Probabilistic hesitant fuzzy decision optimization and Stackelberg games for AI enabled industrial chain collaboration under uncertainty

Abstract The digital and intelligent transformation of industrial chains introduces hierarchical decision-making challenges under stochastic disruptions and imprecise expert judgments. This study proposes an intelligent decision-optimization framework that integrates Stackelberg games with probabilistic hesitant fuzzy sets (PHFSs) to jointly model leader–follower interactions, probabilistic variability, and linguistic hesitation. The resulting bilevel model incorporates a fixed single-leader/multiple-follower hierarchy, event-triggered distributed learning and communication, and PHFS-based multi-objective aggregation in which scalarization is retained while uncertainty is represented in the objective weights. Theoretical analysis characterizes equilibrium existence under the PHFS transformation and gives conditional uniqueness under strict concavity and a contractive follower best-response mapping; experimental convergence is reported by iterations required to satisfy the common stopping tolerance. The framework is evaluated on three computational benchmark configurations derived from previously reported laboratory-scale industrial systems representing bulk-goods transportation, modular manufacturing, and serial–parallel production. Compared with state-based potential-game and fuzzy decision-making baselines, the proposed method achieves 45% faster convergence, improves leader utility by 34–37%, reduces energy consumption by 10–14%, and decreases utility variance under noise by 52%. Ablation experiments further demonstrate that the Stackelberg and PHFS components provide complementary improvements in convergence, robustness, and decision quality. These findings establish the proposed framework as an effective approach for uncertainty-aware coordination and self-optimization in digitally connected industrial-chain and smart-manufacturing systems.