Determining group judgement
in the case of tied and incomparable alternatives
The purpose of the thesis is to determine the group judgement when indifference and/
or incomparability of alternatives is allowed.
The problem of preference aggregation is frequently encountered in various aspects of people activity. Most popular examples are voting systems, group decision support, rankings of consumer goods or services, resource allocation etc. Generally it is assumed that there is a set of experts (decision makers) who evaluate alternatives from a given set, the decision makers are able to evaluate all the alternatives considered and no ties are allowed in group judgement.
However, in real life frequently this is not the case. Hence considering weak and/ or partial preferences is of importance.
The approach presented in the thesis is as follows.
Experts’ opinions are given in order scale, in the form of binary pairwise comparisons matrices. This makes it possible to apply relational analysis approach to model experts’
preferences.
The binary relations paradigm makes it possible to examine transitivity of expert’s opinions and to consider indifference and/ or incomparability of alternatives in experts’ opinions as well as in group judgement.
The criterion of preference aggregation has been formulated in the form of distance – between an aggregated preference to be solved and the set of experts’ preferences – minimization. The optimization problem has been formulated and solved. Some numerical examples have been given.
The group judgement – in the form of a binary relation of an aggregated preference – has been determined as a solution of a constrained binary optimization problem.
This approach makes it possible to explicitly impose some necessary constraints on the form of the group judgement, namely transitivity and/ or completeness or antisymmetry. It has been shown that in some cases the aggregated preference (i.e. its binary pairwise matrix) can be easily expressed in the form of a preference ordering.
Some numerical aspects of determining the group judgement are mentioned in the thesis and several methods have been described. Special attention has been paid to the standard as well as the generalized (i.e. taking into account indifferent and incomparable alternatives) method of Kemeny’s median.
Key words: weak partial preferences, transitivity, constrained binary optimization problem.
