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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).
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