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alternatives
A
k
and
A
l
. Then a
preference intensity measure
(
PIM
k
) is derived for each
alternative
A
k
(step 3) as the sum of the preference intensities of alternative
A
k
regarding the
others alternatives. This is used as the measure of the strength of preference.
DME2
can be implemented as follows:can be implemented as follows:
1.
Compute dominance matrix
D
from the paired dominance values
D'
kl
(2).
2.
If
D
kl
0, then alternative
A
k
is preferred to alternative
A
l
, i.e., the intensity with which
alternative
A
k
is preferred to
A
l
is 1,
PI
kl
=1.
Else (
D
kl
< 0):
If
D
lk
0, then alternative
A
l
dominates alternative
A
k
, therefore, the intensity
with which alternative
A
k
is preferred to
A
l
is 0, i.e.,
PI
kl
=0.
Else note that alternative
A
l
is preferred to alternative
A
k
for those values in
W
kl
(constraints of the optimization problem (2)) that satisfy
D
kl
≤ Σ
i
w
i
u
i
(
x
i
k
) - Σ
i
w
i
u
i
(
x
i
l
) ≤ 0, and
A
k
is preferred to
A
l
for those values in
W
kl
that satisfy 0
≤ Σ
i
w
i
u
i
(
x
i
k
) - Σ
i
w
i
u
i
(
x
i
l
) ≤ -
D
lk
the intensity
A
k
is preferred to
A
l
is
PI
kl
=
D
DD
lk
.
lk
kl
3.
Compute a preference intensity measure for each alternative
A
k
m
PIM
k
=
DP
kl
l
1,
l
k
Rank alternatives according to the
PIM
values, where the best (rank 1) is the alternative with
greatest
PIM
and the worst is the alternative with the least
PIM
.
DME1
and
DME2
, like
AP1
and
AP2
, considered ordinal relations regarding attribute
weights, i.e., DMs ranked attributes in descending order of importance. For this scenario,
Monte Carlo simulation techniques were carried out to analyze their performance and to
compare them with other approaches, such as
surrogate weighting methods
(Stillwell et al.,
1981; Barron & Barrett, 1996) and adapted
classical decision rules
(Salo & Hämäläinen, 2001).
The results showed that
DME2
performs better in terms of the identification of the best
alternative and the overall ranking of alternatives than other dominance measuring methods
proposed by different authors. Also,
DME2
outperforms the adaptation of classical decision
rules and comes quite close to the r
ank-order centroid weights
method, which was identified as
the best approach.
Different cases with incomplete information about weights are considered in (Mateos et al.,
2011b). Specifically, we consider weight intervals, weights fitting independent normal
probability distributions or weights represented by fuzzy numbers (triangular and
trapezoidal). A simulation study was also performed to compare the proposed methods
with the measures reported in Ahn and Park, with classical decision rules and with the
SMAA
and
SMAA-2
methods in the above situations. The results show that
DME2
and
SMAA-2
outperform the other methods in terms of the identification of the best alternative
and the overall ranking of alternatives.
5. Conclusions
Many complex decision-making problems have multiple conflicting objectives in the sense
that further achievement in terms of one objective can occur only at the expense of some
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