Download Representation of trapezoidal fuzzy numbers with shape function

Annals of Fuzzy Mathematics and Informatics
Volume x, No. x, (Month 201y), pp. 1–xx
ISSN: 2093–9310 (print version)
ISSN: 2287–6235 (electronic version)
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c Kyung Moon Sa Co.
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Representation of trapezoidal fuzzy numbers with
shape function
Salim Rezvani, Mohammad Molani
Received 25 11 2013; Revised 06 01 2014; Accepted 08 01 2014
In this paper, the representation of fuzzy numbers with
shape function in α-cut has been proved. Also, we shows how to treat the
defuzzification, ranking and distance of fuzzy numbers with shape function
by modified concept of Graded Mean Integration Representation method.
Abstract.
2010 AMS Classification: 06D72; 08A72, 47S40
Keywords: Graded Mean Integration Representation, Trapezoidal Fuzzy Numbers, Shape Function, α-cut.
Corresponding Author: Salim Rezvani (salim− [email protected] )
1. Introduction
In1965, Zadeh [14] introduced the concept of fuzzy set theory to meet those problems. In 1978, Dubois and Prade [4] defined any of the fuzzy numbers as a fuzzy
subset of the real line. The graded mean integration representation of generalized
fuzzy number was introduced, by Chen and Hsieh [1], it also had been compared
with some other different methods for representation with several different representation methods. He find that the graded mean representation not only can treat
n generalized fuzzy numbers, but also do not change the results of representation
values after increase (or decrease) a generalized fuzzy number into (from) original
generalized fuzzy numbers group. Chen and Chen [2] presented a method for ranking generalized trapezoidal fuzzy numbers. Rezvani [5]-[12] introduced ranking of
fuzzy numbers and Yong Sik Yun [13] presented a method for generalized triangular fuzzy sets. The method for representation of multiplication operation on fuzzy
numbers was proposed by Ch-Ch Chou [3]. We would like to counter their argument by proving that α-cut method is general enough to deal with different type
of fuzzy arithmetic including exponentiation, extracting nth root, taking logarithm.
In fact we illustrate with examples to show that α-cut method is simpler than their
proposed method. However we do acknowledge that the proposed method has more
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
mathematical beauty than the existing alpha-cut method.
In this paper, the representation of fuzzy numbers with shape function in α-cut has
been proved. Also, we shows how to treat the defuzzification, ranking and distance of
fuzzy numbers with shape function by modified concept of Graded Mean Integration
Representation method.
2. Preliminaries
Definition 2.1. Generally, a generalized fuzzy number A is described as any fuzzy
subset of the real line R, whose membership function µA (x) satisfies the following
conditions,
(i) µA (x) is a continuous mapping from R to the closed interval [0, w], 0 < w ≤ 1,
(ii) µA (x) = 0, for all x ∈ (−∞, a],
(iii) µL (x) = L(x) is strictly increasing on [a, b],
(iv) µA (x) = w, for all [b, c], as w is a constant and 0 < w ≤ 1,
(v) µR (x) = R(x) is strictly decreasing on [c, d],
(vi) µA (x) = 0, for all x ∈ [d, ∞),
where a,b,c,d are real numbers such that a < b ≤ c < d.
Definition 2.2. A normal fuzzy number A with shape function
 x−a n
when x ∈ [a, b),
( b−a )








when x ∈ [b, c]
 w
(2.1)
µA =

n

when x ∈ (c, d]
 ( d−x

d−c )





0
otherwise
where n > 0, will be denoted by
(2.2)
A = (a, b, c, d)n .
If A be non-normal fuzzy number, it will be denoted by
(2.3)
A = (a, b, c, d; w)n .
If n = 1, we simply write A = (a, b, c, d), which is known as a normal trapezoidal
fuzzy number.
Definition 2.3. The set Aα = {x ∈ X | µA (x) ≥ α} is said to be the α-cut of a
fuzzy set A.
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The membership function of a fuzzy set A can be expressed in terms of the characteristic functions of its α-cut according to the formula
(2.4)
µA (x) = supα∈(0,1] min(α, µAα (x)),
where
(2.5)
1
0
µAα (x) =
if
if x ∈ Aα
otherwise.
Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal fuzzy number and
Aα , Bα are the α-cuts of A and B, respectively, we have
(2.6)
1
1
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
and
(2.7)
Bα = [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )] ,
2.1. Arithmetic Operations. In this section, addition and subtraction between
two trapezoidal fuzzy numbers, defined on universal set of real numbers R.
Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal fuzzy number,
then
i) A ⊕ B = (a1 + a2 , b1 + b2 , c1 + c2 , d1 + d2 )
ii) A B = (a1 − d2 , b1 − c2 , c1 − b2 , d1 − a2 ).
3. Addition of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal fuzzy number,
Suppose the normal shape function of A,B is
 x−a1 n
when x ∈ [a1 , b1 ),
( b1 −a1 )








when x ∈ [b1 , c1 ]
 1
(3.1)
µA =

d1 −x n

when x ∈ (c1 , d1 ]

 ( d1 −c1 )





0
otherwise
(3.2)
µB =
 x−a2 n
( b2 −a2 )








 1
when x ∈ [a2 , b2 ),

−x n

( dd22−c
)


2





0
when x ∈ (c2 , d2 ]
when x ∈ [b2 , c2 ]
otherwise
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Then
(3.3)
1
1
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
and
(3.4)
Bα = [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )] .
Therefore Aα , Bα are the α-cuts of A and B, respectively. To calculate addition of
fuzzy numbers A and B we first add the α-cuts of A and B using interval arithmetic.
1
1
1
1
Aα + Bα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] + [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )]
(3.5)
1
1
= [(a1 + a2 ) + α n (b1 + b2 − a1 − a2 ), (d1 + d2 ) − α n (d1 + d2 − c1 − c2 )] .
Now, we find the shape function µA+B (x)
1
x = (a1 + a2 ) + α n (b1 + b2 − a1 − a2 )
(3.6)
⇒α=
(x − (a1 + a2 ))n
(b1 + b2 ) − (a1 + a2 )
a1 + a2 ≤ x ≤ b1 + b2
and
1
x = ((d1 + d2 ) − α n (d1 + d2 − c1 − c2 )
(3.7)
⇒α=
((d1 + d2 ) − x)n
(d1 + d2 ) − (c1 + c2 )
Therefore shape function µA+B (x) is
 (x−(a +a ))n
1
2


 (b1 +b2 )−(a1 +a2 )






 1
(3.8)
µA+B =

((d1 +d2 )−x)n


 (d1 +d2 )−(c1 +c2 )






0
c1 + c2 ≤ x ≤ d1 + d2
a1 + a2 ≤ x ≤ b1 + b2
b1 + b2 ≤ x ≤ c1 + c2
c1 + c2 ≤ x ≤ d1 + d2
otherwise
Example 3.1. Let A = (0.2, 0.33, 0.45, 0.5) and B = (0.15, 0.2, 0.3, 0.4) be two
trapezoidal fuzzy number, Suppose the normal shape function of A,B is
 x−0.2 n
( 0.13 )
when 0.2 ≤ x ≤ 0.35,








when 0.33 ≤ x ≤ 0.45
 1
µA =

0.5−x n

when 0.45 ≤ x ≤ 0.5

 ( 0.05 )





0
otherwise
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µB =
 x−0.15 n
( 0.05 )








 1

n

( 0.4−x

0.1 )






0
when 0.15 ≤ x ≤ 0.2,
when 0.2 ≤ x ≤ 0.3
when 0.3 ≤ x ≤ 0.4
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
and
1
1
Bα = [0.15 + 0.05α n , 0.4 − 0.1α n ] .
Therefore
1
1
1
1
Aα + Bα = [0.2 + 0.13α n , 0.5 − 0.05α n ] + [0.15 + 0.05α n , 0.4 − 0.1α n ]
1
1
= [0.35 + 0.18α n , 0.9 − 0.15α n ]
Now, we get
1
x = 0.35 + 0.18α n ⇒ α =
(x − 0.35)n
0.18
0.35 ≤ x ≤ 0.53
(0.9 − x)n
0.15
0.75 ≤ x ≤ 0.9
and
1
x = 0.9 − 0.15α n ⇒ α =
which gives
µA+B =
 (x−0.35)n

0.18







 1









(0.9−x)n
0.15
0
when 0.35 ≤ x ≤ 0.53,
when 0.53 ≤ x ≤ 0.75
when 0.75 ≤ x ≤ 0.9
otherwise
4. Subtraction of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal fuzzy number,
Suppose the normal shape function of A,B is
 x−a1 n
( b1 −a1 )
when x ∈ [a1 , b1 ),








when x ∈ [b1 , c1 ]
 1
(4.1)
µA =

−x n

)
when x ∈ (c1 , d1 ]
 ( dd11−c

1





0
otherwise
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(4.2)
µB =
 x−a2 n
( b2 −a2 )








 1
when x ∈ [a2 , b2 ),

−x n

( dd22−c
)


2





0
when x ∈ (c2 , d2 ]
when x ∈ [b2 , c2 ]
otherwise
Then
(4.3)
1
1
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
and
(4.4)
Bα = [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )] .
Therefore Aα , Bα are the α-cuts of A and B, respectively. To calculate subtraction of
fuzzy numbers A and B we first add the α-cuts of A and B using interval arithmetic.
1
1
1
1
Aα − Bα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] − [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )]
(4.5)
1
1
= [(a1 − d2 ) + α n (b1 − a1 + d2 − c2 ), (d1 − a2 ) − α n (d1 − c1 + b2 − a2 )] .
Now, we find the shape function µA−B (x)
1
x = (a1 − d2 ) + α n (b1 − a1 + d2 − c2 )
(4.6)
⇒α=
(x − (a1 − d2 ))n
(b1 − a1 + d2 − c2 )
a1 − d2 ≤ x ≤ b1 − c2
and
1
x = (d1 − a2 ) − α n (d1 − c1 + b2 − a2 )
(4.7)
⇒α=
((d1 − a2 ) − x)n
(d1 − c1 + b2 − a2 )
Therefore shape function µA−B (x) is
 (x−(a −d ))n
1
2


(b1 −a1 +d2 −c2 )







 1
(4.8)
µA−B =

((d1 −a2 )−x)n



(d
1 −c1 +b2 −a2 )






0
c1 − b2 ≤ x ≤ d1 − a2
a1 − d2 ≤ x ≤ b1 − c2
b1 − c2 ≤ x ≤ c1 − b2
c1 − b2 ≤ x ≤ d1 − a2
otherwise
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Example 4.1. Let A = (0.2, 0.33, 0.45, 0.5) and B = (0.15, 0.2, 0.3, 0.4) be two
trapezoidal fuzzy number, Suppose the normal shape function of A,B is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )








when 0.33 ≤ x ≤ 0.45
 1
µA =

n

when 0.45 ≤ x ≤ 0.5
 ( 0.5−x
0.05 )






0
otherwise
µB =
 x−0.15 n
( 0.05 )








 1

n

( 0.4−x

0.1 )






0
when 0.15 ≤ x ≤ 0.2,
when 0.2 ≤ x ≤ 0.3
when 0.3 ≤ x ≤ 0.4
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
and
1
1
Bα = [0.15 + 0.05α n , 0.4 − 0.1α n ] .
Therefore
1
1
1
1
Aα − Bα = [0.2 + 0.13α n , 0.5 − 0.05α n ] − [0.15 + 0.05α n , 0.4 − 0.1α n ]
1
1
= [0.23α n − 0.2, 0.35 − 0.1α n ]
Now, we get
1
x = 0.23α n − 0.2 ⇒ α =
(x − 0.2)n
0.23
− 0.2 ≤ x ≤ 0.03
and
1
x = 0.35 − 0.1α n ⇒ α =
(0.0.35 − x)n
0.1
0.25 ≤ x ≤ 0.35
which gives
µA−B =
 (x−0.35)n

0.18







 1









(0.9−x)n
0.15
when − 0.2 ≤ x ≤ 0.03,
when 0.03 ≤ x ≤ 0.25
when 0.25 ≤ x ≤ 0.35
0
otherwise
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5. Multiplication of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal fuzzy number,
Suppose the normal shape function of A,B is
 x−a1 n
when x ∈ [a1 , b1 ),
( b1 −a1 )








when x ∈ [b1 , c1 ]
 1
(5.1)
µA =

d1 −x n

when x ∈ (c1 , d1 ]

 ( d1 −c1 )





0
otherwise
(5.2)
µB =
 x−a2 n
( b2 −a2 )








 1
when x ∈ [a2 , b2 ),

−x n

( dd22−c
)


2





0
when x ∈ (c2 , d2 ]
when x ∈ [b2 , c2 ]
otherwise
Then
1
1
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
(5.3)
and
Bα = [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )] .
(5.4)
Therefore Aα , Bα are the α-cuts of A and B, respectively. To calculate multiplication of fuzzy numbers A and B we first add the α-cuts of A and B using interval
arithmetic.
1
1
1
1
Aα .Bα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )].[a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )]
2
1
2
1
= [α n (b1 − a1 )(b2 − a2 ) + α n (a1 (b2 − a2 ) + a2 (b1 − a1 )) + a1 a2 ,
(5.5)
α n (d1 − c1 )(d2 − c2 ) − α n (d1 (d2 − c2 ) + d2 (d1 − c1 )) + d1 d2 ] .
Now, we find the shape function µA.B (x)
2
1
x = α n (b1 − a1 )(b2 − a2 ) + α n (a1 (b2 − a2 ) + a2 (b1 − a1 )) + a1 a2
⇒α=
[
−(a1 (b2 − a2 ) + a2 (b1 − a1 )) +
p
(a1 (b2 − a2 ) + a2 (b1 − a1 ))2 − 4(b1 − a1 )(b2 − a2 )(a1 a2 − x) n
]
2(b1 − a1 )(b2 − a2 )
a1 a2 ≤ x ≤ b1 b2
(5.6)
and
2
1
x = α n (d1 − c1 )(d2 − c2 ) − α n (d1 (d2 − c2 ) + d2 (d1 − c1 )) + d1 d2
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⇒α=
[
(d1 (d2 − c2 ) + d2 (d1 − c1 )) +
p
(d1 (d2 − c2 ) + d2 (d1 − c1 ))2 − 4(d1 − c1 )(d2 − c2 )(d1 d2 − x) n
]
2(d1 − c1 )(d2 − c2 )
c1 c2 ≤ x ≤ d1 d2
(5.7)
Therefore shape function µA.B (x) is
(5.8) 
√
−(a (b −a )+a (b −a ))+ (a1 (b2 −a2 )+a2 (b1 −a1 ))2 −4(b1 −a1 )(b2 −a2 )(a1 a2 −x) n

 [ 1 2 2 2 1 1
]

2(b1 −a1 )(b2 −a2 )







when a1 a2 ≤ x ≤ b1 b2








 1
µA.B =
√


(d1 (d2 −c2 )+d2 (d1 −c1 ))+ (d1 (d2 −c2 )+d2 (d1 −c1 ))2 −4(d1 −c1 )(d2 −c2 )(d1 d2 −x) n


]
[

2(d1 −c1 )(d2 −c2 )








when c1 c2 ≤ x ≤ d1 d2






0
Example 5.1. Let A = (0.2, 0.33, 0.45, 0.5) and B = (0.15, 0.2, 0.3, 0.4) be two
trapezoidal fuzzy number, Suppose the normal shape function of A,B is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )








when 0.33 ≤ x ≤ 0.45
 1
µA =

n

( 0.5−x
when 0.45 ≤ x ≤ 0.5

0.05 )






0
otherwise
 x−0.15 n
( 0.05 )
when 0.15 ≤ x ≤ 0.2,








when 0.2 ≤ x ≤ 0.3
 1
µB =

n

when 0.3 ≤ x ≤ 0.4
 ( 0.4−x
0.1 )






0
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
and
1
1
Bα = [0.15 + 0.05α n , 0.4 − 0.1α n ] .
Therefore
1
1
1
1
Aα .Bα = [0.2 + 0.13α n , 0.5 − 0.05α n ].[0.15 + 0.05α n , 0.4 − 0.1α n ]
9
b1 b2 ≤ x ≤ c1 c2
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1
2
1
2
= [0.03 + 0.0295α n + 0.0065α n , 0.2 − 0.07α n + 0.005α n ]
Now, we get
1
n
x = 0.03 + 0.0295α + 0.0065α
2
n
⇒α=[
−0.0295 +
√
0.00009025 + 0.026x n
]
0.013
0.03 ≤ x ≤ 0.066
and
1
n
x = 0.2 − 0.07α + 0.005α
2
n
⇒α=[
0.07 +
√
0.0009 + 0.02x n
]
0.01
0.135 ≤ x ≤ 0.2
which gives
µA.B =
 −0.0295+√0.00009025+0.026x n
]
[

0.013







 1
√



[ 0.07+ 0.0009+0.02x
]n

0.01





0
when 0.03 ≤ x ≤ 0.066,
when 0.066 ≤ x ≤ 0.135
when 0.135 ≤ x ≤ 0.2
otherwise
6. Division of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal fuzzy number,
Suppose the normal shape function of A,B is
 x−a1 n
when x ∈ [a1 , b1 ),
( b1 −a1 )







 1
when x ∈ [b1 , c1 ]

(6.1)
µA =

−x n

( dd11−c
)
when x ∈ (c1 , d1 ]


1





0
otherwise
(6.2)
µB =
 x−a2 n
( b2 −a2 )








 1
when x ∈ [a2 , b2 ),

−x n

( dd22−c
)


2





0
when x ∈ (c2 , d2 ]
when x ∈ [b2 , c2 ]
otherwise
Then
(6.3)
1
1
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
and
(6.4)
Bα = [a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )] .
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Therefore Aα , Bα are the α-cuts of A and B, respectively. To calculate division of
fuzzy numbers A and B we first add the α-cuts of A and B using interval arithmetic.
1
1
1
1
Aα /Bα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )]/[a2 + α n (b2 − a2 ), d2 − α n (d2 − c2 )]
1
(6.5)
=[
1
a1 + α n (b1 − a1 ) d1 − α n (d1 − c1 )
].
,
1
1
d2 − α n (d2 − c2 ) a2 + α n (b2 − a2 )
Now, we find the shape function µA/B (x)
1
x=
(6.6)
⇒α=[
a1 + α n (b1 − a1 )
1
n
d2 − α (d2 − c2 )
d2 x − a1
]n
(b1 − a1 ) + x(d2 − c2 )
a1 /d2 ≤ x ≤ b1 /c2
and
1
x=
(6.7)
⇒α=[
d1 − α n (d1 − c1 )
1
a2 + α n (b2 − a2 )
d1 − a2 x
]n
x(b2 − a2 ) + (d1 − c1 )
c1 /b2 ≤ x ≤ d1 /a2
Therefore shape function µA/B (x) is
(6.8)
µA/B =

x−a1
]n
[ (b1 −ad12)+x(d

2 −c2 )







 1
a1 /d2 ≤ x ≤ b1 /c2

d1 −a2 x

[ x(b2 −a
]n


2 )+(d1 −c1 )





0
c1 /b2 ≤ x ≤ d1 /a2
b1 /c2 ≤ x ≤ c1 /b2
otherwise
Example 6.1. Let A = (0.2, 0.33, 0.45, 0.5) and B = (0.15, 0.2, 0.3, 0.4) be two
trapezoidal fuzzy number, Suppose the normal shape function of A,B is
 x−0.2 n
( 0.13 )
when 0.2 ≤ x ≤ 0.35,








when 0.33 ≤ x ≤ 0.45
 1
µA =

0.5−x n

when 0.45 ≤ x ≤ 0.5

 ( 0.05 )





0
otherwise
11
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
µB =
 x−0.15 n
( 0.05 )








 1
when 0.15 ≤ x ≤ 0.2,
when 0.2 ≤ x ≤ 0.3

n

( 0.4−x

0.1 )






0
when 0.3 ≤ x ≤ 0.4
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
and
1
1
Bα = [0.15 + 0.05α n , 0.4 − 0.1α n ] .
Therefore
1
1
1
1
Aα /Bα = [0.2 + 0.13α n , 0.5 − 0.05α n ]/[0.15 + 0.05α n , 0.4 − 0.1α n ]
1
1
=[
0.2 + 0.13α n
,
1
0.4 − 0.1α n
0.5 − 0.05α n
1
0.15 + 0.05α n
]
Now, we get
1
0.2 + 0.13α n
x=
0.4 − 0.1α
⇒α=[
1
n
0.4x − 0.2 n
]
0.13 + 0.1x
0.5 ≤ x ≤ 1.1
and
1
x=
0.5 − 0.05α n
0.15 + 0.05α
1
n
⇒α=[
10 − 3x n
]
x+1
2.25 ≤ x ≤ 3.33
which gives
µA/B =
 0.4x−0.2 n
[ 0.13+0.1x ]








 1

n

[ 10−3x

x+1 ]






0
when 0.5 ≤ x ≤ 1.1,
when 1.1 ≤ x ≤ 2.25
when 2.25 ≤ x ≤ 3.33
otherwise
7. Inverse of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) be a
function of A is










(7.1)
µA =









trapezoidal fuzzy number, Suppose the normal shape
1 n
( bx−a
)
1 −a1
when x ∈ [a1 , b1 ),
1
when x ∈ [b1 , c1 ]
−x n
)
( dd11−c
1
when x ∈ (c1 , d1 ]
0
otherwise
12
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
Then
(7.2)
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
Therefore Aα , is the α-cut of A . To calculate inverse of fuzzy numbers A we first
add the α-cut of A using interval arithmetic.
1/Aα = [
(7.3)
=[
1
1
n
1
a1 + α (b1 − a1 ), d1 − α n (d1 − c1 )
1
,
1
n
1
1
n
d1 − α (d1 − c1 ) a1 + α (b1 − a1 )
]
].
Now, we find the shape function µ1/A (x)
x=
(7.4)
⇒α=[
1
1
d1 − α n (d1 − c1 )
xd1 − 1 n
]
x(d1 − c1 )
1/d1 ≤ x ≤ 1/c1
and
x=
(7.5)
⇒α=[
1
1
n
a1 + α (b1 − a1 )
1 − xa1 n
]
x(b1 − a1 )
Therefore shape function µ1/A (x) is
 xd1 −1 n
[ x(d1 −c1 ) ]








 1
(7.6)
µ1/A =

1−xa1 n

]
 [ x(b

1 −a1 )





0
1/b1 ≤ x ≤ 1/a1
1/d1 ≤ x ≤ 1/c1
1/c1 ≤ x ≤ 1/b1
1/b1 ≤ x ≤ 1/a1
otherwise
Example 7.1. Let A = (0.2, 0.33, 0.45, 0.5) be a trapezoidal fuzzy number, Suppose
the normal shape function of A is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )








when 0.33 ≤ x ≤ 0.45
 1
µA =

n

( 0.5−x
when 0.45 ≤ x ≤ 0.5

0.05 )






0
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
13
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
Therefore
1/Aα = [
=[
1
1
n
1
0.2 + 0.13α , 0.5 − 0.05α n
1
0.5 − 0.05α
1
n
,
1
1
0.2 + 0.13α n
]
]
Now, we get
x=
1
1
0.5 − 0.05α n
⇒α=[
0.5x − 1 n
]
0.05x
2 ≤ x ≤ 2.22
⇒α=[
1 − 0.2x n
]
0.13x
2.86 ≤ x ≤ 5
and
x=
1
0.2 + 0.13α
1
n
which gives
µ1/A =
 0.5x−1 n
[ 0.05x ]








 1
when 2 ≤ x ≤ 2.22
when 2.22 ≤ x ≤ 2.86

n

[ 1−0.2x

0.13x ]






0
when 2.86 ≤ x ≤ 5
otherwise
8. Exponential of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) be a
function of A is










(8.1)
µA =









trapezoidal fuzzy number, Suppose the normal shape
1 n
)
( bx−a
1 −a1
when x ∈ [a1 , b1 ),
1
when x ∈ [b1 , c1 ]
−x n
)
( dd11−c
1
when x ∈ (c1 , d1 ]
0
otherwise
Then
(8.2)
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
Therefore Aα , is the α-cut of A . To calculate exponential of fuzzy numbers A we
first add the α-cut of A using interval arithmetic.
1
1
exp(Aα ) = exp([a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )])
(8.3)
1
1
= [exp(a1 + α n (b1 − a1 )), exp(d1 − α n (d1 − c1 ))] .
Now, we find the shape function µ1/A (x)
1
x = exp(a1 + α n (b1 − a1 ))
14
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
(8.4)
⇒α=[
ln x − a1 n
]
b1 − a1
exp(a1 ) ≤ x ≤ exp(b1 )
and
1
x = exp(d1 − α n (d1 − c1 ))
(8.5)
⇒α=[
d1 − ln x n
]
d1 − c1
exp(c1 ) ≤ x ≤ exp(d1 )
Therefore shape function µexp(A) (x) is
(8.6)
µexp(A) =
 ln x−a1 n
[ b1 −a1 ]








 1
exp(a1 ) ≤ x ≤ exp(b1 )

x n

[ dd11−ln


−c1 ]





0
exp(c1 ) ≤ x ≤ exp(d1 )
exp(b1 ) ≤ x ≤ exp(c1 )
otherwise
Example 8.1. Let A = (0.2, 0.33, 0.45, 0.5) be a trapezoidal fuzzy number, Suppose
the normal shape function of A is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )








when 0.33 ≤ x ≤ 0.45
 1
µA =

0.5−x n

when 0.45 ≤ x ≤ 0.5

 ( 0.05 )





0
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
Therefore
1
1
exp(Aα ) = [exp(0.2 + 0.13α n ), exp(0.5 − 0.05α n )]
Now, we get
1
x = exp(0.2 + 0.13α n ) ⇒ α = [
ln x − 0.2 n
]
0.13
exp(0.2) ≤ x ≤ exp(0.33)
0.5 − ln x n
]
0.05
15
exp(0.45) ≤ x ≤ exp(0.5)
and
1
x = exp(0.5 − 0.05α n ) ⇒ α = [
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
which gives
µexp(A) =
 ln x−0.2 n
[ 0.13 ]








 1
when exp(0.2) ≤ x ≤ exp(0.33)
when exp(0.33) ≤ x ≤ exp(0.45)

x n

[ 0.5−ln


0.05 ]





0
when exp(0.45) ≤ x ≤ exp(0.5)
otherwise
9. Logarithm of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) be a
function of A is










(9.1)
µA =









trapezoidal fuzzy number, Suppose the normal shape
1 n
)
( bx−a
1 −a1
when x ∈ [a1 , b1 ),
1
when x ∈ [b1 , c1 ]
−x n
( dd11−c
)
1
when x ∈ (c1 , d1 ]
0
otherwise
Then
(9.2)
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
Therefore Aα , is the α-cut of A . To calculate logarithm of fuzzy numbers A we first
add the α-cut of A using interval arithmetic.
1
1
ln(Aα ) = ln([a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )])
(9.3)
1
1
= [ln(a1 + α n (b1 − a1 )), ln(d1 − α n (d1 − c1 ))] .
Now, we find the shape function µ1/A (x)
1
x = ln(a1 + α n (b1 − a1 ))
(9.4)
⇒α=[
exp x − a1 n
]
b1 − a1
ln(a1 ) ≤ x ≤ ln(b1 )
and
1
x = ln(d1 − α n (d1 − c1 ))
(9.5)
⇒α=[
d1 − exp x n
]
d1 − c1
ln(c1 ) ≤ x ≤ ln(d1 )
16
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
Therefore shape function µln(A) (x) is
(9.6)
µln(A) =
 exp x−a1 n
[ b1 −a1 ]








 1
ln(a1 ) ≤ x ≤ ln(b1 )

x n


[ d1d−exp
]

1 −c1





0
ln(c1 ) ≤ x ≤ ln(d1 )
ln(b1 ) ≤ x ≤ ln(c1 )
otherwise
Example 9.1. Let A = (0.2, 0.33, 0.45, 0.5) be a trapezoidal fuzzy number, Suppose
the normal shape function of A is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )








when 0.33 ≤ x ≤ 0.45
 1
µA =

n

( 0.5−x
when 0.45 ≤ x ≤ 0.5

0.05 )






0
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
Therefore
1
1
ln(Aα ) = [ln(0.2 + 0.13α n ), ln(0.5 − 0.05α n )]
Now, we get
1
x = ln(0.2 + 0.13α n ) ⇒ α = [
exp x − 0.2 n
]
0.13
ln(0.2) ≤ x ≤ ln(0.33)
0.5 − exp x n
]
0.05
ln(0.45) ≤ x ≤ ln(0.5)
and
1
x = ln(0.5 − 0.05α n ) ⇒ α = [
which gives
µln(A) =
 exp x−0.2 n
[ 0.13 ]








 1

x n

[ 0.5−exp
]


0.05





0
when ln(0.2) ≤ x ≤ ln(0.33)
when ln(0.33) ≤ x ≤ ln(0.45)
when ln(0.45) ≤ x ≤ ln(0.5)
otherwise
17
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10. Square root of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) be a
function of A is










(10.1)
µA =









trapezoidal fuzzy number, Suppose the normal shape
1 n
)
( bx−a
1 −a1
when x ∈ [a1 , b1 ),
1
when x ∈ [b1 , c1 ]
−x n
( dd11−c
)
1
when x ∈ (c1 , d1 ]
0
otherwise
Then
(10.2)
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
Therefore Aα , is the α-cut of A . To calculate square root of fuzzy numbers A we
first add the α-cut of A using interval arithmetic.
q
p
1
1
(Aα ) = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )]
(10.3)
q
q
1
1
= [ a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] .
Now, we find the shape function µ1/A (x)
q
1
x = a1 + α n (b1 − a1 )
(10.4)
⇒α=[
x2 − a1 n
]
b1 − a1
√
a1 ≤ x ≤
p
b1
and
x=
(10.5)
⇒α=[
q
1
d1 − α n (d1 − c1 )
d1 − x2 n
]
d1 − c1
Therefore shape function µ√A (x) is
 2
−a1 n

[ xb1 −a
]

1







 1
√
(10.6)
µ A=


d1 −x2 n


 [ d1 −c1 ]





0
18
√
√
c1 ≤ x ≤
a1 ≤ x ≤
√
√
b1 ≤ x ≤
c1 ≤ x ≤
p
d1
√
√
√
b1
c1
d1
otherwise
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
Example 10.1. Let A = (0.2, 0.33, 0.45, 0.5) be a trapezoidal fuzzy number, Suppose the normal shape function of A is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )







 1
when 0.33 ≤ x ≤ 0.45

µA =

n

( 0.5−x
when 0.45 ≤ x ≤ 0.5

0.05 )






0
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
Therefore
q
q
p
1
1
Aα = [ 0.2 + 0.13α n , 0.5 − 0.05α n ]
Now, we get
x2 − 0.2 n
]
0.13
√
q
0.5 − x2 n
1
0.5 − 0.05α n ⇒ α = [
]
0.05
√
q
x=
1
0.2 + 0.13α n ⇒ α = [
0.2 ≤ x ≤
√
0.33
and
x=
0.45 ≤ x ≤
√
0.5
which gives
µ√A =
 x2 −0.2 n
[ 0.13 ]








 1
when

2

n

[ 0.5−x

0.05 ]





0
when
when
√
√
√
0.2 ≤ x ≤
√
0.33 ≤ x ≤
0.45 ≤ x ≤
0.33
√
√
0.5
0.5
otherwise
11. nth root of fuzzy numbers with using α-Cut
Let A = (a1 , b1 , c1 , d1 ) be a
function of A is










(11.1)
µA =









trapezoidal fuzzy number, Suppose the normal shape
1 n
)
( bx−a
1 −a1
when x ∈ [a1 , b1 ),
1
when x ∈ [b1 , c1 ]
−x n
( dd11−c
)
1
when x ∈ (c1 , d1 ]
0
otherwise
Then
(11.2)
1
1
Aα = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] ,
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Therefore Aα , is the α-cut of A . To calculate nth root of fuzzy numbers A we first
add the α-cut of A using interval arithmetic.
1
1
1
1
1
1
(Aα ) n = [a1 + α n (b1 − a1 ), d1 − α n (d1 − c1 )] n
1
1
= [(a1 + α n (b1 − a1 )) n , (d1 − α n (d1 − c1 )) n ] .
(11.3)
Now, we find the shape function µ
1
An
(x)
1
1
x = (a1 + α n (b1 − a1 )) n
⇒α=[
(11.4)
xn − a1 n
]
b1 − a1
√
n
a1 ≤ x ≤
p
n
b1
and
1
1
x = (d1 − α n (d1 − c1 )) n
d1 − xn n
]
d1 − c1
n
Therefore shape function µ √
A (x) is
 xn −a1 n
[ b1 −a1 ]








 1
(11.6)
µ√A =
n

n


[ dd11−x

−c1 ]





0
⇒α=[
(11.5)
√
n
√
n
c1 ≤ x ≤
a1 ≤ x ≤
√
n
√
n
b1 ≤ x ≤
c1 ≤ x ≤
p
n
d1
√
n
√
n
√
n
b1
c1
d1
otherwise
Example 11.1. Let A = (0.2, 0.33, 0.45, 0.5) be a trapezoidal fuzzy number, Suppose the normal shape function of A is
 x−0.2 n
when 0.2 ≤ x ≤ 0.35,
( 0.13 )








when 0.33 ≤ x ≤ 0.45
 1
µA =

n

when 0.45 ≤ x ≤ 0.5
( 0.5−x

0.05 )






0
otherwise
1
1
Aα = [0.2 + 0.13α n , 0.5 − 0.05α n ] ,
Therefore
q
q
p
n
n
1
1
n
Aα = [ 0.2 + 0.13α n , 0.5 − 0.05α n ]
Now, we get
q
x=
n
1
0.2 + 0.13α n ⇒ α = [
xn − 0.2 n
]
0.13
20
√
n
0.2 ≤ x ≤
√
n
0.33
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
and
q
x=
n
1
0.5 − 0.05α n ⇒ α = [
which gives
n
µ√
A =
 xn −0.2 n
[ 0.13 ]








 1
n

n


[ 0.5−x

0.05 ]





0
√
n
0.5 − xn n
]
0.05
when
when
when
√
n
√
n
√
n
0.45 ≤ x ≤
0.2 ≤ x ≤
√
n
0.33 ≤ x ≤
0.45 ≤ x ≤
√
n
0.5
0.33
√
n
√
n
0.5
0.5
otherwise
12. Representation of fuzzy number with Normal Shape Function
Definition 12.1. Let A = (a1 , b1 , c1 , d1 ; wa ) be a trapezoidal fuzzy number, then
the graded mean integration representation of A is defined by
Z wa
Z wa
L−1 (h) + R−1 (h)
) dh/
h dh .
P (A) =
h(
2
0
0
Theorem 12.2. Let A = (a, b, c, d) be a trapezoidal fuzzy number with normal shape
function, where a,b,c,d are real numbers such that a < b ≤ c < d. Then the graded
mean integration representation of A is
P (A) =
(a + d)
n
+
(b − a − d + c)
2
2n + 1
Proof.
L(x) = (
x−a n
)
b−a
R(x) = (
d−x n
)
d−c
So
1
L−1 (h) = a + (b − a)h n
1
R−1 (h) = d − (d − c)h n
1
P (A) =
2
Z
1
1
n
1
n
Z
1
h[(a + (b − a)h ) + (d − (d − c)h )] dh/
0
h dh
0
Z
Z 1
1
1 1
n
=
h dh
h[(a + d) + (b − a − d + c)h ] dh/
2 0
0
Z
Z 1
n+1
1 1
=
[(a + d)h + (b − a − d + c)h n ] dh/
h dh
2 0
0
=
1 (a + d)
n
1
(a + d)
n
+
(b − a − d + c) .
[
+
(b − a − d + c)]/ =
2
2
2n + 1
2
2
2n + 1
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Theorem 12.3. Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal
fuzzy number with normal shape function and P (A) and P (B) are the representation
of A and B respectively. Then
i) P (m ⊕ A) = mP (A),
ii) P (A ⊕ B) = P (A) ⊕ P (B),
iii) P (A B) = P (A) − P (B).
Proof. i) Since m ⊕ A = (ma1 , mb1 , mc1 , md1 ), Then
(ma + md)
n
+
(mb − ma − md + mc) = mP (A) .
2
2n + 1
ii) We have A ⊕ B = (a1 + a2 , b1 + b2 , c1 + c2 , d1 + d2 ), Then
P (m ⊕ A) =
P (A ⊕ B) =
=[
((a1 + a2 ) + (d1 + d2 ))
n
+
((b1 + b2 ) − (a1 + a2 ) − (d1 + d2 ) + (c1 + c2 ))
2
2n + 1
n
(a2 + d2 )
n
(a1 + d1 )
+
(b1 − a1 − d1 + c1 )] + [
+
(b2 − a2 − d2 + c2 )]
2
2n + 1
2
2n + 1
= P (A) ⊕ P (B) .
iii) We have A B = (a1 − d2 , b1 − c2 , c1 − b2 , d1 − a2 ), Then
P (A B) =
((a1 − a2 ) + (d1 − d2 ))
n
+
((b1 − b2 ) − (a1 − a2 ) − (d1 − d2 ) + (c1 − c2 ))
2
2n + 1
P (A B) = [
n
(a2 + d2 )
n
(a1 + d1 )
+
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )]
2
2n + 1
2
2n + 1
= P (A) P (B) .
Definition 12.4. Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal
fuzzy numbers, then A > B if and only if P (A) > P (B) .
Definition 12.5. Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal
fuzzy numbers, then A = B if and only if P (A) = P (B) .
Theorem 12.6. Suppose there are n alternative fuzzy numbers of modified fuzzy
numbers and E = {Ai | i = 1, 2, ..., n} . We can prove that
i) every Ai , Aj ∈ E, then Ai ≥ Aj or Aj ≥ Ai ,
ii) if Ai ≥ Aj and Aj ≥ Ai , then Ai = Aj , ∀Ai , Aj ∈ E,
iii) if Ai ≥ Aj and Aj ≥ Ak , then Ai > Ak , ∀Ai , Aj , Ak ∈ E,
Now, we define distance of trapezoidal fuzzy numbers.
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S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
Definition 12.7. Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) be two trapezoidal
fuzzy numbers, then the distance measure of A and B is
(12.1)
d(A, B) =| P (A) − P (B) | .
Theorem 12.8. Let A = (a1 , b1 , c1 , d1 ) and B = (a2 , b2 , c2 , d2 ) and C = (a3 , b3 , c3 , d3 )
be three trapezoidal fuzzy numbers, then their distance measure satisfies the following
relations
i) d(A, A) = 0 ,
ii) d(A, B) = d(B, A) ,
iii) if A ≤ B ≤ C, then d(A, B) ≤ d(A, C) ,
iv) d(A, B) ≤ d(A, C) + d(C, B) .
Proof. i)
d(A, A) =| P (A) − P (A) |
=| [
n
(a1 + d1 )
n
(a1 + d1 )
+
(b1 − a1 − d1 + c1 )] − [
+
(b1 − a1 − d1 + c1 )] |
2
2n + 1
2
2n + 1
=| [
n
n
(a1 + d1 ) (a1 + d1 )
−
]+[
(b1 − a1 − d1 + c1 ) −
(b1 − a1 − d1 + c1 )] |= 0
2
2
2n + 1
2n + 1
ii)
d(A, B) =| P (A) − P (B) |
=| [
(a1 + d1 )
n
(a2 + d2 )
n
+
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )] |
2
2n + 1
2
2n + 1
=| [
(a2 + d2 )
n
(a1 + d1 )
n
+
(b2 − a2 − d2 + c2 )] − [
+
(b1 − a1 − d1 + c1 )] |
2
2n + 1
2
2n + 1
=| P (B) − P (A) |= d(B, A)
iii) We have B ≤ C, So
P (B) ≤ P (C)
⇒[
(a2 + d2 )
n
(a3 + d3 )
n
+
(b2 − a2 − d2 + c2 )] ≤ [
+
(b3 − a3 − d3 + c3 )]
2
2n + 1
2
2n + 1
⇒[
(a2 + d2 )
n
(a1 + d1 )
n
+
(b2 − a2 − d2 + c2 )] − [
+
(b1 − a1 − d1 + c1 )]
2
2n + 1
2
2n + 1
≤[
(a3 + d3 )
n
(a1 + d1 )
n
+
(b3 − a3 − d3 + c3 )] − [
+
(b1 − a1 − d1 + c1 )]
2
2n + 1
2
2n + 1
⇒| [
(a2 + d2 )
n
(a1 + d1 )
n
+
(b2 − a2 − d2 + c2 )] − [
+
(b1 − a1 − d1 + c1 )] |
2
2n + 1
2
2n + 1
23
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
≤| [
(a3 + d3 )
n
(a1 + d1 )
n
+
(b3 − a3 − d3 + c3 )] − [
+
(b1 − a1 − d1 + c1 )] |
2
2n + 1
2
2n + 1
Of (ii), we have
⇒| [
(a1 + d1 )
n
(a2 + d2 )
n
+
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )] |
2
2n + 1
2
2n + 1
=| P (A) − P (B) |
≤| [
(a1 + d1 )
n
(a3 + d3 )
n
+
(b1 − a1 − d1 + c1 )] − [
+
(b3 − a3 − d3 + c3 )] |
2
2n + 1
2
2n + 1
=| P (A) − P (C) |
⇒| P (A) − P (B) |≤| P (A) − P (C) |⇒ d(A, B) ≤ d(A, C) .
iv)
d(A, B) ≤ d(A, B) =| P (A) − P (B) |≤| P (A) − P (B) |⇒
|[
n
(a2 + d2 )
n
(a1 + d1 )
+
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )] |
2
2n + 1
2
2n + 1
≤| [
n
(a2 + d2 )
n
(a1 + d1 )
+
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )] |
2
2n + 1
2
2n + 1
⇒| [
≤| [
(a1 + d1 )
n
(a2 + d2 )
n
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )] |
+
2
2n + 1
2
2n + 1
(a1 + d1 )
n
(a2 + d2 )
n
+
(b1 − a1 − d1 + c1 )] − [
+
(b2 − a2 − d2 + c2 )]±
2
2n + 1
2
2n + 1
[
≤| [
|[
(a3 + d3 )
n
+
(b3 − a3 − d3 + c3 )] |
2
2n + 1
(a1 + d1 )
n
(a3 + d3 )
n
(b3 − a3 − d3 + c3 )] | +
+
(b1 − a1 − d1 + c1 )] − [
+
2
2n + 1
2
2n + 1
(a3 + d3 )
n
(a2 + d2 )
n
+
(b3 − a3 − d3 + c3 )] − [
+
(b2 − a2 − d2 + c2 )] |
2
2n + 1
2
2n + 1
≤| P (A) − P (A) | + | P (C) − P (B) |= d(A, C) + d(C, B) ⇒ d(A, B) ≤ d(A, C) + d(C, B) .
24
S. Rezvani./Ann. Fuzzy Math. Inform. x (201y), No. x, xx–xx
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Salim Rezvani (salim− [email protected])
Department of Industrial Engineering, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran
Mohammad Molani ([email protected])
Department of Industrial Engineering, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran
25