Transformations of mixed solution types of interval linear equations system with boundaries on its left-hand side to linear inequalities with binary variables

Abstract

To determine the solution set of an interval linear equations system Ax=b, it is necessary to know the semantics of the system. In addition to strong and weak semantics, there are four other important semantics: tolerance, control, left-localized, and right-localized. A solution x with one of these semantics provides that particular behavior in every equation of the system. It could happen that there is a solution x with one semantic in some equations and another semantic in the remaining equations of the system. This solution generates another solution type. We provide fifteen combinations of solution types generated by tolerance, control, left-localized, and right-localized semantics and prove that the solution set containing all four semantics is the weak (united) solution set. We show that each of these fifteen nonnegative solution types with some given boundaries on Ax can be transformed to be the solution set of linear inequalities with binary variables. The transformed system is deterministic and can be used in a linear programming problem with interval linear equation constraints to find an optimistic solution satisfying the semantics of the interval linear equations. Task management and course assignment examples are provided to show the necessity of these solution types.