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The linguistic q-rung orthopair fuzzy sets (Lq-ROFSs) can fully represent linguistic evaluation information by adjusting the parameter $q$ to describe the range of uncertain information. In this paper, we first propose a cosine similarity measure between Lq-ROFSs based on the concept of the cosine similarity measure between intuitionistic fuzzy sets. Considering the cosine similarity measure defined by this method does not satisfy the axiom of similarity measure, we propose the improved similarity measures of Lq-ROFSs according to the corresponding cosine similarity measure and Minkowski distance measure. The improved similarity measures of Lq-ROFSs can deal with the decision information not only from the point of view of algebra but also from the point of view of geometry, which are also proved satisfy the axiom of similarity measure. Furthermore, the similarity measures of Lq-ROFSs based on different linguistic scale functions are defined for considering the semantics of linguistic terms. In addition, we apply the proposed similarity measures of Lq-ROFSs to multi-criteria group decision making problems where the decision makers’ weights are determined by the projection of individual decision on the ideal decision results. Finally, a numerical example is applied to illustrate the feasibility of the proposed method and its effectiveness is verified by comparison with some existing methods. The sensitivity analysis and stability analysis of the proposed method are also given.

The Similarity Measures for Linguistic q-Rung Orthopair Fuzzy Multi-Criteria Group Decision Making Using Projection Method