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ill-known value [ 10 ]. There are several types of fuzzy numbers, commonly used are
triangular and trapezoidal ones, however other shapes are possible as well [ 10 ].
Triangular and trapezoidal fuzzy numbers are often used because calculations with
them and their comparison can be done relatively easily, but it is much better if
calculations and comparisons can be done for any shape of fuzzy numbers.
The most general type of fuzzy numbers that can be utilized for calculations are
so called piecewise linear fuzzy numbers, these fuzzy numbers are de
ned as a set
of
-cuts [ 10 ]. They can approximate any given shape and in their most simple
representation are equal to triangular or trapezoidal fuzzy numbers.
If there is need to combine fuzzy numbers with classic crisp values then crisp
numbers are treated as special case of fuzzy number, where all
α
-cuts are the same
α
degenerative interval [ 10 ].
2.2 Fuzzy Arithmetic
In order to perform basic arithmetic operations with fuzzy numbers there is need for
apparatus that allows and speci
es such operations. The most general form of such
rule is speci
ed by so called extension principle [ 14 ], however this particular
de
nition is complicated in terms of implementation, so alternative approaches that
utilize decomposition theorem and interval arithmetic are used [ 10 ]. The decom-
position theorem states that every fuzzy number (or generally any fuzzy set) A
can
be described by associated sequence of
a
-cuts. An
a
-cut is an interval where all the
objects have membership at
least equal
to
a
. Formally it can is written as:
cut a A ¼ A a ¼ x 2 X j l A ð x Þ a g
[ 10 ]. Such
a
-cut of a fuzzy number is always
closed interval A a ¼½a a ; a a
. The only necessary arithmetic operation for deter-
mination of shortest path is addition, using decomposition theorem and interval
arithmetic the addition of fuzzy number A ; B is [ 15 ]:
A a þ B a ¼½a a þ b a ; a a þ b a
ð 1 Þ
for each
a 2 ½
0
1
. Using this approach the addition of any two fuzzy numbers is
;
possible.
2.3 Possibility Theory
To allow decision making based on fuzzy numbers there is a need for a system that
will allow ranking of fuzzy numbers. There are several such systems however most
of them consider only one point of view on the problem [ 11 ]. The complete set
of ranking indices in the framework of possibility theory was proposed in [ 11 ].
This ranking system uses possibility and necessity measures to determine relation
of two fuzzy numbers.
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