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With the increasing application of (probabilistic) hesitant fuzzy sets in decision-making, the existing integrated methods of hesitant fuzzy information have become too complicated to meet the needs of increasingly complex practical decision-making problems. Therefore, this paper combines the related knowledge of probability theory to firstly introduce the concept of continuous hesitant fuzzy element (C-HFE). Subsequently, the concept of uniform hesitant fuzzy element (U-HFE) is proposed, and discrete (probabilistic) hesitant fuzzy information is transferred to continuous one, benefited from the connection between U-HFEs and C-HFEs with uniform distribution. After then, integration methods of C-HFEs based on mathematical derivation are developed, which lays a theoretical foundation for the continuity of hesitant fuzzy information. Further, facing the problem that the method of mathematical derivation is too tedious, based on computer simulation, this paper proposes another integration method of C-HFEs, which is more concise and easier to apply. Finally, an example of the evaluation of water resources emergency management plans is given to apply the above method to practical decision-making problems.

Integrations of Continuous Hesitant Fuzzy Information in Group Decision Making With a Case Study of Water Resources Emergency Management