Development of Fuzzy Extreme Value Theory Control Charts Using ...

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* Corresponding author. Applied Mathematical Sciences, Vol. 6, 2012, no. 117, 5811 – 5834 Development of Fuzzy Extreme Value Theory Control Charts Using α -cuts for Skewed Populations Rungsarit Intaramo Department of Mathematics , Faculty of Science King Mongkut’s University of Technology Thonburi Bangkok, Thailand [email protected] Adisak Pongpullponsak* Department of Mathematics , Faculty of Science King Mongkut’s University of Technology Thonburi Bangkok, Thailand [email protected] Abstract Recent studies have demonstrated that the adaptive (i.e., variable sample sizes, sampling intervals, and/or action limit coefficients) x charts are quicker than standard Shewhart (SS) x control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In this paper fuzzy extreme value (FEV) theory control charts have been developed from extreme value (EV) control charts using α -cuts with the uncertain data which is evaluated under non-normality. For many problems, control charts come from uncertain data such as human, measurement devices or environmental conditions. In this context, fuzzy set theory is useful to help in solving the data problems caused by this uncertainty. The data for the experiment will be transformed to fuzzy control charts by using membership functions. The efficiency of control charts are determined by average run length (ARL).

Transcript of Development of Fuzzy Extreme Value Theory Control Charts Using ...

Page 1: Development of Fuzzy Extreme Value Theory Control Charts Using ...

*Corresponding author.

Applied Mathematical Sciences, Vol. 6, 2012, no. 117, 5811 – 5834

Development of Fuzzy Extreme Value Theory

Control Charts Using α -cuts for Skewed

Populations

Rungsarit Intaramo

Department of Mathematics , Faculty of Science King Mongkut’s University of Technology Thonburi

Bangkok, Thailand [email protected]

Adisak Pongpullponsak*

Department of Mathematics , Faculty of Science

King Mongkut’s University of Technology Thonburi Bangkok, Thailand

[email protected]

Abstract

Recent studies have demonstrated that the adaptive (i.e., variable sample sizes, sampling intervals, and/or action limit coefficients) x charts are quicker than standard Shewhart (SS) x control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In this paper fuzzy extreme value (FEV) theory control charts have been developed from extreme value (EV) control charts using α -cuts with the uncertain data which is evaluated under non-normality. For many problems, control charts come from uncertain data such as human, measurement devices or environmental conditions. In this context, fuzzy set theory is useful to help in solving the data problems caused by this uncertainty. The data for the experiment will be transformed to fuzzy control charts by using membership functions. The efficiency of control charts are determined by average run length (ARL).

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5812 R. Intaramo and A. Pongpullponsak

Keywords: non - normality distribution data, α-cuts, α-level fuzzy midrange

1 Introduction

The control chart originated in the early 1920s, it has become a powerful tool

in statistical process control (SPC) that is the most widely used in industrial

processes. Control charts are designed to monitor the process of change in mean

and variance, they also reflect the ability of the process. Control charts have two

types: variable and attribute. Techniques of statistical process control are widely

used by the manufacturing industry to detect and eliminate defects during

production. Control chart technique is well-known as a key step in production

process monitoring. The control chart has a major function in detecting the

occurence of assignable causes, so that the necessary correction can be made

before non-conforming products are manufactured in a large amount. The control

chart technique may be considered as both the graphical expression and operation

of statistical hypothesis testing. It is recommended that if a control chart is

employed to monitor process, some test parameters should be determined such as

the sample size, the sampling interval between successive samples, and the

control limits or critical regions of the chart. SPC is an efficient technique for

improvement of a firm’s quality and productivity. The main objective of SPC is

similar to that of the control chart technique, that is, to rapidly examine the

occurrence of assignable causes or process shifts.

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Many studies were done to combine statistical methods and fuzzy set theory.

Fuzzy sets theory was first introduced by Zadeh (1965). In 2005 Zadeh outlined

generalized theory of uncertainty (GTU) which presented a change of perspective

and direction in thinking about the system and uncertainties. Buckley and Eslami

(2004) introduced the theory of estimation of the mean and variance of the

confidence intervals using triangular numbers as the estimator. M. B. Vermaat ET

AL(2003) studied the comparison of control charts based on normal,

non-parametric control charts and extreme value (EV) control charts .

A.Pongpullponsak, W. Suracherkiati and R. Intaramo, (2006) used the concept of

EV theory of M. B. Vermaat ET AL(2003) to develop EV theory control charts

which data are Weibull , lognormal and Burr’s distributions by comparing with its

efficiency of weighted variance method (WV), scaled weighted variance method

(SWV) control charts of A.Pongpullponsak, W. Suracherkiati and

P.Kriweradechachai (2004).

There is limited information available on fuzzy attribute control charts and

their applications: Wang and Raz (1990) proposed some approaches by assigning

a fuzzy set to each linguistic term and then combining these for each sample using

the rules of fuzzy arithmetic. Kanagawa et al. (1993) introduced a control chart

based on the probability density function for linguistic data. Gullbay et al. (2004)

suggested the α -cut fuzzy control charts for linguistic data. Gullbay and

Kahraman (2006) developed fuzzy c control charts for determining the unnatural

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patterns. Information on fuzzy variable control charts and their applications are

also limited: Roland and Wang (2000) introduced fuzzy SPC theory based on the

application of fuzzy logic to the SPC-zone rules. El-Shal and Morris (2000)

modified SPC-zone rules to reduce false alarm and detect the real error. Zarandi et

al. (2008) presented a new hybrid method based on a combination of fuzzified

sensitivity criteria and fuzzy adaptive sampling rules to determine the sample size

and sample interval of the control charts in order to determine the sample size and

sample interval of the control charts. In fact, the problem with control charts is

caused by uncertain data i.e. human, measurement devices or environmental

conditions. The studies of A. Pongpullponsak, W. Suracherkiati and and R.

Intaramo, (2006) are important as they indicate the ambiguity data of the chart.

Thus, fuzzy set theory is useful in helping to solve the problems caused by

uncertain data by applying fuzzy to EV theory to develop a new chart (FEV), in

order to control and improve process efficiency at its best. It was discovered by

Senturk and Erginel (2009) that control charts could be used to solve the problem

of uncertain data by using fuzzy theory. The topic of the research studied was

fuzzy ~

X R− % and ~ ~X S− control charts using α -cut.

The methods used in the transformation of fuzzy sets into scalars are fuzzy

mode, fuzzy median and α -level fuzzy midrange. Which one you choose to use

depends on the difficulty of the computation or preference as in Wang (1990).

The aim of this study is to introduce the framework of FEV theory control

charts which are Weibull, lognormal and Burr’s distributions, using α -cut with

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the methods of α -level fuzzy midrange. First of all, we transform EV theory

control charts to FEV theory control charts. To obtain FEV theory control chart,

triangular fuzzy numbers (a,b,c) are used. Secondly α -cut FEV control charts are

developed by using α -cut approach. Thirdly α -level fuzzy midrange for FEV

control charts are calculated by using α -level fuzzy midrange transformation

techniques. Finally, we can use the ARL to determine the efficiency of the chart.

This paper is organized as follows: non-normal distributions as Weibull,

lognormal and Burr’s, EV control charts andα -level fuzzy midrange are

introduced in the second section. FEV control charts are developed in section 3.

The efficiency of FEV control charts are examined in section 4. The conclusions

are presented in the final section.

2 Model Consideration

In this study, we will consider FEV theory control charts which are

developed from EV theory control charts studied by Pongpullponsak, A.,

Suracherkiati, W. and Intaramo, R. (2006). These charts have non-normal

distribution data which are Weibull, lognormal and Burr’s.

2.1 Weibull distribution

Weibull is continuous distribution that is used widely. Let X be continuous

random variables that are Weibull distribution with 0θ > and 0β > .

Density function

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1 ( / )( ; , ) xf x x eββ θ

β

βθ βθ

− −= 0x >

Cumulative distribution function

( / )( ; , ) 1 xF x eβθθ β −= − 0x >

Where

θ is scale parameter

β is shape parameter

In this study 1θ = and β are relevant, with a coefficient of skewness at

{ }3 0.1,0.5,1, 2,3, 4,5,6,7,8,9α ∈ shown in table 1.

Table 1 represents a coefficient of skewness and shape parameter of Weibull

distribution

Coefficient of skewness 3( )α Shape parameter ( )β

0.1 3.2219

0.5 2.2110

1 1.5630

2 1.0000

3 0.7686

4 0.6478

5 0.5737

6 0.5237

7 0.4873

8 0.4596

9 0.4376

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2.2 Lognormal distribution

Lognormal is correlated with normal distribution but random variables

have positive values. Let X equal continuous random variables that are

lognormal distribution.

Density function

2

2

ln1( ; , ) exp

2 2

xf x

x

μμ σ

σ π σ

∧ ∧

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟=

⎜ ⎟⎜ ⎟⎝ ⎠

0x >

where

μ is scale parameter

σ is shape parameter

In this study 0.1,0.5,1,2,3,4,5,6,7,8,9μ = and σ are relevant with a

coefficient of skewness at { }3 0.1,0.5,1, 2,3, 4,5,6,7,8,9α ∈ shown in table 2.

Table 2 represents a coefficient of skewness and shape parameter of lognormal

distribution

Coefficient of skewness 3( )α Shape parameters ( )σ 0.1 0.0334 0.5 0.1641 1 0.3142 2 0.5513 3 0.7156 4 0.8326 5 0.9202 6 0.9889

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Table 2 (continued)

Coefficient of skewness 3( )α Shape parameters ( )σ 7 1.0446 8 1.0911 9 1.1307

2.3 Burr’s distribution

Burr’s is a type of continuous distribution. Let X equal continuous random

variables that are Burr’s distribution with parameter c and m .

Density function

( )1

1 0( ) 1

0 0

c

mc

mcx xf x x

x

+

⎧≥⎪

= +⎨⎪

<⎩

Cumulative distribution function

( ) 1 (1 )c mF X x −= − + 0x >

where , 1c m ≥

Burr’s distribution can be represented as Weibull distribution when m

increases. 3α and 4α can be obtained by 3cm > and 4cm > respectively, where

3α is the coefficient of skewness and 4α is the coefficient of kurtosis.

This study is configured as shown in table 3.

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Table 3 represents constants 3, , , ,c mα μ σ ′ for a coefficient of skewness of Burr’s

distribution

Coefficient of

skewness 3( )α

Coefficient of

kurtosis 4( )α

c m μ σ ′

0.1 2.9282 4 7.7400 0.54545 0.16157

0.5 3.4277 3 4.2130 0.51865 0.20614

1 4.6410 2 6.7500 0.34802 0.19855

2 18.7740 4.2707 1.0000 1.05309 0.35910

3 20.7609 1 7.3370 0.13102 0.14957

4 46.6350 1 4.2267 0.21134 0.26259

*5 - 1 3.1453 0.26236 0.34434

*6 - 1 3.5580 0.29661 0.40569

*7 - 1 3.9707 0.33087 0.46704

**8 - - - - -

**9 - - - - -

Note μ denotes mean of the population.

σ ′ denotes standard deviation of the population.

* denotes no coefficient of kurtosis because 4cm < .

** denotes without any constant.

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5820 R. Intaramo and A. Pongpullponsak

2.4 Extreme value control chart : EV control chart

EV theory that deals with the tail behaviour of distribution, can be

modelled using EV distribution by Dekker (1989), which can be monitored as an

index of extreme values. Because we can’t make assumptions regarding the value

of kγ , we can use the moment estimator to calculate an approximate value as

follows :

1(1) 2

(1)(2)

( )11 12

kk k

k

MMM

γ−

∧ ⎧ ⎫= + − −⎨ ⎬

⎩ ⎭ (1)

and

1(1)_(1) 2_

(2)_

1 ( )1 12

kkk

k

MMM

γ

−⎧ ⎫⎪ ⎪= + − −⎨ ⎬⎪ ⎪⎩ ⎭

(2)

The study of Dekker (1989), F is the q-quantile of the distribution function, so

1(1)

( ) ( )( / )) 1(1 ; ) (1 ( 0))

k

k

k k kk m k m km kqF q X X M

γ

γγ γ

−∧ ∧ ∧

− −

−− = + − ∧ (3)

with 0 1q< < . x y∧ and x y∨ denotes the minimum and maximum

respectively.

Define

( )( 1) ( )

1

1 (log log )m

r rk k n k m

nM X X

m − + −=

= −∑ (4)

and

( )_

( ) ( 1)1

1 (log log )r m

rk n m

nM X X

m +=

= −∑ (5)

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where an integer takes the values r = 1or 2, and m is the number of upper and

lower order statistics respectively used in the estimation of the control limits.

From EV theory control charts are

(1)( ) ( )

( / )) 1(1 ( 0))k

kkk m k m k

m kqUCL X X Mγ

γγ

− −

−= + − ∧ (6)

(1)_

( 1) ( 1)( / / 2)) 1(1 ( 0))

k

k

kkm mm kLCL X X M

γ

γ

α γ

+ +−

= + − ∧ (7)

where

( )( )_

1

nr

r kk

k

MM

n==∑

n is the number of class, m is number of sample, k is the number of sample size

and ( )rkM is the moment estimator.

Hence, we must approximate the value of ( )rkM by using binomial theorem of

skewed populations which are Weibull, lognormal and Burr’s distributions, see

equation (9), (10) and (11) respectively.

Estimator of ( )rkM of Weibull distribution

( ) ( )rr k

kM E x μ⎡ ⎤= −⎣ ⎦

By binomial theorem so

( )

0( ) ( )

rk

r i k ik

i

kM E x

iμ −

=

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∑ (8)

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5822 R. Intaramo and A. Pongpullponsak

Find 1 ( / )

0

( )k i k i xE X x x e dxββ θ

β

βθ

∞− − − −= ∫

1 ( / )

0

k i xx e dxββ θ

β

βθ

∞+ − − −= ∫

Let x xyβ β

βθ θ⎛ ⎞= =⎜ ⎟⎝ ⎠

x yβ βθ=

1/x y βθ=

1 11dx y dyβ θβ

−=

11 1 1

0

1( )k i

k i yE X y e y dyβ

β ββ

β θ θθ β

+ − −∞ −

− −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠∫

11 1 1

10

1k i

yy e y dyβ

β ββ θ

θ

+ − −∞ −

−−

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠∫

11 11 1

10

k ik i

yy e y dyβ

ββ β

β

θθ

+ − −∞+ − − −

−−

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠∫

1 1 1

0

k ik i yy y e dy

ββ βθ

+ − −∞ −− −

⎛ ⎞= ×⎜ ⎟⎜ ⎟

⎝ ⎠∫

1 1 1

0

k ik i yy e dy

ββ βθ

+ − −∞ + −− −

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠∫

1

0

k ik i yy e dy

ββθ+ −∞ −

− −⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠∫

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Development of fuzzy extreme value theory control charts 5823

k i k iβθ τβ

− ⎛ ⎞+ −= ⎜ ⎟

⎝ ⎠

From equation (8) so

( )

0( )

rk

r i k ik

i

k k iMi

βμ θ τβ

=

⎡ ⎤⎛ ⎞ ⎛ ⎞+ −= −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎣ ⎦∑ (9)

Estimator of ( )rkM of lognormal distribution

( ) ( )rr k

kM E x μ⎡ ⎤= −⎣ ⎦

By binomial theorem so

( )

0

( ) ( )r

kr i k i

ki

kM E x

iμ −

=

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∑

Find

2

2

ln

2

0

1( )2

x

k i k iE X x e dxx

μ

σ

σ π

⎛ ⎞⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠−⎜ ⎟⎜ ⎟∞⎜ ⎟− − ⎝ ⎠

∧= ∫

Let lny x=

yx e=

ydx e dy=

2

22

0

1( ) ( )2

y

k i y k i

yE X e e dx

e

μ

σ

σ π

⎛ ⎞⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠−⎜ ⎟⎜ ⎟∞⎜ ⎟− − ⎝ ⎠

∧= ∫

( )yM k i= −

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5824 R. Intaramo and A. Pongpullponsak

2 2( )( )

2k ik i

eσμ

∧∧ −

− +=

From equation (8) so

2 2( )( )( ) 2

0( )

rk ik k ir i

ki

kM e

i

σμμ

∧∧ −

− +

=

⎡ ⎤⎛ ⎞⎢ ⎥= −⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦

∑ (10)

Estimator of ( )rkM of Burr’s distribution

( ) ( )rr k

kM E x μ⎡ ⎤= −⎣ ⎦

By binomial theorem so

( )

0

( ) ( )r

kr i k i

ki

kM E x

iμ −

=

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∑

Find ( )

1

10

( )1

ck i k i

mc

mcxE X x dxx

∞ −− −

+=+

Let k i v− =

1 cy x= + ,0 1y< <

1

1 cyxy

⎛ ⎞−= ⎜ ⎟⎝ ⎠

1 1

2

1 1 1cyJ dx dyc y y

−⎛ ⎞−

= = ⎜ ⎟⎝ ⎠

1 1( 1) 111

20

1 1 1 1v c

c cm y ymcy dy

y c y y

+ − −

+ ⎛ ⎞ ⎛ ⎞− −= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫

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Development of fuzzy extreme value theory control charts 5825

1

12

0

1 1vc

m ymy dyy y

+ ⎛ ⎞−= ⎜ ⎟

⎝ ⎠∫

1

1 2

0

(1 )v vmc cmy y dy

+ − −= −∫

11 1 1

0

(1 )v vmc cmy y dy

− − + −= −∫

,1v vm mc c

β ⎛ ⎞= − +⎜ ⎟⎝ ⎠

from v k i= −

,1k i k im mc c

β − −⎛ ⎞= − +⎜ ⎟⎝ ⎠

1

( 1)

k i k im mc cm

τ τ

τ

− −⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠=

+

From equation (8) so

( )

0

1( )

( 1)

r

kr i

ki

k i k im mk c cMi m

τ τμ

τ=

⎡ − − ⎤⎛ ⎞ ⎛ ⎞− +⎜ ⎟ ⎜ ⎟⎢ ⎥⎛ ⎞ ⎝ ⎠ ⎝ ⎠⎢ ⎥= −⎜ ⎟ +⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦

∑ (11)

2.5 α -level fuzzy midrange

Fuzzy transformation techniques have four types : fuzzy mode , fuzzy

median , fuzzy average and α -level fuzzy midrange . In this study, the α -level

fuzzy midrange transformation technique is used for FEV theory control charts.

The α -level fuzzy midrange mrf α is defined as the midpoint of the α -level

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5826 R. Intaramo and A. Pongpullponsak

cuts. Let Aα isα -level cuts, nonfuzzy sets that consist of any elements whose

membership is greater than or equal toα . If aα and bα are end points of Aα then

1 ( )2mrf a cα α α= + (12)

In fact the fuzzy mode is a special case of α -level fuzzy midrange when 1α = .

α -level fuzzy midrange of sample ,, mr jj Sα is determined by

,

( ) ( ) ( )2

j j j j j jmr j

a c b a c bSα

α ⎡ ⎤+ + − − −⎣ ⎦= (13)

The definition of α -level fuzzy midrange of sample j for fuzzy ~

x control chart is

,

( ) ( ) ( )

2j j j j j ja c b a c b

mr x j

x x x x x xSα

α−

⎡ ⎤+ + − − −⎣ ⎦= (14)

Then, the condition of process control for each sample can be defined as

,Pr mr x mr x j mr xin control for LCL S UCLocess control

out of control for otherwise

α α α− − −

⎧ ⎫− ≤ ≤⎪ ⎪=⎨ ⎬−⎪ ⎪⎩ ⎭

(15)

3 Fuzzy extreme value theory control chart

3.1 Fuzzy extreme value theory control chart

By studying EV theory control charts, it was discovered that uncertain

data was a problem, so we use fuzzy theory to solve these problems. The studies

of Senturk, S., Erginel N. (2009) used fuzzy theory in control charts. Then we

modified EV theory control charts to FEV theory control charts, which use

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Development of fuzzy extreme value theory control charts 5827

membership represented by a triangular fuzzy number (a,b,c) as shown in Fig 1.

Therefore, the FEV theory control limits are

(1)( ) ( )

( / )) 1(1 ( 0))k

kkk m k m k

m kqUCL X X Mγ

γγ

− −

−= + − ∧ (16)

, ,

, ,

,

,

(1) (1), ,( ), ( ), , ( ), ( ), ,

(1),( ), ( ), ,

( / )) 1 ( / )) 1(1 ( 0)) , (1 ( 0))

( / )) 1, (1 ( 0))

k a k b

k a k b

k c

k c

k a k bk m a k m a k a k m b k m b k b

k ck m c k m c k c

m kq m kqX X M X X M

m kqX X M

γ γ

γ γ

γ

γ

γ γ

γ

∧ ∧

∧ ∧

∧ ∧

− − − −

− −

− −= + − ∧ + − ∧

−+ − ∧

(1)_

( 1) ( 1)( / / 2)) 1(1 ( 0))

k

k

kkm mm kLCL X X M

γ

γ

α γ

+ +

−= + − ∧ (17)

, ,

, ,

,

,

(1) (1)_ _

, ,, ,( 1), ( 1), ( 1), ( 1),

(1)_

,,( 1), ( 1),

( / / 2) 1 ( / / 2) 1(1 ( 0)) , (1 ( 0))

( / / 2) 1, (1 ( 0))

k a k b

k a k b

k c

k c

k a k bk a k bm a m a m b m b

k ck cm c m c

m k m kX X M X X M

m kX X M

γ γ

γ γ

γ

γ

α αγ γ

α γ

− −

− −

− −

+ + + +

+ +

− −= + − ∧ + − ∧

−+ − ∧

Fig 1 represents of a sample of triangular fuzzy number

μ

α

0 a b c aα cα

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5828 R. Intaramo and A. Pongpullponsak

3.2 α -cut fuzzy extreme value theory control chart

An α - cut consists of any elements whose membership is greater than or

equal to α . Applying α - cut of fuzzy sets, the values of ( ),k m aX − , ( ),k m cX − , (1),k aM ,

(1),k cM are determined as follows:

( ), ( ), ( ), ( ),( )k m a k m a k m b k m aX X X Xα α− − − −= + − (18)

( ), ( ), ( ), ( ),( )k m c k m c k m c k m bX X X Xα α− − − −= − − (19)

(1) (1) (1) (1), , , ,( ) ( ) (( ) ( ))k a k a k b k aM M M Mα α= + − (20)

(1) (1) (1) (1), , , ,( ) ( ) (( ) ( ))k c k c k c k bM M M Mα α= − − (21)

Therefore, the α -cut FEV theory control limits are

(1)( ) ( )

( / )) 1(1 ( 0)) ( )k

kkk m k m k

m kqU CL X X Mγ

α α α α

γγ

− −

−= + − ∧ (22)

(1)_

( 1) ( 1)( / / 2)) 1(1 ( 0)) ( )

k

kkm m k

m kLCL X X Mγ

α α α α

γ

α γ

+ +

−= + − ∧ (23)

α -cut control limits are shown in Fig 2.

3.3 α -level fuzzy midrange for α -cut FEV theory control chart

An α -level fuzzy midrange is one of four transformation techniques used to

determine the FEV control charts. In this study α -level fuzzy midrange is used

as the fuzzy transformation method while calculating α -level fuzzy midrange

forα -cut FEV theory control limits

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Development of fuzzy extreme value theory control charts 5829

(1) (1)

( ), ( ), , ,( )

( ) ( )( / )) 1(1 ( 0)) ( )2 2

k

k

k m a k m c k a k ckmr k m

X X M Mm kqUCL Xα α α αγ

α α

γγ

∧− −

⎛ ⎞+ +−= + − ∧⎜ ⎟⎜ ⎟⎝ ⎠

% (24)

(1) (1), ,( 1), ( 1),

( 1)( / / 2) 1 ( ) ( )(1 ( 0)) ( )

2 2

k

k

k a k cm a m ckmr m

X X m k M MLCL Xα α γ α α

α α

γ

α γ

−+ +

+

⎛ ⎞+ − += + − ∧⎜ ⎟⎜ ⎟⎝ ⎠

% (25)

For an approximation of ( )rkM of Weibull, lognormal and Burr’s distributions

see equations (9),(10) and (11) respectively.

Fig 2 α-cut control chart ( ,CL LCL%% and )UCL%

4 Simulation studies

The purpose of this study is to compare the efficiency of FEV theory control

charts for skewed populations i,e., Weibull, lognormal and Burr’s distributions

which have various values of the coefficient of skewness which are 0.1, 0.5, 1.0,

2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, number of class 300n = , number of sample

size 10k = are randomly generated from Weibull, lognormal and Burr’s

distributions with 1,θ β= relevant with a coefficient of skewness shown in table

0

μ

3CLα

2UCL 2CL 2LCL

3UCLα 3UCL 3CL

3LCLα1UCLα 1UCL 1CLα

1CL 3LCL 1LCL1UCLα

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5830 R. Intaramo and A. Pongpullponsak

1, 0,μ σ= relevant with a coefficient of skewness shown in table 2 and ,c m is

shown in table 3 respectively. The procedure is repeated 10,000 times for shift

sizes of 0.5 ,1.0 ,2.0 ,2.5σ σ σ σ and3.0σ . From this study, the results are as :

Table 4 represents the ARL corresponding to a different coefficient of skewness.

Coeffcient of

skewness 3( )α

Weibull

distribution

Lognormal

distribution

Burr’s

distribution

0.1 212.01 215.13 206.76

0.5 189.23 202.20 204.60

1.0 123.17 185.67 195.56

2.0 76.23 52.26 81.16

3.0 52.20 40.70 23.26

4.0 40.17 35.26 22.10

5.0 35.21 23.50 15.23

6.0 32.10 18.61 2.31

7.0 32.06 5.75 1.16

8.0 31.72 3.23 -

9.0 31.62 1.21 -

After determining the UCL and LCL, using equations (24) and (25), the ARL

results are given in Table 4, it shows that right skew increases and the ARL

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Development of fuzzy extreme value theory control charts 5831

decreases. Lognormal distribution is most efficient at a coefficient of skewness

0.1 and the ARL is maximum. Burr’s distribution is most efficient at a coefficient

of skewness 0.5, 1.0 and 2.0. Weibull distribution is most efficient at a coefficient

of skewness 3.0,4.0,5.0,6.0,7.0,8.0 and 9.0 , see Figure 3.

Fig 3 represents comparision of ARL of Weibull,

Lognormal and burr’s distribution.

4.2 If data is shifted, right skew increases and the ARL decreases. In this study,

Weibull distribution is most efficient at a coefficient of skewness 2.0. Burr’s

distribution is most efficient at a coefficient of skewness 0.1, 0.5, 1.0, 3.0, 4.0, 5.0,

6.0, 7.0, 8.0 and 9.0.

5 Conclusions This study is to calculate the ARL of FEV theory control charts, using α -cut

with the methods of α -level fuzzy midrange for skewed populations which are

Weibull, lognormal and Burr’s distributions. The result of the study is, the ARL of

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5832 R. Intaramo and A. Pongpullponsak

FEV theory control charts which have lognormal distribution is most efficient at a

coefficient of skewness 0.1. Burr’s distribution is most efficient at a coefficient of

skewness 0.5, 1.0 and 2.0. Weibull distribution is most efficient at a coefficient of

skewness 3.0,4.0,5.0,6.0,7.0,8.0 and 9.0. The results of the ARL calculation of

FEV theory control charts at a coefficient of skewness 0.1 of Weibull , lognormal

and Burr’s distributions are, ARL = 212.01, 215.13 and 206.76 respectively. In

this study, the ARL using FEV theory is greater than when using EV theory

studied by A.Pongpullponsak, W. Suracherkiati and R. Intaramo, (2006). It shows

that when fuzzy theory is applied to control charts, the performance is better. For

further research, we may be able to develop control charts by using other methods

such as weighted variance method, scaled weighted variance method and

empirical quantile method. These could then be compared with the results in this

study, or, we may study data under other distributions such as student’s t

distribution etc.

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Received: June, 2012