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The expression of fuzzy information under multi-attribute decision making (MADM) is constantly expanded by scholars to solve the problem of uncertain decision making in various application fields. Such as select project private partners, which is difficult to make an appropriate select in a complex and changeable environments. Although the aggregation operator under interval-valued hesitant Pythagorean fuzzy environment is an effective method to solve uncertain decision-making problems, there are still some drawbacks in aggregating operators that do not consider the information loss. In this paper, we develop Hamacher operations and Choquet integral-based method to solve select project private partner problem under probabilistic interval-valued hesitant Pythagorean fuzzy information, which could express decision-makers’ preference information more flexibly and consider the significance and the correlations among the elements. Firstly, we define the probabilistic interval-valued hesitant Pythagorean fuzzy set (PIVHPFS) as an extended mathematical expression of fuzzy sets (FS). Afterward, the Hamacher algorithm concepts are given under the PIVHPFS environment. Besides, we utilize Hamacher operations and Choquet interval-based method to develop the probabilistic interval-valued hesitant Pythagorean fuzzy Hamacher Choquet integral geometric (PIVHPFHCIG) operator. At the same time, some definitions and theorems based on PIVHPFHCIG operator are proposed. After that, we utilize the PIVHPFHCIG operator to develop an approach to solve the MADM problems under the PIVHPFS situation. The new method is feasible to overcome the drawback of information loss, and it is more reasonable for obtaining a better decision result. Finally, the introduction of the best project private partner selecting problem proves the effectiveness and feasibility of PIVHPFS, and the comparison between PIVHPFS and other similar techniques decision methods are also provided.

The Probabilistic Interval-Valued Hesitant Pythagorean Fuzzy Set and Its Application in Selecting Processes of Project Private Partner