Theorem
2 Let b be a |J|-dimensional vector, then for
all
Suppose
that is defined by
for all
then
has the same direction as
.
Proof: For every |J|-dimensional vector b,
for all It follows from Lemma 1 and
that
for all .
From Theorem 2，Kinoshita and Nakanishi mention that the pair of evaluation
rules
and
provides
a consistent overall evaluation vector among all alternatives.
Under the assumption that alternative has the evaluation vector
of
criteria,we
apprlyto
estimating alternative
evaluation vector of criteria from
and then obtain
.Hence,
can be considered as an inverse function of
in the sense as follows:Theorem3
Let b be a|J|-dimensional vector,then
for all
Suppose that
is
defined by (1) for all
then
has
the same direction as
.
Proof:
It follows from definitions of and
that