A Direct Approach to Compute Triangular Fuzzy Banzhaf Values of Cooperative Games With Coalitions' Values Represented by Triangular Fuzzy Numbers

The primary goal of this article is to develop a direct and simplified approach for computing triangular fuzzy Banzhaf values for a special class of cooperative games with coalitions' values represented by triangular fuzzy numbers, which are called triangular fuzzy cooperative games for short. Combining with the beta-cut sets of coalitions' triangular fuzzy values, in this article, we can obtain a series of associated interval-valued cooperative games with the triangular fuzzy cooperative games. Hereby, we construct associated cooperative games with the interval-valued cooperative games. It is proven that the Banzhaf values of the associated cooperative games with interval-valued cooperative games are monotonic and nondecreasing functions under some size monotonicity-like conditions. Hence, the mean and the lower and upper limits of the triangular fuzzy Banzhaf values can be directly and explicitly attained through using only the mean, and the lower and upper limits of the coalitions' triangular fuzzy values, respectively. The developed approach does not use the subtraction of triangular fuzzy numbers and, hereby, can effectively avoid the irrational issues resulting from it. Furthermore, some important properties of the triangular fuzzy Banzhaf values are proven and the applicability and validity of the developed approach are illustrated with a numerical example.