annals of the university of petroşani ∼ economics ∼ vol. xi - part i ...

Decisions in Negotiations Using Expert Systems and … 155
f(V i ) =
n
∑
j=
1
n
∑
j=
1
p
j
p
r
j
ij
, i = 1, m
(2)
The optimum variant will be that for which f(V i ) takes the maximum value.
The method uses the normalized matrix.
2.4. The diameter method
The diameter method has the advantage that in the hierarchy of variants it
takes into consideration the homogeneity or heterogeneity of the data in respect to the
different criteria. A variant will be homogeneous if it takes close values for all criteria
and will be heterogeneous if it takes very big values for some criteria and very small
values for others, with the presumption that all criteria are alike (minimum or
maximum).
We can build the matrix:
⎛ C1
C
2
... C
n ⎞
⎜
⎟
L=
⎜ 1 L11
L12
... L1n
⎟
(3)
⎜...
... ... ... ... ⎟
⎜
⎟
⎝ m Lm
1
Lm2
... Lmn
⎠
in which the column j, j=1,n contains the variants corresponding to the ordering of the
elements of the assemblage {a 1j , a 2j , ..., a mj } in ascending (descending) order if the
criterion C j is a maximum (minimum).
If a i1j , a i2j , ..., a imj , i=1,m, j=1,n is the ordering of the assemblage {a 1j , a 2j , ...,
a mj }, j=1,n, then L 1j =V i1 , L 2j =V i2 , ..., L mj =V im .
For i=1..m and j=1..n are defined the following functions:
• The estimate function:
n
n
∑[ (
i j
)] j ∑
A:V→R, A(V i ) = m − loc V , C p / p
(4)
• The diameter function:
j=
1
d:V→N d(V i ) = max[
loc(
V , C )] − min[ loc(
V , C )]
(5)
j
i
where
loc : V x C →{1, 2, ..., m} loc(V i ,C i )=k, k = 1 , m ⇔ V i = L kj (6)
• The aggregation function:
Aggr : V→ R , Aggr(V i ) =
j
j
j=
1
i
( V ) + m d( Vi)
A i
−
2
j
j
(7)