Occurrence probability derivation considering different behavior strategies and decision making under the probabilistic hesitant fuzzy environment

Compared with the general fuzzy decision making, the calculation process under the probabilistic hesitant fuzzy environment involves two new issues that need further consideration, namely quantitative element probability and qualitative element preference. The contributions of this study are to address them in the same modeling framework. For the quantitative element probability, also element occurrence probability, we propose an optimization method by developing the probabilistic hesitant fuzzy envelopment rate and introducing six behavior strategies. This method deals with two complex scenarios, namely, the occurrence probabilities are completely unknown or partially missing. For the qualitative element preference given by decision makers, we design a fusion technique and propose the probabilistic hesitant fuzzy element preference fusion (PHFEF) model based on the above optimization method. Thus, the PHFEF model is used to address the above issues in the same modeling framework. To apply the above models, we further provide a step-by-step decision-making process to reasonably use the PHFEF model in the probabilistic hesitant fuzzy environment. Finally, an illustrative example is used to show the feasibility and further comparisons are given to present the rationality of the proposed methods.