As a generalization of fuzzy set, hesitant probabilistic fuzzy set and pythagorean triangular fuzzy set have their own unique advantages in describing decision information. As modern socioeconomic decision-making problems are becoming more and more complex, it also becomes more and more difficult to appropriately depict decision makers' cognitive information in decision-making process. In order to describe the decision information more comprehensively, we define a pythagorean probabilistic hesitant triangular fuzzy set (PPHTFS) by combining the pythagorean triangular fuzzy set and the probabilistic hesitant fuzzy set. Firstly, the basic operation and scoring function of the pythagorean probabilistic hesitant triangular fuzzy element (PPHTFE) are proposed, and the comparison rule of two PPHTFEs is given. Then, some pythagorean probabilistic hesitant triangular fuzzy aggregation operators are developed, and their properties are also studied. Finally, a multi-attribute decision-making (MADM) model is constructed based on the proposed operators under the pythagorean probabilistic hesitant triangular fuzzy information, and an illustration example is given to demonstrate the practicability and validity of the proposed decision-making method.
Distance and similarity measures are very important in clustering, pattern recognition, decision-making and other scientific fields. For the existing hesitant fuzzy distance, most of them do not consider the hesitance degree. Even if the hesitance degree is considered, only the degree of dispersion or the number of hesitant fuzzy values are considered. Aiming at these shortages, a new hesitance degree is defined, which has better accuracy and applicability. Then, some hesitant fuzzy distance measures based on the proposed hesitance degree are proposed, which can overcome some shortcomings of the existing distance measures. Finally, the new hesitant fuzzy distance is applied to the hierarchical hesitant fuzzy k-means clustering algorithm, and an illustration example is given to illustrate the effectiveness of the proposed method.
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