0,1 · · · Q0,p r0 Q1,0 Q1,1 · · · Q1,p r1 . . . . . . . . . . . . Qp,0 Qp,1 · · · Qp,p rp · · · 1        Transient probabilities

Q+

i,0

= Fˆ

γ2

µ0(ˆ γ2) − (1 − λ+)Hi λ+

• n, γ1
• Q−

i,0

= 1 − Fˆ

γ2

µ0(ˆ γ2) − (1 − λ−)Hi λ−

• n, γ1
• Qi,j

= Fˆ

γ2

Hj + δ − (1 − λ)Hi λ

• n, γ1
• − Fˆ

γ2

Hj − δ − (1 − λ)Hi λ

• n, γ1
• Philippe CASTAGLIOLA , Giovanni CELANO , Stelios PSARAKIS

MONITORING THE COEFFICIENT OF VARIATION USING EWMA CHART

ARL “local” optimization (Markov chain)

The transition probability matrix P =   Q r 0T 1   =        Q0,0 Q0,1 · · · Q0,p r0 Q1,0 Q1,1 · · · Q1,p r1 . . . . . . . . . . . . Qp,0 Qp,1 · · · Qp,p rp · · · 1        Transient probabilities

Q+

i,0

= Fˆ

γ2

µ0(ˆ γ2) − (1 − λ+)Hi λ+

• n, γ1
• Q−

i,0

= 1 − Fˆ

γ2

µ0(ˆ γ2) − (1 − λ−)Hi λ−

• n, γ1
• Qi,j

= Fˆ

γ2

Hj + δ − (1 − λ)Hi λ

• n, γ1
• − Fˆ

γ2

Hj − δ − (1 − λ)Hi λ

• n, γ1
• Vector of initial probabilities q = (1, 0, . . . , 0)T.

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

ARL “local” optimization (Markov chain)

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

ARL “local” optimization (Markov chain)

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

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

ARL “local” optimization (Markov chain)

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

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

ARL “local” optimization (Markov chain)

Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters (Q, q). ARL, SRDL ν1(L) = qT(I − Q)−11

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

ARL “local” optimization (Markov chain)

Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters (Q, q). ARL, SRDL ν1(L) = qT(I − Q)−11 ν2(L) = 2qT(I − Q)−2Q1

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

ARL “local” optimization (Markov chain)

Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters (Q, q). ARL, SRDL ν1(L) = qT(I − Q)−11 ν2(L) = 2qT(I − Q)−2Q1 and ARL = ν1(L)

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

ARL “local” optimization (Markov chain)

Definition The number of steps L until the process reaches the absorbing state is a Discrete PHase-type (or DPH) random variable of parameters (Q, q). ARL, SRDL ν1(L) = qT(I − Q)−11 ν2(L) = 2qT(I − Q)−2Q1 and ARL = ν1(L) SDRL =

• ν2(L) − ν2

1(L) + ν1(L)

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

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

n = 7, ARL0 = 370.4 τ 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)

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

(λ∗, K ∗) nomograms

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=5 n=7 n=10 n=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=5 n=7 n=10 n=15

K τ γ0 = 0.05 K−∗ K+∗

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=5 n=7 n=10 n=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=5 n=7 n=10 n=15

K τ K−∗ K+∗ γ0 = 0.1

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

(λ∗, K ∗) nomograms

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=5 n=7 n=10 n=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=5 n=7 n=10 n=15

K τ γ0 = 0.15 K−∗ K+∗

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=5 n=7 n=10 n=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=5 n=7 n=10 n=15

K τ K−∗ K+∗ γ0 = 0.2

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

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

n = 5 τ 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 = 7 τ 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)

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

ARL “global” optimization

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

ARL “global” optimization

Drawback of “local” optimization

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

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

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

EARL(γ0, τγ0, λ, K, n)

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

EARL(γ0, τγ0, λ, K, n) with EARL =

• fτ(τ)ARL(γ0, τγ0, λ, K, n)dτ.

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

EARL(γ0, τγ0, λ, K, n) with EARL =

• fτ(τ)ARL(γ0, τγ0, λ, K, n)dτ.

subject to the constraint EARL(γ0, γ0, λ, K, n) = ARL(γ0, γ0, λ, K, n) = ARL0,

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

EARL(γ0, τγ0, λ, K, n) with EARL =

• fτ(τ)ARL(γ0, τγ0, λ, K, n)dτ.

subject to the constraint EARL(γ0, γ0, λ, K, n) = ARL(γ0, γ0, λ, K, n) = ARL0, fτ(τ) is the p.d.f. of the shift τ

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

EARL(γ0, τγ0, λ, K, n) with EARL =

• fτ(τ)ARL(γ0, τγ0, λ, K, n)dτ.

subject to the constraint EARL(γ0, γ0, λ, K, n) = ARL(γ0, γ0, λ, K, n) = ARL0, fτ(τ) is the p.d.f. of the shift τ → uniform distribution over [0.5, 1) (decreasing case)

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

ARL “global” optimization

Drawback of “local” optimization Usually the quality practitioner does not know in advance the entity

• f the next shift size because of the lack of related historical data.

The shift size is not deterministic and varies accordingly to some unknown stochastic model. New objective function and constraint Find out the optimal couples (λ∗, K ∗) such that : (λ∗, K ∗) = argmin

(λ,K)

EARL(γ0, τγ0, λ, K, n) with EARL =

• fτ(τ)ARL(γ0, τγ0, λ, K, n)dτ.

subject to the constraint EARL(γ0, γ0, λ, K, n) = ARL(γ0, γ0, λ, K, n) = ARL0, 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).

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

ARL “global” optimization

n = 7, ARL0 = 370.4 γ0 = 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)

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

ARL “global” optimization

n = 7, ARL0 = 370.4 γ0 = 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)

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

ARL “global” optimization

n = 7, ARL0 = 370.4 γ0 = 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)

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

ARL “global” optimization

n = 7, ARL0 = 370.4 γ0 = 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.

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

An illustrative example

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec.

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean.

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean. ⇒ the coefficient of variation γpd will be monitored.

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean. ⇒ the coefficient of variation γpd will be monitored. Phase I dataset

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean. ⇒ the coefficient of variation γpd will be monitored. Phase I dataset m = 20 sample data, each having sample size n = 5.

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean. ⇒ the coefficient of variation γpd will be monitored. Phase I dataset m = 20 sample data, each having sample size n = 5. Estimation of the nominal coefficient of variation ˆ γ0 = 0.417.

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean. ⇒ the coefficient of variation γpd will be monitored. Phase I dataset m = 20 sample data, each having sample size n = 5. Estimation of the nominal coefficient of variation ˆ γ0 = 0.417. Control limits of the SH-γ chart

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

An illustrative example

A sintering (frittage) process manufacturing mechanical parts Produced parts are required to guarantee a pressure test drop time Tpd from 2 bar to 1.5 bar larger than 30 sec. A Regression study demonstrated the presence of a constant proportionality σpd = γpd × µpd between the standard deviation of the pressure drop time and its mean. ⇒ the coefficient of variation γpd will be monitored. Phase I dataset m = 20 sample data, each having sample size n = 5. Estimation of the nominal coefficient of variation ˆ γ0 = 0.417. Control limits of the SH-γ chart LCLSH = F −1

ˆ γ

0.0027

2

|5, 0.417

• = 0.064725,

UCLSH = F −1

ˆ γ

• 1 − 0.0027

2

|5, 0.417

• = 1.216527.

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

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

0.2 0.4 0.6 0.8 1 1.2 1.4 5 10 15 20 Sample Number LCL=0.0647 UCL=1.2165 γ0 = 0.417 ˆ γk SH-γ chart

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

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

0.2 0.4 0.6 0.8 1 1.2 1.4 5 10 15 20 Sample Number LCL=0.0647 UCL=1.2165 γ0 = 0.417 ˆ γk SH-γ chart, The sintering process seems in-control.

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

An illustrative example (EWMA-γ2 chart)

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

An illustrative example (EWMA-γ2 chart)

Optimal parameters

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

An illustrative example (EWMA-γ2 chart)

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

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

An illustrative example (EWMA-γ2 chart)

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

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

An illustrative example (EWMA-γ2 chart)

Optimal parameters Accordingly to the process engineer experience, an increase of 25% in the coefficient of variation should be interpreted as a signal that something is going wrong. ⇒ τ = 1.25. Optimizing algorithm yields λ+∗ = 0.0793 and K +∗ = 4.3699.

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

An illustrative example (EWMA-γ2 chart)

Optimal parameters Accordingly to the process engineer experience, an increase of 25% in the coefficient of variation should be interpreted as a signal that something is going wrong. ⇒ τ = 1.25. Optimizing algorithm yields λ+∗ = 0.0793 and K +∗ = 4.3699. Upper Control Limit of the EWMA-γ2 chart

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

An illustrative example (EWMA-γ2 chart)

Optimal parameters Accordingly to the process engineer experience, an increase of 25% in the coefficient of variation should be interpreted as a signal that something is going wrong. ⇒ τ = 1.25. Optimizing algorithm yields λ+∗ = 0.0793 and K +∗ = 4.3699. Upper Control Limit of the EWMA-γ2 chart Approximations yield µ0(ˆ γ2) = 0.1557 and σ0(ˆ γ2) = 0.1643.

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

An illustrative example (EWMA-γ2 chart)

Optimal parameters Accordingly to the process engineer experience, an increase of 25% in the coefficient of variation should be interpreted as a signal that something is going wrong. ⇒ τ = 1.25. Optimizing algorithm yields λ+∗ = 0.0793 and K +∗ = 4.3699. Upper Control Limit of the EWMA-γ2 chart Approximations yield µ0(ˆ γ2) = 0.1557 and σ0(ˆ γ2) = 0.1643. UCLEWMA−γ2 = 0.1557 + 4.3699 ×

• 0.0793

2 − 0.0793 × 0.1643 = 0.3016.

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

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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 10 15 20 Sample Number UCL=0.3016 ˆ γ2

k

EWMA-γ2 chart

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

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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 10 15 20 Sample Number UCL=0.3016 ˆ γ2

k

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

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

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

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

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

Phase II : 20 new samples of size n = 5 taken from the process after the

• ccurrence of a special cause increasing process variability.

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

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

Phase II : 20 new samples of size n = 5 taken from the process after the

• ccurrence of a special cause increasing process variability.

0.2 0.4 0.6 0.8 1 1.2 1.4 5 10 15 20 Sample Number LCL=0.0647 UCL=1.2165 γ0 = 0.417 ˆ γk SH-γ chart

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

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

Phase II : 20 new samples of size n = 5 taken from the process after the

• ccurrence of a special cause increasing process variability.

0.2 0.4 0.6 0.8 1 1.2 1.4 5 10 15 20 Sample Number LCL=0.0647 UCL=1.2165 γ0 = 0.417 ˆ γk SH-γ chart, The sintering process seems in-control...

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

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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 10 15 20 Sample Number UCL=0.3016 ˆ γ2

k

EWMA-γ2 chart

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

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

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5 10 15 20 Sample Number UCL=0.3016 ˆ γ2

k

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

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

An illustrative example (¯ Xk, Phase II)

200 400 600 800 1000 1200 1400 1600 5 10 15 20 Sample Number 823.55

¯ X

¯ Xk

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

An illustrative example (Sk, Phase II)

200 400 600 800 1000 1200 1400 1600 1800 5 10 15 20 Sample Number 331.5

Abnormal pattern

S

Sk

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

Conclusions

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

Conclusions

Many situations in which the sample mean and standard deviation vary naturally in a proportional manner when the process is in-control

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

Conclusions

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

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

Conclusions

Many situations in which the sample mean and standard deviation vary naturally in a proportional manner when the process is in-control → ¯ X and S control charts cannot be implemented ! Alternative : monitor the coefficient of variation.

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

Conclusions

Many situations in which the sample mean and standard deviation vary naturally in a proportional manner when the process is in-control → ¯ X and S control charts cannot be implemented ! Alternative : monitor the coefficient of variation. Proposition of the EWMA-γ2 chart (two one-sided EWMA charts).

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

Conclusions

Many situations in which the sample mean and standard deviation vary naturally in a proportional manner when the process is in-control → ¯ X and S control charts cannot be implemented ! Alternative : monitor the coefficient of variation. Proposition of the EWMA-γ2 chart (two one-sided EWMA charts). Outperforms both the SH-γ and EWMA − γ charts.

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

Conclusions

Many situations in which the sample mean and standard deviation vary naturally in a proportional manner when the process is in-control → ¯ X and S control charts cannot be implemented ! Alternative : monitor the coefficient of variation. Proposition of the EWMA-γ2 chart (two one-sided EWMA charts). Outperforms both the SH-γ and EWMA − γ charts. We provide tables and nomograms in order to select the optimal chart parameters.

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

Conclusions

Many situations in which the sample mean and standard deviation vary naturally in a proportional manner when the process is in-control → ¯ X and S control charts cannot be implemented ! Alternative : monitor the coefficient of variation. Proposition of the EWMA-γ2 chart (two one-sided EWMA charts). Outperforms both the SH-γ and EWMA − γ charts. We provide tables and nomograms in order to select the optimal chart parameters. Application on real industrial data. To be published in Journal of Quality Technology.

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