Some neutrosophic uncertain linguistic number Heronian mean operators and their application to multi-attribute group decision making

Heronian mean (HM) is a useful aggregation operator which can deal with the interrelations of the aggregated arguments, and the neutrosophic uncertain linguistic set is a good tool to express the incomplete, indeterminate, and inconsistent information. In this paper, we combine the HM and the neutrosophic uncertain linguistic set and propose some HM operators based on neutrosophic uncertain linguistic numbers. Firstly, we introduce some definition and properties of uncertain linguistic numbers, the single-valued neutrosophic set, and some HM operators including the generalized weighted Heronian mean operator, the improved generalized weighted Heronian mean operator, and the improved generalized geometric weighted Heronian mean operator. Then, we propose the single-valued neutrosophic uncertain linguistic set by combining the uncertain linguistic numbers and the single-valued neutrosophic set. Further, the neutrosophic uncertain linguistic number improved generalized weighted Heronian mean operator and the neutrosophic uncertain linguistic number improved generalized geometric weighted Heronian mean operator are developed, and the properties of them are analyzed. Furthermore, we develop the decision making methods for multi-attribute group decision making problems with neutrosophic uncertain linguistic information and give the detailed decision steps. At last, an illustrate example is given to show the process and the effectiveness of the decision-making method.