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in the undirected case and
0
1
0
P v i 2V w i .d.x;v i / C d.v i ;x//
: : :
P v i 2V w i .d.x;v i / C d.v i ;x//
1
f 1 .x/
: : :
f Q .x/
@
A
@
A
f.x/ WD
WD
in the directed case.
Let S P.G/and W
Q .WedefineW par Df f.x/ 2 W W
R
f.y/ 2 W such
X par WD f x 2 S W f.x/ 2
that f.y/dominates f.x/in the objective space} and
X par . A point x 2
X par .S/ is called a Pareto
W par g .IfS D P.G/we simply write
X par .V/ are called Pareto nodes or
location with respect to S, and the elements of
Pareto vertices.
Computing
X par .V/ can simply be done by pairwise comparison of the nodes.
X par we first have to check if a multicriteria version of Hakimi's node domi-
nance result holds (Hakimi 1964 ). For the directed case we even have
For
X par .V/ D
X par . The proof relies on the concavity of the distance functions among the edges
and also on the fact that in the directed casewehavenochoiceonwhichsideto
exit or enter an edge. This implies that the objective function is strictly concave and
therefore the nodes always dominate the edges. For the technical details and the
proofs the reader is referred to Hamacher et al. ( 1999 ). In the case of undirected
networks, this aspect is slightly more complicated as shown in the next example
(Fig. 9.13 ).
Example 9.7 Consider the following network N D .G;`/ with n D 6 nodes and a
distance matrix D D .d ij / i;jD1;:::;6 given by
0
@
1
A
011432
102341
120323
433052
342503
213230
D D
:
Fig. 9.13
Network of Example 9.7
 
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