Preference levels for clusters of alternatives

In the decision-making tasks for ranking the alternatives and choosing the best ones,procedures on the digraphs are often used.A digraph of the aggregated relation on a set of alternatives for information of experts or criteria is preconstructed.If the digraph does not contain any cycles,then the Demukron algorithm for partitioning the digraph into levels can be used to order the alternatives by preference.This algorithm cannot be applied if there are clusters consisting of equivalent alternatives.In the paper the algorithm for partitioning an arbitrary digraph of into preference levels is proposed.In contrast to the standard procedure,the digraph of the aggregated relation admits the presence of cycles,and,consequently,of equivalent vertexes-alternatives.The vertexes in any cycle of digraph belong to one level of preference.