Determining the type of a solution to the fully pythagorean fuzzy linear equations system: exact, restricted, or relaxed approximate solution

Abstract In engineering and social research, linear systems are commonly used to address real-life problems of various dimensions. Therefore, many studies start by developing linear systems and then finding their solutions. Recent studies have demonstrated the effectiveness of Pythagorean fuzzy sets in capturing and representing complex forms of uncertainty, particularly when understanding the distinctions between membership and non-membership is crucial. This paper pioneers the finding a solution for a general (square or nonsquare) Fully Pythagorean Fuzzy Linear Equations System (FPFLES) with arbitrary triangular Pythagorean fuzzy numbers and fills a critical gap in the existing literature. Since an FPFLES consists of the sum of the multiplications of each arbitrary parameter and variable, and the fuzzy multiplication operation includes the min and max operators, a nonlinearity situation is observed in each equation. To overcome this situation, a transformation from fuzzy multiplication to inequalities is applied, and thus, a mixed integer programming (MIP) problem is formed. Depending on whether the MIP problems created by changing the constraints have an optimal solution, FPFLES has an exact solution or an approximate solution. The types of solutions are examined using a distance measure definition available in the literature. This paper also defines restricted and relaxed approximate solutions for FPFLES by determining whether the left-hand sides obtained from the substitution of solutions are completely covered by the right-hand sides of the equations. The approach is illustrated with some numerical examples, and the numerical results are analyzed within the distance measure to determine the closeness between the left-hand and right-hand sides of the system.