Geoscience Reference
In-Depth Information
will identify supported Pareto-optimal solutions of the linear relaxation of Prob-
lem (
9.6
)-(
9.10
). However, these Pareto-optimal solutions may result in fractional
location variables since Problem (
9.11
)-(
9.14
) is a scalarization of the continuous
version of our original multiobjective location problem. To avoid this inconvenience
we shall solve the binary version of (
9.11
)-(
9.14
), namely
X
M
X
N
Minimize
c
ij
.
B
/x
ij
(9.16)
iD1
jD1
X
N
subject to
x
ij
D
1; i
2
I;
(9.17)
jD1
x
ij
y
j
; i
2
I; j
2
J;
(9.18)
X
N
y
j
D
p;
(9.19)
jD1
x
ij
2f
0;1
g
;y
j
2f
0;1
g
; i
2
I; j
2
J:
(9.20)
Any optimal binary solution of (
9.16
)-(
9.20
) gives a supported Pareto-optimal
solution of our original multiobjective location problem. Repeating the above
process for all feasible basis of Problem (
9.6
)-(
9.10
) will result in a set of supported
Pareto-optimal solutions for the problem.
9.5
Conclusions
In this chapter we have presented and analyzed some of the most important models
of multicriteria location problems considering three different decision spaces:
continuous, networks and discrete. This material provides a general overview of
the state-of-the-art of the field as well as a number of references that can be used by
the interested readers to go for a further analysis of the topic. Emphasis was put on
an efficient (if possible) description of the whole set of Pareto locations.
Acknowledgements
The authors were partially supported by projects FQM-5849 (Junta de
AndalucíanFEDER), Fundación Séneca, grant number 08716/PI/08, the Interuniversity Attraction
Poles Programme initiated by the Belgian Science Policy Office and MTM2010-19576-C02-01/02
(Ministry of Economy and CompetitivenessnFEDER, Spain).