A Generalized Distance Operator and Its Applications under Probabilistic Uncertain Linguistic T-Spherical Fuzzy Environment

To address complex decision-making problems under uncertainty, this paper integrates probabilistic uncertain linguistic term sets (PULTS) and T-spherical fuzzy sets (TSFS) to propose a novel model termed probabilistic uncertain linguistic T-spherical fuzzy sets (PULTSFS). This model comprehensively characterizes fuzzy linguistic evaluation information with probability from three dimensions: membership, neutrality, and non-membership. Furthermore, by extending the ordered weighted distance (OWD) measure, the probabilistic uncertain linguistic T-spherical fuzzy ordered weighted distance (PULTSFOWD) operator is defined, and its mathematical properties along with derived forms—PULTSFOWHD, PULTSFOWED and PULTSFOWGD, those are systematically investigated. Based on this operator, a multi-criteria decision-making method is constructed, in which alternatives are ranked by calculating the aggregated distance between each alternative and the positive ideal solution. The proposed method is validated through a case study on green supplier selection, demonstrating its effectiveness and feasibility. Moreover, the PULTSFOWHD and PULTSFOWED operators exhibit relatively stronger robustness under parameter variations, offering a new tool for complex uncertain decision-making problems.