A Non-Linear Generalization of Optimization Problems Subjected to Continuous Max-T-norm Fuzzy Relational Inequalities

Recently, the latticized linear programming problems subjected to max-min and max-product fuzzy relational inequalities (FRI) have been studied extensively and have been utilized in many interesting applications. In this paper, we introduce a new generalization of the latticized optimization problems whose objective is a non-linear function defined by an arbitrary continuous s-norm (t-conorm), and whose constraints are formed as an FRI defined by an arbitrary continuous t-norm. Firstly, the feasible region of the problem is completely characterized and two necessary and sufficient conditions are proposed to determine the feasibility of the problem. Also, a general method is proposed for finding the exact optimal solutions of the non-linear model. Then, to accelerate the general method, five simplification techniques are provided that reduce the work of computing an optimal solution. Additionally, a polynomial-time method is presented for solving general latticized linear optimization problems subjected to the continuous FRI. Moreover, an application of the proposed non-linear model is described where the objective function and the FRI are defined by the well-known Lukasiewicz s-norm and product t-norm, respectively. Finally, a numerical example is provided to illustrate the proposed algorithm.