Correlation coefficients between normal wiggly hesitant fuzzy sets and their applications.

The multi-criteria decision-making (MCDM) field has long sought tools capable of adeptly capturing the intricacies of human decision-making amidst uncertainty. Hesitant fuzzy sets (HFS) have become a cornerstone in the MCDM field due to their ability to capture the intricacies of human decision-making under uncertainty. Nonetheless, we identified a significant gap in traditional HFS formulations, which often fail to fully harness the nuanced and implicit preferences of decision-makers (DMs). This shortcoming can lead to suboptimal decision outcomes in complex and uncertain environments. We introduce the normal wiggly hesitant fuzzy set (NWHFS), a novel construct that encapsulates both explicit and implicit preferences within a more representative framework. This study pioneers the development of new correlation coefficients for NWHFSs, offering a robust quantitative measure to elucidate the intricate relationships between variables. Our findings demonstrate that NWHFSs significantly enhance the MCDM process, providing a nuanced perspective that traditional HFS models cannot match. The proposed correlation coefficients not only reveal the concealed preferences of DMs but also broaden the decision-making spectrum, offering a more profound understanding of the relationships between alternatives and criteria. We illustrate the superiority of our approach through comparative analysis with existing methods, highlighting its ability to discern subtleties that other models overlook. Moreover, we integrate NWHFSs into clustering analysis, showcasing their potential to classify data sources with shared attributes effectively. This integration is particularly noteworthy for its ability to navigate complex datasets, offering a new dimension in data mining and resource retrieval. In essence, our study redefines the MCDM paradigm by introducing NWHFSs and their correlation coefficients, setting a new standard for decision-making accuracy and insight. The implications of our work extend beyond theory, offering practical solutions to real-world decision-making challenges.