Double Uncertainty Driving and Integrated Decision-Making Under the Mixed Probabilistic Hesitant Environment

Probabilistic hesitant fuzzy sets are an emerging and popular tool in the decision-making field, skilled at describing subjective information. However, it does not take into account the cognitive limitations of decision makers and the complexity of the actual environment can lead to situations such as multiple values or missing probabilistic information. Considering these issues, we first develop the uncertain probabilistic hesitant fuzzy set (UPHFS) to propose the mixed hesitant fuzzy set (MHFS). Moreover, we further give the computational rules and prove the generalization and robustness of the MHFS. Then, we design two new ratio models, namely the mixed probabilistic hesitant crossover ratio (MPHCR) model and mixed preference probabilistic hesitant crossover ratio (MPPHCR) model, and construct an integrated decision-making model based on them. The integrated decision-making model is a collection of integrated decision models for deriving unknown and mixed probabilities and computing optimal decision outcomes, which is different from the previous studies than the similar studies. Therefore, the model results could be more robust and reliable. Further, based on the proposed fuzzy environments and new decision-making models, we give the whole calculation process and integrated decision-making steps. Lastly, an illustrated example of project investment is provided to apply the above methods, processes, and steps and shows their effectiveness.