Solving Fuzzy Nonlinear Optimization Problems Using Null Set Concept

Abstract

In the present paper, we propose a new method for minimizing the fuzzy single-objective function under fuzzy constraints. The algorithm of the method is based on the use of the null set concept. The null set concept allows us to use partial ordering for subtraction between fuzzy numbers, such as simple subtraction and the Hukuhara difference. From this, we have defined the types of solutions for a single-objective optimization problem, namely optimal solutions and H-optimal solutions. In practice, the method starts by turning the initial optimization problem into a deterministic nonlinear bi-objective optimization problem. Then, it uses Karush–Kuhn–Tucker’s optimality conditions to find the best solution of the bi-objective optimization problem. Finally, it deduces the solution to the initial problem using fuzzy algebraic operations to convert the deterministic solution into a fuzzy solution. Through some theorems, we have demonstrated that the obtained solutions by our method are optimal or H-optimal. Furthermore, the resolution of five examples of which a real-world problem has allowed us to compare our algorithm to other algorithms taken into the literature. With these results, our method can be seen as a good choice for solving a single-objective optimization problem where the objective and constraint functions are fuzzy.