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

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). *

Corresponding author.

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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.

Development of fuzzy extreme value theory control charts

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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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R. Intaramo and A. Pongpullponsak

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

Development of fuzzy extreme value theory control charts

5815

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 and α -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 θ > 0 and β > 0 . Density function

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R. Intaramo and A. Pongpullponsak

f ( x;θ , β ) =

β β −1 − ( x /θ )β x e θβ

x>0

Cumulative distribution function F ( x;θ , β ) = 1 − e − ( x /θ )

β

x>0

Where

θ is scale parameter

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

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

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

distribution Coefficient of skewness (α 3 )

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

Development of fuzzy extreme value theory control charts

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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 ⎞ ⎛⎛ ⎞ ⎜ ⎜ ln x − μ ⎟ ⎟ 1 ⎠ ⎟ x>0 f ( x; μ , σ ) = ∧ exp ⎜ ⎝ ∧ ⎜ ⎟ x σ 2π 2σ 2 ⎜ ⎟ ⎝ ⎠

where

μ is scale parameter σ is shape parameter In this study μ = 0.1, 0.5,1, 2,3, 4,5, 6, 7,8,9 and σ are relevant with a coefficient of skewness at α 3 ∈ {0.1, 0.5,1, 2,3, 4,5, 6, 7,8,9} shown in table 2.

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

Coefficient of skewness (α 3 ) 0.1 0.5 1 2 3 4 5 6

Shape parameters (σ ) 0.0334 0.1641 0.3142 0.5513 0.7156 0.8326 0.9202 0.9889

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R. Intaramo and A. Pongpullponsak

Table 2 (continued)

Shape parameters (σ )

Coefficient of skewness (α 3 ) 7 8 9

1.0446 1.0911 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 ⎧ mcx c −1 ⎪ m +1 f ( x) = ⎨ (1 + x c ) ⎪ ⎩0

x≥0 x 0

where c, m ≥ 1 Burr’s distribution can be represented as Weibull distribution when m increases. α 3 and α 4 can be obtained by cm > 3 and cm > 4 respectively, where

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

This study is configured as shown in table 3.

Development of fuzzy extreme value theory control charts

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

distribution Coefficient of

Coefficient of

skewness (α 3 )

kurtosis (α 4 )

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 cm < 4 .

**

denotes without any constant.

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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 γ k , we can use the moment estimator to calculate an approximate value as follows : ∧

γk =M

(1) k

1 ⎧ ( M k(1) ) 2 ⎫ + 1 − ⎨1 − ⎬ M k(2) ⎭ 2⎩

−1

(1)

and _ (1) ⎧ ⎫ − 1 ⎪ ( M k )2 ⎪ γ k = M k + 1 − ⎨1 − _ (2) ⎬ 2⎪ M k ⎭⎪ ⎩

−1

_ (1)

(2)

The study of Dekker (1989), F is the q-quantile of the distribution function, so ∧ −1

∧

∧

F k (1 − q; γ k ) = X ( k − m ) +

(m / kq ))γ k − 1 ∧

γk

with 0 < q < 1 . x ∧ y and x ∨ y

∧

(1 − (γ k ∧ 0)) X ( k − m ) M k(1)

(3)

denotes the minimum and maximum

respectively. Define M k( r ) =

1 m (log X ( k − n +1) − log X ( k − m ) ) r ∑ m n =1

(4)

1 m (log X ( n ) − log X ( m +1) ) r ∑ m n =1

(5)

and _ (r )

Mk =

Development of fuzzy extreme value theory control charts

5821

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 ∧

UCL = X ( k − m ) +

(m / kq))γ k − 1 ∧

γk

∧

(1 − (γ k ∧ 0)) X ( k − m ) M k(1)

−

LCL = X ( m +1) +

(m / kα / 2))γ k − 1 −

γk

−

(6)

_ (1)

(1 − (γ k ∧ 0)) X ( m +1) M k (7)

where n

_ (r )

Mk =

∑M k =1

(r ) k

n

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

and M k( r ) is the moment estimator. Hence, we must approximate the value of M k( r ) by using binomial theorem of skewed populations which are Weibull, lognormal and Burr’s distributions, see equation (9), (10) and (11) respectively. Estimator of M k( r ) of Weibull distribution

M k( r ) = ⎡⎣ E ( x − μ ) k ⎤⎦

r

By binomial theorem so

M

(r ) k

⎡ k ⎛k ⎞ ⎤ = ⎢ ∑ ⎜ ⎟ ( − μ )i E ( x k −i ) ⎥ ⎣ i =0 ⎝ i ⎠ ⎦

r

(8)

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R. Intaramo and A. Pongpullponsak

∞

Find

E ( X k −i ) = ∫ x k −i 0

β β −1 − ( x /θ )β x e dx θβ

β ∞ k + β −i −1 − ( x /θ )β = β ∫x e dx θ 0 β

Let

xβ ⎛x⎞ y=⎜ ⎟ = β θ ⎝θ ⎠ x β = yθ β x = y1/ β θ dx =

1

1

−1

y β θ dy

β

β E ( X k −i ) = β θ

⎛ β1 ⎞ ∫0 ⎜⎜ y θ ⎟⎟ ⎝ ⎠

∞

k + β − i −1

⎛ β1 ⎞ = β −1 ∫ ⎜ y θ ⎟ ⎟ θ 0 ⎜⎝ ⎠ 1

∞

θ k + β −i −1 ∞ ⎛ β = ⎜y θ β −1 ∫0 ⎜⎝

1

=θ

k −i

=θ

k −i

=θ

k −i

e

1

−y

β

k + β −i −1

y

1 −y

e y

⎞ ⎟⎟ ⎠

1

k + β − i −1

β

dy

1

⎛ k + ββ−i −1+ β1 −1 ⎞ − y ⎟⎟ e dy ∫0 ⎜⎜ y ⎝ ⎠ ⎛ k +ββ −i −1 ⎞ − y ⎟⎟ e dy ∫0 ⎜⎜ y ⎝ ⎠

θ dy

−1

1 −1 ⎞ ⎛ k + ββ−i −1 −y β × y y ⎜ ⎟⎟ e dy ∫0 ⎜ ⎝ ⎠

∞

−1

−1

e − y y β dy

∞

∞

β

Development of fuzzy extreme value theory control charts

5823

⎛ k + β −i ⎞ = θ k −iτ ⎜ ⎟ β ⎝ ⎠

From equation (8) so

M

⎡ k ⎛k ⎞ ⎛ k + β − i ⎞⎤ = ⎢ ∑ ⎜ ⎟ (− μ )i θ k −iτ ⎜ ⎟⎥ β ⎝ ⎠⎦ ⎣ i =0 ⎝ i ⎠

(r ) k

r

Estimator of M k( r ) of lognormal distribution M k( r ) = ⎡⎣ E ( x − μ ) k ⎤⎦

r

By binomial theorem so

M

Find

(r ) k

E( X

⎡ k ⎛k ⎞ ⎤ = ⎢ ∑ ⎜ ⎟ ( − μ )i E ( x k −i ) ⎥ ⎣ i =0 ⎝ i ⎠ ⎦

k −i

∞

) = ∫ x k −i 0

Let

1 ∧

x σ 2π

e

r

⎛⎛ ∧ ⎞2 ⎞ ⎜ ⎜ ln x − μ ⎟ ⎟ ⎜ ⎠ ⎟ −⎜ ⎝ ⎟ ∧ ⎜ 2σ 2 ⎟ ⎜ ⎟ ⎝ ⎠

dx

y = ln x

x = ey dx = e y dy

∞

E ( X k − i ) = ∫ (e y ) k −i 0

1 ∧

e σ 2π

= M y (k − i )

y

e

⎛ ⎛ ∧ ⎞2 ⎞ ⎜ ⎜ y−μ ⎟ ⎟ ⎜ ⎟ −⎜ ⎝ ∧ ⎠ ⎟ ⎜ 2σ 2 ⎟ ⎜ ⎟ ⎝ ⎠

dx

(9)

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R. Intaramo and A. Pongpullponsak

∧

∧

=e

μ ( k −i ) +

σ 2 ( k −i )2 2

From equation (8) so ∧ ⎡ k σ 2 ( k −i )2 ⎤ μ ( k −i ) + k ⎛ ⎞ i ⎥ 2 = ⎢ ∑ ⎜ ⎟ (− μ ) e ⎢ i =0 ⎝ i ⎠ ⎥ ⎣⎢ ⎦⎥ ∧

M k( r )

r

(10)

Estimator of M k( r ) of Burr’s distribution M k( r ) = ⎡⎣ E ( x − μ ) k ⎤⎦

r

By binomial theorem so

M

Find

(r ) k

E( X

⎡ k ⎛k ⎞ ⎤ = ⎢ ∑ ⎜ ⎟ ( − μ )i E ( x k −i ) ⎥ ⎣ i =0 ⎝ i ⎠ ⎦ k −i

∞

) = ∫ x k −i 0

Let

mcx c −1

(1 + x )

c m +1

r

dx

k −i = v

y = 1 + xc , 0 < y < 1 1

⎛ 1− y ⎞c x=⎜ ⎟ ⎝ y ⎠

1

1 ⎛ 1− y ⎞c J = dx = ⎜ ⎟ c⎝ y ⎠

−1

1

⎛ 1− y ⎞c = ∫ mcy m +1 ⎜ ⎟ ⎝ y ⎠ 0 1

1 dy y2

( v + c −1)

1

1 ⎛ 1− y ⎞c ⎜ ⎟ c⎝ y ⎠

−1

1 dy y2

Development of fuzzy extreme value theory control charts

5825

v

1

= ∫ my

m +1

0

1

= ∫ my

⎛ 1− y ⎞c 1 ⎜ ⎟ 2 dy ⎝ y ⎠ y

v m +1− − 2 c

v

(1 − y ) c dy

0

1

= ∫ my

v m − −1 c

v

(1 − y ) c

+1−1

dy

0

v v⎞ ⎛ = mβ ⎜ m − ,1 + ⎟ c c⎠ ⎝ from

v = k −i

k −i k −i ⎞ ⎛ = mβ ⎜ m − ,1 + ⎟ c c ⎠ ⎝ k −i ⎞ ⎛ k −i ⎞ ⎛ mτ ⎜ m − ⎟τ ⎜1 + ⎟ c ⎠ ⎝ c ⎠ ⎝ = τ (m + 1)

From equation (8) so

M k( r )

⎡ k −i ⎞ ⎛ k −i ⎞⎤ ⎛ mτ ⎜ m − ⎟τ ⎜ 1 + ⎟ ⎢ k ⎛k ⎞ c ⎠ ⎝ c ⎠⎥ ⎝ i ⎥ = ⎢ ∑ ⎜ ⎟ (− μ ) τ (m + 1) ⎢ i =0 ⎝ i ⎠ ⎥ ⎢⎣ ⎥⎦

r

(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 f mrα is defined as the midpoint of the α -level

5826

R. Intaramo and A. Pongpullponsak

cuts. Let Aα is α -level cuts, nonfuzzy sets that consist of any elements whose membership is greater than or equal to α . If aα and bα are end points of Aα then

1 f mrα = (aα + cα ) 2

(12)

In fact the fuzzy mode is a special case of α -level fuzzy midrange when α = 1 . α α -level fuzzy midrange of sample j , Smr , j is determined by

α S mr ,j =

(a j + c j ) + α ⎡⎣(b j − a j ) − (c j − b j ) ⎤⎦

(13)

2 ~

The definition of α -level fuzzy midrange of sample j for fuzzy x control chart is α = S mr − x, j

( x a j + x c j ) + α ⎡⎣( x b j − x a j ) − ( x c j − x b j ) ⎤⎦ 2

(14)

Then, the condition of process control for each sample can be defined as α α α ⎪⎧in − control for LCLmr−x ≤Smr−x, j ≤UCLmr−x ⎪⎫ Pr ocess control = ⎨ ⎬ out −of control for otherwise ⎪⎩ ⎪⎭

(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

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 ∧

U CL = X ( k − m ) + ∧

= X ( k − m ),a +

(m / kq ))

γ k ,a

∧

γ k ,a ∧

, X ( k − m ),c +

−1

(m / kq))

γ k ,c

−1

∧

γ k ,c

(m / kq))γ k − 1 ∧

γk

∧

(1 − (γ k ∧ 0)) X ( k − m ) M k(1) ∧

∧

(1 − (γ k ,a ∧ 0)) X ( k − m ),a M

(1) k ,a

, X ( k − m ),b +

(m / kq ))

γ k ,b

∧

γ k ,b

∧

LCL = X ( m+1) +

(m / kα / 2))γ k −1 −

γk

−

γ

(m / kα / 2) k ,a −1 −

γ k ,a

−

(m / kα / 2) k ,c −1 −

γ k ,c

−

_ (1)

−

(1 − (γ k ∧ 0)) X ( m+1) M k

(17)

−

_ (1)

(1− (γ k ,a ∧ 0)) X (m+1),a M k ,a , X(m+1),b +

−

γ

, X(m+1),c +

∧

(1 − (γ k ,b ∧ 0)) X ( k − m ),b M k(1),b

(1 − (γ k ,c ∧ 0)) X ( k − m ),c M k(1),c

−

= X (m+1),a +

−1

(16)

γ

(m / kα / 2) k ,b −1 −

γ k ,b

−

_ (1)

(1− (γ k ,c ∧ 0)) X (m+1),c M k ,c

μ

α

0

α a a b

cα

_ (1)

(1− (γ k ,b ∧ 0)) X (m+1),b M k ,b

c

Fig 1 represents of a sample of triangular fuzzy number

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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 X ( k − m ),a , X ( k − m ),c , M k(1),a , M k(1),c are determined as follows: X α ( k − m ),a = X ( k − m ),a + α ( X ( k − m ),b − X ( k − m ), a )

(18)

X α ( k − m ),c = X ( k − m ),c − α ( X ( k − m ),c − X ( k − m ),b )

(19)

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

(20)

( M k(1),c )α = ( M k(1),c ) − α (( M k(1),c ) − ( M k(1),b ))

(21)

Therefore, the α -cut FEV theory control limits are ∧

U CLα = X α ( k − m ) +

(m / kq ))γ k − 1 ∧

γk

∧

(1 − (γ k ∧ 0)) X α ( k − m ) ( M k(1) )α

−

LCLα = X α ( m +1) +

(m / kα / 2))γ k − 1 −

γk

−

_ (1)

(1 − (γ k ∧ 0)) X α ( m +1) ( M k )α

(22)

(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 for α -cut FEV theory control limits

Development of fuzzy extreme value theory control charts

5829

∧

α γk ∧ ⎛ α ⎞ (Mk(1),a )α + (Mk(1),c )α % α = ⎜ X (k −m),a + X (k −m),c ⎟ + (m / kq∧ )) −1 (1− (γ ∧ 0)) X α ( ) (24) UCL k (k −m) mr ⎜ ⎟ γk 2 2 ⎝ ⎠ −

(1)

(1)

− ⎛ X α (m+1),a + X α (m+1),c ⎞ (m / kα / 2)γ k −1 (M k,a )α + (M k,c )α α % (1− (γ k ∧ 0)) X (m+1) ( ) (25) LCLmr = ⎜ ⎟⎟ + − ⎜ γk 2 2 ⎝ ⎠

α

For an approximation of

M k( r ) of Weibull, lognormal and Burr’s distributions

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

μ UCL2

UCL3

UCLα3

CL2

CL3 UCLα1 CLα3 UCL1

LCL2

CLα1

α CL1 LCL3 LCL3

UCLα1

LCL1

0

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

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 n = 300 , number of sample size k = 10 are randomly generated from Weibull, lognormal and Burr’s distributions with θ = 1, β relevant with a coefficient of skewness shown in table

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R. Intaramo and A. Pongpullponsak

1, μ = 0, 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 0.5σ ,1.0σ , 2.0σ , 2.5σ and 3.0σ . From this study, the results are as :

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

Coeffcient of

Weibull

Lognormal

Burr’s

skewness (α 3 )

distribution

distribution

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

Development of fuzzy extreme value theory control charts

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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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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