An integrated VIKOR–AHP method for green energy systems based on q-fractional hesitant fuzzy multi-criteria decision-making

The q-fractional hesitant fuzzy sets (q-FHFSs) extend the conventional hesitant and intuitionistic fuzzy sets by providing a richer framework for representing uncertainty. This paper proposes an integrated VIKOR-AHP framework based on q-FHFSs to enhance multi-criteria decision-making (MCDM) under uncertain conditions. To address the challenge of comparing fractional and heterogeneous hesitant elements, this study develops an LCM-based generalized distance measure for q-fractional hesitant fuzzy numbers, enabling consistent comparison among hesitant elements with different cardinalities. As criteria weights are inherently subjective, the AHP method is used to systematically derive objective priority weights. The proposed VIKOR-AHP hybrid model improves flexibility in handling fractional hesitancy and conflicting criteria, while preserving mathematical rigor in distance quantification. To demonstrate its applicability, the model is applied to the selection of Green Energy Systems, evaluated under selected subcriteria from Cavallaro and Ciraolo's list of sustainability criteria. A sensitivity analysis is conducted to examine the influence of parameter q. Finally, a comparative evaluation against q-fractional hesitant fuzzy TOPSIS (q-FHF TOPSIS), hesitant q-rung orthopair fuzzy TOPSIS (Hq-ROF TOPSIS), hesitant q-rung orthopair fuzzy VIKOR (Hq-ROF VIKOR) and hesitant fuzzy VIKOR (HF VIKOR) across varying q values confirms the effectiveness and robustness of the proposed decision-making method.