A new simplex algorithm for interval-valued Fermatean fuzzy Linear programming problems

Today, in most cases, such as transportation, management, artificial intelligence, industries, decision-making, and in many real-life situations, we deal with uncertainty, imprecise boundaries, and qualitative information. In order to deal with these kinds of critical circumstances, we need to solve an optimization problem with interval-valued Fermatean fuzzy (IVFF) information. A few works have been done on transportation problems with Interval-valued Fermatean fuzzy sets (IVFFSs). In this paper, we first introduce a few new arithmetic operations on the set of IVFFSs. Second, we propose a linear programming problem (LPP) model under an interval-valued Fermatean fuzzy environment and study its Mathematical properties by establishing various theorems. Third, we propose a new simplex algorithm to solve a fully interval-valued Fermatean fuzzy Linear programming (IVFFLPP) problem and solve a few numerical problems to show the applicability of our proposed algorithm. Fourth, we discuss the formulation of the interval-valued Fermatean fuzzy assignment problem (IVFFAP) as a particular case of IVFFLPP and study its properties. Fifth, we establish a new algorithm for solving IVFFAP and solve two numerical problems to discuss the applicability, importance of the proposed algorithms.