# Multiplicative Integral Theory of Generalized Orthopair Fuzzy Sets and Its Applications

## Abstract

There are two main issues of fuzzy multi-attribute decision-making: determine the weight of each attribute and choose an appropriate aggregation method to integrate the evaluation information of different attributes. In order to solve the multi-attribute decision-making problem in generalized orthopair fuzzy environment with unknown attribute weights more effectively, we give a decision-making method based on generalized orthopair fuzzy definite integrals. To be specific, we first introduce the complement operations of q-rung orthopair fuzzy numbers, and then investigate the multiplicative q-rung orthopair fuzzy calculus. Through the complement operations, we establish the mutual conversion formula between additive and multiplicative q-rung orthopair fuzzy calculus theory. Then, we give a multiplicative integral-based q-rung orthopair fuzzy multi-attribute decision-making method, and discuss the relationship between the q-rung orthopair fuzzy definite integrals and the q-rung orthopair fuzzy weighted geometric operator. Compared with traditional decision-making methods, this method does not rely on subjective weight information, which is especially important when dealing with large sample data. Finally, the application of election is studied to verify the feasibility and effectiveness of the proposed method. With the introduction of generalized orthopair fuzzy sets, the expression form of election evaluation information has been expanded. We also provide some examples to compare the obtained results with the results generated by the addition operation and reveal the correlation between them.

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## Acknowledgments

The authors would like to thank the referees for their help to improve the quality of the paper.

## Author information

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### Corresponding author

Correspondence to Zeshui Xu.