  In multiple attribute decision analysis (MADA) problems, one often needs to
deal with assessment information with uncertainty. The evidential reasoning
approach is one of the most effective methods to deal with such MADA problems.
As kernel of the evidential reasoning approach, an original evidential
reasoning (ER) algorithm was firstly proposed by Yang et al, and later they
modified the ER algorithm in order to satisfy the proposed four synthesis
axioms with which a rational aggregation process needs to satisfy. However, up
to present, the essential difference of the two ER algorithms as well as the
rationality of the synthesis axioms are still unclear. In this paper, we
analyze the ER algorithms in Dempster-Shafer theory (DST) framework and prove
that the original ER algorithm follows the reliability discounting and
combination scheme, while the modified one follows the importance discounting
and combination scheme. Further we reveal that the four synthesis axioms are
not valid criteria to check the rationality of one attribute aggregation
algorithm. Based on these new findings, an extended ER algorithm is proposed to
take into account both the reliability and importance of different attributes,
which provides a more general attribute aggregation scheme for MADA with
uncertainty. A motorcycle performance assessment problem is examined to
illustrate the proposed algorithm.
