Information Technology Reference
In-Depth Information
Membership functions can be constructed so that to satisfy the side conditions
formulated in advance which can be imposed either to form of obtained
information, or to processing procedure. Examples of such side conditions are as
follows:
•
Membership function shall reflect affinity to a pre-selected standard, and
evaluation objects are points in parametrical space [65];
•
Membership function shall satisfy the conditions of an interval scale [66];
•
Under paired comparison of objects, if one object is estimated as being
β
-times stronger than the other one, the latter shall be only estimated as being
β
1
/
- times stronger than the first one [67];
•
Restrictions on elements of paired comparison matrix should be introduced
when it is not obviously possible to obtain all its values [68], etc.
Example 1.8.
Building of a fuzzy set on the basis of paired comparisons of objects
[69—70]. An expert estimates the public importance of four innovative projects
under a scale presented in table 1.1.
The matrix of paired comparisons is formed as follows: diagonal elements are
equal to 1, and for the elements which are symmetric versus a diagonal,
1
a
=
,
i
=
1
4
j
=
1
4
ij
a
Element
i
a
of a paired comparison matrix is an evaluation of preference of
i
-
th project over
j
-th by their public importance with its effect on an investor's
image taken into account.
The expert composed a matrix of paired comparisons in the following form:
ji
1
5
6
7
⎛
⎞
⎜
⎟
1
1
4
6
⎜
⎟
5
⎜
⎟
1
1
A
=
.
⎜
⎟
1
4
⎜
⎟
6
4
⎜
⎜
⎟
⎟
1
1
1
1
⎝
7
6
4
⎠
Then, it is necessary to define eigenvalues of the matrix A and to select a
maximum eigenvalue. Let us equate to zero a determinant
Α
−
λ
E
=
0
:
1
−
λ
5
6
7
1
1
−
λ
4
6
5
1
1
4
3
=
λ
−
4
λ
−
6
914
λ
−
2
715
=
0
.
1
−
λ
4
6
4
1
1
1
1
−
λ
7
6
4
