The Extended TOPSIS Method for MAGDM with Probabilistic Hesitant Fermatean Fuzzy Linguistic Term Set

To tackle the uncertainties in language-based evaluation and the limitations of traditional methods in capturing expert preferences within complex decision-making scenarios, this study introduces an extended TOPSIS method for MAGDM with PHFFLTSs. This novel method leverages the advantages of probabilistic linguistic term sets, hesitant fuzzy linguistic term sets, and Fermatean fuzzy sets by constructing a probabilistic hesitant Fermatean fuzzy linguistic term decision matrix. This matrix enables experts to articulate their confidence in various linguistic terms while retaining a hesitant stance towards multiple terms, offering a more nuanced and adaptable reflection of decision-making uncertainties and fuzziness. The paper details the methodological steps, encompassing the creation of a "seven-value" linguistic term set, formulation of the probabilistic hesitant Fermatean fuzzy linguistic term decision matrix, integration of multiple decision matrices, normalization, attribute weight calculation, identification of positive and negative ideal solutions, and proximity coefficient computation. These steps collectively enable a thorough evaluation and ranking of alternatives. Example analysis and comparative analysis reveal that this method surpasses the extended TOPSIS method for MAGDM with PHIFLTSs in both discrimination power and decision quality. It presents a new research avenue and theoretical framework for fuzzy multi-attribute decision-making, enriching the field’s theoretical underpinnings and providing a robust tool for addressing increasingly complex and uncertain decision challenges.