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
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).
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.
Development of fuzzy extreme value theory control charts 5813
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
5814 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 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
5816 R. Intaramo and A. Pongpullponsak
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
Development of fuzzy extreme value theory control charts 5817
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
5818 R. Intaramo and A. Pongpullponsak
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.
Development of fuzzy extreme value theory control charts 5819
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.
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)
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
(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)
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+ −∞ −
− −⎛ ⎞
= ⎜ ⎟⎜ ⎟⎝ ⎠∫
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= −
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
+ − −
+ ⎛ ⎞ ⎛ ⎞− −= ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠∫
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
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
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
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
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
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
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
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.
References [1] Buckley J.J., Eslami E, Uncertain Probabilities II, The Continuous Case,Soft
Computing 8(2004), 193-199.
[2] Dekkers A.L.M., Einmahl J.H.J., De Haan L, A Moment Estimator for the Index of an Extreme-value Distribution, Annals of Statistics 17(1989), 1833–1855.
[3] El-shal S.M., Morris A.S, A Fuzzy Rule-based Algorithm to Improve the Performance of Statistical Process Control In Quality Systems, Journal of Intelligence Fuzzy Systems 9(2000), 207-223.
Development of fuzzy extreme value theory control charts 5833
[4] Gullbay M., Kahraman C., Ruan D, α-cuts Fuzzy Control Charts for Linguistic Data, International Journal of Intelligent Systems 19(2004), 1173-1196.
[5] Gullbay M., Kahraman C, Development of Fuzzy Process Control Charts and Fuzzy Unnatural Pattern Analyses, Computational Statistics and Data Analysis 51(2006), 434-451.
[6] Gullbay M., Kahraman C, An Alternative Approach to Fuzzy Control Charts : Direct Fuzzy Approach, Information Science 77(2006), 1463-1480.
[7] Juran J. M, Quality Control Handbook (4th ed.), NY : Mc Graw Hill. (1998).
[8] Kanagawa A., Tamaki F., Ohta H, Control Charts for Process Average and Variability Based on Linguistic Data, Intelligent Journal of Production Research 31(4) (1993), 913-922.
[9] Lin, Y.C., Chou C.Y, Non-normality and the variable parameters x control charts, European Journal of Operational Research 176 (2007), 361–373.
[10] Pongpullponsak,A., Suracherkiati W. and Itsarangkurnnaayuttya K, A Comparison of Robust of Exponential Weighted Moving Average Control Chart, Shewhart Control Chart and Synthetic Control Chart for Non-normal Distribution, Proceeding: 4 th Applied Statistics Conference of Northern Thailand, Chiang Mai, Thailand, (2002), May 23-24.
[11] Pongpullponsak,A., Suracherkiati W. and Kriweradechachai P, The Comparison of Efficiency of Control Chart by Weighted Variance Method, Nelson Method, Shewhart Method for Skewed Populations, Proceeding: 5 th Applied Statistics Conference of Northern Thailand, Chiang Mai, Thailand, (2004), May 27-29.
[12] Pongpullponsak, A. , Suracherkiati, W. and Intaramo, R, The Comparison of Efficiency of Control Chart by Weighted Variance Method, Scaled Weighted Variance Method,Empirical Quantiles Method and Extreme-value Theory for Skewed Populations, Kmitl Science Journal, 6 (2006), 456-465.
[13] Pongpullponsak, A., Charongrattanasakul, P, Minimizing the Cost of Integrated Systems Approach to Process Control and Maintenance Model by EWMA Control Chart Using Genetic Algorithm, Expert Systems with Applications, 38 (2011), 5178-5186.
5834 R. Intaramo and A. Pongpullponsak
[14] Raz T., Wang H, Probabilistic and Memberships Approaches in the Construction of Control Charts for Linguistic Data, Production Planning and Control,1(1990), 147-157.
[15] Rolands, H., Wang L.R, An Approach of Fuzzy Logic Evaluation and Control in SPC, Quality Reliability Engineering Intelligent, 16 (2000), 91-98.
[16] Senturk, S., Erginel N, Development of Fuzzy X R− and X S− Control Charts Using α-cuts, Information Science 179(2009), 1542-1551.
[17] Shewhart W.A, Economic Control of Quality of Manufactured Product. NY: Van Nostrand, (1931).
[18] Vermaat M. B., Roxana A. Ion., Ronald JMM, A Comparison of Shewhart Individual Control Charts Based on Normal, Non-parametric, and Extreme-value Theory, Quality and Reliability Engineering International 19(2003), 337-353.
[19] Wang, J.H., Raz T, On The Construction of Control Charts Using Linguistic Variables, Intelligent Journal of Production Research 28(1990), 477-487.
[20] Wang, H, Exact Confidence Coefficient of Confidence Intervals for a Binomial Proportion, Stat. Sin 17(2007), 361-368.
[21] Wang, H, Coverage Probability of Prediction Interval for Discrete Random Variables, Computational Statistics and Data Analysis 53(2008), 17-26.
[22] Wang, H, Exact Average Coverage Probabilities and Confidence Coefficients of Confidence Intervals for Discrete Distributions, Stat. Comput 19(2009), 139-148.
[23] Zadeh, L.A, Fuzzy Sets, Information and Control 8(1965), 338-353.
[24] Zadeh, L.A, Toward A Generalized Theory of Uncertainty (GTU)-an Outline, Information Sciences 172(2005), 1-40.
[25] Zarandi, M.H., Alaeddini A., Turksen I.B, A Hybrid Fuzzy Adaptive Sampling Run Rules for Shewhart Control Charts, Information Sciences 178(2008), 1152-1170.
Received: June, 2012