Multiple Criteria Threshold Decision Making Algorithms

In practice it is quite customary that an alternative is evaluated by means of n ≥ 2 grades x1, ..., xn, each of which taking an integer value from 1 («bad») to m ≥ 3 («perfect»). Thus, a problem arises to rank the set X of all n-dimensional vectors x with integer components from 1 to m. Under the assumption that a low grade in the vector x = (x1, ..., xn) cannot be compensated by (any number of) high grades, in this paper we introduce a notion of the enumerating preference function for the weak order on X, generated by the threshold rule, and evaluate this function explicitly. This permits us also to evaluate all equivalence classes and indifference classes of the weak order. An algorithm of ordering of monotone representatives of indifference classes is given, which corresponds to the weak order on X. A dual model to that considered above is presented including an explicit dual enumerating preference function and the ordering algorithm of corresponding monotone representatives