A probabilistic hesitant fuzzy Choquet integral-based TODIM method for multi-attribute group decision-making

Abstract

A probabilistic hesitant fuzzy set (PHFS) is a generalization of the hesitant fuzzy set (HFS) that adds the probability of occurrence to each possible value of the hesitant fuzzy element (HFE). It is a powerful tool for handling uncertain information, especially when there is no complete consensus or full dissensus among decision-makers. The TODIM (an acronym in Portuguese for interactive and multiple attribute decision-making) approach is a multi-attribute decision-making (MADM) technique based on prospect theory. In this paper, we developed a Choquet integral-based TODIM method under a probabilistic hesitant fuzzy environment. Firstly, we introduced both a geometric score function and a geometric variance function of the probabilistic hesitant fuzzy element (PHFE). Furthermore, we proposed a novel comparison law between two PHFEs, and we defined a new distance measure of PHFEs. Considering the interaction between the criteria, we calculated the degree of importance of the criteria with the Shapley value. Next, we introduced the principles and steps of the developed TODIM method. One of the prominent components of the TODIM method is its risk aversion coefficient. However, it has not been sufficiently addressed in the research literature. In this regard, a novel optimization-based method is proposed to calculate the attenuation factor of the TODIM method. Finally, we verify the effectiveness of the Choquet integral-based probabilistic hesitant fuzzy-TODIM (PHF-TODIM) method using a supplier selection problem in the dairy industry.