New preference order relationships and their application to multiobjective interval and fuzzy interval optimization problems

Recently, a method to generate preference ordering relationships in interval space, which is based on a family of automorphisms in bidimensional Euclidean space, and the family of φ-interval optimization problems were introduced in the literature. In this article both the method to generate preference ordering relationships in interval space and the family of φ-interval optimization problems are generalized, providing frameworks for dealing with families of multiobjective interval and multiobjective fuzzy interval optimization problems called the family of φ-multiobjective interval optimization problems and the family of φ-multiobjective fuzzy interval optimization problems, respectively. It is shown that those multiobjective interval and multiobjective fuzzy interval optimization problems presented in the literature, whose formulations are based on a given preference order relationship, can be seen as particular cases of a φ-multiobjective interval optimization problem and of a φ-multiobjective fuzzy interval optimization problems, respectively. Some optimality conditions for both these two families of optimization problems are provided and the methods, concepts and results presented herein are illustrated by several examples.