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Linear fractional vector optimization problems with many components in the solution sets
Linear fractional vector optimization (LFVO) problems form a
special class of nonconvex multiobjective optimization problems
which has a significant role both in the management science and in
the theory of vector optimization. Up to now, only LFVO problems
with at most two connected components in the solution sets have
been discussed in the literature. We propose some examples of LFVO
problems with three or more connected components in the solution
sets. It is proved that for any integer $m$ there exist LFVO
problems with $m$ objective criteria whose solution sets have
exactly $m$ connected components. Besides, we have solved the
conjecture saying that
$\chi(E(\mbox{P}))\leq \min\{m,\mbox{dim}0^+D+1\},$ where
$\chi(E(\mbox{P}))$ is the number of connected components in the
efficient solution set of a LFVO problem $(\mbox{P})$, $m$ is the
number of the objective criteria of $(\mbox{P})$, and
$\mbox{dim}0^+D$ is the dimension of the recession cone $0^+D$ of
the feasible domain $D$ of $(\mbox{P})$. These new facts are
useful for analyzing the practical problems which can be modeled
as quasiconcave vector maximization problems in general, and as
LFVO problems on unbounded feasible domains in particular.