In a groundbreaking advancement poised to transform the landscape of sustainable energy deployment, researchers have unveiled an innovative decision-making framework engineered to optimize green energy systems with unprecedented precision. This novel methodology merges two powerful multi-criteria decision-making (MCDM) techniques, VIKOR and AHP, enhanced further by the integration of q-fractional hesitant fuzzy sets. This fusion not only addresses the intrinsic uncertainties and hesitations that abound in energy planning but also elevates decision quality in contexts riddled with complex trade-offs and competing priorities.
At the heart of this research lies the challenge of selecting optimal green energy configurations amidst a plethora of environmental, economic, technological, and social criteria. Traditional decision-making tools often falter in capturing the ambiguity and hesitation decision-makers experience when weighing multiple criteria that carry uncertain or imprecise values. To surmount this, the study pioneers the application of q-fractional hesitant fuzzy sets, a cutting-edge mathematical tool that quantifies the spectrum of hesitant evaluations, allowing decision-makers to express varying degrees of confidence and preference in an articulate quantitative manner.
The integration begins with the Analytic Hierarchy Process (AHP), a method celebrated for its systematic decomposition of complex decisions into hierarchical structures and pairwise comparisons. AHP synthesizes expert opinions by assigning relative weights to each criterion, effectively illuminating their respective importance. However, it traditionally assumes crisp input data, which may not accurately mirror the nebulous nature of real-world evaluations in green energy scenarios.
Amplifying the robustness of this approach, the VIKOR method is employed as a complementary tool focusing on ranking and selecting from a family of alternatives with conflicting criteria. VIKOR emphasizes a compromise solution strategy, identifying options closest to the ideal point and naturally balancing group utility against individual regret. Its implementation within this framework ensures the resulting selection embodies a consensus-driven compromise, crucial for stakeholder acceptance and practical viability.
The hallmark of this integrated approach is the utilization of q-fractional hesitant fuzzy sets to encapsulate the inherent uncertainty. Unlike classical fuzzy sets, which assign a single membership degree, hesitant fuzzy sets accommodate multiple possible membership values reflecting the hesitant judgment of experts. The q-fractional aspect extends this concept by employing fractional calculus to refine the uncertainty representation, capturing deeper layers of vagueness and fluctuation present in expert assessments of green energy alternatives.
This trio of methodologies harmonizes to produce a decision-support tool capable of navigating the multifaceted challenges of green energy system design. Its ability to process complex, ambiguous input data empowers policymakers and engineers to explore energy configurations that optimize environmental benefits while balancing cost-effectiveness and technological feasibility.
Significantly, the application of this integrated VIKOR-AHP method transcends theoretical constructs by being tailored to practical green energy planning scenarios. It readily assimilates various criteria such as carbon footprint, installation costs, energy output reliability, maintenance demands, and societal acceptance. Each criterion is meticulously weighted and analyzed through the q-fractional hesitant fuzzy lens, yielding a nuanced prioritization scheme that reflects real-world complexities often neglected by conventional models.
Implementation of this approach can revolutionize energy infrastructure decisions at multiple scales, from selecting localized renewable installations to designing national energy portfolios. Given the heightened urgency to transition toward sustainable and resilient energy systems globally, tools that enhance decision clarity and confidence are invaluable.
Moreover, the capacity of this integrated technique to capture hesitant expert judgments fosters inclusivity and robust consensus in multi-stakeholder environments. In green energy projects, where diverse interests converge—from environmental advocates to economic planners—embracing a decision-making model that honors hesitation and ambiguity can bridge gaps and accelerate agreement on sustainable pathways.
The model’s application is expected to improve not only strategic planning but also real-time adaptability. By accommodating evolving data and uncertain inputs, it can dynamically adjust rankings and recommendations as technology costs fluctuate, policies change, or emergent environmental data becomes available, ensuring continued relevance over project lifespans.
Another crucial contribution of this research is the incorporation of fractional calculus within fuzzy logic, a relatively unexplored combination in energy decision-making literature. Fractional calculus offers powerful means to model memory and hereditary properties in complex systems, which, when intertwined with fuzzy hesitant sets, translate into a sophisticated representation of human judgment dynamics over time.
From a computational perspective, the study meticulously details the algorithmic procedures for integrating these techniques, providing reproducible frameworks that can be readily adopted or adapted within existing decision-support systems. This transparency supports scalability and encourages cross-disciplinary applications beyond energy, including environmental management and urban planning.
The anticipated impact of this decision methodology is profound, positioning it as a leading candidate for adoption by governments, industry stakeholders, and researchers tackling the multifactorial challenges of energy transition. As renewable technologies proliferate, decision-makers confront increasingly intricate landscapes requiring tools that reconcile competing objectives under uncertainty—this research rises to meet that need.
Forward-looking implications suggest potential extensions incorporating artificial intelligence and machine learning to further refine input uncertainties and automate criterion weighting processes. Such advancements could yield self-optimizing systems capable of evolving alongside technological innovations and shifting societal values.
This integrated VIKOR-AHP-q-fractional hesitant fuzzy framework emerges as a vital addition to the MCDM arsenal, anchoring green energy decisions in mathematical rigor while remaining attuned to human judgmental subtleties. Its deployment could accelerate sustainable energy adoption by delivering transparent, equitable, and evidence-based guidance to the pivotal decision-makers shaping the planet’s energy future.
In summary, this pioneering research delivers a sophisticated, elastic, and practical decision-support system aligning with the urgent global imperative for sustainable energy solutions. Through a seamless amalgamation of hierarchical structuring, compromise ranking, and nuanced fuzzy modelling, it presents a new epoch in green energy system optimization, promising to facilitate smarter, more resilient, and widely supported energy transitions worldwide.
The study, authored by M. Salih, H. F., Z. A. Ameen, and B. Alharbi alongside colleagues, appears in Scientific Reports in 2026 and represents a seminal contribution to the theoretical and practical toolkit addressing the complex realm of green energy system design under uncertainty.
Subject of Research:
Integrated multi-criteria decision-making methodology combining VIKOR, AHP, and q-fractional hesitant fuzzy sets for optimizing green energy systems.
Article Title:
An integrated VIKOR–AHP method for green energy systems based on q-fractional hesitant fuzzy multi-criteria decision-making.
Article References:
M.Salih, H.F., Ameen, Z.A., Alharbi, B. et al. An integrated VIKOR–AHP method for green energy systems based on q-fractional hesitant fuzzy multi-criteria decision-making. Sci Rep (2026). https://doi.org/10.1038/s41598-026-46076-x
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Tags: advanced fuzzy set theory in MCDMcomplex trade-off analysis in energydecision-making under uncertaintyenvironmental and economic criteria in energygreen energy decision frameworkhandling uncertainty in energy planninghierarchical decision-making methodsinnovative energy policy toolsmulti-criteria decision-making in green energyq-fractional hesitant fuzzy sets applicationsustainable energy system optimizationVIKOR and AHP integration
