The mathematical foundation of the study rests on the evolution of fuzzy set theory since Lotfi Zadeh introduced it in 1965. Zadeh’s original formulation captured uncertainty through a single membership degree between zero and one. Krassimir Atanassov later extended the idea with intuitionistic fuzzy sets, which pair a membership degree with a non-membership degree under the constraint that their sum cannot exceed one. Ronald Yager pushed the boundary further with Pythagorean fuzzy sets, allowing the squared sum of the two degrees to reach one, and then with the q-rung orthopair fuzzy set, in which the sum of the q-th powers of membership and non-membership is bounded by one. Each generalization widened the space of uncertainty that could be represented. Yet the authors argue that even q-rung orthopair sets fall short for many real-world situations, because real decision-making is often multi-stage, conditional, and fractional rather than expressible through crisp scalar pairs. The q-fractional fuzzy set, introduced previously by Muhammad Gulistan and Witold Pedrycz, relaxes the classical constraint to the form in which the fractional sum of membership and non-membership, each divided by the parameter q, must not exceed one, with q greater than or equal to two. This fractional structure allows expert judgments such as a membership of 1.00 paired with a non-membership of 0.67, which would violate the constraints of intuitionistic, Pythagorean, and orthopair models, to be represented faithfully.

On top of this representational layer, the researchers constructed a family of aggregation operators based on the Maclaurin symmetric mean, a classical construct credited to the eighteenth-century mathematician Colin Maclaurin. The distinguishing virtue of the Maclaurin symmetric mean is that it captures interrelationships among multiple input arguments, averaging products over all k-tuple combinations of the inputs. This matters because standard aggregation techniques such as the arithmetic and geometric means implicitly assume that decision criteria are independent of one another, an assumption that rarely holds in complex environmental systems where water quality, habitat condition, disaster resilience, and resource use are deeply intertwined. The team defined four new operators: the q-fractional fuzzy Maclaurin symmetric mean, its weighted variant that incorporates the relative importance of each criterion, an ordered weighted variant that accounts for the positional priority of aggregated values, and a hybrid weighted operator that combines both weighting and ordering information. Each operator was proven to return a valid q-fractional fuzzy number, and the basic operator was shown to satisfy the classical desiderata of aggregation theory: idempotency, commutativity, monotonicity, and boundedness. The authors verified the machinery with worked numerical examples, showing, for instance, that aggregating the q-fractional fuzzy numbers (0.6, 0.2) and (0.8, 0.1) with q equal to 2 yields an aggregated value of approximately (0.245, 0.313).

The practical test of the framework came through a multi-attribute group decision-making exercise evaluating five representative watershed regions of Pakistan: the Northern Mountains, with steep glacial terrain fed by the Indus and Jhelum rivers and highly vulnerable to glacier melt; the Uplands of Northern Punjab, an arable zone challenged by soil erosion and deforestation; the Western Mountain Ranges, a semi-arid area where watershed management is critical for water recharge and flood control; the Southwest Baluchistan Plateau, where scarce water and depleted aquifers demand rainwater harvesting and check dams; and the Coastal Belt along the Arabian Sea, where saline intrusion, coastal erosion, and mangrove protection dominate the management agenda. These five sites were chosen precisely because their contrasting climatic and geographical conditions stress-test the versatility of the model.

Four criteria guided the evaluation, defined in consultation with domain expertise. Water quality served as the primary indicator of ecosystem health and the safety of water for human, animal, and agricultural use. Habitat provision measured the capacity of each watershed to support aquatic and terrestrial biodiversity, with knock-on benefits for fisheries and ecotourism. Protection in crisis situations quantified resilience against floods and droughts, reflecting how well management maintains stable water levels under extreme weather. Effective use of related resources assessed whether land and water were being exploited sustainably, without environmental degradation. The criteria were weighted at 0.145, 0.135, 0.355, and 0.365 respectively, with the heaviest weights assigned to crisis protection and resource efficiency. Four decision-makers contributed their assessments, each representing a distinct discipline: a hydrology expert focused on surface water and groundwater systems, an environmental scientist responsible for habitat and pollution evaluation, an agricultural and land-use specialist versed in soil and irrigation, and a resource and policy management analyst with expertise in environmental economics and policy viability. Each expert encoded their judgments as q-fractional fuzzy numbers, producing individual decision matrices that the framework then fused.

The algorithmic procedure proceeded in a structured sequence. First, the four individual decision matrices were combined into a single collective matrix using the weighted Maclaurin symmetric mean operator, which preserves both the fractional uncertainty of each judgment and the interdependencies among criteria. Second, a score function, defined as the fractional difference between membership and non-membership degrees, was applied to compute the score of every aggregated value, and the alternatives were ranked accordingly to prepare for ordered aggregation. Third, both the weighted and ordered weighted operators were used to derive comprehensive evaluation values for each watershed alternative. Finally, overall performance scores were calculated and the alternatives arranged in descending order to identify the best-managed region. The results revealed a striking consistency: both operators placed alternatives three and four, corresponding to the Western Mountain Ranges and the Southwest Baluchistan Plateau, at the top of the ranking, while both identified alternative two, the Uplands of Northern Punjab, as the least appropriate option. The slight discrepancy at the very top, with the weighted operator favoring one site and the ordered weighted operator the other, reflects the different perspectives embedded in the two aggregation mechanisms, one emphasizing attribute importance and the other positional influence, and the authors present this as evidence of the framework’s adaptability rather than a weakness.

To probe the reliability of the rankings, the team conducted a systematic sensitivity analysis, independently varying the two parameters that govern the model’s behavior. The parameter q controls the strictness of the fuzziness constraint, with smaller values imposing tighter bounds on permissible membership and non-membership degrees, while the parameter k determines how many attributes interact within each combination term of the Maclaurin mean. When q was varied across a reasonable range while all other settings were held constant, the overall ranking order of the watershed alternatives barely changed. The same held true when k was varied: minor numerical differences appeared in the aggregated scores, but the best- and worst-performing regions remained unchanged. This double stability demonstrates that the model balances the trade-off between attribute dominance and inter-criteria interaction without becoming hostage to arbitrary parameter choices, a property the authors regard as essential for deployment in real hydrological settings where uncertainty levels fluctuate.

A comparative analysis against existing approaches further underscored the advantages of the hybrid design. The bare q-fractional fuzzy set offers a rich representation of uncertainty but lacks any aggregation mechanism and cannot capture interactions among criteria. The basic Maclaurin operator adds attribute associations but treats all criteria as equally important. The weighted and ordered weighted variants each add one further layer of realism, while the hybrid operator, capturing attribute importance, positional influence, and interrelationships simultaneously, achieved the best discrimination among alternatives and the most stable ranking outputs. In decision problems where the scores of the top alternatives are close, as they were here, that discriminative capacity is what separates a usable planning tool from an ambiguous one.

The authors are candid about limitations. Expert opinion remains subjective, parameter tuning can influence results, and validation so far rests primarily on a single national case study that does not model interdependencies among all relevant variables. They propose several avenues for extending the work: incorporating large real-world datasets from environmental and hydrological monitoring systems, developing higher-dimensional q-fractional structures, building hybrid operators based on Dombi, Einstein, Hamacher, and Choquet integral approaches, and ultimately coupling the framework with machine learning and data-driven optimization for large-scale, dynamic decision environments. Funded in part by a large research project grant from the Deanship of Research and Graduate Studies at King Khalid University, the study suggests that fractional fuzzy mathematics, a seemingly abstract corner of decision theory, may become a practical instrument for prioritizing environmental investments in water-stressed regions around the world.