Interval order relationships based on automorphisms and their application to interval optimization

This paper presents a method to generate preference ordering relations on interval space based on a family of automorphisms on the bidimensional Euclidean space. This method generates a family of order relation with which many order relations presented in the literature can be obtained as particular cases. This family of preference order relations is used to provide a formulation for a family of interval optimization problems that unifies those formulations whose solution concepts are a Pareto-type. The elements belonging to this family are called φ-interval optimization problems. An advantage of the proposed method is that decision makers can consider a suitable interval optimization problem, choosing an appropriate order relation, which is obtained by choosing an automorphism. Moreover, this paper shows that each φ-interval optimization problem is equivalent to a biobjective optimization problem. Some optimality conditions for the φ-interval optimization problems are obtained. The method, concepts and results presented herein are illustrated by several examples.