Normal Wiggly Probabilistic Hesitant Fuzzy Information for Environmental Quality Evaluation

As a valuable tool for representing uncertain information, probabilistic hesitant fuzzy sets (PHFS) have gained considerable recognition and in-depth discussion in recent years to increase the flexibility and manifest hesitant information in decision-making problems. However, decision-makers (DMs) cannot express all preferences only through a few probabilistic terms in actual decision-making. Much critical information is hidden behind the original information provided by the DMs. Keeping that in mind, we are interested in mining deeper uncertain information from the original probabilistic hesitant fuzzy evaluation data. To achieve the target, we put forward a novel representation tool called the normal wiggly probabilistic hesitant fuzzy set (NWPHFS) to extract deeper uncertain preferences from original probabilistic information. NWPHFS retains the original evaluation information and carries and assesses the potential uncertain details for increasing the rationality of decision-making outcomes. Herein, we propose some fundamental concepts of NWPHFS, for instance, some elementary operational laws, distance measures between two NWPHFSs, and score function. We also suggest two new aggregation operators, that is, the normal wiggly probabilistic hesitant fuzzy weighted averaging (NWPHFWA) and normal wiggly probabilistic hesitant fuzzy weighted geometric (NWPHFWG). A novel mechanism is proposed here to work out multiattribute decision-making (MADM) in solving normal wiggly probabilistic decision-making problems. Through a practical example of environmental quality assessment, the specific calculation steps of this method are epitomized. Finally, we have demonstrated the feasibility and advancement of the proposed approach via a comprehensive comparative study.