Circular q-rung orthopair fuzzy modeling using Schweizer-Sklar weighted aggregation for water purification strategy selection - Jawad Ali; Ahmad N. Al-Kenani; Ioan-Lucian Popa (Scientific Reports)
Abstract Selecting an optimal water purification strategy presents a complex multi-criteria decision-making problem due to inherent uncertainty, imprecise information, and conflicting evaluation criteria. Traditional fuzzy set-based approaches often lack the flexibility to accurately represent such uncertainty. To address these limitations, this study proposes a novel decision-support framework based on circular q-rung orthopair fuzzy ( $$C_{r}q$$ -ROF) set, which extend conventional q-rung orthopair fuzzy sets by incorporating a circular representation to model uncertainty more expressively. A key contribution of this work is the development of an improved score function that overcomes the limited discriminative capability of existing formulations. Moreover, Schweizer–Sklar operational laws are introduced within the $$C_{r}q$$ -ROF environment, and their fundamental algebraic properties are rigorously examined. Two novel aggregation operators—the $$C_{r}q$$ -ROF Schweizer–Sklar weighted average and geometric operators—are constructed and validated to enhance flexibility and accuracy in aggregating expert opinions. To further improve decision reliability, the well-established multi-objective optimization by ratio analysis plus full multiplicative form (MULTIMOORA) method is extended under the $$C_{r}q$$ -ROF framework. This integrated model simultaneously considers evaluation scores, ranking stability, and consistency across multiple ranking strategies, addressing key limitations of prior dominance-based integration techniques. The practicality and effectiveness of the proposed approach are demonstrated through a real-world case study focused on selecting the optimal commercial water purification method, with comparative analysis against existing decision-making techniques.