Aggregation of preferences in multi-criteria problems

Аuthors

Smerchinskaya S. O.^*, Yashina N. P.^*

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: dep805@mai.ru

Abstract

Traditional algorithms of multicriterion search of optimal solutions use the numerical evaluation of alternatives based on various criteria so that reduce to difficulties in accurately estimating using the criteria such as usability, reliability etc. On the other hand the representation of all the criteria on the basis of the unified scale is a difficult problem.
Here a method allowing making the best choice based on the aggregation of information about the preferences on the set of alternatives without specifying their quantitative estimates of the criteria is proposed. For all of the algorithms the software is developed.
Let us consider a set of alternatives A = {a₁, a₂,..., a_n} and a set of criteria K = {K₁, K₂,...,K_m} . For each criteria K_i (i = 1, m) an arbitrary binary relation characterizing the decision maker preferences is on the set of alternatives A defined. Let us denote these relations as p₁, p₂,..., p_m. We need to construct a transitive and reflexive aggregation quasiorder; the choice of optimal solutions will be based on it.
Let us define the preference relations by two ways.If the first one is used the preferences matrix R^t=|| r_ij^t || of the order n where n is a number of alternatives having the elements defined as follows:

if i ≠ j . Here r_ii^t=1 (i = 1,..., n) if p_t is a reflexive relation, otherwise r_ii^t=0. The second approach consists in the representation of the graphs G₁, G₂,..., G_m corresponding to the relations p₁, p₂,..., p_m by its adjacency matrix.
The aggregate relation denoted by ^p must be non-contradictory and be in keeping with the preferences for each used criteria. The non-contradictory of the relation ^p consists in absence contradictory contours containing the alternatives being not equal; also this relation must be transitive.
Minimum sum of distances between the aggregate relation of ñ and the relations p₁, p₂,...,p_m allows taking into account the preferences for each criterion. Let us impose the next condition:

where the distance between the relations is introduced by the following formula:

To find the aggregated preference relation a weighted majority graph introduced in [3] is proposed. A weight defined below is assigned to each arc of this graph:

The algorithm of constructing of the aggregate relation consists in two stages: the destruction of the contradictory contours in the majority graph by removing arcs having the smallest positive weight and construction of the desired quasiorder that can be found on the basis of the transitive closure of the relation p that contains no contradictory contours:


identity relation e provides reflexivity.
The proposed algorithm allows to introduce coefficients of importance k₁, k₂,...,k_m for each quality criterion K₁, K₂,...,K_m.
All algorithms have polynomial complexity.
The proposed method can be used to create an aggregate preference relation for the problems of group choice. In this case the relations p₁, p₂,...,p_m are the individual preferences of experts.

Conclusions

The proposed algorithm allows to construct a noncontradictory aggregated preference relation which is a quasiorder and takes maximum consideration of preferences for each criterion. Based on the aggregated preferences a choice the best alternatives can be made. It can be also noted that making of preferences based on criteria does not require evaluations of alternatives based on the scale of the criteria.

Keywords:

decision-making, optimal choice, majority graph, preference relations, ranging, aggregate preference

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