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In this paper, we introduce a new type of fuzzy set, called Pythagorean linguistic sets (PLSs), to address the preferred and nonpreferred degrees of linguistic variables. Moreover, it allows decision makers to offer effectively handle uncertain information more flexible than intuitionistic linguistic sets (ILSs) when one compares two alternatives in the process of decision making. Some of the fundamental operational laws, score, accuracy, and aggregation operators are defined, and their properties are investigated. Preference relation (PR) is a useful and efficient tool for decision making that only requires the decision makers to compare two alternatives at one time. Taking the advantages of PLSs and PRs, this paper also introduces Pythagorean linguistic preference relations (PLPRs) and studies their application. We propose an approach for group decision making using group recommendations based on consistency matrices and feedback mechanism. First, the proposed method constructs the collective consistency matrix, the weight collective PRs, and the group collective PRs. Then, it constructs a consensus relation for each expert and determines the group consensus degree (GCD) for all experts. If the GCD is smaller than a predefined threshold value, then a feedback mechanism is activated to update the PLPRs. Finally, after the GCD is greater than or equal to the predefined threshold value, we calculate the arithmetic mathematical average values of the updated group collective PR to select the most appropriate alternative. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of fuzzy preference structure, called Pythagorean fuzzy preference relations (PFPRs), to describe uncertain evaluation information in group decision‐making process. Moreover, it allows decision makers to offer effectively handle uncertain information more flexibly than intuitionistic fuzzy preference relations when one compares two alternatives in the process of decision making. Using PFPRs, we propose an approach for group decision making based on group recommendations and consistency matrices. First, the proposed approach constructs the collective consistency matrix, the weight collective preference relations (PRs), and the group collective PR. Then, it construct a consensus relation for each expert and determinate the group consensus degree for all experts. If the group consensus degree is smaller than a predefined threshold value, then it identify the consensus values in each consensus relation which are smaller than the group consensus degree and updates the Pythagorean fuzzy preference values corresponding to the identified consensus values. The above process is continued, until the group consensus degree is larger than or equal to the predefined threshold value. Finally, based on the group collective PR, we calculate the row arithmetic mathematical average values and with respect to that values the various methods are applied for ranking the preference order of the alternatives. Numerical example are provided to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]

This article introduces general framework of multi-granulation bipolar-valued fuzzy (BVF) probabilistic rough sets (MG-BVF-PRSs) models in multi-granulation BVF probabilistic approximation space over two universes. Four types of MG-BVF-PRSs are established, by the four different conditional probabilities of BVF event. For different constraints on parameters, we obtain four kinds of each type MG-BVF-PRSs over two universes. To find a suitable way of explaining and determining these parameters in each kind of each type MG-BVF-PRS, three-way decisions (3WDs) are studied based on Bayesian minimum-risk procedure, i.e., the multi-granulation BVF decision-theoretic rough set (MG-BVF-DTRS) approach. The main contribution of this paper is twofold. One is to extend the fuzzy probabilistic rough set (FPRS) to MG-BVF-PRS model over two universes. Another is to present an approach to select parameters in MG-BVF-PRS modeling by using the process of decision making under conditions of risk. [ABSTRACT FROM AUTHOR]