Identification of Multiple Criteria Decision Making

Abstract

Assuming that m alternatives given by A = {a ₁,..., a _m } are evaluated by n criteria, called Z = {z ₁,..., z _n }, we obtain the utility function

$$ u:A \times Z \to R\,with\,u\left( {{a_i},{z_j}} \right) = {u_{{ij}}} $$

(1.1)

In the classical utility theory u is a linear function. If we are able to set weights w ₁,..., w _n with w _k ≧ 0, ${w_k}\; \geqq \;0,\quad \sum\limits_{{k = 1}}^n {{w_k}} = 1$ representing the importance of z ₁,..., z _n , we get a complete preordering ≳ on pairs of alternatives:

$$ {a_i}\begin{array}{*{20}{c}} > \\ {\left( \sim \right)} \\ \end{array} {a_j} \Leftrightarrow \sum\limits_{{k = 1}}^n {{u_{{ik}}}{w_k}} \begin{array}{*{20}{c}} > \\ {\left( \sim \right)} \\ \end{array} \sum\limits_{{k = 1}}^n {{u_{{jk}}}{w_k}} $$

(1.2)

Obviously each preordering depends on the weights of criteria. Changing the weights we may obtain other preorderings.