Multi-attribute decision making methods based on reference ideal theory with probabilistic hesitant information

Abstract In real world, the best choice may not be the maximum or the minimum but between them, such as the befitting temperature for plants to grow, the suitable pH value for the human health, etc. Meanwhile, since the decision makers’ evaluations may be different and none of them can be ignored, the probabilistic hesitant fuzzy sets were proposed to help the decision makers express their opinions in a simple and comprehensive way. Therefore, the traditional multi-attribute decision making methods cannot deal with this situation and a series of new decision-making methods based on probabilistic hesitant fuzzy sets are developed in this paper. Firstly, we investigate the relationships between the probabilistic hesitant fuzzy elements and the reference ideal values, and propose two distance measures which take all the possible situations into consideration. Then, by combining the reference ideal method and the probabilistic hesitant fuzzy sets, we develop three different decision-making methods to process the reference ideal multi-attribute decision making problem. Then, we apply these methods in the water conservancy project evaluation problem to verify the operability of the methods. We also conduct a comparison analysis to study the reasonability and the stability of these methods. Based on the comparison results, we find that the methods have their own advantages and disadvantages, and different methods are suitable for different decision-making environments and demands. These proposed methods extend the reference ideal methods to the probabilistic hesitant fuzzy environments and provide new angles of decision making in the research field of expert and intelligent systems.