Mean quantities: a multicriteria approach

A new approach to defining the concept of the mean quantity for a fixed finite set $X$ of numbers $x_{1}, x _{2},\dots,x_{n}$ is proposed: the distance of an arbitrary point $x$ from each individual point $x_{i}$ is estimated by the distance $f_{i}(x)$ between them, and the distance of a point $x$ from the entire set $X$ is characterized by a vector criterion $(f_{1}(x), f_{2}(x),\dots,f_{n}(x))$; using this criterion, the preference relation in distance is introduced; the mean value is the point $x^*$, non-dominated with regard to this relation. Properties and structure of such averages for several preference relations, including the Pareto relation and the relation generated by information about the equal importance of criteria, are investigated. The relationship between the introduced mean quantities and the main statistical averages (arithmetic mean and median) is clarified. The issues of constructing sets of such averages are considered and an effective method of construction is proposed for the case when equally important criteria have the first ordinal metric scale. The directions of possible generalizations of the introduced concept for the multidimensional case are discussed.