Solving the minimal solutions of max-min fuzzy relation equation by graph method and branch method

The problem of solving max-min fuzzy relation equation is further discussed in this paper. The main matter of solving max-min fuzzy relation equation is to find its whole minimal solutions. Based on the analysis of the characteristic of minimal solution and previous results, this paper transforms the problem of seeking the whole minimal solutions into constructing a bipartite fuzzy graph and finding the various covering manners for all equations, at the same time an algorithm of generating the whole minimal solutions is given. Furthermore, to remove useless computations in the process of solving the equation, a new method, which is called branch method, is given. Both the two methods are simple, explicit for using, and are helpful for understanding the feature of the equation's minimal solution. Lastly, an example is given to illustrate the application of the two methods.