An extended hesitant fuzzy set for modeling multi-source uncertainty and its applications in multiple-attribute decision-making

Handling uncertainty has always been a pervasive and formidable challenge in the realm of multiple-attribute decision-making (MADM). Hesitant fuzzy set (HFS) can retain several different values, making it a useful tool for expressing the uncertainty experienced by decision-makers (DMs). Recently, the introduction of the normal wiggly hesitant fuzzy set (NWHFS) has aimed to uncover potential uncertain information within HFS, and it has been empirically demonstrated to enhance the rationality of MADM. However, it is essential to acknowledge the role of probabilistic information, which assesses the likelihood of evaluation values occurring and is a factor that cannot be ignored in MADM. In this current study, we propose an innovative approach known as the mixed-normal-based hesitant fuzzy set (MNHFS), considering both probabilistic information and the potential uncertainty. To distinguish MNHFSs in MADM, we defined a score function and distance measures. Subsequently, we apply the proposed MNHFS to TODIM (an acronym in Portuguese for interactive and multiple attribute decision-making) and investigate its validity and superiority in a case study involving the selection of a promising venture capital investment. Finally, we provide a visualization method to illustrate the internal uncertainty of the decision results and offer insights into investment risk.