A Probabilistic Linguistic Three-Way Group Consensus Framework Integrating Bayesian Best–Worst Method and Regret Theory for Age-Friendliness Evaluation of Aging Urban Residential Communities

Multi-criteria group decision making (MCGDM) under linguistic uncertainty remains a fundamental challenge in applied mathematics, where decision makers seldom assign crisp numerical evaluations and frequently exhibit heterogeneous risk attitudes shaped by behavioural factors. An integrated mathematical framework, hereafter PLR-3WBC (Probabilistic Linguistic Regret-driven Three-Way Bayesian Consensus), is developed to systematically integrate four methodological components that have each been individually validated in the MCGDM literature: representation of decision information with explicit probability mass on linguistic terms; quantification of decision-maker regret and rejoice psychology under linguistic uncertainty; classification of alternatives into three actionable decision regions rather than a single-valued ranking; and group consensus reaching with credal weight aggregation. Each component has demonstrated its effectiveness in its respective domain; the present framework capitalises on their complementary strengths by embedding them within a single pipeline equipped with formal guarantees, an integration that has not been previously reported. The framework integrates five methodological components: probabilistic linguistic term sets (PLTS) for information representation; the Bayesian best–worst method (BBWM) for credal criterion weighting; a regret–rejoice value function adapted to the linguistic domain for behavioural evaluation; three-way decision (3WD) thresholds derived from a loss-function model for actionable classification; and a distance-based consensus reaching process with feedback mechanism for group convergence. A case study on age-friendliness evaluation of twelve aging urban residential communities under an indicator system of five dimensions and eighteen criteria, with four expert decision makers, demonstrates that PLR-3WBC delivers an actionable three-way classification, recovers a transparent group consensus, and produces rankings broadly consistent with classical TOPSIS, VIKOR, PROMETHEE-II, and BWM-TOPSIS (Spearman rank correlation exceeding 0.97), thereby confirming that the integrated framework preserves the ordinal reliability of these established methods, while additionally delivering three outputs that arise from the methodological integration: an actionable three-way classification enabling discrete budget-aligned decisions, credal weight intervals quantifying the depth of expert agreement on criterion importance, and a behavioural reordering of borderline non-dominated alternatives that reflects the loss-averse psychology of the decision panel and would remain hidden under single-method deployment. Sensitivity analyses with respect to the regret aversion coefficient, the loss function parameters, and the consensus threshold confirm that the qualitative classification is stable across a wide parameter envelope, supporting the practical deployment of PLR-3WBC in age-friendly community renewal programmes.