An introduction to set-valued fractional linear programming based on the null set concept

Abstract

Set theory is a generalization of interval theory. However, this theory has shortcomings due to the lack of an inverse element for addition. The concept of null sets was therefore introduced to address this issue. Nonetheless, in set-valued optimization, the use of this concept remains largely insufficient. This article, therefore, introduces a linear fractional set-valued optimization problem, the solution to which is based on the concept of null sets. This concept enables a partial order to be established between sets for simple differences and the Hukuhara difference. On this basis, the notions of optimal and H-optimal solutions have been defined. To solve the proposed set-valued linear fractional optimization problem, it is first transformed into a set-valued linear optimization problem. To make this conversion, we have proposed an adapted version of the Charnes and Cooper method applicable to set-valued linear fractional optimization problems. Subsequently, the obtained set-valued linear optimization problem is transformed into a deterministic linear bi-objective optimization problem using the vectorization technique. To apply a classical method for resolution, the bi-objective problem is converted into a single-objective linear optimization problem using the scalarization technique. Finally, an algorithm has been proposed, and two didactic examples have been solved to better illustrate the steps of the proposed procedure.