Some two-dimensional uncertain linguistic Heronian mean operators and their application in multiple-attribute decision making

Heronian mean (HM) is an important aggregation operator which has the characteristic of capturing the correlations of the aggregated arguments. In this paper, we first analyze the shortcomings of the existing weighted HM operators which do not feature reducibility and idempotency, and then, we propose the new weighted generalized Heronian mean operator and weighted generalized geometric Heronian mean operator, and prove that they can satisfy some desirable properties, such as reducibility, idempotency, monotonicity, and boundedness, and discuss some special cases of these operators. Further, because two-dimensional uncertain linguistic information can easily express the fuzzy information, we propose two-dimensional uncertain linguistic weighted generalized Heronian mean (2DULWGHM) operator and the two-dimensional uncertain linguistic weighted generalized geometric Heronian mean (2DULWGGHM) operator, and some desirable properties and special cases of 2DULWGHM and 2DULWGGHM operators are discussed. Moreover, for multiple-attribute decision-making problems in which attribute values take the form of two-dimensional uncertain linguistic variables, some approaches based on the developed operators are proposed. Finally, we gave an illustrative example to explain the steps of the developed methods and to discuss the influence of different parameters on the decision-making results.