A Stochastic Fractional Fuzzy Tensor Framework for Robust Group Decision-Making in Smart City Renewable Energy Planning

Modern smart cities face increasing pressure to invest in sustainable and reliable energy systems while navigating uncertainties arising from fluctuating market conditions, evolving technology landscapes, and diverse expert opinions. Traditional multi-criteria decision-making (MCDM) approaches often fail to fully represent these uncertainties as they typically rely on crisp inputs, lack temporal memory, and do not explicitly account for stochastic variability. To address these limitations, this study introduces a novel Stochastic Fractional Fuzzy Tensor (SFFT)-based Group Decision-Making framework. The proposed approach integrates three dimensions of uncertainty within a unified mathematical structure: fuzzy representation of subjective expert assessments, fractional temporal operators (Caputo derivative, α=0.85) to model the influence of historical evaluations, and stochastic diffusion terms (σ=0.05) to capture real-world volatility. A complete decision algorithm is developed and applied to a realistic smart city renewable energy selection problem involving six alternatives and six criteria evaluated by three experts. The SFFT-based evaluation identified Geothermal Energy as the optimal choice with a score of 0.798, followed by Offshore Wind (0.722) and Waste-to-Hydrogen (0.713). Comparative evaluation against benchmark MCDM methods—TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), VIKOR (VIšekriterijumsko KOmpromisno Rangiranje), and WSM (Weighted Sum Model)—demonstrates that the SFFT approach yields more robust and stable rankings, particularly under uncertainty and model perturbations. Extensive sensitivity analysis confirms high resilience of the top-ranked alternative, with Geothermal retaining the first position in 82.4% of 5000 Monte Carlo simulations under simultaneous variations in weights, memory parameter (α∈[0.25,0.95]), and noise intensity (σ∈[0.01,0.10]). This research provides a realistic, mathematically grounded, and decision-maker-friendly tool for strategic planning in uncertain, dynamic urban environments, with strong potential for deployment in wider engineering, management, and policy applications.