Extended 2-tuple linguistic hybrid aggregation operators and their application to multi-attribute group decision making

The aim of this paper is to develop some new 2-tuple linguistic hybrid aggregation operators, which are called the extended 2-tuple linguistic hybrid arithmetical weighted (ET-LHAW) operator, the extended 2-tuple linguistic hybrid geometric mean (ET-LHGM) operator, the induced ET-LHAW (IET-LHAW) operator and the induced ET-LHGM (IET-LHGM) operator. These operators do not only consider the importance of the elements but also reflect the importance of their ordered positions. Meantime, some desirable properties are studied, such as idempotency, boundary, etc. When the information about linguistic weight vectors is partly known, the models for the optimal linguistic weight vectors on an expert set, on an attribute set and on their ordered sets are established, respectively. Moreover, an approach to multi-attribute group decision making under linguistic environment is developed. Finally, a numerical example is offered to verify the developed method and to demonstrate its practicality and feasibility.