Hesitant triangular multiplicative aggregation operators and their application to multiple attribute group decision making

The hesitant triangular fuzzy set (HTFS) is a generalization of the hesitant fuzzy set, and it permits the membership degree of an element to a set to be represented as several possible triangular fuzzy numbers. However, we find that the HTFS uses the symmetrical 0.1-0.9 scale to express the membership degree information and the results derived by using the traditional hesitant triangular fuzzy aggregation operators based on hesitant triangular fuzzy sets are inconsistent with our intuition in some situations. To overcome this issue, we use the unsymmetrical 1-9 scale to express the membership degree information instead of the symmetrical 0.1-0.9 scale in the HTFS, and then a new concept is introduced, which we call the hesitant triangular multiplicative set reflecting our intuition more objectively. Then, we discuss their operational laws and some desirable properties. Based on these operational laws, we develop a series of hesitant triangular multiplicative aggregation operators for aggregating hesitant triangular multiplicative information and then apply them to present an approach to multiple attribute group decision making under hesitant triangular multiplicative environments. Finally, several practical examples are provided to demonstrate the validity and effectiveness of the developed aggregation operators and decision making approach.