A Novel Picture Fuzzy Linguistic Aggregation Operator and Its Application to Group Decision-making

The intuitionistic fuzzy set (IFS) is an effective tool to express uncertain and incomplete cognitive information with the membership, non-membership, and hesitance degrees. The picture fuzzy set (PFS) is an extension of IFS, and it can contain more information than IFS. Linguistic variables (LVs) can easily describe the complex cognitive information provided by the decision makers. So it is necessary to combine the PFS with the linguistic term and define the picture fuzzy linguistic set (PFLS), then extend Archimedean triangular norms (t-norms) and triangular conorms (s-norms), which are the generalizations of a number of other t-norms and s-norms, such as Einstein, Hamacher, and Frank s-norms and t-norms, to process PFLS. By combining the PFS and the linguistic term, we define the picture fuzzy linguistic set (PFLS) and the operations of the picture fuzzy linguistic numbers (PFLNs). Further, we develop the Archimedean picture fuzzy linguistic weighted arithmetic averaging (A-PFLWAA) operator and discuss a number of properties and special cases of this operator. Besides, we propose an approach to deal with multiple attributes group decision-making (MAGDM) problem on the basis of the developed A-PFLWAA operator. An approach to solve MAGDM problem is proposed on the basis of the A-PFLWAA operator, and the validity and reliability of the proposed method are proven by an illustrative example. The PFLNs are the generalizations of the intuitionistic linguistic numbers (ILNs), and they contain more information than ILNs. In addition, our method cannot only be utilized to solve the problems with ILNs but also be used to deal with the problems with PFLNs, and it is a generalization of a number of the existing methods.