Interval-Based Multicriteria Decision Making

Abstract

This chapter presents a simple computation scheme, combining the Choquet integral with interval arithmetic that allows giving intervals of preferences over multidimensional alternatives. Interval arithmetic is an arithmetic over sets of real numbers called intervals that has been proposed to model uncertainty, and to tackle rounding errors of numerical computations. Interval arithmetic is particularly appropriate to represent outer approximations of real quantities. The range of a real function f over a domain D, denoted by fu(D), can be computed by interval extensions. The most common extension is known as the natural extension. Natural extensions are obtained from the expressions of real functions, and are inclusion monotonic. Because natural extensions are defined by the syntax of real expressions, two equivalent expressions of a given real function f generally lead to different natural interval extensions. This chapter also presents how the interval information can be integrated in the scheme of computation of the Choquet integral, by extending its definition to interval arithmetic.