Weighted picture fuzzy aggregation operators and their applications to multi-criteria decision-making problems

Picture fuzzy sets (PFSs) generally involve the positive membership, neutral membership and negative membership, and have an advantage of accurately describing decision-makers' preference information. In this paper, we develop a picture fuzzy multi-criteria decision-making (MCDM) approach that can consider the interrelationship among criteria and reduce the effects of abnormal preference information simultaneously. Firstly, a weighted picture fuzzy power Choquet ordered geometric (WPFPCOG) operator and a weighted picture fuzzy power Shapley Choquet ordered geometric (WPFPSCOG) operator are respectively developed based on the fuzzy measure, Shapley fuzzy measure and power aggregation operator, and the corresponding special cases and properties are discussed briefly. Secondly, based on the developed WPFPSCOG operator and the Shapley fuzzy measure, an integrated MCDM method is constructed to deal with picture fuzzy problems where the weights of criteria are incompletely known. Then, a mathematical programming model based on the Shapley fuzzy measure is established to obtain the weights of criteria, and the WPFPSCOG operator is used to aggregate the evaluation value of each alternative. Finally, an illustration example is provided to demonstrate the effectiveness and credibility of the developed method.