A variable and a fixed ordering of intervals and their application in optimization with interval-valued functions

In this study, we introduce and analyze the concepts of a fixed ordering structure and a variable ordering structure on intervals. The fixed ordering structures on intervals are defined with the help of a pointed convex cone of intervals. A variable ordering is defined by a set-valued map whose values are convex cones of intervals. In the sequel, a few properties of a cone of intervals are derived. It is shown that a binary relation, defined by a convex cone of intervals, is a partial order relation on intervals; further, the relation is antisymmetric if the convex cone of intervals is pointed. Several results under which a variable ordering map of intervals satisfies the conditions of a partial ordering relation of intervals are provided. The introduced fixed and variable ordering of intervals are applied to define and characterize optimal elements of an optimization problem with interval-valued functions. Finally, we propose a numerical technique and present its algorithmic implementation to obtain the set of optimal elements of an interval optimization problem. We also provide illustrative examples to support the study.