Multi-attribute decision making based on prioritized operators under probabilistic hesitant fuzzy environments

Methodologies and Application


Probabilistic hesitant fuzzy sets (PHFSs) are currently attracting the attention of numerous scholars due to their efficiency in representing uncertain and fuzzy information. PHFSs are more convenient than hesitant fuzzy sets for decision makers to provide their preference information. However, several important issues in PHFSs utilization remain to be addressed. The shortcomings of the operations in PHFSs are discussed in this paper. Probabilistic hesitant fuzzy prioritized weighted average (PHFPWA) and probabilistic hesitant fuzzy prioritized weighted geometric (PHFPWG) operators are developed on the basis of idea of prioritized aggregation operators, and their properties are described. The relationship between PHFPWA and PHFPWG operators is investigated. A multi-attribute decision-making method based on the proposed operators is presented. Two practical examples are provided to illustrate the practicality and effectiveness of the proposed method. Comparison analysis and discussion with other aggregation operators based on the same examples are provided.


Probabilistic hesitant fuzzy sets Multi-attribute decision making Prioritized aggregation operators PHFPWA operator PHFPWG operator 



The authors thank the editors and anonymous reviewers for their helpful comments and suggestions that have led to this improved version of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of XingJian College of Science and Liberal ArtsGuangxi UniversityNanningPeople’s Republic of China
  2. 2.School of Mathematics and Information ScienceGuangxi UniversityNanningPeople’s Republic of China

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