Geoscience Reference
In-Depth Information
(1.8)
a
-cut provides a convenient way of performing arithmetic operations on fuzzy sets and
fuzzy numbers including in applying the EP. Let us consider a fuzzy number (a
membership function) as shown in Figure I.1. Let a
!
-cut level intersects at two points
a
and
b
on the membership function. The values of the variable
x
corresponding to points
a
and
b
are
x
1
and
x
2
(
x
1
,
x
2
X
), respectively. Then the set
A
"
contains all possible values
of the variable
X
including and between
x
1
and
x
2
. The
x
1
and
x
2
are commonly referred
as lower and upper bounds of the
!
-cut.
Figure I.1.
Illustration of an
!
-cut of a fuzzy set
I.2
Fuzzy arithmetic
Fuzzy arithmetic is based on the EP introduced by Zadah (1975) and elaborated by Yager
(1986). Application of the EP to a function of fuzzy variables, also called fuzzy numbers
(Kaufmann and Gupta, 1991), has been explained in Subsection 3.2.1. The
implementation of the EP can be distinguished for monotonic and non-monotonic
functions. For a non-monotonic function its implementation requires an algorithm for the
determination of maximum and minimum values of the function to be evaluated. For a
monotonic function the EP can be greatly simplified as mentioned in Subsection 3.2.1.
The EP for a monotonic function can be applied to perform the basic operations
(addition, subtraction, multiplication and division) between two fuzzy numbers, which
are commonly referred to as fuzzy arithmetic. This section presents the fuzzy arithmetic
for some basic operations. Readers are referred to Kaufmann and Gupta (1991) for a
comprehensive coverage of fuzzy arithmetic.
Let and are two fuzzy numbers represented by membership functions as shown
in Figure I.2. The fuzzy arithmetic consists in finding the upper bound (UB) and the
lower bound (LB) of the resulting fuzzy number where the operator (o)
[(+),(!),(x),(:)] at any value of the
!
-cut (
!
[0,1]). The lower and upper bound
values of
and
are shown in Figure I.2.
Search WWH ::
Custom Search