Solution procedure for the best approximate solution of a particular fuzzy relational equations with max-Łukaseiwicz composition

Fuzzy relational equations have played an important role in fuzzy modeling and applied to many practical problems. Most theories of fuzzy relational equations based on a premise that the solution set is not empty. However, this is often not the case in practical applications. A broader theory of fuzzy relational equations that would allow us to determine an adequate approximate solution when no real solution exists is necessary to study. Subject to different composition of fuzzy relational equations and measure norm, the best approximate solution of fuzzy relational equations cannot yield precisely from many approximate solutions. Therefore, the presented algorithm in the literature for solving the inconsistent fuzzy relation problem usually based on the genetic algorithm (GA) or heuristic algorithm. However, these algorithms are expected to yield good results in most cases but are not guaranteed to yield the best approximate solution. For the purpose of providing a precisely solution procedure, an algorithm for finding the best approximate solution to the fuzzy relational equation with max-Łukaseiwicz composition is proposed in this study. An example is also provided to illustrate how the solution procedure can be applied to find the best approximate solution for the studied problem.