Aggregation Functions for Decision Making

Abstract

Aggregation functions are generally defined and used to combine several numerical values into a single one, so that the final result of the aggregation takes into account all the individual values in a given manner. Such functions are widely used in many well-known disciplines such as statistics, economics, finance, and computer science. For general background, see Grabisch et al. [52]. For instance, suppose that several individuals form quantifiable judgements either about a measure of an object (weight, length, area, height, volume, importance or other attributes) or about a ratio of two such measures (how much heavier, longer, larger, taller, more important, preferable, more meritorious etc. one object is than another). In order to reach a consensus on these judgements, classical aggregation functions have been proposed: arithmetic mean, geometric mean, median and many others. In multicriteria decision making, values to be aggregated are typically preference or satisfaction degrees. A preference degree reveals to what extent an alternative a is preferred to an alternative b, and thus is a relative appraisal. By contrast, a satisfaction degree expresses to what extent a given alternative is satisfactory with respect to a given criterion. It is an absolute appraisal. We assume that the values to be aggregated belong to numerical scales, which can be of ordinal or cardinal type. On an ordinal scale, numbers have no meaning other than defining an order relation on the scale; distances or differences between values cannot be interpreted. On a cardinal scale, distances between values are not quite arbitrary. There are actually several kinds of cardinal scales. On an interval scale, where the position of the zero is a matter of convention, values are defined up to a positive linear transformation i.e. φ(x) = rx + s, with r > 0 and s ∈ R ∗Chapter 17 of Decision-making Process – Concepts and Methods (ISTE/John Wiley, 2009).